Parallel numerical continuation of periodic responses of local nonlinear systems

The aim of this paper is to develop an efficient and robust parallel numerical continuation method for periodic solutions of local nonlinear systems. In this method, a component mode synthesis method is first employed to reduce the nonlinear system with local nonlinearity; the parallel principle is implemented in the correction part of the classic pseudo-arc length continuation, and the step control strategy is dependent on historical and current information from multiple cores. In addition, a numerical continuation (NC) jumping phenomenon may occur alongside the periodic solution tracking process; a geometrical indicator algorithm based on the frequency–amplitude curve is utilized to prevent NC jumping and enhance the robustness of the tracking behavior. By applying the methodology to a nonlinear energy sink and a rod-fastened rolling bearing rotor system, it is shown that numerical continuation can be employed to calculate periodic solutions efficiently and robustly.

[1]  Svetozar Margenov,et al.  Numerical methods and parallel algorithms for computation of periodic responses of plates , 2017, J. Comput. Appl. Math..

[2]  Svetozar Margenov,et al.  Scalable parallel implementation of shooting method for large-scale dynamical systems. Application to bridge components , 2016, J. Comput. Appl. Math..

[3]  Aydin Boyaci.  Numerical Continuation Applied to Nonlinear Rotor Dynamics , 2016 .

[4]  Norio Hasebe,et al.  An Efficient Analysis of High-Order Dynamical System with Local Nonlinearity , 1999 .

[5]  Minqing Jing,et al.  Global nonlinear dynamic characteristics of rod-fastening rotor supported by ball bearings , 2015 .

[6]  Yu Wang,et al.  A 3D finite element-based model order reduction method for parametric resonance and whirling analysis of anisotropic rotor-bearing systems , 2015 .

[7]  Sitae Kim,et al.  Bifurcation Analysis of a Rotor Supported by Five-Pad Tilting Pad Journal Bearings Using Numerical Continuation , 2018 .

[8]  Alan Palazzolo,et al.  Shooting/continuation based bifurcation analysis of large order nonlinear rotordynamic systems , 2018 .

[9]  Christoph W. Schwingshackl,et al.  Continuation techniques for analysis of whole aeroengine dynamics with imperfect bifurcations and isolated solutions , 2016 .

[10]  Willy Govaerts,et al.  MATCONT: A MATLAB package for numerical bifurcation analysis of ODEs , 2003, TOMS.

[11]  Jean-Jacques Sinou,et al.  Nonlinear vibrations of a mechanical system with non-regular nonlinearities and uncertainties , 2013, Commun. Nonlinear Sci. Numer. Simul..

[12]  E. Allgower,et al.  Introduction to Numerical Continuation Methods , 1987 .

[13]  R. Sghir,et al.  Nonlinear stability analysis of a flexible rotor-bearing system by numerical continuation , 2016 .

[14]  Danny C. Sorensen,et al.  A new algorithm for continuation and bifurcation analysis of large scale free surface flows , 2005 .

[15]  P. Sundararajan,et al.  An algorithm for response and stability of large order non-linear systems : Application to rotor systems , 1998 .

[16]  Kiyotaka Yamamura,et al.  A modified predictor-corrector method for tracing solution curves , 2016, 2016 IEEE Asia Pacific Conference on Circuits and Systems (APCCAS).

[17]  P. Sundararajan,et al.  Dynamics of Forced Nonlinear Systems Using Shooting/Arc-Length Continuation Method—Application to Rotor Systems , 1997 .

[18]  M. Friswell,et al.  Using linear model reduction to investigate the dynamics of structures with local non-linearities , 1995 .

[19]  Luc Masset,et al.  The harmonic balance method for bifurcation analysis of large-scale nonlinear mechanical systems , 2015, 1604.05621.

[20]  Lei Hou,et al.  Review for order reduction based on proper orthogonal decomposition and outlooks of applications in mechanical systems , 2019, Mechanical Systems and Signal Processing.

[21]  G. Adiletta.  An insight into the dynamics of a rigid rotor on two-lobe wave squeeze film damper , 2017 .

[22]  Marian Wiercigroch,et al.  Bifurcation analysis of periodic orbits of a non-smooth Jeffcott rotor model , 2013, Commun. Nonlinear Sci. Numer. Simul..

[23]  Dhavide A. Aruliah,et al.  Algorithm 956 , 2013, ArXiv.

[24]  Bo Dong,et al.  Robust continuation methods for tracing solution curves of parameterized systems , 2013, Numerical Algorithms.

[25]  Heng Liu,et al.  Nonlinear Dynamic Analysis of a Flexible Rod Fastening Rotor Bearing System , 2010 .

[26]  L. Xie,et al.  Bifurcation tracking by Harmonic Balance Method for performance tuning of nonlinear dynamical systems , 2017 .

[27]  L. Cherfils.  A parallel and adaptive continuation method for semilinear bifurcation problems , 1998 .

[28]  Justin R. Dobbs,et al.  STRUCTURED BUILDING MODEL REDUCTION TOWARD PARALLEL SIMULATION , 2013 .

[29]  R. Seydel.  Practical bifurcation and stability analysis : from equilibrium to chaos , 1994 .

[30]  S. Léger,et al.  A Moore-Penrose continuation method based on a Schur complement approach for nonlinear finite element bifurcation problems , 2015 .

[31]  Neil L. Book,et al.  A robust path tracking algorithm for homotopy continuation , 1996 .

[32]  Rhb Rob Fey,et al.  Long term structural dynamics of mechanical systems with local nonlinearities , 1996 .