This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: Physics 325 Lecture 12 Twodimensional oscillators The twodimensional 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 = ). 10.5 0.5 11.510.5 0.5 1 1.5 10 y = 170 y = 90 y = 45 y =10.5 0.5 11.510.5 0.5 1 1.5 10 y = 170 y = 90 y = 45 y =1.510.5 0.5 1 1.51.510.5 0.5 1 1.5 170 y =1.510.5 0.5 1 1.51.510.5 0.5 1 1.5 170 y =1.510.5 0.5 1 1.51.510.5 0.5 1 1.5 90 y =1.510.5 0.5 1 1.51.510.5 0.5 1 1.5 90 y =1.510.5 0.5 1 1.51.510.5 0.5 1 1.5 10 y =1.510.5 0.5 1 1.51.510.5 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....
View
Full
Document
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.
 Fall '08
 Staff
 Physics, mechanics

Click to edit the document details