#!/usr/bin/env python
# geotecha - A software suite for geotechncial engineering
# Copyright (C) 2018 Rohan T. Walker (rtrwalker@gmail.com)
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# This program is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU General Public License for more details.
#
# You should have received a copy of the GNU General Public License
# along with this program. If not, see http://www.gnu.org/licenses/gpl.html.
"""
Finite elastic Euler-Bernoulli beam resting on viscoelastic foundation
with piecewise-linear properties, subjected to multiple moving point loads.
"""
from __future__ import division, print_function
import numpy as np
import matplotlib.pyplot as plt
import matplotlib.animation as animation
import matplotlib
from geotecha.mathematics.root_finding import find_n_roots
from scipy import integrate
from scipy.integrate import odeint
import matplotlib.style
import matplotlib as mpl
import time
from datetime import timedelta
import datetime
from collections import OrderedDict
import os
import geotecha.speccon.speccon1d as speccon1d
import geotecha.piecewise.piecewise_linear_1d as pwise
from geotecha.piecewise.piecewise_linear_1d import PolyLine
import geotecha.speccon.integrals as integ
import geotecha.inputoutput.inputoutput as inputoutput
from geotecha.inputoutput.inputoutput import SimpleTimer
from geotecha.plotting.one_d import save_figure
from numpy.testing import assert_allclose
from geotecha.plotting.one_d import MarkersDashesColors
DEBUG=True
[docs]class SpecBeam(object):
"""Finite elastic Euler-Bernoulli beam resting on
viscoelastic foundation subjected to a moving load, piecewise-linear
properties.
An extension of Ding et al. (2012) [1]_ with piecewise linear
material properties and non-linear foundation stiffness k3=0 in k3*w**3.
You don't need all the parameters. Basically if normalised values are
'None' (i.e. default) then those properties will be calculated from
the non normalised quantities. All calculations are done in normalised
space. You only need the non-normalised variables if you want
non-normalised output.
Note: some of the code is a relic of
geotecha.beam_on_foundation.dingetal2012 which is a straight up
implementation of [1]_ . Sorry for any confusion.
Parameters
----------
BC : ["SS","CC","FF"], optional
Boundary condition. Can only have Simply Supported (SS).
Clamped Clamped (CC) and Free Free (FF) are a relic from a differnet
program. Default BC="SS".
nterms : integer, optional
Number of terms for Galerkin truncation. Default nterms=50
E : float, optional
Young's modulus of beam.(E in [1]_).
rho : float, optional
Mass density of beam.
I : float, optional
Second moment of area of beam (I in [1]_).
A : float, optional
Cross sectional area of beam
L : float, optional
Length of beam. (L in [1]_).
k1 : float, optional
Mean stiffness of foundation.
k3 : float, optional
Non linear stiffness of foundation.
mu : float, optional
Viscous damping of foundation.
# Fz : float, optional
# Load. This is a relic; loads now specified with moving_loads_...
v : list of float, optional
Velocity of each group of moving point loads.
moving_loads_x : list of lists of floats, optional
Each sublist contains x-coords of each point load in the moving load
group. The coords are relative to a reference point which will
enter the beam at the corresponding time value in `moving_loads_t0`.
Coords are usually negative for vehicle facing left to right.
moving_loads_Fz : list of lists of floats, optional
Each sublist contains magnitude of of each point load in moving load
group.
moving_loads_v : list of float, optional
Velocity of each moving point load group. When velocity is negative
the x-coords in the corresponding member of `moving_loads_x` will
be multiplied by -1 before the beam enters the beam from the right.
moving_loads_offset : list of float, optional
An initial distance by which the corresponding `moving_loads_x` is
offset. Will default to zero if not inputted.
moving_loads_t0 : list of lists, optional
Time that moving load group enters the beam (strictyly speaking it
is when the 0 coord in `moving_loads_x` enters the beam). If v>0
moving load group starts at x=0, t=0 and moves to the right.
If v<0 then moving load group starts at x=L,t=0 and moves to the
left. Defaults to 0.
moving_loads_x0 : list of float, optional
Position where moving load group appears on the beam. Defaults to
zero if ommited.
moving_loads_L : list of float, optional
Distance the load group travels on beam. Defaults to L if omitted.
Used in conjunction with moving loads_x0.
tvals : 1d array of float, optional
Output times
xvals : 1d array of float, optional
Output x-coord
v_norm : list of float, optional
normalised velocity = V * sqrt(rho / E) for each group of moving
point loads. An example of a consistent
set of units to get the correct v_norm is rho in kg/m^3, L in m, and
E in Pa.
moving_loads_x_norm : list of lists of floats, optional
As per `moving_loads_x` but nomalised.
x_norm = x / L
moving_loads_Fz_norm : list of lists of floats, optional
As per `moving_loads_Fz` but nomalised.
Fz_norm = Fz / (E * A).
moving_loads_v_norm : list of float, optional
As per `moving_loads_v` but nomalised.
v_norm= v * sqrt(rho / E) for each group of moving
point loads. An example of a consistent
set of units to get the correct v_norm is rho in kg/m^3, L in m, and
E in Pa.
moving_loads_offset_norm : list of float, optional
As per `moving_loads_offset` but nomalised.
offset_norm = offet / L
moving_loads_t0_norm : list of lists, optional
As per `moving_loads_t0` but nomalised.
t0_norm = t0 / L * np.sqrt(E / rho)
moving_loads_x0_norm : list of float, optional
As per `moving_loads_x0`. x0_norm=x0/L.
moving_loads_L_norm : list of float, optional
As per `moving_loads_L_norm` but normalised.L_norm = L' / L
tvals_norm : 1d array of float, optional
Normalised output times.
t_norm = t / L * np.sqrt(E / rho)
xvals_norm : 1d array of float, optional
Normalised output coords.
x_norm = x / L
stationary_loads_x : list of float, optional
Location of stationary point loads.
stationary_loads_vs_t : list of Polyline, optional
Stationary load magnitude variation with time. PolyLine(time, magnitude).
stationary_loads_omega_phase : list of 2 element tuples, optional
(omega, phase) to define cyclic variation of stationary point loads. i.e.
mag_vs_time * cos(omega*t + phase). If stationary_loads_omega_phase is None
then cyclic component will be ignored. If stationary_loads_omega_phase is a
list then if any member is None then cyclic component will not be
applied for that load combo.
stationary_loads_x_norm : list of float, optional
As per stationary_loads_x but normalised
stationary_loads_vs_t_norm : list of Polyline, optional
As per stationary_loads_vs_t but normalised
stationary_loads_omega_phase_norm : list of 2 element tuples, optional
As per stationary_loads_omega_phase but normalised.
kf : float, optional
Normalised modulus of elasticity = 1 / L * sqrt(I / A).
# Fz_norm : float, optional
# Normalised load = Fz / (E * A). This is a relic; loads now specified
# with moving_loads_...
mu_norm : float, optional
Normalised damping = mu / A * sqrt(L**2 / (rho * E)).
k1_norm : float, optional
Normalised mean stiffness = k1 * L**2 / (E * A).
k3_norm : float, optional
Normalised non-linear stiffness = k3 * L**4 / (E * A)
nquad : integer, optional
Number of quadrature points for numerical integration of non-linear
k3*w**3*w_ term. Default nquad=30. Note I've had errors when n>35.
For the special case of nquad=None then integration will be performed
using scipy.integrate.quad; this is slower.
Ebar : PolyLine, optional
PolyLine object representing piecewise linear E/Eref vs x.
Default Ebar = PolyLine([0],[1],[1],[1]) i.e. uniform.
rhobar : PolyLine, optional
PolyLine object representing piecewise linear rho/rhoref vs x.
Default rhobar = PolyLine([0],[1],[1],[1]) i.e. uniform.
Ibar : PolyLine, optional
PolyLine object representing piecewise linear I/Iref vs x.
Default Ibar = PolyLine([0],[1],[1],[1]) i.e. uniform.
Abar : PolyLine, optional
PolyLine object representing piecewise linear A/Aref vs x.
Default Abar = PolyLine([0],[1],[1],[1]) i.e. uniform.
k1bar : PolyLine, optional
PolyLine object representing piecewise linear k1/k1ref vs x.
Default k1bar = PolyLine([0],[1],[1],[1]) i.e. uniform.
k3bar : PolyLine, optional
PolyLine object representing piecewise linear k3/k3ref vs x.
Default k3bar = PolyLine([0],[1],[1],[1]) i.e. uniform.
mubar : PolyLine, optional
PolyLine object representing piecewise linear mu/muref vs x.
Default mubar = PolyLine([0],[1],[1],[1]) i.e. uniform.
use_analytical : [False,true]
If true then system of second order ode resulting from spectral
galerkin method will be solved analytically. If False then the
ode system will be solved numerically (potentially very slow).
analytical approach is only avaialble when BC='SS' and
k3=0 (or k3_norm=0).
file_stem : string, optional
File stem for saving deflection data. Default file_stem="specbeam_".
force_calc : [False, True]
If True then calcualtion will happen regardless of presence of output
file (file_stem)
Attributes
----------
phi : function
Relevant Galerkin trial function. Depends on `BC`. See [1]_ for
details.
beta : 1d ndarray of `nterms` float
beta terms in Galerkin trial function.
xj : 1d ndarray of `nquad` float
Quadrature integration points.
Ij : 1d ndarray of `nquad` float
Weighting coefficients for numerical integration.
BC_coeff : int
I don't think this is relevant anymore.
Coefficent to multiply the Fz and k3 terms in the ode.
For `BC`="SS" BC_coeff=2, for `BC`="CC" or "FF" BC_coeff=2.
See [1]_ Equation (31) and (32).
t : 1d ndarray of float
Raw time values. Only valid after running calulate_qk.
t_norm : 1d ndarray of float
Normlised time values = t / L * sqrt(E / rho).
Only valid after running calulate_qk.
qsol : 2d ndarray of shape(len(`t`), 2* `nterms`) float
Values of Galerkin coefficients at each time i.e. qk(t) in [1]_.
w(x) = sum(qk * phi_k(x)).
Only valid after running calulate_qk.
lam_qdotdot : 2d array of shape (nterms, nterms)
spectral coefficeient matrix for qdotdot
psi_q : 2d array of shape (nterms, nterms)
spectral coefficeient matrix for q
psi_qdot : 2d array of shape (nterms, nterms)
spectral coefficeient matrix for qdot.
k3barj : 1d array of float with len(nquad)
value of k3 interpolated from k3bar at the quadrature points. Used
in the k3*w**3 term.
L_norm : float
Normalised length. Always =1.
defl_norm : array of float, size (len(xvals_norm), len(tvals_norm))
normalised defelction at xval,tval
defl : array of float, size (len(xvals_norm), len(tvals_norm))
deflection at xval,tval. ONly if self.L is not None.
beta_block : 1d array float with size (2*nterms)
Block column vector of repeated beta values.
has_moving_loads : boolean
If True then there are moving point loads.
has_stationary_loads : boolean
If True then there are stationary loads.
moving_load_w_norm : list of ndarray
Normalised deflections under each moving load at each time value.
Each element of the list corresponds to each moving load. The array
is of size (len(mvpl.x), len(tvals)) i.e. for each axle in the moving
load group.
moving_load_w : list of ndarray
Non normalised deflections under each moving load.
Notes
-----
defl and def_norm will be 1d arrays if len(xvals)=1.
References
----------
.. [1] Ding, H., Chen, L.-Q., and Yang, S.-P. (2012).
"Convergence of Galerkin truncation for dynamic response of
finite beams on nonlinear foundations under a moving load."
Journal of Sound and Vibration, 331(10), 2426-2442.
.. [2] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
def __init__(self,
BC="SS",
nterms=50,
E=None,
I=None,
rho=None,
A=None,
L=None,
k1=None,
k3=None,
mu=None,
# Fz=None, #This is a relic; loads now specified with moving_loads_...
# v=None, #This is a relic; loads now specified with moving_loads_...
# v_norm=None,
moving_loads_x=None,
moving_loads_Fz=None,
moving_loads_v=None,
moving_loads_offset=None,
moving_loads_t0=None,
moving_loads_x0=None,
moving_loads_L=None,
tvals=None,
xvals=None,
v_norm=None,
moving_loads_x_norm=None,
moving_loads_Fz_norm=None,
moving_loads_v_norm=None,
moving_loads_offset_norm=None,
moving_loads_t0_norm=None,
moving_loads_x0_norm=None,
moving_loads_L_norm=None,
tvals_norm=None,
xvals_norm=None,
stationary_loads_x=None,
stationary_loads_vs_t=None,
stationary_loads_omega_phase=None,
stationary_loads_x_norm=None,
stationary_loads_vs_t_norm=None,
stationary_loads_omega_phase_norm=None,
kf=None,
# Fz_norm=None, #This is a relic; loads now specified with moving_loads_...
mu_norm=None,
k1_norm=None,
k3_norm=None,
nquad=30,
Ebar=PolyLine([0],[1],[1],[1]),
rhobar=PolyLine([0],[1],[1],[1]),
Ibar=PolyLine([0],[1],[1],[1]),
Abar=PolyLine([0],[1],[1],[1]),
k1bar=PolyLine([0],[1],[1],[1]),
k3bar=PolyLine([0],[1],[1],[1]),
mubar=PolyLine([0],[1],[1],[1]),
implementation="vectorized",
use_analytical=False,
file_stem="specbeam_",
force_calc=False):
self.BC = BC
self.nterms = nterms
self.E = E
self.I = I
self.rho = rho
self.I = I
self.A = A
self.L = L
self.k1 = k1
self.k3 = k3
self.mu = mu
# self.Fz = #Fz This is a relic; loads now specified with moving_loads_...
# self.v = v #This is a relic; loads now specified with moving_loads_...
# self.v_norm = v_norm #This is a relic; loads now specified with moving_loads_...
self.moving_loads_x=moving_loads_x
self.moving_loads_Fz=moving_loads_Fz
self.moving_loads_v=moving_loads_v
self.moving_loads_offset=moving_loads_offset
self.moving_loads_t0=moving_loads_t0
self.moving_loads_x0=moving_loads_x0
self.moving_loads_L=moving_loads_L
self.tvals=tvals
self.xvals=xvals
#self.v_norm=v_norm #This is a relic; loads now specified with moving_loads_...
self.moving_loads_x_norm=moving_loads_x_norm
self.moving_loads_Fz_norm=moving_loads_Fz_norm
self.moving_loads_v_norm=moving_loads_v_norm
self.moving_loads_offset_norm=moving_loads_offset_norm
self.moving_loads_t0_norm=moving_loads_t0_norm
self.moving_loads_x0_norm=moving_loads_x0_norm
self.moving_loads_L_norm=moving_loads_L_norm
self.tvals_norm=tvals_norm
self.xvals_norm=xvals_norm
self.stationary_loads_x=stationary_loads_x
self.stationary_loads_vs_t=stationary_loads_vs_t
self.stationary_loads_omega_phase=stationary_loads_omega_phase
self.stationary_loads_x_norm=stationary_loads_x_norm
self.stationary_loads_vs_t_norm=stationary_loads_vs_t_norm
self.stationary_loads_omega_phase_norm=stationary_loads_omega_phase_norm
self.kf = kf
# self.Fz_norm = Fz_norm
self.mu_norm = mu_norm
self.k1_norm = k1_norm
self.k3_norm = k3_norm
self.nquad = nquad
self.Ebar = Ebar
self.Ibar = Ibar
self.Abar = Abar
self.k1bar = k1bar
self.k3bar = k3bar
self.mubar = mubar
self.rhobar = rhobar
self.implementation = implementation
self.use_analytical=use_analytical
self.file_stem=file_stem
self.force_calc=force_calc
if (self.k3_norm is None) and self.k3 is None:
self.k3=0
self.k3_norm=0
if (((self.BC.upper() in ["CC","FF"]) or
(sum([v==0 for v in [self.k3, self.k3_norm]]) != 2))
and self.use_analytical):
raise ValueError ("If use_analytical=True, cannot have BC='CC, BC='FF', or either k3!=None or k3_norm!=None")
# normalised parameters
self.L_norm = 1
if kf is None:
self.kf = 1 / self.L * np.sqrt(self.I / self.A)
# if v_norm is None: #This is a relic; loads now specified with moving_loads_...
# self.v_norm = self.v * np.sqrt(self.rho / self.E) #This is a relic; loads now specified with moving_loads_...
if all([v is None for v in
[self.moving_loads_x_norm,
self.moving_loads_Fz_norm,
self.moving_loads_v_norm,
self.moving_loads_x,
self.moving_loads_Fz,
self.moving_loads_v]
]):
#no moving loads
self.has_moving_loads=False
else:
# at least one moving loads
self.has_moving_loads=True
if self.moving_loads_x_norm is None:
self.moving_loads_x_norm = [[vv/self.L for vv in v] for v in self.moving_loads_x]
if self.moving_loads_Fz_norm is None:
self.moving_loads_Fz_norm = [[vv / (self.E * self.A) for vv in v] for v in self.moving_loads_Fz]
if self.moving_loads_v_norm is None:
self.moving_loads_v_norm = [v * np.sqrt(self.rho / self.E) for v in self.moving_loads_v]
#optional moving load variables
###################
if (self.moving_loads_offset is None):
if not self.moving_loads_x is None:
#raw input values used:
self.moving_loads_offset = np.zeros(len(self.moving_loads_x))
elif self.moving_loads_offset_norm is None:
#nomalised values unsed but default behaviour
self.moving_loads_offset_norm = np.zeros(len(self.moving_loads_x_norm))
if self.moving_loads_offset_norm is None:
self.moving_loads_offset_norm = [v /self.L for v in self.moving_loads_offset]
###################
if (self.moving_loads_t0 is None):
if not self.moving_loads_x is None:
#raw input values used:
self.moving_loads_t0 = np.zeros(len(self.moving_loads_x))
elif self.moving_loads_t0_norm is None:
#nomalised values used but default behaviour
self.moving_loads_t0_norm = np.zeros(len(self.moving_loads_x_norm))
if self.moving_loads_t0_norm is None:
self.moving_loads_t0_norm = [v / self.L * np.sqrt(self.E / self.rho) for v in self.moving_loads_t0]
###################
if self.moving_loads_x0 is None:
if not self.moving_loads_x is None:
#raw input values used for moving loads but default for x
self.moving_loads_x0 = np.zeros(len(self.moving_loads_x))
elif self.moving_loads_x0_norm is None:
#nomalised values used but default behaviour
self.moving_loads_x0_norm = np.zeros(len(self.moving_loads_x_norm))
if self.moving_loads_x0_norm is None:
self.moving_loads_x0_norm = [v /self.L for v in self.moving_loads_x0]
###################
if self.moving_loads_L is None:
if not self.moving_loads_x is None:
#raw input values used for moving loads but default for x
self.moving_loads_L = self.L * np.ones(len(self.moving_loads_x), dtype=float)
elif self.moving_loads_L_norm is None:
#nomalised values used but default behaviour
self.moving_loads_L_norm = np.ones(len(self.moving_loads_x_norm),dtype=float)
if self.moving_loads_L_norm is None:
self.moving_loads_L_norm = [v /self.L for v in self.moving_loads_L]
###################
self.moving_loads_norm = [MovingPointLoads(x=xv,
p=pv,
offset=offv,
v=vv,
t0=t0v,
L=Lv,
x0=x0v)
for xv,pv,offv,vv,t0v,Lv,x0v in
zip(self.moving_loads_x_norm,
self.moving_loads_Fz_norm,
self.moving_loads_offset_norm,
self.moving_loads_v_norm,
self.moving_loads_t0_norm,
self.moving_loads_L_norm,
self.moving_loads_x0_norm)]
if not self.moving_loads_x is None:
self.moving_loads = [MovingPointLoads(x=xv,
p=pv,
offset=offv,
v=vv,
t0=t0v,
L=self.L)
for xv,pv,offv,vv,t0v in
zip(self.moving_loads_x,
self.moving_loads_Fz,
self.moving_loads_offset,
self.moving_loads_v,
self.moving_loads_t0)]
if self.tvals_norm is None:
self.tvals_norm = self.tvals / self.L * np.sqrt(self.E / self.rho)
if self.xvals_norm is None:
self.xvals_norm = self.xvals / self.L
if all([v is None for v in
[self.stationary_loads_x_norm,
self.stationary_loads_omega_phase_norm,
self.stationary_loads_vs_t_norm,
self.stationary_loads_x,
self.stationary_loads_omega_phase,
self.stationary_loads_vs_t_norm,]
]):
self.has_stationary_loads=False
else:
self.has_stationary_loads=True
#check if stationary loads omega phase is missing
if self.stationary_loads_omega_phase is None:
if not self.stationary_loads_x is None:
#use default omega_phase i.e. NONE
self.stationary_loads_omega_phase = len(self.stationary_loads_x) * [None]
if self.stationary_loads_x_norm is None:
self.stationary_loads_x_norm = [v / self.L for v in self.stationary_loads_x]
if self.stationary_loads_vs_t_norm is None:
#PolyLine(time, mag)
#what if
self.stationary_loads_vs_t_norm = (
[PolyLine(v.x / self.L * np.sqrt(self.E / self.rho), v.y / (self.E * self.A)) for v in self.stationary_loads_vs_t])
if self.stationary_loads_omega_phase_norm is None:
if self.stationary_loads_omega_phase is None:
#use default omega_phase_norm, i.e. None
self.stationary_loads_omega_phase_norm = len(self.stationary_loads_x_norm) * [None]
else:
#only omega is effected, phase remains unchanged
self.stationary_loads_omega_phase_norm = (
[(v[0] * self.L / np.sqrt(self.E / self.rho),v[1]/ (self.E * self.A))
if not v is None else None for v in self.stationary_loads_omega_phase])
if self.kf is None:
self.kf = 1 / self.L * np.sqrt(self.I / self.A)
# if Fz_norm is None: #This is a relic; loads now specified with moving_loads_...
# self.Fz_norm = self.Fz / (self.E * self.A) #This is a relic; loads now specified with moving_loads_...
if self.mu_norm is None:
self.mu_norm = self.mu / self.A * np.sqrt(self.L**2 / (self.rho * self.E))
if self.k1_norm is None:
self.k1_norm = self.k1 * self.L**2 / (self.E * self.A)
if self.k3_norm is None:
self.k3_norm = self.k3 * self.L**4 / (self.E* self.A)
# phi, beta, and quadrature points
if self.BC == "SS":
self.phi = self.phiSS
self.beta = np.pi * (np.arange(self.nterms) + 1)
#note self.beta is the same as rtrwalker's more usual self.m
if not self.nquad is None:
self.xj = 0.5 * (1 - np.cos(np.pi * np.arange(self.nquad) / (self.nquad - 1)))
self.BC_coeff = 1# for DingEtal2012 it is two but it is not relevant anymore because i don't multiply the whole equation by 2.
elif self.BC == "CC" or self.BC == "FF":
#This is a relic; can only have "SS"
raise ValueError("'BC' can only be 'SS'")
def _f(beta):
return 1 - np.cos(beta) * np.cosh(beta)
self.beta = find_n_roots(_f, n=self.nterms, x0=0.001, dx=3.14159 / 10, p=1.1)
if not self.nquad is None:
self.xj = 0.5*(1 - np.cos(np.pi * np.arange(self.nquad)/(self.nquad - 3)))
self.xj[0] = 0
self.xj[1] = 0.0001
self.xj[-2] = 0.0001
self.xj[-1] = 0
self.BC_coeff = 1 # Coefficeint to multiply Fz(vt) and k3*Integral terms by
if self.BC == "CC":
self.phi = self.phiCC
if self.BC == "FF":
self.phi = self.phiFF
self.beta[1:] = self.beta[0:-1]
self.beta[0] = 0.0
else:
raise ValueError("only BC='SS', 'CC' and 'FF' have been implemented, you have {}".format(self.BC))
# quadrature weighting
if not self.nquad is None:
rhs = np.reciprocal(np.arange(self.nquad, dtype=float) + 1)
lhs = self.xj[np.newaxis, :] ** np.arange(self.nquad)[:, np.newaxis]
self.Ij = np.linalg.solve(lhs, rhs)
self.vsvdot = np.zeros(2 * self.nterms) #vector of state values for odeint
[docs] def phiSS(self, x, beta):
return np.sin(beta * x)
[docs] def phiCC(self, x, beta):
def _xi(beta):
return (np.cosh(beta) - np.cos(beta))/(np.sinh(beta) - np.sin(beta))
return (np.cosh(beta * x)
- np.cos(beta * x)
+ _xi(beta) * (np.sin(beta * x) - np.sinh(beta * x))
)
[docs] def phiFF(self, x, beta):
def _xi(beta):
return (-(np.cos(beta) - np.cosh(beta))/(np.sin(beta) - np.sinh(beta)))
return (np.cos(beta * x)
+ np.cosh(beta * x)
+ _xi(beta) * (np.sin(beta * x) + np.sinh(beta * x))
)
[docs] def w(self, qk, x):
"""Nomalised vertcal deformation at x.
Parameters
----------
qk : 1d ndarray of nterm floats
Galerkin coefficients.
x : float
nomalised distances to calculate deflection at.
Returns
-------
w : float
vertical deformation at x value.
"""
return np.sum(qk * self.phi(x, self.beta))
[docs] def vectorfield(self, vsv, tnow, p=()):
"""
Parameters
----------
vsv : float
Vector of the state variables.
vsv = [q1, q2, ...qk, q1dot, q2dot, ..., qkdot]
where qk is the kth galerkin coefficient and qkdot is the time
derivative of the kth Galerkin coefficient.
tnow : float
Current time.
p : various
Vector of parameters
Returns
-------
vsvdot : vector of state variables first derivatives
vsvdot = [q1dot, q2dot, ...qkdot, q1dotdot, q2dotdot, ..., qkdotdot]
"""
q = vsv[:self.nterms]
qdot = vsv[self.nterms:]
self.vsvdot[self.nterms:] = np.dot(self.psi_q, q)
self.vsvdot[self.nterms:] += np.dot(self.psi_qdot, qdot)
for i in range(self.nterms):
self.vsvdot[i] = qdot[i]
# self.vsvdot[self.nterms + i] = - self.mu_norm * qdot[i]
#
# self.vsvdot[self.nterms + i] -= (self.k1_norm + self.kf**2 * self.beta[i]**4) * q[i]
#
#TODO: this is where the moving loads need to go
# self.vsvdot[self.nterms + i] += self.BC_coeff * self.Fz_norm * self.phi(self.v_norm * tnow, self.beta[i])
if tnow>0:
np.random.rand()
if self.has_moving_loads:
for mvl in self.moving_loads_norm:
mvl_x, mvl_p = mvl.point_loads_on_beam(t=tnow)
self.vsvdot[self.nterms + i] += np.sum(self.BC_coeff * mvl_p * self.phi(mvl_x, self.beta[i]))
if self.has_stationary_loads:
for j in range(len(self.stationary_loads_x_norm)):
x=self.stationary_loads_x_norm[j]
mag_vs_time = self.stationary_loads_vs_t_norm[j]
mag = pwise.pinterp_x_y(a=mag_vs_time,xi=tnow)
omega_phase = self.stationary_loads_omega_phase_norm[j]
if omega_phase is None:
self.vsvdot[self.nterms + i] += self.BC_coeff * mag*np.sin(x*self.beta[i])
else:
omega,phase = omega_phase
self.vsvdot[self.nterms + i] += self.BC_coeff * mag*np.sin(x*self.beta[i])*np.cos(omega*tnow+phase)
if 1:
# DIY quadrature
Fj = np.sum(q[:,None] * self.phi(self.xj[None, :], self.beta[:,None]), axis=0)**3
y = np.sum(self.Ij * Fj * self.k3barj * self.phi(self.xj, self.beta[i]))
else:
raise ValueError("Haven't checked this yet")
# scipy
# print("yquad = {:12.2g}".format(y))
y, err = integrate.quad(self.w_cubed_wi, 0, 1, args=(q,i))
# print("yscip = {:12.2g}".format(y))
#maybe the quad integrate is not great for the oscillating function
self.vsvdot[self.nterms + i] -= self.BC_coeff * self.k3_norm * y
self.vsvdot[self.nterms:] = np.linalg.solve(self.lam_qdotdot, self.vsvdot[self.nterms:])
return self.vsvdot
[docs] def w_cubed_wi(self, x, q, i):
"""Non-linear cube term for numerical integration"""
return self.w(q, x)**3 * self.phi(x, self.beta[i])
[docs] def calulate_qk(self, t=None, t_norm=None, **odeint_kwargs):
"""Calculate the nterm Galerkin coefficients qk at each time value
Parameters
----------
t : float or array of float, optional
Raw time values.
t_norm : float or array of float, optional
Normalised time values. If t_norm==None then it will be
calculated from raw t values and other params
t_norm = t / L * sqrt(E / rho).
Notes
-----
This method determines initializes self.t and self.t_norm and
calculates `self.qsol`.
"""
if t_norm is None:
self.t_norm = t / self.L * np.sqrt(self.E / self.rho)
else:
self.t_norm = t_norm
self.t = t
vsv0 = np.zeros(2*self.nterms) # initial conditions
self.time_independant_matrices()
self.qsol = odeint(self.vectorfield,
vsv0,
self.t_norm,
args=(),
**odeint_kwargs)
[docs] def time_independant_matrices(self):
"""Make all time independent matrices"""
self.lam_qdotdot = integ.pdim1sin_abf_linear(m=self.beta,
a=self.rhobar,
b=self.Abar,
implementation=self.implementation)
self.psi_q = - self.kf**2 * integ.pdim1sin_DD_abDDf_linear(
m=self.beta,
a = self.Ebar,
b = self.Ibar,
implementation=self.implementation)
self.psi_q -= self.k1_norm * integ.pdim1sin_af_linear(
m=self.beta,
a = self.k1bar,
implementation=self.implementation)
self.psi_qdot = -self.mu_norm * integ.pdim1sin_af_linear(
m=self.beta,
a=self.mubar,
implementation=self.implementation)
self.k3barj = pwise.pinterp_x_y(self.k1bar, self.xj)
[docs] def wofx(self, x=None, x_norm=None, tslice=slice(None, None, None), normalise_w=True):
"""Deflection at distance x, and times t[tslice]
Parameters
----------
x : float or ndarray of float, optional
Raw values of x to calculate deflection at. Default x=None.
x_norm : float or array of float, optional
Normalised x values to calc deflection at. If x_norm==None then
it will be calculated frm `x` and other properties : x_norm = x / L.
Default x_norm=None.
tslice : slice object, optional
slice to select subset of time values. Default
tslice=slice(None, None, None) i.e. all time values.
Note the array of time values is already in the object (it was
used to calc the qk galerkin coefficients).
normalise_w : True/False, optional
If True then output is normalised deflection. Default nomalise_w=True.
Returns
-------
w : array of float, shape(len(x),len(t))
Deflections at x and self.t_norm[tslice]
"""
if x_norm is None:
x_norm = x / self.L
x_norm = np.atleast_1d(x_norm)
v = np.zeros((len(x_norm), len(self.tvals_norm[tslice])))
for i, xx in enumerate(x_norm):
for j, qq in enumerate(self.qsol[tslice, :self.nterms]):
v[i, j] = self.w(qq, xx)
if not normalise_w:
v *= self.L
if len(x_norm)==1:
return v[0]
else:
return v
[docs] def runme(self):
if (self.force_calc) or (not self.load_defl()):
if self.use_analytical:
self.time_independant_matrices()
#adjust matrices so that thaey are all onthe left hand side of the
# equation. i.e. analytical equation is arranged differently to
# numerical equations (and I did the numerical formulation before
# the analyitcal).
self.lam_qdotdot*=1 # rho*A
self.psi_q*=-1 #kf**2 * E*I +++++ k1
self.psi_qdot*=-1 #mu
# make the block beta. ONly the second nterms elements will be relevant in calcs.
self.beta_block = np.empty(2 * self.nterms, dtype=float)
self.beta_block[:self.nterms] = self.beta
self.beta_block[self.nterms:] = self.beta
self._make_gam()
self._make_psi()
self._make_eigs_and_v()
self.make_time_dependent_arrays()
self.make_output()
else:
#using numerical answer
self.calulate_qk(t_norm=self.tvals_norm)
self.defl_norm = self.wofx(x_norm=self.xvals_norm, normalise_w=True)
#Should be array of shape (len(self.xvals_norm), len(self.tvals_norm))
if not self.L is None:
self.defl = self.defl_norm * self.L
def _make_gam(self):
"""Make the 2n =*2n block gamma matrix
Used with `use_analytical`=True
"""
#self.m_block = np.empty(2 * self.neig, dtype=float)
#self.m_block[:self.neig] = self.m
#self.m_block[self.neig:] = self.m
self.gam = np.zeros((2*self.nterms, 2*self.nterms), dtype=float)
self.gam[:self.nterms, :self.nterms] = np.eye(self.nterms) #top left
self.gam[:self.nterms, self.nterms:] = 0 #top right
self.gam[self.nterms:, :self.nterms] = 0 #bottom left
self.gam[self.nterms:, self.nterms:] = self.lam_qdotdot[:,:]#botom right
#self.gam[np.abs(self.gam)<1e-8] = 0.0
# self.lam_qdotdot #+ve
#
# self.psi_q #-ve
#
# self.psi_qdot*=1 #-ve
return
def _make_psi(self):
"""Make the 2n =*2n block psi matrix
Used with `use_analytical`=True"""
self.psi = np.zeros((2*self.nterms, 2*self.nterms), dtype=float)
self.psi[:self.nterms, :self.nterms] = 0 #top left
self.psi[:self.nterms, self.nterms:] = -np.eye(self.nterms) #top right
self.psi[self.nterms:, :self.nterms] = self.psi_q #bottom left
self.psi[self.nterms:, self.nterms:] = self.psi_qdot#botom right
#self.psi[np.abs(self.psi)<1e-8] = 0.0
return
def _make_eigs_and_v(self):
"""make Igam_psi, v and eigs, and Igamv
#TODO: make sure this is all still relevatn after copyint from speccon1d_unsat
Finds the eigenvalues, `self.eigs`, and eigenvectors, `self.v` of
inverse(gam)*psi. Once found the matrix inverse(gamma*v), `self.Igamv`
is determined.
Notes
-----
From the original equation
.. math:: \\mathbf{\\Gamma}\\mathbf{A}'=\\mathbf{\\Psi A}+loading\\:terms
`self.eigs` and `self.v` are the eigenvalues and eigenvegtors of the matrix `self.Igam_psi`
.. math:: \\left(\\mathbf{\\Gamma}^{-1}\\mathbf{\\Psi}\\right)
"""
# self.psi[np.abs(self.psi) < 1e-8] = 0.0
Igam_psi = np.dot(np.linalg.inv(self.gam), self.psi)
self.eigs, self.v = np.linalg.eig(Igam_psi)
self.v = np.asarray(self.v)
self.Igamv = np.linalg.inv(np.dot(self.gam, self.v))
if False:
for i, eig in enumerate(self.eigs):
print("{:s}, {:12.6f}".format(str(i+1).zfill(3),eig))
[docs] def make_time_dependent_arrays(self):
"""make all time dependent arrays
See Also
--------
self.make_E_Igamv_the()
"""
self.make_E_Igamv_the()
self.v_E_Igamv_the = np.dot(self.v, self.E_Igamv_the)
[docs] def make_E_Igamv_the(self):
"""sum contributions from all loads
Calculates all contributions to E*inverse(gam*v)*theta part of solution
defl=phi*vE*inverse(gam*v)*theta. i.e. surcharge, vacuum, top and bottom
pore pressure boundary conditions. `make_E_Igamv_the` will create
`self.E_Igamv_the`. `self.E_Igamv_the` is an array
of size (nterms, len(tvals)). So the columns are the column array
E*inverse(gam*v)*theta calculated at each output time. This will allow
us later to do defl = phi*v*self.E_Igamv_the
See Also
--------
_make_E_Igamv_the_mvl : moiving point load contribution
"""
self.E_Igamv_the = np.zeros((2*self.nterms, len(self.tvals_norm)), dtype=complex)
if self.has_moving_loads:
self._make_E_Igamv_the_mvpl() #moving point loads
self.E_Igamv_the += self.E_Igamv_the_mvpl
if self.has_stationary_loads:
self._make_E_Igamv_the_stationary() #moving point loads
self.E_Igamv_the += self.E_Igamv_the_stationary
return
def _make_E_Igamv_the_mvpl(self):
"""Moving point load contribution to self.E_Igam_the
Makes `self.E_Igamv_mvpl` which is an array
of size (nterms, len(tvals))
"""
self.E_Igamv_the_mvpl = speccon1d.dim1sin_E_Igamv_the_mvpl(
self.beta_block,
self.eigs,
self.tvals_norm,
self.Igamv,
self.moving_loads_norm,
dT=1.0,
theta_zero_indexes=slice(None, self.nterms),
implementation=self.implementation)
return
def _make_E_Igamv_the_stationary(self):
"""Stationary point load contribution to self.E_Igam_the
Makes `self.E_Igamv_stationary` which is an array
of size (nterms, len(tvals))
"""
pseudo_k = [1 for v in self.stationary_loads_x_norm]
self.E_Igamv_the_stationary = speccon1d.dim1sin_E_Igamv_the_deltamag_linear(
self.beta_block,
self.eigs,
self.tvals_norm,
self.Igamv,
self.stationary_loads_x_norm,
pseudo_k,
self.stationary_loads_vs_t_norm,
self.stationary_loads_omega_phase_norm,
dT=1,
theta_zero_indexes=slice(None, self.nterms),
implementation=self.implementation)
return
[docs] def make_output(self):
"""make all output"""
self._make_defl()
def _make_defl(self):
"Make deflection and deflection rate"""
self.defl_norm = speccon1d.dim1sin_f(self.beta, self.xvals_norm,
self.tvals_norm,
self.v_E_Igamv_the[:self.nterms, :],
drn=0)
self.defl_norm = np.atleast_2d(self.defl_norm)
#check if complex part is small
if np.allclose(np.imag(self.defl_norm),0,atol=1e-12):
self.defl_norm = np.real(self.defl_norm)
else:
raise ValueError('Imaginary parts of self.defl are > 1e-12')
if not self.L is None:
self.defl = self.defl_norm * self.L
return
[docs] def saveme(self):
"""Save deflection vs time to file
Deflection will be saved to :
'self.file_stem + "_defl.csv'
Normalised deflection will be saved to:
'self.file_stem + "_defl_norm.csv"
Notes:
-----
Might not handle singel x value and single t vlue.
