/
piecewise_uniform.py
297 lines (259 loc) · 12.7 KB
/
piecewise_uniform.py
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
"""
Piecewise uniform model for exponentially tapered circular rods
Author: Michal K. Kalkowski
M.Kalkowski@soton.ac.uk
Copyright (c) 2016 by Michal K. Kalkowski
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included
in all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS
OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL
THE AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING
FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS
IN THE SOFTWARE.
Anyone who uses/copies/reuses/modifies/merges/publishes this code
is asked to give attribution to the author by citing the following publication:
Michał K. Kalkowski, Jen M. Muggleton, Emiliano Rustighi, An experimental approach for the determination of axial and flexural wavenumbers in circular exponentially tapered bars, Journal of Sound and Vibration, Volume 390, 3 March 2017, Pages 67-85, ISSN 0022-460X, http://dx.doi.org/10.1016/j.jsv.2016.10.018.
(http://www.sciencedirect.com/science/article/pii/S0022460X1630551X)
"""
from __future__ import print_function
import numpy as np
def piecewise_uniform_beam(f, E, nu, rho, R0, L, beta, n_el, x_Ls, G=0,
kappa='a', nat=False, nat_only=False):
"""
Calculates the response of an exponentially tapered circular beam
according to the piecewise uniform model
abd the Timoshenko beam theory.
Parameters:
-----------
f : frequency vector (1D array)
E : Young's modulus
nu : Poisson's ratio
rho : density
R0 : starting radius of the tapered beam
L : length of the beam
beta : flare constant
n_el : number of elements in the piecewise formulation
x_Ls : locations at which the response is requested (1D array)
G : shear modulus (if not related to E via Poisson's ratio)
kappa: Timoshenko shear correction factor
(if different from the standard one)
nat : boolean, specifies whether the natural frequencies
should also be computed
nat_only : boolean, if True, only natural frequencies are computed (no FRF)
Returns:
--------
resp : receptances at the desired locations
nat_frq : natural frequencies, if requested
or
nat_frq, if nat_only is True
"""
def matmul_tup(matrices):
"""
Helper function for numpy.matmul. Allows for performing a matrix
multiuplication on a series
of matrices. As in matmul, the multiplication is performed on
the last two dimensions.
Parameters:
-----------
matrices : list of matrices to multiply
"""
for i in range(len(matrices) - 1):
if i == 0:
product = np.matmul(matrices[0], matrices[1])
else:
