Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reducedorder models for nonlinear unforced mechanical vibrations. Here we extend these results to periodically or quasiperiodically forced mechanical systems, obtaining analytic expressions for forced responses and backbone curves on modal (i.e. two-dimensional) time dependent SSMs. A judicious choice of the parameterization of these SSMs allows us to simplify the reduced dynamics considerably. We demonstrate our analytical formulae on three numerical examples and compare them to results obtained from available normal form methods.
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...
While periodic responses of periodically forced dissipative nonlinear mechanical systems are commonly observed in experiments and numerics, their existence can rarely be concluded in rigorous mathematical terms. This lack of a priori existence criteria for mechanical systems hinders definitive conclusions about periodic orbits from approximate numerical methods, such as harmonic balance. In this work, we establish results guaranteeing the existence of a periodic response without restricting the amplitude of the forcing or the response. Our results provide a priori justification for the use of numerical methods for the detection of periodic responses. We illustrate on examples that each condition of the existence criterion we discuss is essential.
Abstract. Parametric excitation can lead to instabilities as well as to an improved stability behavior, depending on whether a parametric resonance or anti-resonance is induced. In order to calculate the stability domains and boundaries, the method of averaging is applied. The problem is reformulated in state space representation, which allows a general handling of the averaging method especially for systems with non-symmetric system matrices. It is highlighted that this approach can enhance the first order approximation significantly. Two example systems are investigated: a generic mechanical system and a flexible rotor in journal bearings with adjustable geometry.
Forced responses of mechanical systems are crucial design and performance criteria. Hence, their robust and reliable calculation is of vital importance. While numerical computation of periodic responses benefits from an extensive mathematical basis, the literature for quasi-periodically forced systems is limited. More specifically, the absence of applicable and general existence criterion for quasi-periodic orbits of nonlinear mechanical systems impedes definitive conclusions from numerical methods such as harmonic balance. In this work, we establish a general existence criterion for quasi-periodically forced vibratory systems with nonlinear stiffness terms. Our criterion does not rely on any small parameters and hence is applicable for large response and forcing amplitudes. On explicit numerical examples, we demonstrate how our existence criterion can be utilized to justify subsequent numerical computations of forced responses.
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