# Frequency Response Analysis

## Frequency Response Analysis

### Formulation

The motion equation of frequency response analysis when damping is not considered becomes as follows:

If this is expanded for each eigenmode, it becomes

If this is substituted into Eq.$\eqref{eq:2.6.1}$, the following equation is obtained:

The following is the proof that this eigenfrequency is real. By defining $\omega_j^2 = \lambda_j$ removing the complex conjugate of Eq.$\eqref{eq:2.6.3}$, Eq.$\eqref{eq:2.6.4}$, the following equation is obtained:

If this multiplied by $\overline{U_J}^T$, the following equation is obtained:

From Eq.$\eqref{eq:2.6.5}$, it becomes

In this case, the mass matrix is a positive-definite symmetric matrix; thus,

holds for eigenvectors that are not zero vectors. Therefore,

and ${\omega_j}^2 = \lambda_j$ becomes a real number. In this case, two different modes are analyzed.

From this, the following is obtained:

If the eigenvalue is different, it becomes

That is, different eigenmodes are orthogonal to the mass matrix. The advantage of same modes is that if they are normalized for the mass matrix Eq.$\eqref{eq:2.6.12}$, the handling becomes easier.

Further, the frequency response analysis is formulated when damping is considered. The motion equation to be analyzed is expressed in Eq. Eq.$\eqref{eq:2.6.13}$.

The damping term, assuming a Rayleigh-type damping, can be expressed as Eq.$\eqref{eq:2.6.13}$.

With the eigenvector obtained in eigenvalue analysis, the displacement vector can be expanded at time t as in Eq.$\eqref{eq:2.6.15}$:

where the external force term,

defines $b_j(t)$ in the harmonic oscillator equation. The following motion equation Eq.$\eqref{eq:2.6.13}$ acquires the form of forced vibration holds:

If the real and imaginary parts of the expansion coefficient of $b_i(t)$ are determined, it becomes Eq.$\eqref{eq:2.6.18}$ and Eq.$\eqref{eq:2.6.19}$: