python - 使用 scipy.integrate.odeint 时如何正确实现时间因变量

Question

我正在尝试解决一个基本上类似于这个的颂歌系统，但多了一个 Spring 和阻尼器 ==> http://scipy-cookbook.readthedocs.io/items/CoupledSpringMassSystem.html

不过，我有一个小问题，因为我要实现的参数之一是时间相关的。我的第一次尝试如下:

import scipy as sci
import numpy as np
import matplotlib.pyplot as plt

def bump(t):    
    if t <= (0.25 / 6.9):
        return (0.075 * (1 - np.cos(np.pi * 8 * 6.9 * t)))
    else:
        return 0 


def membre_droite(w, t, p):
    x1,y1,x2,y2,x3,y3 = w
    m1,m2,m3,k1,k2,k3,l1,l2,l3,c2,c3 = p
    f = [y1,
         (-k1 * (x1 - l1 - bump(t)) + k2 * (x2 - x1 - l2) + c2 * (y2 - y1)) / m1,
          y2,
          (-c2 * (y2 - y1) - k2 * (x2 - x1 - l2) + k3 * (x3 - x2 - l3) + c3 * (y3 - y2)) / m2,
          y3,
          (-c3 * (y3 - y2) - k3 * (x3 - x2 - l3)) / m3]
    return f



# Initial values
x11 = 0.08
y11 = 0
x22 = 0.35
y22 = 0
x33 = 0.6
y33 = 0

# Parameters
m1 = 90
m2 = 4000
m3 = 105
k1 = 250000
k2 = 25000
k3 = 30000
l1 = 0.08
l2 = x22-x11
l3 = x33-x22
c2 = 2500
c3 = 850

# Initial paramaters regrouped + time array 
time=np.linspace(0.0, 5, 1001)
w0 = [x11,y11,x22,y22,x33,y33]
p0 = [m1,m2,m3,k1,k2,k3,l1,l2,l3,c2,c3]
x1,y1,x2,y2,x3,y3 = sci.integrate.odeint(membre_droite, w0, time, args=(p0,)).T

plt.plot(time,x1,'b')
plt.plot(time,x2,'g')
plt.plot(time,x3,'r') 
plt.plot(time,y2,'yellow')
plt.plot(time,y3,'black')                                      
plt.xlabel('t')
plt.grid(True)
plt.legend((r'$x_1$', r'$x_2$', r'$x_3$', r'$y_2$', r'$y_3$'))
plt.show()

我得到的错误是:

if t <= (0.25 / 6.9):
ValueError: The truth value of an array with more than one element is ambiguous. Use a.any() or a.all()

我寻找过类似的案例并且遇到了这个主题 ==> Solving a system of odes (with changing constant!) using scipy.integrate.odeint?

然后我尝试将我的代码调整为这种格式:

import scipy as sci
import numpy as np
import matplotlib.pyplot as plt

def bump(t):    
    if t <= (0.25 / 6.9):
        return (0.075 * (1 - np.cos(np.pi * 8 * 6.9 * t)))
    else:
        return 0


def membre_droite(w, t, bump):
    x1,y1,x2,y2,x3,y3 = w
    f = [y1,
         (-250000 * (x1 - x11 - bump(t)) + 25000 * (x2 - x1 - x22 + x11) + 2500 * (y2-y1)) / 90,
         y2,
         (-2500 * (y2 - y1) - 25000 * (x2 - x1 - x22 + x11) + 30000 * (x3 - x2 - x33 + x22) + 850 * (y3 - y2)) / 4000,
         y3,
         (-850 * (y3 - y2) - 30000 * (x3 - x2 - x33 + x22)) / 105]
    return f



# Initial values
x11 = 0.08
y11 = 0
x22 = 0.35
y22 = 0
x33 = 0.6
y33 = 0



# Initial paramaters regrouped + time array 
time = np.linspace(0.0, 5, 1001)
w0 = [x11,y11,x22,y22,x33,y33]

x1,y1,x2,y2,x3,y3 = sci.integrate.odeint(membre_droite, w0, time, args=(bump,)).T

plt.plot(time,x1,'b')
plt.plot(time,x2,'g')
plt.plot(time,x3,'r') 
plt.plot(time,y2,'yellow')
plt.plot(time,y3,'black')                                      
plt.xlabel('t')
plt.grid(True)
plt.legend((r'$x_1$', r'$x_2$', r'$x_3$', r'$y_2$', r'$y_3$'))
plt.show()

阅读前面的链接，它应该有效，但我收到另一个错误:

(-250000 * (x1 - x11 - bump(t)) + 25000 * (x2 - x1 - x22 + x11) + 2500 * (y2 - y1)) / 90,
TypeError: 'list' object is not callable

Answer 1

更改为:

from scipy.integrate import odeint 

和

x1,y1,x2,y2,x3,y3 = odeint(membre_droite, w0, time, args=(bump,)).T

完整代码:

from scipy.integrate import odeint
import numpy as np
import matplotlib.pyplot as plt

def bump(t):    
    if t <= (0.25 / 6.9):
        return (0.075 * (1 - np.cos(np.pi * 8 * 6.9 * t)))
    else:
        return 0

def membre_droite(w, t, bump):
    x1,y1,x2,y2,x3,y3 = w
    f = [y1,
         (-250000 * (x1 - x11 - bump(t)) + 25000 * (x2 - x1 - x22 + x11) + 2500 * (y2 - y1)) / 90,
         y2,
         (-2500 * (y2 - y1) - 25000 * (x2 - x1 - x22 + x11) + 30000 * (x3 - x2 - x33 + x22) + 850 * (y3 - y2)) / 4000,
         y3,
         (-850 * (y3 - y2) - 30000 * (x3 - x2 - x33 + x22)) / 105]
    return f



# Initial values
x11 = 0.08
y11 = 0
x22 = 0.35
y22 = 0
x33 = 0.6
y33 = 0



# Initial paramaters regrouped + time array 
time = np.linspace(0.0, 5, 1001)
w0 = [x11,y11,x22,y22,x33,y33]

x1,y1,x2,y2,x3,y3 = odeint(membre_droite, w0, time, args=(bump,)).T

plt.plot(time,x1,'b')
plt.plot(time,x2,'g')
plt.plot(time,x3,'r') 
plt.plot(time,y2,'yellow')
plt.plot(time,y3,'black')                                      
plt.xlabel('t')
plt.grid(True)
plt.legend((r'$x_1$', r'$x_2$', r'$x_3$', r'$y_2$', r'$y_3$'))
plt.show()