Non-linear Oscillations of a Pendulum on a Flexible Stretchable String

Mechanics Physics
non-linear oscillations flexible string asymptotic method resonance energy transfer

Academic Background Introduction

The motion of a pendulum is one of the fundamental problems in classical mechanics, while the vibration of a pendulum on a flexible and stretchable string involves more complex nonlinear dynamic phenomena. Such problems have broad significance in practical applications, such as in engineering structures, biomechanics, and materials science. However, due to the flexibility and stretchability of the string, the vibration of the pendulum involves not only transverse vibrations but also longitudinal vibrations, making the problem particularly complex. Traditional research methods often struggle to handle such multi-degree-of-freedom, nonlinear coupled vibration systems.

The authors of this paper aim to study the nonlinear vibration phenomena of a pendulum on a flexible and stretchable string, particularly the interaction between transverse and longitudinal vibrations and their energy transfer mechanisms. By proposing an analytical asymptotic method, the authors successfully separate high-frequency and low-frequency motions and reveal resonance phenomena in the system. This research not only provides new theoretical tools for understanding complex nonlinear vibration systems but also offers important references for related engineering applications.

Source of the Paper

This paper is co-authored by A. A. Malashin, A. D. Ostromogilskiy, D. A. Khramov, and P. A. Diakov. They are affiliated with the Faculty of Mechanics and Mathematics at Lomonosov Moscow State University (MSU), the Federal Science Center Scientific Research Institute for System Analysis of the Russian Academy of Sciences, and the Bauman Moscow State Technical University, respectively. The paper was accepted on January 27, 2025, and published in the journal Nonlinear Dynamics.

Research Process and Detailed Content

a) Research Process

  1. Problem Modeling and Equation Establishment
    The authors first established the motion equations for a pendulum on a flexible and stretchable string. The motion of the string is described by a set of coupled nonlinear partial differential equations, which take into account both transverse and longitudinal vibrations of the string. To simplify the problem, the authors introduced dimensionless variables and divided the equations into two parts describing high-frequency and low-frequency motions.

  2. Proposal and Application of the Asymptotic Method
    The authors proposed a new asymptotic method to separate high-frequency and low-frequency motions. By introducing a small parameter, they transformed the problem into a system with “slow time” and “fast time.” For low-frequency motion, they used elliptic equations for description, while for high-frequency motion, hyperbolic equations were employed. This method simplifies the complex nonlinear problem and allows for approximate solutions through asymptotic series.

  3. Numerical Simulation and Experimental Verification
    To verify the correctness of the theoretical analysis, the authors conducted numerical simulations. They designed a finite difference method to solve the high-frequency motion equations and validated the numerical simulation results through experiments. In the experiments, a flexible rubber string was used with a mass attached to one end. Using high-speed video equipment, the authors recorded the vibration process of the string and processed and analyzed the data.

b) Main Results

  1. Analytical Solution for Low-Frequency Motion
    The authors obtained the analytical solution for low-frequency motion using the asymptotic method and found that the pendulum’s motion resembles that of a “swinging spring” system. They also discovered that when the string’s tension is low, the amplitude of transverse vibrations significantly increases.

  2. Energy Transfer in High-Frequency Motion
    In high-frequency motion, the authors revealed the phenomenon of energy transfer from longitudinal to transverse vibrations. Under certain initial conditions, the amplitude of transverse vibrations can increase severalfold. This finding provides a new perspective for understanding energy transfer mechanisms in complex vibration systems.

  3. Experimental Verification
    The experimental results showed good agreement with the theoretical analysis and numerical simulations. Particularly in resonance cases, the authors observed significant energy transfer phenomena, further validating the correctness of the theoretical model.

Conclusions and Research Value

By proposing a new asymptotic method, this paper successfully addresses the nonlinear vibration problem of a pendulum on a flexible and stretchable string. The research not only reveals the energy transfer mechanism between transverse and longitudinal vibrations but also provides important theoretical support for related engineering applications. For example, in the vibration analysis of structures such as bridges and cables, this research can help engineers better understand and control vibration phenomena, thereby improving the safety and stability of structures.

Research Highlights

  1. Novel Asymptotic Method
    The asymptotic method proposed in this paper provides new tools for handling complex nonlinear vibration systems, particularly innovative in separating high-frequency and low-frequency motions.

  2. Revelation of Energy Transfer
    The study is the first to reveal the phenomenon of energy transfer from longitudinal to transverse vibrations in a pendulum system on a flexible and stretchable string, offering a new perspective for understanding complex vibration systems.

  3. Rigorous Experimental Verification
    Through meticulous experimental design and data processing, the authors verified the correctness of the theoretical analysis and numerical simulations, further enhancing the reliability of the research results.

Other Valuable Information

This research not only has theoretical value but also broad application prospects. For example, in fields such as aerospace, civil engineering, and mechanical engineering, this research can help engineers better understand and control vibration phenomena in complex structures, thereby improving system performance and safety. Additionally, the asymptotic method proposed in this paper can be applied to the study of other nonlinear dynamic problems, demonstrating high universality and potential for widespread use.