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Global detection of detached periodic solution branches of friction-damped mechanical systems

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Abstract

Two novel methods to determine detached periodic solution branches of low-dimensional and large-scale friction-damped mechanical systems have been developed. The approach for low-dimensional systems is an extension of the global terrain method and consists of three steps: First, the non-smooth elastic Coulomb slider is temporarily modified to pose a \(C^2\) continuous energy surface in the frequency domain. Second, the global terrain method is extended to rescale the search space consecutively along multiple eigendirections of the Hessian at the solution points, facilitating the determination of disconnected solutions. Finally, found solutions are used as the starting points for a homotopy strategy to detect the solutions of the original non-smooth problem. The method for large-scale systems based on an invariant manifold approach consists of three steps: First, the system’s dimension is decreased following a nonlinear modal reduction leading to a two-dimensional surrogate problem. Second, various subspaces for direct extrema, direct bifurcation, constant frequency, and constant amplitude solution detection are defined. Finally, a deterministic line search, a stochastic global solution method, and a newly developed deterministic line deflation are applied to the problem. The proposed methods are applied to low-dimensional and large-scale friction-damped mechanical systems simultaneously subjected to external and self-excitation as well as a low-dimensional mechanical system with shape-memory alloy nonlinearity subjected to external excitation. Their ability to find all solutions is discussed and compared with existing solution procedures such as bifurcation tracking, deflation, homotopy, and the global terrain method. Depending on the chosen subspace for the global search, the proposed methods are capable of providing additional detached solution branches (isola).

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  1. Institute of Dynamics and Vibration Research, Leibniz University Hannover, Appelstraße 11, 30167, Hannover, Germany

    Torsten Heinze, Lars Panning-von Scheidt & Jörg Wallaschek

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Heinze, T., Panning-von Scheidt, L. & Wallaschek, J. Global detection of detached periodic solution branches of friction-damped mechanical systems. Nonlinear Dyn 99, 1841–1870 (2020). https://doi.org/10.1007/s11071-019-05425-4

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