Development Of Biorthonormal Eigenvectors For Modal Analysis Of Linear Discrete Non-classically Damped Systems

✍ Scribed by S.F. Felszeghy

• Publisher: Elsevier Science
• Year: 1994
• Tongue: English
• Weight: 525 KB
• Volume: 176
• Category: Article
• ISSN: 0022-460X
• DOI: 10.1006/jsvi.1994.1372

✦ Synopsis

For linear discrete non-classically damped systems, it is shown that truncated orthonormal and biorthonormal eigenvector matrices can be developed for the purpose of uncoupling the physical equations of motion in a truncated ordered set of normal co-ordinates. This approach avoids for large systems the impractical step of having to find the entire modal matrix and its inverse and then apply these matrices as a similarity transformation to the system matrix. The development focuses on systems that possess simultaneously both classical and non-classical natural modes of free vibration. It is shown that the forced motion can be expanded partially in the classical modal vectors and partially in the non-classical modal vectors. The biorthonormal eigenvectors needed for expanding the forced motion are associated with asynchronous non-classical natural modes. These modes arise whenever the algebraic multiplicity of a non-classical eigenvalue exceeds the eigenvalue's geometric multiplicity in the non-classical eigenvector subspace. The development takes advantage of the simplifying properties that come with a symmetric formulation of the equations of motion in state space. Two examples illustrate the analytical results.