Harmonic Oscillator with Modified Damping (2)

Harmonic Oscillator with Modified Damping (2) - Kevin Brent...

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Kevin Brent Akshaya Annavajhala Professor Gurarie Math 224 Lab 2.3: The Harmonic Oscillator with Modified Damping Abstract: This lab analyzed the effects of various modifications to the damping term of the standard harmonic oscillator model: { y’ = v, mv’ + bv + ky = 0}. The first variation was an undamped oscillator, where the damping coefficient b is equal to 0. The other constants were m = 2, and k = 5. The solutions were found to have a period of 3.97 using both analytic methods as well as the pplane7 package for MATLAB, with the graphs shown below. The second variation was the damped harmonic oscillator, where b = 2. This system produced a periodic solution, which repeated with a value of 4.19, once again using pplane7. An analytic solution is also present in the appendix. The next variation was a harmonic oscillator with nonlinear damping, where the system was modified such that mv’ + b|v|v + ky = 0, using the same parameters as above. The period of the solutions was found using the NDSolve function in Mathematica to be 3.97, the same as for the first model. The final variant was the harmonic oscillator with second-order damping, or mv’ + (y 2 – α)v + ky = 0. Alpha was given the value of 3, whereas the other constants remained the same as before. Once again, pplane7 and NDSolve independently confirmed the period of 4.19 for the solutions to this model. This is the same as the oscillator with linear damping.
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The general model for a harmonic oscillator is 0 2 2 = + + ky dt dy b dt y d m , (1) where y represents the position of the oscillator at time t , m is the mass, b is the damping coefficient, and k is the spring constant. In order to work with this model numerically, it was separated into two first-order differential equations: v dt dy = , and (2) 0 = + + ky bv dt dv m , (3) where v represents the velocity of the oscillator at time t . The first case considered was an undamped harmonic oscillator, or one for which the value of b in Equation 3 is zero, and all other constants are non-zero. The second case was that of a harmonic oscillator with linear damping. For this model, the value of b , along with all other constants, was non-zero. The third case was a harmonic oscillator with non-linear damping. For this model, Equation 3 was altered: 0 = + + ky v v b dt dv m . (4) Finally, the last case considered was a harmonic oscillator with second-order damping, for which Equation 3 was altered yet again: ( 29 0 2 = + - + ky v y dt dv m α , (5) where α is a constant related to damping. The values of the constants used in these models were as follows: m = 2, k = 5, b = 2, and α = 3. Because this project is taken from chapter two of the textbook, and analytic solutions for systems of differential equations are looked at in chapter
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three, all of these models were analyzed numerically using MATLAB 7.4.0 and the package pplane7.m. Later, analytic solutions using Mathematica were attempted, and the notebook is presented as the appendix.
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Harmonic Oscillator with Modified Damping (2) - Kevin Brent...

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