Iterative harmonic balance for period-one rotating solution of parametric pendulum
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O R I G I N A L PA P E R
Iterative harmonic balance for period-one rotating solution of parametric pendulum Hui Zhang · Tian-Wei Ma
Received: 4 January 2012 / Accepted: 26 September 2012 / Published online: 11 October 2012 © Springer Science+Business Media Dordrecht 2012
Abstract In this study, an iterative method based on harmonic balance for the period-one rotation of parametrically excited pendulum is proposed. Based on the definition of the period-one rotating orbit, the exact form of the solution can be obtained using the Fourier series. An iterative harmonic balance process is proposed to estimate the coefficients in the exact solution form. The general formula for each iteration step is presented. The method is evaluated using two criteria, which are the system energy error and the global residual error. The performance of the proposed method is compared with the results from multiscale method and perturbation method. The numerical results obtained with the Dormand–Prince method (ODE45 in MATLAB©) are used as the baseline of the evaluation. Keywords Parametric pendulum · Iteration · Rotating orbits · Harmonic balance · Nonlinear systems
This research was supported by National Science Foundation (Grant No. CMMI 0758632). H. Zhang · T.-W. Ma () Department of Civil and Environmental Engineering, University of Hawaii at M¯anoa, Honolulu, HI 96822, USA e-mail: [email protected]
1 Introduction Early investigation of purely rotating orbits of a parametrically excited pendulum were reported in the 1980s [1, 2]. In a later investigation, a lower bound of the excitation amplitude, required for the periodone rotating orbit, was numerically observed [3]. It was found in later studies [4, 5] that pure rotations exist only in narrow strips inside the first resonance zone in the parameter space. More recently, the multiscale method [6] and the perturbation method [7] were respectively employed resulting in two approximate solutions for period-one rotations. In [6], an analytical approximation was obtained using the multiscale procedure and only the first-order solution was developed. In [7], an alternative solution was proposed by reformulating the governing equation as an integral equation, which yields an approximate solution in implicit form. Besides the multiscale and the perturbation methods, harmonic balance is another effective tool for analyzing nonlinear dynamical systems, which has been widely applied to a variety of engineering problems [8–12]. Once a pendulum is in a period-one rotation orbit, the magnitude of its angular velocity periodically fluctuates around the excitation frequency. Using the Fourier series, therefore, the exact form of the solution can be established. As there exist infinite number of harmonic terms in the exact solution form and the generation of beat frequencies due to the parametric excitation, which prohibits the exact solution of the co-
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efficients, an iterative procedure of harmonic balance is proposed to estimate the coefficients in the solution.
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