article · Scientific Reports
This research investigates the dynamical behaviour of a two degree-of-freedom system comprising a mass attached to a nonlinear damped harmonic spring pendulum coupled with a piezoelectric device. The model experiences a parametric excitation force alongside an operating moment. To suppress harmful vibrations that compromise efficiency during resonance, a negative-velocity-feedback controller is integrated into the system. Lagrangian mechanics provide the foundational equations of motion, which are solved analytically to the third order using a multiple-scales strategy and verified against fourth-order Runge-Kutta numerical methods. System stability, resonance states, and transitions are evaluated through bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums. Simultaneously, the attached piezoelectric element captures energy from the oscillatory motion, converting mechanical vibration into electrical power.
Unchecked mechanical vibrations can cause structural fatigue, noise, and failure in engineered systems. Developing methods that simultaneously dampen dangerous resonance while harvesting the dissipated energy as electricity offers a double benefit. It helps improve the durability and energy efficiency of mechanical components operating under continuous dynamic excitation.
The abstract highlights potential applications across commercial, industrial, aerospace, automotive, and medical sectors seeking vibration mitigation and energy harvesting. However, the research is at an early theoretical stage, relying on mathematical modelling and computer simulations. Significant hardware development, prototype fabrication, and physical bench testing will be required before these control and harvesting dynamics can be applied in real-world devices.
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This study aims to demonstrate the behaviors of a two degree-of-freedom (DOF) dynamical system consisting of attached mass to a nonlinear damped harmonic spring pendulum with a piezoelectric device. Such a system is influenced by a parametric excitation force on the direction of the spring's elongation and an operating moment at the supported point. A negative-velocity-feedback (NVF) controller is inserted into the main system to reduce the undesired vibrations that affect the system's efficiency, especially at the resonance state. The equations of motion (EOM) are derived by using Lagrangian equations. Through the use of the multiple-scales-strategy (MSS), approximate solutions (AS) are investigated up to the third order. The accuracy of the AS is verified by comparing them to the obtained numerical solutions (NS) through the fourth-order Runge-Kutta Method (RK-4). The study delves into resonance cases and solvability conditions to provide the modulation equations (ME). Graphical representations showing the time histories of the obtained solutions and frequency responses are presented utilizing Wolfram Mathematica 13.2 in addition to MATLAB software. Additionally, discusses the bifurcation diagrams, Poincaré maps, and Lyapunov exponent spectrums to show the various behavior patterns of the system. To convert vibrating motion into electrical power, a piezoelectric sensor is connected to the dynamical model, which is just one of the energy harvesting (EH) technologies with extensive applications in the commercial, industrial, aerospace, automotive, and medical industries. Moreover, the time histories of the obtained solutions with and without control are analyzed graphically. Finally, resonance curves are used to discuss stability analysis and steady-state solutions.
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DOI: 10.1038/s41598-024-75342-z
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