The Harmonic Oscillator with Modified Damping Autonomous second-order differential equa...

The Harmonic Oscillator with Modified Damping

Autonomous second-order differential equations are studied numerically by reducing them to first-order systems with two dependent variables. In this lab you will use the computer to analyze three somewhat related second-order equations. In particular, you will analyze phase planes and y(t)- and v(t)-graphs to describe the long-term behavior of the solutions.

In Sections 2.1 and 2.3, we discuss the most classic of all second-order equations, the harmonic oscillator. The harmonic oscillator is

It is an example of a second-order, homogeneous, linear equation with constant coefficients. In the text we explain how this equation is used to model the motion of a spring. The force due to the spring is assumed to obey Hooke’s law (the force is proportional to the amount the spring is compressed or stretched). The force due to damping is assumed to be proportional to the velocity. In your report you should describe the motion of the spring assuming certain values of m, b, and k. (A table of values of the parameters is given below. Your instructor will tell you what values of m, b, and k to consider.) Your report should discuss the following:

Your report: Address the questions in each item above in the form of a short essay. Be particularly sure to describe the behavior of the solution and the corresponding behavior of the mass-spring system. You may use the phase planes and graphs of y(t) to illustrate the points you make in your essay. (However, please remember that, although one good illustration may be worth 1000 words, 1000 illustrations are usually worth nothing.)

(Harmonic oscillator with nonlinear damping) Repeat Part 1 using the equation

in place of the usual harmonic oscillator equation. (Note that even with the same value of the parameter b, the drag forces in this equation and the equation in Part 2 have the same magnitude only for velocity ±1. Also, note that the sign of the term

is the same as the sign of dy/dt, hence this damping force is always directed opposite the direction of motion. The difference between this equation and that in Part 2 is the size of the damping for small and large velocities. One of the many examples of situations for which this is a better model than linear damping is the drag on airplane tires from wet snow or slush. Drag from only four inches of slush was enough to cause the 1958 crash during take-off of the plane carrying the Manchester United soccer team. Currently, large airplanes are allowed to take off and land in no more than one-half inch of wet snow or slush.∗