Assessing the Local Stability of Periodic Motions for Large Multibody Non-Linear Systems Using Proper Orthogonal Decomposition

The eigenvalues of the monodromy matrix, known as Floquet characteristic multipliers, are used to study the local stability of periodic motions of a non-linear system of differential-algebraic equations (DAE). When the size of the underlying system is large, the cost of computing the monodromy matrix and its eigenvalues may be too high. In addition, for non-minimal set equations, such as those of a DAE system, there is a certain number of spurious eigenvalues associated with the algebraic constraint equations, which are meaningless for the assessment of the stability of motions. An approach to extract the dominant eigenvalues of the transition matrix without explicitly computing it is presented. The selection of the eigenvalues is based on a proper orthogonal decomposition (POD), which extracts the minimal set of dominant local modes of the transient dynamics on an “energy” contents basis. To make the procedure applicable to both numerical and experimental tests, a unifying experimental philosophy is pursued for the analysis of complex multidisciplinary multibody models. Some applications are outlined, and comparisons with other techniques are presented to demonstrate the accuracy of the proposed procedure.