Homework #5
Contents
import numpy as np
from numpy import sin,cos,pi
import matplotlib.pyplot as plt
from scipy.integrate import solve_ivp # import the ordinary differential equation integrator in Python
plt.style.use('fivethirtyeight')
Homework #5¶
A roller coaster is being designed on a parabolic track that rotates at a constant speed as seen in the figure above. Assume the cart rolls on the track as a frictionless point-mass of 100-kg. Determine the equations of motion in terms of the distance from the lowest point, \(q_1=x_2\).
a. What is the kinetic energy of the cart?
b. What is the potential energy of the cart?
c. What is the equation of motion for the cart?
1. Create a function, cart_ode, that represents the equation of motion for the car in terms of \(x_2\)¶
def cart_ode(t,r,w):
'''
cart_ode(t,r,w)
Set of 2 ODEs that return dx2/dt and d^2x2/dt^2 with input
x2 and dx2/dt, dr/dt = f(t,r)
Parameters
----------
t: current time
r: current state [x, dx]
w: system rotation rate [rad/s]
Returns
-------
dy: derivative of current state [dx, ddx]
'''
dr=np.zeros(np.shape(r))
dr[0] = r[1]
dr[1] #= ... your equation here
return dr
2. Solve the cart_ode initial value problem for x(0)=10 m, dx/dt(0)=0 m/s and \(\omega\)=0 rad/s¶
x0=10
v0=0
w=0 # rad/s
end_time=10 # choose an end time that displays one full period
r0 = solve_ivp(lambda t,r: cart_ode(t,r,w),[0, end_time],[x0,v0])
3. Solve the cart_ode initial value problem for x(0)=3 m, dx/dt(0)=0 m/s and \(\omega\)=1 rad/s¶
x0=10
v0=0
w=3
end_time=10 # choose an end time that displays one full period
r1 = solve_ivp(lambda t,r: cart_ode(t,r,w),[0, end_time],[x0,v0])
4. Solve the cart_ode initial value problem for x(0)=3 m, dx/dt(0)=0 m/s and \(\omega\)=2 rad/s¶
x0=10
v0=0
w=6
end_time=10 # choose an end time that displays one full period
r2 = solve_ivp(lambda t,r: cart_ode(t,r,w),[0, end_time],[x0,v0])