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Magnetic coupling nonlinear internal resonance: An inquiry-based experimental teaching design
Experimental Technology and Management 2026, 43(5): 282-287
Published: 20 May 2026
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Objective

In line with recent research advances, this work develops an exploratory experimental teaching platform specifically designed to investigate nonlinear internal resonance phenomena, focusing on magnetic coupling effects as the key mechanism. The main goal of this platform is to help students systematically and thoroughly learn the fundamental theories of nonlinear vibration and to improve their skills in independent critical thinking, experimental planning, and problem-solving design. Additionally, the platform guides students through the entire process of structural modeling, theoretical analysis, and experimental validation of internal resonance phenomena, fostering a deeper understanding of the involved dynamic behaviors.

Methods

For the physical experimental setup, double cantilever beam specimens are made using 3D printing technology, and an internal resonance testing platform is built by integrating these specimens with a vibration testing bench, laser displacement sensors, and an LMS data acquisition system. Identical permanent magnets are attached to the free ends of the two cantilever beams, with their stiffness adjusted to experimentally tune and control the internal resonance occurrence. In the theoretical modeling, a magnetic dipole model is used to describe and calculate the nonlinear magnetic forces between the magnets. Based on this, the equations governing the magnetically coupled double cantilever beam system are formulated following Newton's second law. During testing, the system is excited with a constant-amplitude sinusoidal sweep, and its time-domain responses are recorded using an LMS data acquisition system. These signals are then transformed into the frequency domain via fast Fourier transform, allowing clearer observation and detailed analysis of internal resonance features.

Results

Focusing on the 2 : 1 internal resonance, key experimental observations include: (1) sweep-frequency tests show that the low-frequency cantilever beam, with a natural frequency around 17.5 Hz, exhibits distinct amplitude peaks when excited within the 48–55 Hz range;(2) frequency spectrum analysis indicates that under 48 Hz excitation, the response frequency of the low-frequency beam stabilizes near 24 Hz, confirming a 2 : 1 internal resonance between the two modes;(3) when the excitation is close to 50 Hz, the system initially does not display internal resonance under steady conditions; however, applying an external perturbation reliably triggers a clear 2 : 1 resonant response, demonstrating the phenomenon's sensitivity to initial conditions and external factors. Collectively, these results illustrate the characteristic frequency coupling and activation threshold associated with 2 : 1 internal resonance in the magnetically coupled cantilever system.

Conclusions

Practical teaching experience and implementation results confirm that the proposed experimental platform is easy to assemble, follows standard procedures, and is highly practical and user-friendly in educational settings. Furthermore, the platform greatly enhances students' understanding of nonlinear dynamics fundamentals and encourages their development in scientific inquiry, engineering insight, and hands-on experimental skills, supporting a more inquiry-based and applied approach to teaching.

Open Access Full Length Article Issue
Bandgap formation and low-frequency structural vibration suppression for stiffened plate-type metastructure with general boundary conditions
Chinese Journal of Aeronautics 2023, 36(10): 210-228
Published: 13 May 2023
Abstract Collect

Metastructures with unique mechanical properties have shown attractive potential application in vibration and noise reduction. Typically, most of the metastructures deal with the vibration bandgap properties of infinite structures without considering specific boundary condition and dynamic behaviors, which cannot be directly applied to the engineering structures. In this research, we design a Stiffened Plate-type Metastructure (SPM) composed of a plate with periodic stiffeners and cantilever beam-type resonators subjected to general boundary conditions for low-frequency vibration suppression. The effects of boundary conditions and the number and orientation of the stiffeners on Locally Resonant (LR) type bandgap properties in SPM are further investigated. An analytical modeling framework is developed to predict the bandgap formations and vibration behaviors of SPMs in finite-size configuration. The governing equations of the SPM reinforced by various arrangements of stiffeners are derived based on the first-order shear deformation theory and Hamilton’s principle, and a Fourier series combined with auxiliary functions is employed to satisfy the arbitrary boundary conditions. Finite element analysis and experimental investigations of vibration behaviors for the SPM are carried out to validate the accuracy and reliability of the present analytical model. For practical designs of the SPMs with specific boundary conditions, it is found that there exist optimal numbers of stiffeners and resonators which can produce the significant LR-type bandgap behaviors. Furthermore, various arrangements of stiffeners and resonators are explored for different boundary conditions by breaking the requirement of spatially periodicity. It is shown that for the designed SPM, the vibration modes of its host structure should be considered to widen the frequency range in which the resonators transfer and store energy, and hence improve the performance of low-frequency vibration suppression. The present work can provide a significant theoretical guidance for the engineering application of metamaterial stiffened structures.

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