Unformatted text preview: ME563 – Fall 2011
Purdue University
West Lafayette, IN Homework Set No. 8
Assignment date: Friday, October 14
Due date: Friday, October 21
Please attach this cover sheet to your completed homework assignment. Name
PUID Problem 8.1
Problem 8.2
Problem 8.3 TOTAL Homework 8.1
The EOMs for an undamped twoDOF system are given by: 100
2 −1
0
0
¨ 0 1 0 + −1
2 −1 = 0 sinΩt
x
x
001
0 −1
1
1
The particular solution of these EOMs is to be written as: P (t) = XsinΩt.
x
1. Determine all resonance and antiresonance frequencies for X1 (Ω), X2 (Ω) and X3 (Ω). Also,
determine the values of X1 (0), X2 (0) and X3 (0).
2. Make hand sketches of X1 (Ω), X2 (Ω) and X3 (Ω) vs. Ω. These sketches should clearly
show the resonance and antiresonance frequencies, as well as the values for X1 (0), X2 (0) and
X3 (0) found above.
You may use Matlab to determine the resonance frequencies. All other work (including your
sketches) should be done by hand. Homework 8.2
Consider a damped twoDOF with the following EOMs:
¨ + [C ] + [K ] = 2 sinΩt + 0 cosΩt
˙
[M ]
x
x
x
0
1 (N ewtons) The real and imaginary parts of three elements of the complex transfer matrix H (Ω) vs. Ω are
shown on the following pages.
Determine the steadystate responses xP 1 (t) and xP 2 (t) corresponding to Ω = 0.5 rad/sec. Write
your solutions as xP j (t) = Xj sin(Ωt − φj ). 50
40 real(H11) mm/N 30
20
10
0
10
20
30
40 0 0.5 1 1.5
rad/sec 2 2.5 3 0 0.5 1 1.5
rad/sec 2 2.5 3 20
10
0 imag(H11) mm/N 10
20
30
40
50
60
70
80 50
40 real(H12) mm/N 30
20
10
0
10
20
30
40 0 0.5 1 1.5
rad/sec 2 2.5 3 0 0.5 1 1.5
rad/sec 2 2.5 3 20
10
0 imag(H12) mm/N 10
20
30
40
50
60
70
80 50
40 real(H22) mm/N 30
20
10
0
10
20
30
40 0 0.5 1 1.5
rad/sec 2 2.5 3 0 0.5 1 1.5
rad/sec 2 2.5 3 20
10
0 imag(H22) mm/N 10
20
30
40
50
60
70
80 Homework 8.3
The damped singleDOF shown below experiences a Tperiodioc excitation, g (t). The excitation
g (t) is deﬁned as being equal to the harmonic function shown below for −T /4 < t < T /4 and equal
to zero elsewhere over −T /< t < T /2. Since g (t) is Tperiodic, it can be written as the following
2
Fourier series: g (t) = g0 + ∞ [gcj cosj Ωt + gsj sinj Ωt], where Ω = 2π /T .
j =1
1. Determine the Fourier coeﬃcients g0 , gcj and gsj for g (t). 2. If the steadystate response is written in its Fourier series as xP (t) = X (0) +
φj ), determine the response Fourier amplitudes X (j ) . ∞ j =1 X (j ) sin(j Ωt− 3. Numerically evaluate kX (j ) /A for j = 1, 2, 3, 4. Use damping ratio and natural frequency
values of ζ = 0.1 and ωn = π /T , respectivley. ...
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This document was uploaded on 12/23/2011.
 Fall '09

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