Principal component analysis

Principal component analysis (PCA) lets you study a complex noise and vibration problem associated with multiple, partially correlated sources (references) by reducing the problem to one with uncorrelated references. It allows you to describe the global problem as a superposition of individual single reference phenomena (interpret this as a single excitation). This is analogous to converting a problem from the physical domain to the modal domain. As with modes, only a few principal components are needed to express the physical phenomenon. The principal components represent variables that dominate the variation of results.

Singular value decomposition of a multi-reference cross-power matrix yields the principal components. The software generates as many principal components as there are references. The new references are the principal components, which the software then transforms to virtual referenced spectra of responses and references. These virtual data act as virtual loads for deterministic analyses, and you can use them in a Load Identification meta-solution or a load recipe for the following analyses:

• Transfer path analysis (TPA) to compute the distribution of vibro-acoustic responses from a source to a microphone or response node through both structural and acoustic pathways.

• Simcenter Nastran SOL 108 or SOL 111 vibro-acoustic analysis.

• MSC Nastran SOL 108 or SOL 111 vibro-acoustic analysis.

You can use a post-processing scenario to display the imported auto-powers and cross-powers, singular values from the decomposition, and so on, and because the PCA process creates a diagonal matrix, the Averaging post-processing scenario can easily compute an RMS of the virtual, random loads across subcases and plot them.

A typical application is road noise analysis, where you import measurement data taken at multiple locations from a vehicle driving over rough pavement at a constant speed, identify the operating forces at the different suspension connections, and use PCA computed virtual loads in a TPA analysis. You can then use the TPA analysis results to determine the main contributors of the complex noise and vibration problem, for example, heavy excitation, intense transfer, or both.

For more information on TPA, see Transfer Path Analysis.

Road noise theory

Road noise analysis for a vehicle in this software consists of PCA and TPA analysis, and you must consider the following:

• Multiple, partially incoherent forces acting on suspensions generate the noise.

• No fixed phase relations exist between the response degrees of freedom.

• To capture the various sources, you need multiple reference signals.

Typically, you make measurements at multiple locations. You choose a set of references, which are representative of the problem. That is, you can assume that the tire patch excitation can be represented by a few reference measurements, for example, wheel center vibrations. When R sources exist, then the number of references N are selected such that N, \ge ,R.

You also take measurements for multiple target locations M.

Schematic of PCA process

Forces on vehicle suspensions

When N references (X1...XN) and M responses (Y1...YM) exist, the computation of the cross power matrix is as follows:

\left[\begin{matrix}X\left(\omega\right)\Y\left(\omega\right)\\end{matrix}\right]\left\lfloor\begin{matrix}X^\ast\left(\omega\right)&Y^\ast\left(\omega\right)\\end{matrix}\right\rfloor=\left[\begin{matrix}XX^\ast\left(\omega\right)&XY^\ast\left(\omega\right)\YX^\ast\left(\omega\right)&YY^\ast\left(\omega\right)\\end{matrix}\right]\

where:

• The superscript * represents a complex conjugate.

• X\left(\omega\right)=\left{\begin{matrix}X_1\left(\omega\right)\\vdots\X_N\left(\omega\right)\\end{matrix}\right}

• Y\left(\omega\right)=\left{\begin{matrix}Y_1\left(\omega\right)\\vdots\Y_N\left(\omega\right)\\end{matrix}\right}

Note:

In the following equations, the function of frequency \left(\omega\right) are omitted for convenience. Frequency dependence is implicit in the equations.

As you can see, the cross-power matrix is no longer diagonal for partially correlated signals. When the cross-power matrix is not diagonal, response RMS must include all the cross terms.

Given the cross-power matrix of the *XX ** references, you can obtain a set of perfectly uncorrelated signals X' \left(\omega\right) such that a linear combination of these signals can describe the original set of signals.

