=========================preview======================
(mech300g)[2007](s)midterm~PPSpider^midterm_10520.pdf
Back to MECH300G Login to download
======================================================
Solution to MECH300G Midterm Exam
Spring 2007

Date: 29 March 2006 Time: 1:25 pm C 2:55 pm Venue: Rm 2306 (2/F, Lift 17-18)
Student Name:____________
Student ID:_______________
Student Signature:__________
Problem 1 (20 pts)
A spring-mass system with m=10 kg and k = 5000 N/m is subjected to a harmonic force of amplitude 250 N and frequency . If the maximum amplitude of the mass is observed to be 100 mm, find the value of driving frequency in Hz.
Ans:
n =

k / m =
5000 /10 = 22.3607 rad/s

st = F0/ k = 250 /5000 = 0.05m
X = st (1/(1. ( /n )2 )
1/2 1/2
i.e. =n (1. st / X ) = 22.3607(1. 0.05/ 0.10) = 15.8114 rad/s
f =15.8114/(2)=2.52 Hz
Problem 2 (30 pts)
2.1 Please describe how to use Matlab to determine the forced vibration response (frequency
response) of a two degree of freedom system with the following equations of motion: .10. r . 2 .1. r .F10 cost.
2..&x&(t) + 8..x(t) =..
01 .12 0
..... .
2.2 Please describe how to use Matlab to calculate the natural frequencies and vibration modes?
rr
2.3 If F10 = 0 and x(0) = [1 0]T , x&= 0 , please describe how to use Matlab to determine the forced vibration using Modal Analysis.
Ans:

2.1 Forced Vibration Response
xt=[ ]; om=[ ];
m=[2 0; 0 2]; k=8*[2 -1; -1 2];
for i=1: 101

w= 5*(i-1)/100;
om=[om w];
x=inv(k-w^2*m)*[1; 0];
xt =[xt x];

end
om=om;

xt=xt;
plot(om,xt(:,1),om,xt(:,2))

2.2 Natural Freq and Vibration Modes
a=inv(k);
D=a*m;
lambda=eig(D)
% natural frequency: freq is a vector
freq=sqrt(1./lambda)
% dd is the vibration modes; dd=[X1 X2]
[vv,dd]=eig(D)


2.3 Modal Analysis
w1=freq(1); w2=freq(2);

v1=vv(:,1)/vv(1,1);
v2=vv(:,2)/vv(1,2);
% X1, X2 are orthonormalized eigenvector/vibration mode
M1=v1'*m*v1; norm1=1/sqrt(M1);
X1=v1*norm1;
M2=v2'*m*v2; norm2=1/sqrt(M2);
X2=v2*norm2;

% X is the Modal Matrix
X=[X1 X2];

% x0 = initial displacement vector
% xd0 = initial vel. vector
xd0=[0 0]';
x0 =[1 0]';

% q0 = initial displacement in modal space
% qd0= initial vel. Vector in modal space
q0=X'*m*x0;
qd0=X'*m*xd0;

% note qd0 = zero vector
q10=q0(1); q20=q0(2);
t=0:.1:20;

q1t=q10*cos(w1*t);
q2t=q20*cos(w2*t);

%
xt_f=kron(q1t,X1)+kron(q2t,X2)

% final vibration amplitude as a function of time
% x1m(t), x2m(t)
x1m=xt_f(1,:);
x2m=xt_f(2,:);

Problem 3 (30 pts)
Two identical pendulums, each with mass m and length l, are connected by a spring of stiffness k at a distance d from the fixed end as shown in the following figure.

3.1 Assume small vibration, derive the equations of motion of the two masses.
3.2 Find the natural frequencies and mode shapes of the system. m = 1, k =1, l = 1, d = ..
3.3 Find the fundamental natural frequency using Rayleighs method (Assume the trial vector (1,2) has the same componentsC one). m = 1,