Question

Consider an undamped system where the vector-matrix form of the system model is: F(t) [8 olx Mx + Kx = 0 18X, + 2000 -1800 x -1800 4500 I:1-[0] The system is initially at rest with x(0) = 0 and x,0)=0 when input F(t) = 84 sin15t is applied to the system. Use the modal decomposition method described in chapter 5 to find the system response. Some intermediate results (find these as part of your solution) are: The system's two natural frequencies and mode shapes are: m2 = 10 s with v, = and 0, = 20 s with v, = . [-] 1 % The mass-normalized modal matrix and its inverse are: v= 2,1 mm and V-1 2 3 2 -3

Answer 1

let X1 = 0 -1800 + 2700 X2 - 0 X2 = 1800 2700 X2 - 2/3 1 Vi 2/3 2nd mode - 0 [K-WML Xi X2 -1200 -1800 XI = 0 1.11 -1800 -2 Too X2 let Xi=1 - 1800 2700 X2 = 0 ; X2 : 2 3 p2 • (-275) Vz V= [Vi vz] - ve [ 143 2/ 3-2/3) cs Scanned with CamScanner

84 Sin 15 t - . 15 ija le conda 10 a 84 Sin 15 t 84 Sin 1st 84 sin 15t q, + loo q. Öz + 40092 84 Sinist - Above ean of the form - is qi + Wir qi Fosin Wt i=1,2,..... qi(t)- Ai Sin Wt q.(t)- Ai Sinut 92(t) = Az Sin wt äi(t): - A W? sin wt 92 Ct): - A2W2 Sin wt the Values in eqn © ¢ © , we eget substitusting + loo (A, sin 15 t)- 84 Sin 15t - AI X (15)? sin 15t (-152 +100) Ai - 84 ALE -0.672 A - Az (15)2 Sin ist + 400 (A2 sin ist) = 84 sin ist Az = 84 A2 = 0.48 400-152 CS Scanned with CamScanner