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Oscillation and Reaction Board Techniques for Estimating Inertial Properties of a Below-knee Prosthesis
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An innovative methodology in analyzing certain pendulum oscillators.

Galal M Moatimid1, T S Amer2, Abdallah A Galal3

  • 1Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt.

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|May 7, 2025
PubMed
Summary
This summary is machine-generated.

This study analyzes three types of simple pendulums (SPs) using a non-perturbative approach (NPA). The NPA offers a novel method for analyzing nonlinear dynamics and stability in mechanical systems.

Keywords:
He’s frequency formulaNon-perturbative approachNonlinear oscillationsPhase portraitsSimple pendulumStability diagrams

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Area of Science:

  • Physics
  • Mechanical Engineering
  • Applied Mathematics

Background:

  • Pendulum oscillators are fundamental to understanding harmonic motion, energy conservation, and nonlinear dynamics.
  • Their applications span diverse fields including engineering, seismology, and quantum mechanics.

Purpose of the Study:

  • To analyze three distinct simple pendulum (SP) systems: a charged magnetic SP, a rolling cylinder SP, and a damped SP in fluid flow.
  • To apply the non-perturbative approach (NPA) for a novel analysis of nonlinear dynamics and stability criteria.

Main Methods:

  • Utilized the non-perturbative approach (NPA), based on He's frequency formula (HFF), to linearize nonlinear ordinary differential equations (ODEs).
  • Employed Mathematica Software (MS) for numerical comparison and validation of the linearized models against nonlinear ODEs.
  • Generated time history plots and phase plane plots to analyze system behavior and stability.

Main Results:

  • The NPA demonstrated advantages over traditional perturbation techniques, avoiding Taylor expansions and enabling stability analysis.
  • Numerical comparisons showed strong concordance between the NPA's linearized solutions and the original nonlinear ODEs.
  • Phase portraits revealed system stability and instability near equilibrium points, influenced by magnetic fields and angular velocities.

Conclusions:

  • The non-perturbative approach (NPA) provides a precise and effective method for analyzing complex nonlinear dynamics in simple pendulum systems.
  • The study highlights the influence of system parameters, such as magnetic fields, on pendulum motion and stability.
  • This research offers valuable insights into the behavior of mechanical vibrations and nonlinear systems.