I am a fourth year PhD student working on nonlinear dynamical systems supervised by Prof. George Haller.
I analyze complex dynamical systems. My research goal is to understand the fundamental dynamic behavior of nonlinear mechanical systems. For this purpose I extend existing methods, program computational tools and perform fundamental analytical research. My applications focus mainly on mechanical vibrations, such as vibration supressing, stability analysis and calculation of the forced response (FRF plots).
Spectral submanifolds (SSMs) have recently been shown to provide exact and unique reduced-order models for nonlinear unforced mechanical vibrations. We have extended these results to periodically or quasi-periodically forced mechanical systems, and obtained analytic expressions for forced responses and backbone curves on modal (i.e. two dimensional) time-dependent SSMs. A judicious choice of the parametrization of these SSMs allows us to simplify the reduced dynamics considerably. We have demonstrated our analytical formulae on three numerical examplesand compare them to results obtained from availablenormal-form methods.Reference:
T. Breunung & G. Haller, Explicit backbone curves from spectral submanifolds of forced-damped nonlinear mechanical systems. Proc. R. Soc. A 474 (2018) 20180083. doi
We have develped an integral equation approach that to enable fast computation of the response of non-linear multi-degree-of-freedom mechanical systemsunder 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 theintegral 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 robustconvergence. 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 com-puting steady-state response.Reference:
S. Jain, T. Breunung, & G. Haller, Fast computation of steady-state response for high-degree-of-freedom nonlinear systems Nonlinear Dynamics 97, 1 (2019) 313-341. doi
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. We have established 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.Reference:
T. Breunung & G. Haller, When does a periodic response exist in a periodically forced multi-degree-of-freedom mechanical system? Nonlinear Dynamics 98, 3 (2019) 1761-1780 doi
We have developed a method to clarify the role of deterministic invariant manifolds if small white noise excitation is added. To this end we have extended the notion of a transport barrier from fluid dynamics to the mechanical vibrations setting. Thereby, we can calrify the relevance of normally hyperbolic invariant manifolds and spectral submanifolds under small whithe noise perturbations.Work in progress with
F. Kogelbauer & G. Haller