Two groups of piezoelectric ceramic segments bonded to the underside of a metal stator ring are driven by voltages in electrical quadrature (90° out of time-phase) and bonded 90° apart in space. Each group alone excites a flexural standing wave with n wavelengths around the ring (n = 9 here, as in the classic Shinsei-type motor). Superposing them gives the vertical surface deflection:
w(θ,t) = A · [ sin(nθ)·cos(ωt) + sin(φ)·cos(nθ)·sin(ωt) ]
φ is the electrical phase offset between the two drive channels. At φ = ±90° the bracket collapses to A·sin(nθ ± ωt) — a pure traveling flexural wave. At φ = 0 it stays a pure standing wave with no net transport, which is why the rotor stalls at the "Standing" preset.
A traveling bending wave does not just move surface points up and down — for a beam/ring of half-thickness h, the tangential (in-plane) displacement is tied to the slope of the bend:
u(θ,t) = −h · (1/R) · ∂w/∂θ
Because u is 90° out of phase with w in space, every surface point traces a small ellipse (visible as the trailing orange loop on the ring) instead of a straight line. Near the wave crest the point moves mostly tangentially and pushes forward through friction contact with the rotor teeth; on the return half of the loop it dips below the mean surface and lifts clear — a one-way ratchet that rectifies a purely oscillatory input into continuous rotor rotation.
- f, A — set the wave's angular frequency ω = 2πf and vertical amplitude, which set the maximum tip velocity h·(n/R)·A·ω.
- φ — sets both the direction (sign) and the fraction sin(φ) of the wave that is genuinely traveling vs. standing.
- μ — friction/preload coupling efficiency between stator tip velocity and rotor surface velocity, modeled with saturation μ/(μ+μ₀).
This is the operating principle of real traveling-wave ultrasonic motors (e.g. camera autofocus and robotic-joint actuators): no windings, no magnets — just piezoceramic bending waves rectified by friction, giving high holding torque at zero drive power.