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285lect28 - 1 Lecture 29 1.1 Damped forced oscillation...

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1 Lecture 29 1.1 Damped forced oscillation Recall the equation of the damped forced oscillation: mx 00 + cx 0 + kx = F ( t ) : We assume that F ( t ) = F 0 cos !t . Recall also that for complimentary solu- tion mx 00 + cx 0 + kx = 0 the following cases are possible: ° c > c cr : x c ( t ) = C 1 e r 1 t + C 2 e r 2 t ; where r 1 < r 2 < 0 : ° c = c cr : x c ( t ) = ( C 0 + C 1 t ) e ° pt , where p = c 2 m : ° c < c cr : x c ( t ) = Ce ° pt cos ( ! 1 t ± ° ) , where ! 1 = q ! 2 0 ± p 2 ; where c = c cr = 2 p km . Let us °nd a particular solution of the non-homogeneous equation mx 00 + cx 0 + kx = F 0 cos !t: Set x p ( t ) = A cos !t + B sin !t: We have x 0 p ( t ) = ± !A sin !t + B! cos !t ; x 00 p ( t ) = ± ! 2 A cos !t ± ! 2 B sin !t: 1
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mx 00 + cx 0 + kx = m ° ± ! 2 A cos !t ± ! 2 B sin !t ± + c ( ± !A sin !t + B! cos !t ) + k ( A cos !t + B sin !t ) = cos !t ² ± m! 2 A + cB! + kA ³ + sin !t ² ± m! 2 B ± c!A + kB ³ = F 0 cos !t: We get the following system of linear equations: ± m! 2 A + cB! + kA = F 0 ± m! 2 B ± c!A + kB = 0 ) ( k ± m! 2 ) A + c!B = F 0 I ± c!A + ( k ± m! 2 ) B = 0 II Ic! + II ( k ± m! 2 ) : h ( c! ) 2 + ° k ± m! 2 ± 2 i B = F 0 c! ) B = F 0 c!
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