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

$$\mathbf{X}_{m} = \left\lbrack x_{1} x_{2} x_{3} \cdots x_{m} \right\rbrack$$

Let

$$\phi = \mathbf{X}_{m}s = \sum_{i = 1}^{m}{x_{i}s_{i}}$$

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

$$\mathbf{r} = \mathbf{K}\boldsymbol{\phi} - \Lambda\mathbf{M}\boldsymbol{\phi}$$

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

$${\mathbf{X}_{m}}^{T}\mathbf{r} = {\mathbf{X}_{m}}^{T}\mathbf{K}\phi - \lambda{\mathbf{X}_{m}}^{T}\mathbf{M}\phi = 0$$

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

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

or

$$\mathbf{K}_{m}\mathbf{s} - \lambda\mathbf{M}_{m}\mathbf{s} = 0$$

with

$$\mathbf{K}_{m} = {\mathbf{X}_{m}}^{T}\mathbf{K}\mathbf{X}_{m}$$

$$\mathbf{M}_{m} = {\mathbf{X}_{m}}^{T}\mathbf{M}\mathbf{X}_{m}$$

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

$$\phi = \mathbf{X}_{m}s = \sum_{i = 1}^{m}{x_{i}s_{i}}$$

Ritz trial vectors

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

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

Solve for first vector

$$\begin{array}{lll} \mathbf{K}{\mathbf{X}_{1}}^{*} = \mathbf{f} & \text{solve for} & {\mathbf{X}_{1}}^{*} \\ {\mathbf{X}_{1}}^{T}\mathbf{M}\mathbf{X}_{1} = 1 & \text{normalize} & \mathbf{M} \end{array}$$

Solve for additional vectors

$$\begin{array}{lll} \mathbf{K}{\mathbf{X}_{i}}^{*} = \mathbf{M}\mathbf{X}_{i - 1}&\text{solve for }&{\mathbf{X}_{1}}^{*}\\ c_{j} = {\mathbf{X}_{j}}^{T}\mathbf{M}{\mathbf{X}_{i}}^{*}&\text{for}&j = 1,\ldots,i - 1 \\ X_{i} = {X_{i}}^{*} - \sum_{j = 1}^{i - 1}{c_{j}\mathbf{X}_{j}}&\text{orthogonalize} & \mathbf{M}\\ {\mathbf{X}_{m}}^{T}\mathbf{M}\mathbf{X}_{m} = 1&\text{normalize}&\mathbf{M} \end{array}$$

Ritz P-delta analysis

As for modal analysis where we can use modal P-delta analysis, we can adapt ritz analysis to consider P-delta effects so we can account for loading on the structure that affects its natural frequencies and mode shapes. A first pass is carried out from which the geometric stiffness matrix, $\mathbf{K}_{g}$, can be calculated. This is used to modify the stiffness matrix, and the method runs the same as above, but using $\mathbf{K} + \mathbf{K}_{g}$ in place of $\mathbf{K}$.