Question

B A o

The bar ABC is attached to the vertical rod with a horizontal pin. The essembly is free to rotate about the axis of the rod. in the absence of friction, the equations of motion of the system are

$\phi = -2\theta \phi \cot \theta$

If the system is set into motion with the initial conditions and $\phi (0)=20 rad/s$ , obtain a numerical solution with the adaptive Runger-Kutta method from $t=0$ to and plot $\phi$ vs.

Answer 1

#!/usr/bin/env python # import numpy as np; from numpy import asarray, zeros, sin, cos, tan, pi from run_kut5 import integrate from printSoln import printSoln import pylab def cot(x): return 1./tan(x) def F(x,y): theta, theta_dot, phi, phi_dot = y # vector y is = [ theta(t), theta'(t), phi(t), phi'(t) ] F = zeros( (len(y),), dtype=float ) F[0] = theta_dot F[1] = phi_dot**2 * sin(theta) * cos(theta) F[2] = phi_dot F[3] = - 2 * theta_dot * phi_dot * cot(theta) return F x0 = 0.0 # start of the integration xStop = 1.5 # end of the integration h = 0.1 freq = 0 y0 = asarray( [ pi/12, 0, 0, 20 ], dtype=float ) X,Y = integrate(F,x0,y0,xStop,h) printSoln(X,Y,freq) if True: pylab.plot( X, Y[:,3] ) pylab.xlabel('t'); pylab.ylabel("phi'(t)") pylab.grid('on') pylab.axis('tight') pylab.show() 

the above solution will give the graph and the adaptive numerical solution !