Ch4-4(6)c - 1 MAE351 Mechanical Vibrations Lecture 18(Chap...

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2009-03-17 1 1 MAE351 Mechanical Vibrations Lecture 18 (Chap. 4.6) 2 k 1 x 1 x 2 k 2 4.6 Forced Response by Modal Analysis ) ( 1 t F m 1 m 2 F 1 F 2 c 1 c 2 ) ( ) ( ) ( 2 t F t F t F Kx x C x M n diagonal. is , ~ ~ eq., d transforme the in i.e., able, diagonaliz is that Assume 2 1 F M q K q C q C
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2009-03-17 2 3 Forced Response (cont) : tion transforma coordinate another using system the Decouple r P q ) ( 2 be should equation the so F 2 2 th 2 1 t f r r r i M P r r r I i i i i i i i T i i i Responses to harmonic, periodic, or general forces can be obtained as in Chapters 2 and 3. • Note that the forcing function in a modal coordinate is a linear combination of many physical forces. 4 Forced Response (cont) • An excitation on a single physical DOF may EOM. decoupled the for ) F( ) f( t M P t 2 1 T “spread” to all modal DOFs • It is theoretically possible to drive a Multi-DOF system at one of its natural frequencies but not to allow resonant response by choosing the modal excitation force at the corresponding coordinate to be zero be zero. ? by shape mode i the to related be to happens if What time. of fuction a is ) ( and vector spatial a is where Let th i t g Mu b b b
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2009-03-17 3 5 Forced Response (cont) 11 The force in the decoupled EOM's becomes TT T  22 th () remembering the transformation or . But through orthogonality, , where 00 010 0 ii i T i tg t g t g t   fP M M u PM u P v qM x vM u Pv e e T  1 in the position i th only Since ( ) excites only the mode, ( ) may
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This note was uploaded on 02/11/2010 for the course MECHANICAL ms316 taught by Professor Abduljaba during the Spring '09 term at Kalamazoo.

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Ch4-4(6)c - 1 MAE351 Mechanical Vibrations Lecture 18(Chap...

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