product = np.matmul(product, matrices[i + 1])
return product
if G == 0:
G = E/(2*(1 + nu))
if kappa == 'a':
kappa = 6*(1 + nu)/(7 + 6*nu)
# configure piecewise elements
x0_discrete = np.linspace(0, L, n_el, False)
element_length = x0_discrete[1] - x0_discrete[0]
mid_x_discrete = np.linspace(element_length/2, L-element_length/2,
n_el, True)
r_discrete = R0*np.exp(-beta*mid_x_discrete)
A = np.pi*r_discrete**2
I = A*r_discrete**2/4
Cs = (G*kappa/rho)**0.5
Cb = (E*I/(rho*A.astype('complex')))**0.5
Cr = (I/A.astype('complex'))**0.5
omega = 2*np.pi*f.reshape(-1, 1)
# calculating wavenumbers
k1 = np.sqrt(omega**2/2*(1/Cs**2 + Cr**2/Cb**2) +
np.sqrt(omega**4/4*(1/Cs**2 - Cr**2/Cb**2)**2 +
omega**2/Cb**2))
k2 = np.sqrt(omega**2/2*(1/Cs**2 + Cr**2/Cb**2) -
np.sqrt(omega**4/4*(1/Cs**2 - Cr**2/Cb**2)**2 +
omega**2/Cb**2))
inds = np.where(abs(np.exp(-1j*k2*0.1)) > 1)
k2[inds[0], inds[1]] *= -1
# calculate propagation matrices
tau = np.zeros([len(f), n_el, 4, 4], 'complex')
tau[:, :, 0, 0] = np.exp(-1j*k1*element_length)
tau[:, :, 1, 1] = np.exp(-1j*k2*element_length)
tau[:, :, 2, 2] = np.exp(1j*k1*element_length)
tau[:, :, 3, 3] = np.exp(1j*k2*element_length)
# calculate P coefficients (see paper)
P1 = k1*(1 - omega**2/(k1**2*Cs**2))
P2 = k2*(1 - omega**2/(k2**2*Cs**2))
# define wave mode shapes
phi_q_pos = np.ones([len(f), n_el, 2, 2], 'complex')
phi_q_neg = np.ones([len(f), n_el, 2, 2], 'complex')
phi_f_pos = np.zeros([len(f), n_el, 2, 2], 'complex')
phi_f_neg = np.zeros([len(f), n_el, 2, 2], 'complex')
phi_q_pos[:, :, 1, 0] = -1j*P1
phi_q_pos[:, :, 1, 1] = -1j*P2
phi_q_neg[:, :, 1, 0] = 1j*P1
phi_q_neg[:, :, 1, 1] = 1j*P2
phi_f_pos[:, :, 0, 0] = G*A*kappa*1j*(P1 - k1)
phi_f_pos[:, :, 0, 1] = G*A*kappa*1j*(P2 - k2)
phi_f_pos[:, :, 1, 0] = -k1*P1*E*I
phi_f_pos[:, :, 1, 1] = -k2*P2*E*I
phi_f_neg[:, :, 0, 0] = -G*A*kappa*1j*(P1 - k1)
phi_f_neg[:, :, 0, 1] = -G*A*kappa*1j*(P2 - k2)
phi_f_neg[:, :, 1, 0] = -k1*P1*E*I
phi_f_neg[:, :, 1, 1] = -k2*P2*E*I
# calculate reflection matrices
R_L = np.zeros([len(omega), 2, 2], 'complex')
R_R = np.zeros([len(omega), 2, 2], 'complex')
scalar_L = -1/(P2[:, 0]*k2[:, 0]*(P1[:, 0] - k1[:, 0]) + \
P1[:, 0]*k1[:, 0]*(k2[:, 0] - P2[:, 0]))
R_L[:, 0, 0] = scalar_L*(-(P1[:, 0]*k1[:, 0]*(P2[:, 0] - k2[:, 0]) + \
P2[:, 0]*k2[:, 0]*(P1[:, 0] - k1[:, 0])))
R_L[:, 0, 1] = scalar_L*(-2*k2[:, 0]*P2[:, 0]*(P2[:, 0] - k2[:, 0]))
R_L[:, 1, 0] = scalar_L*(2*P1[:, 0]*k1[:, 0]*(P1[:, 0] - k1[:, 0]))
R_L[:, 1, 1] = scalar_L*(P1[:, 0]*k1[:, 0]*(P2[:, 0] - k2[:, 0]) + \
P2[:, 0]*k2[:, 0]*(P1[:, 0] - k1[:, 0]))
scalar_R = -1/(P2[:, -1]*k2[:, -1]*(P1[:, -1] - k1[:, -1]) + \
