Structure-preserving isospectral transformation for total or partial decoupling of self-adjoint quadratic pencils

Quadratic pencils λ2M + λC + K, where M, C, and K are n × n real matrices, arise in a broad range of important applications. Its spectral properties affect the vibration behavior of the underlying system which often consists of many elements coupled together through an intricate network of inter-con...

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Bibliographic Details
Published in:Journal of sound and vibration 2019-06, Vol.449, p.157-171
Main Authors: Jiang, Nan, Chu, Moody T., Shen, Jihong
Format: Article
Language:English
Subjects:
Decoupling
Degrees of freedom
Eigenvalues
Floating point arithmetic
Floating structures
Gradient flow
Isospectrality
Lancaster structure
Numerical integration
Quality assessment
Rescaling
Self-adjoint quadratic pencil
Sound
Structure preserving transformation
Subsystems
Transformations
Vibration
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Summary:Quadratic pencils λ2M + λC + K, where M, C, and K are n × n real matrices, arise in a broad range of important applications. Its spectral properties affect the vibration behavior of the underlying system which often consists of many elements coupled together through an intricate network of inter-connectivities. It is known that an n-degree-of-freedom system with semi-simple eigenvalues can be reduced to, without tampering with the innate vibration properties, n mutually independent single-degree-of-freedom subsystems, referred to as total decoupling. This paper revisits the problem with the additional constraint that the masses should stay invariant throughout the reduction process. Rescaling the masses if necessary, M is assumed to be the identity matrix. Isospectral flows are derived to either totally or partially decouple C and K to independent units of modules. Indeed, the same framework can be tailored to handle any kinds of desired structure. Two new results are obtained. First, the global convergence is guaranteed by using the Łojasiewicz gradient inequality. Second, bounds on errors due to numerical integration and floating-point arithmetic calculation are derived, which can be used for assessing the quality of the transformation. Numerical experiments on four distinct scenarios are given to demonstrate the capabilities of the framework of handling the decoupling problem.
ISSN:0022-460X
1095-8568
DOI:10.1016/j.jsv.2019.01.009