About
Developed by K. Roos
In this set of exercises the student builds a computational model of a hanging massspring system that is constrained to move in 1D, using the simple Euler and the EulerCromer numerical schemes. The student is guided to discover, by using the model to produce graphs of the position, velocity, and energy of the mass as a function of time, that the Euler algorithm does not conserve energy, and that for this simple oscillatory system, a modified algorithm (EulerCromer) is necessary to avoid artificial behavior in the model.
Subject Area  Mechanics 

Levels  First Year and Beyond the First Year 
Available Implementations  C/C++, Fortran, IPython/Jupyter Notebook, Mathematica, Octave*/MATLAB, Python, and Spreadsheet 
Learning Objectives 
Students who complete this set of exercises will

# Written by:
# Kelly Roos
# Engineering Physics
# Bradley University
# This email address is being protected from spambots. You need JavaScript enabled to view it.  309.677.2997
import numpy as np
import matplotlib.pyplot as plt
import math
# Input parameters for model
dt = 0.01 # time step (s)
vi = 0 # initial velocity (m/s)
yi = 0.1 # initial position (m), relative to eq position yeq (m)
m = 1.0 # mass (kg)
g = 9.8 # gravity (m/s^2)
k = 10 # spring constant (N/m)
t_steps = 2000 # number of time steps
# Defines the 1D arrays to be used in the computation and
# sets all values in the arrays to zero
time = np.zeros(t_steps)
y_Euler = np.zeros(t_steps)
y_EC = np.zeros(t_steps)
y_exact = np.zeros(t_steps)
v_Euler = np.zeros(t_steps)
v_EC = np.zeros(t_steps)
v_exact = np.zeros(t_steps)
energyEuler = np.zeros(t_steps);
energyEC = np.zeros(t_steps);
# Initial conditions
time[1] = 0
y_Euler[1] = yi
y_EC[1] = yi
y_exact[1] = yi
v_Euler[1] = vi
v_EC[1] = vi
v_exact[1] = vi
energyEuler[1]=0.5*m*v_Euler[1]**2+0.5*k*(y_Euler[1]+m*g/k)**2m*g*(y_Euler[1]+m*g/k)
energyEC[1]=0.5*m*v_EC[1]**2+0.5*k*(y_EC[1]+m*g/k)**2m*g*(y_EC[1]+m*g/k)
# Main loop: Euler algorithm, eulerCromer, and evaluation of exact solutions for v and y
for i in range(2, t_steps):
time[i] = time[i1] + dt
# Euler
v_Euler[i]=v_Euler[i1]k*y_Euler[i1]*dt/m
y_Euler[i]=y_Euler[i1]+v_Euler[i1]*dt
# EulerCromer
v_EC[i]=v_EC[i1]k*y_EC[i1]*dt/m
y_EC[i]=y_EC[i1]+v_EC[i]*dt
# Exact (assuming vi=0)
v_exact[i]=yi*math.sqrt(k/m)*math.sin(math.sqrt(k/m)*time[i])
y_exact[i]=yi*math.cos(math.sqrt(k/m)*time[i])
# Energies
energyEuler[i]=0.5*m*v_Euler[i]**2+0.5*k*(y_Euler[i]+m*g/k)**2m*g*(y_Euler[i]+m*g/k)
energyEC[i]=0.5*m*v_EC[i]**2+0.5*k*(y_EC[i]+m*g/k)**2m*g*(y_EC[i]+m*g/k)
# Plotting Results
plt.plot(time, v_Euler, time, v_EC)
plt.ylim((0.3, 0.3))
plt.xlim((0, 20))
plt.xlabel('time (s)')
plt.ylabel('position (m)')
plt.title('position vs. time')
plt.show()
Exercise 1: Euler Algorithm Model of a SHO
Build a computational model of a simple hanging harmonic oscillator using the Euler method. Use realistic values for the parameters (i.e., spring constant and attached mass , such as would be encountered in a typical introductory mechanics laboratory exercise. Also, assume that the mass of the spring is negligible compared to the attached mass, and that the harmonic oscillator has been stretched vertically downward a distance , relative to its hanging equilibrium position and released from rest. Use the model to produce graphs of the position and velocity of the mass as a function of time, and compare these with the exact functions for the position and velocity,
and
that result from solving Newton’s 2nd Law analytically. Does the angular frequency match that expected for a simple harmonic oscillator of mass an spring constant ?
Exercise 2: Artificial Behavior with the Euler Algorithm
You may (should!) have noticed that something is not right with the Euler model of your hanging oscillator. Describe in detail the artificial behavior you observe in your model, and explain why it doesn’t represent a realistic oscillating mass. Recall that in the Euler method, the accuracy of the solution can be increased by using a smaller value of . Can you get rid of the artificial behavior by making smaller?
Exercise 3: Energy in the Euler Algorithm Model of a SHO
Modify your model to produce a graph of the total energy of the oscillator as a function of time. Describe in detail what happens to the energy, and the artificial behavior observed. Can this artificial behavior in the energy be corrected by making smaller? What can you conclude about using the Euler method to model a simple harmonic oscillator?
Exercise 4: EulerCromer Algorithm Model of a SHO
Build a model of the hanging oscillator using the modified Euler, or EulerCromer, numerical method. Compare the results you obtain (i.e. position and velocity vs. time) with those obtained from the simple Euler method, and with the exact solution. Comment in detail on your results.
Exercise 5: Energy in the EulerCromer Algorithm Model of a SHO
Modify your model to produce a graph of the total energy as a function of time. Is energy conserved for the EulerCromer algorithm?
For Teachers
Translations
Software Requirements
Android  iOS  Windows  MacOS  
with best with  Chrome  Chrome  Chrome  Chrome 
support fullscreen?  Yes. Chrome/Opera No. Firefox/ Sumsung Internet  Not yet  Yes  Yes 
cannot work on  some mobile browser that don't understand JavaScript such as.....  cannot work on Internet Explorer 9 and below 
Credits
Fremont Teng; Loo Kang Wee
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Sample Learning Goals
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For Teachers
Hanging Simple Harmonic Oscillator JavaScript Simulation Applet HTML5
Instructions
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