我在 python 中使用 odeint 來求解微分方程，為什么我的質量的 position 發散到無窮大？

Question

對於一個涉及三個彈簧和兩個質量的問題，我已經獲得了我的運動方程。 每側的兩個彈簧是非線性的，而中間的 spring 是線性的。 它們也沒有質量。 看圖片清楚

我還把它放在了一個能夠求解二階微分方程的代碼中，我的代碼如下：

   import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
initial=[-5,0,5,0] # [x,xdot,x2,x2dot]
t = np.linspace(0,5,10000) # Creating a vector which will represent time. 10 seconds divided in to 10000 intervals

def func(initials,t): # Defining function which is used in the odeint for solving diff.eq
    m1=5                    #Mass of M_1
    m2=5                    #Mass of M_2
    k12=10                  #Spring constant of spring connected and inbetween M_1 and M_2
    k1=10                   #Spring constant of spring connected to M_1
    k2=10                   #Spring constant of spring connected to M_2
    c1=2                    #constant of C for eq related to M_1
    c2=2                    #constant of C for eq related to M_2
    L1=5                    #Length of spring connected to M_1
    L2=5                    #Length of spring connected to M_2
    x1=initials[0]         #Initial values as chosen in row 5
    x2=initials[2]           #Initial values as chosen in row 5
    x1dotdot=(-k12*(x1-x2)/m1)-(k1*x1/m1)+(c1*2*np.pi/(L1*m1)*np.sin(2*np.pi*x1/L1))
    x2dotdot=(k12/m2*(x1-x2))-(k2*x2/m2)+(c2*2*np.pi/(L2*m2)*np.sin(2*np.pi*x2/L2))
    return(initials[1],x1dotdot,initials[3],x2dotdot)
output = odeint(func,initial,t)


plt.plot(t,output[:,0],'g:',linewidth = 2, label = 'M_1')
plt.plot(t,output[:,2],'y:',linewidth = 2, label = 'M_2')

plt.legend()
plt.xlabel('Time')
plt.ylabel('Velocities of the masses M_1 and M_2')
plt.show()

這里的問題是，當我通過返回首字母（1）和首字母（3）選擇 plot速度時，它們以初始值 -5 和 5 而不是我從一開始就給它們的 0 開始。 當我 plot 位置時，我期望 -5 和 5。

另一個問題也是職位。 當我選擇 plot 通過返回 initials(0) 和 initials(2) 他們發散到無窮大。

我不知道如何理解這一點。

Answer 1

我認為您唯一的問題是混淆從odeint返回的output中會發生什么。 它將按照定義的順序返回 state 向量（此處： X = [x1, x1dot, x2, x2dot] ） - 而不是 state 向量的導數。

我在下面添加了一些額外的標簽/評論，希望能夠澄清，並返回以下內容

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

# State vector: X = [x1, x1dot, x2, x2dot]
initial=[-5,0,5,0] # [x,xdot,x2,x2dot]
t = np.linspace(0,5,10000) # Creating a vector which will represent time. 10 seconds divided in to 10000 intervals

def func(X,t): # Defining function which is used in the odeint for solving diff.eq
    m1=5                    #Mass of M_1
    m2=5                    #Mass of M_2
    k12=10                  #Spring constant of spring connected and inbetween M_1 and M_2
    k1=10                   #Spring constant of spring connected to M_1
    k2=10                   #Spring constant of spring connected to M_2
    c1=2                    #constant of C for eq related to M_1
    c2=2                    #constant of C for eq related to M_2
    L1=5                    #Length of spring connected to M_1
    L2=5                    #Length of spring connected to M_2
    x1=X[0]                 #Current value of x1
    x2=X[2]                 #Current value of x2
    x1dotdot=(-k12*(x1-x2)/m1)-(k1*x1/m1)+(c1*2*np.pi/(L1*m1)*np.sin(2*np.pi*x1/L1))
    x2dotdot=(k12/m2*(x1-x2))-(k2*x2/m2)+(c2*2*np.pi/(L2*m2)*np.sin(2*np.pi*x2/L2))

    # Need to return derivate of State vector X,
    # X    = [x1,    x1dot,    x2,    x2dot]
    # Xdot = [x1dot, x1dotdot, x2dot, x2dotdot]
    return(X[1], x1dotdot, X[3], x2dotdot)

output = odeint(func,initial,t)

fig, (ax1, ax2) = plt.subplots(2 ,1)

# Positions are in columns 0 and 2, X = [x1, x1dot, x2, x2dot]
ax1.plot(t,output[:,0],'g:',linewidth = 2, label = 'M_1')
ax1.plot(t,output[:,2],'y:',linewidth = 2, label = 'M_2')
ax1.set_xlabel('Time')
ax1.set_ylabel('Positions of the masses')
ax1.legend()

# Velocities are in columns 1 and 3, X = [x1, x1dot, x2, x2dot]
ax2.plot(t,output[:,1],'g:',linewidth = 2, label = 'M_1')
ax2.plot(t,output[:,3],'y:',linewidth = 2, label = 'M_2')
ax2.set_xlabel('Time')
ax2.set_ylabel('Velocities of the masses')
ax2.legend()

plt.show()