Integral de convolución

\begin{equation*} u = \frac{1}{m \omega_{d}} \int_{0}^{t} F(\tau) \, e^{-\beta \omega \bigl( t - \tau \bigr)} \sin \big[ \omega_{d} \bigl( t - \tau \bigr) \bigr] \, d \tau = \frac{1}{m \omega_{d}} \, F \ast G \end{equation*}

Vibración amortiguada con carga impulsiva rectangular

Usando los datos de un ejemplo anterior

import numpy as np
import matplotlib.pyplot as plt

m = 42.0/981
omega = 15.6604597634
beta = 0.05
omega_damping = omega*np.power(1-beta**2,1.0/2.0)
F0 = 2.0
t1 = 1.0

delta = 0.001
t = np.arange(3000)*delta
F = np.zeros(3000)

for i in range(3000):
    if t[i] <= t1:
        F[i] = F0
    else:
        F[i] = 0.0
        
G = np.exp(-beta*omega*t)*np.sin(omega_damping*t)

plt.figure(figsize=(19,8.5))
plt.plot(t,F)
plt.plot(t,G)
plt.xlabel(r'$t$  seg.')
plt.ylabel(r'$F_{0}$')
plt.grid(True)
plt.show()

H = np.convolve(F,G)*delta/(m*omega_damping)
H = H[:3000]

plt.figure(figsize=(19,8.5))
plt.plot(t, H)
plt.xlabel(r'$t$  seg.')
plt.ylabel(r'$u$  cm.')
plt.grid(True)
plt.show()

Vibración amortiguada con carga impulsiva senoidal

Usando los datos de un ejemplo anterior

import numpy as np
import matplotlib.pyplot as plt

m = 25.0/981
omega = 16.5734727803
beta = 0.05
omega_damping = omega*np.power(1-beta**2,1.0/2.0)
F0 = 2.0
t1 = 0.5

delta = 0.001
t = np.arange(3000)*delta
F = np.zeros(3000)

for i in range(3000):
    if t[i] <= t1:
        F[i] = F0*np.sin(np.pi*t[i]/t1)
    else:
        F[i] = 0.0
        
G = np.exp(-beta*omega*t)*np.sin(omega_damping*t)

plt.figure(figsize=(19,8.5))
plt.plot(t,F)
plt.plot(t,G)
plt.xlabel(r'$t$  seg.')
plt.ylabel(r'$F_{0}$')
plt.grid(True)
plt.show()

H = np.convolve(F,G)*delta/(m*omega_damping)
H = H[:3000]

plt.figure(figsize=(19,8.5))
plt.plot(t, H)
plt.xlabel(r'$t$  seg.')
plt.ylabel(r'$u$  cm.')
plt.grid(True)
plt.show()

Vibración amortiguada con carga impulsiva triangular simétrica

Usando los datos de un ejemplo anterior

import numpy as np
import matplotlib.pyplot as plt

m = 30.0/981
omega = 9.90454441153
beta = 0.05
omega_damping = omega*np.power(1-beta**2,1.0/2.0)
F0 = 4.5
t1 = 0.3
t2 = 0.6

delta = 0.001
t = np.arange(3000)*delta
F = np.zeros(3000)

for i in range(3000):
    if t[i] <= t1:
        F[i] = (F0*t[i])/t1
    elif t1 < t[i] <= t2:
        F[i] = -(F0*(t[i] - t1))/(t2 - t1) + F0
    else:
        0
        
G = np.exp(-beta*omega*t)*np.sin(omega_damping*t)

plt.figure(figsize=(19,8.5))
plt.plot(t,F)
plt.plot(t,G)
plt.xlabel(r'$t$  seg.')
plt.ylabel(r'$F_{0}$')
plt.grid(True)
plt.show()

H = np.convolve(F,G)*delta/(m*omega_damping)
H = H[:3000]

plt.figure(figsize=(19,8.5))
plt.plot(t, H)
plt.xlabel(r'$t$  seg.')
plt.ylabel(r'$u$  cm.')
plt.grid(True)
plt.show()