Ch4-3(4-5)c - 2009-03-17 1 MAE351 Mechanical Vibrations...

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2009-03-17 1 1 MAE351 Mechanical Vibrations Lecture 17 (Chap. 4.4-4.5) 2 Section 4.4 More than 2 DOF (Multi-DOF Systems) Extending previous Extending previous section to any number of degrees of freedom
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2009-03-17 2 3 Many systems have large numbers of DOF 43 section as same the stays Process 0 0 0 3 2 3 3 2 2 2 2 1 2 1 1 ) ( ) ( ) ( ) ( ) ( ) ( t r t r t r t r t r t r 4.3 Just get more modal equations, one for each degree of freedom ( n is the number of DOF) 4 Mode Summation Approach Based on the idea that any possible time response is just a linear combination of the eigenvectors  i n i t j i t j i n i i e b e a t t t K t i i 1 1 ) ( = ) ( let ) ( ~ ) ( with Starting v q q 0 q q t ) ( as this write also can term. each for solutions t independen linearly two q
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2009-03-17 3 5 Mode Summation Approach (cont) Find the constants and from the I.C. ii d   11 (0) sin and (0) cos Assuming eigenvectors normalized such that (0)= sin = sin sin nn i i i i i T j j TT T jj i i i i i j i j j dd d     qv q v vv vq v v Simila rly for the initial velocities, (0)= cos T j j d 6 Mode Summation Approach (cont) Solve for and from the two IC equations (0) (0) d 1 and tan sin about (0)= if you just crank it through the above expressions you might conclude that 0 i d d IMPORTANT NOTE q 0 ie thetrivialsoln you might conclude that i , i.e., the trivial soln. Be careful with (0) as well. q
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2009-03-17 4 7 Mode Summation Approach (cont) Instead, return to (0) 0 and realize that 0 instead of 0. Get from the expression i ii i T di d d    (0) =c o s (0) i i i i T i i i dd d   vq 8 Mode Summation Approach (cont) Watch out for rigid-body modes!! if = 0 if 1 = 0, q 1 ( t ) a 1 e j 0 t b 1 e j 0 t  v i a 1 b 1 v i does not give two linearly independent solutions.
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Ch4-3(4-5)c - 2009-03-17 1 MAE351 Mechanical Vibrations...

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