Approximate Methods for Multiple Degree of Freedom Systems: Chapter 9 Examples
Example 9-1
At some instant in time, the masses in the system shown below are in positions and
and have velocities
and
respectively.
- Determine the potential and kinetic energies in the system in terms of
,
,
and
from first principles.
- Show that the potential and kinetic energies can also be expressed as
where
and
Example 9-2
- For the system shown above, estimate the fundamental natural frequency using Rayleigh’s Quotient. Assume
and use trial vectors of
and comment on the results.
It was previously determined that - Repeat the calculations in part (1) assuming
,
, and
.
Note that the exact solution in this case is
Example 9-3
Use Rayleigh’s method to estimate the fundamental natural frequency for a light simply supported beam supporting three equally spaced identical masses.
Assume trial vectors of
Note that for this problem
Example 9-4
Estimate the fundamental natural frequency for the system shown below using Dunkerley’s Method.
Consider the following cases:
i)
ii)
iii)
Note that the stiffness matrix for this problem is
Example 9-5
Use Dunkerley’s method to estimate the fundamental natural frequency for a light simply supported beam supporting three equally spaced identical masses.
Note that for this system
Example 9-6
For the system shown below:
-
- Use matrix iteration to find the lowest and highest natural frequencies and associated mode shapes.
- Use Rayleigh’s quotient to find an upper bound to the lowest natural frequency. Use a trial vector of
- Use Dunkerley’s method to find a lower bound for the lowest natural frequency.
- Use Dunkerley’s method to find an upper bound for the highest natural frequency.
Note that for this problem,