Modeling and analyzing the motion of a 2DOF dynamical tuned absorber system close to resonance

Abstract

This work investigates the planar motion of a dynamical model with two degrees-of-freedom (DOF) consisting of a connected tuned absorber with a simple pendulum. It is taken into account that the pendulum’s pivot moves in a Lissajous trajectory with stationary angular velocity in the presence of a harmonic excitation moment. In terms of the model’s generalized coordinates, Lagrange’s equations are used to derive the motion’s controlling system. The approximate solutions of this system, up to a higher order of approximation, are achieved utilizing the approach of multiple scales (AMS). Resonance cases are all classified, in which two of them are examined simultaneously to gain the corresponding equations of modulation. The solutions at the steady-state are studied in terms of solvability conditions. According to the Routh-Hurwitz criteria, all potential fixed points at steady and unsteady states are determined and graphed. The dynamical behavior of the motion's time-histories and the curves of resonance are drawn. Regions of stability are examined by inspecting their graphs in order to assess the favorable impact of various parameters on the motion. The achieved outcomes are regarded as novel because the used methodology is applied to a specific dynamical system. The importance of this model under study can be seen from its numerous applications in disciplines like engineering and physics. Furthermore, pendulum vibration absorbers are commonly employed to reduce the vibrations in engineering constructions such as chimneys, bridges, television towers, high buildings, auto-balancing shafts, and antennas.

Dynamical analysis of a forced vibrating planar motion of a spring pendulum

“…perform specific time scales, where 0  is a fast time scales and 12 ,  are slow ones. To formulate () n z  and () n  , we can write  in termers of n  as follows [37] , ( ).…”

Evaluation of the stability of a two degrees-of-freedom dynamical system

“…9,10 Vibrational motions can be studied by transforming the motion into equations that may be solved, studying their stability areas and the responsibilities of various parameters of the solution to find the best possible solutions. [11][12][13] The vibrational motions are noted in the movement of some pendulums on various trajectories and many other examples in daily life. [14][15][16][17][18] It should be highlighted that many researchers have become interested in the movement of damped elastic pendulums along various pathways.…”