Dynamic Analysis of Rotating Pendulum by Hamiltonian Approach

A conservative system always admits Hamiltonian invariant, which is kept unchanged during oscillation. This property is used to obtain the approximate frequency-amplitude relationship of the governing equation with sinusoidal nonlinearity. Here, we applied Hamiltonian approach to obtain natural frequency of the nonlinear rotating pendulum. The problem has been solved without series approximation and other restrictive assumptions. Numerical simulations are then conducted to prove the efficiency of the suggested technique. 1. Introduction The rotational pendulum equation [1, 2] arises in a number of models describing the phenomenon in engineering. This equation has been described in the wind-excited vibration absorber [3] and mechanical and civil structure [4, 5] and has received much attention recently. To improve the understanding of dynamical systems, it is important to seek their exact solution. Most dynamical systems can not be solved exactly; numerical or approximate methods must be used. Numerical methods are often costly and time consuming to get a complete dynamics of the problem. Little progress was made on the integrability of the rotational pendulum by Lai et al. [6]. Lai and his colleagues used Taylor’s series and Chebyshev’s polynomials to convert the trigonometric nonlinearity to algebraic nonlinearity. They used Mickens iteration method [7] and found the approximate explicit formulas. Various alternative approaches have been proposed for solving nonlinear dynamical system, parameter-expanding method [8], frequency-amplitude formulation [9], max-min approach [10], harmonic balance method [11], variational approach [12], homotopy perturbation method [13–15], Lindstedt-Poincare method [16], and Hamiltonian approach [17–19]. 2. Governing Equation of a Rotational Pendulum Let us consider a pendulum revolving about a vertical axis and swinging horizontally as shown in Figure 1. The rotational pendulum is assumed to have a length and a lumped mass and turn at constant speed . The kinetic energy and potential energy are where is the angular displacement of the pendulum in the vertical direction. The equation of rotational pendulum can be derived using the Lagrange equation. From the Lagrange equation of motion where . ？We have The second-order differential equation of the rotational pendulum system with initial conditions is where , . Figure 1: Rotational pendulum at a constant speed. According to (1), we have Consequently the rotational pendulum equation has a conservative behavior and a periodic solution. The variational principle for (4) can be