The motion equation of frequency response analysis when damping is not considered becomes as follows:
\[\begin{equation}
M \ddot{U} + K U = 0
\label{eq:2.6.1}
\end{equation}\]
If this is expanded for each eigenmode, it becomes
\[\begin{equation}
U = U_j e^{i \omega_j t}
\label{eq:2.6.2}
\end{equation}\]
If this is substituted into Eq.\(\eqref{eq:2.6.1}\), the following equation is obtained:
\[\begin{equation}
K U_j = \omega_j^2 M U_j
\label{eq:2.6.3}
\end{equation}\]
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:
\[\begin{align}
K U_j &= \lambda_j M U_j \nonumber \\\
K \overline{U_J} &= \overline{\lambda_J} M \overline{U_J}
\label{eq:2.6.4}
\end{align}\]
If this multiplied by \(\overline{U_J}^T\), the following equation is obtained:
\[\begin{align}
{U_j}^T K \overline{U_J} &= \overline{\lambda_J} {U_j}^T M \overline{U_J} \nonumber \\\
{\overline{U_J}}^T K U_j &= \lambda_j \overline{U_J}^T M U_j
\label{eq:2.6.5}
\end{align}\]
\[\begin{equation}
{U_j}^T M U_i = 0
\label{eq:2.6.11}
\end{equation}\]
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.
\[\begin{equation}
{U_i}^T M U_i = 1
\label{eq:2.6.12}
\end{equation}\]
Further, the frequency response analysis is formulated when damping is considered. The motion equation to be analyzed is expressed in Eq. Eq.\(\eqref{eq:2.6.13}\).
\[\begin{equation}
M \ddot{U} + C \dot{U} + K U = F
\label{eq:2.6.13}
\end{equation}\]
The damping term, assuming a Rayleigh-type damping, can be expressed as Eq.\(\eqref{eq:2.6.13}\).
\[\begin{equation}
C = \alpha M + \beta K
\label{eq:2.6.14}
\end{equation}\]
With the eigenvector obtained in eigenvalue analysis, the displacement vector can be expanded at time t as in Eq.\(\eqref{eq:2.6.15}\):
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.\(\eqref{eq:2.6.19}\):