An Algorithm for Melnikov Functions and Application to a Chaotic Rotor

Abstract

Abstract. In this work we study a dynamical system with a complicated nonlinearity, which describes oscillation of a turbine rotor, and give an algorithm to compute Melnikov functions for analysis of its chaotic behavior. We first derive the rotor model whose nonlinear term brings difficulties to investigating the distribution and qualitative properties of its equilibria. This nonlinear model provides a typical example of a system for which the homoclinic and heteroclinic orbits cannot be analytically determined. In order to apply Melnikov's method to make clear the underlying conditions for chaotic motion, we present a generic algorithm that provides a systematic procedure to compute Melnikov functions numerically. Substantial analysis is done so that the numerical approximation precision at each phase of the computation can be guaranteed. Using the algorithm developed in this paper, it is straightforward to obtain a sufficient condition for chaotic motion under damping and periodic external excitation, whenever the rotor parameters are given.Key words. mathematical modeling, Melnikov function, chaos, modulus of continuity, oscillatory integrand AMS subject classifications. 34C28, 34C37, 34C45 DOI. 10.1137/S10648275034207261. Introduction. The oscillation phenomenon of the turbine rotor has long been observed in industry. Many efforts [7,8] have been made to analyze and control the mode and scale of such oscillations, which are caused by a number of factors, e.g., unbalanced centrifugal force. However, to eliminate such vibrations effectively, which are harmful to the quality and life span of the system, it is necessary to examine the underlying nature of those nonlinear oscillations and to find the relationship between the oscillatory behavior and the physical parameters of the real system.Consider the shaft of a rigid rotor supported symmetrically by four equal bearings at the ends on a low-speed balance platform, as shown in Figure 1. The elasticity in each bearing can be modeled equivalently as a spring. Let L denote the length of the shaft, and let c and K denote the length and stiffness of each relaxed spring, respectively. The following linear model is often used [3,4,18,27]:

A Unified Approach to Chaos Suppressing and Inducing in a Periodically Forced Family of Nonlinear Oscillators

“…14-24]. has been confirmed in (14). The homoclinic orbits have been solved in (16), so also satisfies.…”

“…However, this method applies only to Hamiltonian systems in simple forms, with the curves of homoclinic (heteroclinic) orbits available analytically. In some cases, the nonlinear function is highly complicated, such as the chaotic rotor model in [14], Melnikov's analytical method does not work because it is difficult to solve and in analytical forms for the two homoclinic orbits, aside from the technical difficulties in the method of residues. In other case, the analytical expressions of these orbits can be solved, but the orders of singular points are fractional-order [39]; method of residues loses effectiveness.…”

“…Numerical simulations for trajectories of perturbed system (11) can hardly display convincing evidence for chaotic behavior because the system is non-autonomous [14]. Theoretically, iteration of its Poincaré mapping on the stable manifold and the unstable manifold can describe chaos as well.…”

Simulation to chaos in a rotor

“…The oscillation of the shaft of a rigid rotor described in Figure 1 is modeled by the nonlinear differential equation ( [19])…”

“…Even if such expressions were given, as in [10,11,12], the computation of integrals in the Melnikov functions would remain a very difficult task. In [19] a numerical algorithm is given to compute the Melnikov function ( [6]) with no need of the closed-form of the heteroclinic (homoclinic) orbits. The algorithm produces numerical approximations V + (n) and 2I + (n, k) to the integrals V + and I + respectively for any given precision ε > 0.…”