Analytic approximate solutions of the cubic–quintic Duffing–van​ der Pol equation with two-external periodic forcing terms: Stability analysis

Journal of Low Frequency Noise, Vibration and Active Control

et al. 2021

The current study examines the hybrid Rayleigh–Van der Pol–Duffing oscillator (HRVD) with a cubic–quintic nonlinear term and an external excited force. The Poincaré–Lindstedt technique is adapted to attain an approximate bounded solution. A comparison between the approximate solution with the fourth-order Runge–Kutta method (RK4) shows a good matching. In case of the autonomous system, the linearized stability approach is employed to realize the stability performance near fixed points. The phase portraits are plotted to visualize the behavior of HRVD around their fixed points. The multiple scales method, along with a nonlinear integrated positive position feedback (NIPPF) controller, is employed to minimize the vibrations of the excited force. Optimal conditions of the operation system and frequency response curves (FRCs) are discussed at different values of the controller and the system parameters. The system is scrutinized numerically and graphically before and after providing the controller at the primary resonance case. The MATLAB program is employed to simulate the effectiveness of different parameters and the controller on the system. The calculations showed that NIPPF is the best controller. The validations of time history and FRC of the analysis as well as the numerical results are satisfied by making a comparison among them.

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller

Tian

Journal of Low Frequency Noise, Vibration and Active Control

et al. 2021

“…Similar derivations of Eq. ( 8 ) was earlier given by Nayfeh 19 in several situations for various mechanical problems, and by El-Dib and Moatimid 21 for a rocking rigid rod over a circular surface.…”

Section: Organization Of the Modelmentioning

confidence: 83%

“…6 and 7 . The details are provided by Galeb et al 21 in our previous study. It really should be noted that the current framework in addition the stability analysis, departs from those shown by Galeb et al 21 .…”

Section: Linearized Stabilitymentioning

confidence: 99%

Analytical solution for the motion of a pendulum with rolling wheel: stability analysis

Amer

2022

Sci Rep

The current work focuses on the motion of a simple pendulum connected to a wheel and a lightweight spring. The fundamental equation of motion is transformed into a complicated nonlinear ordinary differential equation under restricted surroundings. To achieve the approximate regular solution, the combination of the Homotopy perturbation method (HPM) and Laplace transforms is adopted in combination with the nonlinear expanded frequency. In order to verify the achievable solution, the technique of Runge–Kutta of fourth-order (RK4) is employed. The existence of the obtained solutions over the time, as well as their related phase plane plots, are graphed to display the influence of the parameters on the motion behavior. Additionally, the linearized stability analysis is validated to understand the stability in the neighborhood of the fixed points. The phase portraits near the equilibrium points are sketched.

“…To characterize the stability and instability maps, it is much more useful to study a set of randomly selected systems. The details may be found in our previous work [38]. The numerical calculations are carried out throughout Table 1.…”

Section: Linearization Of Stabilitymentioning

confidence: 99%

Analytical Approximate Solutions of a Magnetic Spherical Pendulum: Stability Analysis

Amer

2022

J. Vib. Eng. Technol.

Purpose Under certain conditions, the governing equation of motion of magnetic spherical pendulum results in a cubic-quintic Duffing equation. The current work aims to achieve an analytical bounded procedure of this equation. Methods This may be accomplished by grouping nonlinear expanded frequency, Homotopy perturbation method (HPM), and Laplace transforms. Therefore, this technique helps disregard the appearance of the source of secular terms. Results To validate the obtained explanation, based on the method of Runge–Kutta of the fourth order (RK4), the numerical calculation is performed. On the other hand, the linearized stability analysis is carried out to explore stability neighbouring the fixed points. Moreover, the time history of the attained solution and the corresponding phase plane plots are obtained to expose the influence of the affecting factors in the behavior of motion. Conclusions A comparison between both solutions gives a good matching between them, which explores the worthy accuracy of the approach in question. Several phase portraits are planned toward illustrating the different types of stability and instability near the equilibrium points, where the relation between the expanded and the cyclotron frequency (that are generated by the magnetic field) is characterized for diverse standards of the azimuthal angular velocity.