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Unformatted text preview: MAE294B/SIO203B: Methods in Applied Mechanics Winter Quarter 2010 http://maecourses.ucsd.edu/mae294b Solution II 1 Skip the naive expansion. The leading-order solution is x = A ( T ) e i t + c . c . At O ( ) , one finds x 1 tt + x 1 + 2 x tT +( 4cos t ) x = . Secular terms look like e i t so look at ( 4cos t ) x = 2 ( e i t + e- i t )( A e i t + A * e i t ) = 2 A [ e i ( 1 + ) t + e i ( 1- ) t ]+ 2 A * [ e i (- 1 + ) t + e i (- 1- ) t ] . Secular terms are hence possible for =- 2 , , 2. Cosine is even so =- 2 is the same as = 2. The equation can be solved exactly for = 0 and gives A e i t + c.c with = 1 + 4 , so there is no growth. For = 2, the amplitude equation is 2i A T + 2 A * = Writing A = u + i v gives i u T- v T + u- i v = 0, i.e. u T = v , v T = u , so u and v are proportional to e T , corresponding to exponential growth (except for very special initial conditions). 2 Skip the naive expansion. Expand the exponential as e- x 3 = 1- x 3 + . The leading-order solution is x = A ( T ) e i t + A * e- i t- 1 + c . c . At O ( ) , one finds x 1 tt + x 1 + 2 x tT- x 3 t x = ....
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This note was uploaded on 03/16/2010 for the course MAE 294b taught by Professor Young,w during the Winter '08 term at UCSD.
- Winter '08