Dynamics of diamagnetically levitating resonators
X. Chen (TU Delft - Mechanical Engineering)
P.G. Steeneken – Promotor (TU Delft - Applied Sciences, TU Delft - Mechanical Engineering)
F. Alijani – Copromotor (TU Delft - Mechanical Engineering)
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Abstract
Microelectromechanical systems (MEMS), enabled by advances in silicon technology, are widely used in everyday devices such as smartphones, gaming controllers, and printers. By exploiting resonance phenomena, MEMS can function as ultra-sensitive sensors. Achieving low noise in these systems requires minimizing energy dissipation, with clamping losses often representing a dominant damping mechanism. Levitation offers a promising approach to eliminate such losses.
Among different levitation techniques, diamagnetic levitation is unique because it is passive and does not require continuous energy input. Its relatively strong magnetic potential also enables the levitation of macroscopic objects, which is advantageous for developing ultra-sensitive accelerometers, gravimeters, and sensors for probing macroscopic limits of quantum mechanics. Although diamagnetic levitation has been widely studied, its dynamic behavior is not yet fully understood. This thesis investigates both the rigid-body dynamics and elastodynamics of diamagnetically levitating resonators through theoretical analysis and experiments.
Chapter 2 examines the linear rigid-body dynamics of diamagnetically levitating resonators. By modeling and measuring the levitation height and resonance frequencies of millimeter-scale graphite resonators, the influence of magnetic fields on system stiffness is analyzed. Measurements of quality factors in high vacuum reveal that eddy current damping is the dominant source of energy dissipation, and that the quality factor strongly depends on resonator size.
To address this limitation, Chapter 3 introduces a method to suppress eddy current losses by developing diamagnetic composite materials. Micro-scale graphite particles are dispersed within an epoxy matrix to create macroscopic composites. These materials achieve quality factors exceeding 450,000, more than 400 times higher than those of graphite plates of similar dimensions. The influence of particle size, volume fraction, and resonator geometry on damping is investigated. The resulting composites enable acceleration noise floors comparable to superconducting levitation systems operating at cryogenic temperatures.
Chapter 4 explores the nonlinear dynamics of levitating graphite resonators. Using base excitation to drive the resonator into resonance, nonlinear stiffness effects caused by magnetic forces are studied. The system exhibits softening-type nonlinearity that deviates from classical Duffing behavior due to the asymmetric magnetic force field. Nonlinear damping mechanisms are also examined. While eddy current damping remains largely linear, squeeze-film air damping significantly influences the nonlinear response when large vibration amplitudes occur.
In Chapter 5, the elastodynamics of levitating resonators are investigated and applied to the development of a mass sensor based on resonance frequency shifts. The first ten bending modes are experimentally characterized and compared with analytical models. Using the third bending mode, a levitating mass sensor is developed and calibrated with micro glass beads. The sensor is validated by measuring liquid densities and monitoring the real-time evaporation of water droplets. Frequency stability analysis demonstrates that the system can detect mass changes as small as 4 ng.
Overall, this thesis advances the understanding of the dynamics of diamagnetically levitating resonators and demonstrates their potential for high-performance sensing applications.