Abstract
In this work we apply and compare two numerical path continuation algorithms for solving algebraic equations arising when applying the Harmonic Balance Method to compute periodic regimes of nonlinear dynamical systems. The first algorithm relies on a predictor-corrector scheme and an Alternating Frequency-Time approach. This algorithm can be applied directly also to non-analytic nonlinearities. The second algorithm relies on a high-order Taylor series expansion of the solution path (the so-called Asymptotic Numerical Method) and can be formulated entirely in the frequency domain. The series expansion can be viewed as a high-order predictor equipped with inherent error estimation capabilities, which permits to avoid correction steps. The second algorithm is limited to analytic nonlinearities, and typically additional variables need to be introduced to cast the equation system into a form that permits the efficient computation of the required high-order derivatives. We apply the algorithms to selected vibration problems involving mechanical systems with polynomial stiffness, dry friction and unilateral contact nonlinearities. We assess the influence of the algorithmic parameters of both methods to draw a picture of their differences and similarities. We analyze the computational performance in detail, to identify bottlenecks of the two methods.
Highlights
Harmonic Balance (HB) permits the efficient approximation of periodic solutions of nonlinear ordinary differential equations
The results suggest that Classical Harmonic Balance with Asymptotic Numerical Method (cHB-ANM) becomes more efficient for larger numbers of DOFs
The number of solution points was larger in the case of the cHB-ANM, which overall lead to approximately the same total number of Jacobian factorizations for the entire solution path
Summary
Harmonic Balance (HB) permits the efficient approximation of periodic solutions of nonlinear ordinary differential equations. Substitution into the ordinary differential equation system gives a residual term, which is made orthogonal to the Fourier basis function (Fourier-Galerkin projection). This corresponds to requiring that the Fourier coefficients of the residual are zero, for those harmonics retained in the ansatz. When we apply this method to the equation of motion of a mechanical system, with generalized coordinates q, subjected to periodic forcing, we obtain the algebraic equation system, r(q , Ω ) := S(Ω ) q + ˆf nl(q , Ω ) − ˆf ex =! Woiwode; Narayanaa Balaji; Kappauf; Tubita; Guillot; Vergez; Cochelin; Grolet; Krack
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