"""
#TODO: Need to account for single xvalue and single y value
two_d_defl_norm = np.atleast_2d(self.defl_norm)
inputoutput.save_grid_data_to_file(data_dicts=dict(
name="_defl_norm",
data=two_d_defl_norm,#self.defl_norm,
row_labels=self.xvals_norm,
row_labels_label="x_norm",
column_labels=self.tvals_norm),
file_stem=self.file_stem,
create_directory=False)
if not self.L is None:
two_d_defl = np.atleast_2d(self.defl)
inputoutput.save_grid_data_to_file(data_dicts=dict(
name="_defl",
data=two_d_defl,#self.defl,
row_labels=self.xvals,
row_labels_label="xvals",
column_labels=self.tvals),
file_stem=self.file_stem,
create_directory=False)
[docs] def load_defl(self):
"""Loads defl_norm from txt file if it exists
Returns
-------
Loaded : boolean
If True then self.defl either already exists or has been
successfully loaded.
"""
# fname = self.file_stem + "_defl_norm.csv"
if hasattr(self, "defl_norm"):
#defl_norm already exists in memory
return True
#check if out file exists
fname = self.file_stem + "_defl_norm.csv"
if not os.path.isfile(fname):
return False
try:
with open(fname,'r') as f:
lines = f.readlines()
t_norm = np.array([float(v) for v in lines[0].split(",")[2:]])
a = np.atleast_2d(np.genfromtxt(fname=fname,skip_header=1, dtype=float,delimiter=","))
x_norm = a[:, 1]
defl_norm = a[:, 2:]
self.defl_norm = defl_norm
if not self.L is None:
self.defl = self.defl_norm * self.L
self.xvals = self.xvals_norm * self.L
print('loaded it')
return True
except:
return False
[docs] def onClick(self, event):
if self.pause:
self.ani.event_source.stop()
else:
self.ani.event_source.start()
self.pause ^= True
[docs] def plot_w_vs_x_overtime(self, ax=None, norm=True):
"""Plot the normalised deflection vs distance for all times
"""
if ax is None:
fig = plt.figure()
ax = fig.add_subplot(111)
self.load_defl()
if norm:
for t, defl in zip(self.tvals_norm, self.defl_norm.T):
ax.plot(self.xvals_norm, defl,
label = "t={:5.2g}".format(t))
ax.set_xlabel("x_norm")
ax.set_ylabel("w_norm")
else:
for t, defl in zip(self.tvals, self.defl.T):
ax.plot(self.xvals, defl,
label = "t={:5.2g}".format(t))
ax.set_xlabel("x")
ax.set_ylabel("w")
leg = ax.legend()
leg.draggable()
return ax
[docs] def animateme(self, xlim=None, ylim=None, norm=True, saveme=False,interval=50):
"""Animate the deflection vs distance over time plot
Will display beam osciallations and evolving max and min deflection
envelopes. Position and magnitude of load will also be shown.
Parameters
----------
xlim : tuple of 2 floats, optional
Limits of x-axis , default xlim=None, i.e. use Beam ends
ylim : tuple of 2 floats, optional
Limits of y-axis , default ylim=None, i.e. limits will be
calculated ylim=(1.5*defl_min,1.1*defl_max)
norm : True/False, optional
If norm=True normalised values will be plotted. Default norm=True.
saveme : False/True
Whether to save the animation to disk. Default saveme=False.
interval : float
Number of miliseconds for each frame. Default interval=50.
"""
self.pause = False
self.load_defl()
fig = plt.figure()
ax = fig.add_subplot(111)
ax2 =ax.twinx()
if norm:
xx = self.xvals_norm
tt = self.tvals_norm
defl = self.defl_norm
if self.has_moving_loads:
moving_loads = self.moving_loads_norm
if self.has_stationary_loads:
stationary_loads_x = self.stationary_loads_x_norm
stationary_loads_vs_t = self.stationary_loads_vs_t_norm
stationary_loads_omega_phase = self.stationary_loads_omega_phase_norm
ax.set_xlabel("x_norm")
ax.set_ylabel("w_norm")
ax2.set_ylabel("Fz_norm")
else:
xx = self.xvals
tt = self.tvals
defl = self.defl
if self.has_moving_loads:
moving_loads = self.moving_loads
if self.has_stationary_loads:
stationary_loads_x = self.stationary_loads_x_norm
stationary_loads_vs_t = self.stationary_loads_vs_t_norm
stationary_loads_omega_phase = self.stationary_loads_omega_phase_norm
ax.set_xlabel("x")
ax.set_ylabel("w")
ax2.set_ylabel("Fz")
defl_min = np.min(defl)
defl_max = np.max(defl)
x_min = np.min(xx)
x_max = np.max(xx)
if not xlim is None:
ax.set_xlim(xlim)
else:
ax.set_xlim(x_min,x_max)
if not ylim is None:
ax.set_ylim(ylim)
else:
ax.set_ylim(2*defl_min,1.5*defl_max)
ax.set_ylim(1.5*defl_min,1.1*defl_max)
max_defl_evolve = np.zeros_like(defl)
min_defl_evolve = np.zeros_like(defl)
max_defl_evolve[:,0] = defl[:,0]
min_defl_evolve[:,0] = defl[:,0]
for i in range(len(tt)):
if i==0:
continue
max_defl_evolve[:,i] = np.maximum(defl[:,i,], max_defl_evolve[:,i-1])
min_defl_evolve[:,i] = np.minimum(defl[:,i,], min_defl_evolve[:,i-1])
ax.grid()
ax.invert_yaxis()
#moving point load
data_line, = ax.plot((np.nan,np.nan),(np.nan,np.nan),
linestyle="-",
color='blue')
min_line, = ax.plot((np.nan,np.nan),(np.nan,np.nan),
linestyle="-",
color='orange')
max_line, = ax.plot((np.nan,np.nan),(np.nan,np.nan),
linestyle="-",
color='cyan')
pmax=1e-50
pmin=1e50
p_scale_mvl=1e-50
if self.has_moving_loads:
# mvpl1_line1,mvpl1_line2,..., mvpl2_line1, mvpl2_line2, mvpl_line3, ...
mvpl_lines = [ax2.annotate("", xy=(np.nan,np.nan), xytext=(np.nan, np.nan),
arrowprops=dict(facecolor='red', shrink=0.0)
#arrowstyle="->",
#edgecolor='red',
#shrinkline=0
#)
) for mvpl in moving_loads for j in mvpl.xt]
p_scale_mvl = abs(max([p for mvpl in moving_loads for p in mvpl.p]))
pmax = max(pmax,max([p for mvpl in moving_loads for p in mvpl.p]))
pmin = min(pmin,min([p for mvpl in moving_loads for p in mvpl.p]))
else:
mvpl_lines=[]
p_scale_stationary=1e-50
if self.has_stationary_loads:
x_map = np.zeros(len(stationary_loads_x), dtype=int)
x_map[0] = 0
for i in range(1,len(stationary_loads_x)):
for j in range(i+1):
if np.isclose(stationary_loads_x[i], stationary_loads_x[j]):
x_map[i]=j
break
stationary_lines=[ax2.annotate("", xy=(np.nan,np.nan), xytext=(np.nan, np.nan),
arrowprops=dict(facecolor='green', shrink=0.0)
#arrowstyle="->",
#edgecolor='red',
#shrinkline=0
#)
) for v in range(max(x_map)+1)]#stationary_loads_x]
p_scale_stationary = abs(max([p for mag_vs_time in stationary_loads_vs_t for p in mag_vs_time.y]))
pmin_temp=[]
pmax_temp=[]
for i in range(max(x_map)+1):
sum_mag_max=0
sum_mag_min=0
for j in range(len(stationary_loads_x)):
if x_map[j]!=i:
continue
mag_max = max(stationary_loads_vs_t[j].y)
mag_min = min(stationary_loads_vs_t[j].y)
sum_mag_max +=mag_max
sum_mag_min +=mag_min
pmin_temp.append(sum_mag_min)
pmax_temp.append(sum_mag_max)
pmax = max(pmax, max(pmax_temp))
pmin = min(pmin, min(pmin_temp))
else:
stationary_lines=[]
ax2.set_ylim(-2*max(abs(pmin) , abs(pmax)),1.5*max(abs(pmin) , abs(pmax)))
# align_yaxis(ax, 0, ax2, 0)
align_yaxis(ax,ax2)
p_scale = max(p_scale_mvl, p_scale_stationary)
p_scale *=2/abs(defl_min)/3 # max load will plot to half the scale
time_template = 'time = {:.4f}'
time_text = ax.text(0.05, 0.9, '', transform=ax.transAxes)
fig.tight_layout()
def init():
i=0
if self.has_moving_loads:
for mvpl in moving_loads:
for j in range(len(mvpl.xt)):
mvpl_lines[i].xy = (np.nan,np.nan)
mvpl_lines[i].xyann = (np.nan,np.nan)
i=i+1
if self.has_stationary_loads:
for j in range(max(x_map)+1):#len(stationary_loads_x)):
stationary_lines[j].xy = (np.nan,np.nan)
stationary_lines[j].xyann = (np.nan,np.nan)
data_line.set_data([], [])
min_line.set_data([], [])
max_line.set_data([], [])
time_text.set_text('')
return tuple(stationary_lines) + tuple(mvpl_lines) + (data_line,) + (time_text,) + (min_line,)+(max_line,)
def animate(frame):
i=0
if self.has_moving_loads:
for mvpl in moving_loads:
if mvpl.v<0:
x0 = mvpl.x0 + mvpl.L
else:
x0 = mvpl.x0
thisx = mvpl.axle_positions(x0=x0, t=tt[frame],v0=mvpl.v,t0=mvpl.t0)
#thisy = -mvpl.p / p_scale#defl_max/5
thisy=mvpl.p
for j in range(len(mvpl.xt)):
mvpl_lines[i].xy = (thisx[j],0)
mvpl_lines[i].xyann = (thisx[j], thisy[j])
i=i+1
if self.has_stationary_loads:
for j in range(max(x_map)+1):#len(stationary_loads_x)):
sum_mag = 0
for k in range(len(stationary_loads_x)):
if x_map[k] != j:
continue
mag = pwise.pinterp_x_y(a=stationary_loads_vs_t[k],xi=tt[frame])
if stationary_loads_omega_phase[k] is None:
pass
else:
omega, phase = stationary_loads_omega_phase[k]
mag*=np.cos(omega*tt[frame]+phase)
sum_mag +=mag
stationary_lines[j].xy = (stationary_loads_x[j],0)
#stationary_lines[j].xyann = (stationary_loads_x[j], -mag/p_scale)
stationary_lines[j].xyann = (stationary_loads_x[j], sum_mag)
min_line.set_data(xx, min_defl_evolve[:,frame])
max_line.set_data(xx, max_defl_evolve[:,frame])
data_line.set_data(xx, defl.T[frame])
time_text.set_text(time_template.format(tt[frame]))
return tuple(stationary_lines) + tuple(mvpl_lines) + (data_line,) + (time_text,) + (min_line,)+(max_line,)
self.ani = animation.FuncAnimation(fig, animate, len(tt),
interval=interval, blit=True, init_func=init,repeat=True)
fig.canvas.mpl_connect('button_press_event', self.onClick)
# anim_running = True
#
# def onClick(event):
# nonlocal anim_running
# if anim_running:
# ani.event_source.stop()
# anim_running = False
# else:
# ani.event_source.start()
# anim_running = True
if saveme:
pass
self.ani.save(self.file_stem+"_anim.mp4", fps=15,codec="libx264")#changing the codec seems to work for me.
# ani.save(self.file_stem+"_anim", fps=15)#changing the codec seems to work for me.
return self.ani
[docs] def defl_vs_time_under_loads(self, xlim_norm=(0,1)):
"""Deflection vs time under each moving load
i.e. What is deflection under each axle
Creates and populates self.moving_load_w_norm .
Normalised deflections under each moving load at each time value.
Each element of the list corresponds to each moving load. The array
is of size (len(mvpl.x), len(tvals)) i.e. for each axle in the moving
load group.
Parameters
----------
xlim_norm : tuple, optional
won't show defelction untill within these normalised bounds
Default xlim_norm=(0,1)
"""
self.moving_load_w_norm = []
for i, mvpl in enumerate(self.moving_loads_norm):
if mvpl.v<0:
x0 = mvpl.x0 + mvpl.L
else:
x0 = mvpl.x0
wvals = np.zeros((len(mvpl.x), len(self.tvals_norm)))
for j, t in enumerate(self.tvals_norm):
xnow = mvpl.axle_positions(x0=x0, t=t,v0=mvpl.v, t0=mvpl.t0)
wnow = np.interp(xnow, self.xvals_norm, self.defl_norm[:,j])
#remove points where load is not on beam.
wnow[xnow<xlim_norm[0]] = np.nan
wnow[xnow>xlim_norm[1]] = np.nan
wvals[:,j] = wnow
self.moving_load_w_norm.append(wvals)
if not self.L is None:
self.moving_load_w = [v*self.L for v in self.moving_load_w_norm]
return
[docs]class MovingPointLoads(object):
"""Multi-axle vehicle represented by point loads in a line
Parameters
----------
x : 1d array or list
x-coord of point loads. Vehicle assumed to start at 0.
p : 1d array or list
Magnitude of point loads at corresponding x value.
offset : float, optional
x-coords will be offset by this distance. Positive values will move
x to the right (i.e. increase in x).
v : float, optional
Velocity. Default v = None, i.e. you'll need to enter in various
functions.
L : float, optional
Length of Beam. Default L=None i.e. you'll need to explicitly
enter in various functions. You can use a psuedo L with x0 to have the
loads begin and end midspan.
t0 : float, optional
Time the point loads begin at x0 on the beam. default t0=0 ie. you'll
have to explicitly enter in various functions.
x0 : float, optional
Start position of loads on beam. Defualt x0=0.0 .
Attributes
----------
xt : 1d array
transformed x coords after mirror and offset are applied.
start_time, end_time : 1d array
start and end time for each point load traversing length L.
"""
def __init__(self, x, p, offset=0,v=None,L=None,t0=0,x0=0.0):
self.x = np.array(x, dtype=float)#array(x.copy())
self.p = np.array(p, dtype=float)#np.asarray(p.copy())
self.offset = offset
self.xt = self.x.copy()
self.xt += offset
self.L = L
self.t0 = t0
self.v = v
self.x0 = x0
self.xnow = np.zeros_like(self.xt)
self.pnow = np.zeros_like(self.xt)
return
[docs] def setv(self, v):
"""Set the vehicle's velocity, then calc start_time and end_time when
vehicle's axles enter and leave the beam"""
self._v = v
if not v is None:
if v > 0:
self.start_time = self.t0 + (0 - self.xt) / v
self.end_time = self.t0 + (self.L - self.xt) / v
else:
#train is travelling in negative direction
# the (-1) is to mirror around reference point
self.start_time = self.t0 - (-1)*self.xt / v
self.end_time = self.t0 - (self.L + (-1)*self.xt) / v
else:
self.start_time = None
self.end_time = None
return
[docs] def getv(self):
return self._v
[docs] def delv(self):
del self._v
del self.start_time
del self.end_time
return
v = property(getv, setv, delv, "Velocity")
[docs] def point_loads_on_beam(self, t):
"""Position and magnitude of point loads between [0, L] at time t
Parameters
----------
t : float
Current time.
Returns
-------
x : 1d array of float
position of each load at time t.
p : 1d array of float
Load at time t. Will be set to zero if load is not beteen
0 and L.
Notes
-----
Position can be offset by self.x0
"""
if self.v >= 0:
self.xnow[:] = self.axle_positions(t=t, v0=self.v, x0=self.x0)
else:
self.xnow[:] = self.axle_positions(t=t, v0=self.v, x0=self.L)
self.pnow[:] = self.p
self.pnow[((t< self.start_time) | (t>self.end_time))] = 0
return self.xnow, self.pnow
[docs] def convert_to_specbeam(self, v=None, L=None, t0=None, x0=None):
"""Convert vehicle at velocity v into
surcharge_vs_time and surcharge_omega_phase PolyLines for use with
spectral method.
Parameters
----------
v : float, optional
Velocity of vehicle. Default v=None in which case the v used
during object initialization will be uses
L : float, optional
Length of beam. Default L=None in which case the L used
during object initialization will be uses
t0 : float, optional
time when vehicle reference point begins entering the beam. Default t0=0.
If v>0 vehicle will enter from the right. If v<0 vehicle
will enter from the right. v=0 will result in an error.
x0 : float, optional
Position where loads start. Default x0=none, i.e. value used
during initialization will be used.
Notes
-----
Vehicle is made up of axles at a relative distance dxi from a
reference point. dxi should be negative. If velocity v is positive
then the reference point enters the beam (at x=x0) at time t0. The
position of the ith axle is:
xi = x0 + dxi + v * (t - t0)
The ith axle enters the beam at position x0 at time
tstarti = t0 - dxi / v
and leaves the beam (x = x0 + L) at time
tendi = t0+(L-dxi)/v
If v is negative then the reference point enters the beam from
the right, then we first flip the vehicle by making dx_i=-dxi. The
positiion and times entering and leaving the beam are
xi = x0 + L + dx_i + v * (t - t0)
and
tstart = t0 - dx_i / v
tend = t0 - (L + dx_i)/v
For use in spec beam we will need sin(M*xi) in a form
sin(omega*t+phase).
for v>0
xi = v * t + (x0 + dxi - v * t0)
so omega/M = v and phase/M = x0 + dxi-v*t0
for v<0
xi = v * t + (x0 + L + dx_i - v * t0)
so omega/M = v and phase/M = x0 + L + dx_i - v*t0
Returns
-------
plines : list of PolyLine
omega_phase : list of two element tuples
frequency and phase angle for use in spec beam. as per the notes
above will need to multiply both omega and phase by M for use
in specbeam.
"""
if v is None:
v = self.v
if L is None:
L = self.L
if t0 is None:
t0 = self.t0
if x0 is None:
x0 = self.x0
plines=[]
omega_phase=[]
if v==0:
raise ValueError("Can not have v=0 i.e. no static loads.")
if v > 0:
for xi, p in zip(self.xt, self.p):
start_time = t0 + (0 - xi) / v
end_time = t0 + (L - xi) / v
plines.append(PolyLine([start_time,start_time,end_time,end_time],
[0, p, p,0]))
omega_phase.append((v, xi + x0 - v * t0))
else:
#train is travelling in negative direction
for xi, p in zip(self.xt, self.p):
xi *= -1 #mirror around reference point.
start_time = t0 - xi / v
end_time = t0 - (L + xi) / v
plines.append(PolyLine([start_time,start_time,end_time,end_time],
[0, p, p, 0]))
omega_phase.append((v, x0 + L + xi - v*t0))
return plines, omega_phase
[docs] def plot_specbeam_PolyLines(self, ax=None, v=1, L=1, t0=0, x0=0):
"""Plot load_vs_time for each vehicle axis.
Parameters
----------
ax : Matplotlib.Axes, optional
Axes to plot on. Default ax=None i.e. a new fig and ax will be
created.
v : float, optional
velocity, default v=1.
L : float, optional
Lenght of beam default L=1
x0, t0 : float, optional
Vehicle starts at positon x0 at time t0. Default x0=0, t0=0
"""
plines, omega_phases = self.convert_to_specbeam(v=v, L=L, t0=0, x0=x0)
if ax is None:
fig, ax = plt.subplots()
for i in range(len(plines)):
omega,phase = omega_phases[i]
xx = plines[i].x
yy=plines[i].y
label = "$i={},\\omega={},\\phi={}$".format(i,omega,phase)
label = "${},{},{}$".format(i,omega,phase)
ax.plot(xx,yy,label=label)
ax.set_title("L={},v={},t0={},x0={}".format(L,v,t0,x0))
ax.set_xlabel("time")
ax.set_ylabel("magnitude")
leg = ax.legend(title ="$i,\\omega,\\phi$",
labelspacing=0)
leg.draggable(0)
return ax
[docs] def axle_positions(self, t=0, v0=0, t0=0, x0=0):
"""Positions of axles at given time
Parameters
----------
t : float
Time of interest
v0 : float, optional
Initial speed of vehicle. Default v0=0.
If v0<0 then axle coords will be multiplied by -1 before moving.
t0 : float, optional
Time when vehicle starts moving at velocity v0. Default t0=0
x0 : float, optional
Start postion of vehicle
Returns
-------
xnew : 1d array
xcoords of axles at time t.
"""
self.xnow[:]=self.xt[:]#xnew = self.xt.copy()
if v0<0:
self.xnow*=-1
self.xnow+= x0 + v0*(t-t0)
return self.xnow
[docs] def plotme(self, ax=None,t=0, v0=0, t0=0, x0=0, xlim=None, ylim=None):
"""Plot the load at time t
Each axle load is represented by an arrow, with head at 0 and tail
at the load magnitude.
Parameters
----------
ax : pyplot.Axes, optional
Axes object to plot on. Default ax=None i.e. new figure and
axes will be created.
t : float, optional
Time at which to plot the vehicle positoin. Default t=0.
v0 : float, optional
Initial velocity of vehicle. default v0=0. If v0<0 then
positions will be multiplied by -1 before adjusting position.
t0 : float, optional
Time the vehicle starts moving. Default t0=0.
x0 : float, optional
Start position of vehicle
xlim, ylim : tuple of size 2, default xlim=ylim=None.
axes limits.
Returns
-------
ax : pyplot.Axes opbect
Axes object with plotted vehicle position.
"""
if ax is None:
fig = plt.figure()
ax = fig.add_subplot(111)
xnew = self.axle_positions(t=t, v0=v0, t0=t0, x0=x0)
lines = [ax.annotate("", xy=(xnew[i], 0), xytext=(xnew[i], self.p[i]),
arrowprops=dict(facecolor='red', shrink=0.0)
#arrowstyle="->",
#edgecolor='red',
#shrinkline=0
#)
) for i in range(len(self.xt))]
if not xlim is None:
ax.set_xlim(xlim)
if not ylim is None:
ax.set_ylim(ylim)
return ax
[docs] def animateme(self, x0=0, t0=0, tf=1,v0=1, xlim=None, ylim=None):
"""Animate moving vehicle
Parameters
----------
x0 : float, optional
Start position of vehicle, default=0.
t0 : float, optional
Start time of animation. Default t0=0.
tf : float, optional
End time of animation. Default tf=1.
v0 : float, optional
Initial velocity of vehicle. Default v0=1. If v0<0 then
positions will be multiplied by -1 before adjusting position.
xlim, ylim : tuple of size 2, default xlim=ylim=None.
axes limits.
Returns
-------
ani : matplotlib animation
Animation of moving vehicle
"""
#http://stackoverflow.com/questions/15887820/animation-by-using-matplotlib-errorbar
fig = plt.figure()
ax = fig.add_subplot(111)
if not xlim is None:
ax.set_xlim(xlim)
if not ylim is None:
ax.set_ylim(ylim)
ax.grid()
lines = [ax.annotate("", xy=(np.nan,np.nan), xytext=(np.nan, np.nan),
arrowprops=dict(facecolor='red', shrink=0.0)
#arrowstyle="->",
#edgecolor='red',
#shrinkline=0
#)
) for i in range(len(self.xt))]
time_template = 'time = {:.1f}'
time_text = ax.text(0.05, 0.9, '', transform=ax.transAxes)
def init():
time_text.set_text("")
for i in range(len(self.xt)):
lines[i].xy = (np.nan,np.nan)
lines[i].xyann = (np.nan,np.nan)
time_text.set_text('')
return tuple(lines)+(time_text,)
def animate(t):
thisx = self.axle_positions(t=t,v0=v0,t0=t0,x0=x0)
thisy = self.p
for i in range(len(self.xt)):
lines[i].xy = (thisx[i], 0)
lines[i].xyann = (thisx[i],thisy[i])
time_text.set_text(time_template.format(t))
return tuple(lines) + (time_text,)
ani = animation.FuncAnimation(fig, animate, np.linspace(t0,tf,100),
interval=50, blit=True, init_func=init,repeat=True)
return ani
[docs] def time_axles_pass_point(self, x, v0=0, t0=0, x0=0):
"""time values when each moving load passes a certain point
Parameters
----------
x : float
x-coord of interest
v0 : float, optional
Initial speed of vehicle. Default v0=0.
If v0<0 then axle coords will be multiplied by -1 before moving.
t0 : float, optional
Time when vehicle starts moving at velocity v0. Default t0=0
x0 : float, optional
Start postion of vehicle
L : float, optional
Length of beam. Default L=None in which case the L used
during object initialization will be uses.
Returns
-------
t_pass : 1d array of float
list of times corresponding to when each axle passes the given
x value.
Notes
-----
"""
self.xnow[:]=self.xt[:]
if v0<0:
self.xnow*=-1
return (x-self.xnow-x0)/v0 + t0
[docs] def time_axles_pass_beam_point(self,x):
"""Time values when each axle passes a particular point on a beam
Parameters
----------
x : float
x coord of interest
Returns
-------
t_pass : 1d aray of float
time values when each axle passes the particualr point
Notes
-----
If self.v < 0 then loads enter beam from the left, i.e. at time
t0 axles will be at L+x0+xt
Not really sure if I've accountered for self.x0!=0 for the
negative velocity. Just make self.x0 =0 for simplicity
"""
if self.v>0:
t_pass = self.time_axles_pass_point(x, v0=self.v, t0=self.t0, x0=self.x0)
elif self.v<0:
t_pass = self.time_axles_pass_point(x, v0=self.v, t0=self.t0, x0=(self.L+self.x0))
else:
raise ValueError("v can't be zero")
return t_pass
##***Don't think I need this combo of align_yaxis & adjust_yaxis, I use a
##***simpler version of align_yaxis that is hardcoded to align at y=0 on both
##***axes
#def align_yaxis(ax1, v1, ax2, v2):
# """Adjust y-axis bounds so that value v1 in ax1 is aligned to value v2
# in ax2.
#
# Should keep keep all data visible.
#
# Parameters
# ----------
# ax1, ax2 : matplotlib.Axes
# Axes objects in which to align particular y-axis values
# v1, v2 : float
# y-axis values on which to align.
#
# See Also
# --------
# adjust_yaxis : helper routine to move shift yaxis
#
# Notes
# -----
# Copied from http://stackoverflow.com/questions/10481990/matplotlib-axis-with-two-scales-shared-origin
#
# """
#
# _, y1 = ax1.transData.transform((0, v1))
# _, y2 = ax2.transData.transform((0, v2))
# adjust_yaxis(ax2,(y1-y2)/2,v2)
# adjust_yaxis(ax1,(y2-y1)/2,v1)
#
# return
#
#
#def adjust_yaxis(ax, ydif, v):
# """Adjust y-axis bounds by shifting ydif but keeping value v in the same
# visual location.
#
# Parameters
# ----------
# ax : Matplotlib.Axes
# Axes object to change
# ydif : float
#
# See Also
# --------
# align_yaxis : parent routine
#
# Notes
# -----
# Copied from: http://stackoverflow.com/questions/10481990/matplotlib-axis-with-two-scales-shared-origin
#
# """
#
# inv = ax.transData.inverted()
# _, dy = inv.transform((0, 0)) - inv.transform((0, ydif))
# miny, maxy = ax.get_ylim()
# miny, maxy = miny - v, maxy - v
# if -miny>maxy or (-miny==maxy and dy > 0):
# nminy = miny
# nmaxy = miny*(maxy+dy)/(miny+dy)
# else:
# nmaxy = maxy
# nminy = maxy*(miny+dy)/(maxy+dy)
# ax.set_ylim(nminy+v, nmaxy+v)
#
# return
[docs]def align_yaxis(ax1, ax2):
"""Adjust y-axis limits so zeros of the two axes align, zooming them out
by same ratio.
Parameters
----------
ax1, ax2 : matpolotlib.Axes
Axes to align.
"""
axes = (ax1, ax2)
extrema = [ax.get_ylim() for ax in axes]
tops = [extr[1] / (extr[1] - extr[0]) for extr in extrema]
# Ensure that plots (intervals) are ordered bottom to top:
if tops[0] > tops[1]:
axes, extrema, tops = [list(reversed(l)) for l in (axes, extrema, tops)]
# How much would the plot overflow if we kept current zoom levels?
tot_span = tops[1] + 1 - tops[0]
b_new_t = extrema[0][0] + tot_span * (extrema[0][1] - extrema[0][0])
t_new_b = extrema[1][1] - tot_span * (extrema[1][1] - extrema[1][0])
axes[0].set_ylim(extrema[0][0], b_new_t)
axes[1].set_ylim(t_new_b, extrema[1][1])
[docs]def DAFrelative(alp, bet, k2_k1):
"""Relative Dynamic Amplification Factor, based on ratio of beam
stiffness to some reference beam stiffness.
Parameters
----------
alp : float
velocity ratio. alp=v/vcr; vcr= SQRT(2*SQRT(k*EI)/(rho*A))
bet : float
damping ratio. bet=c/ccr; ccr=2*SQRT(rho*A*k)
k2_k1 : float
stiffness ratio of beam to current beam
Returns
-------
DAF : float
Relative Dynamic Amplification Factor
Notes
-----
DAFrelative is a terse implementation for an appendix of code in
Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
import numpy as np
alp /= k2_k1**(0.25)
bet *= (1/k2_k1)**0.5
gam = np.roots([1.0, 0.0, 4.0 * alp**2, -8 * alp * bet, 4])
cmat = np.zeros((4, 4), dtype=complex)
cmat[0, :] = [1, 1, -1, -1]
cmat[1, :] = [gam[0], gam[1], -gam[2], -gam[3]]
cmat[2, :] = [gam[0]**2, gam[1]**2, -gam[2]**2, -gam[3]**2]
cmat[3, :] = [gam[0]**3, gam[1]**3, -gam[2]**3, -gam[3]**3]
rhs = np.array([0, 0, 0, 8.0], dtype=complex)
A = np.linalg.solve(cmat, rhs)
s = np.linspace(-8,8,1001, dtype=complex)
disp = np.empty(len(s), dtype=complex)
s_neg = s[s<0]
s_pos = s[s>=0]
disp[s>=0] = A[0]*np.exp(s_pos*gam[0])+A[1]*np.exp(s_pos*gam[1])
disp[s<0] = A[2]*np.exp(s_neg*gam[2])+A[3]*np.exp(s_neg*gam[3])
DAF = np.real(np.max(disp) * (1 / k2_k1)**0.75)
return DAF
[docs]def point_load_static_defl_shape_function(x, L, xc=0.0):
"""Deflection shape function for static point load on beam on foundation.
Parameters
----------
x : float
x value
L : float
Characteristc length (1/lam)
x0 : float, optional
position of load.
Returns
-------
nu : float
shape function at x
"""
return np.exp(-np.abs(x-xc)/L)*(np.cos(np.abs(x-xc)/L)+np.sin(np.abs(x-xc)/L))
[docs]def multi_static(xc, frel, wsingle=1, xlim=(-10,10), n=500, L=1):
"""Deflection shape for multiple static point loads
Parameters
----------
xc : float
x-coord of each point load
frel : float
relative magnitude of each point load
wsingle : float, optional
maximum deflection under unit load
xlim : tuple with two values
xmin, xmax of evaluation
L : float, optional
Characterisic length
Returns
-------
x,w : 1d ndarray
x coords, and static deflection
See Also
--------
point_load_static_defl_shape_function : single load function
"""
x = np.linspace(xlim[0], xlim[1], n)
w = np.zeros_like(x)
for i, (xc_,frel_) in enumerate(zip(xc, frel)):
w+= frel_*point_load_static_defl_shape_function(x=x, L=L, xc=xc_)
return x, w*wsingle
[docs]def SpecBeam_const_mat_midpoint_defl_runme_analytical_play():
"""Test SpecBeam for constant mat: close to Ding et al Figure 8, displacement vs time at
beam midpoint (using the runme method) but with k3=0"""
start_time0 = time.time()
ftime = datetime.datetime.fromtimestamp(start_time0).strftime('%Y-%m-%d %H%M%S')
# expected values are digitised from Ding et al 2012.
expected_50terms_t = np.array(
[ 3.511, 3.553, 3.576, 3.605, 3.621, 3.637, 3.651, 3.661,
3.676, 3.693, 3.701, 3.717, 3.733, 3.743, 3.761, 3.777,
3.79 , 3.804, 3.822, 3.833, 3.848, 3.862, 3.877, 3.899,
3.919, 3.94 , 3.956, 3.97 , 3.978, 3.99 , 4.001, 4.014,
4.025, 4.041, 4.052, 4.064, 4.076, 4.086, 4.094, 4.104,
4.112, 4.123, 4.134, 4.146, 4.159, 4.172, 4.18 , 4.192,
4.212, 4.221, 4.234, 4.255, 4.273, 4.292, 4.324, 4.357,
4.373, 4.399, 4.418, 4.445])
#old values for k3~=0
# expected_50terms_displacement = np.array(
# [ -1.30900000e-04, 8.85900000e-05, 2.63900000e-04,
# 3.30200000e-04, 3.30500000e-04, 2.87100000e-04,
# 2.00000000e-04, 9.09500000e-05, -8.36000000e-05,
# -3.01800000e-04, -4.32800000e-04, -6.07300000e-04,
# -7.59900000e-04, -8.90900000e-04, -9.99800000e-04,
# -1.02100000e-03, -9.99100000e-04, -8.67700000e-04,
# -5.61300000e-04, -2.11300000e-04, 2.04300000e-04,
# 7.29200000e-04, 1.42900000e-03, 2.41300000e-03,
# 3.46300000e-03, 4.29400000e-03, 5.08100000e-03,
# 5.62800000e-03, 5.86800000e-03, 6.19600000e-03,
# 6.39300000e-03, 6.52500000e-03, 6.59100000e-03,
# 6.48200000e-03, 6.32900000e-03, 6.02300000e-03,
# 5.69600000e-03, 5.45500000e-03, 5.06200000e-03,
# 4.75600000e-03, 4.40700000e-03, 3.90400000e-03,
# 3.35800000e-03, 2.87800000e-03, 2.35300000e-03,
# 1.82900000e-03, 1.52300000e-03, 9.98700000e-04,
# 5.18300000e-04, 2.12500000e-04, -4.95200000e-05,
# -2.67600000e-04, -3.76500000e-04, -3.76100000e-04,
# -2.00600000e-04, 1.87300000e-05, 1.06500000e-04,
# 1.94500000e-04, 1.94900000e-04, 1.30000000e-04])
expected_50terms_displacement = np.array(
[ -1.74400116e-05, 2.56595529e-04, 3.64963938e-04,
3.96163257e-04, 3.52239419e-04, 2.55908617e-04,
1.36296573e-04, 3.12193756e-05, -1.48035583e-04,
-3.73050108e-04, -4.80427236e-04, -6.84847774e-04,
-8.61705766e-04, -9.44268811e-04, -1.02333177e-03,
-1.00753661e-03, -9.18192599e-04, -7.32771117e-04,
-3.72382730e-04, -7.79760498e-05, 4.06998900e-04,
9.37679667e-04, 1.57072235e-03, 2.58323632e-03,
3.52928135e-03, 4.47726082e-03, 5.12406706e-03,
5.61329890e-03, 5.84231170e-03, 6.13309619e-03,
6.32505191e-03, 6.46203546e-03, 6.49032925e-03,
6.39329339e-03, 6.23960924e-03, 5.99242067e-03,
5.67177967e-03, 5.35598822e-03, 5.07919623e-03,
4.69552191e-03, 4.37030991e-03, 3.90678952e-03,
3.43034780e-03, 2.90774639e-03, 2.35231514e-03,
1.82145021e-03, 1.51724930e-03, 1.09014092e-03,
4.86434682e-04, 2.61200867e-04, -5.02752907e-06,
-2.99906282e-04, -4.26188536e-04, -4.46910518e-04,
-3.09328219e-04, -6.53887601e-05, 4.77786782e-05,
1.84483472e-04, 2.30390682e-04, 2.18090273e-04])
expected_200terms_t = np.array(
[ 3.503, 3.519, 3.539, 3.556, 3.576, 3.595, 3.613, 3.632,
3.651, 3.667, 3.69 , 3.708, 3.727, 3.746, 3.766, 3.782,
3.801, 3.822, 3.84 , 3.856, 3.878, 3.896, 3.915, 3.933,
3.951, 3.97 , 3.99 , 4.009, 4.027, 4.046, 4.067, 4.085,
4.102, 4.122, 4.141, 4.159, 4.178, 4.197, 4.217, 4.234,
4.252, 4.273, 4.291, 4.31 , 4.329, 4.347, 4.368, 4.386,
4.405, 4.423, 4.442, 4.46 , 4.479, 4.495])
#wrong : displ are with k3~=0
expected_200terms_displacement = np.array(
[ 7.04000000e-08, 2.22800000e-05, 2.27000000e-05,
1.23200000e-06, 2.35100000e-05, 2.39300000e-05,
2.46400000e-06, -1.89700000e-05, -6.22700000e-05,
-1.27500000e-04, -1.70700000e-04, -2.57800000e-04,
-3.44800000e-04, -4.53600000e-04, -5.40600000e-04,
-5.84000000e-04, -5.61700000e-04, -4.73800000e-04,
-2.54900000e-04, 1.17100000e-04, 6.85900000e-04,
1.51700000e-03, 2.50100000e-03, 3.70300000e-03,
5.01500000e-03, 6.26200000e-03, 7.20200000e-03,
7.61800000e-03, 7.39900000e-03, 6.74400000e-03,
5.91400000e-03, 5.01800000e-03, 4.10100000e-03,
3.22700000e-03, 2.50600000e-03, 1.85000000e-03,
1.32600000e-03, 9.11400000e-04, 5.84000000e-04,
3.00200000e-04, 1.47600000e-04, 3.87500000e-05,
-4.82900000e-05, -1.13400000e-04, -1.13000000e-04,
-1.34500000e-04, -1.34000000e-04, -1.33600000e-04,
-1.11400000e-04, -6.72600000e-05, -6.68400000e-05,
-6.64500000e-05, -4.41700000e-05, -4.38200000e-05])
vfact=2
t = np.linspace(0, 4.5*2.5/vfact, 500)
x_norm = np.linspace(0,1,2001)
#see Dingetal2012 Table 2 for material properties
pdict = OrderedDict(
E = 6.998*1e9, #Pa
rho = 2373, #kg/m3
L = 160, #m
#v_norm=0.01165,
kf=5.41e-4,
#Fz_norm=1.013e-4,
mu_norm=39.263,
k1_norm=97.552,
# k3_norm=2.497e6,
nterms=200,
BC="SS",
nquad=20,
# k1bar=PolyLine([0,0.5],[0.5,1],[1,1],[1,.25]),
# k1bar=PolyLine([0,0.25,.75,1],[1.0,1,4,4]),
# k1bar=PolyLine([0,0.4,.6,1],[1.0,1,.25,.25]),
k1bar=PolyLine([0,0.4,.6,1],[1.0,1,4,4]),
# k1bar=PolyLine([0,0.5],[0.5,1],[2,1],[2,1]),
moving_loads_x_norm=[[0,-0.05,-0.1]],
moving_loads_Fz_norm=[3*[1.013e-4]],
moving_loads_v_norm=[0.01165*vfact],
tvals=t,
# tvals=np.array([4.0]),
# xvals_norm=np.array([0.5]),
xvals_norm=x_norm,
use_analytical=True,
# implementation="vectorized",
implementation="fortran",
file_stem="specbeam200moving_soft_to_stiff",
force_calc=True,
)
a = SpecBeam(**pdict)
a.runme()
a.saveme()
a.animateme(norm=True,saveme=True,interval=18)
# a.plot_w_vs_x_overtime(norm=True)
plt.show()
# a.calulate_qk(t=t)
#
# yall = a.wofx(x_norm=0.5, normalise_w=False)
ycompare = np.interp(expected_50terms_t, t, a.defl)
print("n=", pdict['nterms'])
end_time0 = time.time()
elapsed_time = (end_time0 - start_time0); print("Total run time={}".format(str(timedelta(seconds=elapsed_time))))
if DEBUG:
title ='SpecBeam, const mat prop vs Ding et al (2012) \nFigure 8 Displacement at Midpoint of beam but with k3=0'
print(title)
print(' '.join(['{:>9s}']*4).format('t', 'expect', 'calc', 'diff'))
for i, j, k in zip(expected_50terms_t, expected_50terms_displacement, ycompare):
print(' '.join(['{:9.4f}']*4).format(i, j, k, j-k))
print()
fig = plt.figure(figsize=(10,6))
ax = fig.add_subplot("111")
ax.set_xlabel("time, s")
ax.set_ylabel("w, m")
ax.set_xlim(3.5, 4.5)
ax.set_title(title)
ax.plot(expected_50terms_t, expected_50terms_displacement,
label="expect n=50", color='green', marker='o', ls='-')
ax.plot(expected_200terms_t, expected_200terms_displacement,
label="expect n=200", color='black', marker=None, ls='-')
ax.plot(t, a.defl, label="calc n=50")
ax.plot(expected_50terms_t, ycompare, label="calc, n=50 (compare)",
color='red', marker='s', ms=4, ls=':')
leg = ax.legend(loc='upper left')
leg.draggable()
plt.show()
assert_allclose(expected_50terms_displacement, ycompare, atol=2.4e-4)
[docs]def SpecBeam_stationary_load_const_mat_midpoint_defl_runme_analytical_play():
"""Test SpecBeam for constant mat: close to Ding et al Figure 8, displacement vs time at
beam midpoint (using the runme method) but with k3=0"""
start_time0 = time.time()
ftime = datetime.datetime.fromtimestamp(start_time0).strftime('%Y-%m-%d %H%M%S')
