[英]Runge Kutta 4 and pendulum simulation in python
我正在嘗試制作一個使用Runge kutta 4繪制擺擺的python程序。我擁有的方程是角加速度= -(m*g*r/I) * np.sin(y)
。
請找到我的代碼。 我是python的新手。
import numpy as np
import matplotlib.pyplot as plt
m = 3.0
g = 9.8
r = 2.0
I = 12.0
h = 0.0025
l=2.0
cycle = 10.0
t = np.arange(0, cycle, h)
n = (int)((cycle)/h)
initial_angle = 90.0
y=np.zeros(n)
v=np.zeros(n)
def accel(theta):
return -(m*g*r/I)*np.sin(theta)
y[0] = np.radians(initial_angle)
v[0] = np.radians(0.0)
for i in range(0, n-1):
k1y = h*v[i]
k1v = h*accel(y[i])
k2y = h*(v[i]+0.5*k1v)
k2v = h*accel(y[i]+0.5*k1y)
k3y = h*(v[i]+0.5*k2v)
k3v = h*accel(y[i]+0.5*k2y)
k4y = h*(v[i]+k3v)
k4v = h*accel(y[i]+k3y)
y[i+1] = y[i] + (k1y + 2 * k2y + 2 * k3y + k4y) / 6.0
v[i+1] = v[i] + (k1v + 2 * k2v + 2 * k3v + k4v) / 6.0
plt.plot(t, y)
plt.title('Pendulum Motion:')
plt.xlabel('time (s)')
plt.ylabel('angle (rad)')
plt.grid(True)
plt.show()
我現在更新了代碼。 我正正弦。 謝謝..
您已應用RK4步驟,就好像在求解一階方程式一樣。 您需要將二階方程轉換為一階系統,然后求解該耦合系統。
v = dy/dt
acceleration = dv/dt
因此RK4的每個步驟都有兩個組成部分
k1y = h*v
k1v = h*accel(y)
k2y = h*(v+0.5*k1v)
k2v = h*accel(y+0.5*k1y)
等等
完整的代碼給出了一個很好的正弦波
import numpy as np
import matplotlib.pyplot as plt
m = 3.0
g = 9.8
r = 2.0
I = 12.0
h = 0.0025
l=2.0
cycle = 10.0
t = np.arange(0, cycle, h)
# step height h
n = len(t)
initial_angle = 90.0
y=np.zeros(n)
v=np.zeros(n)
def accel(theta): return -(m*g*r/I)*np.sin(theta)
y[0] = np.radians(initial_angle)
v[0] = np.radians(0.0)
for i in range(0, n-1):
k1y = h*v[i]
k1v = h*accel(y[i])
k2y = h*(v[i]+0.5*k1v)
k2v = h*accel(y[i]+0.5*k1y)
k3y = h*(v[i]+0.5*k2v)
k3v = h*accel(y[i]+0.5*k2y)
k4y = h*(v[i]+k3v)
k4v = h*accel(y[i]+k3y)
# Update next value of y
y[i+1] = y[i] + (k1y + 2 * k2y + 2 * k3y + k4y) / 6.0
v[i+1] = v[i] + (k1v + 2 * k2v + 2 * k3v + k4v) / 6.0
plt.plot(t, y)
plt.title('Pendulum Motion:')
plt.xlabel('time (s)')
plt.ylabel('angle (rad)')
plt.grid(True)
plt.show()
import numpy as np
import matplotlib.pyplot as plt
m = 3.0
g = 9.8
r = 2.0
I = 12.0
h = 0.0025
l=2.0
cycle = 10.0
t = np.arange(0, cycle, h)
# step height h
n = len(t)
initial_angle = 90.0
y=np.zeros(n)
v=np.zeros(n)
def accel(theta): return -(m*g*r/I)*np.sin(theta)
y[0] = np.radians(initial_angle)
v[0] = np.radians(0.0)
for i in range(0, n-1):
k1y = h*v[i]
k1v = h*accel(y[i])
k2y = h*(v[i]+0.5*k1v)
k2v = h*accel(y[i]+0.5*k1y)
k3y = h*(v[i]+0.5*k2v)
k3v = h*accel(y[i]+0.5*k2y)
k4y = h*(v[i]+k3v)
k4v = h*accel(y[i]+k3y)
# Update next value of y
y[i+1] = y[i] + (k1y + 2 * k2y + 2 * k3y + k4y) / 6.0
v[i+1] = v[i] + (k1v + 2 * k2v + 2 * k3v + k4v) / 6.0
plt.plot(t, y)
plt.title('Pendulum Motion:')
plt.xlabel('time (s)')
plt.ylabel('angle (rad)')
plt.grid(True)
plt.show()
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