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Frequency Response Analysis

Problem Formulation

Frequency response analysis evaluates the steady-state response amplitude and phase of a system in the frequency domain when the external force varies harmonically with time. Because the governing equation of motion is treated differently with and without damping, this chapter first derives the eigenmodes from undamped free vibration and then uses them as a modal basis to expand the damped harmonic response.

Undamped Free Vibration and Eigenmodes

When damping is neglected, the equation of motion is

\[\begin{equation} M \ddot{U} + K U = 0 \label{eq:2.6.1} \end{equation}\]

Expanding the displacement in each eigenmode gives

\[\begin{equation} U = U_j e^{i \omega_j t} \label{eq:2.6.2} \end{equation}\]

Substituting this into Eq.\(\eqref{eq:2.6.1}\) gives

\[\begin{equation} K U_j = \omega_j^2 M U_j \label{eq:2.6.3} \end{equation}\]

This is a generalized eigenvalue problem. Numerical solution methods for it, including shifted inverse iteration and the Lanczos method, are covered in Modal Analysis. This section summarizes the properties of the resulting natural frequencies and eigenmodes that are used in the subsequent harmonic-response expansion.

Reality of the Natural Frequencies

The natural frequency can be shown to be real as follows. Let \(\omega_j^2 = \lambda_j\). Taking the complex conjugate of Eq.\(\eqref{eq:2.6.3}\) gives Eq.\(\eqref{eq:2.6.4}\).

\[\begin{equation} K U_j = \lambda_j M U_j K \overline{UJ} = \overline{\lambda_J} M \overline{U_J} \label{eq:2.6.4} \end{equation}\]

Multiplying by \(\overline{U}_J^T\) gives

\[\begin{equation} U_j^T K \overline{U}_J = \overline{\lambda}_J U_j^T M \overline{U}_J \overline{U}_J^T K U_j = \lambda_j \overline{U}_J^T M U_j \label{eq:2.6.5} \end{equation}\]

From Eq.\(\eqref{eq:2.6.5}\),

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

Because the mass matrix is symmetric positive definite, for a nonzero eigenvector we have

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

Therefore,

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

Thus, \(\omega_j^2 = \lambda_j\) is real.

Orthogonality and Normalization of Eigenmodes

Consider two different modes.

\[\begin{equation} K U_i = \lambda_i M U_i K U_j = \lambda_j M U_j \label{eq:2.6.9} \end{equation}\]

It follows that

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

and, when the eigenvalues are different,

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

Thus, different eigenmodes are orthogonal with respect to the mass matrix. For the same mode, normalizing with respect to the mass matrix as in Eq.\(\eqref{eq:2.6.12}\) makes the subsequent treatment simpler.

\[\begin{equation} U_i^T M U_i = 1 \label{eq:2.6.12} \end{equation}\]

Harmonic Response with Damping

Next, the formulation of frequency response analysis with damping is presented. The governing equation of motion is given by 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 is assumed to be of Rayleigh type and is expressed as Eq.\(\eqref{eq:2.6.14}\).

\[\begin{equation} C = \alpha M + \beta K \label{eq:2.6.14} \end{equation}\]

Using the eigenvectors obtained from modal analysis, the displacement vector at time t can be expanded as Eq.\(\eqref{eq:2.6.15}\).

\[\begin{equation} U(t) = \sum_i b_i(t) U_i \label{eq:2.6.15} \end{equation}\]

Suppose the external-force term is the harmonic form

\[\begin{equation} F(t) = ( F_R + i F_I )e^{i \Omega t} \label{eq:2.6.16} \end{equation}\]

We then determine \(b_{j}(t)\). Since the equation of motion in Eq.\(\eqref{eq:2.6.13}\) has the form of forced vibration,

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

holds. The real and imaginary parts of the expansion coefficient \(b_{j}(t)\) are given by Eqs.\(\eqref{eq:2.6.18}i\) and \(\eqref{eq:2.6.19}\), respectively.

\[\begin{equation} b_{jR} = \frac{ U^T_j F_R (\omega^2_j - \Omega^2) + U^T_j F_I (\alpha + \beta \omega_j^2) \Omega}{ (\omega^2_j - \Omega^2)^2 + (\alpha + \beta \omega_j^2)^2 \Omega^2} \label{eq:2.6.18} \end{equation}\]
\[\begin{equation} b_{jI} = \frac{ U^T_j F_I(\omega^2_j - \Omega^2) - U^T_j F_R(\alpha + \beta \omega_j^2) \Omega}{ (\omega^2_j - \Omega^2)^2 + (\alpha + \beta \omega_j^2)^2 \Omega^2} \label{eq:2.6.19} \end{equation}\]

This gives the desired harmonic-response coefficients.