Part I Problems and Solutions
Problem 1: A driven spring-mass-dashpot system is modeled by the DE
..
.
mx + cx + kx = F0 cos ωt
with m = 1, c = 6, and k = 45. F0 = 50. Find the amplitude A(ω ) of the response as a
function of the input frequency ω and find the frequency which gives the largest system
response. Is this a system for which ‘practical resonance’ occurs?
Solution: Using the formulas derived in this session, we have
�
k
m
1
2
� c �2 � 12
A(ω ) = F0
��
A(ω ) = 50
��
�2
+ c2 ω 2
�2
+ 36ω 2
k − mω 2
45 − ω 2
�− 12
�− 12
ωmax =
− m
= the frequency which gives practical resonance if
√
√
√
c < 4km. In this case,
� c = 6 < √ 4 · 45 · 1 = 6 5. So the maximum system response
1 36
occurs when ωmax = 45
27 ≈ 5.196( rad
1 − 2 1 =
sec ).
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18.03SC Differential Equations��
Fall 2011 ��
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