Modal analysis is the study of the natural characteristic of structures.

Every piping system has natural characteristics. When we impose vibration on a piping system, the piping system takes deformation patterns when the vibration frequency approaches the natural frequencey of the piping system. The deformation pattern at the natural frequencies takes on variety of shapes depending on which frequency is used to excite the piping system.

The deformation patterns are referred to as mode shapes of the piping system.

The natural frequency and mode shape depend on the mass and stiffness distribution in the piping system.

f = Natural frequency (cycles/sec)

p = Period (seconds/cycle)

f = 1/p

The absolute magnitude of the displacement in a mode shape computed in an eigensolution is unknown, only the shape of the mode is known, the maximum displacement is indeterminate.

For a cantilever, the natural frequency of the various modes of vibration can be calculated as :

**ω _{n} = (0.597 n Π)^{2} √ [ E I / (m L^{4}) ]**

where:

- ω
_{n}= natural frequency of mode n, rad/sec - n = mode number
- E = modulus of elasticity of cantiliver psi
- I = moment of Inertia of cantilever in
^{4} - m = unit mass per length of cantilever slug/in
- L = Length of cantilever

From the above equation it is obvious that the natural frequency of the second mode of vibration of the cantilever is four times that of the first mode of vibration.

A modal analysis of piping system can be carried out using Caesar pipe stress analysis program. The modal of the piping system is first generated and the first 5 to 10 modes of vibration are extracted.

With ascending frequencies, more points in a system are tied to their original position, which means that higher order modes are tied more tightly to their original position. Hence, higher modes are less likely to vibrate than the lower modes.

to be continued .......