Physics 325 Lecture 9 - 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 x y m x x my y k x x k y y + = 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 y y x t A t y t A t x y = = 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 , and x y x y x A A = = = for several different values of 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 = ). -1-0.5 0.5 1-1.5-1-0.5 0.5 1 1.5 10 y = 170 y = 90 y = 45 y =-1-0.5 0.5 1-1.5-1-0.5 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 1.5 170 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 90 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 10 y =-1.5-1-0.5 0.5 1 1.5-1.5-1-0.5 0.5 1 1.5 10 y = When the frequency ratios are not rational fractions, the f s do igure not close on amped Oscillations themselves (i.e. the path is not retraced). D s we all know, a real mass on a spring does not satisfy the equation of motion we wrote A 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: x d F c = (12.1) w ses the motion (resistive force). The here the minus sign specifies that the force oppo equation of motion for a mass on a spring is now mx kx cx = (12.2) earranging this a little by defining R 2 and 2 k c m m = = (12.3) 2.2) becomes (1 2 2 x x x + + = (12.4) quation (12.4) is another homogeneous, linear differquation (12....
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This note was uploaded on 10/06/2011 for the course PHYS 325 taught by Professor Staff during the Fall '08 term at University of Illinois, Urbana Champaign.

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Physics 325 Lecture 9 - Physics 325 Lecture 12...

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