# expected values are digitised from Ding et al 2012.
expected_50terms_t = np.array(
[ 3.511, 3.553, 3.576, 3.605, 3.621, 3.637, 3.651, 3.661,
3.676, 3.693, 3.701, 3.717, 3.733, 3.743, 3.761, 3.777,
3.79 , 3.804, 3.822, 3.833, 3.848, 3.862, 3.877, 3.899,
3.919, 3.94 , 3.956, 3.97 , 3.978, 3.99 , 4.001, 4.014,
4.025, 4.041, 4.052, 4.064, 4.076, 4.086, 4.094, 4.104,
4.112, 4.123, 4.134, 4.146, 4.159, 4.172, 4.18 , 4.192,
4.212, 4.221, 4.234, 4.255, 4.273, 4.292, 4.324, 4.357,
4.373, 4.399, 4.418, 4.445])
#old values for k3~=0
# expected_50terms_displacement = np.array(
# [ -1.30900000e-04, 8.85900000e-05, 2.63900000e-04,
# 3.30200000e-04, 3.30500000e-04, 2.87100000e-04,
# 2.00000000e-04, 9.09500000e-05, -8.36000000e-05,
# -3.01800000e-04, -4.32800000e-04, -6.07300000e-04,
# -7.59900000e-04, -8.90900000e-04, -9.99800000e-04,
# -1.02100000e-03, -9.99100000e-04, -8.67700000e-04,
# -5.61300000e-04, -2.11300000e-04, 2.04300000e-04,
# 7.29200000e-04, 1.42900000e-03, 2.41300000e-03,
# 3.46300000e-03, 4.29400000e-03, 5.08100000e-03,
# 5.62800000e-03, 5.86800000e-03, 6.19600000e-03,
# 6.39300000e-03, 6.52500000e-03, 6.59100000e-03,
# 6.48200000e-03, 6.32900000e-03, 6.02300000e-03,
# 5.69600000e-03, 5.45500000e-03, 5.06200000e-03,
# 4.75600000e-03, 4.40700000e-03, 3.90400000e-03,
# 3.35800000e-03, 2.87800000e-03, 2.35300000e-03,
# 1.82900000e-03, 1.52300000e-03, 9.98700000e-04,
# 5.18300000e-04, 2.12500000e-04, -4.95200000e-05,
# -2.67600000e-04, -3.76500000e-04, -3.76100000e-04,
# -2.00600000e-04, 1.87300000e-05, 1.06500000e-04,
# 1.94500000e-04, 1.94900000e-04, 1.30000000e-04])
expected_50terms_displacement = np.array(
[ -1.74400116e-05, 2.56595529e-04, 3.64963938e-04,
3.96163257e-04, 3.52239419e-04, 2.55908617e-04,
1.36296573e-04, 3.12193756e-05, -1.48035583e-04,
-3.73050108e-04, -4.80427236e-04, -6.84847774e-04,
-8.61705766e-04, -9.44268811e-04, -1.02333177e-03,
-1.00753661e-03, -9.18192599e-04, -7.32771117e-04,
-3.72382730e-04, -7.79760498e-05, 4.06998900e-04,
9.37679667e-04, 1.57072235e-03, 2.58323632e-03,
3.52928135e-03, 4.47726082e-03, 5.12406706e-03,
5.61329890e-03, 5.84231170e-03, 6.13309619e-03,
6.32505191e-03, 6.46203546e-03, 6.49032925e-03,
6.39329339e-03, 6.23960924e-03, 5.99242067e-03,
5.67177967e-03, 5.35598822e-03, 5.07919623e-03,
4.69552191e-03, 4.37030991e-03, 3.90678952e-03,
3.43034780e-03, 2.90774639e-03, 2.35231514e-03,
1.82145021e-03, 1.51724930e-03, 1.09014092e-03,
4.86434682e-04, 2.61200867e-04, -5.02752907e-06,
-2.99906282e-04, -4.26188536e-04, -4.46910518e-04,
-3.09328219e-04, -6.53887601e-05, 4.77786782e-05,
1.84483472e-04, 2.30390682e-04, 2.18090273e-04])
expected_200terms_t = np.array(
[ 3.503, 3.519, 3.539, 3.556, 3.576, 3.595, 3.613, 3.632,
3.651, 3.667, 3.69 , 3.708, 3.727, 3.746, 3.766, 3.782,
3.801, 3.822, 3.84 , 3.856, 3.878, 3.896, 3.915, 3.933,
3.951, 3.97 , 3.99 , 4.009, 4.027, 4.046, 4.067, 4.085,
4.102, 4.122, 4.141, 4.159, 4.178, 4.197, 4.217, 4.234,
4.252, 4.273, 4.291, 4.31 , 4.329, 4.347, 4.368, 4.386,
4.405, 4.423, 4.442, 4.46 , 4.479, 4.495])
#wrong : displ are with k3~=0
expected_200terms_displacement = np.array(
[ 7.04000000e-08, 2.22800000e-05, 2.27000000e-05,
1.23200000e-06, 2.35100000e-05, 2.39300000e-05,
2.46400000e-06, -1.89700000e-05, -6.22700000e-05,
-1.27500000e-04, -1.70700000e-04, -2.57800000e-04,
-3.44800000e-04, -4.53600000e-04, -5.40600000e-04,
-5.84000000e-04, -5.61700000e-04, -4.73800000e-04,
-2.54900000e-04, 1.17100000e-04, 6.85900000e-04,
1.51700000e-03, 2.50100000e-03, 3.70300000e-03,
5.01500000e-03, 6.26200000e-03, 7.20200000e-03,
7.61800000e-03, 7.39900000e-03, 6.74400000e-03,
5.91400000e-03, 5.01800000e-03, 4.10100000e-03,
3.22700000e-03, 2.50600000e-03, 1.85000000e-03,
1.32600000e-03, 9.11400000e-04, 5.84000000e-04,
3.00200000e-04, 1.47600000e-04, 3.87500000e-05,
-4.82900000e-05, -1.13400000e-04, -1.13000000e-04,
-1.34500000e-04, -1.34000000e-04, -1.33600000e-04,
-1.11400000e-04, -6.72600000e-05, -6.68400000e-05,
-6.64500000e-05, -4.41700000e-05, -4.38200000e-05])
vfact=2
t = np.linspace(0, 4.5*3/vfact, 200)
x_norm = np.linspace(0,1,2001)
#see Dingetal2012 Table 2 for material properties
pdict = OrderedDict(
E = 6.998*1e9, #Pa
rho = 2373, #kg/m3
L = 160, #m
#v_norm=0.01165,
kf=5.41e-4,
#Fz_norm=1.013e-4,
mu_norm=39.263,
k1_norm=97.552,
# k3_norm=2.497e6,
nterms=200,
BC="SS",
nquad=20,
# k1bar=PolyLine([0,0.5],[0.5,1],[1,1],[1,.25]),
# k1bar=PolyLine([0,0.25,.75,1],[1.0,1,4,4]),
k1bar=PolyLine([0,0.4,.6,1],[1.0,1,.25,.25]),
# k1bar=PolyLine([0,0.5],[0.5,1],[2,1],[2,1]),
# moving_loads_x_norm=[[0,-0.05,-0.1]],
# moving_loads_Fz_norm=[3*[1.013e-4]],
# moving_loads_v_norm=[0.01165*vfact],
# stationary_loads_x=None,
# stationary_loads_vs_t=None,
# stationary_loads_omega_phase=None,
stationary_loads_x_norm=[0.5, 0.5],
stationary_loads_vs_t_norm=2*[PolyLine([0,t[-1]/160*np.sqrt(6.998*1e9/2373.)],[0.5*1.013e-4,0.5*1.013e-4])],
stationary_loads_omega_phase_norm=[(.5,0),(0,0)],#None,
tvals=t,
# tvals=np.array([4.0]),
# xvals_norm=np.array([0.5]),
xvals_norm=x_norm,
use_analytical=True,
# implementation="vectorized",
implementation="fortran",
file_stem="specbeam200stationary",
force_calc=True,
)
a = SpecBeam(**pdict)
a.runme()
a.saveme()
a.animateme(norm=True)
# a.plot_w_vs_x_overtime(norm=True)
plt.show()
[docs]def moving_load_on_beam_on_elastic_foundation(alp, bet,slim=(-6,6),npts=1001):
"""Displacement vs distance from moving point load on infinite beam on
elastic foundation.
This is to generate data for a a single subplot of esveld Figure 6.17
Esveld, C. (2001). Modern railway track. MRT-productions Zaltbommel, The Netherlands, Netherlands.0
s is in multiples of lambda.
Parameters
----------
alp : float
velocity ratio.
bet : float
damping ratio
slim : two element tuple of float, optional
Distance, in multiples of characteristic length, on either side
of moving load to assess deflection at Default slim = (-6,6).
npts : int, optional
number of points witin slim to evaluate deflection. Use an odd
value to capture the mid point directly under the mmoving load.
Default npts=1001.
Returns
-------
s : 1D numpy array of float
Distances, in multiples of characteristic length,
from moving point load.
w.r.t. normalised by characteristic lengths
disp : 1D numpy array of float
Displacements at s.
"""
gam = np.roots([1.0, 0.0, 4.0*alp**2,-8*alp*bet,4])
#print(gam)
cmat = np.zeros((4,4), dtype=complex)
cmat[0,:]=[1,1,-1,-1]
cmat[1,:]=[gam[0], gam[1], -gam[2], -gam[3]]
cmat[2,:]=[gam[0]**2, gam[1]**2, -gam[2]**2, -gam[3]**2]
cmat[3,:]=[gam[0]**3, gam[1]**3, -gam[2]**3, -gam[3]**3]
rhs = np.array([0, 0, 0, 8.0], dtype=complex)
A=np.linalg.solve(cmat, rhs)
#print(A)
smin,smax=slim
s=np.linspace(smin,smax,npts, dtype=complex)
disp = np.empty(len(s), dtype=complex)
s_neg = s[s<0]
s_pos = s[s>=0]
#assume that gam0 and gam1 are have negative real components
disp[s>=0]=A[0]*np.exp(s_pos*gam[0])+A[1]*np.exp(s_pos*gam[1])
disp[s<0]=A[2]*np.exp(s_neg*gam[2])+A[3]*np.exp(s_neg*gam[3])
return s, disp
[docs]def esveld_fig6p17():
"""Displacement shapes before and after a moving
point load on infinite beam on elastic foundation for
a sub crtical , critical, super critcal velocity ratio (alpha) and
damping ratios (beta) , Esveld Figure 6p17.
Esveld, C. (2001). Modern railway track. MRT-productions Zaltbommel, The Netherlands, Netherlands.
Returns
-------
fig : matplotlib figure
the deflection shape plot.
"""
fig, axs = plt.subplots(4, 3, sharex=True, sharey=True, figsize=(12,11))
for i,alp in enumerate([0,.5,1,2]):
for j,bet in enumerate([0,.1,1.1]):
s, disp =moving_load_on_beam_on_elastic_foundation(alp=alp, bet=bet)
axs[i,j].plot(s,disp)
axs[i,j].set_ylim([-2,3])
axs[i,j].invert_yaxis()
axs[i,j].set_ylabel("rel vert displ")
axs[i,j].set_xlabel("Rel dist, s $\\alpha="+str(alp)+",\\beta=" +str(bet) +"$")
axs[i,j].grid()
return fig
[docs]def esveld_fig6p18():
"""Dynamic Amplification Factor vs
velocity ratio (alpha) for variuos damping ratios (beta) for moving
point load on infinite beam on elastic foundation, Esveld Figure 6p18.
Esveld, C. (2001). Modern railway track. MRT-productions Zaltbommel, The Netherlands, Netherlands.
Returns
-------
fig : matplotlib figure
the Dynamic amplification plotplot of
"""
fig, ax = plt.subplots(figsize=(6,8))
alpha=np.linspace(0,3,100)
#DAF = np.empty_like(alp)
for j,bet in enumerate([0,0.05,0.1,0.3,1.1,2.0]):
DAF = np.empty_like(alpha)
for i, alp in enumerate(alpha):
s, disp=moving_load_on_beam_on_elastic_foundation(alp=alp, bet=bet)
mx = np.max(disp)
DAF[i] = mx
ax.plot(alpha, DAF, label="$\\beta=" + str(bet) + "$")
leg = ax.legend()
leg.draggable()
ax.set_ylim([0,4])
ax.set_xlim([0,3])
ax.grid()
ax.set_ylabel("Deflection amplification factor")
ax.set_xlabel("Rel dist, s $\\alpha="+str(alp)+",\\beta=" +str(bet) +"$")
return fig
[docs]def case_study_PaixaoEtAl2014(
saveas=None,
nterms=100,
force_calc=True,
nx=100, nt=100):
"""Try and replicate Paixao et al 2014 for 6 rail cars going over
transition using specbeam.
Some eplots will be produced.
You'll have to read the source code for this one.
Parameters
----------
saveas : string, optional
Filename stem (not including extension) to save figure to.
Default saveas=None, i.e. figure won't be saved to disk.
nx : int, optional
Number of points between `xlim[0]` and `xlim[1]` at which to
calculate deflection. Default nx = 100.
nt : int, optional
Number of evaluation times from when load enters `xlim[0]` and
leaves `xlim[1]`. Default nt=100
nterms : int, optional
Number of series terms in approximation. More terms leads to higher
accuracy but also longer computation times. Start of with a
small enough value to observe behaviour and then increase
to confirm. Default nterms=100 .
force_calc : [True, False], optional
When True all calcuations will be done from scratch. When False
Current working directory will be searched for a results file
named according to "saveas" and
if found it will be loaded. Be careful as name of results file may
not be unique to your analysis parameters. Safest to use
force_calc = True, but if you are simply regenerating graphs then
consider force_calc=False. Default force_calc=True.
"""
####
start_time0 = time.time()
ftime = datetime.datetime.fromtimestamp(start_time0).strftime('%Y-%m-%d %H%M%S')
#raw digitized data,
#time in seconds but not sure what time zero represents, probably just
# offset the specbeam data to line up with the peaks, which is what
# paxior seems to do
#Displacemtn in mm positive up
fig12top_t_raw = np.array(
[ 0.12883271, 0.13973975, 0.1478486 , 0.15608597, 0.17517319,
0.18598029, 0.19146229, 0.20236935, 0.21608873, 0.22693857,
0.22956522, 0.24040088, 0.24591183, 0.25674748, 0.26498448,
0.27853267, 0.28120274, 0.29483585, 0.30842746, 0.31379463,
0.3274287 , 0.34114769, 0.35198335, 0.35465343, 0.35733701,
0.37639615, 0.39545432, 0.4090884 , 0.43087358, 0.45003211,
0.46359477, 0.48263944, 0.49891464, 0.51260468, 0.5317497 ,
0.5398294 , 0.55340654, 0.56148624, 0.5697242 , 0.5860129 ,
0.59135209, 0.59961803, 0.60769869, 0.61592217, 0.62133277,
0.62955528, 0.63769288, 0.64029155, 0.64558731, 0.64814352,
0.65066981, 0.65316812, 0.65569538, 0.65826509, 0.66657349,
0.67219831, 0.67506813, 0.67793795, 0.68077883, 0.6836197 ,
0.68901582, 0.69165694, 0.6942836 , 0.69688226, 0.69922328,
0.69942303, 0.70452001, 0.71281393, 0.71319992, 0.71572621,
0.71609773, 0.71899553, 0.73024516, 0.73308604, 0.73861146,
0.74690539, 0.75243081, 0.75796974, 0.76623568, 0.77981282,
0.785152 , 0.79336005, 0.80695166, 0.81506031, 0.81775837,
0.82867991, 0.83953004, 0.85042263, 0.85580427, 0.86131522,
0.86952423, 0.87773227, 0.88585539, 0.89944701, 0.90757013,
0.91572124, 0.91839132, 0.93740703, 0.94564499, 0.9509697 ,
0.95630888, 0.95873578, 0.95890755, 0.96363204, 0.96388969,
0.96605991, 0.96614483, 0.9715699 , 0.97710883, 0.97999313,
0.98287646, 0.98563241, 0.98576076, 0.99131416, 0.99141355,
0.99695345, 1.00222123, 1.0050621 , 1.00738865, 1.00756041,
1.00991591, 1.01001626, 1.0181529 , 1.02092236, 1.02377771,
1.02656165, 1.02944595, 1.02955982, 1.03237271, 1.03525604,
1.03809691, 1.03819727, 1.04367927, 1.0437362 , 1.04923268,
1.05201662, 1.05755555, 1.06298062, 1.06562174, 1.06823392,
1.07081811, 1.07335888, 1.07588614, 1.0783555 , 1.08092521,
1.08352388, 1.09192202, 1.09198957, 1.09481693, 1.09765395,
1.10324113, 1.10332798, 1.10622289, 1.11435759, 1.11699196,
1.11945264, 1.11961669, 1.12198086, 1.12455734, 1.12716276,
1.13274994, 1.13314558, 1.1356159 , 1.135751 , 1.13599224,
1.1388582 , 1.14713765, 1.15000362, 1.15285993, 1.15573554,
1.16401499, 1.17223654, 1.17773688, 1.18053529, 1.18878579,
1.19957417, 1.19964172, 1.20222784, 1.21588217, 1.23223843,
1.24307505, 1.25395028, 1.25933481, 1.2648062 , 1.27838334,
1.28658559, 1.29747046, 1.30295149, 1.31105725, 1.32193247,
1.33287524, 1.33550961, 1.35997162, 1.37088544, 1.37355841,
1.37621208, 1.37865346, 1.37880785, 1.3839222 , 1.38636357,
1.38651797, 1.39147792, 1.39164196, 1.39413159, 1.39693 ,
1.39988281, 1.40250753, 1.4053928 , 1.40826841, 1.40836491,
1.41119227, 1.41950067, 1.42218329, 1.42756783, 1.4302022 ,
1.43263392, 1.43280762, 1.43530689, 1.44065283, 1.4434802 ,
1.44635581, 1.44915422, 1.45475106, 1.4548765 , 1.45767492,
1.45781966, 1.46336825, 1.46351299, 1.46639825, 1.47737962,
1.48291855, 1.48825484, 1.48834169, 1.49352358, 1.49608076,
1.49862828, 1.50096351, 1.5011758 , 1.50361718, 1.506184 ,
1.51171328, 1.51190628, 1.51747416, 1.52304204, 1.52586941,
1.52874502, 1.53422605, 1.53962024, 1.54212916, 1.54224496,
1.54717596, 1.5473979 , 1.552464 , 1.55556156, 1.56085925,
1.56121629, 1.56417875, 1.56708331, 1.56711226, 1.57543996,
1.57555575, 1.58654677, 1.58937414, 1.59763429, 1.60313462,
1.60323112, 1.6087604 , 1.61959702, 1.62764488, 1.62772208,
1.64390464, 1.65211654, 1.65759757, 1.66575158, 1.68204993,
1.69026184, 1.69568497, 1.70390652, 1.70666634, 1.71480104,
1.72028208, 1.73385922, 1.74471514, 1.75010933, 1.75823438,
1.76086875, 1.77181152, 1.77727326, 1.78542726, 1.79084075,
1.7989465 , 1.80154227, 1.80411875, 1.80644432, 1.80668557,
1.80895325, 1.81151042, 1.81681776, 1.82501036, 1.82515511,
1.83075194, 1.8336372 , 1.83659001, 1.8421579 , 1.85278222,
1.85296557, 1.85806062, 1.86060814, 1.86316531, 1.86878144,
1.87171495, 1.8719369 , 1.88026459, 1.88313056, 1.88330425,
1.88616057, 1.88904583, 1.89452686, 1.90278701, 1.90284491,
1.9082584 , 1.91638345, 1.91896957, 1.9215943 , 1.92398742,
1.92420937, 1.9264867 , 1.92903422, 1.93440911, 1.94273681,
1.9428526 , 1.94854593, 1.94861348, 1.95417171, 1.95972029,
1.9677199 , 1.97011303, 1.97024812, 1.97540107, 1.97798719,
1.9810365 , 1.98362262, 1.98389281, 1.98681667, 1.98961509,
1.99518297, 2.0007798 , 2.00094385, 2.00651173, 2.01473328,
2.01480083, 2.02035906, 2.03406164, 2.04493687, 2.0504179 ,
2.05577349, 2.07209115, 2.09383194, 2.09923577, 2.11281292,
2.12378463, 2.13738107, 2.14829489, 2.16183343, 2.17272795,
2.17809319, 2.18892981, 2.2025745 , 2.20807484, 2.22157478,
2.22669877, 2.22684352, 2.23196751, 2.23433169, 2.23449574,
2.23961973, 2.24493672, 2.25042741, 2.25327407, 2.25338987,
2.25626548, 2.26182371, 2.26193951, 2.26751704, 2.27278578,
2.27287263, 2.27808347, 2.28065029, 2.28319781, 2.2885148 ,
2.29419848, 2.29714164, 2.30002691, 2.30292182, 2.3085669 ,
2.31140391, 2.31431812, 2.31449182, 2.32820406, 2.33360789,
2.33894418, 2.3415303 , 2.34682799, 2.34943341, 2.35177829,
2.35196164, 2.35710493, 2.36538438, 2.36822139, 2.36837579,
2.37394367, 2.37407877, 2.38239682, 2.39029028, 2.39045432,
2.3928957 , 2.39531778, 2.39546252, 2.40054792, 2.40895281,
2.40904931, 2.41211792, 2.41771475, 2.42050352, 2.42056141,
2.42070616, 2.42633194, 2.43185157, 2.44012137, 2.44292944,
2.45385291, 2.45665132, 2.47026706, 2.47836317, 2.49467117,
2.51105638, 2.51909458, 2.53537364, 2.5435759 , 2.54636466,
2.55726883, 2.56274022, 2.57635595, 2.58449066, 2.59809675,
2.60619285, 2.61155809, 2.62244296, 2.62796259, 2.63335678,
2.6441934 , 2.64952004, 2.65208686, 2.65717226, 2.65738456,
2.6623831 , 2.66488238, 2.66747815, 2.67304603, 2.6758734 ,
2.67602779, 2.68161497, 2.68444234, 2.68729865, 2.69284723,
2.69293408, 2.69831862, 2.70093369, 2.70350051, 2.70581644,
2.70600943, 2.7138643 , 2.71949008, 2.72247184, 2.72264553,
2.72830991, 2.73125307, 2.73417693, 2.7370622 , 2.73983166,
2.75361145, 2.75900563, 2.76430332, 2.76687015, 2.76928257,
2.76938872, 2.77458026, 2.779849 , 2.78241583, 2.78542654,
2.78803196, 2.79375423, 2.79664915, 2.80223633, 2.80769806,
2.81575557, 2.81835134, 2.82073482, 2.82086026, 2.82099536,
2.82335954, 2.82884057, 2.83448565, 2.83740951, 2.84023688,
2.84035268, 2.84318969, 2.8460653 , 2.84896021, 2.85185513,
2.85742301, 2.86298124, 2.87659698, 2.87941469, 2.89031886,
2.89854042, 2.90934809, 2.92025226, 2.9339452 , 2.94472392,
2.96654191, 2.98289817, 2.99927372, 3.01290876, 3.03735147,
3.04829424, 3.06466014, 3.0726501 , 3.07786095, 3.08304284,
3.08557106, 3.08813789, 3.09345487, 3.09911925, 3.10474503,
3.10487048, 3.10781364, 3.11340082, 3.11348767, 3.12694901,
3.12704551, 3.12952548, 3.13210196, 3.13473633, 3.1373707 ,
3.13749614, 3.14315087, 3.14607473, 3.14621948, 3.14908544,
3.14914334, 3.15474017, 3.15765438, 3.16053965, 3.16337666,
3.17162716, 3.17998381, 3.19369604, 3.20467741, 3.21547544,
3.23462046, 3.24272621, 3.26179404, 3.27815994, 3.28909306,
3.29174673, 3.30531422, 3.30805474, 3.32164153, 3.33527657,
3.35438299, 3.36251769, 3.37613343, 3.38425849, 3.3952688 ,
3.40890384, 3.42254853, 3.43348165, 3.44437617, 3.45804015,
3.47168484, 3.48261796, 3.49624335, 3.50985909, 3.51528222,
3.52077291, 3.53442725, 3.5425716 , 3.55618734, 3.56709151,
3.57801498, 3.58348636, 3.59440018])
fig12top_w_raw = np.array(
[ 4.18764000e-02, 4.18600000e-02, 1.56698000e-02,
3.65997000e-02, 3.65710000e-02, -9.45000000e-05,
1.03684000e-02, 1.03520000e-02, 4.17449000e-02,
2.07862000e-02, -1.58670000e-02, -4.20614000e-02,
-2.11273000e-02, -4.73216000e-02, -2.63916000e-02,
-5.78256000e-02, -7.87721000e-02, -7.87926000e-02,
-9.45199000e-02, -1.25942000e-01, -1.25962000e-01,
-9.45692000e-02, -1.20764000e-01, -1.41710000e-01,
-1.57421000e-01, -1.67921000e-01, -1.78421000e-01,
-1.78441000e-01, -1.88945000e-01, -1.62796000e-01,
-1.88995000e-01, -2.04730000e-01, -2.36168000e-01,
-2.15246000e-01, -1.94333000e-01, -2.30994000e-01,
-2.51957000e-01, -2.88619000e-01, -2.67689000e-01,
-2.93891000e-01, -3.35784000e-01, -3.04383000e-01,
-3.41044000e-01, -3.25350000e-01, -3.41065000e-01,
-3.25370000e-01, -3.41089000e-01, -3.88214000e-01,
-4.45813000e-01, -5.08644000e-01, -5.81947000e-01,
-6.65720000e-01, -7.39022000e-01, -7.96618000e-01,
-7.49510000e-01, -6.86691000e-01, -6.34339000e-01,
-5.81988000e-01, -5.40107000e-01, -4.98227000e-01,
-5.19177000e-01, -5.50595000e-01, -5.87248000e-01,
-6.34372000e-01, -7.75737000e-01, -7.02439000e-01,
-8.33337000e-01, -7.91464000e-01, -6.50104000e-01,
-7.23406000e-01, -5.87281000e-01, -5.24458000e-01,
-3.98821000e-01, -3.56940000e-01, -3.30770000e-01,
-2.88898000e-01, -2.62728000e-01, -2.31323000e-01,
-1.99922000e-01, -2.20885000e-01, -2.62778000e-01,
-2.52319000e-01, -2.68046000e-01, -2.94236000e-01,
-3.04712000e-01, -2.99492000e-01, -3.20451000e-01,
-3.25703000e-01, -3.51889000e-01, -3.30955000e-01,
-3.20496000e-01, -3.10038000e-01, -3.30992000e-01,
-3.46720000e-01, -3.67674000e-01, -3.78158000e-01,
-3.99104000e-01, -4.25311000e-01, -4.04381000e-01,
-4.51509000e-01, -4.93402000e-01, -6.03353000e-01,
-5.40526000e-01, -8.07549000e-01, -7.13309000e-01,
-9.17501000e-01, -8.86087000e-01, -8.96566000e-01,
-8.65161000e-01, -8.07574000e-01, -7.49987000e-01,
-7.39520000e-01, -6.92399000e-01, -6.55759000e-01,
-6.19109000e-01, -5.87704000e-01, -6.55775000e-01,
-6.13894000e-01, -7.60495000e-01, -6.97668000e-01,
-8.33797000e-01, -7.97148000e-01, -8.12867000e-01,
-7.97164000e-01, -7.50048000e-01, -7.29110000e-01,
-6.71523000e-01, -6.29638000e-01, -5.98229000e-01,
-5.40642000e-01, -4.98761000e-01, -4.62112000e-01,
-4.51649000e-01, -4.30707000e-01, -4.15008000e-01,
-3.94070000e-01, -3.62665000e-01, -3.73144000e-01,
-4.04562000e-01, -4.46450000e-01, -4.98810000e-01,
-5.66877000e-01, -6.40179000e-01, -7.34424000e-01,
-7.92019000e-01, -8.39144000e-01, -7.60622000e-01,
-7.34444000e-01, -6.97799000e-01, -6.55919000e-01,
-6.08807000e-01, -5.77393000e-01, -5.14570000e-01,
-5.30290000e-01, -5.66943000e-01, -6.61187000e-01,
-6.03596000e-01, -7.34489000e-01, -7.92085000e-01,
-8.33974000e-01, -7.86862000e-01, -6.40265000e-01,
-7.34510000e-01, -6.87390000e-01, -5.98385000e-01,
-5.46033000e-01, -5.09397000e-01, -4.57045000e-01,
-4.09929000e-01, -3.57577000e-01, -3.20940000e-01,
-3.05246000e-01, -2.89547000e-01, -2.63374000e-01,
-2.37208000e-01, -2.79109000e-01, -2.52931000e-01,
-3.05291000e-01, -3.00076000e-01, -3.00101000e-01,
-3.26295000e-01, -3.36783000e-01, -3.62969000e-01,
-3.57741000e-01, -3.78704000e-01, -3.68245000e-01,
-3.78733000e-01, -3.68270000e-01, -3.94460000e-01,
-4.04948000e-01, -3.94493000e-01, -4.25911000e-01,
-4.57361000e-01, -4.52142000e-01, -4.73089000e-01,
-4.99271000e-01, -6.03986000e-01, -5.46395000e-01,
-6.72057000e-01, -7.76773000e-01, -7.19181000e-01,
-9.02435000e-01, -8.39608000e-01, -9.28617000e-01,
-9.02443000e-01, -8.18678000e-01, -8.55331000e-01,
-7.97744000e-01, -7.45392000e-01, -7.08743000e-01,
-6.72098000e-01, -6.24990000e-01, -6.40701000e-01,
-6.66887000e-01, -6.98305000e-01, -8.08256000e-01,
-7.45429000e-01, -8.29203000e-01, -8.65860000e-01,
-8.29215000e-01, -7.76863000e-01, -7.50689000e-01,
-6.98342000e-01, -6.51222000e-01, -6.25048000e-01,
-5.72692000e-01, -5.36051000e-01, -4.83695000e-01,
-4.26108000e-01, -3.99947000e-01, -3.68541000e-01,
-4.10434000e-01, -3.79021000e-01, -4.78505000e-01,
-5.41336000e-01, -6.04167000e-01, -7.50767000e-01,
-6.72234000e-01, -7.76949000e-01, -8.34545000e-01,
-8.08375000e-01, -7.35077000e-01, -6.93201000e-01,
-6.51324000e-01, -6.14679000e-01, -5.62328000e-01,
-5.51865000e-01, -5.72815000e-01, -6.51353000e-01,
-6.09468000e-01, -8.03193000e-01, -7.19424000e-01,
-8.60793000e-01, -7.24672000e-01, -7.82271000e-01,
-6.51382000e-01, -5.67617000e-01, -4.99558000e-01,
-4.89087000e-01, -4.36744000e-01, -3.94859000e-01,
-3.63462000e-01, -3.26817000e-01, -2.95416000e-01,
-2.79717000e-01, -2.43068000e-01, -2.16899000e-01,
-2.43093000e-01, -2.90225000e-01, -2.64048000e-01,
-3.26899000e-01, -3.16440000e-01, -3.05977000e-01,
-3.16461000e-01, -3.37428000e-01, -3.26969000e-01,
-3.37448000e-01, -3.21754000e-01, -3.11287000e-01,
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6.34187000e-02, 2.15177000e-02, 4.24312000e-02,
1.62410000e-02, 1.09766000e-02, 1.09520000e-02,
2.14067000e-02, -4.77531000e-03, -3.09738000e-02,
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-1.56862000e-02, -1.04671000e-02])
fig13top_t_raw = np.array(
[ 0.179243, 0.183404, 0.195889, 0.208374, 0.214617, 0.22294 ,
0.229183, 0.239587, 0.24791 , 0.258314, 0.264557, 0.27288 ,
0.279123, 0.289527, 0.29785 , 0.306174, 0.312416, 0.316578,
0.333225, 0.335305, 0.351952, 0.37068 , 0.374841, 0.376922,
0.385246, 0.387326, 0.393569, 0.399811, 0.399811, 0.403973,
0.408135, 0.416458, 0.418539, 0.422701, 0.426862, 0.433105,
0.439347, 0.443509, 0.443509, 0.451832, 0.453913, 0.455994,
0.462237, 0.464317, 0.466398, 0.468479, 0.474722, 0.476802,
0.478883, 0.483045, 0.487207, 0.491368, 0.493449, 0.49553 ,
0.505934, 0.510096, 0.514257, 0.5205 , 0.522581, 0.526743,
0.532985, 0.541308, 0.549632, 0.557955, 0.57044 , 0.574602,
0.587087, 0.591249, 0.593329, 0.603734, 0.612057, 0.614138,
0.62038 , 0.628704, 0.637027, 0.64535 , 0.647431, 0.653674,
0.659916, 0.664078, 0.674482, 0.678644, 0.686967, 0.689048,
0.69321 , 0.697371, 0.699452, 0.701533, 0.705695, 0.709856,
0.711937, 0.716099, 0.71818 , 0.726503, 0.732746, 0.734826,
0.738988, 0.74315 , 0.749392, 0.751473, 0.753554, 0.755635,
0.759796, 0.761877, 0.766039, 0.770201, 0.772281, 0.774362,
0.776443, 0.780605, 0.784766, 0.786847, 0.791009, 0.795171,
0.801413, 0.809737, 0.813898, 0.815979, 0.820141, 0.822222,
0.826383, 0.828464, 0.830545, 0.834707, 0.840949, 0.84303 ,
0.84303 , 0.845111, 0.859677, 0.863838, 0.865919, 0.870081,
0.872162, 0.878404, 0.884647, 0.886728, 0.888808, 0.890889,
0.895051, 0.897132, 0.901293, 0.903374, 0.905455, 0.91794 ,
0.920021, 0.928344, 0.94291 , 0.947072, 0.955395, 0.957476,
0.963719, 0.965799, 0.974123, 0.980365, 0.984527, 0.990769,
0.994931, 0.997012, 1.00534 , 1.0095 , 1.02198 , 1.02614 ,
1.03447 , 1.04071 , 1.04695 , 1.04903 , 1.05528 , 1.0636 ,
1.06776 , 1.07192 , 1.07816 , 1.08441 , 1.09065 , 1.09897 ,
1.11146 , 1.11354 , 1.11562 , 1.11978 , 1.12394 , 1.12602 ,
1.1281 , 1.13019 , 1.13227 , 1.13851 , 1.14059 , 1.14475 ,
1.15099 , 1.15307 , 1.15724 , 1.15724 , 1.16764 , 1.17388 ,
1.17596 , 1.18013 , 1.18221 , 1.18637 , 1.18845 , 1.19469 ,
