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Analytical Mech Homework Solutions 184

# Analytical Mech Homework Solutions 184 - 21 2...

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21 2. Anti-symmetric oscillation of 2&3, while 1 remains fixed 13 21 0 0; 1 1 or θθ θ   == =  a 3. Anti-symmetric oscillation of 1&2 together with respect to 3 1 1; 1 1 or = a Thus, and ± AA = 10 1 1 1 1 =11- 1 01- 1 1- 1 1 4 We can now diagonalize K and M diag 4- 2 - 2 -2 2+2K -2K -2 -2K = ± AKA K 111 101 =0 1- 1 11- 1 1 1 00 0 4 0 0 048 020 01 6 0 0 diag diag K Likewise =+ = KM The eigenfrequencies are 2 i diag diag i ω  =  22 2 3 2 4 ; 4 K ωω + = The general solution can be generated from the following table … 12 11 21 2 31 2 0 QQ Q aa aa a −− 3 3 3 3 a where () 1 1 1 1c o s 1 at δ =− Q , 2 2 0 o s 1 Q , 33 3 3 1 o s 1 Q Thus, ( ) 3 cos cos ωδ +− ( ) ( ) 2 2 2 2 2 3 cos cos cos atatat ( ) ( ) cos cos cos
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