P1[:, -1]*k1[:, -1]*(k2[:, -1] - P2[:, -1]))
R_R[:, 0, 0] = scalar_R*(-(P1[:, -1]*k1[:, -1]*(P2[:, -1] - k2[:, -1]) + \
P2[:, -1]*k2[:, -1]*(P1[:, -1] - k1[:, -1])))
R_R[:, 0, 1] = scalar_R*(-2*k2[:, -1]*P2[:, -1]*(P2[:, -1] - k2[:, -1]))
R_R[:, 1, 0] = scalar_R*(2*P1[:, -1]*k1[:, -1]*(P1[:, -1] - k1[:, -1]))
R_R[:, 1, 1] = scalar_R*(P1[:, -1]*k1[:, -1]*(P2[:, -1] - k2[:, -1]) + \
P2[:, -1]*k2[:, -1]*(P1[:, -1] - k1[:, -1]))
# calculate scattering matrices at the junctions as T = inv(Ta).dot(Tb)
phi_q_pos_1 = phi_q_pos[:, :-1, :, :]
phi_q_neg_1 = phi_q_neg[:, :-1, :, :]
phi_f_pos_1 = phi_f_pos[:, :-1, :, :]
phi_f_neg_1 = phi_f_neg[:, :-1, :, :]
phi_q_pos_2 = phi_q_pos[:, 1:, :, :]
phi_q_neg_2 = phi_q_neg[:, 1:, :, :]
phi_f_pos_2 = phi_f_pos[:, 1:, :, :]
phi_f_neg_2 = phi_f_neg[:, 1:, :, :]
Ta = np.zeros([len(f), n_el-1, 4, 4], 'complex')
Ta[:, :, :2, :2] = phi_q_pos_1
Ta[:, :, :2, 2:] = -phi_q_neg_2
Ta[:, :, 2:, :2] = phi_f_pos_1
Ta[:, :, 2:, 2:] = -phi_f_neg_2
Tb = np.zeros([len(f), n_el-1, 4, 4], 'complex')
Tb[:, :, :2, :2] = -phi_q_neg_1
Tb[:, :, :2, 2:] = phi_q_pos_2
Tb[:, :, 2:, :2] = -phi_f_neg_1
Tb[:, :, 2:, 2:] = phi_f_pos_2
Tainv = np.linalg.inv(Ta)
T = np.matmul(Tainv, Tb)
# calculate excited wave amplitudes (Q=1, M=0)
scalar = 1/(k2*P2*(P1 - k1) + k1*P1*(k2 - P2))[:, 0]
e_pos_1 = scalar*-k2[:, 0]*P2[:, 0]*(-1j*1/G/A[0]/kappa)
e_pos_2 = scalar*k1[:, 0]*P1[:, 0]*(-1j*1/G/A[0]/kappa)
e_pos = np.c_[e_pos_1, e_pos_2]
# set up the piecewise unform model (see Appendix B of the paper)
Tau = tau[:, :, :2, :2]
ident1 = np.dstack(n_el*[np.eye(2)]).T
ident2 = np.dstack(len(f)*[np.eye(2)]).T
ident = np.tile(ident1, (len(f), 1, 1, 1))
Tau_prev = np.concatenate((np.zeros([len(f), 1, 2, 2], 'complex'),
tau[:, :-1, :2, :2]), axis=1)
Tau_next = np.concatenate((tau[:, 1:, :2, :2],
np.zeros([len(f), 1, 2, 2], 'complex')), axis=1)
RLs = np.zeros([len(f), n_el, 2, 2], 'complex')
RRs = np.zeros([len(f), n_el, 2, 2], 'complex')
TLs = np.zeros([len(f), n_el, 2, 2], 'complex')
TRs = np.zeros([len(f), n_el, 2, 2], 'complex')
RLs[:, 0] = R_L
RLs[:, 1:] = T[:, :, 2:, 2:]
RRs[:, -1] = R_R
RRs[:, :-1] = T[:, :, :2, :2]
TLs[:, 1:] = T[:, :, 2:, :2]
TRs[:, :-1] = T[:, :, :2, 2:]
A_pos = np.linalg.inv(ident - matmul_tup((RLs, Tau, RRs, Tau)))
C_pos = matmul_tup((A_pos, TLs, Tau_prev))
D_pos = matmul_tup((A_pos, RLs, Tau, TRs, Tau_next))
B_neg = np.linalg.inv(ident - matmul_tup((RRs, Tau, RLs, Tau)))
A_neg = matmul_tup((B_neg, RRs, Tau))
C_neg = matmul_tup((B_neg, RRs, Tau, TLs, Tau_prev))
D_neg = matmul_tup((B_neg, TRs, Tau_next))
H_neg = np.zeros([len(f), n_el, 2, 2], 'complex')
