Stability of the Dynamical Motion of a Damped 3DOF Auto-parametric Pendulum System

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.

show abstract

“…(48) as well as exhibiting its characteristics. Therefore, the following transformations are considered [45,46]…”

Section: Nonlinear Analysismentioning

confidence: 99%

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

et al. 2022

Self Cite

View full text Add to dashboard Cite

show abstract

“…Therefore, we can phrase their solutions as a linear combination of k d e λτ , where λ represents the eigenvalue of these perturbations and k d (d 1, 2, 3, 4) are constants, which are counted from the roots real potions. In this context, if h j0 and θ j0 ( j 1, 2) are stable asymptotically, then the roots real portions of the next characteristic equation must therefore be negative [41] curves of Fig. 13b show that the stability areas are σ 1 < 0.001 and σ 1 < 0.002, while the unstable ones are 0.001 ≤ σ 1 and 0.002 ≤ σ 1 when ω 1 1.5 and ω 1 1.7.…”

Section: Oscillations At the Case Of Steady-statementioning

confidence: 99%

On the solutions and stability for an auto-parametric dynamical system

2022

Self Cite

View full text Add to dashboard Cite

The main goal of this study is to look at the motion of a damped two degrees-of-freedom (DOF) auto-parametric dynamical system. Lagrange’s equations are used to derive the governing equations of motion (EOM). Up to a good desired order, the approximate solutions are achieved utilizing the method of multiple scales (MMS). Two cases of resonance, namely; internal and primary external one are examined simultaneously to explore the solvability conditions of the motion and the corresponding modulation equations (ME). These equations are reduced to two algebraic equations, through the elimination of the modified phases, in terms of the detuning parameters and the amplitudes. The kind of stable or unstable fixed point is estimated. In certain plots, the time histories graphs of the achieved solutions, as well as the adjusted phases and amplitudes are used to depict the motion of the system at any instant. The conditions of Routh–Hurwitz are used to study the various stability zones and their analysis. The achieved outcomes are considered to be novel and original, in which the used strategy is applied on a particular dynamical system. The significance of the studied system can be observed in its applications in a number of disciplines, such as swaying structures and rotor dynamics.

show abstract

“…A further examination of the solvability criteria reveals six nonlinear PDE regarding to the functions B j = B j ( 1 , 2 ), (j = 1, 2, 3) . Consequently, one writes them in their below polar forms Here, bj and j represent the real functions of amplitudes and phases of ỹ, φ, and γ , while b j denote the amplitudes of y, , and as indicated in the hypotheses (7).…”

Section: Resonance's Classifications and Modulation Equationsmentioning

confidence: 99%

“…Over the past few periods, a large number of publications addressing different types of dynamic dampers and associated subjects drew the attention of numerous researchers e.g. [4][5][6][7][8][9]. In [4], the author investigated the vibrational motion of an absorber with a rotating base to reduce the vertical excitations, from which the fundamental frequency has been derived.…”

Section: Introductionmentioning

confidence: 99%

“…The planar movement of a two degrees-of-freedom (2DOF) auto-parametric model, consisting of a movable attached main mass through a damped spring coupled to a simple pendulum of a rigid arm as a secondary part, was discussed in [6]. In [7], the authors replaced the pendulum rigid arm with a spring pendulum with linear stiffness to constitute a 3DOF model and generalize the work in [6]. Recently, the stability of 2DOF auto-parametric systems has been analyzed in [8] and [9] for two dynamical models.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dynamical Stability of a 3-DOF Auto-Parametric Vibrating System

Amer

Moatimid

Amer

2022

J. Vib. Eng. Technol.

Self Cite

View full text Add to dashboard Cite

Purpose This article concentrates on the oscillating movement of an auto-parametric dynamical system comprising of a damped Duffing oscillator and an associated simple pendulum in addition to a rigid body as main and secondary systems, respectively. Methods According to the system generalized coordinates, the controlling equations of motion are derived utilizing Lagrange's approach. These equations are solved applying the perturbation methodology of multiple scales up to higher orders of approximation to achieve further precise unique outcomes. The fourth-order Runge–Kutta algorithm is employed to obtain numerical outcomes of the governing system. Results The comparison between both solutions demonstrates their high level of consistency and highlights the great accuracy of the adopted analytical strategy. Despite the conventional nature of the applied methodology, the obtained results for the studied dynamical system are considered new. Conclusions In light of the solvability criteria, all resonance scenarios are classified, in which two of the fundamental exterior resonances are examined simultaneously with one of the interior resonances. Therefore, the modulation equations are achieved. The conditions of Routh–Hurwitz are employed to inspect the stability/instability regions and to analyze them in accordance with the solutions in the steady-state case. For various factors of the examined structure, the temporary history solutions, the curves of resonance in terms of the adjusted amplitudes and phases, and the stability zones are graphically presented and discussed. Applications The results of the current study will be of interest to wide range experts in the fields of mechanical and aerospace technology, as well as those working to reduce rotors dynamical vibrations and attenuate vibration caused by swinging structures.

show abstract

Stability of the Dynamical Motion of a Damped 3DOF Auto-parametric Pendulum System

Cited by 20 publications

References 29 publications

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

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

On the solutions and stability for an auto-parametric dynamical system

Dynamical Stability of a 3-DOF Auto-Parametric Vibrating System

Contact Info

Product

Resources

About