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

Abstract

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.