The stability of 3-DOF triple-rigid-body pendulum system near resonances

Abstract

In this article, the motion of three degree-of-freedom (DOF) dynamical system consisting of a triple rigid body pendulum (TRBP) in the presence of three harmonically external moments is studied. In view of the generalized coordinates of the system, Lagrange's equations are used to obtain the governing system of equations of motion (EOM). The analytic approximate solutions are gained up to the third approximation utilizing the approach of multiple scales (AMS) as novel solutions. The solvability conditions are determined in accordance with the elimination of secular terms. Therefore, the arising various resonances cases have been categorized and the equations of modulation have been achieved. The temporal histories of the obtained approximate solutions, as well as the resonance curves, are visually displayed to reveal the positive effects of the various parameters on the dynamical motion. The numerical results of the governing system are achieved using the fourth-order Runge–Kutta method. The visually depicted comparison of asymptotic and numerical solutions demonstrates high accuracy of the employed perturbation approach. The criteria of Routh–Hurwitz are used to investigate the stability and instability zones, which are then analyzed in terms of steady-state solutions. The strength of this work stems from its uses in engineering vibrational control applications which carry the investigated system a huge amount of importance.

Analyzing the motion of a forced oscillating system on the verge of resonance

“…where z 01 , :::, z 04 are known initial quantities. The substitution from the solutions ( 17), ( 18), ( 20), ( 21), (23), and (24) into the series (9) and the hypotheses (7), yields the desired approximate solutions y and z up to third approximation.…”

“…Many scientific researchers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16] have looked at dynamical pendulum models with 2DOF or 3DOF. The reason goes back to their importance in various domains of nonlinear dynamical systems such as industrial applications, biology, and medicine.…”

“…The damped motion of a linear spring in the presence of a force along the spring’s direction and two opposite moments at its pivot point to move in a circular path is investigated in Starosta et al 3 The case of the elliptic route is examined in Amer et al 4 Two perturbation approaches are used in Kamińska et al 5 to gain the asymptotic solutions (AS) of the equations of motion (EOM) for a nonlinear spring subjected to a resistance related with the velocity and one torque at the pivot point. The nonlinear motion of a damped elastic rigid-body-pendulum (RBP) is investigated in many works of literature, for example, Awrejcewicz et al, 6 Amer et al, 7 Amer et al, 8 He et al, 9 and He et al, 10 whether its suspension point may be fixed or moving in various routes. Utilizing the TMS, the AS of the EOM are gained and plotted versus time to examine the behavior of the models’ motions.…”

Dynamical analysis of a damped harmonic forced duffing oscillator with time delay

“…As opposed to that, the linearized stability can be investigated near the fixed points (equilibrium points). Therefore, one can use the following consistent assumptions 33,34 : Inserting Eq. ( 38) into Eqs.…”

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

“…The periodic motions of a periodically forced nonlinear elastic pendulum are explored in [33] using the semi-analytical approach, as well as the motion's bifurcation and stability analyses. In [34], RHC are used to explore the stability and instability zones, which are subsequently studied in terms of steady-state solutions. In [35], the authors obtained analytical solutions of an oscillating 2DOF dynamical system using MSA.…”