The Tilting Mode: A New Degree of Freedom for Magneto-Mechanical Resonator Sensors

Magneto-mechanical resonators (MMRs) are an emerging class of passive, wireless sensors. Their torsional oscillation mode has recently been established for sensing and tracking applications. In this work, we report the identification and characterization of a second mechanical mode, the tilting mode, that provides sensitivity along an axis inaccessible to the torsional mode, opening up a new degree of freedom for tracking and sensing with a single MMR sensor. We derive an analytical model predicting the tilting frequency as a function of the geometric and magnetic parameters of the resonator, compare the tilting mode frequency to that of the torsional mode, and obtain a characteristic frequency ratio between the torsional and the tilting mode in the small angle approximation. Experimental characterization using three-axis excitation and detection confirms the mode’s existence and its directional selectivity. Notably, the three-axis frequency response shows no observable cross-coupling between the torsional and the tilting mode. We further show that the tilting mode frequency follows the predicted dependence on magnet distance, confirming the analytical model and the mode’s applicability for sensing, analogous to that of the torsional mode.

The tilting mode is a second mechanical mode in which the rotor oscillates in a plane containing the filament axis. Based on high-speed camera observations, the rotor tilts about its center in the uw-plane while remaining centered above the stator. The analytical model uses holonomic constraints: the rotor with radius r is always centered above the stator, and the inextensible filament with length l is attached at the midpoint between the magnetic poles on the surface of the rotor. Geometric analysis yields the center-to-center distance d between the two magnets as a function of the tilting angle β: d(β) = d0 + r(1 − cos β) + l − sqrt(l2 − r2 sin2 β), where d0 is the center-to-center distance in the equilibrium position. Modeling both magnets as point dipoles, the potential energy U of the rotor in the field of the stator is U(β) = −m1 · B2 = −µ0 m1 m2 / (4π d(β)3) cos β. Applying the torque relation I β̈ = −dU/dβ, the equation of motion is obtained. In the small-angle approximation (β ≪ 1), this simplifies to a harmonic oscillation equivalent to that of the torsional mode derived in [1], without damping, scaled by the factor C = (1 + 3r/d0) · (1 + r/l). The tilting mode natural frequency ftilt is thus given by ftilt = sqrt(C) ftorsion, where ftorsion is the torsional mode natural frequency. The factor C depends solely on the geometric ratios r/d0 and r/l: for larger ratios, the restoring torque increases, yielding a higher tilting frequency relative to the torsional mode.

Two MMRs are used in the experiments. MMR1 consists of two spherical NdFeB magnets (N35, 5 mm diameter) and an ultra-high-molecular-weight polyethylene (UHMWPE, Dyneema) filament of length 20 mm in a fixed housing. The center-to-center distance between rotor and stator is 9 mm. MMR1 is used for three-axis spectral characterization. MMR2 consists of a spherical rotor (N40, 4 mm diameter), a cylindrical stator (N35, 4 mm diameter and 4 mm height), and a UHMWPE filament of length 5 mm. MMR2 is enclosed in a compressible polyamide housing that allows the rotor-stator distance to be varied by compressing the housing in a vise. The excitation and readout system is based on a transmit-receive architecture with a cube of three orthogonal coil pairs enabling independent excitation and signal acquisition in x-, y-, and z-direction. For each measurement, the rotor is excited with a 0.2 s sinusoidal burst, and the free decay signal is recorded after the excitation has stopped.

The three-axis frequency response of MMR1 shows three main resonance frequencies: the torsional mode at 89 Hz (f1 = ftorsion), the second harmonic of the torsional mode at 177 Hz (f2 = 2ftorsion), and the tilting mode at 138 Hz (ftilt). Besides the dominant peaks, a weak resonance close to f1 is visible at 107 Hz (f?) for y-excitation and y-response, which might be attributed to a pendulum mode. The torsional mode is observed mainly for y-excitation and strongest in the y-response as expected. The second harmonic of the torsional mode is observed mainly in x-direction, which is in agreement with the model derived in [1]. The z-components of the torsional mode are attributed to slight misalignment of the MMR in the coil system and fabrication tolerances of the MMR. The tilting mode is observed mainly for z-excitation and strongest in the z-response. However, weaker x- and y-components are also observed, as well as a slight response from y-excitation. Analogous to the torsional mode, these components are caused by misalignment of the MMR and fabrication tolerances. Remarkably, no cross-coupling between the torsional and tilting modes is observed: the tilting mode frequency ftilt does not appear in response to excitation at the torsional frequency f1, and vice versa. The resonance at f? might be attributed to a pendulum mode, which couples to the main torsional mode. In the recorded time signal, the additional mode is visible as a beating between the two frequencies, resulting in amplitude modulation of the signal. The ratio ftilt/f1 yields an experimental value of sqrt(C) = 1.55 for MMR1. Using the values for r, l and d0 from Section III-A leads to a theoretical value of sqrt(C) = 1.39 for MMR1. The discrepancy might be caused by neglecting the filament’s bending stiffness, which provides an additional restoring torque for the tilting mode, and by the unaccounted modification of the rotor’s moment of inertia due to the cap.

