APPM 2360 - Fall 2010 - Lab 2

The Harmonic Oscillator with Modified Damping

Instructions

This lab is due Monday, November 8, 2010 at the beginning of lecture. TA's will hold office hours in the ECCR 143 or ECCR 252 computer labs during that week. The TAs in the lab are there to help you with Mathematica and Matlab syntax; they are not there to debug your files.  Please keep this in mind.

• Three (3) students is the maximum group size.You need to hand in only one lab per group.
• Use a cover page
• List the name, Student ID, recitation number, TA name, and instructor for each student in the group on the cover page
• No loose leaf paper, please use printer paper for all calculations included in appendix
• You are required to submit both, a paper copy, and an electronic copy in MS Word or PDF format to AMESS, the Applied Math online submission website.
• Four decimal point accuracy will suffice for this lab.
• Refer to the lab writing guidelines. They are important and you will be held to them; up to 20% of your final grade can be based on things like writing, introduction, conclusion, etc.
• Late projects are not accepted.

  On your title page, clearly mark

• Name(s),  Put the name of the group member you want to get the graded lab in recitation first.
• Student ID number(s)
• Semester (Fall, 2010)
• Professor
• TA and section
• If you do not include this information, as much as 10% will be deducted from your final score

Format is worth 20% of your grade. Please refer to the following writing guidelines for the expository sections of this report. You are required to know and follow the Writing Guidelines for all labs.

Writing Guidelines for Differential Equations Lab

Take a look at this sample lab with an example good writeup, and bad writeup. Please do NOT simply give a numbered list of answers to the questions, but rather integrate them into a written report.

Please make sure when you are saving your files to either email them to yourself or use a USB flash drive.

• Getting Help: The APPM 2460 webpage has brief tutorials in Matlab and Mathematica that might be handy.

• Start early! Office hours are much less crowded earlier in the week than on Friday afternoon. TA's will be able to give you more individualized help.

• Work in a group: it can reduce your workload.

The lab is due in lecture and on AMESS on Monday, November 8th.

Goal

The goal of this lab is to interpret an ODE model of a harmonic oscillator by:

• Solving an initial value problem analytically

• Examining a phase portrait

• Comparing two models

Model

For the derivation of the harmonic oscillator model with damping, see the following derivation as well as Chapter 4 of your textbook.

Questions

Address the questions in each item below in the form of a report.  Be particularly sure to describe the behavior or the solution and the corresponding behavior of the mass-spring system.  You'll want to use phase portraits and graphs of x(t) to illustrate the points you make in your report.
 

• A.  Harmonic Oscillator Model

In the derivation, we found that a simple harmonic oscillator can be described by the following ODE:

mx''+ax'+kx = F(t)            (1)




1. Describe the physical configuration.

2. Classify the above equation, referring to Chapter 4 of your textbook if needed.

3. Describe the physical quantity represented by each term in the model (including units) in a table. The table should have columns for the term, its units, and a breif description. You do not need to give details of the derivation of equation (1).

4. Rewrite (1) as a first order system. Make sure to connect the physical significance of each term, as described in question A.1, with the terms of the first order system.

• B.  Harmonic Oscillator - Unforced with No Damping

1. We call a harmonic oscillator undamped and unforced if, in equation (1), a = 0 and F(t)=0, respectively.   Find the undamped, unforced analytic solution to (1) with initial data x(0)=x₀,   x'(0)=v₀.     Describe the solutions.  Give the period if they are periodic.

2. Next we'd like to examine the behavior of the system using plots, both of the analytic solution and of the phase plane, to guide our understanding. Although we have both types of information here, it is often the case that we can't find analytic solutions to an ODE as we saw in the Fish Lab. In such cases, the phase portraits can be very useful in understanding stability and other dynamical properties of the system.
   
   Create solution plots (x vs. t and x' vs. t) and phase portraits (x' vs. x)  for the following systems:

   • A 3.4-kilogram mass is attached to the spring with k = 1.25 N/m.  The mass is released from rest at a height of 0.5 m above the equilibrium position.

• A 3.4-kilogram mass is attached to the spring with k = 1.25 N/m.  The mass is started in motion from the equilibrium position with an initial velocity of 0.8 m/sec in the downward direction.

• C.  Harmonic Oscillator - Unforced with Damping

1. Repeat question B.2 for only the second set of initial conditions with damping coefficients, a = 10, sqrt(17) (about 4.123), and 2 Ns/m; these types of motion are classified as overdamped, critically damped and oscillatory damped (or underdamped), respectively.

2. Briefly comment on the similarities and differences between the damped case and the undamped case.

3. Show that the types of motion (overdamped, critically damped and oscillatory damped)  that result from (1) are completely determined by the quantity a² - 4km.

4. Breifly comment on the similarities and differences between these three types of motion.

• D.  Coupled Harmonic Oscillators 

Now consider the two coupled spring-mass oscillators in the figure below.

1. Identify each of the terms in the governing differential equation (provided below the figure).

2. Rewrite the second order system as a system of four first order differential equations. The x₁ and x₂ displacements are measured from the m₁ and m₂ equilibrium positions.

3. Use an ODE integrator to solve the system of four first order differential equations with the mass and spring constant from B.2 for both. Let the system start from rest with each mass displaced by 0.5 meters downward (x₁ (0)=0.5 and x₂(0)=0.5). Assume the equilibrium position of m₁ is 1.0 meter below the ceiling and m₂ is 1.5 meters below m₁ and plot the positions of both masses relative to the ceiling on the same graph. Plot the distance between the masses in another figure.  Explain the motion observed in each plot.

4. Do the masses collide with the ceiling or with each other during your simulation?

5. Solve the system again but allow the second mass to be initially displaced by 0.5 meters upward. Generate the same plots. Discuss how the position plots differ from D.2?

Your report should show that you can relate a physical problem to its mathematical model.  Remember to follow the writing guidelines in your report. DO NOT SIMPLY PROVIDE A LIST OF ANSWERS TO THE QUESTIONS ABOVE!

Some interesting links related to harmonic oscillators:
http://www.enm.bris.ac.uk/research/nonlinear/tacoma/tacoma.html#mpeg

Created by Keith Wojciechowski  .   Modified by David Biagioni. Modified by Steve Chestnut Modified by Andy Guinn (Fall 2010)