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Ritz Analysis

Often the use of modal analysis requires a large number of modes to be calculated in order to capture the dynamic characteristics of the structure. This is particularly the case when the horizontal and vertical stiffnesses of the structure are significantly different (while the mass is the same). One way to circumvent this problem is to use Ritz (or Rayleigh-Ritz) analysis which yield approximate eigenvalues. While these are approximate they have some useful characteristics.

The eigenvalues (natural frequencies) are upper bounds to the true eigenvalues

The mode shapes are linear combinations of the exact eigenvectors

The number of Ritz vectors required to capture the dynamic characteristics of the structure is usually significantly less that that required for a proper eigenvalue analysis.

Ritz analysis method​

A set of trial vectors based initially on gravity loads in each of the x, y and z directions. The subsequent trial vectors are created from these with the condition that they are orthogonal to the previous vectors. The assumption is that we can get approximations to the eigenvectors by taking a linear combination of the trial vectors.

So for trial vectors

Xm=[x1x2x3⋯xm]\mathbf{X}_{m} = \left\lbrack x_{1} x_{2} x_{3} \cdots x_{m} \right\rbrack

Let

ϕ=Xms=∑i=1mxisi\phi = \mathbf{X}_{m}s = \sum_{i = 1}^{m}{x_{i}s_{i}}

and if the approximation to the eigenvalue is λ\lambda, the residual associated with the approximating pair {λ,ϕ}\left\{ \lambda,\phi \right\} is given by

r=Kφ−ΛMφ\mathbf{r} = \mathbf{K}\boldsymbol{\varphi} - \Lambda\mathbf{M}\boldsymbol{\varphi}

The Rayleigh-Ritz method requires the residual vector be orthogonal to each of the trial vectors, so

XmTr=XmTKϕ−λXmTMϕ=0{\mathbf{X}_{m}}^{T}\mathbf{r} = {\mathbf{X}_{m}}^{T}\mathbf{K}\phi - \lambda{\mathbf{X}_{m}}^{T}\mathbf{M}\phi = 0

Substituting for ϕ\phi\mathbf{} from above gives

XmTKXms−λXmTMXms=0{\mathbf{X}_{m}}^{T}\mathbf{K}\mathbf{X}_{m}\mathbf{s} - \lambda{\mathbf{X}_{m}}^{T}\mathbf{M}\mathbf{X}_{m}\mathbf{s} = 0

or

Kms−λMms=0\mathbf{K}_{m}\mathbf{s} - \lambda\mathbf{M}_{m}\mathbf{s} = 0

with

Km=XmTKXm\mathbf{K}_{m} = {\mathbf{X}_{m}}^{T}\mathbf{K}\mathbf{X}_{m}
Mm=XmTMXm\mathbf{M}_{m} = {\mathbf{X}_{m}}^{T}\mathbf{M}\mathbf{X}_{m}

This eigenproblem is then solved for the eigenpairs {λ,s}\left\{ \lambda,\mathbf{s} \right\} and then the approximate eigenvectors are evaluated from

ϕ=Xms=∑i=1mxisi\phi = \mathbf{X}_{m}s = \sum_{i = 1}^{m}{x_{i}s_{i}}

Ritz trial vectors​

The algorithm as applied in a single direction is as follows:

Create a load vector f\mathbf{f} corresponding to a gravity load in the direction of interest

Solve for first vector

KX1∗=fsolve forX1∗X1TMX1=1normalizeM\begin{array}{lll} \mathbf{K}{\mathbf{X}_{1}}^{*} = \mathbf{f} & \text{solve for} & {\mathbf{X}_{1}}^{*} \\ {\mathbf{X}_{1}}^{T}\mathbf{M}\mathbf{X}_{1} = 1 & \text{normalize} & \mathbf{M} \end{array}

Solve for additional vectors

KXi∗=MXi−1solve for X1∗cj=XjTMXi∗forj=1,…,i−1Xi=Xi∗−∑j=1i−1cjXjorthogonalizeMXmTMXm=1normalizeM\begin{array}{lll} \mathbf{K}{\mathbf{X}_{i}}^{*} = \mathbf{M}\mathbf{X}_{i - 1}&\text{solve for }&{\mathbf{X}_{1}}^{*}\\ c_{j} = {\mathbf{X}_{j}}^{T}\mathbf{M}{\mathbf{X}_{i}}^{*}&\text{for}&j = 1,\ldots,i - 1 \\ X_{i} = {X_{i}}^{*} - \sum_{j = 1}^{i - 1}{c_{j}\mathbf{X}_{j}}&\text{orthogonalize} & \mathbf{M}\\ {\mathbf{X}_{m}}^{T}\mathbf{M}\mathbf{X}_{m} = 1&\text{normalize}&\mathbf{M} \end{array}