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

Journal of Low Frequency Noise, Vibration and Active Control

et al. 2022

In this paper, a nonlinear vibrating dynamical system of two degrees-of-freedom is examined. By virtue of the generalized coordinates, the controlling system of motion is constructed using Lagrange’s equations. Among other perturbation approaches, the technique of multiple scales (TMS) is utilized to get the required solutions analytically of this system till the third approximation. These solutions allow us to discuss the system’s behavior and realize both the solvability requirements and the modulation equations in the context of the system’s resonance scenarios. The resonance curves and solution diagrams are drawn to demonstrate the influence of the various parameters of the behavior of the considered system. The stability and instability of the gained possible fixed points are examined. The used method is considered traditional, but it is applied to a certain specified model. Therefore, the obtained outcomes are considered novel. The importance of this work is represented in its various applications in practical life, such as the motion of railway vehicles, for transporting goods in ports, and for facilitating the movement of passengers’ baggage at airports.

Section: Steady-state Solutionsmentioning

confidence: 99%

“…This section’s purpose is to look at the steady-state solutions of the considered model, which correspond to the zero derivative of amplitudes hj and modified phases θj(j=1,2). Therefore, we consider 39 …”

Section: Steady-state Solutionsmentioning

confidence: 99%

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

Ghanem

Journal of Low Frequency Noise, Vibration and Active Control

et al. 2022

“…The functions B j (j = 1, 2, 3) can be calculated in accordance with the previous restrictions ( 14) and ( 16), besides the initial conditions With the help of the assumptions (7), the suggested series (8), and the achieved approximate solutions ( 13), (15), and ( 17), we may simply obtain the appropriate approximate expressions of y, , and up to the third-order of approximation.…”

Section: Order Ofmentioning

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%

“…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. The first one is the vertical movement of a spinning cylinder on a connected circular surface with a nonlinear damped spring, while the horizontal frictional motion of this surface was considered in [9].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

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

Moatimid

2022

J. Vib. Eng. Technol.

Self 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.

Analytical and numerical study of a vibrating magnetic inverted pendulum

2023

The current study investigates the stability structure of the base periodic motion of an inverted pendulum (IP). A uniform magnetic field affects the motion in the direction of the plane configuration. Furthermore, a non-conservative force as one that dampens air is considered. Its underlying equation of motion is derived from traditional analytical mechanics. The mathematical analysis is made simpler by substituting the Taylor theory in order to expand the restoring forces. The modified Homotopy perturbation method (HPM) is employed to achieve a roughly adequate regular result. To support the prior result, a numerical method based on the fourth-order Runge-Kutta method (RK4) is employed. The graphs for both the analytic and numerical solutions are highly consistent with one another, which indicates that the perturbation strategy is accurate. The solution time history curve exhibits a decaying performance and indicates that it is steady and without chaos. The resonance and non-resonance cases are found through the stability study by using the time scale method. In all perturbation approaches, the methodology of multiple time scales is actually regarded as a further standard approach. The time history is used to create a collection of graphs. Some graphical representations are used to illustrate how the typical physical values affect the behavior of the discovered solution. It has been discovered that the statically unstable IP can have its instability reduced by raising the spring torsional constant stiffness as well as the damped coefficient. Moreover, the magnetic field has a significant role in the stability configuration, which explains that at higher values of this field, the decaying waves take much more time than the smaller values of this field. Accordingly, it can be employed in various engineering devices that need a certain period of time to be more stable.