For the distance series, the rotor-stator distance of MMR2 is varied as described in Section III-A. The natural frequencies of both the torsional and the tilting mode are shown as a function of the change in rotor-stator distance. Each data point represents the mean of 20 measurements, with the standard deviation below 1.5 Hz for all data points. Both data sets are fitted with functions of the form a(x + x0)(−3/2), consistent with the expected scaling [9]. The experimental and theoretical results for sqrt(C) are shown in Table I. The center-to-center distance d0 for each measurement is determined from the fit parameter x0 of the tilting mode fit together with the screw pitch of the vise. Consistent with the results for MMR1, the experimental values are systematically higher than the theoretical predictions. In addition to the factors discussed for MMR1, the point-dipole approximation could be inappropriate for small values of d0, since the stator of MMR2 is cylindrical.

In conclusion, the tilting mode was identified and characterized as a new degree of freedom in MMRs. The mode appears to be decoupled from the torsional mode and follows the predicted scaling with magnet distance, confirming its applicability for sensing. This opens the possibility of enhanced spatial tracking without added complexity. Future work will address the model discrepancy by incorporating the rotor’s modified moment of inertia as well as the filament’s bending stiffness, and investigate simultaneous dual-mode operation.

Improvements to AI Systems:

1. Multi-Mode Sensor Fusion for 3D Tracking: Train an AI model to simultaneously process torsional and tilting mode frequency responses from a single MMR. The improved system can estimate 3D position and orientation of a passive sensor in real time, using the decoupled modes to resolve ambiguities (e.g., separating in-plane rotation from out-of-plane tilt) without requiring multiple sensors or complex hardware.

2. Physics-Informed Calibration for Manufacturing Tolerances: Use the analytical model (including the factor C = (1 + 3r/d0)(1 + r/l)) as a prior in a Bayesian neural network. The AI can automatically calibrate each MMR by learning the effective geometric parameters (r, l, d0) from measured frequency responses, compensating for fabrication tolerances (e.g., filament bending stiffness, moment of inertia changes) that cause deviations from theory.

3. Predictive Maintenance and Anomaly Detection: Develop a recurrent neural network (RNN) that monitors the ratio ftilt/ftorsion over time. The system can detect drifts in this ratio (e.g., due to filament degradation, magnet demagnetization, or distance changes) and flag anomalies, enabling proactive sensor replacement or recalibration in industrial or biomedical implants.

4. Adaptive Excitation and Readout Optimization: Implement a reinforcement learning agent that controls the three-axis excitation coil system. The agent learns to choose optimal excitation frequencies and directions to maximize signal-to-noise ratio for each mode, while avoiding cross-coupling artifacts, improving measurement accuracy in noisy or constrained environments.

5. Simultaneous Dual-Mode Sensing with Mode Separation: Build a signal-processing AI (e.g., a convolutional neural network on time-frequency representations) that can separate overlapping torsional and tilting mode responses from a single free-decay signal. This allows the system to measure two independent physical quantities (e.g., temperature and pressure) simultaneously from one MMR, doubling the sensing throughput.

6. Digital Twin for Design Optimization: Create a generative model (e.g., a variational autoencoder) trained on the analytical and experimental data to predict the frequency ratio sqrt(C) for arbitrary geometric and magnetic parameters. The AI can rapidly propose optimal MMR designs (e.g., r/d0, r/l ratios) for target sensing ranges or to achieve a desired frequency separation, reducing trial-and-error prototyping.

7. Misalignment Correction in Coil Systems: Use a supervised learning model trained on the observed cross-axis components (e.g., x- and y-components of the tilting mode) to estimate the MMR’s misalignment angles relative to the coil cube. The AI can then digitally correct the measured signals, providing accurate directional data without requiring physical realignment.

8. Autonomous Distance Sensing for Implantable Devices: Deploy a lightweight neural network on edge hardware that maps the measured tilting mode frequency (scaling as d(-3/2)) to absolute rotor-stator distance. This enables wireless, passive monitoring of joint compression or tissue swelling in biomedical implants, with the AI compensating for nonlinearities at small distances (e.g., point-dipole approximation errors).