Modal Perturbation Method and its Applications in Structural Systems

A new perturbation method is developed to solve any eigenvalue equation of the form $({\mathbf{A}}_0+\mathbf{\Delta }\mathbf{A}){\mathbf{X}}^{*}=({\mathbf{B}}_0+\mathbf{\Delta }\mathbf{B}){\mathbf{X}}^{*}{\mathbf{\Lambda }}^{*}$ based on the solution of an original system described by ${\mathbf{A}}_0\mathbf{X}={\mathbf{B}}_0\mathbf{X}\mathbf{\Lambda }\text{.}$ The eigenvectors of the modified system are expanded in a subspace spanned with a small number of vibration modes of the original system. In doing so, the former eigenvalue equation of the modified system is transformed into a set of algebraic equations, which require a significantly less computational effort to solve for the eigensolutions of complex structural systems. Four numerical examples show that the developed technique gives rise to the eigensolution of high accuracy and it is an effective approach for dynamic reanalysis of the structures with numerous degrees of freedom. In comparison with the conventional small parameter perturbation, the developed technique is applicable to a wider range of problems, and only m mode shapes are used based on the Ritz expansion so that the final solution can be derived efficiently. The technique also extends laboratory model tests for complex structures with the concept of dynamic hybrid tests numerically and experimentally.