Phys 325 Spring 2011 Lecture 14 - Physics 325 Lecture 14...

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Physics 325 Lecture 14 Superposition & Fourier Series We now investigate driven oscillations with non-sinusoidal driving forces. We’ll start with a simple superposition of two sinusoidal driving forces with different frequencies, and then show how to generalize this result to an arbitrary driving force. We take a driving force to be F ( t ) F 1 cos 1 t F 2 cos 2 t (14.1) and the equation of motion, which follows from Equation (13.2), is 2 12 0 1 2 2 cos cos FF x x x t t mm (14.2) This equation, as before is linear and therefore the sum of solutions is also a solution. This inspires us to write two equations with solutions x 1 ( t ) and x 2 ( t ): 2 1 1 1 0 1 1 2 2 2 2 0 2 2 2 cos 2 cos F x x x t m F x x x t m We can add these two equations to find d 2 dt 2 x 1 x 2 2 d dt x 1 x 2 0 2 x 1 x 2 F 1 m cos 1 t F 2 m cos 2 t Therefore, x 1 ( t ) x 2 ( t ) is a solution to Equation (14.2). We’ve demonstrated this for a superposition of two terms, but it obviously works for an arbitrary number of terms. Let 1 ( ) ( ) ( ) cos N n n n n n F t F t F t F t  (14.3) Then for each n we can write 2 0 2 cos n n n n n F x x x t m (14.4) and 1 ( ) ( ) N n n x t x t (14.5) satisfies the differential equation
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2 0 () 2 Ft x x x m  (14.6) This is the principle of superposition , and it works because the differential equation (14.6) is linear. In solving the problem of a sinusoidal driving force in Lecture 13, we have already found solutions to the differential equation (14.4). Therefore, we can state the following: If the driving force F ( t ) has the form   ( ) cos n n n n F t F t   (14.7) and the oscillator system obeys Equation (14.6), then the solution for the resulting steady- state motion (long times) is       2 2 22 0 1 ( ) cos 2 n p n n n nn F x t t m  (14.8) with 1 0 2 tan n n n  (14.9) Fourier’s Theorem The realization that we can superpose the solutions to sinusoidal driving forces to find the solution to a superposition of forces becomes quite powerful as a result of Fourier’s theorem.
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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 14 - Physics 325 Lecture 14...

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