Frequency Response Analysis

Frequency Response Analysis

Formulation

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:

$$\begin{align} {U_j}^T K \overline{U_J} &= \overline{\lambda_J} {U_j}^T M \overline{U_J} \nonumber \\\ {\overline{U_J}}^T K U_j &= \lambda_j \overline{U_J}^T M U_j \label{eq:2.6.5} \end{align}$$

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

$$\begin{equation} 0 = (\lambda_j - \overline{\lambda_J}) \overline{U_J}^T M U_j \label{eq:2.6.6} \end{equation}$$

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

$$\begin{equation} \overline{U_J} M U_j > 0 \label{eq:2.6.7} \end{equation}$$

holds for eigenvectors that are not zero vectors. Therefore,

$$\begin{equation} \lambda_j = \overline{\lambda_J} \label{eq:2.6.8} \end{equation}$$

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

$$\begin{align} K U_i &= \lambda_i M U_i \nonumber \\\ K U_j &= \lambda_j M U_j \label{eq:2.6.9} \end{align}$$

From this, the following is obtained:

$$\begin{equation} (\lambda_i - \lambda_j) {U_j}^T M U_j = 0 \label{eq:2.6.10} \end{equation}$$

If the eigenvalue is different, it becomes

$$\begin{equation} {U_j}^T M U_i = 0 \label{eq:2.6.11} \end{equation}$$

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.$\eqref{eq:2.6.15}$:

$$\begin{equation} U(t) = \sum_{i} b_i(t) U_i \label{eq:2.6.15} \end{equation}$$

where the external force term,

$$\begin{equation} F(t) = \lbrace F_R + iF_I \rbrace e^{i\Omega t} \label{eq:2.6.16} \end{equation}$$

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:

$$\begin{equation} b_j(t) = (b_{jR} + b_{jI}) e^{j\Omega t} \label{eq:2.6.17} \end{equation}$$

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.$\eqref{eq:2.6.19}$:

$$\begin{equation} b_{jR} = \frac{ {U_j}^T F_R({\omega_j}^2 - \Omega^2) + {U_j}^T F_I(\alpha+\beta{\omega_j}^2)\Omega } { ({\omega_j}^2 - \Omega^2)^2 + (\alpha + \beta{\omega_j}^2)^2 \Omega^2 } \label{eq:2.6.18} \end{equation}$$

$$\begin{equation} b_{jI} = \frac{ {U_j}^T F_I({\omega_j}^2 - \Omega^2) + {U_j}^T F_R(\alpha+\beta{\omega_j}^2)\Omega } { ({\omega_j}^2 - \Omega^2)^2 + (\alpha + \beta{\omega_j}^2)^2 \Omega^2 } \label{eq:2.6.19} \end{equation}$$