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Unit I. Non-Linear Oscillations and Chaos

A. Review Chapter 3, sec. 1-2, and sec. 4-6 of Marion, paying particular attention to sec. 4 dealing with phase diagrams and their use to describe the motion of a onedimensional system. Be sure that you can construct and recognize the phase diagram plot for a simple harmonic oscillator, and for a damped harmonic oscillator (under all three conditions of damping: underdamped, critically damped and overdamped). Toward that end, you should attempt to work the following problems. 1. Correctly draw the phase diagram for an undamped, undriven simple harmonic oscillator with maximum amplitude of E. Derive the equation for this phase diagram. How does its shape change for various values of E and =9 ? 2. Is it possible for one portion of a phase path to overlap another? Justify your answer. 3. Use MatLab to plot the phase diagram for an underdamped simple harmonic oscillator. 4. Now consider the critically damped harmonic oscillator whose equation is given by B oe E F> /-#> Draw the phase diagram for this oscillator and show that the phase path asymptotically approaches the line B oe # B. B. Read Chapter 4, sec. 1-3. These sections deal with non-linear oscillations and the technique of successive approximations to solve these equations. Section 4.3 deals with phase diagrams for non-linear systems and how they can be constructed from an understanding of the potential energy of the system. Work the following problems: 1. Problem 4.1: Here you develop the potential energy function for a nonlinear oscillator. 2. Problem 4.2: Here you construct the phase diagram for the potential defined in problem 4.1. 3. Problem 4.8: This problem investigates the solution of a simple problem by solving for the equations of motion and constructing the phase diagram for a simple system. Once the phase diagram is constructed the period of oscillation can be determined. 4. Problem 4.5: The solution to equations for non-linear oscillators often cannot be found exactly (or simply). One must often be content with approximate solutions. In this problem we investigate the technique of

finding an approximate solution to a non-linear equation using the technique of successive approximations. C. Read section 4.4 on the plane pendulum and work problem 4.6 where you will derive expressions for the phase paths of this pendulum. As you work this problem, you should consider the following: 1. Calculate the potential energy curve for the simple pendulum and compare it with the potential energy curve for the harmonic oscillator. 2. Plot phase diagrams for the simple pendulum just as you did for the nonlinear oscillator in problem 4.2. Plot these for three cases: 1) E = 1.5 MgL, 2) 2.0 MgL, 3) 2.5 MgL, where M is the mass of the pendulum, L is the length of the pendulum, and g is the acceleration of gravity. 3. Now we want to extend this last problem by looking at the solution of the "damped" simple pendulum. For this particular case, let's write the equation in terms of the moment of inertial of the suspended mass (M oe Q P# ). This gives M .# ) .) oe , Q 1P=38Ð)Ñ # .> .> =! oe È1ÎP

If you divide by the moment of inertia, and define the "natural frequency" by the equation

this last equation can be written in a dimensionless form as .# ) .) oe =38Ð)Ñ # .7 .7 where 7 oe =! >, and - oe ,Î7P# =! . If we let ) oe B, this last equation has the form of a damped harmonic oscillator, with the non-linear force term =38Ð)Ñ. Note: The total energy for this oscillator can also be expressed in dimensionless form as &oe " .) OE # .7


" -9=)

where the first term is the dimensionless kinetic energy, and the second term is the dimensionless potential energy. We wish to solve the equation of motion for the damped pendulum, and we will make use of the dimensionless form of the equation. To understand how to use MatLab to solve differential equations, read the attached handout.

D. Read sec. 4.5 and 4.6, an introduction to the strange and unpredictable motion of non-linear systems and work problem 4.10 which illustrates the unpredictable nature of non-linear systems. E. Work supplimental problems 1 and 2 which deals with self-limiting oscillators. F. Read sec. 4.7-4.8 and work problem 4.14. G. Be prepared to discuss the meaning of chaos and its relation to classical deteministic mechanics.

Supplimentary Problems 1. Some nonlinear oscillators exhibit self-limiting behavior. One example of this is the van der Pol oscillator, described by the equation C oe + ^C# C# C =# C oe ! 9 # C9 9 Many of these equations can only be solved using numberical techniques. However, there is an equation, similar in form to the van der Pol equation, for which its self-limiting nature is easy to calculate explicitly. This equation has the form # B # OEE# B# B B =# B oe ! 9 "# (a) (b) Make the substitution C oe B and rewrite the equation above in terms of C, C , and B. Show that the nonlinear damping terms is negative for all values points Bß C inside of an ellipse, zero for all points on the ellipse, and positive for all points outside the ellipse, causing the the oscillator to be driven toward phase points on the ellipse. In other words, no matter how the system is started, the system will ultimately vibrate with amplitude E, its behavior being self-limiting. The ellipse in phase space toward which the particle motion tends to gravitate is called the limit cycle. The van der Pol equation also exhibits this type of behavior, but it is not quite so easily demonstrated. B # ^E# B# B =# B oe ! 9


The equation of motion for the van der Pol oscillator can be expressed in the form

Let E oe " and =9 oe " and solve this equation numerically, making a phase space plot of its motion. Let the motion evolve for 10 periods (1 period oe #1Î=9 ). Assume the following conditions: (a) # oe !Þ!&ß B9 ß B 9 oe "Þ&ß ! . (b) # oe !Þ!&ß B9 ß B 9 oe 0Þ&ß ! . (c,d) Repeat (a) and (b) with # oe !Þ&. Does the motion exhibit a limit cycle? Describe it.


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