Solving equations of motion

import numpy as np
import matplotlib.pyplot as plt
from scipy.optimize import fsolve
plt.style.use('fivethirtyeight')

Hide code cell source

 
from IPython.lib.display import YouTubeVideo
YouTubeVideo('Je-qT2FQrrM')

Solving equations of motion#

In this notebook, you will plot the solutions to three 1-DOF equations of motion. Each starts at the origin with an initial velocity, \(v = 5~m/s\). The three equations of motion and solutions are derived in the video above

system

equation of motion

solution

a.

\(m\ddot{x} = -\mu mg\)

\(\rightarrow x(t) = v_0t - \frac{\mu gt^2}{2}\)

b.

\(m\ddot{x} = -b \dot{x}\)

\(\rightarrow x(t) = \frac{v_0 m}{b}\left(1 - e^{-\frac{b}{m} t}\right)\)

c.

\(m\ddot{x} = -k x\)

\(\rightarrow x(t) = \frac{v_0}{\omega}\sin\omega t\)

Coulomb friction on a sliding block#

This first example, has a small trick. The acceleration is constant, \(-\mu g\), until the velocity is zero. At this point, the block stops moving. To solve for \(x(t)\)

  • calculate \(x(t)\) and \(v(t)\) if acceleration is constant

  • set the values of \(v(t)<0\) to 0

  • set the values of \(x(t)\) given \(v(t)=0\) as the maximum \(x\)

Here, \(\mu=0.3\) and m = 0.5 kg

t = np.linspace(0, 3)
xa = 5*t - 0.5*0.3*9.81*t**2
va = 5 - 0.3*1*9.81*t
va[va < 0] = 0
xa[va == 0] = xa.max()
plt.plot(t, xa)
plt.xlabel('time (s)')
plt.ylabel('position (m)')

Text(0, 0.5, 'position (m)')
../_images/c041a27d4f0bebcf9915634dfe62e243f2ec561718b3951d918de0ef70006391.png

Viscous friction#

This second example has a exponentially decaying speed. This type of motion is common in door dampers and shock absorbers. The faster the object moves, the faster it decelerates.

  • \(v(t) = v_0 e^{-\frac{b}{m}t}\)

  • \(x(t) = \frac{v_0 m}{b}\left(1 - e^{-\frac{b}{m} t}\right)\)

Here, b = 1 kg/s and m = 0.5 kg

m = 0.5
b = 1
xb = 5*m/b*(1-np.exp(-b/m*t))
plt.plot(t, xb)
plt.xlabel('time (s)')
plt.ylabel('position (m)')

Text(0, 0.5, 'position (m)')
../_images/4f4baa54a241a37e6a14b26640a7c1e1c78a13f34a1ea0500a072bed243f7885.png

Linear spring and the harmonic oscillator#

This third example is a harmonic oscillator. Any object that has a restoring force e.g. a spring attached to a mass, a pendulum swinging, object hanging from a rubber band. The harmonic oscillator is described by the general equation

\(\ddot{x} = -\omega^2 x\)

where \(\omega = \sqrt{\frac{k}{m}}\) for a spring mass. Here, \(k=2~N/m\) and m=0.5 kg.

w = np.sqrt(2/0.5)
xc = 5/w*np.sin(w*t)
plt.plot(t, xc)
plt.xlabel('time (s)')
plt.ylabel('position (m)');
../_images/1dcb463b0ced84a87643ece0c1303a73ec94507f4f39ec85da37cc62926b156e.png

Wrapping up - comparing all three examples#

You have plotted three solutions

  1. sliding with friction

  2. viscous friction

  3. harmonic oscillator

Now, you can plot all three together.

plt.plot(t, xa, label = 'friction')
plt.plot(t, xb, label = 'viscous') 
plt.plot(t, xc, label = 'harmonic')
plt.legend();
plt.xlabel('time (s)')
plt.ylabel('position (m)');
../_images/6e04567c7feaeb7750e672d35ecf9ffea4659c02bca10b3eb9b3c65c2000dd62.png

Some similiraties between the three plots

  • each plot begins at 0 m

  • each plot has the same initial slope

Some differences between the three plots

  • the friction and viscous friction have a final position, but the harmonic plot continues to move

  • the blue friction plot has two distinct functions: \(\propto t^2\) and \(\propto constant\), but the other plots are continuous functions

Contents