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Unformatted text preview: 18.03 Problem Set 6: Solutions 22. [Part I problems 4] (a)  g ( t ) = π 2- 4 π (cos( t )+ cos(3 t ) 3 2 + · · · ) according to L21 or II.21(f). Since x p = A cos( ωt ) ω 2 n- ω 2 is a solution to ¨ x + ω 2 n x = A cos( ωt ), by Superposition III x p = π 2 ω 2 n- 4 π cos( t ) ω 2 n- 1 + cos(3 t ) 3 2 ( ω 3 n- 3 2 ) + · · · is a solution to ¨ x + ω 2 n x = g ( t ) (provided none of the denominators vanishes). (b)  The system is in resonance for ω n = 1 , 3 , 5 , . . . . (c)  For ω n near to 1, the dominant term in x p is- 4 π cos( t ) ω 2 n- 1 . This is a large positive multiple of cos( t ) for ω n < 1 and a large negative multiple of cos( t ) for ω n > 1. (d) , (e)  When ω n is an integer, the homogeneous solutions A cos( ω n t- φ ) have period 2 π , so if some particular solution is periodic of period 2 π then all solutions are. This occurs when ω n is an even integer. More generally, if the particular solution is periodic (necessarily of period 2 π ) and some integer multiple of its period coincides with an integer multiple of the period of the nonzero homogeneous solutions ( 2 π ω n ), then all solutions are periodic of that common period. This happens when ω n is any nonnegative rational number other than an odd integer.integer....
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This note was uploaded on 02/06/2009 for the course 18 18.03 taught by Professor Unknown during the Spring '09 term at MIT.
- Spring '09