Periodic Property and Instability of a Rotating Pendulum System

The current paper investigates the dynamical property of a pendulum attached to a rotating rigid frame with a constant angular velocity about the vertical axis passing to the pivot point of the pendulum. He’s homotopy perturbation method is used to obtain the analytic solution of the governing nonlinear differential equation of motion. The fourth-order Runge-Kutta method (RKM) and He’s frequency formulation are used to verify the high accuracy of the obtained solution. The stability condition of the motion is examined and discussed. Some plots of the time histories of the gained solutions are portrayed graphically to reveal the impact of the distinct parameters on the dynamical motion.

[1]  Caixia Liu.  Periodic solution of fractal Phi-4 equation , 2021, Thermal Science.

[2]  S. S. Ganji,et al.  VARIATIONAL APPROACH METHOD FOR NONLINEAR OSCILLATIONS OF THE MOTION OF A RIGID ROD ROCKING BACK AND CUBIC , 2008 .

[3]  Y. El‐Dib.  Stability approach for periodic delay Mathieu equation by the He- multiple-scales method , 2018, Alexandria Engineering Journal.

[4]  Daniel Olvera Trejo,et al.  He's frequency–amplitude formulation for nonlinear oscillators using Jacobi elliptic functions , 2020 .

[5]  G. Moatimid,et al.  Stability Configuration of a Rocking Rigid Rod over a Circular Surface Using the Homotopy Perturbation Method and Laplace Transform , 2019, Arabian Journal for Science and Engineering.

[6]  Shaowen Yao,et al.  He's fractional derivative for the evolution equation , 2020 .

[7]  M. Hosen,et al.  Accurate approximations of the nonlinear vibration of couple-mass-spring systems with linear and nonlinear stiffnesses , 2021, Journal of Low Frequency Noise, Vibration and Active Control.

[8]  Homotopy Perturbation–Based Dynamic Analysis of Structural Systems , 2020 .

[9]  I. Khan,et al.  Micropolar mixed convective flow with Cattaneo-Christov heat flux: Non-fourier heat conduction analysis , 2020 .

[10]  I. Pavlović,et al.  DYNAMIC BEHAVIOR OF TWO ELASTICALLY CONNECTED NANOBEAMS UNDER A WHITE NOISE PROCESS , 2020 .

[11]  M. Hosen.  Analysis of nonlinear vibration of couple-mass–spring systems using iteration technique , 2020 .

[12]  Claude Brezinski,et al.  Numerical Methods for Engineers and Scientists , 1992 .

[13]  Y. El‐Dib.  Periodic solution of the cubic nonlinear Klein–Gordon equation and the stability criteria via the He-multiple-scales method , 2018, Pramana.

[14]  G. Feng.  He’s frequency formula to fractal undamped Duffing equation , 2021, Journal of Low Frequency Noise, Vibration and Active Control.

[15]  T. Amer,et al.  The Vibrational Motion of a Dynamical System Using Homotopy Perturbation Technique , 2020, Applied Mathematics.

[16]  Chun-Hui He,et al.  Homotopy perturbation method coupled with the enhanced perturbation method , 2018, Journal of Low Frequency Noise, Vibration and Active Control.

[17]  Cai-Cheng Liu.  A short remark on He’s frequency formulation , 2020, Journal of Low Frequency Noise, Vibration and Active Control.

[18]  N. S. Elgazery,et al.  A Periodic Solution of the Newell-Whitehead-Segel (NWS) Wave Equation via Fractional Calculus , 2020 .

[19]  Naveed Anjum,et al.  Application of he’s fractional derivative and fractional complex transform for time fractional Camassa-Holm equation , 2020 .

[20]  N. Anjum,et al.  FRACTAL N/MEMS: FROM PULL-IN INSTABILITY TO PULL-IN STABILITY , 2020 .

[21]  J. Niu,et al.  He’s frequency formulation for nonlinear vibration of a porous foundation with fractal derivative , 2021, GEM - International Journal on Geomathematics.

[22]  Galal M. Moatimid,et al.  Nonlinear instability of two streaming-superposed magnetic Reiner-Rivlin Fluids by He-Laplace method , 2021 .

[23]  Ji-Huan He,et al.  Homotopy perturbation method for N/MEMS oscillators , 2020 .

[24]  R. T. Matoog,et al.  The Rank Upgrading Technique for a Harmonic Restoring Force of Nonlinear Oscillators , 2021 .

[25]  Haiyan Song.  A thermodynamic model for a packing dynamical system , 2020 .

[26]  Yuting Zuo.  A gecko-like fractal receptor of a three-dimensional printing technology: a fractal oscillator , 2021, Journal of Mathematical Chemistry.

[27]  A. Koochi,et al.  Nonlinear Oscillations of CNT Nano-resonator Based on Nonlocal Elasticity: The Energy Balance Method , 2021 .

[28]  Dan Tian,et al.  Hybrid rayleigh–van der pol–duffing oscillator: Stability analysis and controller , 2021, Journal of Low Frequency Noise, Vibration and Active Control.

[29]  Y. El‐Dib.  Periodic Solution and Stability Behavior for Nonlinear Oscillator Having a Cubic Nonlinearity Time-Delayed , 2018 .

[30]  T. Amer,et al.  On the motion of a damped rigid body near resonances under the influence of harmonically external force and moments , 2020 .

[31]  Yue Wu,et al.  Residual calculation in He’s frequency–amplitude formulation , 2020 .

[32]  Ji-Huan He.  Homotopy perturbation technique , 1999 .

[33]  Shaowen Yao.  Variational principle for non-linear fractional wave equation in a fractal space , 2021, Thermal Science.

[34]  Shaowen Yao,et al.  A variational principle for the photocatalytic NOx abatement , 2020, Thermal Science.

[35]  Ji-Huan He,et al.  The fastest insight into the large amplitude vibration of a string , 2021, Reports in Mechanical Engineering.

[36]  U. Filobello-Niño,et al.  The study of heat transfer phenomena by using modified homotopy perturbation method coupled by Laplace transform , 2020 .

[37]  Y. El‐Dib.  Homotopy Perturbation for Excited Nonlinear Equations , 2017 .

[38]  Ji-Huan He.  The simpler, the better: Analytical methods for nonlinear oscillators and fractional oscillators , 2019, Journal of Low Frequency Noise, Vibration and Active Control.

[39]  I. A. Yeasmin,et al.  The modified Lindstedt–Poincare method for solving quadratic nonlinear oscillators , 2020, Journal of Low Frequency Noise, Vibration and Active Control.