How to Fix Index Error in Differential Equation?

Question

I am trying to create a program that solves the mass-spring-damper system using backward differentiating, the only problem is that I am running into an index error that I am not sure how to solve:

import numpy as np 
import matplotlib.pyplot as plt

def MSD_Solver(m,b,K):
    #input: m = mass, b = damping ratio, K = spring constant
    #output: (t,x) time vs position

    tinitial = 0
    tfinal = 15
    step = .005

    t = np.linspace(tinitial,tfinal,step)

    x = np.zeros_like(t)

    x[0]=0
    x[1]=0
    for k in range (len(t)-1):            # extra element so subtract by 1
        x[k] = (t**2)/((m+b)*t+(t**2)*k) + (x[k-2](-m))/((m+b)*t+(t**2)*k) + (x[k-1]((2*m)+(b*t)))/((m+b)*t+(t**2)*k)
    return plt.plot(t,x)

print(MSD_Solver(1,.5,5)),MSD_Solver(1,1,5),MSD_Solver(1,2,5)
plt.show()

Answer 1

The linspace doc shows that the third argument is the number of items, not the step. Your step value got truncated to 0, so the returned array for t was empty. As a result, x has no elements, and x[0] is out of range.

Try this:

tinitial = 0
tfinal = 15
step = .005
num = (tfinal - tinitial) / step + 1

t = np.linspace(tinitial,tfinal,num)

This will get you to the semantic errors in your complex computation.

Answer 2

You want, probably(?), use first and second order difference quotients to discretize

m*x''(t) + b*x'(t) + K*x(t) = 1

to

m*(x[j+1]-2*x[j]+x[j-1]) + 0.5*dt*b*(x[j+1]-x[j-1]) + dt^2*K*x[j] = dt**2

so that

x[j+1] = ( dt**2 + (2*m-K*dt**2)*x[j] - (m-0.5*dt*b)*x[j-1] ) / (m+0.5*dt*b)

In code

def MSD_Solver(m,b,K):
    #input: m = mass, b = damping ratio, K = spring constant
    #output: (t,x) time vs position

    tinitial = 0
    tfinal = 15
    step = .005

    t = np.arange(tinitial,tfinal,step)
    x = np.zeros_like(t)

    dt = t[1]-t[0]  # use the actual time step
    x[0:2] = [ 0, 0]
    for j in range(1,len(t)-1):
        x[j+1] = ( dt**2 + (2*m-K*dt**2)*x[j] - (m-0.5*dt*b)*x[j-1] ) / (m+0.5*dt*b)
    return t,x

t,x = MSD_Solver(1,.5,5)        
plt.plot(t,x); plt.show();