Modal Analysis - FrontISTR Documentation

Generalized Eigenvalue Problem¶

For free-vibration analysis of a continuum, the domain is spatially discretized and modeled as a multi-degree-of-freedom system of lumped masses, as shown in Figure 2.3.1. For an undamped free-vibration problem, the governing equation (equation of motion) is as follows.

\[\begin{equation} M u + K \ddot{u} = 0 \label{eq:2.3.1} \end{equation}\]

Here, $u$ is the generalized displacement vector, $M$ is the mass matrix, and $K$ is the stiffness matrix. Let the natural angular frequency be $\omega$, let $a$, $b$, and $c$ be arbitrary constants, and let $x$ be a vector. Define the function

\[\begin{equation} u(t) = (a \sin \omega t + b \cos \omega t ) x \label{eq:2.3.2} \end{equation}\]

The second derivative of this expression is

\[\begin{equation} \ddot{u}(t) = -\omega^2 (a \sin \omega t + b \cos \omega t) x \label{eq:2.3.3} \end{equation}\]

Substituting these expressions into Equation $\eqref{eq:2.3.1}$ gives

\[\begin{equation} M u + K \ddot{u} = (a \sin \omega t + b \cos \omega t) (- \omega^2 M + K x ) = ( -\lambda M + K x) = 0 \label{eq:2.3.4} \end{equation}\]

That is,

\[\begin{equation} K x = \lambda M x \label{eq:2.3.5} \end{equation}\]

is obtained.

Thus, if Equation $\eqref{eq:2.3.5}$ is satisfied by the coefficient $\lambda = \omega^2$ and the vector $x$ can be found, the function $u(t)$ is a solution of Equation $\eqref{eq:2.3.1}$.

The coefficient $\lambda$ is called an eigenvalue and the vector $x$ an eigenvector. The problem of determining them from Equation $\eqref{eq:2.3.1}$ is called a generalized eigenvalue problem.

Example of a multi-degree-of-freedom system in undamped free vibration

Figure 2.3.1 Example of a multi-degree-of-freedom system in undamped free vibration

Matrix Properties and Assumptions¶

For the generalized eigenvalue problem $K x = \lambda M x$ obtained in the previous section, this manual assumes the following matrix properties. These assumptions underlie the convergence and range of applicability of the shifted inverse iteration and Lanczos methods described below. For a complex matrix, the transpose is the complex conjugate, while a real matrix is symmetric. Specifically, for matrix $K$, if its $ij$ component is $k_{ij}$, and the complex conjugate of $k$ is denoted by $\bar{k}$, then

\[\begin{equation} k_{ij} = \bar{k}_{ji} \label{eq:2.3.6} \end{equation}\]

the following relation holds.

In this manual, the matrices are assumed to be symmetric and positive definite. Positive definite means that all eigenvalues are positive; equivalently, the matrix always satisfies Equation $\eqref{eq:2.3.7}$ below.

\[\begin{equation} x^{t} A x > 0 \label{eq:2.3.7} \end{equation}\]

Shifted Inverse Iteration¶

In finite element structural analysis, all eigenvalues are generally unnecessary in practice; in many cases, only a few low-order eigenvalues are sufficient. HEC-MW is intended for large-scale problems, for which the matrices are large and very sparse (contain many zero entries). It is therefore important to compute the eigenvalues of low-order modes efficiently with this property in mind.

Let $\sigma$ be the lower bound of the eigenvalues. Equation $\eqref{eq:2.3.5}$ can then be transformed into the following mathematically equivalent form.

\[\begin{equation} (K - \sigma M)^{-1} M x = \frac{1}{(\lambda-\sigma)} x \label{eq:2.3.8} \end{equation}\]

This transformation has the following useful properties for computation.

The mode ordering is inverted.
Eigenvalues near $\rho$ are mapped to the largest values.

In actual computation, the largest eigenvalues are often obtained first. Therefore, rather than Equation $\eqref{eq:2.3.5}$, the main convergence calculation is applied to Equation $\eqref{eq:2.3.8}$, with the aim of obtaining eigenvalues near $\rho$ first. This technique is called shifted inverse iteration.

Lanczos Method¶

Rationale for Use (Comparison with the Jacobi Method)¶

Among classical methods, the Jacobi method is well known.

This method is effective when the matrix is small and dense. However, because the matrices handled by HEC-MW are large and sparse, the Jacobi method is not used; instead, the iterative Lanczos method is employed.

