Analyzing the Stability for the Motion of an Unstretched Double Pendulum near Resonance

Tarek S Amer,Mohamed A Bek,Abdelkarim S Elameer,Roman Starosta

doi:10.3390/app11209520

Tarek S Amer, Mohamed A Bek + Show 2 more

Open Access

https://doi.org/10.3390/app11209520

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Abstract

This work looks at the nonlinear dynamical motion of an unstretched two degrees of freedom double pendulum in which its pivot point follows an elliptic route with steady angular velocity. These pendulums have different lengths and are attached with different masses. Lagrange’s equations are employed to derive the governing kinematic system of motion. The multiple scales technique is utilized to find the desired approximate solutions up to the third order of approximation. Resonance cases have been classified, and modulation equations are formulated. Solvability requirements for the steady-state solutions are specified. The obtained solutions and resonance curves are represented graphically. The nonlinear stability approach is used to check the impact of the various parameters on the dynamical motion. The comparison between the attained analytic solutions and the numerical ones reveals a high degree of consistency between them and reflects an excellent accuracy of the used approach. The importance of the mentioned model points to its applications in a wide range of fields such as ships motion, swaying buildings, transportation devices and rotor dynamics.

Highlights

Since applied mechanics is considered as a section of physical science that describes a response of the bodies’ system, which started from rest or motion under the influence of external forces [4], it has been used in many fields of engineering, especially electrical and mechanical engineering, in the areas of engineering machines, rotor dynamics, pumps and compressors
This paper studied the dynamical motion of the 2-degrees of freedom (DOF) double pendulum system with two different massless rods of constant lengths
It has been considered that its pivot point moves in an elliptic trajectory with a steady angular velocity

Summary

Introduction

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. The dynamic response of a spring pendulum was investigated in [14,15], in which the MST was applied to gain the asymptotic solutions of the analyzed systems. The time histories and the resonance curves were plotted to reveal the impact of the body’s parameters on the motion Another trajectory for the motion of the pivot point was considered in [20], in which the authors elaborated the dynamical motion of the 2-DOF damped elastic pendulum in a Lissajous curve. The curves of the time histories of the attained solutions and resonance ones were plotted with various selected values of the used parameters to reveal the good impact of these parameters on the dynamical behaviour of the investigated system. The importance of the model selected lies in its diverse set of applications in different fields, for example, in swaying buildings

Dynamical Modelling

Analysis of the Solution

Vibrations and Conditions of Resonance

Steady-State Case

Conclusions