Normal Modes Analysis, also called eigenvalue analysis or eigenvalue extraction, is a technique used to calculate the vibration shapes and associated frequencies that a structure will exhibit. It is important to know these frequencies because if cyclic loads are applied at these frequencies, the structure can go into a resonance condition that will lead to catastrophic failure. It is also important to know the shapes in order to make sure that loads are not applied at points that will cause the resonance condition.

Normal modes analysis is also required for modal frequency response and modal transient analysis. In these analyses the problem is transformed from the direct mesh coordinates, where the number of degrees-of-freedom can be in the millions, to the modal coordinates where the number of degrees-of-freedom is just the number of modes used. Typically, the upper bound frequency in this case is 1.5 times the highest loading frequency or response frequency of interest.

In OptiStruct, normal modes analysis can be performed using one of two algorithms: Lanczos or the automated multi-level sub-structuring eigenvalue solution (AMSES). The eigenvalue extraction data for Lanczos is specified on the EIGRL data and for the automated multi-level sub-structuring eigenvalue solution method, the EIGRA data is used.

In addition, OptiStruct has an interface to the AMLS software developed at the University of Texas. AMLS uses the automated multi-level sub-structuring method for eigenvalue extraction. The use of AMLS is triggered by using the PARAM, AMLS set to YES input data in conjunction with the EIGRL card (only).

The Lanczos Method

The Lanczos method has the advantage that the eigenvalues and associated mode shapes are calculated exactly. This method is efficient for calculations in which the number of modes is small and the full shape of each mode is required. The disadvantage of the Lanczos method is that it is slow for large problems with millions of degrees-of-freedom for which hundreds of modes are required. The run times for these types of problems can easily stretch into days. In these cases, the AMSES or AMLS method must be used.

The Automated Multi-level Sub-structuring Eigenvalue Solution Method (AMSES)

The AMSES method has the advantage that only a portion of the eigenvector needs to be calculated. Since only a portion of the eigenvector is calculated, the disk space and disk I/O is greatly reduced. This leads to much shorter run times. For a typical NVH frequency response analysis there is only about 100 degrees-of-freedom of interest. In these cases, solutions of thousands of modes for meshes of millions of degrees-of- freedom can be solved in just a few hours. The disadvantage of the AMSES method is that the calculations are not exact. However, the modal frequencies are still accurate to a few digits. Also, for NVH analysis it is important that the mode shapes form modal space that covers all possible deformation patterns, but not so important that each individual mode shape is accurate.

AMSES Usage Guidelines

The following guidelines list the factors affecting AMSES usage:

The Governing Equations

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Normal Modes Analysis

The equilibrium equation for a structure performing free vibration appears as the eigenvalue problem:

Where, K is the stiffness matrix of the structure and M is the mass matrix. Damping is neglected.

The solution of the eigenvalue problem yields n eigenvalues , where n is the number of degrees-of-freedom. The vector A is the eigenvector corresponding to the eigenvalue.

The eigenvalue problem is solved using Lanczos, AMSES, or AMLS.

The natural frequency fi follows directly from the eigenvalue .

Input Specification

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In order to run a normal modes analysis, an EIGRL or EIGRA bulk data entry needs to be given to define the number of modes to be extracted. EIGRL or EIGRA data needs to be referenced by a METHOD statement in a SUBCASE in the subcase information section.

It is not necessary to define boundary conditions using an SPC statement. If no boundary conditions are applied, a zero eigenvalue is computed for each rigid body degree of freedom of the model.

It is possible to request the computation of residual vectors in conjunction with a normal modes analysis. Residual vectors are static displacements ortho-normalized with the eigenvectors to be used in an external frequency response analysis. In order to get this output, users have to define degrees-of-freedom using USET and USET1. The degrees-of-freedom are then used to define loads in the unit load method to compute the residual vectors. RESVEC = YES needs to be defined in the normal modes subcase, if the Lanczos eigensolver is used. Residual vectors associated with USET and USET1 data are always created, if the AMSES or AMLS eigensolvers are used. Boundary conditions defined using SPC or inertia relief must be applied to create residual vectors.

Subcase Definition

A normal modes subcase may be explicitly identified by setting ANALYSIS=MODES, but it is also implicitly chosen for any subcase containing the METHOD data selector (when the ANALYSIS entry is not present).

The following data selectors are recognized for an normal modes subcase definition.

 

Bulk Data

Bulk data entries which have particular significance for normal modes analysis include:

 

Sample Input

SUBCASE 100

 SPC = 5

 METHOD = 24

$

BEGIN BULK

$

EIGRL, 24, 0.0, 1000.

ENDDATA

$

Output

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Results of interest from eigenvalue extraction include maximum displacement, modal stresses, energies and multi-point constraint forces. These are requested via the I/O Options DISPLACEMENT, EKE, ESE, STRESS, GPSTRESS and MPCFORCE, respectively.

See Also:

Saving and Retrieving Normal Modes Analysis

AMLS (Automatic Multi-Level Sub-structuring) Eigensolver