Problem 2
(see Ex 9.6)
# Remove
import ema as em
import matplotlib.pyplot as plt
import numpy as np
%config InlineBackend.figure_format = 'svg' # used to make plots look nicer
from ema_examples.dynamics import E09_06
from ema.utilities.ipyutils import disp_sbs
from scipy.linalg import eigL = 1
mass = 1
EI = 1/12
mdl = E09_06(L=L, m = mass, EI = EI)
fig, ax = plt.subplots()
em.plot_structure(mdl, ax)
mdl.DOF[[7, 8, 9], [1, 2, 3], [4, 5, 6]]
m, K = em.Mass_matrix(mdl), em.K_matrix(mdl)
m[0,0] = 3.*mass
m[1,1] = 1.*mass
m[2:,2:] = 0*mass
disp_sbs(m.f.df, K.f.df)| $u_{{1}}$ | $u_{{2}}$ | $u_{{3}}$ | $u_{{4}}$ | $u_{{5}}$ | $u_{{6}}$ | |
|---|---|---|---|---|---|---|
| $u_{{1}}$ | 3.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| $u_{{2}}$ | 0.0 | 1.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| $u_{{3}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| $u_{{4}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| $u_{{5}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| $u_{{6}}$ | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | |
|---|---|---|---|---|---|---|
| $P_{1}$ | 8334.333333 | 0.000000 | 0.500000 | -8333.333333 | 0.0 | 0.000000 |
| $P_{2}$ | 0.000000 | 8334.333333 | 0.500000 | 0.000000 | -1.0 | 0.500000 |
| $P_{3}$ | 0.500000 | 0.500000 | 0.666667 | 0.000000 | -0.5 | 0.166667 |
| $P_{4}$ | -8333.333333 | 0.000000 | 0.000000 | 8333.333333 | 0.0 | 0.000000 |
| $P_{5}$ | 0.000000 | -1.000000 | -0.500000 | 0.000000 | 1.0 | -0.500000 |
| $P_{6}$ | 0.000000 | 0.500000 | 0.166667 | 0.000000 | -0.5 | 0.333333 |
# k, m = em.analysis.StaticCondensation(k.f, m.f, idxs=[3,4])
k = em.analysis.kStaticCondensation(K.f, idxs=[3,4])
# disp_sbs(k.df)
k| $4$ | $5$ | |
|---|---|---|
| $4$ | 0.571384 | 0.214267 |
| $5$ | 0.214267 | 0.142849 |
m = m[0:2,0:2]
freq2, Phi = eig(m, k)
Phi/0.57293852array([[-1. , -0.75123369],
[-1.4305171 , 1.57544505]])M = Phi.T@m@Phi
K = Phi.T@k@Phi
MStructural_Vector([[ 1.65651710e+00, -1.11022302e-16],
[ 0.00000000e+00, 1.37050677e+00]])a) Modal expansion of earthquake forces
\[\mathrm{p}_{\mathrm{eff}}(t)=-\mathrm{m} \iota \ddot{u}_{g}(t)\]
\[\mathbf{m} \iota=\sum_{n=1}^{N} \mathbf{s}_{n}=\sum_{n=1}^{N} \Gamma_{n} \mathbf{m} \phi_{n}\]
I = N = 2
iota = np.array([0, 1])
Ln = np.array([sum(Phi.T[n,i]*sum(m[i,j]*iota[j] for j in range(I)) for i in range(I)) for n in range(N)])
Ln = Phi.T@m@iota
LnStructural_Vector([-0.81959835, 0.90263316])gamma = np.array([Ln[n]/M[n,n] for n in range(N)])
gammaarray([-0.49477204, 0.65861269])sn = [gamma[n]*(m@Phi[:,n]) for n in range(N)]
sn[Structural_Vector([0.85042189, 0.40551435]),
Structural_Vector([-0.85042189, 0.59448565])]