Magnetic coupling nonlinear internal resonance: An inquiry-based experimental teaching design

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.

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.

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.

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.