Keywords  Pendulum, nonlinear absorber, parametric excitation

Introduction To reduce the oscillations of a pendulum type structure, Matsuhisa et al. [3] proposed a linear damper. Since then, nonlinear absorbers have been studied and developped as the nonlinear energy sink (NES) [4]. Hurel et al. [1] studied the behavior of a two degree-of-freedom pendulum coupled with a NES under a generalized force. Here we study the two dimensional model of the same system under a parametric excitation corresponding to the imposed vertical movement of the attached point of the pendulum.

Description of the system The main system is a pendulum of masses M in the gravitational field which can oscillate in the plan around its attachment point O with an angle phi. A nonlinear absorber is coupled to the pendulum at a distance a from O. The absorber is a mass m << M linked to the pendulum with a purely nonlinear cubic restoring force and a damping. The relative displacement of the mass is in the orthogonal direction of the main axis of the pendulum. The displacement of the point O in the vertical direction is imposed and is assumed to be periodic. It is expressed with Fourier series. The governing equations of the system are determined with the Lagrange equations.

Analytic development The system is treated with a multiple scales method where the small parameter epsilon is the ratio of mass m and M. We assume the variables are very small and we perform a rescaling with epsilon. Then, the complex variables of Manevitch are introduced Phi and U [2]. The different time scales are linked to each other with the parameter epsilon. A Galerkin method is used to keep only the first harmonic of the response of the system. We assume that the fundamental frequency of excitation is close to the natural frequency of the pendulum. At the first time scale, the system equations describe an asymptotic state called slow invariant manifold (SIM) with its stable and unstable zones. At the second time scale, the equations show that the amplitude of oscillations depends on the second coefficient of the Fourrier series. For a given value of this coefficient, the equilibrium and singular points can be represented as a function of the frequency of excitation. For some frequencies, several equilibrium points exist. To better understand the behavior of the system, we need to trace the phase portrait of the variable Phi. The analysis of the basins of attraction identifies the domain where the amplitudes are decreasing to zero and the domain where the oscillation are maintained.

Numerical integrations In order to validate the analytic developments, the governing equations of the system are numerically solved.

Conclusion First, the analytic study is coherent with the numerical results and it gives us keys to understand the behavior of the system. This study shows that there are possibilities of multiple states of equilibrium (stable or unstable) which should be traced carefully in design.

References
[1] Hurel, G., Ture Savadkoohi, A., Lamarque, C.H.: Nonlinear vibratory energy exchanges between a two degrees-of-freedom pendulum and a nonlinear absorber. Journal of Engineering Mechanics (2018). DOI 10.1061/(ASCE)EM.1943-7889.0001620. (in press)
[2] Manevitch, L.I.: The Description of Localized Normal Modes in a Chain of Nonlinear Coupled Oscillators Using Complex Variables. Nonlinear Dynamics 25(1), 95–109 (2001). DOI 10.1023/A:1012994430793
[3] Matsuhisa, H., Gu, R., Wang, Y., Nishihara, O., Sato, S.: Vibration Control of a Ropeway Carrier by Passive Dynamic Vibration Absorbers. JSME international journal. Ser. C, Dynamics, control, robotics, design and manufacturing 38(4), 657–662 (1995). DOI 10.1299/jsmec1993.38.657
[4] Vakakis, A.F., Gendelman, O.: Energy Pumping in Nonlinear Mechanical Oscillators: Part II. Resonance Capture. Journal of Applied Mechanics 68(1), 42 (2001). DOI 10.1115/1.1345525