Numerical scheme for period-m motion of second-order nonlinear dynamical systems based on generalized harmonic balance method
- 199 Downloads
Prediction of periodic motion plays key roles in identifying bifurcations and chaos for nonlinear dynamical systems. In this paper, a semi-analytical and semi-numerical scheme is developed as a combination of the analytical generalized harmonic balance method and the Newton–Raphson iteration for period-m solution of second-order nonlinear systems. The nonlinear external loading is approximated by the Taylor’s expansion of displacement and velocity, and is expressed as summations of many orders of Fourier harmonics pairs. A set of nonlinear algebraic equations are solved iteratively for the coefficients of harmonic pairs until the convergence of solution is achieved. The periodic solutions for period-2 motion in a periodically forced Duffing oscillator and period-3 motion in a buckled, nonlinear Jeffcott rotor system are obtained from the present scheme, and the corresponding stability and bifurcation are evaluated through eigenvalue analysis. The results from the present scheme are found in good agreement with the existent analytical solutions. The present scheme can be used as a general purpose numerical realization of the generalized harmonic balance method in evaluating periodical nonlinear dynamical systems since it is not involved with analytical derivation of Fourier expansion of external loading.
KeywordsPeriodic-m motion Generalized harmonic balance method Fourier series Stability Bifurcation
The authors are grateful for the sponsorships by State Key Laboratory of Structural Analysis for Industrial Equipment (Grant S14204), Liaoning Provincial Program for Science and Technology (Grant 2014028004), the Collaborative Innovation Center of Major Machine Manufacturing in Liaoning, and the State Key Development Program for Basic Research of China (Grant 2015CB057300).
- 2.Evensen, D.A.: Nonlinear Flexural Vibrations of Thin-Walled Circular Cylinders. NASA TN D-4090 (1967)Google Scholar
- 18.Luo, A.C., Yu, B.: Analytical routes of period-m motions to chaos in a parametric, quadratic nonlinear oscillator. Int. J. Dyn. Control (2014). doi: 10.1007/s40435-014-0112-7
- 19.Huang, J.Z., Luo, A.C.: Analytical periodic motions and bifurcations in a nonlinear rotor system. Int. J. Dyn. Control 2(3), 425–459 (2014)Google Scholar
- 20.Huang, J.Z., Luo, A.C.: Periodic motions and bifurcation trees in a buckled, nonlinear Jeffcott rotor system. Int. J. Bifurc. Chaos 25(1), 1550002 (2015)Google Scholar
- 21.Huang, J.Z., Luo, A.C.: Analytical solutions of period-1 motions in a buckled, nonlinear Jeffcott rotor system. Int. J. Dyn. Control (2015). doi: 10.1007/40435-015-0149-2
- 22.Wang, Y.F., Liu, Z.W.: A matrix-based computational scheme of generalized harmonic balance method for periodic solutions of nonlinear vibratory systems. J. Appl. Nonlinear Dyn 4(4), 379–389 (2015)Google Scholar
- 23.Kantorovich, L.V.: On Newton’s method for functional equations. Dokl. Akad. Nauk. SSSR 59(7), 1237–1240 (1948)Google Scholar