Fast computation of steady-state response for high-degree-of-freedom nonlinear systems

Abstract

We discuss an integral equation approach that enables fast computation of the response of nonlinear multi-degree-of-freedom mechanical systems under periodic and quasi-periodic external excitation. The kernel of this integral equation is a Green's function that we compute explicitly for general mechanical systems. We derive conditions under which the integral equation can be solved by a simple and fast Picard iteration even for non-smooth mechanical systems. The convergence of this iteration cannot be guaranteed for near-resonant forcing, for which we employ a Newton-Raphson iteration instead, obtaining robust convergence. We further show that this integral-equation approach can be appended with standard continuation schemes to achieve an additional, significant performance increase over common approaches to computing steady-state response. arXiv:1810.10103v1 [math.DS] 23 Oct 2018 ods such as the Craig-Bampton method [54] (cf. Theodosiou et al. [47]), proper orthogonal decomposition [55] (cf. Kerchen et al. [48]), reduction using natural modes (cf. Amabili [49], Touzé et al. [50]) and the modal-derivative method of Idelsohn & Cardona [53] (cf. Sombroek et al. [51], Jain et al. [52]).A common feature of these methods is their local nature: they seek to approximate nonlinear steadystate response in the vicinity of an equilibrium. Thus, high-amplitude oscillations are generally missed by these approaches.On the analytic side, perturbation techniques relying on a small parameter have been widely used to approximate the steady-state response of nonlinear systems. Nayfeh et al. [24,25] give a formal multiple-scales expansion applied to a system with small damping, small nonlinearities and small forcing. Their results are detailed amplitude equations to be worked out on a case-by-case basis. Mitropolskii and Van Dao [23] apply the method of averaging (cf. Bogoliubov and Mitropolsky [7] or, more recently, Sanders and Verhulst [30]) after a transformation to amplitude-phase coordinates in the case of small damping, nonlinearities and forcing. They consider single as well as multi-harmonic forcing of multi degree of freedom systems and obtain the solution in terms of a multi-frequency Fourier expansion. Their formulas become involved even for a single oscillator, thus condensed formulas or algorithms are unavailable for general systems. As conceded by Mitroposkii and Van Dao [23], the series expansion is formal, as no attention is given to the actual existence of a periodic response. Existence is indeed a subtle question in this context, since the envisioned periodic orbits would perturb from a non-hyperbolic fixed point.Vakakis [36] relaxes the small nonlinearity assumption and describes a perturbation approach for obtaining the periodic response of a single-degree-of-freedom Duffing oscillator subject to small forcing and small damping. A formal series expansion is performed around a conservative limit, where periodic solutions are explicitly known (elliptic Duffing oscillator). This approach only works for perturbatio...

When does a periodic response exist in a periodically forced multi-degree-of-freedom mechanical system?

“…which satisfies the negative mean-forcing requirement (20). Further, note that the forcing (76) is T /2 periodic for the case ρ 1 = 1 and T periodic in the case ρ 1 = −1.…”

“…is unstable for some parameter values a, Ω, c 2 > 0, c 1 > 0, ω 1 , ω 2 and κ, then we can find a T -periodic forcing f 1 (t) satisfying condition (20), such that system (22) has no periodic orbit.…”

“…where the sign depends on the choice of the forcing (20). Clearly, the orthogonality condition (77) is not satisfied and therefore, by theorem B.1 system (22), has no periodic solution.…”

Integral equations and model reduction for fast computation of nonlinear periodic response

“…Computing the steady‐state response of periodically forced nonlinear systems is a challenging task for contemporary engineering problems comprising high‐dimensional finite element models. A number of methods are nominally available in the literature for nonlinear periodic response calculation, ranging from analytical perturbation techniques 1,2 to standalone computational packages (AUTO, 3 coco , 4 NLvib 5 ) that perform numerical continuation (see References 6,7 for a review). Despite today's advances in computing, however, a good approximation to nonlinear forced response curves in complex structural vibration problems remains challenging to obtain and hence model reduction is still required 8 …”

Duffing-type oscillator under harmonic excitation with a variable value of excitation amplitude and time-dependent external disturbances

“…From tools that are used in the analysis of nonlinear systems, the most general tools are the frequency response curve 4 , 5 , 11 , 15 , 17 – 21 , 27 , 33 , 34 , 38 , the backbone curve 11 , 12 , 19 and, when the numerical approach is used, time histories 16 , 35 , 43 . In the group of more sophisticated tools, however, more specific techniques can also be mentioned: phase portraits 7 , 8 , 35 – 37 , bifurcation diagrams 8 , 38 , 39 , basins of attraction 40 and Poincaré maps 8 , 35 , 41 , 42 .…”