Abstract

This paper addresses the forced response analysis of systems whose eigenvalue loci veer when plotted against a system parameter. It builds on the study by Chen and Ginsberg [“On the relationship between veering of eigenvalue loci and parameter sensitivity of eigenfunctions,” ASME Journal of Vibration and Acoustics, 114, pp. 141–148 (1992)], which established a singular perturbation solution for the eigensolution as a function of a system parameter. It is shown here that if the position of the boundary depends on the system parameter, then the modes predicted by the earlier work will fail to satisfy geometric boundary conditions. This error is corrected by introducing a coordinate straining transformation. The modes obtained from the modified perturbation expansion are used as basis functions for a Ritz series expansion of the displacement. These steps lead to a highly efficient methodology for evaluating forced response for any value of the system parameter in the veering range. The development employs the classic two-span beam with a torsional spring at the intermediate pin support as the prototype illustrating the general steps. The parameter associated with the veering phenomena is the offset of the intermediate pin from the symmetric position. The adequacy of the solution is demonstrated by comparisons to results of an analytical Sturm-Liouville solution. The generality of the perturbation solution is further demonstrated by comparing the perturbation and analytical solutions for two other configurations featuring two-span beams: two simply-supported beams coupled by a weak torsional spring and a beam supporting an attached dead mass near its midspan.

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