The three-dimensional contact model of Ran et al. (2020) describes the drive of a toothless, 2-mm-radius, micro-electro-mechanical-systems (MEMS) fabricated traveling-wave piezoelectric ultrasonic micromotor through a smooth-surface, continuum-pressure formulation: the friction layer is treated as a Winkler-type elastic foundation (a bed of independent springs, i.e., local pressure proportional to local compression) whose normal pressure over the instantaneous contact zone follows p(r,θ)=(E/h)(R(r)cos⁡(nθ)−H)p(r,\theta)=(E/h)(R(r)\cos(n\theta)-H), with a spatially uniform Coulomb friction coefficient and with micro-scale inertia and air damping neglected. Three linked extensions of this formulation are addressed neither in the source paper nor in the subsequent literature on this device class: (i) the effect of stator/rotor surface roughness, asperity-scale contact mechanics, and fabrication-induced geometric imperfections on the contact boundary, local pressure distribution, and net frictional torque; (ii) a predictive analysis of torque capability under load as a function of preload, friction-layer properties, and operating conditions; and (iii) quantification of the neglected inertial and air-damping contributions and of the uniform-friction assumption. Adjacent literature has since produced Greenwood-Williamson-type (statistical asperity-height-distribution) and multi-scale rough-contact models for macro-scale ultrasonic motors and linear piezoelectric actuators, but no published work transfers these to the toothless MEMS traveling-wave micromotor, where the vibration amplitude is itself of the order of the surface roughness and the contact area is preload-determined and radially varying.

Background

An ultrasonic motor converts the resonant, high-frequency vibration of a piezoelectric stator into macroscopic rotor motion through friction at a pressed contact interface. In a traveling-wave ultrasonic motor (TWUM), two phase-shifted standing flexural waves superpose into a wave that travels around an annular stator; surface points then move on elliptical trajectories and drag a spring-loaded rotor through a friction layer bonded to one of the mating surfaces. Because essentially all energy and momentum transfer occurs within a micrometer-scale contact zone whose position and size change synchronously with the traveling wave, the contact model — how normal pressure is distributed and how tangential friction integrates into net torque — determines every prediction about efficiency, speed, and start-up.

Conventional TWUMs carry teeth on the stator, which pin the contact geometry, and classical contact treatments (point-contact, line-contact, and viscoelastic-layer models) exploit this. Ran et al. (2020) considered the opposite, micro-scale situation: a smooth, toothless silicon ring stator (2 mm outer radius) with sputtered lead zirconate titanate (PZT) piezoelectric film, driven in its B13B_{13} bending mode (a flexural mode family indexed by nodal circles and nodal diameters), fabricated with MEMS (micro-electro-mechanical systems) technology on silicon-on-insulator wafers. With no teeth, the contact zone is set entirely by the interplay of preload and the radially varying wave amplitude. Their model works in cylindrical coordinates: the stator out-of-plane displacement is w(r,θ,t)=R(r)cos⁡(nθ−ωt)w(r,\theta,t)=R(r)\cos(n\theta-\omega t), where R(r)R(r) is the measured radial amplitude profile of the B13B_{13} mode; the friction layer of thickness hh and Young's modulus EE is treated as a Winkler-type elastic foundation (a bed of independent springs, i.e., local pressure proportional to local compression), giving the normal pressure

p(r,θ)=Eh(R(r)cos⁡(nθ)−H)p(r,\theta)=\frac{E}{h}\big(R(r)\cos(n\theta)-H\big)

over the instantaneous contact domain DD where the compression is positive, with HH the contact-boundary height fixed by the preload. The tangential velocity field of stator surface points follows from plate kinematics, and the net driving torque is

τ=μ∬Dsign⁡(ωs(r,θ)−ωr) p(r,θ) r dr dθ,\tau=\mu\iint_{D}\operatorname{sign}\big(\omega_s(r,\theta)-\omega_r\big)\,p(r,\theta)\,r\,dr\,d\theta,

with a single, spatially uniform Coulomb friction coefficient μ\mu, ωs\omega_s the local stator angular velocity, and ωr\omega_r the rotor angular velocity. Integrating this torque over the rotor dynamics reproduces the observed two-stage start-up (constant acceleration while all contact points drive, then decreasing acceleration as part of the contact zone begins to resist), and the model was validated on the fabricated 2 mm motor by free start-up and steady-speed measurements with a high-speed camera.