1.19677 , 1.19677 , 1.20093 , 1.20301 , 1.21134 , 1.21342 ,
1.21966 , 1.2259 , 1.23215 , 1.23423 , 1.23631 , 1.23839 ,
1.24255 , 1.24463 , 1.24671 , 1.24879 , 1.25296 , 1.2592 ,
1.26336 , 1.26544 , 1.26752 , 1.2696 , 1.27376 , 1.27793 ,
1.28209 , 1.28625 , 1.28833 , 1.29041 , 1.29249 , 1.29665 ,
1.29873 , 1.30081 , 1.30914 , 1.31122 , 1.31122 , 1.3133 ,
1.31746 , 1.32162 , 1.3237 , 1.32787 , 1.34035 , 1.34243 ,
1.34867 , 1.35284 , 1.357 , 1.3674 , 1.37781 , 1.39237 ,
1.40278 , 1.4111 , 1.4215 , 1.42358 , 1.43399 , 1.44439 ,
1.4548 , 1.46104 , 1.47144 , 1.47352 , 1.48393 , 1.49225 ,
1.50057 , 1.50474 , 1.50682 , 1.51722 , 1.52554 , 1.53595 ,
1.54011 , 1.54219 , 1.54635 , 1.54843 , 1.5526 , 1.55676 ,
1.563 , 1.56716 , 1.5734 , 1.57549 , 1.57549 , 1.57965 ,
1.58173 , 1.58589 , 1.59005 , 1.59629 , 1.59837 , 1.60046 ,
1.60462 , 1.6067 , 1.61294 , 1.61918 , 1.62126 , 1.62334 ,
1.62959 , 1.63167 , 1.63375 , 1.63791 , 1.63999 , 1.65248 ,
1.65456 , 1.65664 , 1.6608 , 1.66288 , 1.66496 , 1.66912 ,
1.6712 , 1.67328 , 1.67537 , 1.67953 , 1.68577 , 1.68785 ,
1.69201 , 1.69409 , 1.69617 , 1.70034 , 1.7045 , 1.70866 ,
1.71282 , 1.7149 , 1.71698 , 1.71906 , 1.72322 , 1.73363 ,
1.73363 , 1.73987 , 1.74195 , 1.74403 , 1.74611 , 1.75236 ,
1.75444 , 1.76068 , 1.76692 , 1.77316 , 1.78565 , 1.79397 ,
1.80022 , 1.80854 , 1.81686 , 1.82727 , 1.83767 , 1.84183 ,
1.84599 , 1.85848 , 1.86472 , 1.87721 , 1.88137 , 1.88761 ,
1.89801 , 1.90634 , 1.91674 , 1.92299 , 1.92923 , 1.93755 ,
1.94796 , 1.96044 , 1.96252 , 1.9646 , 1.96876 , 1.97084 ,
1.97501 , 1.97917 , 1.98125 , 1.98541 , 1.98749 , 1.99165 ,
1.9979 , 1.99998 , 2.00206 , 2.00622 , 2.01454 , 2.01662 ,
2.0187 , 2.02078 , 2.02495 , 2.02703 , 2.02911 , 2.03535 ,
2.03743 , 2.04159 , 2.04367 , 2.04575 , 2.04784 , 2.05408 ,
2.05824 , 2.06032 , 2.0624 , 2.06656 , 2.07697 , 2.07905 ,
2.08321 , 2.08529 , 2.08737 , 2.09153 , 2.09361 , 2.09569 ,
2.09778 , 2.10194 , 2.1061 , 2.11026 , 2.11026 , 2.11234 ,
2.11442 , 2.12066 , 2.12483 , 2.12899 , 2.13315 , 2.13939 ,
2.14147 , 2.14355 , 2.1498 , 2.15604 , 2.15812 , 2.16436 ,
2.16852 , 2.17269 , 2.17477 , 2.17685 , 2.18517 , 2.19141 ,
2.19557 , 2.20182 , 2.2039 , 2.21014 , 2.2143 , 2.21846 ,
2.22263 , 2.22679 , 2.23511 , 2.23719 , 2.24343 , 2.25384 ,
2.26008 , 2.26632 , 2.27673 , 2.28713 , 2.29337 , 2.3017 ,
2.3121 , 2.32251 , 2.33083 , 2.33915 , 2.36204 , 2.3662 ,
2.37661 , 2.38493 , 2.39117 , 2.39534 , 2.39742 , 2.3995 ,
2.40574 , 2.40782 , 2.4099 , 2.41198 , 2.41614 , 2.42239 ,
2.42447 , 2.42863 , 2.43071 , 2.43903 , 2.44319 , 2.44528 ,
2.44944 , 2.4536 , 2.45568 , 2.45984 , 2.464 , 2.46608 ,
2.46816 , 2.47233 , 2.47649 , 2.47857 , 2.48273 , 2.48481 ,
2.48897 , 2.50562 , 2.50978 , 2.51394 , 2.5181 , 2.52019 ,
2.52227 , 2.52435 , 2.52851 , 2.53267 , 2.53683 , 2.53891 ,
2.54099 , 2.54516 , 2.54932 , 2.55556 , 2.55764 , 2.55972 ,
2.56388 , 2.56804 , 2.57013 , 2.57429 , 2.57637 , 2.58053 ,
2.58261 , 2.58677 , 2.59093 , 2.5951 , 2.59718 , 2.59926 ,
2.60134 , 2.60342 , 2.61382 , 2.6159 , 2.62007 , 2.63671 ,
2.64712 , 2.65752 , 2.66584 , 2.67209 , 2.68249 , 2.68873 ,
2.69706 , 2.71578 , 2.72619 , 2.73243 , 2.73867 , 2.75532 ,
2.76781 , 2.77197 , 2.78237 , 2.79069 , 2.79902 , 2.80526 ,
2.80942 , 2.81566 , 2.81775 , 2.81983 , 2.82399 , 2.82607 ,
2.82815 , 2.83023 , 2.83439 , 2.83647 , 2.83855 , 2.8448 ,
2.84688 , 2.85104 , 2.8552 , 2.86144 , 2.86769 , 2.86977 ,
2.87393 , 2.87601 , 2.87809 , 2.88225 , 2.88849 , 2.89266 ,
2.89474 , 2.90098 , 2.90306 , 2.90514 , 2.90722 , 2.91138 ,
2.91346 , 2.92387 , 2.93219 , 2.93843 , 2.95092 , 2.95716 ,
2.97173 , 2.98005 , 2.98837 , 2.99254 , 2.99878 , 3.00086 ,
3.00294 , 3.0071 , 3.01126 , 3.01334 , 3.01751 , 3.02791 ,
3.03623 , 3.04456 , 3.05288 , 3.06953 , 3.07161 , 3.08825 ,
3.10074 , 3.10282 , 3.11947 , 3.12987 , 3.14236 , 3.15484 ,
3.159 , 3.16941 , 3.17981 , 3.19022 , 3.19646 , 3.20686 ,
3.22143 , 3.23391 , 3.24432 , 3.25888 , 3.26513 , 3.27553 ,
3.28177 , 3.29634 , 3.30674 , 3.31715 , 3.32547 , 3.33171 ,
3.3442 , 3.3546 , 3.36293 , 3.36917 , 3.37749 , 3.38581 ,
3.38998 , 3.3983 , 3.40662 , 3.41703 , 3.42327 , 3.43784 ,
3.44408 , 3.45448 , 3.46489 , 3.47737 , 3.48569 , 3.49194 ,
3.51066 , 3.52107 , 3.52731 , 3.54396 , 3.54604 , 3.55436 ,
3.56477 , 3.57309 , 3.58141 , 3.58557 , 3.59598 , 3.6043 ,
3.61263 , 3.62095 , 3.63343 , 3.64176 ])
fig13top_w_raw = np.array(
[ 0. , -0.0198413 , 0. , -0.015873 , 0. ,
-0.015873 , 0.015873 , -0.0198413 , -0.00396825, -0.0198413 ,
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-0.031746 , -0.0119048 , -0.031746 , -0.0277778 , -0.00793651,
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-0.47619 , -0.365079 , -0.31746 , -0.234127 , -0.18254 ,
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0.0198413 , -0.0714286 , 0.0119048 , -0.0436508 , -0.0595238 ,
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-0.0277778 , 0.00396825, -0.0277778 , -0.0119048 , 0.00396825,
0.0238095 , 0.0119048 , -0.0119048 , -0.0357143 , -0.0793651 ,
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-0.0357143 , 0.00396825, -0.0357143 , -0.0436508 , 0.00396825,
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-0.015873 , -0.0396825 , -0.00396825, -0.047619 , -0.0238095 ,
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-0.27381 , -0.464286 , -0.34127 , -0.551587 , -0.646825 ,
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-0.246032 , -0.166667 , -0.0873016 , -0.103175 , -0.047619 ,
0. , -0.0674603 , -0.00793651, -0.047619 , 0.015873 ,
-0.103175 , 0.0396825 , 0.186508 , 0.0833333 , 0.031746 ,
-0.142857 , -0.0277778 , 0.0198413 , -0.0357143 , -0.0634921 ,
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-0.0277778 , -0.0198413 , -0.0238095 , -0.0277778 ])
fig13bot_t_raw = np.array(
[ 0.14382 , 0.14955 , 0.15719 , 0.16674 , 0.17247 , 0.1973 ,
0.21067 , 0.22022 , 0.229769, 0.241229, 0.254599, 0.266059,
0.273699, 0.283249, 0.290889, 0.298529, 0.306169, 0.315719,
0.321449, 0.336729, 0.348189, 0.355829, 0.357739, 0.359649,
0.363469, 0.365379, 0.367289, 0.371109, 0.378749, 0.384479,
0.388299, 0.390209, 0.394029, 0.399759, 0.407398, 0.411218,
0.416948, 0.424588, 0.426498, 0.428408, 0.434138, 0.436048,
0.439868, 0.443688, 0.445598, 0.449418, 0.455148, 0.457058,
0.460878, 0.462788, 0.472338, 0.479978, 0.487618, 0.497168,
0.504808, 0.514358, 0.521998, 0.533458, 0.539188, 0.556378,
0.564018, 0.571658, 0.579297, 0.590757, 0.598397, 0.606037,
0.617497, 0.619407, 0.634687, 0.644237, 0.649967, 0.657607,
0.667157, 0.669067, 0.674797, 0.676707, 0.680527, 0.682437,
0.688167, 0.697717, 0.703447, 0.705357, 0.711087, 0.716817,
0.718727, 0.720637, 0.724457, 0.728277, 0.737827, 0.741647,
0.745467, 0.747377, 0.749286, 0.756926, 0.760746, 0.762656,
0.768386, 0.772206, 0.776026, 0.779846, 0.783666, 0.789396,
0.793216, 0.795126, 0.798946, 0.802766, 0.808496, 0.812316,
0.816136, 0.821866, 0.829506, 0.833326, 0.837146, 0.840966,
0.844786, 0.850516, 0.858156, 0.861976, 0.861976, 0.863886,
0.867706, 0.875346, 0.879166, 0.882986, 0.886806, 0.894446,
0.903996, 0.911636, 0.921186, 0.928825, 0.938375, 0.946015,
0.961295, 0.976575, 0.980395, 0.989945, 0.999495, 1.0205 ,
1.03196 , 1.03578 , 1.04342 , 1.04915 , 1.06061 , 1.06825 ,
1.07971 , 1.08353 , 1.08926 , 1.09499 , 1.0969 , 1.10072 ,
1.10263 , 1.10645 , 1.11409 , 1.11982 , 1.12173 , 1.12364 ,
1.12746 , 1.12937 , 1.13701 , 1.14274 , 1.14656 , 1.15038 ,
1.15229 , 1.1542 , 1.15993 , 1.16375 , 1.16757 , 1.17139 ,
1.17521 , 1.18094 , 1.18285 , 1.18667 , 1.18858 , 1.19049 ,
1.20386 , 1.20768 , 1.20959 , 1.21341 , 1.21723 , 1.21914 ,
1.21914 , 1.22678 , 1.2306 , 1.23251 , 1.23442 , 1.23824 ,
1.24015 , 1.24397 , 1.24779 , 1.25543 , 1.25734 , 1.26116 ,
1.26307 , 1.2688 , 1.27262 , 1.27453 , 1.28026 , 1.28217 ,
1.28599 , 1.28981 , 1.29554 , 1.29745 , 1.30509 , 1.31082 ,
1.31655 , 1.32419 , 1.32992 , 1.33947 , 1.34902 , 1.35475 ,
1.36048 , 1.37194 , 1.37767 , 1.38149 , 1.38722 , 1.39868 ,
1.40632 , 1.41396 , 1.41969 , 1.42924 , 1.44261 , 1.45216 ,
1.46171 , 1.46935 , 1.47699 , 1.48463 , 1.49991 , 1.50373 ,
1.50946 , 1.51519 , 1.5171 , 1.52092 , 1.52474 , 1.52856 ,
1.53047 , 1.53238 , 1.53429 , 1.54002 , 1.54575 , 1.54766 ,
1.55148 , 1.56103 , 1.56103 , 1.56867 , 1.57058 , 1.57631 ,
1.58204 , 1.58586 , 1.59159 , 1.59541 , 1.59732 , 1.59923 ,
1.60878 , 1.61069 , 1.61451 , 1.62024 , 1.62215 , 1.62597 ,
1.6317 , 1.63361 , 1.63743 , 1.64316 , 1.64507 , 1.64889 ,
1.65462 , 1.66035 , 1.66417 , 1.66608 , 1.6699 , 1.67563 ,
1.67754 , 1.68327 , 1.689 , 1.69091 , 1.69282 , 1.69855 ,
1.70237 , 1.70428 , 1.70619 , 1.71574 , 1.71956 , 1.71956 ,
1.72147 , 1.73102 , 1.73484 , 1.74057 , 1.75012 , 1.75203 ,
1.75394 , 1.75776 , 1.76158 , 1.7654 , 1.76922 , 1.77113 ,
1.77304 , 1.77877 , 1.7845 , 1.78641 , 1.79023 , 1.79787 ,
1.8036 , 1.80742 , 1.81124 , 1.81697 , 1.82461 , 1.83607 ,
1.84371 , 1.85708 , 1.87618 , 1.88191 , 1.89528 , 1.90292 ,
1.9182 , 1.92393 , 1.93157 , 1.9373 , 1.93921 , 1.94112 ,
1.94303 , 1.94876 , 1.95067 , 1.95258 , 1.9564 , 1.96022 ,
1.96595 , 1.96977 , 1.97168 , 1.97359 , 1.98887 , 1.99269 ,
1.9946 , 1.99651 , 2.00224 , 2.00606 , 2.00988 , 2.01561 ,
2.01752 , 2.02134 , 2.02516 , 2.03089 , 2.0328 , 2.03662 ,
2.04426 , 2.04999 , 2.05572 , 2.05763 , 2.05954 , 2.06527 ,
2.071 , 2.07673 , 2.08246 , 2.08437 , 2.08437 , 2.09201 ,
2.09583 , 2.09774 , 2.10538 , 2.1092 , 2.11111 , 2.11684 ,
2.12257 , 2.12639 , 2.13021 , 2.13212 , 2.13594 , 2.13785 ,
2.14167 , 2.14358 , 2.1474 , 2.15122 , 2.15504 , 2.16077 ,
2.16841 , 2.17414 , 2.17605 , 2.17987 , 2.18942 , 2.19324 ,
2.20279 , 2.20852 , 2.21807 , 2.2238 , 2.22571 , 2.22953 ,
2.23526 , 2.24099 , 2.24481 , 2.24863 , 2.25436 , 2.26009 ,
2.26391 , 2.27346 , 2.28301 , 2.29447 , 2.29829 , 2.30402 ,
2.30784 , 2.31357 , 2.3193 , 2.32503 , 2.33267 , 2.34031 ,
2.35177 , 2.3575 , 2.36132 , 2.36705 , 2.37087 , 2.37469 ,
2.37851 , 2.38042 , 2.38233 , 2.38997 , 2.39379 , 2.39952 ,
2.40143 , 2.40716 , 2.4148 , 2.41671 , 2.42244 , 2.42626 ,
2.43199 , 2.43581 , 2.43772 , 2.43963 , 2.44345 , 2.44727 ,
2.45109 , 2.453 , 2.45491 , 2.45873 , 2.46637 , 2.46828 ,
2.47592 , 2.47974 , 2.48547 , 2.4912 , 2.49884 , 2.50266 ,
2.50457 , 2.50839 , 2.51603 , 2.52367 , 2.52749 , 2.53131 ,
2.53322 , 2.53704 , 2.53895 , 2.54659 , 2.55232 , 2.55614 ,
2.55996 , 2.5676 , 2.56951 , 2.57333 , 2.58288 , 2.58479 ,
2.59816 , 2.60007 , 2.60389 , 2.60962 , 2.61344 , 2.61726 ,
2.62108 , 2.63063 , 2.63636 , 2.644 , 2.64973 , 2.65164 ,
2.65928 , 2.66501 , 2.67265 , 2.68029 , 2.68602 , 2.68984 ,
2.69748 , 2.7013 , 2.71085 , 2.7204 , 2.73377 , 2.73759 ,
2.74714 , 2.75286 , 2.75668 , 2.76814 , 2.77196 , 2.77578 ,
2.78151 , 2.78533 , 2.78724 , 2.79106 , 2.79297 , 2.79488 ,
2.80061 , 2.80252 , 2.80634 , 2.81016 , 2.81398 , 2.81589 ,
2.81971 , 2.82162 , 2.82926 , 2.83499 , 2.8369 , 2.84072 ,
2.84263 , 2.84263 , 2.84645 , 2.84836 , 2.85218 , 2.85409 ,
2.856 , 2.85791 , 2.86173 , 2.86555 , 2.86746 , 2.87319 ,
2.88083 , 2.88274 , 2.88656 , 2.88847 , 2.89611 , 2.89993 ,
2.90375 , 2.91139 , 2.91903 , 2.92667 , 2.93431 , 2.94004 ,
2.94386 , 2.94959 , 2.95341 , 2.96296 , 2.96678 , 2.96869 ,
2.97633 , 2.98206 , 2.98397 , 2.99161 , 3.00116 , 3.00307 ,
3.01453 , 3.01835 , 3.02599 , 3.03554 , 3.05082 , 3.0661 ,
3.07183 , 3.07756 , 3.0852 , 3.09857 , 3.11003 , 3.11958 ,
3.1234 , 3.13677 , 3.15014 , 3.15778 , 3.16924 , 3.17306 ,
3.18261 , 3.19025 , 3.20935 , 3.23036 , 3.24182 , 3.24373 ,
3.25901 , 3.27047 , 3.28002 , 3.28766 , 3.29912 , 3.31058 ,
3.32013 , 3.32777 , 3.33732 , 3.34687 , 3.35833 , 3.37552 ,
3.38507 , 3.39462 , 3.40608 , 3.41372 , 3.41945 , 3.429 ,
3.43855 , 3.4481 , 3.46529 , 3.46911 , 3.47866 , 3.49012 ,
3.49967 , 3.50922 , 3.5245 , 3.53405 , 3.53596 , 3.54169 ,
3.54742 , 3.54933 , 3.55506 , 3.56461 , 3.57225 , 3.58562 ,
3.59135 , 3.60663 ])
fig13bot_w_raw = np.array(
[ 3.77358000e-03, -1.13208000e-02, 7.54717000e-03,
-1.13208000e-02, 1.50943000e-02, -3.77358000e-02,
2.22045000e-16, -2.64151000e-02, 7.54717000e-03,
3.77358000e-03, 1.50943000e-02, -1.13208000e-02,
7.54717000e-03, -1.50943000e-02, 3.77358000e-03,
-1.13208000e-02, 3.77358000e-03, -1.50943000e-02,
1.13208000e-02, 1.50943000e-02, -1.50943000e-02,
-9.81132000e-02, -5.28302000e-02, -1.47170000e-01,
-2.79245000e-01, -3.47170000e-01, -4.07547000e-01,
-4.86792000e-01, -6.22642000e-01, -5.66038000e-01,
-5.24528000e-01, -4.26415000e-01, -3.47170000e-01,
-3.01887000e-01, -4.07547000e-01, -3.81132000e-01,
-5.13208000e-01, -6.94340000e-01, -5.43396000e-01,
-6.30189000e-01, -4.83019000e-01, -4.52830000e-01,
-3.66038000e-01, -2.75472000e-01, -1.84906000e-01,
-1.62264000e-01, -1.35849000e-01, -5.66038000e-02,
-1.09434000e-01, 3.01887000e-02, -6.03774000e-02,
4.15094000e-02, -4.90566000e-02, 6.03774000e-02,
-3.77358000e-02, 3.39623000e-02, -4.15094000e-02,
4.15094000e-02, 2.22045000e-16, 3.01887000e-02,
-3.01887000e-02, 4.52830000e-02, -7.54717000e-03,
3.39623000e-02, -7.54717000e-03, 3.77358000e-02,
-2.64151000e-02, 3.77358000e-02, -3.39623000e-02,
4.90566000e-02, -1.13208000e-02, 1.50943000e-02,
-5.66038000e-02, -2.07547000e-01, -2.79245000e-01,
-3.50943000e-01, -4.45283000e-01, -5.32075000e-01,
-5.77358000e-01, -5.09434000e-01, -4.67925000e-01,
-3.81132000e-01, -3.05660000e-01, -3.35849000e-01,
-3.84906000e-01, -4.26415000e-01, -4.67925000e-01,
-5.77358000e-01, -6.60377000e-01, -6.11321000e-01,
-4.03774000e-01, -3.24528000e-01, -2.86792000e-01,
-2.64151000e-01, -6.79245000e-02, -1.62264000e-01,
-3.77358000e-03, -6.03774000e-02, -1.05660000e-01,
-1.50943000e-01, -2.18868000e-01, -3.05660000e-01,
-4.03774000e-01, -4.86792000e-01, -5.50943000e-01,
-6.22642000e-01, -5.66038000e-01, -5.24528000e-01,
-4.07547000e-01, -3.28302000e-01, -3.66038000e-01,
-4.11321000e-01, -4.79245000e-01, -5.39623000e-01,
-6.52830000e-01, -6.26415000e-01, -5.66038000e-01,
-4.71698000e-01, -4.71698000e-01, -2.37736000e-01,
-1.69811000e-01, -1.24528000e-01, -7.92453000e-02,
-3.01887000e-02, 3.77358000e-03, 3.39623000e-02,
3.77358000e-03, 5.28302000e-02, -1.13208000e-02,
5.66038000e-02, 2.22045000e-16, 2.26415000e-02,
3.77358000e-02, 3.01887000e-02, 1.13208000e-02,
3.39623000e-02, 1.88679000e-02, 3.77358000e-02,
1.88679000e-02, 3.39623000e-02, 2.26415000e-02,
5.28302000e-02, 1.13208000e-02, 3.01887000e-02,
3.01887000e-02, 7.54717000e-03, -3.77358000e-02,
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9.81132000e-02, -1.16981000e-01, 3.09434000e-01,
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2.26415000e-02, -7.54717000e-03, 2.26415000e-02,
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1.88679000e-02, 3.77358000e-03, -1.88679000e-02,
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1.13208000e-02, 4.15094000e-02, -3.77358000e-03,
2.26415000e-02, 3.01887000e-02, 1.13208000e-02,
2.64151000e-02, 1.50943000e-02, -7.54717000e-03,
2.26415000e-02, -3.77358000e-03, 1.13208000e-02,
1.13208000e-02, 1.50943000e-02, -1.13208000e-02,
1.50943000e-02, 7.54717000e-03, 1.88679000e-02,
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1.88679000e-02, 3.77358000e-03, 7.54717000e-03,
1.88679000e-02, -3.77358000e-03, 7.54717000e-03,
2.26415000e-02, 3.77358000e-03, 3.77358000e-03,
-3.77358000e-03, 1.88679000e-02, 3.01887000e-02,
1.13208000e-02, -1.13208000e-02, 2.22045000e-16,
7.54717000e-03, 7.54717000e-03, 7.54717000e-03,
1.50943000e-02, 7.54717000e-03])
fig14top_t_raw = np.array(
[ 0.747111, 0.75739 , 0.761501, 0.769724, 0.777947, 0.78617 ,
0.794393, 0.804672, 0.808783, 0.817006, 0.825229, 0.82934 ,
0.837563, 0.84373 , 0.851953, 0.856065, 0.862232, 0.870455,
0.893068, 0.903346, 0.911569, 0.917736, 0.921848, 0.938293,
0.942405, 0.944461, 0.946516, 0.950628, 0.952683, 0.956795,
0.960906, 0.962962, 0.965018, 0.969129, 0.973241, 0.975296,
0.977352, 0.983519, 0.985575, 0.987631, 0.991742, 0.993798,
1.00202 , 1.00819 , 1.0123 , 1.01641 , 1.02052 , 1.02258 ,
1.02875 , 1.0308 , 1.03491 , 1.03697 , 1.04108 , 1.04519 ,
1.04725 , 1.0493 , 1.05136 , 1.05547 , 1.06164 , 1.08014 ,
1.09042 , 1.10275 , 1.10892 , 1.11509 , 1.12742 , 1.13564 ,
1.14592 , 1.1562 , 1.16237 , 1.17059 , 1.17676 , 1.18498 ,
1.1932 , 1.20348 , 1.20759 , 1.21376 , 1.22198 , 1.24665 ,
1.25899 , 1.26104 , 1.26515 , 1.26721 , 1.26926 , 1.27132 ,
1.27749 , 1.27954 , 1.2816 , 1.28365 , 1.28777 , 1.29393 ,
1.29599 , 1.29804 , 1.3001 , 1.30216 , 1.30421 , 1.30627 ,
1.31243 , 1.32066 , 1.32271 , 1.32477 , 1.33094 , 1.3371 ,
1.34121 , 1.34327 , 1.34533 , 1.34944 , 1.35149 , 1.35355 ,
1.35972 , 1.36383 , 1.37616 , 1.38027 , 1.38233 , 1.38438 ,
1.38644 , 1.3885 , 1.39261 , 1.39466 , 1.40083 , 1.407 ,
1.41316 , 1.41522 , 1.41728 , 1.41933 , 1.42755 , 1.43167 ,
1.43372 , 1.43578 , 1.44194 , 1.44606 , 1.45222 , 1.45633 ,
1.45839 , 1.46045 , 1.4625 , 1.46456 , 1.46867 , 1.47278 ,
1.47484 , 1.47689 , 1.48511 , 1.49128 , 1.49334 , 1.49951 ,
1.50156 , 1.50978 , 1.52006 , 1.52829 , 1.5324 , 1.53856 ,
1.54062 , 1.5509 , 1.55501 , 1.56529 , 1.57351 , 1.57968 ,
1.58173 , 1.58996 , 1.59407 , 1.60435 , 1.60846 , 1.61463 ,
1.62079 , 1.62696 , 1.63107 , 1.63724 , 1.64135 , 1.64752 ,
1.65368 , 1.66191 , 1.67219 , 1.67835 , 1.68246 , 1.68863 ,
1.69069 , 1.69274 , 1.69685 , 1.70097 , 1.70508 , 1.70713 ,
1.71124 , 1.71741 , 1.71947 , 1.72358 , 1.72769 , 1.73386 ,
1.74002 , 1.74208 , 1.74414 , 1.74825 , 1.7503 , 1.75441 ,
1.76058 , 1.76264 , 1.76469 , 1.76675 , 1.77086 , 1.77292 ,
1.77497 , 1.77703 , 1.78114 , 1.78525 , 1.78936 , 1.79142 ,
1.79758 , 1.8017 , 1.80581 , 1.80786 , 1.80992 , 1.81403 ,
1.81609 , 1.81814 , 1.82431 , 1.82842 , 1.83048 , 1.83459 ,
1.84075 , 1.84898 , 1.8572 , 1.85926 , 1.86337 , 1.86748 ,
1.87159 , 1.8757 , 1.87776 , 1.87981 , 1.88187 , 1.88392 ,
1.88804 , 1.89009 , 1.8942 , 1.89626 , 1.90243 , 1.90859 ,
1.91065 , 1.91476 , 1.92709 , 1.93532 , 1.94148 , 1.94765 ,
1.95587 , 1.96204 , 1.9641 , 1.97026 , 1.97643 , 1.98054 ,
1.99699 , 2.00316 , 2.00521 , 2.01549 , 2.02988 , 2.04016 ,
2.05044 , 2.07305 , 2.07922 , 2.08744 , 2.09566 , 2.10183 ,
2.10594 , 2.108 , 2.11211 , 2.11622 , 2.11828 , 2.12444 ,
2.1265 , 2.12856 , 2.13472 , 2.14089 , 2.14295 , 2.145 ,
2.14911 , 2.15117 , 2.15734 , 2.16145 , 2.16556 , 2.16761 ,
2.17173 , 2.17378 , 2.17789 , 2.182 , 2.18406 , 2.18817 ,
2.19023 , 2.19434 , 2.19639 , 2.20051 , 2.20462 , 2.20667 ,
2.21284 , 2.2149 , 2.22106 , 2.22723 , 2.22929 , 2.23134 ,
2.23545 , 2.23751 , 2.23956 , 2.24368 , 2.24573 , 2.2519 ,
2.25601 , 2.25601 , 2.26012 , 2.26218 , 2.26629 , 2.27246 ,
2.27657 , 2.28068 , 2.28273 , 2.28479 , 2.29096 , 2.29918 ,
2.30124 , 2.30535 , 2.30946 , 2.31151 , 2.31563 , 2.31974 ,
2.32179 , 2.32385 , 2.33413 , 2.33618 , 2.34235 , 2.35057 ,
2.35263 , 2.3588 , 2.36291 , 2.37524 , 2.3773 , 2.38346 ,
2.38758 , 2.39785 , 2.40402 , 2.4143 , 2.42047 , 2.42869 ,
2.43486 , 2.44514 , 2.45336 , 2.46158 , 2.48008 , 2.49036 ,
2.49859 , 2.50681 , 2.51298 , 2.52531 , 2.52942 , 2.53148 ,
2.53559 , 2.53764 , 2.54176 , 2.54792 , 2.54998 , 2.55203 ,
2.55409 , 2.56231 , 2.56642 , 2.56848 , 2.57259 , 2.5767 ,
2.58287 , 2.58904 , 2.59109 , 2.5952 , 2.59726 , 2.60137 ,
2.60548 , 2.60959 , 2.61165 , 2.61371 , 2.61782 , 2.61987 ,
2.62193 , 2.62398 , 2.62604 , 2.6281 , 2.63221 , 2.64043 ,
2.64865 , 2.65276 , 2.65482 , 2.65893 , 2.66304 , 2.6651 ,
2.66921 , 2.67743 , 2.67949 , 2.68154 , 2.68771 , 2.69182 ,
2.69593 , 2.70005 , 2.7021 , 2.70621 , 2.71032 , 2.71444 ,
2.71855 , 2.7206 , 2.72471 , 2.72883 , 2.73088 , 2.73499 ,
2.73705 , 2.7391 , 2.74322 , 2.74733 , 2.75761 , 2.75966 ,
2.76583 , 2.77816 , 2.78227 , 2.79255 , 2.80078 , 2.809 ,
2.81928 , 2.82956 , 2.84806 , 2.85422 , 2.86656 , 2.87889 ,
2.88917 , 2.89945 , 2.92617 , 2.94056 , 2.9529 , 2.95495 ,
2.95907 , 2.96112 , 2.96523 , 2.96729 , 2.97346 , 2.97551 ,
2.97962 , 2.98785 , 2.99196 , 2.99401 , 2.99607 , 3.00018 ,
3.01252 , 3.01663 , 3.01868 , 3.02279 , 3.03307 , 3.03718 ,
3.03924 , 3.04952 , 3.05157 , 3.05363 , 3.0598 , 3.06391 ,
3.07008 , 3.07213 , 3.07624 , 3.08035 , 3.08241 , 3.08652 ,
3.08858 , 3.09269 , 3.10091 , 3.10297 , 3.10502 , 3.10913 ,
3.11325 , 3.11941 , 3.12558 , 3.12558 , 3.12969 , 3.1338 ,
3.13791 , 3.14614 , 3.14819 , 3.15025 , 3.15436 , 3.15847 ,
3.16053 , 3.16669 , 3.16875 , 3.17081 , 3.17492 , 3.17697 ,
3.18931 , 3.19547 , 3.2037 , 3.21192 , 3.2222 , 3.22631 ,
3.23042 , 3.23453 , 3.24276 , 3.25098 , 3.25509 , 3.26126 ,
3.26948 , 3.2777 , 3.28593 , 3.29209 , 3.30237 , 3.30854 ,
3.33321 , 3.34554 , 3.35582 , 3.36404 , 3.37432 , 3.37843 ,
3.37843 , 3.38254 , 3.38666 , 3.38871 , 3.39077 , 3.39488 ,
3.39899 , 3.40105 , 3.40516 , 3.40927 , 3.41132 , 3.41544 ,
3.41749 , 3.4216 , 3.42366 , 3.43188 , 3.43394 , 3.43599 ,
3.4401 , 3.44216 , 3.44422 , 3.44833 , 3.45449 , 3.46066 ,
3.46477 , 3.46683 , 3.46889 , 3.473 , 3.47505 , 3.47916 ,
3.48533 , 3.48944 , 3.49767 , 3.50383 , 3.51206 , 3.51617 ,
3.52233 , 3.5285 , 3.53467 , 3.54289 , 3.547 , 3.55111 ,
3.55728 , 3.55934 , 3.56345 , 3.58401 , 3.58606 , 3.58812 ,
3.59223 , 3.5984 , 3.60456 , 3.61073 , 3.61895 , 3.62512 ,
3.63129 , 3.6354 , 3.64362 , 3.6539 , 3.66418 , 3.66829 ,
3.67446 , 3.6909 , 3.69501 , 3.70324 , 3.70735 , 3.71968 ,
3.72379 , 3.73202 , 3.74435 , 3.75257 , 3.76285 , 3.77108 ,
3.78135 , 3.80191 , 3.8163 , 3.82452 , 3.83891 , 3.85125 ,
3.86153 , 3.88003 , 3.88414 , 3.89647 , 3.91498 , 3.9232 ,
3.93553 , 3.94787 , 3.95609 , 3.97254 , 3.98282 , 3.98898 ,
3.99721 , 4.00132 , 4.01571 , 4.02599 , 4.03832 , 4.05065 ,
4.06093 , 4.06916 , 4.0856 , 4.09588 , 4.1041 , 4.11438 ,
4.13083 , 4.13699 , 4.14522 , 4.1555 , 4.16372 , 4.18016 ,
4.20278 ])
fig14top_w_raw = np.array(
[ 2.22045000e-16, 2.22045000e-16, -1.18110000e-02,
2.22045000e-16, 2.22045000e-16, -2.36220000e-02,
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2.36220000e-02, 1.57480000e-02, 1.96850000e-02,
1.96850000e-02, 3.54331000e-02, 3.14961000e-02,
1.57480000e-02, 2.36220000e-02, 1.57480000e-02,
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2.75591000e-02, 4.33071000e-02, 2.75591000e-02,
2.36220000e-02, 3.54331000e-02, 1.57480000e-02,
2.75591000e-02, 1.18110000e-02, 1.57480000e-02,
3.54331000e-02, 1.96850000e-02, 3.54331000e-02,
1.96850000e-02, 3.93701000e-02, 4.72441000e-02,
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3.54331000e-02, 4.72441000e-02, 3.14961000e-02,
2.75591000e-02, 2.75591000e-02, 3.93701000e-02,
3.14961000e-02, 3.93701000e-02, 3.54331000e-02,
2.36220000e-02])
fig14bot_t_raw = np.array(
[ 0.578169, 0.584338, 0.592563, 0.600788, 0.613126, 0.621351,
0.629576, 0.639857, 0.648082, 0.654251, 0.662476, 0.670701,
0.67687 , 0.683039, 0.689208, 0.691264, 0.695377, 0.699489,
0.705658, 0.709771, 0.713883, 0.720052, 0.728277, 0.738559,
0.74884 , 0.759121, 0.763234, 0.771459, 0.773515, 0.777628,
0.785853, 0.787909, 0.794078, 0.796134, 0.800247, 0.80436 ,
0.808472, 0.812585, 0.818754, 0.824922, 0.826979, 0.831091,
0.839316, 0.845485, 0.847541, 0.851654, 0.85371 , 0.857823,
0.861935, 0.866048, 0.870161, 0.872217, 0.876329, 0.880442,
0.882498, 0.888667, 0.890723, 0.894836, 0.905117, 0.911286,
0.919511, 0.927736, 0.938018, 0.946243, 0.956524, 0.970918,
0.977087, 0.985312, 0.993537, 1.00382 , 1.00999 , 1.01616 ,
1.02438 , 1.04289 , 1.05111 , 1.06962 , 1.07579 , 1.07784 ,
1.08401 , 1.09224 , 1.09635 , 1.09841 , 1.10252 , 1.10458 ,
1.11075 , 1.1128 , 1.11691 , 1.11897 , 1.12514 , 1.12925 ,
1.13336 , 1.13542 , 1.13748 , 1.14159 , 1.14776 , 1.15187 ,
1.15393 , 1.15804 , 1.1601 , 1.16215 , 1.16421 , 1.16626 ,
1.17243 , 1.17449 , 1.17655 , 1.18066 , 1.18683 , 1.18888 ,
1.19094 , 1.193 , 1.193 , 1.19711 , 1.20122 , 1.20945 ,
1.21356 , 1.21562 , 1.21767 , 1.22178 , 1.22384 , 1.2259 ,
1.23207 , 1.23412 , 1.23823 , 1.2444 , 1.24646 , 1.25057 ,
1.25057 , 1.25263 , 1.25674 , 1.2588 , 1.26497 , 1.26702 ,
1.27319 , 1.2773 , 1.27936 , 1.28142 , 1.28964 , 1.2917 ,
1.29375 , 1.29581 , 1.30198 , 1.30404 , 1.3102 , 1.31432 ,
1.31843 , 1.32871 , 1.33488 , 1.33899 , 1.34927 , 1.35339 ,
1.36367 , 1.36984 , 1.37806 , 1.38629 , 1.39451 , 1.40274 ,
1.41096 , 1.41919 , 1.43152 , 1.43769 , 1.44592 , 1.44798 ,
1.45209 , 1.4562 , 1.46031 , 1.46237 , 1.46854 , 1.47676 ,
1.48499 , 1.4891 , 1.50349 , 1.51583 , 1.51994 , 1.522 ,
1.52611 , 1.52817 , 1.53228 , 1.5364 , 1.53845 , 1.54256 ,
1.54462 , 1.54668 , 1.5549 , 1.55901 , 1.56724 , 1.57341 ,
1.57546 , 1.57752 , 1.57958 , 1.58163 , 1.58369 , 1.58575 ,
1.59191 , 1.59808 , 1.60014 , 1.60631 , 1.60836 , 1.61042 ,
1.61453 , 1.6207 , 1.62893 , 1.63715 , 1.63921 , 1.64127 ,
1.64332 , 1.64743 , 1.65155 , 1.6536 , 1.65566 , 1.65977 ,
1.66388 , 1.668 , 1.67005 , 1.67211 , 1.67828 , 1.6865 ,
1.68856 , 1.69062 , 1.69473 , 1.69884 , 1.7009 , 1.70501 ,
1.71118 , 1.71324 , 1.71529 , 1.71735 , 1.72352 , 1.72557 ,
1.72969 , 1.73585 , 1.74202 , 1.74614 , 1.7523 , 1.75642 ,
1.76053 , 1.76259 , 1.7667 , 1.77492 , 1.78315 , 1.79343 ,
1.7996 , 1.81194 , 1.82016 , 1.8325 , 1.84484 , 1.85306 ,
