## Advanced Dynamics and Vibrations: Numerical methods for vibrating systems

[Jason’s part goes here]

where we can have a different mass as the equation is different. For SHM

If we choose then

Therefore:

Note:

If

Now using the result above

Consider

Note:

We can apply this idea to various dynamical systems which are more complicated than the simple spring/mass. Consider an approximation to longitudinal vibration of a bar

Consider a beam of total mass and cross section modulus which we approximate as

if , go to

if

Therefore:

Therefore:

N = 4

To calculate the mode shapes one can start at the free end e.g.:

Assume

Now we use the recursion relationship

Therefore:

For the 4 storey building for the lowest natural frequency is

Therefore:

Check

Therefore:

if ( is low) then:

for the lower frequencies

where is density

For other boundary conditions, the expression for is determined then the natural frequencies are known. For fixed ends.

Consider the fixed boundaries:

Therefore:

Therefore:

Therefore: