3.8.6.2. Damped Method-Real and Imaginary Parts of the Eigenvector
In a damped system, the response at different nodes can be out of phase. At any given node, the
amplitude will be the vector sum of the real and imaginary components of the eigenvector.
3.8.7. QR Damped Method
The QR damped method (MODOPT,QRDAMP) combines the advantages of the Block Lanczos method
with the complex Hessenberg method. The key concept is to approximately represent the first few
complex damped eigenvalues by modal transformation using a small number of eigenvectors of the
undamped system. Aft er the undamped mode shapes are evaluated by using the real eigensolution
(Block Lanczos method), the equations of motion are transformed to these modal coordinates.
Using the QR algorithm, a smaller eigenvalue problem is then solved in the modal subspace. This ap-
proach gives good results for lightly damped systems and can also apply to any arbitrary damping type
(proportional or non-proportional symmetric damping or nonsymmetrical gyroscopic damping matrix).
This approach also supports nonsymmetrical stiffness if present in the model.
The QRDAMP eigensolver applies to models having an unsymmetrical global stiffness matrix where only
a few elements contribute nonsymmetrical element stiffness matrices. For example, in a brake-friction
problem, the local part of a model with friction contacts generat es a nonsymmetrical stiffness matrix
in contact elements. When a non-symmetric stiffness matrix is encountered the eigenfrequencies and
mode shapes obtained by the QRDAMP eigensolver must be verified by rerunning the analysis with the
non-symmetric eigensolver. If a non-symmetric stiffness matrix is encountered a warning message
cautioning the user is output by the QRDAMP eigensolver right after the completion of the Block Lanczos
eigensolution.
The QRDAMP eigensolver works best when there is a larger “modal subspace” to converge and is
therefore the best option for larger models. Because the accuracy of this method is dependent on the
number of modes used in the calculations, a sufficient number of fundamental modes should be present
(especially for highly damped systems) to provide good results. The QR damped method is not recom-
mended for critically damped or overdamped systems. This method outputs both the real and imaginary
eigenvalues (frequencies), but outputs only the real eigenvectors (mode shapes). When requested
however, complex mode shapes of damped systems are computed.
In general, ANSYS, Inc. recommends using the Damp eigensolver for small models. It produces more
accurate eigensolutions than the QR Damped eigensolver for damped systems.
3.8.8. Storage of Complex Results
While the UNSYM, DAMP, and QRDAMP methods all generat e complex eigensolutions, there is a difference
in the way eigenresults are stored in the result files, particularly the eigenvalues.
Eigensolver Eigenvalue Eigenvector
Real Part: Mode
shape associated toUNSYM Real Part: natural frequency
Imaginary Part: measure of
the stability of the system the Real Part of the ei-
genvalue. (Natural fre-
A negative imaginary part means quency)
the system is stable, whereas a Imaginary Part: Mode
positive value means the system shape associated to
is unstable the imaginary part ofRelease 15.0 - © SAS IP, Inc. All rights reserved. - Contains proprietary and confidential informationComparing Mode-Extraction Methods