## 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.

1. Determine the potential and kinetic energies in the system in terms of , , and from first principles.
2. Show that the potential and kinetic energies can also be expressed as

where

and

### Example 9-2

1. 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

2. 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:

1. Use matrix iteration to find the lowest and highest natural frequencies and associated mode shapes.
2. Use Rayleigh’s quotient to find an upper bound to the lowest natural frequency. Use a trial vector of

3. Use Dunkerley’s method to find a lower bound for the lowest natural frequency.
4. Use Dunkerley’s method to find an upper bound for the highest natural frequency.

Note that for this problem,