Computation of Nonlinear Normal Modes through Shooting and Pseudo-Arclength Computation

Abstract

Most existing constructive techniques for computing NNMs are based on asymptotic approaches and rely on fairly involved mathematical developments. In this context, algorithms for the numerical continuation of periodic solutions are really quite sophisticated and advanced (see, e.g., (1; 2), and the AUTO and MATCONT softwares). These algorithms have been extensively used for computing the forced response and limit cycles of nonlinear dynamical systems. Interestingly, there have been very few attempts to compute the periodic solutions of conservative mechanical structures (i.e., NNM motions) using numerical continuation techniques. One of the first approaches was proposed by Slater in (3) who combined a shooting method with sequential continuation to solve the nonlinear boundary value problem that defines a family of NNM motions. Similar approaches were considered in Lee et al. (4) and Bajaj et al. (5). A more sophisticated continuation method is the so-called asymptotic-numerical method. It is a semi-analytical technique that is based on a power series expansion of the unknowns parameterized by a control parameter. It is described in the next chapter. In this study, a shooting procedure is combined with the so-called pseudo-arclength continuation method for the computation of NNM motions. We show that the NNM computation is possible with limited implementation effort, which holds promise for a practical and accurate method for determining the NNMs of nonlinear vibrating structures. This chapter is organized as follows. In the next section, the proposed algorithm for NNM computation is presented. Its theoretical background is first recalled, and the numerical implementation is then described. Improvements are also presented for the reduction of the computational burden. The proposed algorithm is then demonstrated using three different nonlinear vibrating systems in Section 3.