On the Modeling and Simulation of Variable-Length Pendulum Systems: A Review

Abstract

A comprehensive review of variable-length pendulums is presented. An attempt at a unique evaluation of current trends in this field is carried out in accordance with mathematical modeling, dynamical analysis, and original computer simulations. Perspectives of future trends are also noted on the basis of various concepts and possible theoretical and engineering applications. Some important physical concepts are verified using dedicated numerical procedures and assessed based on dynamical analysis. At the end of the review, it is concluded that many variable-length pendulums are very demanding in the modeling and analysis of parametric dynamical systems, but basic knowledge about constant-length pendulums can be used as a good starting point in providing much accurate mathematical description of physical processes. Finally, an extended model for a variable-length pendulum’s mechanical application being derived from the Swinging Atwood Machine is proposed. The extended SAM presents a novel SAM concept being derived from a variable-length double pendulum with a suspension between the two pendulums. The results of original numerical simulations show that the extended SAM’s nonlinear dynamics presented in the current work can be thoroughly studied, and more modifications can be achieved. The new technique can reduce residual vibrations through damping when the desired level of the crane is reached. It can also be applied in simple mechatronic and robotic systems.

Employing Electromagnetic Actuator to Harvest Energy Through a Variable-Length Pendulum

“…Examples include its integration into robotics, electromechanical setups such as induction motors, purely electrical networks like dc-dc power converters, and lifting devices like mine elevators or cranes. Moreover, its relevance extends to unconventional domains, contributing to seismic activity detection through inverted pendulum concepts and even playing a role in the development of wave energy converters (WEC), as elaborated by Yakubu, Yurchenko, and others [24,25,26]. This versatility underscores the variable-length pendulum's significance in both theoretical and practical contexts within the realm of dynamic systems and mechatronics.…”

Dynamics and non-integrability of the double spring pendulum

“…Currently, there is great activity studying variable-length pendulum systems, such as the swinging Atwood machine [27] and its generalizatins [28], the variable-length coupled pendulums recently studied in [24], or the double variable-length pendulum with counterweight mass [71][72][73]. Variable-length pendulum systems are excellent examples for studying nonlinear dynamics, chaos, and bifurcations.…”

Variable-Length Pendulum-Based Mechatronic Systems for Energy Harvesting: A Review of Dynamic Models

“…According to a review article [2], the variable-length pendulum system is recognized as highly demanding due to its complex modeling and analysis requirements. However, the authors suggested that a solid understanding of constant-length pendulums can serve as a foundation for creating a more accurate mathematical description of physical processes.…”

“…Variable-length pendulums refer to those whose length can be adjusted either manually or automatically [1]. These pendulums find applications across diverse fields, serving as energy harvesters (e.g., in wave energy converters), load-lifting mechanisms (used in cranes), components in robotics, timekeeping devices, experimental tools, and even attractions in amusement parks [2]. Due to their dynamic or manual length adjustment capability, variable-length pendulums have proven versatile in scientific research, education, and practical contexts [3].…”