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(mech300g)[2006](s)midterm~PPSpider^midterm_10518.pdf
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Solution to MECH300G Midterm Exam Spring 2006
Student Name:____________ Student ID:_______________ Student Signature:__________
Problem 1 (30 pts) 2DOF Vibration Analysis using Matlab
1.1 Please describe how to use Matlab to determine the free vibration response (time response) of a two degree of freedom system with equations of motion:
10

2

.
1 12
.

.

.

.

r

r
&&
r

2

x(t)
+

8

x(t)
=

0

..

..

..

..

01

.

with initial conditions
...
... ...
10
,
01
1.2 Please describe how to determine the frequency response of this system to the externally applied force vector {F10 cos t 0}T.
Ans:
This vibrating system (M, K) and initial conditions are exactly the same problem in Homework #4 (Problem 6.73). The method to solve this problme is exactly the same as Problem 6.92 in HW#4.
1.1 Modify the beginning of PROGRAM8.M according to the modal analysis results calculated in Homework #4:
n=2;
nvec=2;
nstep=500;
delt=0.05;

xm=[2 0; 0 2];

omf=[5 10];

om=[2 2*sqrt(3)];

z=[0.0 0.0];

x0=[1.0 0.0];
xd0=[0.0 1.0];

ev=[1 -1;1 1]/sqrt(2);

for i=1:nstep
time=i*delt;
f(1,i)=0*cos(3.5*time);

end
for i=1:20
f(2,i)=0*f(1,i);

...

r

r
&
x
(0)

(0)

x
=

=

end

The forced response of 2DOF has been discussed in Slide 54-61 of the lecture note (http://teaching.ust.hk/~mech300/mech300_4_1.pdf) and Section 5.6 of the textbook.
Method 1:
Follow the matlab program similar to Example 5.14 in the textbook.

Method 2 (take advantage of Matlabs matrix and vector operation): om is the frequency vector, xt is the frequency response vector
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))

Problem 2 (30 pts) Vibration of Aircraft
To make a more precise model for the vibration of the aircraft wing, the aerospace engineer is trying to create a two-degree-of-freedom model for the wing. If the wing of an aircraft can be modeled as a cantilever beam of uniform cross section, a b, as shown in the following figure. We assume the mass of the wing, M, can be lumped into two concentrated masses: one is right at the position of the engine, the other is located at the free end of the wing (cantilever beam). Note that the mass of engine, m, should be considered. The aircraft engine has a rotating unbalance mass m at radius r. Assume damping and the effect of the wing between the engine and the free end to be negligible.
2.1 Derive the 22 flexibility matrix of this system. Assume l = L/2.
2.2 The equations of motion
2.3 Find the natural frequencies and the vibration mode(s) of this system. Assume m = M/5, E = 70GPa, a = 1 m, b = 10 m, L = 50 m, M = 1000kg, = 2700 kg/m3.
2.4 Determine the maximum deflection of the system at an angular velocity of f rad/s. Assume f = 1000 rad/s, r =1 cm.