1.8654 , 1.87568 , 1.88185 , 1.88802 , 1.89624 , 1.90036 ,
1.91064 , 1.92092 , 1.92503 , 1.93326 , 1.94148 , 1.94354 ,
1.94765 , 1.95176 , 1.95588 , 1.95793 , 1.96205 , 1.9641 ,
1.96616 , 1.97027 , 1.9785 , 1.98055 , 1.98261 , 1.98878 ,
1.99495 , 1.99906 , 2.00317 , 2.00523 , 2.00728 , 2.00934 ,
2.01345 , 2.02168 , 2.02373 , 2.0299 , 2.03196 , 2.03401 ,
2.03607 , 2.04224 , 2.04635 , 2.05252 , 2.05663 , 2.0628 ,
2.06692 , 2.07103 , 2.07308 , 2.07514 , 2.0772 , 2.07925 ,
2.08337 , 2.08953 , 2.09159 , 2.09365 , 2.09776 , 2.10187 ,
2.10598 , 2.11215 , 2.11832 , 2.12038 , 2.12243 , 2.12243 ,
2.1286 , 2.13477 , 2.13683 , 2.13889 , 2.143 , 2.14711 ,
2.14917 , 2.15122 , 2.15328 , 2.15534 , 2.1615 , 2.16562 ,
2.16973 , 2.18207 , 2.18412 , 2.19235 , 2.19852 , 2.20263 ,
2.21085 , 2.21908 , 2.22731 , 2.23347 , 2.23759 , 2.24581 ,
2.25198 , 2.26226 , 2.27049 , 2.28077 , 2.28694 , 2.29722 ,
2.30133 , 2.30544 , 2.31367 , 2.31984 , 2.32806 , 2.33834 ,
2.34451 , 2.35479 , 2.36302 , 2.36713 , 2.37124 , 2.37536 ,
2.37947 , 2.38153 , 2.38564 , 2.38975 , 2.39181 , 2.40003 ,
2.40415 , 2.4062 , 2.41237 , 2.4206 , 2.42265 , 2.42676 ,
2.43088 , 2.43293 , 2.4391 , 2.44733 , 2.45144 , 2.45555 ,
2.45966 , 2.46172 , 2.46378 , 2.46583 , 2.472 , 2.47817 ,
2.48023 , 2.4864 , 2.49051 , 2.49668 , 2.49873 , 2.50079 ,
2.50696 , 2.51313 , 2.51724 , 2.51724 , 2.52135 , 2.53369 ,
2.5378 , 2.53986 , 2.54192 , 2.54397 , 2.54808 , 2.55631 ,
2.55837 , 2.56248 , 2.56453 , 2.56865 , 2.57276 , 2.57482 ,
2.57893 , 2.5851 , 2.59127 , 2.59538 , 2.60772 , 2.61594 ,
2.62828 , 2.63445 , 2.64062 , 2.6509 , 2.65707 , 2.66735 ,
2.67969 , 2.6838 , 2.68997 , 2.70025 , 2.70847 , 2.71876 ,
2.72904 , 2.74754 , 2.75577 , 2.76194 , 2.77428 , 2.78456 ,
2.78867 , 2.79278 , 2.79484 , 2.79895 , 2.80306 , 2.80512 ,
2.80718 , 2.81129 , 2.81334 , 2.82157 , 2.82568 , 2.82774 ,
2.8298 , 2.83596 , 2.84213 , 2.84625 , 2.8483 , 2.85036 ,
2.85241 , 2.85653 , 2.85858 , 2.86475 , 2.87092 , 2.87298 ,
2.87709 , 2.8812 , 2.88531 , 2.88737 , 2.89354 , 2.89971 ,
2.90382 , 2.90588 , 2.90999 , 2.9141 , 2.91616 , 2.91822 ,
2.92027 , 2.92233 , 2.9285 , 2.93261 , 2.93878 , 2.94083 ,
2.94495 , 2.94906 , 2.95112 , 2.95523 , 2.95728 , 2.9614 ,
2.96345 , 2.96551 , 2.96757 , 2.96962 , 2.97373 , 2.9799 ,
2.98196 , 2.98402 , 2.98813 , 2.99018 , 2.9943 , 2.99635 ,
2.99841 , 3.00458 , 3.0128 , 3.01486 , 3.0272 , 3.03954 ,
3.05187 , 3.05599 , 3.07449 , 3.08066 , 3.08683 , 3.10122 ,
3.10739 , 3.11562 , 3.11973 , 3.1259 , 3.13001 , 3.14441 ,
3.15469 , 3.16086 , 3.16291 , 3.17319 , 3.17731 , 3.18759 ,
3.19581 , 3.20404 , 3.21226 , 3.21432 , 3.21843 , 3.22049 ,
3.22254 , 3.22666 , 3.23077 , 3.23283 , 3.23488 , 3.23694 ,
3.24311 , 3.24928 , 3.24928 , 3.25133 , 3.25339 , 3.2575 ,
3.26573 , 3.27395 , 3.27806 , 3.28218 , 3.28423 , 3.28629 ,
3.28835 , 3.29451 , 3.29657 , 3.30274 , 3.30685 , 3.30891 ,
3.31096 , 3.31508 , 3.3233 , 3.33564 , 3.34798 , 3.3562 ,
3.36854 , 3.37882 , 3.39322 , 3.4035 , 3.41378 , 3.42406 ,
3.43228 , 3.4364 , 3.44051 , 3.44257 , 3.44874 , 3.46313 ,
3.47135 , 3.48369 , 3.48575 , 3.49397 , 3.49809 , 3.50837 ,
3.51248 , 3.5207 , 3.53099 , 3.53921 , 3.55361 , 3.568 ,
3.57417 , 3.58239 , 3.58856 , 3.59473 , 3.60296 , 3.61118 ,
3.62146 , 3.62969 , 3.64408 , 3.65025 , 3.65642 , 3.66053 ,
3.68109 , 3.69138 , 3.69755 , 3.70166 , 3.70988 , 3.72428 ,
3.73456 , 3.75101 , 3.76335 , 3.79213 , 3.81064 , 3.82298 ,
3.83532 , 3.84148 , 3.85177 , 3.85588 , 3.85999 , 3.87233 ,
3.87644 , 3.88467 , 3.89084 , 3.897 , 3.90317 , 3.9114 ,
3.92168 , 3.93402 , 3.94224 , 3.95458 , 3.96075 , 3.96692 ,
3.97103 , 3.97926 , 3.98748 , 3.99571 , 4.00393 , 4.02038 ,
4.03066 , 4.04094 ])
fig14bot_w_raw = np.array(
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1.62048000e-02, 3.24746000e-02, 8.10359000e-03,
2.84408000e-02, 2.84529000e-02])
fig15top_t_raw = np.array(
[ 1.01862, 1.04636, 1.05823, 1.06221, 1.07013, 1.07807,
1.08598, 1.08997, 1.09985, 1.10979, 1.11571, 1.12365,
1.14149, 1.15932, 1.16919, 1.17913, 1.18899, 1.19684,
1.2047 , 1.20672, 1.21058, 1.22052, 1.22055, 1.22458,
1.22662, 1.23065, 1.23078, 1.23095, 1.23717, 1.23938,
1.24354, 1.24761, 1.25735, 1.25922, 1.26111, 1.26297,
1.26684, 1.27479, 1.27686, 1.28095, 1.28511, 1.28735,
1.2915 , 1.29735, 1.30115, 1.30297, 1.3068 , 1.30866,
1.31053, 1.31438, 1.31824, 1.32199, 1.32209, 1.32586,
1.32976, 1.33171, 1.33364, 1.34151, 1.34744, 1.35736,
1.36329, 1.37124, 1.37915, 1.38711, 1.395 , 1.4069 ,
1.41483, 1.43461, 1.44255, 1.44847, 1.4584 , 1.49011,
1.49799, 1.50786, 1.51375, 1.51767, 1.52556, 1.52956,
1.52961, 1.53366, 1.53774, 1.53983, 1.54399, 1.54423,
1.55047, 1.55268, 1.55878, 1.56463, 1.56649, 1.56821,
1.56834, 1.57402, 1.57592, 1.5838 , 1.58781, 1.59194,
1.59421, 1.60038, 1.60812, 1.61198, 1.61384, 1.6157 ,
1.61755, 1.6214 , 1.62527, 1.62903, 1.62914, 1.63287,
1.63476, 1.63865, 1.64656, 1.65053, 1.65062, 1.65276,
1.65698, 1.65924, 1.66344, 1.66561, 1.6736 , 1.6775 ,
1.67928, 1.68318, 1.68502, 1.68691, 1.69279, 1.70274,
1.70283, 1.70498, 1.7072 , 1.71137, 1.71543, 1.71931,
1.72118, 1.72305, 1.72688, 1.7287 , 1.73251, 1.73435,
1.73826, 1.74214, 1.74591, 1.74602, 1.74979, 1.75173,
1.75564, 1.75755, 1.76345, 1.76942, 1.77732, 1.78528,
1.79915, 1.813 , 1.81897, 1.82488, 1.83282, 1.8467 ,
1.8546 , 1.86252, 1.87642, 1.91208, 1.91995, 1.92588,
1.92982, 1.93967, 1.94357, 1.94753, 1.95151, 1.95352,
1.95555, 1.95957, 1.96166, 1.96377, 1.96792, 1.97017,
1.97436, 1.97654, 1.98057, 1.98844, 1.99021, 1.99232,
1.996 , 1.99787, 2.00374, 2.0117 , 2.01377, 2.01595,
2.02019, 2.0223 , 2.02432, 2.03021, 2.03208, 2.03392,
2.03579, 2.03964, 2.04349, 2.04532, 2.04718, 2.05104,
2.05491, 2.05679, 2.05865, 2.06256, 2.06444, 2.0704 ,
2.07445, 2.07657, 2.0787 , 2.0808 , 2.08295, 2.08718,
2.08735, 2.09151, 2.09359, 2.10519, 2.10897, 2.11279,
2.11867, 2.1227 , 2.12274, 2.12682, 2.129 , 2.13122,
2.13537, 2.13936, 2.14324, 2.14705, 2.14894, 2.15082,
2.15464, 2.1565 , 2.15829, 2.1584 , 2.16217, 2.16607,
2.16796, 2.17183, 2.17372, 2.17763, 2.17956, 2.18348,
2.18937, 2.19931, 2.20523, 2.21123, 2.22308, 2.22905,
2.23497, 2.24093, 2.24882, 2.25677, 2.2627 , 2.28055,
2.29242, 2.29636, 2.30232, 2.30432, 2.31622, 2.3202 ,
2.32612, 2.33997, 2.34786, 2.35178, 2.35569, 2.36552,
2.36946, 2.37748, 2.37953, 2.38357, 2.38765, 2.38977,
2.39202, 2.39435, 2.3966 , 2.40267, 2.41053, 2.41242,
2.4143 , 2.41618, 2.41801, 2.42185, 2.42574, 2.43168,
2.43765, 2.43776, 2.43993, 2.44218, 2.44828, 2.45414,
2.45601, 2.45987, 2.46171, 2.46354, 2.46538, 2.46923,
2.47113, 2.47687, 2.47698, 2.47874, 2.48457, 2.49042,
2.49646, 2.50055, 2.50264, 2.50472, 2.50692, 2.509 ,
2.50922, 2.51143, 2.51353, 2.52143, 2.52528, 2.52914,
2.53099, 2.53286, 2.53674, 2.54459, 2.54664, 2.54879,
2.55498, 2.55716, 2.56122, 2.5651 , 2.56704, 2.57089,
2.57279, 2.57662, 2.57845, 2.5803 , 2.58222, 2.58613,
2.59002, 2.59385, 2.59573, 2.59763, 2.60155, 2.60347,
2.60738, 2.61533, 2.62524, 2.63119, 2.63512, 2.64701,
2.651 , 2.6609 , 2.66484, 2.67475, 2.68465, 2.69456,
2.70448, 2.71238, 2.71833, 2.72624, 2.7322 , 2.74011,
2.75003, 2.76189, 2.77177, 2.77768, 2.77962, 2.78556,
2.78945, 2.79338, 2.80139, 2.80343, 2.80548, 2.8095 ,
2.81158, 2.81574, 2.81593, 2.81831, 2.82251, 2.82857,
2.83437, 2.83443, 2.84006, 2.84021, 2.84387, 2.84576,
2.85165, 2.86158, 2.8617 , 2.86392, 2.86816, 2.8722 ,
2.87999, 2.88181, 2.88559, 2.88727, 2.88741, 2.89313,
2.89503, 2.89888, 2.90074, 2.90462, 2.90851, 2.91237,
2.92039, 2.92248, 2.92663, 2.93081, 2.93502, 2.93724,
2.94134, 2.94914, 2.95296, 2.95485, 2.95867, 2.96257,
2.97053, 2.97061, 2.97276, 2.97893, 2.98113, 2.98525,
2.99296, 2.9931 , 2.99873, 3.00041, 3.00054, 3.00615,
3.00625, 3.01002, 3.01387, 3.01772, 3.01963, 3.02353,
3.02742, 3.03134, 3.03527, 3.04121, 3.05115, 3.06104,
3.067 , 3.0749 , 3.08088, 3.09077, 3.09476, 3.12841,
3.13634, 3.14426, 3.15215, 3.16009, 3.17198, 3.17791,
3.1819 , 3.1878 , 3.19965, 3.20356, 3.21144, 3.21533,
3.22528, 3.22533, 3.22739, 3.23145, 3.23357, 3.23774,
3.24193, 3.2442 , 3.24643, 3.25256, 3.2584 , 3.26027,
3.2621 , 3.2678 , 3.26793, 3.27555, 3.28357, 3.28569,
3.28788, 3.29617, 3.302 , 3.30388, 3.30773, 3.30956,
3.31334, 3.31515, 3.32096, 3.32283, 3.32289, 3.32473,
3.33056, 3.33244, 3.33435, 3.34224, 3.34634, 3.34848,
3.35461, 3.35487, 3.35711, 3.36127, 3.36532, 3.3712 ,
3.37508, 3.37695, 3.37881, 3.38066, 3.38073, 3.38651,
3.39447, 3.39849, 3.39864, 3.40284, 3.40506, 3.40716,
3.41688, 3.41873, 3.4206 , 3.42246, 3.42428, 3.43 ,
3.43206, 3.43589, 3.4378 , 3.43968, 3.44552, 3.44743,
3.45129, 3.45134, 3.45521, 3.46315, 3.47705, 3.48694,
3.49886, 3.50678, 3.52063, 3.53058, 3.54046, 3.54638,
3.55627, 3.56019, 3.57605, 3.58401, 3.59192, 3.59788,
3.60581, 3.60975, 3.61565, 3.62355, 3.6255 , 3.63338,
3.63732, 3.63926, 3.64518, 3.64919, 3.65123, 3.6533 ,
3.65537, 3.65746, 3.66357, 3.66379, 3.66607, 3.67034,
3.67645, 3.68227, 3.68409, 3.68592, 3.68971, 3.68979,
3.69357, 3.69942, 3.70543, 3.71144, 3.71157, 3.71372,
3.71591, 3.71796, 3.72382, 3.72944, 3.72959, 3.72968,
3.73525, 3.73909, 3.74297, 3.74489, 3.74874, 3.7526 ,
3.75451, 3.75843, 3.76233, 3.76625, 3.77814, 3.78209,
3.796 , 3.8039 , 3.80985, 3.81774, 3.83163, 3.8376 ,
3.84749, 3.85343, 3.86333, 3.87723, 3.8911 , 3.90695,
3.91883, 3.92673, 3.93266, 3.94658, 3.9743 , 3.98417,
3.98815, 4.00202, 4.00398, 4.01388, 4.01984, 4.03172,
4.04757, 4.05154, 4.06741, 4.07929, 4.08322, 4.09117,
4.09711, 4.111 , 4.11495, 4.12288, 4.13078, 4.13672,
4.14462, 4.1664 , 4.18226, 4.19611, 4.20009, 4.20999,
4.21395, 4.22784, 4.23776, 4.24767, 4.25163, 4.27342,
4.28731, 4.3091 , 4.33091, 4.34478, 4.35664, 4.37645,
4.3824 , 4.39229, 4.40417, 4.4121 , 4.41607, 4.42398,
4.43589, 4.44378, 4.46163, 4.47746])
fig15top_w_raw = np.array(
[ 0. , 0. , 0.00383142, -0.00383142, 0. ,
-0.0114943 , -0.00383142, -0.0153257 , -0.00383142, -0.0191571 ,
-0.00766284, -0.0191571 , -0.0229885 , -0.0268199 , -0.00766284,
-0.0268199 , -0.00383142, 0.0344828 , 0.0689655 , 0.045977 ,
0.0996169 , 0.0842912 , 0.0689655 , 0.0344828 , 0.00383142,
-0.0306513 , -0.0957854 , -0.183908 , -0.321839 , -0.440613 ,
-0.536398 , -0.59387 , -0.509579 , -0.455939 , -0.40613 ,
-0.348659 , -0.298851 , -0.314176 , -0.356322 , -0.425287 ,
-0.524904 , -0.655172 , -0.747126 , -0.701149 , -0.62069 ,
-0.54023 , -0.471264 , -0.409962 , -0.356322 , -0.298851 ,
-0.245211 , -0.141762 , -0.191571 , -0.0957854 , -0.0613027 ,
-0.045977 , -0.0191571 , 0.00766284, 0.0114943 , 0.00383142,
0.0114943 , -0.00383142, 0.00383142, -0.0114943 , 0.00383142,
-0.00383142, -0.00383142, 0.0114943 , 0. , 0.0114943 ,
0. , -0.00766284, 0.0153257 , 0.0344828 , 0.0613027 ,
0.0804598 , 0.0957854 , 0.0766284 , 0.0536398 , 0.00766284,
-0.0498084 , -0.10728 , -0.206897 , -0.329502 , -0.475096 ,
-0.59387 , -0.670498 , -0.624521 , -0.563218 , -0.43295 ,
-0.498084 , -0.363985 , -0.32567 , -0.302682 , -0.32567 ,
-0.409962 , -0.555556 , -0.670498 , -0.578544 , -0.528736 ,
-0.467433 , -0.40613 , -0.340996 , -0.283525 , -0.237548 ,
-0.1341 , -0.191571 , -0.0727969 , -0.0268199 , 0.00766284,
0.0191571 , 0.0114943 , -0.0306513 , -0.114943 , -0.241379 ,
-0.383142 , -0.505747 , -0.597701 , -0.632184 , -0.601533 ,
-0.501916 , -0.467433 , -0.398467 , -0.35249 , -0.318008 ,
-0.344828 , -0.390805 , -0.475096 , -0.59387 , -0.701149 ,
-0.747126 , -0.708812 , -0.651341 , -0.59387 , -0.528736 ,
-0.448276 , -0.371648 , -0.298851 , -0.275862 , -0.233716 ,
-0.1341 , -0.191571 , -0.0957854 , -0.0727969 , -0.045977 ,
-0.0114943 , 0.0114943 , -0.00383142, 0.00766284, -0.00766284,
-0.0114943 , -0.00383142, -0.0153257 , 0. , -0.00766284,
-0.0114943 , 0. , 0. , -0.0153257 , -0.0153257 ,
0.0114943 , 0.0153257 , 0.0268199 , 0.0536398 , 0.0881226 ,
0.0881226 , 0.0766284 , 0.0613027 , 0.0383142 , 0.00766284,
-0.045977 , -0.111111 , -0.206897 , -0.340996 , -0.455939 ,
-0.555556 , -0.590038 , -0.563218 , -0.45977 , -0.521073 ,
-0.37931 , -0.32567 , -0.287356 , -0.306513 , -0.35249 ,
-0.452107 , -0.590038 , -0.655172 , -0.678161 , -0.651341 ,
-0.59387 , -0.524904 , -0.467433 , -0.409962 , -0.35249 ,
-0.279693 , -0.214559 , -0.164751 , -0.118774 , -0.0651341 ,
-0.00766284, 0.0191571 , 0.0727969 , 0.0613027 , 0.0191571 ,
-0.0536398 , -0.126437 , -0.187739 , -0.272031 , -0.409962 ,
-0.494253 , -0.59387 , -0.643678 , -0.498084 , -0.409962 ,
-0.337165 , -0.306513 , -0.337165 , -0.360153 , -0.417625 ,
-0.521073 , -0.639847 , -0.735632 , -0.747126 , -0.704981 ,
-0.632184 , -0.582375 , -0.532567 , -0.45977 , -0.402299 ,
-0.302682 , -0.360153 , -0.264368 , -0.233716 , -0.183908 ,
-0.137931 , -0.0957854 , -0.0651341 , -0.0421456 , -0.0191571 ,
0.00383142, -0.0114943 , 0. , -0.0268199 , -0.0114943 ,
-0.0268199 , -0.0114943 , -0.0229885 , -0.00383142, -0.0191571 ,
-0.0114943 , -0.0229885 , -0.0153257 , -0.00383142, -0.0114943 ,
-0.0229885 , -0.0268199 , -0.0344828 , -0.0268199 , -0.0153257 ,
0. , 0.0229885 , 0.0498084 , 0.0842912 , 0.0957854 ,
0.0498084 , 0.0114943 , -0.0268199 , -0.0842912 , -0.157088 ,
-0.291188 , -0.467433 , -0.601533 , -0.666667 , -0.636015 ,
-0.590038 , -0.54023 , -0.48659 , -0.409962 , -0.348659 ,
-0.314176 , -0.310345 , -0.32567 , -0.37931 , -0.478927 ,
-0.613027 , -0.689655 , -0.651341 , -0.59387 , -0.54023 ,
-0.471264 , -0.394636 , -0.321839 , -0.268199 , -0.226054 ,
-0.126437 , -0.180077 , -0.0689655 , -0.0114943 , 0.0344828 ,
-0.0153257 , -0.0766284 , -0.1341 , -0.183908 , -0.291188 ,
-0.340996 , -0.455939 , -0.570881 , -0.632184 , -0.616858 ,
-0.563218 , -0.509579 , -0.444444 , -0.386973 , -0.344828 ,
-0.306513 , -0.344828 , -0.429119 , -0.551724 , -0.655172 ,
-0.704981 , -0.662835 , -0.639847 , -0.586207 , -0.544061 ,
-0.478927 , -0.402299 , -0.337165 , -0.302682 , -0.279693 ,
-0.241379 , -0.176245 , -0.122605 , -0.0842912 , -0.0613027 ,
-0.0306513 , -0.00766284, -0.0191571 , -0.0191571 , -0.0229885 ,
-0.00766284, -0.0114943 , -0.0229885 , -0.0229885 , -0.0114943 ,
-0.0153257 , -0.0114943 , -0.0153257 , -0.0229885 , -0.00766284,
-0.0153257 , -0.00766284, -0.0153257 , -0.00766284, -0.0153257 ,
0. , 0.0114943 , 0.0268199 , 0.045977 , 0.0498084 ,
0.0842912 , 0.0996169 , 0.0574713 , 0.0268199 , -0.00383142,
-0.0344828 , -0.0881226 , -0.183908 , -0.283525 , -0.482759 ,
-0.605364 , -0.662835 , -0.590038 , -0.62069 , -0.463602 ,
-0.536398 , -0.386973 , -0.340996 , -0.314176 , -0.32567 ,
-0.386973 , -0.505747 , -0.64751 , -0.685824 , -0.62069 ,
-0.536398 , -0.448276 , -0.295019 , -0.363985 , -0.252874 ,
-0.210728 , -0.157088 , -0.0957854 , -0.0536398 , -0.0153257 ,
0.0344828 , -0.0114943 , -0.0689655 , -0.16092 , -0.272031 ,
-0.398467 , -0.521073 , -0.586207 , -0.524904 , -0.452107 ,
-0.40613 , -0.337165 , -0.306513 , -0.32567 , -0.363985 ,
-0.448276 , -0.563218 , -0.67433 , -0.754789 , -0.64751 ,
-0.716475 , -0.559387 , -0.40613 , -0.475096 , -0.306513 ,
-0.356322 , -0.260536 , -0.203065 , -0.145594 , -0.10728 ,
-0.0766284 , -0.0421456 , -0.0191571 , -0.00383142, -0.00383142,
-0.0229885 , -0.0153257 , -0.0229885 , -0.0114943 , -0.0268199 ,
-0.0229885 , -0.0344828 , -0.0229885 , -0.0268199 , -0.0229885 ,
-0.00766284, -0.0153257 , -0.0153257 , -0.00766284, -0.0229885 ,
-0.00383142, 0.0153257 , 0.0421456 , 0.0651341 , 0.0996169 ,
0.0766284 , 0.0536398 , 0.0153257 , -0.0344828 , -0.10728 ,
-0.210728 , -0.32567 , -0.471264 , -0.597701 , -0.693487 ,
-0.639847 , -0.582375 , -0.509579 , -0.386973 , -0.452107 ,
-0.298851 , -0.344828 , -0.417625 , -0.521073 , -0.704981 ,
-0.64751 , -0.597701 , -0.54023 , -0.467433 , -0.371648 ,
-0.287356 , -0.222222 , -0.164751 , -0.195402 , -0.122605 ,
-0.0651341 , -0.0153257 , 0.0191571 , 0.0383142 , -0.0306513 ,
-0.111111 , -0.206897 , -0.337165 , -0.467433 , -0.570881 ,
-0.613027 , -0.582375 , -0.54023 , -0.482759 , -0.425287 ,
-0.356322 , -0.390805 , -0.310345 , -0.329502 , -0.360153 ,
-0.43295 , -0.555556 , -0.67433 , -0.735632 , -0.639847 ,
-0.574713 , -0.521073 , -0.45977 , -0.375479 , -0.264368 ,
-0.306513 , -0.237548 , -0.199234 , -0.149425 , -0.0996169 ,
-0.0613027 , -0.0114943 , -0.0383142 , 0.0114943 , 0. ,
-0.0153257 , -0.00766284, -0.0229885 , -0.0229885 , -0.0153257 ,
-0.0383142 , -0.0229885 , -0.0114943 , -0.00766284, 0.0153257 ,
0.00766284, -0.00766284, 0. , -0.0114943 , -0.0114943 ,
0. , 0.0191571 , 0.0306513 , 0.045977 , 0.0689655 ,
0.0804598 , 0.0996169 , 0.114943 , 0.0881226 , 0.0574713 ,
0.0153257 , -0.0306513 , -0.0881226 , -0.172414 , -0.283525 ,
-0.43295 , -0.586207 , -0.67433 , -0.609195 , -0.528736 ,
-0.452107 , -0.367816 , -0.40613 , -0.314176 , -0.268199 ,
-0.298851 , -0.337165 , -0.398467 , -0.48659 , -0.59387 ,
-0.624521 , -0.586207 , -0.421456 , -0.498084 , -0.544061 ,
-0.35249 , -0.291188 , -0.252874 , -0.218391 , -0.164751 ,
-0.111111 , -0.0766284 , -0.0536398 , -0.0229885 , -0.00383142,
-0.00766284, 0. , -0.0191571 , -0.00766284, -0.0114943 ,
0.00383142, -0.00766284, -0.0229885 , -0.0153257 , -0.0114943 ,
-0.0114943 , -0.0268199 , -0.0268199 , -0.0268199 , -0.0268199 ,
-0.0114943 , -0.00766284, -0.0344828 , -0.0268199 , -0.00766284,
-0.0191571 , -0.0191571 , -0.00766284, -0.00766284, -0.0153257 ,
-0.0114943 , -0.0114943 , -0.0191571 , -0.0268199 , -0.0268199 ,
-0.0114943 , -0.0229885 , -0.0229885 , -0.0344828 , -0.0268199 ,
-0.0306513 , -0.0191571 , -0.0153257 , -0.00383142, 0. ,
-0.00383142, 0.00383142, -0.00766284, -0.00383142, -0.00383142,
-0.0153257 , -0.0229885 , -0.0229885 , -0.0229885 , -0.0229885 ,
-0.0344828 , -0.0344828 , -0.0421456 , -0.045977 , -0.0306513 ,
-0.0306513 , -0.0383142 , -0.0306513 , -0.0268199 , -0.0268199 ,
-0.0344828 , -0.0229885 , -0.0383142 , -0.0229885 , -0.0306513 ,
-0.0229885 ])
fig15bot_t_raw = np.array(
[ 0.817702, 0.825628, 0.835535, 0.839498, 0.849406, 0.853369,
0.863276, 0.869221, 0.879128, 0.891017, 0.896962, 0.910832,
0.918758, 0.928666, 0.936592, 0.940555, 0.944518, 0.952444,
0.958389, 0.968296, 0.976222, 0.980185, 0.990093, 0.996037,
1.00198 , 1.00594 , 1.01189 , 1.01387 , 1.01783 , 1.0218 ,
1.02576 , 1.02972 , 1.0317 , 1.03567 , 1.03765 , 1.04161 ,
1.04557 , 1.04954 , 1.0535 , 1.05548 , 1.06341 , 1.06737 ,
1.07133 , 1.0753 , 1.07728 , 1.07926 , 1.08124 , 1.0852 ,
1.09115 , 1.09511 , 1.09709 , 1.10106 , 1.10304 , 1.10502 ,
1.10898 , 1.12087 , 1.13276 , 1.13871 , 1.14663 , 1.1605 ,
1.16645 , 1.17041 , 1.17834 , 1.18626 , 1.19815 , 1.20608 ,
1.21797 , 1.22985 , 1.24174 , 1.25165 , 1.26552 , 1.27345 ,
1.28336 , 1.29326 , 1.30119 , 1.30713 , 1.31506 , 1.31902 ,
1.321 , 1.32497 , 1.32695 , 1.33091 , 1.33686 , 1.33884 ,
1.3428 , 1.34478 , 1.35073 , 1.35667 , 1.35865 , 1.36063 ,
1.36262 , 1.36658 , 1.37252 , 1.37649 , 1.37847 , 1.38243 ,
1.38639 , 1.38838 , 1.38838 , 1.39432 , 1.39828 , 1.40225 ,
1.40423 , 1.40621 , 1.41017 , 1.41414 , 1.41612 , 1.42008 ,
1.42206 , 1.42602 , 1.43197 , 1.43395 , 1.43989 , 1.44584 ,
1.44782 , 1.4498 , 1.45178 , 1.45377 , 1.45773 , 1.45971 ,
1.46367 , 1.46764 , 1.46962 , 1.4716 , 1.47358 , 1.47754 ,
1.47952 , 1.48349 , 1.48745 , 1.48943 , 1.49736 , 1.50132 ,
1.5033 , 1.50727 , 1.51321 , 1.51519 , 1.51717 , 1.52114 ,
1.52312 , 1.5251 , 1.52906 , 1.53303 , 1.53501 , 1.54293 ,
1.55284 , 1.56275 , 1.56869 , 1.5786 , 1.58454 , 1.58851 ,
1.59247 , 1.59842 , 1.6103 , 1.61427 , 1.62219 , 1.62814 ,
1.63805 , 1.65192 , 1.66182 , 1.66975 , 1.67569 , 1.67768 ,
1.68164 , 1.6856 , 1.69353 , 1.69551 , 1.70344 , 1.71136 ,
1.71731 , 1.71929 , 1.72127 , 1.72523 , 1.73118 , 1.7391 ,
1.74307 , 1.74703 , 1.74901 , 1.75297 , 1.75495 , 1.75694 ,
1.7609 , 1.76486 , 1.76882 , 1.77279 , 1.77477 , 1.7827 ,
1.78468 , 1.78864 , 1.7926 , 1.79458 , 1.80449 , 1.80647 ,
1.80845 , 1.81242 , 1.8144 , 1.82034 , 1.82431 , 1.82629 ,
1.82827 , 1.83223 , 1.8362 , 1.83818 , 1.84214 , 1.84809 ,
1.85007 , 1.85205 , 1.85799 , 1.86196 , 1.86394 , 1.8679 ,
1.86988 , 1.87384 , 1.87583 , 1.87979 , 1.88177 , 1.88375 ,
1.8897 , 1.89366 , 1.89762 , 1.90159 , 1.90753 , 1.91149 ,
1.91347 , 1.91546 , 1.91942 , 1.92536 , 1.92735 , 1.93725 ,
1.93923 , 1.94122 , 1.9432 , 1.94518 , 1.94914 , 1.95112 ,
1.95509 , 1.95707 , 1.96103 , 1.96499 , 1.96698 , 1.97292 ,
1.97886 , 1.99075 , 1.9967 , 2.00462 , 2.01057 , 2.01453 ,
2.02642 , 2.03237 , 2.03633 , 2.04227 , 2.05614 , 2.07001 ,
2.07596 , 2.08587 , 2.1037 , 2.11757 , 2.12946 , 2.1354 ,
2.14333 , 2.14927 , 2.15918 , 2.16314 , 2.16711 , 2.16909 ,
2.17305 , 2.17702 , 2.18098 , 2.18296 , 2.18494 , 2.18692 ,
2.19089 , 2.19485 , 2.19683 , 2.20079 , 2.20476 , 2.20674 ,
2.20872 , 2.21268 , 2.22061 , 2.22457 , 2.22655 , 2.23052 ,
2.23646 , 2.24042 , 2.24439 , 2.24637 , 2.24835 , 2.25231 ,
2.25628 , 2.26024 , 2.26222 , 2.2642 , 2.26816 , 2.27411 ,
2.27807 , 2.28005 , 2.28402 , 2.286 , 2.29591 , 2.29987 ,
2.30185 , 2.30383 , 2.30779 , 2.31176 , 2.3177 , 2.32167 ,
2.32365 , 2.32563 , 2.33554 , 2.33752 , 2.34148 , 2.34346 ,
2.34544 , 2.34941 , 2.35337 , 2.3613 , 2.36526 , 2.36922 ,
2.3712 , 2.37318 , 2.37517 , 2.37913 , 2.38507 , 2.38705 ,
2.39102 , 2.39696 , 2.40291 , 2.41083 , 2.4148 , 2.42669 ,
2.43263 , 2.43857 , 2.45046 , 2.4683 , 2.48019 , 2.48613 ,
2.5 , 2.50396 , 2.51387 , 2.5218 , 2.52972 , 2.54161 ,
2.55152 , 2.56539 , 2.5753 , 2.58322 , 2.58917 , 2.59313 ,
2.59709 , 2.60304 , 2.60502 , 2.607 , 2.60898 , 2.61295 ,
2.61493 , 2.61691 , 2.62087 , 2.62682 , 2.6288 , 2.63078 ,
2.63474 , 2.63672 , 2.64267 , 2.64663 , 2.6506 , 2.65258 ,
2.65852 , 2.66248 , 2.66645 , 2.67239 , 2.67635 , 2.67834 ,
2.6823 , 2.68428 , 2.68626 , 2.69023 , 2.69419 , 2.70013 ,
2.70211 , 2.7041 , 2.70608 , 2.71202 , 2.71797 , 2.71995 ,
2.72193 , 2.72787 , 2.72986 , 2.73382 , 2.74174 , 2.74571 ,
2.74571 , 2.74967 , 2.7576 , 2.76354 , 2.7675 , 2.7675 ,
2.77147 , 2.77345 , 2.77741 , 2.78336 , 2.78732 , 2.7893 ,
2.79128 , 2.79525 , 2.79723 , 2.80119 , 2.80317 , 2.80515 ,
2.81902 , 2.82893 , 2.83289 , 2.8428 , 2.85469 , 2.8646 ,
2.87054 , 2.87252 , 2.87847 , 2.88243 , 2.88838 , 2.89432 ,
2.89828 , 2.90423 , 2.90819 , 2.91414 , 2.92206 , 2.92404 ,
2.9399 , 2.94584 , 2.95178 , 2.95971 , 2.96962 , 2.97358 ,
2.98745 , 2.99538 , 3.00132 , 3.00529 , 3.01321 , 3.01519 ,
3.01916 , 3.02114 , 3.0251 , 3.02708 , 3.03104 , 3.03501 ,
3.03699 , 3.03897 , 3.04293 , 3.05086 , 3.05284 , 3.0568 ,
3.05879 , 3.06077 , 3.06671 , 3.07067 , 3.07266 , 3.07464 ,
3.08058 , 3.08653 , 3.08851 , 3.09643 , 3.09842 , 3.1004 ,
3.10238 , 3.10436 , 3.10832 , 3.1103 , 3.11625 , 3.12418 ,
3.12814 , 3.13012 , 3.13408 , 3.13606 , 3.14399 , 3.14795 ,
3.14994 , 3.1539 , 3.15588 , 3.16579 , 3.16777 , 3.16975 ,
3.17768 , 3.18362 , 3.18758 , 3.19155 , 3.19749 , 3.19749 ,
3.20542 , 3.20938 , 3.21136 , 3.21334 , 3.21731 , 3.21929 ,
3.22325 , 3.22523 , 3.22721 , 3.2292 , 3.2391 , 3.24505 ,
3.25099 , 3.26486 , 3.27081 , 3.27873 , 3.29062 , 3.30053 ,
3.30449 , 3.30846 , 3.31836 , 3.32233 , 3.32827 , 3.3362 ,
3.34214 , 3.35799 , 3.37979 , 3.39564 , 3.40555 , 3.41149 ,
3.41942 , 3.42933 , 3.43527 , 3.43924 , 3.4432 , 3.44716 ,
3.44914 , 3.45112 , 3.45509 , 3.45905 , 3.46103 , 3.46301 ,
3.47094 , 3.4749 , 3.47688 , 3.47887 , 3.48283 , 3.48481 ,
3.49075 , 3.49472 , 3.50066 , 3.50661 , 3.50859 , 3.51453 ,
3.52048 , 3.52246 , 3.52642 , 3.53038 , 3.53435 , 3.53831 ,
3.54029 , 3.54227 , 3.54624 , 3.5502 , 3.55614 , 3.55813 ,
3.56605 , 3.572 , 3.57992 , 3.58587 , 3.59181 , 3.59776 ,
3.6037 , 3.60766 , 3.61955 , 3.63144 , 3.64729 , 3.6572 ,
3.66513 , 3.68296 , 3.69485 , 3.70278 , 3.71268 , 3.72655 ,
3.74042 , 3.77015 , 3.77213 , 3.77609 , 3.786 , 3.79392 ,
3.80185 , 3.80581 , 3.81176 , 3.82563 , 3.83554 , 3.84941 ,
3.87913 , 3.88706 , 3.90885 , 3.91678 , 3.93857 , 3.9465 ,
3.95641 , 3.97226 , 3.98019 , 3.99208 , 4.00991 , 4.03567 ,
4.04558 , 4.06143 , 4.06539 , 4.07332 , 4.08124 , 4.08719 ,
4.10106 , 4.11097 , 4.11889 , 4.12682 , 4.13276 , 4.13673 ,
4.14663 , 4.16447 , 4.17636 , 4.1823 , 4.2041 , 4.20806 ,
4.23184 , 4.24967 , 4.2576 , 4.27543 , 4.28138 ])
fig15bot_w_raw = np.array(
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-0.232558 , -0.170543 , -0.0891473 , -0.0348837 , 0.00387597,
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0.00775194, 0. , -0.0232558 , -0.00775194, 0.0116279 ,
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-0.325581 , -0.360465 , -0.414729 , -0.496124 , -0.577519 ,
-0.434109 , -0.372093 , -0.317829 , -0.275194 , -0.224806 ,
-0.0968992 , -0.0465116 , -0.0155039 , 0.00387597, 0.0503876 ,
0.0271318 , 0.0736434 , 0.0465116 , 0.0155039 , 0. ,
0.0271318 , 0.0542636 , 0.00387597, 0.0310078 , -0.0271318 ,
0.0193798 , -0.0116279 , 0.0193798 , -0.0465116 , 0.00775194,
-0.0310078 , 0.0193798 , -0.0271318 , 0. , -0.0155039 ,
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-0.410853 , -0.294574 , -0.24031 , -0.166667 , -0.112403 ,
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-0.391473 , -0.46124 , -0.523256 , -0.577519 , -0.515504 ,
-0.418605 , -0.317829 , -0.248062 , -0.197674 , -0.158915 ,