The software decomposes any matrix including the cross-power matrix as follows:

\left[ {X; \cdot ;{X^H}} \right] = \left[ U \right]\left[ {{X^\prime }; \cdot ;{X^\prime }^H} \right]\left[ {{U^T}} \right]

where the product \left[X\ \cdot \ X^H\left(\omega\right)\right] is a diagonal matrix:

\left[X^\prime\ \cdot \ {X^\prime}^H\right]=\left[\begin{matrix}\sigma_1&0&\cdots&0\0&\sigma_2&\cdots&0\\cdots&\cdots&\cdots&\cdots\0&\cdots&\cdots&\sigma_n\\end{matrix}\right]

where:

• X' is the set of uncorrelated principal components (also known as virtual references).

• The superscript H denotes a Hermitian matrix, which is the transpose of the complex conjugate matrix.

• The product X^\prime\ \cdot \ X^H is a diagonal matrix of auto powers of principal components (also known as virtual auto-powers).

• U is the unitary matrix such that U \cdot \ U^T=I by definition.

The above decomposition of a matrix is also known as singular value decomposition (SVD). After the SVD \sigma_i’s, which are auto-powers of virtual references, are obtained in such a way that \sigma_1\ >\ \sigma_{2\ }>\ \cdots\ >\ \sigma_n. These auto-powers represent relative weights of the N phenomenon

\left[X\ \cdot \ X^H\right]=\left[U \cdot X^\prime\right]\left[U \cdot X^\prime\right]^H

The transformation from physical to the principal coordinates are then expressed as follows:

X=U \cdot X^\prime;{\ \ \ \ \ \ \ \ \ \ \ \ \ X}^\prime=U^T \cdot X

Note:

This is very similar to the transformation from physical to modal coordinates, where the eigenvectors form the basis vectors for the transformation.

The cross-power of responses in terms of principal coordinates becomes:

Y \cdot {X^\prime}^H=\left[Y \cdot X^H\right]U

The cross-power of references in terms of principal coordinates then becomes:

X \cdot {X^\prime}^H=\left[X \cdot X^H\right]U

When you define a transformation such that:

\left[X"\right]=\left[X\prime\right]\left[\begin{matrix}\frac{1}{\sqrt{\sigma_1}}&0&\cdots&0\0&\frac{1}{\sqrt{\sigma_2}}&\cdots&0\\cdots&\cdots&\cdots&\cdots\0&\cdots&\cdots&\frac{1}{\sqrt{\sigma_n}}\\end{matrix}\right]

It follows that:

\left[X^{\prime\prime} \cdot {X^{\prime\prime}}^H\right]=[I]

The virtual referenced spectra of responses are then given by:

Y \cdot {X^{\prime\prime}}^H

Similarly, virtual referenced spectra of references are then given by:

X \cdot {X^{\prime\prime}}^H

The virtual referenced spectra act as virtual loads for a deterministic solution. This is analogous to application of modal force (physical force projected in modal coordinates) in a modal solution.

When the physical type of the spectra is acceleration, these can either be applied directly as acceleration loads at the reference location or forces computed using a Load Identification meta-solution.

TPA analysis

For TPA analysis for road noise, you do the following:

1. Use test data in PCA analysis to obtain virtual loads.

2. Input the virtual acceleration loads in the Load Identification meta-solution, where the software computes virtual forces based on FRFs and virtual accelerations.

3. Use the virtual forces as loads in the TPA analysis.

Workflow diagram

Post-processing of results

In Scenario Based Data-Visualization, you can post-process the imported auto-powers and cross-powers and singular decomposition values with the Function Plots post-processing scenario, and because the PCA process creates a diagonal matrix, the Averaging post-processing scenario can easily compute an RMS of the virtual, random loads across subcases and plot them.

Where do I find it?

Application	Pre/Post
Prerequisites	A Simulation file as the work and displayed partSimcenter Test.Lab Project File (*.lms) that contains cross and auto power data on the same range of frequencies and where the cross power terms are of the same physical typeA new, empty Principal Component solution process
Command Finder	Principal Component
Simulation Navigator	Expand Principal Component node→right-click Cross Powers node→Edit
Location in dialog box	In Auto Power and Cross Power group, click Browse →select a file→OK→Refresh to populate the data

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Principal component analysis, Simcenter 3D 2021.1 Series

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