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

Abstract

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.

Existence of quasi-periodic responses in quasi-periodically forced nonlinear mechanical systems

“…Since periodic orbits are a special case of quasi-periodic solutions (i.e., K = 1 in Definition 2.1), one can also establish the existence of periodic solutions via Theorem 3.1. Comparing with our previous result [8] conditions (C1) and (C2) are identical. In the quasi-periodic case, however, we require inner product of the coordinates q and the stiffness terms S(q) to grow sufficient far from the origin (cf.…”

“…It seems naturally to expect quasi-periodic vibrations in mechanical systems when the applied external loads are quasi-periodic. This expectation, however, is generally not true even for the periodic case, as we have investigated in depth for mechanical systems [8]. Consequently, the question arises when we can rigorously expect a quasi-periodic steady-state response of a forced nonlinear mechanical system.…”

“…(1) actually exists. Applying the harmonic balance to a system without clarifying the existence of a steady-state response, can lead to erroneous conclusions as we have demonstrated on an explicit, periodically driven mechanical system [8]. To aid the existing numerical tools, it is thus of fundamental importance to guarantee the existence of quasi-periodic solutions to Eq.…”

“…For the periodic case, however, various results can establish the existence of periodic solutions even for high amplitude oscillations. In our previous work [8], we identified a strengthened version of a theorem by Rouche and Mawhin [52] to be the most relevant for periodically driven mechanical systems. Here, we extend our result to the quasiperiodic case.…”

Nonexistence of Periodic Orbits for Forced–Damped Potential Systems in Bounded Domains

“…For general damping magnitude and forcing amplitude, topological techniques, including variants of topological degree theory, have been applied successfully to obtain existence of periodic orbits in mechanical systems, cf. [9] and [3]. As the methods are based on continuous deformation, however, qualitative properties do not immediately follow in general.…”