H_neg[:, -1] = matmul_tup((RRs[:, -1], Tau[:, -1], C_pos[:, -1]))
H_pos = np.zeros([len(f), n_el, 2, 2], 'complex')
H_pos[:, -1] = C_pos[:, -1]
# perform the piecewise uniform reduction
for i in range(n_el-2, 0, -1):
H_pos[:, i] = np.matmul(np.linalg.inv(ident2 - np.matmul(D_pos[:, i],
H_neg[:, i + 1])), C_pos[:, i])
H_neg[:, i] = C_neg[:, i] + matmul_tup((D_neg[:, i], H_neg[:, i + 1],
H_pos[:, i]))
# if only natural frequencies are requested, calculate and return them
if nat_only:
nat_frq = np.linalg.det(np.matmul(D_pos[:, 0], H_neg[:, 1]) - ident2)
return nat_frq
# otherwise, start calculating travelling waves in the elements (element-by-element)
else:
a_pos = np.zeros([len(f), n_el, 2, 1], 'complex')
a_neg = np.zeros([len(f), n_el, 2, 1], 'complex')
a_pos[:, 0] = matmul_tup((np.linalg.inv(ident2 - np.matmul(D_pos[:, 0],
H_neg[:, 1])), A_pos[:, 0], e_pos.reshape(len(f), 2, 1)))
a_neg[:, 0] = np.matmul(A_neg[:, 0], e_pos.reshape(len(f), 2, 1)) + \
matmul_tup((D_neg[:, 0], H_neg[:, 1], a_pos[:, 0]))
for i in range(1, n_el):
if i == n_el - 1:
a_pos[:, i] = np.matmul(H_pos[:, i], a_pos[:, i - 1])
a_neg[:, i] = matmul_tup((R_R, tau[:, -1, :2, :2], a_pos[:, i]))
else:
a_pos[:, i] = np.matmul(H_pos[:, i], a_pos[:, i - 1])
a_neg[:, i] = np.matmul(H_neg[:, i], a_pos[:, i - 1])
section_inds = []
section_loc = []
# calculate the receptance at desired locations in (x_Ls)
for x_L in x_Ls:
# identify in which piecewise uniform are locations in x_Ls
where_is = np.argmin(abs(x0_discrete - x_L))
if x_L < x0_discrete[where_is] or \
x0_discrete[where_is] == element_length:
section_inds.append(where_is - 1)
section_loc.append(x_L - x0_discrete[where_is - 1])
else:
section_inds.append(where_is)
section_loc.append(x_L - x0_discrete[where_is])
# calculate the response according to relevant propagation matrices
resp = np.zeros([len(f), len(section_inds)], 'complex')
for i, j in enumerate(section_inds):
tau_pos = np.zeros([len(f), 2, 2], 'complex')
tau_neg = np.zeros([len(f), 2, 2], 'complex')
tau_pos[:, 0, 0] = np.exp(-1j*k1[:, j]*section_loc[i])
tau_pos[:, 1, 1] = np.exp(-1j*k2[:, j]*section_loc[i])
tau_neg[:, 0, 0] = np.exp(-1j*k1[:, j]*(element_length -
section_loc[i]))
tau_neg[:, 1, 1] = np.exp(-1j*k2[:, j]*(element_length -
section_loc[i]))
resp[:, i] = (matmul_tup((phi_q_pos[:, j, :, :],
tau_pos, a_pos[:, j])) + \
matmul_tup((phi_q_neg[:, j, :, :],
tau_neg, a_neg[:, j])))[:, 0].squeeze()
# if natural frequencies are also of interest
if nat:
nat_frq = np.linalg.det(np.matmul(D_pos[:, 0],
H_neg[:, 1]) - ident2)
return resp, nat_frq
else:
return resp