Phys 325 Spring 2011 Lecture 12 - Physics 325 Lecture 12...

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Unformatted text preview: Physics 325 Lecture 12 Two-dimensional oscillators The two-dimensional harmonic oscillator is quite straightforward. The equations of motion are ˆ ˆ ˆ ˆ x y m x i my j k x i k y j This is really two decoupled equations: x y x x y y where , x x y y k m k m The solutions are cos sin x x x y y y x t A t y t A t The x and y motion have different amplitudes, frequencies and phases. The motion of such an oscillator can be displayed on a Lissajous figure which plots x vs. y as a function of time. An example is shown below for the case A x A y , x y and x 0 for several different values of y -1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 10 y 170 y 90 y 45 y -1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 10 y 170 y 90 y 45 y More interesting plots arise when the frequencies are different. Here are a couple of examples with 2 y x and various phase differences ( x ). When the frequency ratios are not rational fractions, the figures do not close on themselves (i.e. the path is not retraced). Damped Oscillations As we all know, a real mass on a spring does not satisfy the equation of motion we wrote in Equation (11.1). Instead, a real system will encounter friction or other dissipative forces that take energy away from the system. We call this damping . Typically, these forces are velocity dependent, and the simplest form is linear dependence on velocity: d F cx (12.1) where the minus sign specifies that the force opposes the motion (resistive force). The equation of motion for a mass on a spring is now mx kx cx (12.2) Rearranging this a little by defining 2 and 2 k c m m (12.3) Equation (12.2) becomes 2 2 x x x (12.4) -1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 170 y -1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 170 y -1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 10 y -1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 10 y -1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 90 y -1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 90 y Equation (12.4) is another homogeneous, linear differential equation. We can solve it in...
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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Phys 325 Spring 2011 Lecture 12 - Physics 325 Lecture 12...

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