利用python求解物理學中的雙彈簧質能系統詳解

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物理的模型如下：

在這個系統里有兩個物體，它們的質量分別是m1和m2，被兩個彈簧連接在一起，伸縮系統為k1和k2，左端固定。假定沒有外力時，兩個彈簧的長度為L1和L2。

由于兩物體有重力，那么在平面上形成摩擦力，那么摩擦系數分別為b1和b2。所以可以把微分方程寫成這樣：

這是一個二階的微分方程，為了使用python來求解，需要把它轉換為一階微分方程。所以引入下面兩個變量：

這兩個相當于運動的速度。通過運算可以改為這樣：

這時可以線性方程改為向量數組的方式，就可以使用python定義了

代碼如下：    
# Use ODEINT to solve the differential equations defined by the vector field
from scipy.integrate import odeint
 
def vectorfield(w, t, p):
 """
 Defines the differential equations for the coupled spring-mass system.
 
 Arguments:
  w : vector of the state variables:
     w = [x1,y1,x2,y2]
  t : time
  p : vector of the parameters:
     p = [m1,m2,k1,k2,L1,L2,b1,b2]
 """
 x1, y1, x2, y2 = w
 m1, m2, k1, k2, L1, L2, b1, b2 = p
 
 # Create f = (x1',y1',x2',y2'):
 f = [y1,
   (-b1 * y1 - k1 * (x1 - L1) + k2 * (x2 - x1 - L2)) / m1,
   y2,
   (-b2 * y2 - k2 * (x2 - x1 - L2)) / m2]
 return f
 
# Parameter values
# Masses:
m1 = 1.0
m2 = 1.5
# Spring constants
k1 = 8.0
k2 = 40.0
# Natural lengths
L1 = 0.5
L2 = 1.0
# Friction coefficients
b1 = 0.8
b2 = 0.5
 
# Initial conditions
# x1 and x2 are the initial displacements; y1 and y2 are the initial velocities
x1 = 0.5
y1 = 0.0
x2 = 2.25
y2 = 0.0
 
# ODE solver parameters
abserr = 1.0e-8
relerr = 1.0e-6
stoptime = 10.0
numpoints = 250
 
# Create the time samples for the output of the ODE solver.
# I use a large number of points, only because I want to make
# a plot of the solution that looks nice.
t = [stoptime * float(i) / (numpoints - 1) for i in range(numpoints)]
 
# Pack up the parameters and initial conditions:
p = [m1, m2, k1, k2, L1, L2, b1, b2]
w0 = [x1, y1, x2, y2]
 
# Call the ODE solver.
wsol = odeint(vectorfield, w0, t, args=(p,),
    atol=abserr, rtol=relerr)
 
with open('two_springs.dat', 'w') as f:
 # Print & save the solution.
 for t1, w1 in zip(t, wsol):   
  out = '{0} {1} {2} {3} {4}\n'.format(t1, w1[0], w1[1], w1[2], w1[3]);
  print(out)
  f.write(out);

在這里把結果輸出到文件two_springs.dat，接著寫一個程序來把數據顯示成圖片，就可以發表論文了，代碼如下：    
# Plot the solution that was generated
 
from numpy import loadtxt
from pylab import figure, plot, xlabel, grid, hold, legend, title, savefig
from matplotlib.font_manager import FontProperties
 
t, x1, xy, x2, y2 = loadtxt('two_springs.dat', unpack=True)
 
figure(1, figsize=(6, 4.5))
 
xlabel('t')
grid(True)
lw = 1
 
plot(t, x1, 'b', linewidth=lw)
plot(t, x2, 'g', linewidth=lw)
 
legend((r'$x_1$', r'$x_2$'), prop=FontProperties(size=16))
title('Mass Displacements for the\nCoupled Spring-Mass System')
savefig('two_springs.png', dpi=100)

最后來查看一下輸出的png圖片如下：

總結

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