ME 563
HOMEWORK # 6 SOLUTIONS
Fall 2010
PROBLEM 1:
Assume that there is viscous damping distributed along a string as it vibrates.
The damping coefficient
per unit length is c.
The density of the string is
ρ
, the tension is T, and the length is L.
Derive the form
of the free response, u(x,t), of the string assuming the string is fixed at both ends.
The equation of motion for such a string is:
where
The solution u(x,t)=G(t)*H(x) can then be substituted into this equation of motion yielding,
A separation of functions dependent on x and t, respectively, yields,
In order for this expression to be true for all t and x, the following two conditions must be satisfied:
After applying the boundary conditions, the form of the free response is found to be:
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2
PROBLEM 2:
Assume a solution of the form found in Problem #1 with c=0.
Given the triangular displacement initial
condition shown below with zero initial velocity, compute the free response along the string.
Plot the
solution at t=0 using 2, 3, 10, and 20 modes of vibration to determine the accuracy of the response.
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 Fall '09
 Boundary value problem

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