ASE 365 - Lecture 27 - Systems with rigid body modes . of...

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Unformatted text preview: Systems with rigid body modes . of solution nonzero a is there since , Then x x = = K K det . with , i.e., applied, force zero with nt displaceme nonzero a have can They : ed unrestrain physically often are structures Aerospace x x = K . satisfies zero of frequency natural a Then ) det( 2 =- M K n. deformatio no is there so , is for energy strain the since , a is satisfying vector the Then ) ( = = = = x x x x x x T T K V K mode body rigid definite. positive than rather te, semidefini positive is Then K . frequency natural same the with modes different ng representi , satisfying vectors t independen linearly multiple be must there modes, body rigid multiple are there If = = x x K rotation). 3 n, translatio (3 modes body rigid 6 have typically aircraft) t, (spacecraf D- 3 in structures ed Unrestrain . a as known mode body rigid a introduces this ), automobile in drivetrain linkage, steering (e.g., structure, the of rest the to relative freely move can structure a of part If mechanism 3 , 2 , . , , 1 1 2 1 1 1 1 1 1 1 = = + = = = = + r k m m m k k m r r r r : modes flexible so but : mode body rigid : response) (free EOMs Modal 3 , 2 , sin cos ) ( = + = r t b t a t r r r r r : time in harmonic is response free modes, flexible For . represents This : time in is response free , since modes, body rigid for However, motion velocity constant linear t b a t 1 1 1 1 ) ( + = = Modal expansion (decomposition) [ ] [ ] x e e e x e I x x x x x x x x x x n i n n n i n i i n n n i = = = = + + + =...
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ASE 365 - Lecture 27 - Systems with rigid body modes . of...

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