A multi-parametric recursive continuation method for nonlinear dynamical systems

Abstract

The aim of this paper is to provide an efficient multi-parametric recursive continuation method of specific solution points of a nonlinear dynamical system such as bifurcation points. The proposed method explores the topology of specific points found on the frequency response curves by tracking extremum points in the successive codimensions of the problem with respect to multiple system parameters. To do so, the characterization of extremum points by a constraint equation and its associated extended system are presented. As a result, a recursive algorithm is generated by successively appending new constraint equations to the extended system at each new level of continuation. Then, the methodology is applied to a nonlinear tuned vibration absorber (NLTVA). The limit of existence of isolated solutions and extremum points optimizing the region without isolated solution are found and used to improve the NLTVA.

A nonlinear optimization shooting method for bifurcation tracking of nonlinear systems

“…The selection and optimization of system parameters for the tuned vibration absorber can be referred to Bekdaş and Nigdeli (2011), Bekdaş et al (2019, Kayabekir et al (2020), Nigdel and Bekdaş (2014), and Yucel et al (2019. To compare with the numerical results of previous literatures, simulation computation is undertaken with the same values of system parameters in Grenat et al, (2019) which are listed in Table 1. The analysis of nonlinear dynamic behaviors is performed for F = 0.15 and F = 0.75 respectively, whereas other parameters are kept constant.…”

“…For example on the basis of the harmonic balance method with an arclength continuation technique, Xie et al (2017) tackled the bifurcation tracking problem by means of the so-called extended system technique. Grenat et al(2019) applied the extended system technique for multiparametric bifurcation tracking problem. In Huntley et al (2017), several extended system methods are studied to track bifurcations of an aerofoil.…”

Robustness and reliability investigations on a nonlinear energy sink device concept

“…Thus, the optimization problem can be formulated by minimizing the green surface between F t and F N , where F N denotes load-displacement curve associated to the NES system and F t the target curve. To minimize this surface, the mean square error (MSE) is calculated and the objective function is formulated in equation (2). Where x is the vector of the design variables and u is the set of values of u, i.e.…”

“…Vibration mitigation, and energy dissipation in mechanical systems is a rapidly evolving field (e.g. [1][2][3][4][5]. The evolution in this domain comes from the need to design more rigorous devices of vibration mitigation.…”

Conditions for the existence of isolated backbone curves

“…Numerous approaches have been used to detect and trace isolas. One numerical method is continuation which uses special points, such as fold bifurcations and extremum points, to trace the evolution of isolas by varying specific parameters [30,31]. In combination with continuation methods, singularity theory can be used to provide complementary information in the prediction and identification of isolas [32][33][34].…”