Cross-orthogonality (X-Ortho)

An important property of an undamped or proportionally damped multi-degree-of-freedom system is that the mode shapes exhibit orthogonality with respect to the system mass matrix as follows:

[ψ]T[M][ψ] = [mr]	Eq. 1

where:

• [mr] is a diagonal matrix where the rth element, mr, on the diagonal is the modal mass of mode r.

• [M] is the system's mass matrix.

• [ψ] is the mode shape matrix.

After mass normalization by scaling each mode r by 1/√mr, Eq. 1 simplifies to:

[Φ]T[M][Φ] = [I]	Eq. 2

where [I] is the identity matrix and [Φ] is the scaled mode shape matrix.

The cross-orthogonality metric uses the orthogonality property to quantitatively compare work, [ΨA], and reference mode shapes, [ΨX], as follows:

[R] = [ψX]T[Mr][ψA]	Eq. 3

where

• [R] is the resultant cross-orthogonality matrix of size LX x LA.

• [Mr] is the reduced mass matrix of size N x N, reduced to the common work and reference degrees of freedom N. The reduced mass matrix is obtained from the full system mass matrix [M] through model reduction techniques, such as a Guyan reduction.

• [ψA] is the work mode shapes matrix of size N x LA.

• [ψX] is the reference mode shapes matrix of size N x LX.

If [ψX] is exactly the same as [ψA], [R] is diagonal. All off-diagonal terms of matrix [R] are zero. For this reason, when [ψX] is not exactly the same as [ψA], the values of the off-diagonal terms indicate how close [ψX] and [ψA] are to being equal.

Starting in Simcenter 3D 2020.1, the cross-orthogonality matrix [R] contains complex values.

The correlation solver first normalizes the work and reference mode shapes by the square root of the diagonal elements of a self-orthogonality calculation, using the reduced mass matrix. This is called mass normalization. Thus, for a cross-orthogonality matrix computation involving real mode shapes, entries vary between ±1 and 0.

• 0 indicates perfectly orthogonal mode shapes.

• ±1 indicates perfectly matching mode shapes, with a negative sign for the opposite phase.

Learn more

Correlation metrics

Pre-test solution process

Correlation solution process

Look up more details

Modal Analysis

Converting complex modes to real modes

Accounting for repeated modes

Modal Scale Factor (MSF)

Modal Assurance Criteria (MAC)

Coordinate MAC (COMAC)

Frequency Response Assurance Criterion (FRAC)

Min-MAC algorithm

MODMAC algorithm

Normal Mode Indicator Function (NMIF) algorithm

Driving Point Residue algorithm

Scientific literature references for correlation

Quick links

Command reference

Pre/Post video examples

Bulk Entry Descriptions

Simcenter 3D tutorials

Browse Simcenter 3D help by product area

Cross-orthogonality (X-Ortho), Simcenter 3D 2021.1 Series

© 2020 Siemens

window.mainLanguage="en_US"

window.delivId=""

window.projectId=""

MathJax.Hub.Config({ TeX: { extensions: ["autoload-all.js"] }, tex2jax: { displayMath: [ ] }, "SVG": { scale: 125 } });