利用python求解物理学中的双弹簧质能系统详解

2020-01-04 16:32:50

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前言

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物理的模型如下：

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在这个系统里有两个物体，它们的质量分别是m1和m2，被两个弹簧连接在一起，伸缩系统为k1和k2，左端固定。假定没有外力时，两个弹簧的长度为L1和L2。

由于两物体有重力，那么在平面上形成摩擦力，那么摩擦系数分别为b1和b2。所以可以把微分方程写成这样：

实验物理,python,python计算物理,python物理引擎

这是一个二阶的微分方程，为了使用python来求解，需要把它转换为一阶微分方程。所以引入下面两个变量：

实验物理,python,python计算物理,python物理引擎

这两个相当于运动的速度。通过运算可以改为这样：

实验物理,python,python计算物理,python物理引擎

这时可以线性方程改为向量数组的方式，就可以使用python定义了

代码如下：

# 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);

在这里把结果输出到文件two_springs.dat，接着写一个程序来把数据显示成图片，就可以发表论文了，代码如下：

# 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)

最后来查看一下输出的png图片如下：

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