II. Free Vibration

Study of the dynamic characteristics of a structure:

•natural frequencies

•normal modes (shapes)

Let f(t) = 0 and C = 0 (ignore damping) in the dynamic equation (8) and obtain

Assume that displacements vary harmonically with time, that is,

u (t ) = u sin( ωt ), u& (t ) = ω u cos( ωt ),

u&&(t ) = −ω 2 u sin( ωt ),

where u is the vector of nodal displacement amplitudes. Eq. (12) yields,

This is a generalized eigenvalue problem (EVP).

Solutions?

Trivial solution: u = 0 for any values of ω (not interesting).

Nontrivial solutions: u ≠ 0 only if

This is an n-th order polynomial of ω2, from which we can find n solutions (roots) or eigenvalues ωi.

ωi (i = 1, 2, …, n) are the natural frequencies (or characteristic frequencies) of the structure.

ω1 (the smallest one) is called the fundamental frequency.

For each ωi , Eq. (13) gives one solution (or eigen) vector

[K − ω i 2 M ]u i = 0 .

u i (i=1,2,…,n) are the normal modes (or natural modes, mode shapes, etc.).

Properties of Normal Modes

if ωi ≠ωj . That is, modes are orthogonal (or independent) to each other with respect to K and M matrices.

Normalize the modes:

Note:

•Magnitudes of displacements (modes) or stresses in normal mode analysis have no physical meaning.

•For normal mode analysis, no support of the structure is necessary.

ωi = 0 there are rigid body motions of the whole or a part of the structure.

apply this to check the FEA model (check for mechanism or free elements in the models).

•Lower modes are more accurate than higher modes in the FE calculations (less spatial variations in the lower modesfewer elements/wave length are needed).

Example:

We can see that mode 1 is calculated much more accurately than mode 2, with one beam element.