Algorithm and Characteristics¶

Proposed by C. Lanczos in the 1950s, this method is an algorithm for reducing a matrix to tridiagonal form and has the following characteristics.

It is an iterative convergent method and can proceed while keeping the matrix sparse.
The algorithm primarily consists of matrix-vector products and is well suited to parallelization.
It is well suited to geometric domain decomposition associated with finite element meshes.
Efficient computation is possible by limiting the number of eigenvalues and the range of modes to be obtained.

The Lanczos method starts from an initial vector, successively generates orthogonal vectors, and constructs a basis for a subspace. It is said to be faster than the subspace method, another iterative method, and is widely used in finite element programs. However, this method is sensitive to numerical errors, which can destroy vector orthogonality and may cause the computation to break down. Measures against such numerical errors are therefore essential.

Geometric Interpretation (Krylov Subspace)¶

By introducing the following change of variables in Equation $\eqref{eq:2.3.8}$,

\[ A = (K - \sigma M)^{-1} M \]

\[\begin{equation} \frac{1}{\lambda-\sigma}= \zeta \label{eq:2.3.9} \end{equation}\]

the problem can be rewritten as

\[\begin{equation} A x = \zeta x \label{eq:2.3.10} \end{equation}\]

is obtained.

For an appropriate vector $q_0$, apply the linear transformation represented by matrix $A$ (see Figure 2.3.2).

$Linear transformation by matrix $A$ of $q_0$$

Figure 2.3.2 Linear transformation by matrix $A$ of $q_0$

The transformed vector is orthogonalized within the space spanned together with the original vector. That is, Gram-Schmidt orthogonalization is performed as illustrated in Figure 2.3.2. Let the resulting vector be $r_1$. Normalize it to unit length to obtain $q_1$ (Figure 2.3.3). From $q_1$, obtain $q_2$ using the same procedure. At this point, $q_2$ is orthogonal to both $q_1$ and $q_0$ (Figure 2.3.4). Continuing the same calculation yields mutually orthogonal vectors up to the order of the matrix.

$A vector orthogonal to $q_0$: $q_1$$

Figure 2.3.3 A vector orthogonal to $q_0$: $q_1$

$A vector orthogonal to $q_1$ and $q_0$: $q_2$$

Figure 2.3.4 A vector orthogonal to $q_1$ and $q_0$: $q_2$

In particular, the Lanczos algorithm applies to the vector sequence $A q_0$, $A q_1$, $A q_2$

or equivalently $A q_0$, $A^2 q_0$, $A^3 q_0$, ,$A^n q_0$

by Gram-Schmidt orthogonalization. This vector sequence is called a Krylov sequence, and the space it spans is called a Krylov subspace. When Gram-Schmidt orthogonalization is performed in this space, each new vector can be obtained using the two most recent vectors. This is called the Lanczos principle.

Tridiagonalization¶

In the above iteration, the calculation for the (i+1)-th vector can be written as

\[\begin{equation} \beta_{i+1} q_{i+1} + \alpha_{i+1} q_{i} + \gamma_{i+1} q_{i-1} = Aq_{i} \label{eq:2.3.11} \end{equation}\]

where

\[ \beta_{i+1} = \frac{1}{||r_{i+1}||} \]

\[ \alpha_{i+1} = \frac{(q_i, Aq_i)}{(q_i, q_i)} \]

\[\begin{equation} \gamma_{i+1} = \frac{(q_{i-1}, Aq_i)}{(q_{i-1}, q_{i-1})} \label{eq:2.3.12} \end{equation}\]

Using matrix notation, this becomes

\[\begin{equation} AQ_m = Q_m T_m \label{eq:2.3.13} \end{equation}\]

where

\[ Q_m = [q_{1}, q_{2}, q_{3}, \ldots ,q_{m}] \]

\[\begin{equation} T= \begin{pmatrix} \alpha_{1} & \gamma_{1} & & &\\ \beta_{2} & \alpha_{2} & \gamma_{2} & & \\ & \cdots & & &\\ & & & \beta_{m} & \alpha_{m} \end{pmatrix} \label{2.3.14} \end{equation}\]

Thus, the eigenvalues are obtained by solving the eigenvalue problem for the tridiagonal matrix obtained from Equation $\eqref{eq:2.3.13}$.

Frequency Response Analysis — Frequency response using modal-analysis results
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