-0.104651 , -0.0581395 , -0.0155039 , 0.0271318 , 0.00387597,
0.0813953 , 0.0116279 , -0.00775194, -0.0271318 , 0.00775194,
-0.0193798 , 0.0465116 , 0.0155039 , 0.0310078 , -0.00387597,
-0.0232558 , -0.00775194, -0.0155039 , -0.0658915 , -0.0348837 ,
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0. , -0.0155039 , 0. , 0.0581395 , 0.0232558 ,
0.00387597, 0. , 0.0310078 , 0.0465116 , 0.0155039 ,
0.0465116 , 0.0620155 , -0.00387597, 0.0155039 , 0.0348837 ,
0.0658915 , 0.0542636 , 0.0658915 , 0.0503876 , -0.0658915 ,
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-0.0542636 , -0.0232558 , 0.0155039 , 0.0465116 , 0.0813953 ,
0.0658915 , 0.0310078 , 0.0542636 , 0.0387597 , 0.0387597 ,
0.0271318 , -0.0193798 , -0.00387597, 0.0116279 , -0.0232558 ,
0. , -0.0232558 , -0.0232558 , -0.00387597, -0.0271318 ,
-0.00387597, -0.0271318 , -0.0155039 , -0.0310078 , -0.0426357 ,
-0.0193798 , -0.0232558 , 0.0155039 , 0.0271318 , 0.0116279 ,
0.0155039 , 0.00775194, -0.0465116 , -0.0155039 , -0.0465116 ,
-0.0155039 , -0.00387597, -0.00387597, 0.00775194, -0.0310078 ,
0. , 0.00387597, 0.00387597, 0.00387597, -0.0232558 ,
-0.00775194, -0.0116279 , 0.0116279 , -0.0310078 , -0.0155039 ,
-0.0387597 , -0.0155039 , -0.0310078 , 0.00387597, -0.0232558 ,
0.00387597, 0.0232558 , -0.0155039 , -0.00387597, -0.0310078 ,
-0.00387597, -0.0348837 , -0.0193798 , -0.0271318 , 0.00387597,
-0.0387597 , 0.00387597, 0.00775194, -0.00387597, -0.0193798 ,
0. , -0.00775194])
fig19b_bridge_to_emb_x_raw = np.array(
[ -4.92072000e+00, -4.90682000e+00, -4.71917000e+00,
-4.70637000e+00, -4.58438000e+00, -4.50665000e+00,
-4.38320000e+00, -4.37332000e+00, -4.23944000e+00,
-4.18165000e+00, -4.10538000e+00, -3.98010000e+00,
-3.97169000e+00, -3.84549000e+00, -3.77142000e+00,
-3.64413000e+00, -3.63828000e+00, -3.50952000e+00,
-3.43874000e+00, -3.30816000e+00, -3.17208000e+00,
-3.04040000e+00, -3.03821000e+00, -2.83739000e+00,
-2.70351000e+00, -2.63968000e+00, -2.50251000e+00,
-2.43905000e+00, -2.23787000e+00, -2.23494000e+00,
-2.10308000e+00, -1.96755000e+00, -1.90135000e+00,
-1.76674000e+00, -1.69998000e+00, -1.50008000e+00,
-1.36694000e+00, -1.23397000e+00, -1.16722000e+00,
-9.68049000e-01, -9.00379000e-01, -8.35635000e-01,
-6.36282000e-01, -5.67149000e-01, -4.34735000e-01,
-3.70357000e-01, -3.02138000e-01, -2.36114000e-01,
-3.67613000e-02, 2.96285000e-02, 9.56527000e-02,
4.96918000e-01, 7.63941000e-01, 1.16466000e+00,
1.36383000e+00, 1.63176000e+00, 2.03212000e+00,
2.09869000e+00, 2.63219000e+00, 2.63237000e+00,
3.16641000e+00, 3.23207000e+00, 3.49964000e+00,
3.76539000e+00, 3.96620000e+00, 4.29888000e+00,
4.56645000e+00, 5.09922000e+00, 5.23346000e+00,
5.76623000e+00, 5.83371000e+00, 6.43415000e+00,
6.76683000e+00, 7.10134000e+00, 7.63465000e+00,
7.70178000e+00, 8.30221000e+00, 8.83552000e+00,
8.90283000e+00, 9.50345000e+00, 9.90307000e+00,
1.01706000e+01, 1.11049000e+01, 1.11712000e+01,
1.19057000e+01, 1.21726000e+01, 1.26399000e+01,
1.30404000e+01, 1.34409000e+01, 1.43088000e+01,
1.44423000e+01, 1.51098000e+01, 1.59107000e+01,
1.59775000e+01, 1.69788000e+01, 1.69790000e+01,
1.78464000e+01, 1.79132000e+01, 1.85808000e+01,
1.87810000e+01, 1.91814000e+01, 1.92483000e+01,
1.94484000e+01, 1.98478000e+01, 1.99151000e+01,
2.04486000e+01, 2.05141000e+01, 2.08483000e+01,
2.13135000e+01, 2.15147000e+01, 2.20478000e+01,
2.21799000e+01, 2.25144000e+01, 2.28478000e+01,
2.29131000e+01, 2.36480000e+01, 2.37131000e+01,
2.44481000e+01, 2.46469000e+01, 2.52485000e+01,
2.55806000e+01, 2.58485000e+01, 2.65810000e+01,
2.67156000e+01, 2.73161000e+01, 2.73815000e+01,
2.78496000e+01, 2.81822000e+01, 2.84500000e+01,
2.89167000e+01, 2.92496000e+01, 2.95843000e+01,
2.98504000e+01, 3.01180000e+01, 3.05843000e+01,
3.07182000e+01, 3.16521000e+01, 3.17192000e+01,
3.24533000e+01, 3.27869000e+01, 3.31874000e+01,
3.35874000e+01, 3.40551000e+01, 3.46557000e+01,
3.56569000e+01, 3.58573000e+01, 3.65913000e+01,
3.69920000e+01, 3.76595000e+01, 3.77263000e+01,
3.86607000e+01, 3.93281000e+01, 3.99958000e+01,
4.11306000e+01, 4.11308000e+01, 4.19319000e+01,
4.28663000e+01, 4.29330000e+01, 4.40011000e+01,
4.50692000e+01, 4.52697000e+01, 4.64043000e+01,
4.66044000e+01, 4.70053000e+01, 4.72050000e+01,
4.81393000e+01, 4.88737000e+01, 4.90744000e+01,
4.94746000e+01, 4.98087000e+01, 4.98755000e+01])
fig19b_bridge_to_emb_w_raw = np.array(
[-0.712329, -0.711006, -0.693151, -0.691329, -0.673973, -0.66868 ,
-0.660274, -0.659334, -0.646595, -0.641096, -0.633839, -0.621918,
-0.62089 , -0.605479, -0.599433, -0.589041, -0.588326, -0.572603,
-0.566825, -0.556164, -0.550595, -0.545205, -0.54519 , -0.543817,
-0.542902, -0.542466, -0.53872 , -0.536986, -0.523288, -0.522871,
-0.50411 , -0.489385, -0.482192, -0.471203, -0.465753, -0.472331,
-0.476712, -0.482184, -0.484932, -0.486976, -0.487671, -0.489268,
-0.494185, -0.49589 , -0.512329, -0.51898 , -0.526027, -0.536986,
-0.550685, -0.558927, -0.567123, -0.560543, -0.556164, -0.565315,
-0.569863, -0.565469, -0.558904, -0.559816, -0.567121, -0.567123,
-0.567123, -0.568743, -0.575342, -0.581584, -0.586301, -0.590857,
-0.594521, -0.601086, -0.60274 , -0.610035, -0.610959, -0.616438,
-0.619171, -0.621918, -0.626785, -0.627397, -0.632877, -0.635309,
-0.635616, -0.638356, -0.641638, -0.643836, -0.649315, -0.649542,
-0.652055, -0.653051, -0.654795, -0.654795, -0.654795, -0.654795,
-0.654795, -0.654795, -0.657323, -0.657534, -0.657534, -0.657535,
-0.660274, -0.660274, -0.660274, -0.660274, -0.660274, -0.660274,
-0.663014, -0.670048, -0.671233, -0.679452, -0.681696, -0.693151,
-0.704627, -0.709589, -0.723288, -0.726389, -0.734247, -0.739726,
-0.740844, -0.753425, -0.754539, -0.767123, -0.769845, -0.778082,
-0.784148, -0.789041, -0.798299, -0.8 , -0.805479, -0.806488,
-0.813699, -0.816735, -0.819178, -0.827397, -0.827397, -0.827397,
-0.830129, -0.832877, -0.839263, -0.841096, -0.846208, -0.846575,
-0.849315, -0.85056 , -0.852055, -0.853318, -0.854795, -0.855708,
-0.85723 , -0.857534, -0.859306, -0.860274, -0.860274, -0.860274,
-0.86347 , -0.865753, -0.864739, -0.863014, -0.863014, -0.863014,
-0.863014, -0.863014, -0.863014, -0.863014, -0.86273 , -0.861124,
-0.860841, -0.860274, -0.860803, -0.863277, -0.865222, -0.865753,
-0.865753, -0.865753, -0.865753])
fig19b_emb_to_bridge_x_raw = np.array(
[ -4.92072000e+00, -4.90682000e+00, -4.71917000e+00,
-4.70637000e+00, -4.58438000e+00, -4.50665000e+00,
-4.38320000e+00, -4.37332000e+00, -4.23944000e+00,
-4.18165000e+00, -4.10538000e+00, -3.98010000e+00,
-3.97169000e+00, -3.84549000e+00, -3.77142000e+00,
-3.64413000e+00, -3.63828000e+00, -3.50952000e+00,
-3.43874000e+00, -3.30816000e+00, -3.17208000e+00,
-3.04040000e+00, -3.03821000e+00, -2.83739000e+00,
-2.70351000e+00, -2.63968000e+00, -2.50251000e+00,
-2.43905000e+00, -2.23787000e+00, -2.23494000e+00,
-2.10308000e+00, -1.96755000e+00, -1.90135000e+00,
-1.76674000e+00, -1.69998000e+00, -1.50008000e+00,
-1.36694000e+00, -1.23397000e+00, -1.16722000e+00,
-9.68049000e-01, -9.00379000e-01, -8.35635000e-01,
-6.36282000e-01, -5.67149000e-01, -4.34735000e-01,
-3.70357000e-01, -3.02138000e-01, -2.36114000e-01,
-3.67613000e-02, 2.96285000e-02, 9.56527000e-02,
4.96918000e-01, 7.63941000e-01, 1.16466000e+00,
1.36383000e+00, 1.63176000e+00, 2.03212000e+00,
2.09869000e+00, 2.63219000e+00, 2.63237000e+00,
3.16641000e+00, 3.23207000e+00, 3.49964000e+00,
3.76539000e+00, 3.96620000e+00, 4.29888000e+00,
4.56645000e+00, 5.09922000e+00, 5.23346000e+00,
5.76623000e+00, 5.83371000e+00, 6.43415000e+00,
6.76683000e+00, 7.10134000e+00, 7.63465000e+00,
7.70178000e+00, 8.30221000e+00, 8.83552000e+00,
8.90283000e+00, 9.50345000e+00, 9.90307000e+00,
1.01706000e+01, 1.11049000e+01, 1.11712000e+01,
1.19057000e+01, 1.21726000e+01, 1.26399000e+01,
1.30404000e+01, 1.34409000e+01, 1.43088000e+01,
1.44423000e+01, 1.51098000e+01, 1.59107000e+01,
1.59775000e+01, 1.69788000e+01, 1.69790000e+01,
1.78464000e+01, 1.79132000e+01, 1.85808000e+01,
1.87810000e+01, 1.91814000e+01, 1.92483000e+01,
1.94484000e+01, 1.98478000e+01, 1.99151000e+01,
2.04486000e+01, 2.05141000e+01, 2.08483000e+01,
2.13135000e+01, 2.15147000e+01, 2.20478000e+01,
2.21799000e+01, 2.25144000e+01, 2.28478000e+01,
2.29131000e+01, 2.36480000e+01, 2.37131000e+01,
2.44481000e+01, 2.46469000e+01, 2.52485000e+01,
2.55806000e+01, 2.58485000e+01, 2.65810000e+01,
2.67156000e+01, 2.73161000e+01, 2.73815000e+01,
2.78496000e+01, 2.81822000e+01, 2.84500000e+01,
2.89167000e+01, 2.92496000e+01, 2.95843000e+01,
2.98504000e+01, 3.01180000e+01, 3.05843000e+01,
3.07182000e+01, 3.16521000e+01, 3.17192000e+01,
3.24533000e+01, 3.27869000e+01, 3.31874000e+01,
3.35874000e+01, 3.40551000e+01, 3.46557000e+01,
3.56569000e+01, 3.58573000e+01, 3.65913000e+01,
3.69920000e+01, 3.76595000e+01, 3.77263000e+01,
3.86607000e+01, 3.93281000e+01, 3.99958000e+01,
4.11306000e+01, 4.11308000e+01, 4.19319000e+01,
4.28663000e+01, 4.29330000e+01, 4.40011000e+01,
4.50692000e+01, 4.52697000e+01, 4.64043000e+01,
4.66044000e+01, 4.70053000e+01, 4.72050000e+01,
4.81393000e+01, 4.88737000e+01, 4.90744000e+01,
4.94746000e+01, 4.98087000e+01, 4.98755000e+01])
fig19b_emb_to_bridge_w_raw = np.array(
[-0.5043 , -0.50411 , -0.501545, -0.50137 , -0.50639 , -0.509589,
-0.512126, -0.512329, -0.506849, -0.503306, -0.49863 , -0.496063,
-0.49589 , -0.49589 , -0.49589 , -0.501129, -0.50137 , -0.508441,
-0.512329, -0.515012, -0.517808, -0.512419, -0.512329, -0.50411 ,
-0.49863 , -0.49515 , -0.487671, -0.485722, -0.479542, -0.479452,
-0.47675 , -0.473973, -0.471263, -0.465753, -0.467125, -0.471233,
-0.478087, -0.484932, -0.489058, -0.50137 , -0.509771, -0.517808,
-0.531507, -0.53578 , -0.543966, -0.547945, -0.549347, -0.550704,
-0.5548 , -0.556164, -0.556164, -0.556164, -0.555069, -0.553425,
-0.554593, -0.556164, -0.560863, -0.561644, -0.569863, -0.569867,
-0.582062, -0.583562, -0.58906 , -0.594521, -0.597614, -0.60274 ,
-0.606404, -0.613699, -0.615353, -0.621918, -0.622657, -0.629233,
-0.632877, -0.632877, -0.632877, -0.633489, -0.638969, -0.643836,
-0.644354, -0.648978, -0.652055, -0.652633, -0.654651, -0.654795,
-0.654795, -0.654795, -0.654795, -0.654795, -0.654795, -0.654795,
-0.654795, -0.65604 , -0.657534, -0.657363, -0.654795, -0.654795,
-0.659882, -0.660274, -0.660274, -0.660274, -0.663014, -0.664665,
-0.6696 , -0.679452, -0.681389, -0.696745, -0.69863 , -0.708937,
-0.723288, -0.728377, -0.741865, -0.745205, -0.752705, -0.76018 ,
-0.761644, -0.776744, -0.778082, -0.786708, -0.789041, -0.797867,
-0.80274 , -0.806409, -0.816438, -0.81782 , -0.823985, -0.824658,
-0.82786 , -0.830137, -0.832886, -0.837678, -0.841096, -0.841096,
-0.841096, -0.843094, -0.846575, -0.847262, -0.852055, -0.852055,
-0.852055, -0.852055, -0.856167, -0.860274, -0.859075, -0.857534,
-0.860274, -0.860862, -0.863014, -0.861986, -0.860274, -0.860457,
-0.863014, -0.863014, -0.863014, -0.860274, -0.860274, -0.860274,
-0.863014, -0.863014, -0.863014, -0.860707, -0.860274, -0.863014,
-0.865753, -0.867582, -0.868493, -0.873973, -0.871233, -0.871233,
-0.871233, -0.865753, -0.864659])
fig12top_w = fig12top_w_raw
fig12top_t = fig12top_t_raw + 0
fig13top_w = fig13top_w_raw
fig13top_t = fig13top_t_raw + 0
fig13bot_w = fig13bot_w_raw
fig13bot_t = fig13bot_t_raw + 0
fig14top_w = fig14top_w_raw
fig14top_t = fig14top_t_raw + 0
fig14bot_w = fig14bot_w_raw
fig14bot_t = fig14bot_t_raw + 0
fig15top_w = fig15top_w_raw
fig15top_t = fig15top_t_raw + 0
fig15bot_w = fig15bot_w_raw
fig15bot_t = fig15bot_t_raw + 0
fig19b_emb_to_bridge_x = fig19b_emb_to_bridge_x_raw
fig19b_emb_to_bridge_w = fig19b_emb_to_bridge_w_raw
fig19b_bridge_to_emb_x = fig19b_bridge_to_emb_x_raw
fig19b_bridge_to_emb_w = fig19b_bridge_to_emb_w_raw
top_ts = [fig13top_t, fig14top_t, fig15top_t, fig12top_t]
top_ws = [fig13top_w, fig14top_w, fig15top_w, fig12top_w]
top_t_offset=[0.03719, 0.173177, 0.23286-0.01137,-0.456988 ]
with open(ftime + ".txt", "w") as f:
f.write(ftime + os.linesep)
matplotlib.rcParams.update({'font.size': 11})
fig = plt.figure(figsize=(3.54,4.5))
gs0 = matplotlib.gridspec.GridSpec(
2, 1,
# width_ratios=[1, 2],
height_ratios=[1,1],
# hspace=0.05,
)
ax = fig.add_subplot(gs0[0,0])
ax2 = fig.add_subplot(gs0[1,0], sharex=ax)
fig2 = plt.figure(figsize=(8,8))
ax3 = fig2.add_subplot("111")
#fig2 = plt.figure(figsize=(3.54,3.54))
#ax2 = fig2.add_subplot("111")
#fig3 = plt.figure(figsize=(3.54,3.54))
#ax3 = fig3.add_subplot("111")
#t_lengths = np.linspace(25,0,6)
#betas = [0, 0.05, 0.1, 0.3, 1.1, 2.0]
fig4 = plt.figure(figsize=(3.54,8))
gs = matplotlib.gridspec.GridSpec(
4, 1,
# width_ratios=[1, 2],
# height_ratios=[1,4],
# hspace=0.05,
# hspace = 0.05,
# top=0.05,
)
axs =[]
for i in range(4):
axs.append(fig4.add_subplot(gs[i,0]))
# if i==0:
#
# else:
# axs.append(fig4.add_subplot(gs[i,0], sharex=axs[0]))
if i!=3:
axs[i].get_xaxis().set_ticks([])
axs[i].set_ylim(-1.2,0.2)
axs[i].set_xlim(0,4)
axs[i].set_ylabel("Rail displ. (mm)")
axs[3].set_xlabel("Time (s)")
k1 = 160e6*0.5 #N/m
step1 = 2*0.684881633*.9
step2 = 3*0.5*0.9
#1030
#10000
pier_width = 0.5
pierk=2.5#10000/80/4
beamk=2*0.9#10000/80/4
m1=2
m2=2
#x, k1bar, mubar, rho_bar
xp,kp,mup,rhop = np.array(
[[0.0, 1., 1, 1], #start
[30, 1., 1., 1],#start of UGM wedge,
[30+(40-30)*0.25, 1+(step1-1)*0.25**m1, 1, 1],
[30+(40-30)*0.50, 1+(step1-1)*0.50**m1, 1, 1],
[30+(40-30)*0.75, 1+(step1-1)*0.75**m1, 1, 1],
[40, step1, 1., 1], #end of UGM wedge,
[50, step1, 1., 1], #start of CBM,
[50+(57-50)*0.25, step1+(step2-step1)*0.25**m2, 1, 1],
[50+(57-50)*0.50, step1+(step2-step1)*0.50**m2, 1, 1],
[50+(57-50)*0.75, step1+(step2-step1)*0.75**m2, 1, 1],
[57, step2, 1., 1], #end of CBM
[60, step2, 1., 1],#Start abutment pier1
[60, pierk, 1., 1],
[60+2*pier_width, pierk, 1., 1],
[60+2*pier_width, beamk, 1., 1], #end abutment pier 1
[97.5-pier_width, beamk, 1., 1],#Start abutment pier 2
[97.5-pier_width, pierk, 1., 1],
[97.5, pierk, 1., 1],
# [97.5+pier_width, pierk, 1., 1],
# [97.5+pier_width, beamk, 1., 1], #end abutment pier 2
# [135-pier_width, beamk, 1., 1],#Start abutment pier 3
# [135-pier_width, pierk, 1., 1],
# [135+pier_width, pierk, 1., 1],
# [135+pier_width, beamk, 1., 1], #end abutment pier 3
# [172.5-pier_width, beamk, 1., 1],#Start abutment pier 4
# [172.5-pier_width, pierk, 1., 1],
# [172.5, pierk, 1., 1], # end of beam
]).T
# [60-pier_width, step2, 1., 1],#Start abutment pier1
# [60-pier_width, pierk, 1., 1],
# [60+pier_width, pierk, 1., 1],
# [60+pier_width, beamk, 1., 1], #end abutment pier 1
# [97.5-pier_width, beamk, 1., 1],#Start abutment pier 2
# [97.5-pier_width, pierk, 1., 1],
# [97.5, pierk, 1., 1],
#xp,kp,mup,rhop = np.array(
# [[0.0, 1., 1, 1], #start
# [30, 1., 1., 1],#start of UGM wedge,
# [40, step1, 1., 1], #end of UGM wedge,
# [50, step1, 1., 1], #start of CBM,
# [57, step2, 1., 1], #end of CBM
# [60, step2, 1., 1],#Start abutment pier1
# [60, pierk, 1., 1],
# [60+2*pier_width, pierk, 1., 1],
# [60+2*pier_width, beamk, 1., 1], #end abutment pier 1
# [97.5-pier_width, beamk, 1., 1],#Start abutment pier 2
# [97.5-pier_width, pierk, 1., 1],
# [97.5, pierk, 1., 1],
## [97.5+pier_width, pierk, 1., 1],
## [97.5+pier_width, beamk, 1., 1], #end abutment pier 2
## [135-pier_width, beamk, 1., 1],#Start abutment pier 3
## [135-pier_width, pierk, 1., 1],
## [135+pier_width, pierk, 1., 1],
## [135+pier_width, beamk, 1., 1], #end abutment pier 3
## [172.5-pier_width, beamk, 1., 1],#Start abutment pier 4
## [172.5-pier_width, pierk, 1., 1],
## [172.5, pierk, 1., 1], # end of beam
# ]).T
#
#
## [60-pier_width, step2, 1., 1],#Start abutment pier1
## [60-pier_width, pierk, 1., 1],
## [60+pier_width, pierk, 1., 1],
## [60+pier_width, beamk, 1., 1], #end abutment pier 1
## [97.5-pier_width, beamk, 1., 1],#Start abutment pier 2
## [97.5-pier_width, pierk, 1., 1],
## [97.5, pierk, 1., 1],
#6 train cars distance in (m), load in (kN)
train_x = np.array(
[ -3.85, -6.55, -22.85, -25.55, -29.75, -32.45, -48.75,
-51.45, -55.65, -58.35, -74.65, -77.35, -81.55, -84.25,
-100.55, -103.25, -107.45, -110.15, -126.45, -129.15, -133.35,
-136.05, -152.35, -155.05])
train_f = np.array(
[-129.8, -133.2, -132.6, -129.2, -131.5, -134.9, -134.9, -131.5,
-128.8, -128.8, -128. , -128.8, -132.7, -132.7, -132.7, -132.7,
-133.2, -136.6, -136.6, -133.2, -130.2, -133.6, -134.2, -130.8])*1000
L = xp[-1]
train_v = 220*1000/60/60 #m/s
tstart = 0
tend = (64.5 + (-1) * train_x[-1]) / train_v
tvals = np.linspace(tstart, tend, nt)
# x=64.5--> Paix -4.5
# x=18.9--> Paix 41.1
# x=45.3--> Paix 14.7
# x=59.1--> Paix 0.9
#i.e. xrohan = 64.5 - xpaix -4.5 ; need to check
#xpaix = -4.5 + (64.5-xrohan)
x_interest = [18.9, 45.3, 59.1, 64.5]
xvals = np.append(np.linspace(0,64.5,nx), x_interest)
xvals.sort()
i_interest = np.searchsorted(xvals, x_interest)
pdict = OrderedDict(
E = 210e9, #Pa
rho = 7850, #kg/m^3
L = L, #m
A = 0.3*0.12*7.7e-3, # m^2
I = 2*30.55e-6,#m^4
k1 = k1,#160e6,#N/m
mu = 17e3,
# kf=5.41e-4,
#Fz_norm=1.013e-4,
# mu_norm=39.263/4/2/1000,
# mubar=PolyLine([0,0.05,0.05,.95,.95,1],[1,1,1,1,1,1]),
# k1_norm=97.552,
# k1bar=PolyLine([0,0.2,0.3,1],[1.0,1,0.1,0.1]),
# k1bar=PolyLine([0,0.2,0.2,1],[1.0,1,1,1]),
# k3_norm=2.497e6,
k1bar = PolyLine(xp/L, kp),
moving_loads_x = [train_x], #m
moving_loads_Fz = [train_f], #N
moving_loads_v = [train_v], #m/s
nterms=nterms,
BC="SS",
# nquad=20,
# moving_loads_x_norm=[[0]],
# moving_loads_Fz_norm=[[1.013e-4]],
# moving_loads_v_norm=[v,],#[0.01165,],
tvals=tvals,
xvals_norm=xvals/L,
use_analytical=True,
implementation="fortran",
force_calc=force_calc,
)
#zmax = np.zeros((len(alphas), len(t_lengths)), dtype=float)
#zmin = np.zeros((len(alphas), len(t_lengths)), dtype=float)
start_time1 = time.time()
f.write("*" * 70 + os.linesep)
vequals = "Paixao et al 2016"
f.write(vequals + os.linesep); print(vequals)
pdict["file_stem"] = "Paixaoet al_n{:d}".format(nterms)
a = SpecBeam(**pdict)
a.runme()
a.saveme()
a.animateme()
#w_now_max = np.max(a.defl)
#w_now_min = np.min(a.defl)
defl_max_envelope = np.max(a.defl, axis=1)
defl_min_envelope = np.min(a.defl, axis=1)
#xpaix = -4.5 + (64.5-xrohan)
dist_from_abut = -4.5 + (64.5-pdict["xvals_norm"]*L)
# line, =ax.plot(pdict["xvals_norm"]*L, defl_max_envelope*1000,
# label="hello")
# ax.plot(pdict["xvals_norm"]*L, defl_min_envelope*1000,
# color = line.get_color(),
# )
line, =ax.plot(dist_from_abut, defl_min_envelope*1000,
label="specbeam")
ax.plot(fig19b_emb_to_bridge_x,fig19b_emb_to_bridge_w, label="axle FEM (Paixao)",
marker="s", color="green",markerfacecolor="none",
markevery=0.2,ms=5,markeredgecolor="green")
for i, (ii,xx) in enumerate(zip(i_interest,x_interest)):
axs[i].plot(a.tvals, a.defl[ii,:]*1000, label='specbeam')
offset_t = top_t_offset[i]
axs[i].plot(top_ts[i]-offset_t,top_ws[i], label="measured x={:.1f}m".format(-4.5 + (64.5-xx)))
ax3.plot(a.tvals,a.defl[ii,:]*1000,
label="x={}".format(xx))
end_time1 = time.time()
elapsed_time = (end_time1 - start_time1); print("Run time={}".format(str(timedelta(seconds=elapsed_time))))
adjust = dict(top=0.9, bottom=0.13, left=0.19, right=0.95)
#peak envelope plot
###################
#ax.set_title("$\\beta={},k_1/k_2={}$".format(beta, k_ratio))
# ax.set_xlabel("x")
ax.set_ylabel("Peak displacement (mm)")
ax.grid()
# leg = ax3.legend(title="$\\alpha$",
# loc="upper right",
## ncol=2,
# labelspacing=.2,
# handlelength=2,
# fontsize=8
# )
# leg.draggable()
ax.xaxis.labelpad = -1
ax.yaxis.labelpad = 0
ax.set_xlim(50, -5)
leg = ax.legend(fontsize=8,
loc='upper left')
leg.draggable()
fig.subplots_adjust(top=0.95, bottom=0.13, left=0.19, right=0.9)
####################
#Kplot
dist_from_abut = -4.5 + (64.5-xp)
ax2.plot(dist_from_abut, kp)#xp, kp)
ax2.set_xlabel("Distance from abutment (m)")
ax2.set_ylabel("$k/k_{ref}$")
ax2.set_ylim(0.9, 3)
for i, x in enumerate(x_interest):
y = np.interp(x,xp,kp)
if i==0:
ax2.plot(-4.5 + (64.5-x),y,ls=':', marker='o', label="defl. vs time evaluation point")
else:
ax2.plot(-4.5 + (64.5-x),y,ls=':', marker='o')
leg = axs[i].legend(fontsize=8,
loc='lower left')
leg.draggable()
leg = ax2.legend(fontsize=8,
loc='upper left',
numpoints=1)
leg.draggable()
#displacemetn vs tiem at points plot
###################
#ax3.set_title("$\\beta={},k_1/k_2={}$".format(beta, k_ratio))
ax3.set_xlabel("time (s)")
ax3.set_ylabel("Displacement (mm)")
# ax3.grid()
leg = ax3.legend(title="$\\alpha$",
loc="upper right",
# ncol=2,
labelspacing=.2,
handlelength=2,
fontsize=8
)
# leg.draggable()
ax3.xaxis.labelpad = -1
ax3.yaxis.labelpad = 0
#compare with field
fig4.subplots_adjust(top=0.98, bottom=0.08, left=0.19, right=0.9, hspace=0.04)
# fig.subplots_adjust(**adjust)
end_time0 = time.time()
elapsed_time = (end_time0 - start_time0); print("Total run time={}".format(str(timedelta(seconds=elapsed_time))))
f.write("Total run time={}".format(str(timedelta(seconds=elapsed_time))) + os.linesep)
# a.animateme()
if not saveas is None:
save_figure(fig=fig, fname=saveas+"_fig1")
save_figure(fig=fig4, fname=saveas+"_fig2")
# save_figure(fig=fig2, fname=saveas + "_up")
plt.show()
fig.clf()
# fig2.clf()
[docs]def transition_DAF1(saveas=None,
ax=None,
kratios=None,
alpmax=3,
npts=500,
betas=None,
article_formatting=False,
DAFmax=4):
"""Plot of Dynamic Amplification Factor (DAF) vs velocity ratio
for different stiffness and damping values.
Parameters
----------
saveas : string, optional
Filename (not including extension) to save figure to.
Default saveas=None, i.e. figure won't be saved to disk.
ax : matplotlib.Axes object, optional
Axes object to plot on. Default ax=None, i.e. figure and axis
will be created. Nb this might return an error if used as the function
will try and return then figure.
kratios : list of float, optional
striffness ratio values. Default kratios=None which will
mean kratios=[1,2].
alpmax : float, optional
maximum velocit ratio considered (i.e x-axis max). Default alpmax=3
npts : int, optional
velocity ratios (i.e x-axis values) will be npts between
0 and alpmax. Default npts=500.
betas : list of float, optional
list of damping ratios to consider. Default betas=None which will
mean betas=[0,0.05, 0.1, 0.3, 1.1, 2.0]
article_formatting : [False, True], optional
You should never need this. It was a quick hack to do some
custom formatting for a journal article. Dont even try messing with
it. will ony work with 6 beta values or less. Default=False,
i.e. no special formatting.
DAFmax : float, optional
Y-axis max. Default DAFmax=4
Returns
-------
fig : matplotlib figure
the figure
"""
if article_formatting:
mdc = MarkersDashesColors(markersize=6,linewidth=1)
mdc.construct_styles(markers=[2,25,11,5,8,15],
dashes=[0],
marker_colors=[0,1,2,3,4,5],
line_colors=[0,1,2,3,4,5])
styles=mdc(markers=[2,25,11,5,8,15],
dashes=[0],
marker_colors=[0,1,2,3,4,5],
line_colors=[0,1,2,3,4,5])
#a.demo_styles()
start_time0 = time.time()
#ftime = datetime.datetime.fromtimestamp(start_time0).strftime('%Y-%m-%d %H%M%S')
print("Started transition_DAF1")
if ax is None:
fig,ax = plt.subplots(figsize=(3.54,3.54))
matplotlib.rcParams.update({'font.size': 11})
if kratios is None:
kratios=[1,2]
if betas is None:
betas=[0, 0.05, 0.1, 0.3, 1.1, 2.0]
for k, kratio in enumerate(kratios):
start_time1 = time.time()
print("doing kratio={}".format(kratio))
alpha=np.linspace(0,alpmax,npts)
for j,bet in enumerate(betas):
DAF = np.empty_like(alpha)
for i, alp in enumerate(alpha):
DAF[i] = DAFinfinite(alp, bet=bet,kratio=kratio)
labela = "$k_2/k_1={}$".format(kratio)
labelb = "$\\beta={}$".format(bet)
if k==0:
if j==0:
label=labelb+", " + labela
else:
label=labelb
else:
if j==0:
label = labela
else:
label=None
if j==0:
if article_formatting:
line, =ax.plot(alpha, DAF, label=label,markevery=(0.03+[0,0,0,0,0,.06][k],500.0),**styles[k])
else:
line, =ax.plot(alpha, DAF, label=label,
#alpha=1,
# marker='s',ms=2,
)
else:
ax.plot(alpha, DAF, label=label,
#alpha=0.8-0.5*j/len(betas),
#marker='s',ms=2,
color = line.get_color())
end_time1 = time.time()
elapsed_time = (end_time1 - start_time1); print("Run time={}".format(str(timedelta(seconds=elapsed_time))))
leg = ax.legend(
# title="$\\beta$",
# loc="upper center",
# ncol=2,
labelspacing=0,
handlelength=2,
fontsize=8,
numpoints=1,
)
ax.set_ylim([0, DAFmax])
ax.set_xlim([0, alpmax])
ax.grid()
ax.set_ylabel("$DAF_{wrt1}$")
ax.set_xlabel("$\\alpha_1$")
ax.xaxis.labelpad = -1
ax.yaxis.labelpad = 0
fig.subplots_adjust(top=0.95, bottom=0.13, left=0.14, right=0.95)
end_time0 = time.time()
elapsed_time = (end_time0 - start_time0); print("Total run time={}".format(str(timedelta(seconds=elapsed_time))))
if not saveas is None:
save_figure(fig=fig, fname=saveas)
return fig
[docs]def transition_DAF1_at_specific_alp(saveas=None,
ax=None,
alphas=None,
kratios=None,
betas=None,
DAFmax=4):
"""Plot of Dynamic Amplification Factor (DAF) vs stiffness ratio
for different velocity ratios and damping values.