Three simplifications define the open problem. First, both surfaces are assumed geometrically smooth: microscale surface roughness, asperity-scale (individual-peak) contact mechanics, and fabrication-induced geometric imperfections are absent from the continuum-pressure formulation, although the stator vibration amplitude is itself of order micrometers — the same order as the roughness of MEMS-machined surfaces — so the smooth-surface assumption is not obviously harmless. Second, the model was never exercised as a torque-capability predictor: no torque-under-load analysis (torque-speed behavior, stall torque) against preload, friction-layer thickness and modulus, or operating conditions was performed. Third, the consequences of neglecting inertia and air damping at the micro scale, and of the spatially uniform friction coefficient, were left unquantified.

Neighboring results frame, but do not close, this gap. For macro-scale motors, contact models incorporate viscoelastic layers, rotor flexibility, and preload-dependent contact (e.g., Jiang et al. 2023), and preload-based efficiency models exist (Sun et al. 2021). Rough-contact methodology itself is available: Zhang, Piao and Chen (2021) introduced the Greenwood-Williamson (GW) statistical model — summation of elastic micro-contacts over a Gaussian distribution of asperity heights on a nominally flat surface — into the intermittent contact of a linear piezoelectric actuator; He, Yao and Xu (2022) built and validated a multi-point rough-surface contact model for a linear ultrasonic motor; and Liang et al. (2025) built a multi-scale rough-surface contact model (wavelet separation of microscopic roughness from mesoscopic waviness, GW-type statistics, time-varying contact dead zone) for a macro-scale ultrasonic motor with a friction layer, validated against measured mechanical characteristic curves. None of these treats the toothless MEMS traveling-wave micromotor, whose contact geometry is three-dimensional and radially varying, whose wave amplitude is comparable to the roughness, and whose dominant geometric errors come from MEMS fabrication (residual-stress distortion, etched sidewalls) rather than machining.

Problem Statement

For the toothless MEMS traveling-wave ultrasonic micromotor of Ran et al. (2020) — a 2 mm-radius, B13-mode silicon/PZT ring stator driving a rotor through a friction layer — extend the three-dimensional cylindrical-coordinate contact model beyond its smooth-surface, continuum-pressure formulation so that it becomes a predictive description of the real, non-ideal contact interface. The task comprises three jointly supporting parts, all in service of that single objective, and all three are individually mandatory — a partial delivery that omits any one of them does not resolve the task:

  1. Incorporate stator and rotor surface roughness, asperity-scale contact mechanics, and fabrication-induced geometric imperfections into the model, and quantify their effects on the instantaneous contact boundary, the local normal pressure distribution, and the net frictional torque.

  2. On that basis, develop the missing torque-capability analysis: prediction of torque output under load (start-up torque, torque-speed behavior, stall torque) as a function of preload, friction-layer properties such as thickness and Young's modulus, and operating conditions.

  3. Quantify the consequences of the modeling simplifications — the neglect of inertial and air-damping contributions at the micro scale, and the assumption of a spatially uniform friction coefficient.

An answer must deliver the extended model's formulation integrated with the traveling-wave kinematics of the underlying three-dimensional model and instantiated for the MEMS-scale toothless traveling-wave motor, yielding concrete quantitative predictions — start-up transient, steady-state speed, and torque-speed and stall behavior as functions of preload and friction-layer properties — for that device, not merely a formalism or a macro-scale surrogate. Values of physical parameters that the statement does not fix may be taken from Ran et al. (2020) and its citations or, where data are unavailable, chosen as physically plausible values for the MEMS-scale silicon/PZT device class and stated explicitly; throughout, 'quantify' means to give such explicit quantitative results (numerical estimates, scaling relations, or sensitivity ranges), not purely qualitative statements.

References

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