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Problems
for Chapter 5
Analytical
Techniques and Solutions for Linear Elastic Solids
5.10. Solutions
to Dynamic Problems
5.10.1. Use the Rayleigh-Ritz method to obtain the natural
frequency of vibration of the spring-mass system shown (the displacement
associated with the vibration mode is trivial)
5.10.2. Use the Rayleigh-Ritz method to estimate the
fundamental frequency of the spring-mass system shown. You should be able to obtain an exact result,
by describing the mode shape in terms of a single parameter, and minimizing
the frequency appropriately.
5.10.3. Reconsider problem 5.10.2. Try to find the second frequency of vibration for the system by selecting another
approximation to the mode shape, which is (by construction) orthogonal to the
first.
5.10.4. Use the Rayleigh-Ritz method to estimate the
fundamental frequency of the clamped-pinned beam illustrated in the figure.
Assume that the beam has Young’s modulus  and mass density ,
and its cross-section has area A and
moment of area .
5.10.5. Use the Rayleigh-Ritz method to estimate the
fundamental frequency of the pinned-pinned beam illustrated in the figure. Assume
that the beam has Young’s modulus and mass density ,
and its cross-section has area A and
moment of area .
5.10.6. A beam with length L Young’s modulus and mass density ,
and its cross-section has area A and
moment of area is bonded to an elastic foundation, which
exerts a restoring force per unit length on the beam.
The beam is pinned at both ends. Use the Rayleigh-Ritz method to
estimate the natural frequency of vibration of the beam.
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