Parameters
----------
saveas : string, optional
Filename (not including extension) to save figure to.
Default saveas=None, i.e. figure won't be saved to disk.
ax : matplotlib.Axes object, optional
Axes object to plot on. Default ax=None, i.e. figure and axis
will be created. Nb this might return an error if used as the function
will try and return then figure.
alphas : list of float,optional
Specific velocity ratios to calc at. Default alphas=None which will
use [0.1, 0.5, 0.8].
kratios : list of float, optional
striffness ratio values. Default kratios=None which will
mean kratios=np.logspace(np.log10(0.1), np.log10(100), 100), i.e
100 points logspaced between 0.1 and 100.
betas : list of float, optional
list of damping ratios to consider. Default betas=None which will
DAFmax : float, optional
Y-axis max. Default DAFmax=4
Returns
-------
fig : matplotlib figure
the figure
"""
start_time0 = time.time()
print("Started transition_DAF1_at_specific_alp")
if ax is None:
fig,ax = plt.subplots(figsize=(3.54,3.54))
matplotlib.rcParams.update({'font.size': 11})
if kratios is None:
kratios=np.logspace(np.log10(0.1), np.log10(100), 100)
if alphas is None:
alphas = [0.1,0.5,0.8]
if betas is None:
betas = [0, 0.05, 0.1, 0.3, 1.1, 2.0]
for i, alp in enumerate(alphas):
start_time1 = time.time()
print("doing alpha={}".format(alp))
for j,bet in enumerate(betas):
DAF = np.empty_like(kratios)
for k, kratio in enumerate(kratios):
DAF[k] = DAFinfinite(alp/kratio**(0.25), bet*(1/kratio)**0.5)
DAF[k] *= (1/kratio)**(0.75)
labela = "$\\alpha={}$".format(alp)
labelb = "$\\beta={}$".format(bet)
if i==0:
if j==0:
label=labelb+", " + labela
else:
label=labelb
else:
if j==0:
label = labela
else:
label=None
if j==0:
line, =ax.plot(kratios, DAF, label=label,
# marker='s',ms=2,
)
else:
ax.plot(kratios, DAF, label=label,
# marker='s',ms=2,
color = line.get_color())
end_time1 = time.time()
elapsed_time = (end_time1 - start_time1); print("Run time={}".format(str(timedelta(seconds=elapsed_time))))
leg = ax.legend()
leg.draggable()
leg = ax.legend(
# title="$\\beta$",
# loc="upper center",
# ncol=2,
labelspacing=.2,
handlelength=2,
fontsize=8,
# numpoints=1,
)
ax.set_ylim([0, DAFmax])
ax.set_xscale('log')
ax.grid()
ax.set_ylabel("Deflection amplification factor, w.r.t. k1")
ax.set_xlabel("$k_2/k_1$")
end_time0 = time.time()
elapsed_time = (end_time0 - start_time0); print("Total run time={}".format(str(timedelta(seconds=elapsed_time))))
if not saveas is None:
save_figure(fig=fig, fname=saveas)
return ax
[docs]def DAFinfinite(alp, bet, kratio=1.0):
"""Dynamic amplification factor for infinite beam on winkler foundation
Parameters
----------
alp : float
Velocity ratio. alp = v/vcr where critical velocity
vcr=sqrt(2*sqrt(k*E*I)/(rho*A))**0.5
bet : float
Damping ratio. bet=c/ccr where critical damping ccr=2*sqrt(rho*A*k).
The damping ratio is with respect to the reference foundation stiffness.
kratio : float, optional
Ratio of foundation stiffness to reference value i.e. k2/k1 where
k1 is the reference value. Default kratio=1
Returns
-------
DAF : float
Dynamic/deflection amplification factor with respect to the reference
foundation stiffness, k1. i.e.
deflection = DAF*[static deflection of point load on beam on k1 ]
Notes
-----
Only works for scalar input
"""
s, disp = moving_load_on_beam_on_elastic_foundation(
alp/kratio**(0.25),
bet*(1/kratio)**0.5
,slim=(-6,6),npts=1001)
DAF = np.real(np.max(disp)*(1/kratio)**(0.75))
return DAF
[docs]def transition_zones1(
Lt_Lcs=[0.1,4],
alphas=[0.8],
reverses=[False,True],
xi_Lc=None, #ratio of xinterest over characteristic length
kratio=50,
beta=0.2,
alpha_design=0.8,
ntrans=50,
DAF_distribution="linear",
tanhend=0.01,
xlim = (-8, 8),
nx=100, #number of x points
nt=100, #number of time values
t_extend = 1.2, # factor to extend maximum evalutated time.
nterms=150,
ninterp = 15, # number of points to discretize the transition zone
nloads = 1,
load_spacing = 0.03,
animateme=False,
saveas=None,
force_calc=True,
article_formatting=False,
):
"""Exploration of deflection envelopes caused by point loads travelling
through a stiffness transition zone on beam on visco-elastic foundation
using Specbeam.
- Transition between reference beam/foundation with stiffness k to beam on
foundation with stiffness k*`kratio`.
- Transition zone length Lt is a multiple of the characteristic length
Lc of the reference beam/foundation. Lt = `Lt_Lc`*Lc
- Transition zone divided into ntrans points.
- Change in stiffness over transition zone is a calculated value based on
a goal displacement profile. The shape of the goal displacement profile
is controlled be `DAF_distribution` which is usually linear.
- At each evaluation point in transition zone the stiffness needed to
achieve the desired displacement value is back calcualted based on
infinite beam theory with a point load travelling at velocity ratio
of `alpha_design` with damping ratio `beta` (`alpha_design` and `beta`
are w.r.t. the reference beam/foundation).
- for input into specbeam the design stiffness profile piecewise linear
interpolated at `ninterp` points to produce piecewise linear analysis
profile. `ninterp` should be less thatn or equal to `ntrans`
- Moving load will then traverse the design stiffness profile and
depending on dynamic effects the moving load displacement may or may not
match the design displacements. You can only design for one speed so
deflection response might be differnt when actual speed is different
to design speed.
- Note only the stiffness changes. We assume that damping determined
by `beta` of the reference beam/foundation does not change in the
transition zone. Perhaps not realistic but simply an assumption to
enable exploration.
3 plots will be produced:
- Line will be plotted based on permututions of `Lt_Lcs`, `alphas`,
`reverses`. So number of lines can quickly get out of hand. Usually
interested in varyin one of the parameters at a time.
- Plot 1a) shows the design transtion stiffness profile
Plot 1b) shows max deflection experienced at each point
- Plot 2) is deflection vs time at the point which experiences the
maximum deflection. also plots a marker at the time when
load passes the point.
- Plot 3) is the deflection vs time curve under the moving point load.
Parameters
----------
Lt_Lcs : list of float, optional
Transtion zone lengths as a mulitple of characteristic length.
Default Lt_Lcs = [0.1,4].
alphas : list of float, optional
Velocity ratios to analyze. Velocity rations are
with respect to the reference beam/foundation. Default alphas = [0.8]
reverses : list of Bool, optional
Directions of travel. False means left to right from reference
beam/foundation to other beam/foundation. True means the reverse
direction from right to left from the other beam/foundation to the
reference beam foundation. Default reverses=[False,True].
xi_Lc : [False, True], optional
If True then deflection vs time will be plotted at the point where
the maximum deflection was experienced. Default xi_Lc=None i.e. False
and no plot will be created. Also plots a marker at the time when
load passes the point.
kratio : float optional
stiffness ratio k2/k1 where k1 is the stiffness of the reference
beam/foundation. Default kratio = 50.
beta : float, optional
Reference value of damping ratio. Default beta = 0.2
alpha_design : float, optional
Design velocity ratio used to back calculate the design stiffness
ratio. Default alpha_design = 0.8.
ntrans : int, optional
Number of points to approximate the desired displacement
transition profile into, to then back calculate the stiffness
profile used for analysis. Default ntrans = 50 .
DAF_distribution : ["linear", "tanh", "reverse"], optional
Shape of goal transition deflection distrribution.
"linear" isself explanatory and simplest.
"tanh" is a hyperbolic tangent, with steepness of transtion
controlled by `tanend`. Not well tested.
"reverse" is a compund reverse curve of two curclar arcs tangent to
the end points and tangent at the arc join. Not well tested
Default DAF_distribution = "linear".
tanend : float, optional
value controling the steepness of a "tanh" `DAF_distribution`.
Say transitioning between points (-0.5, -1) and (0.5, 1). Tanh
curve will never actually hit valus of -1 and +1, it will approach
those values as x goes to += infinity. Using tanend=0.01 will control
transtion such that the tanh curve evalutes to y=1+0.01 and y=1-0.01
at the start/end points. This evaluated value is vever actually used
because we have given start and end points but it does control the
position of points just inside the start/end points and hence the
steepness of transition.
xlim : two element tuple of float, optional
Distances on either side of the transtion zone midpoint to evaluate
deflection at. The mid point of the transition would be at 0.
For example default xlim=(-8,8) which would be
8 characteristic lenghts on either side of the transition zone
midpoint. Note: Characteristic lenth Lc is with respect to the
reference beam/foundation. Default xlim=(-8, 8).
nx : int, optional
Number of points between `xlim[0]` and `xlim[1]` at which to
calculate deflection. Default nx = 100.
nt : int, optional
Number of evaluation times from when load enters `xlim[0]` and
leaves `xlim[1]`. Default nt=100
t_extend : float, optional
factor to extend maximum evalutated time. As per `nt' evaluation times
are normally between first point entering `xlim[0]' and leaving
`xlim[1]`. However, with mulitple loads, loads can still be on the
beam at that point. max eval time can be extended fby a factor.
t_extend = 1.2 .
ninterp : int, optional
For use in specbeam the back calculated stiffness and damping
transitions profiles are interpoalted at `ninterp` points. A coarser
discretization will improve computation times but might no capture
the changing properties approraitely.
Note: `ninterp` should be less thatn or equal to `ntrans`.
Default ninterp = 15 .
nterms : int, optional
Number of series terms in approximation. More terms leads to higher
accuracy but also longer computation times. Start of with a
small enough value to observe behaviour and then increase
to confirm. Default nterms=150 .
nloads : int, optional
Number of eqully spaced moving point loads. Default nloads = 1 .
load_spacing : float, optional
Spacing between the multiple points loads. Units are multiples of
Lc characteristic length. Default load_spacing = 0.03 .
animateme : [False, True], optional
If True the deflectons and moving loads will be animated.
Default anmateme=False.
saveas : string, optional
Filename stem (not including extension) to save figure to.
Default saveas=None, i.e. figure won't be saved to disk.
force_calc : [True, False], optional
When True all calcuations will be done from scratch. When False
Current working directory will be searched for a results file
named according to "saveas" and
if found it will be loaded. Be careful as name of results file may
not be unique to your analysis parameters. Safest to use
force_calc = True, but if you are simply regenerating graphs then
consider force_calc=False. Default force_calc=True.
article_formatting : [False, True], optional
You should never need this. It was a quick hack to do some
custom formatting for a journal article. Dont even try messing with
it. will ony work with 3 Lt_Lcs values values or less. Default=False,
i.e. no special formatting.
Notes
-----
While all parameters are normalised the raw properties that all others
are based on are:
- E = 6.998*1e9, #Pa
- rho = 2373, #kg/m3
- L = 160, #m
- A = 0.3*0.1, # m^2
- I = 0.00022477900799999998,
- k1 = 800002.6125,
"""
if article_formatting:
mdc = MarkersDashesColors(markersize=4,linewidth=1)
# mdc.demo_options()
mdc.construct_styles(markers=[5,15,2],
dashes=[0],
marker_colors=[0,1,2],
line_colors=[0,1,2])
styles=mdc(markers=[5,15,2],
dashes=[0],
marker_colors=[0,1,2],
line_colors=[0,1,2])
# mdc.demo_styles()
#figure
fig = plt.figure(figsize=(3.54,5))
gs = matplotlib.gridspec.GridSpec(2, 1,
# width_ratios=[1, 2],
height_ratios=[1,4],
hspace=0.05)
matplotlib.rcParams.update({'font.size': 11})
#DAF vs kratio plot
# ax1 = fig.add_subplot(gs[1,0])
# transition_DAF1_at_specific_alp(ax=ax1)
# ax1.set_ylim(0,2)
#DAF vs xtrans plot
ax2 = ax2 = fig.add_subplot(gs[1,0])
ax2.set_ylabel("$w/w_{static1}$")
# ax2.set_xlim(-0.5,1.5)
ax2.set_xlabel('$x/L_c$')
ax2.set_ylim(0,2)
# ax1.set_ylim(-10, 10)
#k/k1 vs xtrans plot
ax3 = fig.add_subplot(gs[0,0], )#sharex=ax2)
# ax3.set_xlabel('$x/L_t$')
ax3.set_ylabel('$k/k_1$')
ax3.grid()
ax3.xaxis.set_ticklabels([])
# xlim = (-.5,1.5)
# xlim = (-8,8)
ax2.set_xlim(*xlim)
ax2.grid()
if not xi_Lc is None:
fig4, ax4 = plt.subplots() # deflection vs time at point
fig5 = plt.figure(figsize=(3.54, 3.54))
ax5 = fig5.add_subplot('111')
pdict = OrderedDict(
E = 6.998*1e9, #Pa
rho = 2373, #kg/m3
L = 160, #m
A = 0.3*0.1, # m^2
I = 0.00022477900799999998,
k1 = 800002.6125,
nterms=nterms,
BC="SS",
# moving_loads_x_norm=[[0-vv*load_spacing for vv in range(nloads)]],
# moving_loads_Fz_norm=[[1.013e-4]*nloads],
# xvals_norm=np.linspace(xeval[0], xeval[1], nx),
use_analytical=True,
implementation="fortran",
force_calc=force_calc,
)
for k, Lt_Lc in enumerate(Lt_Lcs):
print("doing Lt_Lc=", Lt_Lc)
trans_characteristic_factor = Lt_Lc# multiple of k1 characteristic length
trans_middle = 0.5 # midpoint of transition is midpoint of beam
lam = (pdict['k1']/(4*pdict['E']*pdict['I']))**0.25
trans_len_raw = trans_characteristic_factor / lam
print("Characteristic len ={}".format(1/lam))
print("normalised:{}".format(1/lam/pdict["L"]))
trans_len = trans_len_raw / pdict["L"]
# trans_len=0.2 #normalised values w.r.t. L
print("tlen/char_len={}".format(trans_len/(1/lam/pdict["L"])))
trans_start = trans_middle - trans_len/2
trans_end = trans_start + trans_len
print("tstart={:8.3f},tend={:8.3f},tlen={:8.3f}".format(trans_start,trans_end, trans_len))
xeval = (trans_middle - abs(xlim[0])/lam/pdict["L"],
trans_middle + abs(xlim[1])/lam/pdict["L"]) # Points either side of midpoint to evluate between
print("xeval:",xeval)
xtrans = np.linspace(0, 1, ntrans)
if nloads>1:
print("loadspace/char_len={}".format(load_spacing*pdict["L"]/(1/lam)))
# work out goal displacement distribution
DAF1 = DAFinfinite(alp=alpha_design, bet=beta)
DAF2 = DAFinfinite(alp=alpha_design,bet=beta,kratio=kratio)
if DAF_distribution.lower()=="linear":
DAF_goal = DAF1 + (DAF2-DAF1)*xtrans
elif DAF_distribution.lower()=="tanh":
b = tanhtranib(x=0, y=tanhend, a=0.5)
print("b:",b)
DAF_goal = (tanhtran(x=xtrans, a=0.5, b=b) - 0.5)/(1-2*tanhend) + 0.5
DAF_goal = DAF_goal * DAF2 + (1-DAF_goal) * DAF1
elif DAF_distribution.lower()=="reverse":
DAF_goal = reverse_curve_transition(xtrans,0,DAF1, 1, DAF2)
DAF_goal[0] = DAF1
DAF_goal[-1] = DAF2
# figg,axx=plt.subplots()
# axx.plot(xtrans,DAF_goal)
# plt.show()
# back calucatee the stiffness profile needed to get the displacemtn profile
kkk = np.linspace(1, kratio,300,dtype=float)
yyy = np.zeros_like(kkk)
for iii, krat in enumerate(kkk):
yyy[iii] = DAFinfinite(alp=alpha_design,bet=beta,kratio=krat)
# figg,axx=plt.subplots()
# axx.plot(kkk,yyy,marker="o")
# plt.show()
ktrans = np.empty_like(xtrans)
for iii, DAF in enumerate(DAF_goal):
if iii==0:
ktrans[0]=1
elif iii==ntrans-1:
ktrans[-1]=kratio
else:
ktrans[iii]=np.interp(DAF, yyy[::-1], kkk[::-1])
ctrans = np.ones_like(xtrans)
#plot the stiffness profile
xx = [xlim[0]]; xx.extend((xtrans-trans_middle)*trans_len*pdict["L"]*lam); xx.append(xlim[1])
yy = [1]; yy.extend(ktrans); yy.append(kratio)
ax3.plot(xx, yy)
DAFtrans = np.empty_like(xtrans)
for i, alp in enumerate(alphas):
for j, krat in enumerate(ktrans):
DAFtrans[j] = DAFinfinite(alp=alp,bet=beta,kratio=krat)
yy = [DAFtrans[0]]; yy.extend(DAFtrans); yy.append(DAFtrans[-1])
label="$L_t/L_c={}, ideal$".format(Lt_Lc)
if article_formatting:
line,= ax2.plot(xx,yy,label=label,**styles[k])
else:
line,= ax2.plot(xx,yy,ls="-",label=label)
#sample the k value to form a poly line for specbeam
xtrans_spec = np.linspace(0, 1, ninterp)
ktrans_spec = np.interp(xtrans_spec, xtrans, ktrans)
pline_x = [0]
pline_x.extend(trans_start + xtrans_spec * trans_len)
pline_x.append(1)
pline_k1 = [ktrans_spec[0]]
pline_k1.extend(ktrans_spec)
pline_k1.append(ktrans_spec[-1])
k1bar = PolyLine(pline_x, pline_k1)
#sample the c values to form a poly line
ctrans_spec = np.interp(xtrans_spec, xtrans, ctrans)
pline_mu = [1]
pline_mu.extend(ctrans_spec)
pline_mu.append(ctrans_spec[-1])
mubar = PolyLine(pline_x, pline_mu)
v_crit = np.sqrt(2/pdict["rho"]/pdict["A"]*np.sqrt(pdict["k1"]*pdict["E"]*pdict["I"]))
v_raw = v_crit*alp
c_crit = np.sqrt(4*pdict["rho"]*pdict["A"]*pdict["k1"])
c_raw = c_crit * beta
tmax = pdict["L"] / v_raw # time for first load to reach end of beam
pdict["moving_loads_x_norm"]=[[0-vv*load_spacing for vv in range(nloads)]]
pdict["moving_loads_Fz_norm"]=[[1.013e-4]*nloads]
pdict["xvals_norm"]=np.linspace(xeval[0], xeval[1], nx)
pdict["mu"] = c_raw
pdict["tvals"] = np.linspace(tmax*xeval[0],tmax*xeval[1]*t_extend,nt)
pdict['k1bar']=k1bar
pdict['mubar']=mubar
for reverse in reverses:
if reverse:
pdict["moving_loads_v_norm"] = [-1*v_raw*np.sqrt(pdict["rho"]/pdict["E"])]
else:
pdict["moving_loads_v_norm"] = [1*v_raw*np.sqrt(pdict["rho"]/pdict["E"])]
#pdict["file_stem"] = prefix + "DAF_alp{:5.3f}_bet{:5.3f}_n{:d}".format(alpha,beta,nterms)
a = SpecBeam(**pdict)
a.runme()
if animateme:
a.animateme()
#a.saveme()
max_envelope = np.max(a.defl, axis=1)
# xplot = (pdict['xvals_norm']-trans_start)/trans_len
xplot = (pdict['xvals_norm']-trans_middle)*pdict['L']*lam
# print(pdict['xvals_norm'])
# lam = (pdict['k1']/(4*pdict['E']*pdict['I']))**0.25
Q = pdict["moving_loads_Fz_norm"][0][0] * pdict['E'] * pdict['A']
wstatic = Q * lam / (2*pdict['k1'])
# account for multiple loads
# pdict["moving_loads_x_norm"]=[[0-vv*load_spacing for vv in range(nloads)]]
# pdict["moving_loads_Fz_norm"]=[[1.013e-4]*nloads]
xmultim,ymulti = multi_static(xc=[0-vv*(load_spacing*pdict["L"]) for vv in range(nloads)],
frel=[1]*nloads,
L=1/lam,
xlim = (0-load_spacing*pdict["L"]*(nloads + 1), load_spacing*pdict["L"]),
wsingle=wstatic)
wstatic = np.max(ymulti)
# print(lam, Q, wstatic)
# print(max_envelope)
yplot = max_envelope / wstatic
# print(xplot.shape, yplot.shape)
# print(xplot)
if reverse:
label = "$L_t/L_c={},\ \\leftarrow$".format(Lt_Lc)
if k==0:
label = "$spec\ \\leftarrow$".format(Lt_Lc)
else:
label=None
ax2.plot(xplot, yplot, dashes=[4,2,1,2],
color=line.get_color(),
label = label)
else:
label = "$L_t/L_c={},\ \\rightarrow$".format(Lt_Lc)
if k==0:
label = "$spec\ \\rightarrow$".format(Lt_Lc)
else:
label=None
ax2.plot(xplot, yplot, ls='--',
color=line.get_color(),
label = label)
#point vs time plot
if not xi_Lc is None:
max_position=np.unravel_index(a.defl.argmax(), a.defl.shape)[0]
xeval=a.xvals_norm[max_position]
# xeval = trans_middle + xi_Lc / lam/pdict["L"]
xi_Lc_ = (xeval-trans_middle)*lam*pdict["L"]
ieval = np.searchsorted(a.xvals_norm,xeval)
xplot=a.tvals#tvals_norm
yplot = a.defl[ieval-1,:] + (xeval-a.xvals_norm[ieval-1])/(a.xvals_norm[ieval]-a.xvals_norm[ieval-1]) * (a.defl[ieval,:]-a.defl[ieval-1,:])
# yplot = a.defl[ieval,:] + (xeval-a.xvals_norm[ieval])/(a.xvals_norm[ieval+1]-a.xvals_norm[ieval]) * (a.defl[ieval+1,:]-a.defl[ieval,:])
yplot /= wstatic
if article_formatting:
ax4.plot(xplot, yplot,
label="$L_t/L_c={}$".format(Lt_Lc),
markevery=(k/3*0.09,0.15),
**styles[k])
else:
ax4.plot(xplot, yplot, color=line.get_color(),
label="$L_t/L_c={}$".format(Lt_Lc) )
for imvpl, mvpl in enumerate(a.moving_loads_norm):
t_pass = mvpl.time_axles_pass_beam_point(xeval)
t_pass *= pdict["L"] / np.sqrt(pdict["E"] / pdict["rho"])
ax4.plot(t_pass, np.interp(t_pass,xplot,yplot),
color=line.get_color(),
marker="o", ms=7, ls=':',
label="$x/L_c={:.1f}$".format(xi_Lc_))
#under loads
a.defl_vs_time_under_loads(xlim_norm=(a.xvals_norm[0], a.xvals_norm[-1]))
dash_list=[(None,None),[2,2],[3,2,1,2],[4,4]]
for imvpl, mvpl in enumerate(a.moving_loads_norm):
for j in range(len(mvpl.x)):
if len(Lt_Lcs)==1:
label = label="$load {}$".format(j+1)
else:
label = label="$L_t/L_c={},\ load {}$".format(Lt_Lc, j+1)
ax5.plot(a.tvals, a.moving_load_w[imvpl][j] / wstatic,
color=line.get_color(),
label=label,
dashes=dash_list[j])
leg = ax2.legend(
# title="$\\beta$",
loc="upper right",
# ncol=2,
labelspacing=.2,
handlelength=2,
fontsize=8,
numpoints=1,
)
leg.draggable()
ax2.xaxis.labelpad = -1
ax2.yaxis.labelpad = 0
ax3.xaxis.labelpad = -1
ax3.yaxis.labelpad = 0
yloc = plt.MaxNLocator(5)
ax3.yaxis.set_major_locator(yloc)
fig.subplots_adjust(top=0.95, bottom=0.13, left=0.14, right=0.95)
if not xi_Lc is None:
ax4.set_xlabel('$Time (s)$')
ax4.set_ylabel('$w/w_{static1}$')
leg = ax4.legend(
# title="",
labelspacing=.2,
handlelength=2,
fontsize=8,
numpoints=1,
loc="center left",)
leg.draggable()
ax5.set_xlabel('$Time (s)$')
ax5.set_ylabel('$w/w_{static1}$')
leg = ax5.legend(
# title="",
labelspacing=.2,
handlelength=2,
fontsize=8,
numpoints=1,
loc="lower right",)
leg.draggable()
fig5.subplots_adjust(top=0.95, bottom=0.13, left=0.15, right=0.95)
if not saveas is None:
save_figure(fig=fig, fname=saveas + "_fig1")
save_figure(fig=fig4, fname=saveas + "_fig4")
save_figure(fig=fig5, fname=saveas + "_fig5")
return fig
[docs]def transition_zones2(animateme=False,
saveas=None,
xlim = (-8, 8),
ntrans = 50, # number of points to sample analytical trasnstion zone
kratio = 50,
beta = 0.2,
alphas = [0.8],
nx = 100, #number of x points
nt=100, #number of time values
nterms=150,
force_calc=True,
ninterp = 15, # number of points to discretize the transition zone
nloads = 1,
load_spacing = 0.03,
alpha_design = 0.8, #this will be ignored
Lt_Lcs = [0.1,4],
t_extend = 1.1, # factor to extend maximum evalutated time.
DAF_distribution = "linear",
tanhend=0.01,
reverses=[False,True],
xi_Lc=None, #ratio of xinterest over characteristic length
prefix = "",
article_formatting=True):
"""Essentially a duplicate of transition_zones1 to get a DAF* vs Lt/Lc
plot for forward and reverse plots
Be warned that this can produce hundreds of plots and take hours to run
when all you are interested is the single resulting plot.
Basically a transtion_zones1 analysis will be run. The maximum
defelction (w.r.t. to the infinite beam deflection at same speed
and damping) for that run will be recorded. That will be one point for
one alpha value for one Lt/Lc value. Points will be joined to form
a line for for each alphas and reverses combination.
I used this once for a journal article figure.
"""
if article_formatting:
mdc1 = MarkersDashesColors(markersize=4,linewidth=1)
mdc1.construct_styles(markers=[2,25,11,5,8,15],
dashes=[0],
marker_colors=[0,1,2,3,4,5],
line_colors=[0,1,2,3,4,5])
styles1=mdc1(markers=[2,25,11,5,8,15],
dashes=[0],
marker_colors=[0,1,2,3,4,5],
line_colors=[0,1,2,3,4,5])
mdc2 = MarkersDashesColors(markersize=4,linewidth=1)
mdc2.construct_styles(markers=[13,18,21,9,0,3],
dashes=[1],
marker_colors=[0,1,2,3,4,5],
line_colors=[0,1,2,3,4,5])
styles2=mdc1(markers=[13,18,21,9,0,3],
dashes=[1],
marker_colors=[0,1,2,3,4,5],
line_colors=[0,1,2,3,4,5])
#figure
fig = plt.figure(figsize=(3.54,5))
gs = matplotlib.gridspec.GridSpec(2, 1,
# width_ratios=[1, 2],
height_ratios=[1,4],
hspace=0.05)
matplotlib.rcParams.update({'font.size': 11})
#DAF vs kratio plot
# ax1 = fig.add_subplot(gs[1,0])
# transition_DAF1_at_specific_alp(ax=ax1)
# ax1.set_ylim(0,2)
#DAF vs xtrans plot
ax2 = ax2 = fig.add_subplot(gs[1,0])
ax2.set_ylabel("$w/w_{static1}$")
# ax2.set_xlim(-0.5,1.5)
ax2.set_xlabel('$x/L_c$')
ax2.set_ylim(0,2)
# ax1.set_ylim(-10, 10)
#k/k1 vs xtrans plot
ax3 = fig.add_subplot(gs[0,0], )#sharex=ax2)
# ax3.set_xlabel('$x/L_t$')
ax3.set_ylabel('$k/k_1$')
ax3.grid()
ax3.xaxis.set_ticklabels([])
# xlim = (-.5,1.5)
# xlim = (-8,8)
ax2.set_xlim(*xlim)
ax2.grid()
if not xi_Lc is None:
fig4, ax4 = plt.subplots() # deflection vs time at point
fig5, ax5 = plt.subplots()
fig6 = plt.figure(figsize=(3.54,3.54))
ax6 = fig6.add_subplot('111')
pdict = OrderedDict(
E = 6.998*1e9, #Pa
rho = 2373, #kg/m3
L = 160, #m
A = 0.3*0.1, # m^2
I = 0.00022477900799999998,
k1 = 800002.6125,
nterms=nterms,
BC="SS",
# moving_loads_x_norm=[[0-vv*load_spacing for vv in range(nloads)]],
# moving_loads_Fz_norm=[[1.013e-4]*nloads],
# xvals_norm=np.linspace(xeval[0], xeval[1], nx),
use_analytical=True,
implementation="fortran",
force_calc=force_calc,
)
max_ratios=[]
for reverse in reverses:
max_ratios.append(np.zeros((len(alphas), len(Lt_Lcs))))
for k, Lt_Lc in enumerate(Lt_Lcs):
print("doing Lt_Lc=", Lt_Lc)
trans_characteristic_factor = Lt_Lc# multiple of k1 characteristic length
trans_middle = 0.5
# t_extend = 1.1 # factor to extend maximum evalutated time.
lam = (pdict['k1']/(4*pdict['E']*pdict['I']))**0.25
trans_len_raw = trans_characteristic_factor / lam
print("Characteristic len ={}".format(1/lam))
print("normalised:{}".format(1/lam/pdict["L"]))
trans_len = trans_len_raw / pdict["L"]
# trans_len=0.2 #normalised values w.r.t. L
print("tlen/char_len={}".format(trans_len/(1/lam/pdict["L"])))
trans_start = trans_middle - trans_len/2
trans_end = trans_start + trans_len
print("tstart={:8.3f},tend={:8.3f},tlen={:8.3f}".format(trans_start,trans_end, trans_len))
xeval = (min(0.4,trans_start - (0-xlim[0]) * trans_len),
max(0.6,trans_end + (xlim[1] - 1)* trans_len))
xeval = (trans_middle - abs(xlim[0])/lam/pdict["L"], trans_middle + abs(xlim[1])/lam/pdict["L"])
print("xeval:",xeval)
xtrans = np.linspace(0, 1, ntrans)
DAFtrans = np.empty_like(xtrans)
for i, alp in enumerate(alphas):
alpha_design = alp
DAF1 = DAFinfinite(alp=alpha_design, bet=beta)
DAF2 = DAFinfinite(alp=alpha_design,bet=beta,kratio=kratio)
if DAF_distribution.lower()=="linear":
DAF_goal = DAF1 + (DAF2-DAF1)*xtrans
elif DAF_distribution.lower()=="tanh":
#ideal transition
b = tanhtranib(x=0, y=tanhend, a=0.5)
print("b:",b)
DAF_goal = (tanhtran(x=xtrans, a=0.5, b=b) - 0.5)/(1-2*tanhend) + 0.5
DAF_goal = DAF_goal * DAF2 + (1-DAF_goal) * DAF1
elif DAF_distribution.lower()=="reverse":
DAF_goal = reverse_curve_transition(xtrans,0,DAF1, 1, DAF2)
DAF_goal[0] = DAF1
DAF_goal[-1] = DAF2
kkk = np.linspace(1,kratio,300,dtype=float)
yyy = np.zeros_like(kkk)
for iii,krat in enumerate(kkk):
yyy[iii]=DAFinfinite(alp=alpha_design,bet=beta,kratio=krat)
ktrans = np.empty_like(xtrans)
for iii, DAF in enumerate(DAF_goal):
if iii==0:
ktrans[0]=1
elif iii==ntrans-1:
ktrans[-1]=kratio
else:
ktrans[iii]=np.interp(DAF, yyy[::-1], kkk[::-1])
ctrans = np.ones_like(xtrans)
xx = [xlim[0]]; xx.extend((xtrans-trans_middle)*trans_len*pdict["L"]*lam); xx.append(xlim[1])
yy = [1]; yy.extend(ktrans); yy.append(kratio)
for j, krat in enumerate(ktrans):
DAFtrans[j] = DAFinfinite(alp=alp,bet=beta,kratio=krat)
yy = [DAFtrans[0]]; yy.extend(DAFtrans); yy.append(DAFtrans[-1])
label="$L_t/L_c={}, idealized$".format(Lt_Lc)
line,= ax2.plot(xx,yy,ls="-",label=label)
#mymodel
#sample the k value to form a poly line
xtrans_spec = np.linspace(0, 1, ninterp)
# if reverse:
# ktrans_spec = np.interp(xtrans_spec, xtrans, ktrans_reverse)
# else:
ktrans_spec = np.interp(xtrans_spec, xtrans, ktrans)
# print(ktrans_spec)
pline_x = [0]
pline_x.extend(trans_start + xtrans_spec * trans_len)
pline_x.append(1)
pline_k1 = [ktrans_spec[0]]
pline_k1.extend(ktrans_spec)
pline_k1.append(ktrans_spec[-1])
k1bar = PolyLine(pline_x, pline_k1)
# print(k1bar)
#sample the c values to form a poly line
ctrans_spec = np.interp(xtrans_spec, xtrans, ctrans)
pline_mu = [1]
pline_mu.extend(ctrans_spec)
pline_mu.append(ctrans_spec[-1])
mubar = PolyLine(pline_x, pline_mu)
# print(mubar)
# xeval = (trans_start - (0-xlim[0]) * trans_len,
# trans_end + (xlim[1] - 1)* trans_len)
# print(xeval)
#
## xwindow = (xeval[0],xeval[1])
#
# pdict = OrderedDict(
# E = 6.998*1e9, #Pa
# rho = 2373, #kg/m3
# L = 160, #m
# A = 0.3*0.1, # m^2
# I = 0.00022477900799999998,
# k1 = 800002.6125,
# nterms=nterms,
# BC="SS",
# moving_loads_x_norm=[[0-vv*load_spacing for vv in range(nloads)]],
# moving_loads_Fz_norm=[[1.013e-4]*nloads],
# xvals_norm=np.linspace(xeval[0], xeval[1], nx),
# use_analytical=True,
# implementation="fortran",
# force_calc=force_calc,
# )
v_crit = np.sqrt(2/pdict["rho"]/pdict["A"]*np.sqrt(pdict["k1"]*pdict["E"]*pdict["I"]))
v_raw = v_crit*alp
c_crit = np.sqrt(4*pdict["rho"]*pdict["A"]*pdict["k1"])
c_raw = c_crit * beta
tmax = pdict["L"] / v_raw
pdict["moving_loads_x_norm"]=[[0-vv*load_spacing for vv in range(nloads)]]
pdict["moving_loads_Fz_norm"]=[[1.013e-4]*nloads]
pdict["xvals_norm"]=np.linspace(xeval[0], xeval[1], nx)
pdict["mu"] = c_raw
pdict["tvals"] = np.linspace(tmax*xeval[0],tmax*xeval[1]*t_extend,nt)
pdict['k1bar']=k1bar
pdict['mubar']=mubar
for m, reverse in enumerate(reverses):
if reverse:
pdict["moving_loads_v_norm"] = [-1*v_raw*np.sqrt(pdict["rho"]/pdict["E"])]
pdict["file_stem"] = prefix + "tans2_alp{:5.3f}_bet{:5.3f}_n{:d}_LtLc{:5.3f}_rev".format(alp,beta,nterms,Lt_Lc)
else:
pdict["moving_loads_v_norm"] = [1*v_raw*np.sqrt(pdict["rho"]/pdict["E"])]
pdict["file_stem"] = prefix + "tans2_alp{:5.3f}_bet{:5.3f}_n{:d}_LtLc{:5.3f}_for".format(alp,beta,nterms,Lt_Lc)
print(pdict["file_stem"])
a = SpecBeam(**pdict)
a.runme()
if animateme:
a.animateme()
a.saveme()
max_envelope = np.max(a.defl, axis=1)
# xplot = (pdict['xvals_norm']-trans_start)/trans_len
xplot = (pdict['xvals_norm']-trans_middle)*pdict['L']*lam
# print(pdict['xvals_norm'])
# lam = (pdict['k1']/(4*pdict['E']*pdict['I']))**0.25
Q = pdict["moving_loads_Fz_norm"][0][0] * pdict['E'] * pdict['A']
wstatic = Q * lam / (2*pdict['k1'])
# print(lam, Q, wstatic)
# print(max_envelope)
yplot = max_envelope / wstatic
max_ratio = np.max(a.defl)/ wstatic / DAF1
max_ratios[m][i, k] = max_ratio
# print(xplot.shape, yplot.shape)
# print(xplot)
if reverse:
label = "$L_t/L_c={},\ \\leftarrow$".format(Lt_Lc)
if k==0:
label = "$spec\ \\leftarrow$".format(Lt_Lc)
else:
label=None
ax2.plot(xplot, yplot, dashes=[4,2,1,2],
color=line.get_color(),
label = label)
else:
label = "$L_t/L_c={},\ \\rightarrow$".format(Lt_Lc)
if k==0:
label = "$spec\ \\rightarrow$".format(Lt_Lc)
else:
label=None
ax2.plot(xplot, yplot, ls='--',
color=line.get_color(),
label = label)
#point vs time plot
if not xi_Lc is None:
max_position=np.unravel_index(a.defl.argmax(), a.defl.shape)[0]
xevalu=a.xvals_norm[max_position]
# xevalu = trans_middle + xi_Lc / lam/pdict["L"]
xi_Lc_ = (xevalu-trans_middle)*lam*pdict["L"]
ieval = np.searchsorted(a.xvals_norm,xevalu)
xplot=a.tvals_norm
yplot = a.defl[ieval-1,:] + (xevalu-a.xvals_norm[ieval-1])/(a.xvals_norm[ieval]-a.xvals_norm[ieval-1]) * (a.defl[ieval,:]-a.defl[ieval-1,:])
# yplot = a.defl[ieval,:] + (xevalu-a.xvals_norm[ieval])/(a.xvals_norm[ieval+1]-a.xvals_norm[ieval]) * (a.defl[ieval+1,:]-a.defl[ieval,:])
yplot /= wstatic
ax4.plot(xplot, yplot, color=line.get_color(),
label="$L_t/L_c={}$".format(Lt_Lc) )
for imvpl, mvpl in enumerate(a.moving_loads_norm):
t_pass = mvpl.time_axles_pass_beam_point(xevalu)
ax4.plot(t_pass, np.interp(t_pass,xplot,yplot),
color=line.get_color(),
marker="o", ls=':',
label="$x/L_c={:.1f}$".format(xi_Lc_))
#under loads
a.defl_vs_time_under_loads(xlim_norm=(a.xvals_norm[0], a.xvals_norm[-1]))
dash_list=[(None,None),[2,2],[3,2,1,2],[4,4]]
for imvpl, mvpl in enumerate(a.moving_loads_norm):
for j in range(len(mvpl.x)):
ax5.plot(a.tvals, a.moving_load_w[imvpl][j] / wstatic,
color=line.get_color(),
label="$L_t/L_c={},\ load {}$".format(Lt_Lc, j+1),
dashes=dash_list[j])
leg = ax2.legend(
# title="$\\beta$",
loc="upper right",
# ncol=2,
labelspacing=.2,
handlelength=2,
fontsize=8,
numpoints=1,
)
leg.draggable()
ax2.xaxis.labelpad = -1
ax2.yaxis.labelpad = 0
ax3.xaxis.labelpad = -1
ax3.yaxis.labelpad = 0
fig.subplots_adjust(top=0.95, bottom=0.13, left=0.14, right=0.95)
if not xi_Lc is None:
ax4.set_xlabel('$norm _time$')
ax4.set_ylabel('$w/w_{static1}$')
leg = ax4.legend(
# title="",
labelspacing=.2,
handlelength=2,
fontsize=8,
numpoints=1)
leg.draggable()
ax5.set_xlabel('$time$')
ax5.set_ylabel('$w/w_{static1}$')
leg = ax5.legend(
# title="",
labelspacing=.2,
handlelength=2,
fontsize=8,
numpoints=1)
leg.draggable()
for i, alp in enumerate(alphas):
for m, reverse in enumerate(reverses):
xplot = Lt_Lcs
yplot = max_ratios[m][i,:]
if reverse:
label = "$\\alpha_1={},\ \\leftarrow$".format(alp)
# if k==0:
# label = "$spec\ \\leftarrow$".format(Lt_Lc)
# else:
# label=None
# ax2.plot(xplot, yplot, dashes=[4,2,1,2],
# color=line.get_color(),
# label = label)
if article_formatting:
ax6.plot(xplot, yplot,
label=label,
markevery=(i/6*0.1,0.1),
**styles1[i])
else:
ax6.plot(xplot, yplot,
label=label,
ls='--',color=line.get_color())
else:
label = "$\\alpha_1={},\ \\rightarrow$".format(alp)
# if k==0:
# label = "$spec\ \\rightarrow$".format(Lt_Lc)
# else:
# label=None
#
# ax2.plot(xplot, yplot, ls='--',
# color=line.get_color(),
# label = label)
if article_formatting:
line,=ax6.plot(xplot, yplot,
label=label,
markevery=(i/6*0.1,0.1),
**styles2[i])
else:
line,=ax6.plot(xplot, yplot,
label=label,
ls='-')
# ax6.plot(xplot,yplot,label="$\\alpha_1={}$".format(alp))
ax6.set_xlabel('$L_t/L_c$')
ax6.set_ylabel('DAF*')
def flip(items, ncol):
import itertools
return itertools.chain(*[items[i::ncol] for i in range(ncol)])
handles, labels = ax6.get_legend_handles_labels()
leg = ax6.legend(flip(handles, 2), flip(labels, 2),
ncol=2,
labelspacing=.1,
handlelength=2,
fontsize=8,
handletextpad=0.1,
borderpad=0.2,
columnspacing=0.2)
leg.draggable()
# for k, Lt_Lcs in enumerate(Lt_Lcs):
fig6.subplots_adjust(top=0.98, bottom=0.15, left=0.18, right=0.98)
if not saveas is None:
save_figure(fig=fig6, fname=saveas)
return fig
[docs]def tanhtran(x, a, b):
"""Transition between two functions using tanh
t(x) Smoothly transitions between 0 and 1;
Used with:
[transition f(x) to g(x)](x) = g(x)*t(x) + (1-t(x))*f(x)
Parameters
----------
x : float
x values to evaluate at
a : float
Position of tranistion. Often a=0.
b : float,
b close to zero gives a sharp change.
