article · Nonlinear Dynamics
This research examines the dynamics of a novel two-degree-of-freedom double pendulum whose pivot point is constrained to move along a planar Lissajous curve. Using Lagrange equations, governing nonlinear differential equations were formulated, and higher-order analytical approximations were derived through the multiple-scales method. These analytical solutions were subsequently verified using fourth-order Runge-Kutta numerical simulations. The study evaluates system stability and frequency response curves across diverse resonance conditions using Routh-Hurwitz criteria. In addition, comprehensive analyses using bifurcation diagrams and Lyapunov exponent spectra demonstrate the conditions under which the system transitions between stable, quasi-stable, and chaotic behaviours. The resulting models capture how specific parameter values and resonance states govern complex vibrational movements.
Mechanical components frequently undergo complex, unpredictable oscillations when subjected to cyclic or curved motion paths. By clarifying how coupled pendular systems shift from steady oscillation into chaotic instability, this work helps dynamicists and engineers understand the mathematical limits of stability and prevent unintended, damaging vibrations in multi-jointed mechanical systems.
The abstract points to applications in robotics, pump compressors, rotor dynamics, and transportation machinery needing vibration analysis. The work represents early-stage analytical research, providing foundational equations and stability maps rather than a tested hardware prototype or applied software package. Prospective users are engineering design teams and dynamicists who could incorporate these resonance and chaos-boundary models into early design and simulation workflows for multi-body mechanical devices.
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Abstract The focus of this paper is to examine the motion of a novel double pendulum (DP) system with two degrees of freedom (DOF). This system operates under specific constraints to follow a Lissajous curve, with its pivot point moving along this path in a plane. The nonlinear differential equations governing this system are derived using Lagrange's equations. Their analytical solutions (AS) are subsequently calculated using the multiple-scales method (MSM), which provides higher-order approximations. These solutions are considered new, as the traditional MSM has been applied to this novel system for the first time. Additionally, the accuracy of these solutions is validated through numerical results obtained using the fourth-order Runge–Kutta method. The solvability conditions and characteristic exponents are determined based on resonance cases. The Routh–Hurwitz criteria (RHC) are employed to assess the stability of the fixed points corresponding to the steady-state solutions. They are also used to demonstrate the frequency response curves. The nonlinear stability analysis is performed by examining the stability and instability ranges. Resonance curves and time history plots are presented to analyze the behavior of the system for specific parameter values. The investigation delves into a comprehensive analysis of bifurcation diagrams (BDs) and Lyapunov exponent spectra (LEs), aiming to uncover the various types of motion present within the system. Systematic examination of these charts reveals critical insights into transitions between stable, quasi-stable, and chaotic dynamical behaviors. This work has practical applications in various fields, such as robotics, pump compressors, rotor dynamics, and transportation devices. It can be used to study the vibrational motion of these systems.
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DOI: 10.1007/s11071-024-10169-x
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