Question

4.11 Compute the natural frequencies and mode shapes of the following system: 4 0 4 X10 -2 X= 0 1 1 -2 and Calculate the response of the system to the initial conditions: x, 1 2 -2120 20

Answer 1

(2) K-MA2 0 る= 220 [4 0 40 -201 01 0 1 40-4x20 -20 10 -20 -20 10-20 Mi+Kx 0 to find natural frequency det|K -MA =0 -40-20 0 40-4A -20-10x, -20 = 0 10-2 40x-20x 0 if 1 -20 (40-4) (10-2)-400 0 400-402-402+422-400 0 -2 2) 422-802 0 1 42(2-20) 0 2 X 4 0 る= 20 1 1 modal matrix p 2 -2 /20 Let the solution will be x 0q o V20rad /sec O=Orad /sec 1 {4,)2 -2l9,0 mode shapes [K-M2,]x = 0 [K-MA 0 -20 1 10-2j 0 initial condition l(0) l(012] (1) 40-42 -20 |(0)5 = inv 4(0)f 2 -2 (1 1 (1) 40 -20 -20 10 20 40x-20x 0 l4,(0)-2/20 if 1 x2 \(0)5/ 4,(0f 20 = inv 2 (a) (1 2] (-2/20 0 les(0))20 A

(3) Mo+Ko=0 mặ +kg = 0 [(t)).[1 |1 -2|0 12 -2|(t) |1 -2||0 1|-20 10 j|g,(t)| |oJ 2T4 0T1 2 4 040 -20 8 0 0 0 0 160 (t[oJ |D 8||4,(t)] (t)=0 (t) A g(0)= 0 A 0 4(t)=0 q;(t)= B (0)=1 B=1 Į=(1)b 84+160q 0 i+204 0 sin t General solution is q, (t) = q,(0)cos t+ 20 sin20 t 20 4;(t) = 0xcos/20 t+ q(t)= sin /20 t Solution is x=q lの4 lx,() 2 -2lg,()J 1 lの」 2 -2 1 1 sin 20 t 1+sin20 t 2-2sin20 t x(t)= 1+ sinv20 t x.(t)= 2-2sin20 t

2 1.5 1 0.5 0 0 10 9 7 6 5 3 4 2 1 time 3 2.5 2 1.5 9 7 6 5 3 4 2 1 0 time 1 qdot = 0 4.4721 Tm k 160 x1 (t) x1 (t) O O 10

clc;clear all;close all;
M=[4 0;0 1];K=[40 -20;-20 10];
mode=[1 1;2 -2];
q=inv(mode)*[1;2]
qdot=inv(mode)*[sqrt(20);-2*sqrt(20)]
m=mode'*M*mode
k=mode'*K*mode
t=0:0.01:10;
x1=1+sin(sqrt(20)*t);
x2=2-sin(sqrt(20)*t);
subplot 211
plot(t,x1)
xlabel('time')
ylabel('x1(t)')
subplot 212
plot(t,x2)
xlabel('time')
ylabel('x1(t)')