Returns
-------
tfunc: float
0.5 + 0.5 * tanh((x-a)/b)
See Also
--------
tanhtrani : inverse of tanhtran
tanhtranib : get smoothing factor from speccific x,y
"""
return 0.5+0.5*np.tanh((x-0.5)/b)
[docs]def tanhtrani(y, a, b):
"""Inverse of Transition between two functions using tanh
x value that matches y
Parameters
----------
y : float
y value to evaluate at
a : float
Position of tranistion. Often a=0.
b : float,
b close to zero gives a sharp change.
Returns
-------
tfunc: float
b * arctanh((y - 0.5) / 0.5) + a
See Also
--------
tanhtran : inverse of tanhtrani
tanhtranib : get smoothing factor from specific point.
"""
return b * np.arctanh((y - 0.5) / 0.5) + a
[docs]def tanhtranib(x,y,a):
"""Smoothing factor of tansition curve to get known (x,y)
b value such that t(x) goes through (x,y)
Parameters
----------
x,y : float
x values to evaluate at
a : float
Position of tranistion. Often a=0.
Returns
-------
tfunc: float
(x-a) / arctanh((y - 0.5) / 0.5)
See Also
--------
tanhtran : transition function
tanhtrani : inverse of tanhtran
"""
return (x-a) / np.arctanh((y - 0.5) / 0.5)
[docs]def FIG_DAF_envelope_vary_beta(ax=None, betas=None, alphas=None,
saveas=None,
nx=200,nt=200,nterms=100, force_calc=True,
t_extend=1,
ylim=None,
numerical_DAF=False,
article_formatting=False):
"""DAF envelope for whole beam with different alpha & beta
A plot will be produced of DAF vs normalised distance along beam, where
the dynamic deflection in the DAF calculation is the maximum experienced
each x/L point.
Parameters
----------
ax : matplotlib.Axes object, optional
Axes object to plot on. Default ax=None i.e. fig and axis will be
created.
betas : list of float, optional
list of damping ratios, Default betas=None which will use
[0, 0.05, 0.1].
alphas : list of float, optional
list of velocity ratios, Default alphas=None which will use
[0.7].
saveas : string, optional
Filename stem (not including extension) to save figure to.
Default saveas=None, i.e. figure won't be saved to disk.
nx : int, optional
Number of points between `xlim[0]` and `xlim[1]` at which to
calculate deflection. Default nx = 200.
nt : int, optional
Number of evaluation times from when load enters `xlim[0]` and
leaves `xlim[1]`. Default nt=200
nterms : int, optional
Number of series terms in approximation. More terms leads to higher
accuracy but also longer computation times. Start of with a
small enough value to observe behaviour and then increase
to confirm. Default nterms=100 .
force_calc : [True, False], optional
When True all calcuations will be done from scratch. When False
Current working directory will be searched for a results file
named according to "saveas" and
if found it will be loaded. Be careful as name of results file may
not be unique to your analysis parameters. Safest to use
force_calc = True, but if you are simply regenerating graphs then
consider force_calc=False. Default force_calc=True.
t_extend : float, optional
factor to extend maximum evalutated time. As per `nt' evaluation times
are normally between first point entering `xlim[0]' and leaving
`xlim[1]`. However, with mulitple loads, loads can still be on the
beam at that point. max eval time can be extended fby a factor.
t_extend = 1 .
ylim : float, optional
Y axis limit. default ylim=None i.e. auto axis scale
numerical_DAF : [False, True], optional
If True then the static deflection for use in Dynamic amplification
factor calcuations (DAF = w/wstatic) willbe determined from a
very slow moving load travelling over the beam/foundation.
If False then the analytical expression for wstatic will be used.
Default numerical_DAF=False.
article_formatting : [False, True], optional
You should never need this. It was a quick hack to do some
custom formatting for a journal article. Dont even try messing with
it. will ony work with 3 Lt_Lcs values values or less. Default=False,
i.e. no special formatting.
"""
if article_formatting:
mdc = MarkersDashesColors(markersize=3,linewidth=1)
styles=mdc(markers=[5,15,2],
dashes=[0],
marker_colors=[0,1,2],
line_colors=[0,1,2])
mevery = [0.4,0.15,0.1]
#used for figure 3 in publication
mytimer=SimpleTimer()
mytimer.start(0,"FIG_DAF_envelope_vary_beta")
if ax is None:
fig,ax = plt.subplots(figsize=(3.54,3.54))
matplotlib.rcParams.update({'font.size': 11})
pdict = OrderedDict(
E = 6.998*1e9, #Pa
rho = 2373, #kg/m3
L = 160, #m
A = 0.3*0.1, # m^2
I = 0.00022477900799999998,
k1 = 800002.6125,
nterms=nterms,
BC="SS",
# moving_loads_x_norm=[[0-vv*load_spacing for vv in range(nloads)]],
# moving_loads_Fz_norm=[[1.013e-4]*nloads],
# xvals_norm=np.linspace(xeval[0], xeval[1], nx),
moving_loads_x_norm=[[0]],
moving_loads_Fz_norm=[[1.013e-4]],
xvals_norm=np.linspace(0, 1, nx),
use_analytical=True,
implementation="fortran",
force_calc=force_calc,
)
if alphas is None:
alphas = [0.7]
if betas is None:
betas = [0, 0.05, 0.1]
if numerical_DAF:
alphas.insert(0,0.00001)
for i,alp in enumerate(alphas):
for j,bet in enumerate(betas):
mytimer.start(1, "alpha={}, beta={}".format(alp, bet))
# start_time1 = time.time()
# print("doing alpha={}, beta={}".format(alp, bet))
v_crit = np.sqrt(2/pdict["rho"]/pdict["A"]*np.sqrt(pdict["k1"]*pdict["E"]*pdict["I"]))
v_raw = v_crit*alp
c_crit = np.sqrt(4*pdict["rho"]*pdict["A"]*pdict["k1"])
c_raw = c_crit * bet
tmax = pdict["L"] / v_raw
pdict["file_stem"] = "envelope_alp{:5.3f}_bet{:5.3f}_nx{:d}_nt{:d}_n{:d}".format(alp,bet,nx,nt,nterms)
pdict["mu"] = c_raw
pdict["tvals"] = np.linspace(tmax*0,tmax*t_extend, nt)
pdict["moving_loads_v_norm"] = [1*v_raw*np.sqrt(pdict["rho"]/pdict["E"])]
a = SpecBeam(**pdict)
a.runme()
# a.animateme()
a.saveme()
max_envelope = np.max(a.defl, axis=1)
lam = (pdict['k1']/(4*pdict['E']*pdict['I']))**0.25
Q = pdict["moving_loads_Fz_norm"][0][0] * pdict['E'] * pdict['A']
if numerical_DAF:
if i==0:
wstatic = np.max(max_envelope[int(nx*0.4):int(nx*0.6)])
continue
else:
wstatic = Q * lam / (2*pdict['k1'])
xplot = pdict['xvals_norm']
yplot = max_envelope / wstatic
# label = "$\\alpha={:3.2g},\\ \\beta={:3.2g}$".format(alp,bet)
label = "${:3.2g}$".format(bet)
if article_formatting:
line, = ax.plot(xplot, yplot,
label=label,
markevery=(j/3*0.9,mevery[j]),
**styles[j])
else:
line, = ax.plot(xplot, yplot,
label=label,
ls='-',)
# color=line.get_color())
DAF = DAFinfinite(alp, bet)
if j==0:
label="analytical"
else:
label=None
ax.plot(0.6, DAF,
marker='o', ms=6,
ls='None',
color = line.get_color(),
label=label)
mytimer.finish(1)
ax.grid()
ax.set_xlabel("$x/L$")
ax.set_ylabel("$DAF$")
if not ylim is None:
ax.set_ylim(*ylim)
leg=ax.legend()
leg = ax.legend(
title="$\\beta$",
loc="upper center",
# ncol=2,
labelspacing=.2,
handlelength=2,
fontsize=8,
numpoints=1,
)
leg.draggable()
ax.xaxis.labelpad = -1
ax.yaxis.labelpad = 0
mytimer.finish(0)
# end_time0 = time.time()
# elapsed_time = (end_time0 - start_time0); print("Total run time={}".format(str(timedelta(seconds=elapsed_time))))
fig.subplots_adjust(top=0.95, bottom=0.13, left=0.14, right=0.95)
if not saveas is None:
save_figure(fig=fig, fname=saveas)
# save_figure(fig=fig2, fname=saveas + "_up")
return ax
[docs]def static_point_load_shape(x, L):
"""Deflection shape parameter for point load at x=0 on infinite beam on
elastic foundation with characteristic length L.
Deflection is relative to the maximum value
Parameters
----------
x : float
x coordinate(s)
L : float
Beam characteristic Length.
Returns
-------
out : float
Relative deflection at x.
Notes
-----
FYI the defelctions are relative to w_max which is :
L = Characteristic length = k1 / ((4 * E * I)**0.25)
Q = Point load
w_max = static = Q * lam / (2 * k1)
"""
return np.exp(-np.abs(x)/L)*(np.cos(np.abs(x)/L)+np.sin(np.abs(x)/L))
[docs]def specbeam_vs_analytical_single_static_load():
"""Compare specbeam vs analytical solution for single static load
at mid point of 'long' beam.
Will produce a plot and an animation.
Note that you can't really have a static load in specbeam. You can
have a slowly applied load that then remains constant. Thus depending
on the beam properties e.g. damping, you can get a maximum displacement
envelope that is higher than the static case in that the momentum of the
beam carries the deflection past the 'analytical" maximum and then
oscillations are created untill they are damped out.
"""
mytimer=SimpleTimer()
mytimer.start(0,"single_static_load")
vfact=2
t = np.linspace(0, 2, 40)
t[-1]=999
# x = np.linspace(0,160,2001)
#see Dingetal2012 Table 2 for material properties
pdict = OrderedDict(
E = 6.998*1e9, #Pa
rho = 2373, #kg/m3
L = 20, #m
A = 0.3*0.1, # m^2
I = 0.00022477900799999998,
k1 = 800002.6125,
#v_norm=0.01165,
# kf=5.41e-4,
#Fz_norm=1.013e-4,
mu_norm=39.263,#/100,
# k1_norm=97.552,
# k3_norm=2.497e6,
nterms=201,
BC="SS",
nquad=20,
# k1bar=PolyLine([0,0.5],[0.5,1],[1,1],[1,.25]),
# k1bar=PolyLine([0,0.25,.75,1],[1.0,1,4,4]),
# k1bar=PolyLine([0,0.4,.6,1],[1.0,1,.25,.25]),
# k1bar=PolyLine([0,0.5],[0.5,1],[2,1],[2,1]),
# moving_loads_x_norm=[[0,-0.05,-0.1]],
# moving_loads_Fz_norm=[3*[1.013e-4]],
# moving_loads_v_norm=[0.01165*vfact],
# stationary_loads_x=None,
# stationary_loads_vs_t=None,
# stationary_loads_omega_phase=None,
stationary_loads_x_norm=[0.5],
# stationary_loads_vs_t_norm=[PolyLine([0,8],[1,1])],
stationary_loads_vs_t_norm=[PolyLine([0,1,1000],[0,1e-4,1e-4])],
# stationary_loads_omega_phase_norm=[(.5,0),(0,0)],#None,
tvals_norm=t,
# tvals=np.array([4.0]),
# xvals_norm=np.array([0.5]),
# xvals=x,
use_analytical=True,
# implementation="vectorized",
implementation="fortran",
file_stem="single_static_load",
force_calc=True,
)
pdict['xvals'] = np.linspace(0,pdict['L'], 2001)
a = SpecBeam(**pdict)
a.runme()
# a.saveme()
a.animateme(norm=True)
# a.plot_w_vs_x_overtime(norm=True)
max_envelope = np.max(a.defl, axis=1)
min_envelope = np.min(a.defl, axis=1)
max_index = np.argmax(a.defl[int((2001-1)/2),:])
#print(a.defl[int((2001-1)/2),:])
#print(max_index)
maxy = a.defl[:, max_index]
yyy = a.defl[:,-1]
lam = (pdict['k1'] /(4*pdict['E']*pdict['I']))**0.25
Q = pdict["stationary_loads_vs_t_norm"][0].y[1] * pdict['E'] * pdict['A']
wstatic = Q * lam / (2*pdict['k1'])
fig,ax = plt.subplots()
ax.plot(a.xvals, maxy, label='When max reached')
# line,=ax.plot(a.xvals, max_envelope, label='envelope')
# ax.plot(a.xvals, min_envelope, color=line.get_color())
ax.plot(a.xvals, yyy, label='last',lw=4)
xstatic = np.linspace(0,pdict["L"], 5001)
ystatic = wstatic * static_point_load_shape((xstatic-pdict["L"]/2),1/lam)
ax.plot(xstatic, ystatic, label="analytical",
ls='None', marker='o', markevery=0.02)
ax.set_xlabel('x')
ax.set_ylabel('Deflection')
ax.set_title('Single static load at beam midpoint')
leg = ax.legend()
leg.draggable()
mytimer.finish(0)
plt.show()
[docs]def reverse_curve_transition(x, x1, y1, x2, y2):
"""Transition curve between (x1, y1) and (x2, y2)
Connects two parallel horizontal lines by two circular arcs.
Parameters
----------
x : array of float
value(s) at which to caluclate transition curve
x1, y1 : float
start point of transition
x2, y2 : float
end point of transition
Returns
-------
y : float
value of transition curve at x.
Notes
-----
Two circular curves transition in a rectangle h high by L wide. Each
curve has radius R and traces out a curve of delta radians.
h=2*R*(1-cos(delta))
&
L=2*R*(sin(delta))
Solve these in wolfram alpha for delta :Solve[(1-Cos[x])/Sin[x]-h/L,x]
to get : delta = 2*arctan(h/L)
Then use equation of circle in particualr quadrant to get points on curve.
adjust for x1,y1 and x2,y2 starting points.
"""
x = np.atleast_1d(x)
y = np.empty_like(x)
if y1 == y2:
#no transition --> horizontal line
y[:]=y1
return y
xmid = (x2 + x1) / 2
xm = x < xmid #LHS of transition
xp = x >= xmid #RHS of transition
if x2 > x1:
xa, xb = x1, x2
ya, yb = y1, y2
elif x1>x2:
xa, xb= x2, x1
ya, yb= y2, y1
else:
raise ValueError("x1 must be differnet to x2")
xmm = x <= xa #to left of transition
xpp = x >= xb #to right of transition
xm*=~xmm
xp*=~xpp
delta = 2 * np.arctan(abs(yb - ya) / abs(xb - xa))
R = abs(xb - xa) / (2*np.sin(delta))
if yb<ya:
y[xm] = np.sqrt(R**2 - (x[xm]-xa)**2) + ya - R
y[xp] = -np.sqrt(R**2 - (x[xp]-xb)**2) + yb + R
else:
y[xm] = -np.sqrt(R**2 - (x[xm]-xa)**2) + ya + R
y[xp] = +np.sqrt(R**2 - (x[xp]-xb)**2) + yb - R
y[xmm] = ya
y[xpp] = yb
return y
[docs]def article_figure_01():
"""Figure 1 Mid point deflection comparison with Ding et al 2012, figure
from Walker and Indraratna (in press).
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
################################
# FIGURE GENERATING CODE
# Figure 1 Mid point deflection comparison with Ding et al 2012
#################################
mpl.style.use('classic')
#Journal paper figure, verify against ding et al.
FIGURE_verify_against_DingEtAl2012_fig8(v_nterms=(50,75,150,200),saveas="verify_Dingetal2012")#75,150,200))
plt.show()
[docs]def article_figure_02():
"""Figure 2 Deflection amplification factor compared with analytical
infinite beam solution, figure from Walker and Indraratna (in press).
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
#################################
# FIGURE GENERATING CODE
# Figure 2 Deflection amplification factor compared with analytical infinite beam solution
#################################
mpl.style.use('classic')
FIGURE_DAF_constant_prop(saveas="DAF_constant_astatic",
nterms=150,
force_calc=False,
# betas=[0.05],
# alphas=np.linspace(1e-5,3,4),
# alphas=np.linspace(0.01,0.95,10),
nx=2000, nt=100,
prefix="static_a",
# end_damp=0.04,
xwindow=(0.3,0.7),
xeval=(0, 1),
numerical_DAF=False,
article_formatting=True)
plt.show()
[docs]def article_figure_03():
"""Figure 3 Low damping DAF for velocity ratio alp=0p5,
figure from Walker and Indraratna (in press).
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
#################################
# FIGURE GENERATING CODE
# Figure 3 Low damping DAF for velocity ratio alp=0.5
#################################
mpl.style.use('classic')
ax=FIG_DAF_envelope_vary_beta(ax=None,
betas=[0, 0.05,0.5],
alphas=[0.5],
saveas="low_beta_DAF_envelope",
nx=1001,nt=1001,nterms=150,
force_calc=False,
t_extend=1,
ylim=(0.8,1.6),
numerical_DAF=False,
article_formatting=True)
plt.show()
[docs]def article_figure_04():
"""Figure 4 Relative DAF for various stiffness ratios,
figure from Walker and Indraratna (in press).
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
#################################
# FIGURE GENERATING CODE
# Figure 4 Relative DAF for various stiffness ratios
#################################
mpl.style.use('classic')
fig = transition_DAF1(kratios=[0.5, 1, 2, 5, 20, 50],
alpmax=3,
npts=500,
saveas="transitionDAF1",
article_formatting=True)
plt.show()
[docs]def article_figure_05a_05b_05c_07():
"""Figure 5 a),b),&c) Max deflection envelope for bet=0p1 k2_k1=10,
and Figure 7 Deflection vs time at point which experiences maximum
deflection alp=0p8, bet=0p1, k2_k1=10, soft to stiff=forward,
figure from Walker and Indraratna (in press).
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
#################################
# FIGURE GENERATING CODE
# Figure 5 a),b),&c) Max deflection envelope for bet=0.1, k2_k1=10
# will need to comment out code as appropriate for 5a, 5b, 5c.
# Figure 7 Deflection vs time at point which experiences maximum deflection alp=0.8, bet=0.1, k2_k1=10, soft to stiff=forward
#################################
mpl.style.use('classic')
for alpha_design, alphas, reverses, saveas in zip(
[0.3, 0.8, 0.8],
[[0.3], [0.8] , [0.8]],
[[False], [False] , [True]],
["Lt_lc_alp0.3_bet0.1_forward",
"Lt_lc_alp0.8_bet0.1_forward" ,
"Lt_lc_alp0.8_bet0.1_reverse"]
):
transition_zones1(xlim = (-8,8),
ntrans = 50, # number of points to sample analytical trasnstion zone
kratio = 10,
beta = 0.1,
nx = 400, #number of x points
nt=400, #number of time values
nterms=150,
force_calc=True,
ninterp = 50, # number of points to discretize the transition zone
nloads = 1,
load_spacing = 0.03,
Lt_Lcs = [0.01,2,5], #2,4,6],
t_extend = 1.0, # factor to extend maximum evalutated time.
DAF_distribution = "linear",
tanhend=0.01,
alpha_design = alpha_design, #alpha for back calculating k profile
alphas = alphas,
reverses=reverses,
saveas=saveas,
animateme=False,
xi_Lc=True, #ratio of xinterest over characteristic length)
article_formatting=True)
plt.show()
[docs]def article_figure_06():
"""Additional deflection for transition effects, bet=0p1, alp=10, (dashed=stiff to soft, solid=soft to stiff),
figure from Walker and Indraratna (in press).
This takes forever to run.
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
#################################
# FIGURE GENERATING CODE
# Figure 6 Additional deflection for transition effects, bet=0.1, alp=10, (dashed=stiff to soft, solid=soft to stiff)
#################################
#this is for generating DAF* vs Lt_Lc
trans2(xlim = (-10,10),
ntrans = 50, # number of points to sample analytical trasnstion zone
kratio = 10,
beta = 0.1,
alphas = [0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 0.8],
nx = 400, #number of x points
nt=400, #number of time values
nterms=150,
force_calc=False,
ninterp = 50, # number of points to discretize the transition zone
nloads = 1,
load_spacing = 0.03,
#alpha_design = 0.3, #alpha for back calculating k profile, #this will be ignored
Lt_Lcs = np.linspace(0.01, 15, 50), #2,4,6],
t_extend = 1.2, # factor to extend maximum evalutated time.
DAF_distribution = "linear",
tanhend=0.01,
reverses=[False,True],
animateme=False,
saveas="DAf_star_vs_Lt_Lc",
xi_Lc=True, #ratio of xinterest over characteristic length)
)
plt.show()
[docs]def article_figure_08():
"""Figure 8 Deflection vs time under four moving loads alp=0p8, bet=0p1, k2/k1=10, soft to stiff,
figure from Walker and Indraratna (in press).
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
#################################
# FIGURE GENERATING CODE
# Figure 8 Deflection vs time under four moving loads alp=0.8, bet=0.1, k2/k1=10, soft to stiff
#################################
mpl.style.use('classic')
#This is for multi-load
for Lt in [0.01, 4, 8]:
transition_zones1(xlim = (-10,10),
ntrans = 50, # number of points to sample analytical trasnstion zone
kratio = 10,
beta = 0.1,
alphas = [0.8],
nx = 400, #number of x points
nt=400, #number of time values
nterms=150,
force_calc=True,
ninterp=50, # number of points to discretize the transition zone
nloads = 4,
load_spacing = 0.03,
alpha_design = 0.8, #alpha for back calculating k profile
Lt_Lcs = [Lt], #2,4,6],
t_extend = 1.2, # factor to extend maximum evalutated time.
DAF_distribution = "linear",
tanhend=0.01,
reverses=[False],
animateme=True,
saveas="four_loads_Lt_lc{:.2g}_alp0.8_bet0.1_forward".format(Lt),
xi_Lc=True, #ratio of xinterest over characteristic length)
)
plt.show()
[docs]def article_figure_09():
"""Figure 9 is drawing in a word file, this routine is just a marker
to indicated that figure 9 has not been forgotten,
figure from Walker and Indraratna (in press).
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
#################################
# FIGURE GENERATING CODE
# Figure 9 is a drawing in a word file
#################################
print("Figure 9 is drawing in a word file. This routine is just a marker to indicated that figure 9 has not been forgotten")
[docs]def article_figure_10_11():
"""Figure 10 Measured vs modelled deflections at various locations in the
transition zone, Figure 11 Peak displacement and stiffness distribution,
figure from Walker and Indraratna (in press).
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
#################################
# FIGURE GENERATING CODE
# Figure 10 Measured vs modelled deflections at various locations in the transition zone
# Figure 11 Peak displacement and stiffness distribution.
#################################
mpl.style.use('classic')
#generate comparison plots for Paixao case study
case_study_PaixaoEtAl2014(saveas="Paixao", nterms=150, force_calc=False,
nx=500,
nt=500)
[docs]def article_figure_12():
"""Figure 12 python code for relative dynamic amplification factor
calculation, figure from Walker and Indraratna (in press).
References
----------
.. [1] Walker, R.T.R. and Indraratna, B, (in press) "Moving loads on a
viscoelastic foundation with special reference to railway
transition zones". International Journal of Geomechanics.
"""
#################################
# FIGURE GENERATING CODE
# Figure 12 python code for relative dynamic amplification factor calculation
#################################
mpl.style.use('classic')
from geotecha.plotting.one_d import figure_from_source_code
fig = figure_from_source_code(DAFrelative, figsize=(6.5,5))
save_figure(fig, "DAFrelative")
print("done")
plt.show()
if __name__ == "__main__":
import nose
mpl.style.use('classic')
# nose.runmodule(argv=['nose', '--verbosity=3', '--with-doctest', '--doctest-options=+ELLIPSIS'])
# test_DingEtAl2012()
# test_DingEtAl2012_deflection_shape()
# test_SpecBeam_const_mat_midpoint_defl()
# test_SpecBeam_const_mat_deflection_shape()
# test_SpecBeam_const_mat_midpoint_defl_runme()
# test_SpecBeam_const_mat_deflection_shape_runme()
# test_SpecBeam_const_mat_midpoint_defl_runme_analytical_50()
# test_SpecBeam_const_mat_midpoint_defl_runme_analytical_200()
# SpecBeam_const_mat_midpoint_defl_runme_analytical_play()
# test_SpecBeam_const_mat_deflection_shape_stationary_load()
# SpecBeam_stationary_load_const_mat_midpoint_defl_runme_analytical_play()
# dingetal_figure_8()
############################################
# Article figures
############################################
# article_figure_01()
# article_figure_02()
# article_figure_03()
# article_figure_04()
# article_figure_05a_05b_05c_07()
# article_figure_06() # this can take forever to run!!!!!!!! i.e. 5+hours
# article_figure_07()
# article_figure_08()
# article_figure_09()
# article_figure_10_11()
# article_figure_12()
############################################
# Other code
############################################
if 0:
##An example of using the reverse transition curves
x = np.linspace(-4, 4, 1000)
x1, y1 = (-2, 2)
fig, ax = plt.subplots()
for x2, y2 in [(1,1), (1,.5), (1,2.5),(-3,3),(-3,0.7), (3,2)]:
ax.plot(x1, y1, marker='o')
ax.plot(x2, y2, marker='s')
y = reverse_curve_transition(x, x1, y1, x2, y2)
# reverse_curve_transition(xtrans,0,DAF1, 1, DAF2)
ax.plot(x,y)
ax.set_xlabel("x")
ax.set_ylabel("y")
ax.set_title("Transiton curve")
plt.show()
if 0:
#Use SpecBeam to analyse/check a single static load at midpoint
specbeam_vs_analytical_single_static_load()
if 0:
#example of deflection envelope
for alpha_design, alphas, reverses, saveas in zip(
[0.8],
[[0.8]],
[[False]],
["Lt_lc_alp0.3_bet0.1_forward"]
):
transition_zones1(xlim = (-8,8),
ntrans = 50, # number of points to sample analytical trasnstion zone
kratio = 10,
beta = 0.1,
nx = 400, #number of x points
nt=400, #number of time values
nterms=150,
force_calc=False,
ninterp = 50, # number of points to discretize the transition zone
nloads = 1,
load_spacing = 0.03,
Lt_Lcs = [2], #2,4,6],
t_extend = 1.0, # factor to extend maximum evalutated time.
DAF_distribution = "linear",
tanhend=0.01,
alpha_design = alpha_design, #alpha for back calculating k profile
alphas = alphas,
reverses=reverses,
saveas=saveas,
animateme=True,
xi_Lc=True, #ratio of xinterest over characteristic length)
article_formatting=False)
plt.show()
if 0:
esveld_fig6p17()
esveld_fig6p18()
plt.show()
if 0:
#Messing around with deflection shape from multiple static loads
fig, ax = plt.subplots()
x = np.linspace(-4, 4, 100)
#single load, choose the location with xc:
y = point_load_static_defl_shape_function(x, 1, xc=1)
ax.plot(x, y ,label='single load')
ax.set_xlabel("x/L_c")
ax.set_ylabel("relative w")
ax.invert_yaxis()
#load centers equally spaced, with equal magnitude of 1:
for n in [3]:
xx,yy = multi_static(xc=np.arange(n)*3, frel=[1]*n)
ax.plot(xx,yy, label="{} loads".format(n))
ax.grid()
leg = ax.legend()
plt.show()
if 0:
#messing around with moving point loads
a = MovingPointLoads([0.,-3,-4,-5],[20.,15,10,10 ])
a.plotme(ax=None,t=0, v0=0, t0=0, x0=0, xlim = (-10,10), ylim=(0,100))
plt.show()#x0,t0,tf,v
if 0:
#messing around with moving point loads
a = MovingPointLoads([0.,-3,-4,-5],[20.,15,10,10 ])
# a.animateme(x0=-5,t0=0,tf=15,v0=6, xlim = (-10,20), ylim=(0,100))
a.animateme(x0=-5,t0=0,tf=15,v0=6, xlim = (-10,20), ylim=(0,100))
plt.show()#x0,t0,tf,v
if 0:
#messing around with moving point loads
a = MovingPointLoads([0.,-3,-4,-5],[20.,15,10,10 ])
a.animateme(x0=20,t0=0,tf=15,v0=-6, xlim = (-10,20), ylim=(0,100))
plt.show()#x0,t0,tf,v
if 0:
#messing around with moving point loads
a = MovingPointLoads([0.,-3,-4,-5],[20.,15,10,10 ])
a.plot_specbeam_PolyLines(ax=None, v=-2, L=15, t0=0)
plt.show()#x0,t0,tf,v
if 1:
#messing around with moving point loads
a = MovingPointLoads(x=[0.,-3,-4,-5],p=[20.,15,10,10 ],v=3,L=10,t0=0)
fig, ax = plt.subplots()
for t in [0,2,3,6,12]:
x,p = a.point_loads_on_beam(t)
ax.plot(x,p,marker='o',linestyle="None",label= "{}".format(t))
ax.set_title("L={},v={}".format(a.L,a.v))
ax.set_ylim(0,100)
plt.show()
if 0:
#messing around with moving point loads
a = MovingPointLoads(x=[0.,-3,-4,-5],p=[20.,15,10,10 ],v=-3,L=10,t0=0)
fig, ax = plt.subplots()
for t in [0,2,3,6,12]:
x,p = a.point_loads_on_beam(t)
ax.plot(x,p,marker='o',linestyle="None",label= "{}".format(t))
ax.set_title("L={},v={}".format(a.L,a.v))
ax.set_ylim(0,100)
plt.show()
############################################
# Misc code that Rohan has forgotten the use
# of but was no doubt useful at the time.
############################################
if 0:
#k3!=0 so use the numberical, not sure
start_time0 = time.time()
ftime = datetime.datetime.fromtimestamp(start_time0).strftime('%Y-%m-%d %H%M%S')
with open(ftime + ".txt", "w") as f:
f.write(ftime + os.linesep)
fig=plt.figure()
ax = fig.add_subplot("111")
ax.set_xlabel("time, s")
ax.set_ylabel("w, m")
ax.set_xlim(3.5, 4.5)
for v in [50,75, ]:
vstr="nterms"
vfmt="{}"
start_time1 = time.time()
f.write("*" * 70 + os.linesep)
vequals = "{}={}".format(vstr, vfmt.format(v))
f.write(vequals + os.linesep); print(vequals)
pdict = OrderedDict(
E = 6.998*1e9, #Pa
rho = 2373, #kg/m3
L = 160, #m
v_norm=0.01165,
kf=5.41e-4,
Fz_norm=1.013e-4,
mu_norm=39.263,
k1_norm=97.552,
k3_norm=2.497e6,
nterms=v,
BC="CC",
nquad=20,
)
t = np.linspace(0, 4.5, 400)
f.write(repr(pdict) + os.linesep)
for BC in ["SS"]:# "CC", "FF"]:
pdict["BC"] = BC
a = SpecBeam(**pdict)
a.calulate_qk(t=t)
x = t
y = a.wofx(x_norm=0.5, normalise_w=False)
ax.plot(x, y, label="x=0.5, {}".format(vequals))
end_time1 = time.time()
elapsed_time = (end_time1 - start_time1); print("Run time={}".format(str(timedelta(seconds=elapsed_time))))
f.write("Run time={}".format(str(timedelta(seconds=elapsed_time))) + os.linesep)
leg = ax.legend()
leg.draggable()
end_time0 = time.time()
elapsed_time = (end_time0 - start_time0); print("Total run time={}".format(str(timedelta(seconds=elapsed_time))))
f.write("Total run time={}".format(str(timedelta(seconds=elapsed_time))) + os.linesep)
plt.savefig(ftime+".pdf")
plt.show()
if 0:
#Journal paper figure, convergence with step change
FIGURE_convergence_step_change(v_nterms=(50,75,150,200),saveas="convergence_step_change")#75,150,200))
if 0:
#Journal paper figure, vary velocitiesconvergence with step change
# FIGURE_convergence_step_change(v_nterms=(50,75,150),saveas="convergence_step_change")#75,150,200))
FIGURE_defl_envelope_vary_velocity(
velocities_norm=(0.01165/2, 0.01165, 0.01165*2, 0.01165*4, 0.01165*10/1.13),
saveas=None,
nterms=150,
nx=200, nt=200,
force_calc=True)
if 0:
#Journal paper figure, length of transition zone
FIGURE_DAF_transition_length(saveas=None,k_ratio=4,beta=0.1,nterms=100,force_calc=True)
if 0:
FIGURE_DAF_transition_length_esveld(saveas=None,k_ratio=40,beta=0.05,nterms=100,force_calc=True)
if 0:
betas=[0.05, 0.1, 0.5, 1, 1.3]
kratios=[1,2,5,10,20,100,1000,10000]
betas=[1, 1.3]
kratios=[1,2,5,10,20,100,1000,10000]
# betas=[0.05]
# kratios=[40]
for beta in betas:
for kratio in kratios:
saveas="DAFwrtinf_t_len_bet{}_krat{}".format(beta,kratio)
# saveas=None
FIGURE_DAFwrtinf_transition_length(
saveas=saveas,k_ratio=kratio,beta=beta,nterms=150,force_calc=False)
if 0:
betas=[0.05, 0.1, 0.5, 1, 1.3]
kratios=[1,2,5,10,20,100,1000,10000]
betas=[1, 1.3]
# kratios=[1,2,5,10,20,100,1000,10000]
# betas=[0.05]
# kratios=[40]
for beta in betas:
for kratio in kratios:
saveas="revDAFwrtinf_t_len_bet{}_krat{}".format(beta,kratio)
# saveas=None
FIGURE_DAFwrtinf_transition_lengthrev(
saveas=saveas,k_ratio=kratio,beta=beta,nterms=150,force_calc=False)
