Traveling-Wave Ultrasonic Motor — Unrolled Wave & Phase Portrait

Two-Phase Piezo Drive

Drive frequency f 40 kHz

Vibration amplitude A 1.5 µm

Phase A→B offset φ 90°

Preload / friction μ 0.30

Live Readouts

Traveling wave (CCW)

Wave phase speed

— m/s

Contact tip speed

— mm/s

Rotor speed

— rpm

How this 2D view works

Same two-phase superposition as the 3D stator, but "unrolled": the ring's angular coordinate θ becomes the horizontal axis of a strip plot, so the flexural standing/traveling wave is drawn exactly like a wave on a string instead of a tilted 3D ring:

w(θ,t) = A · [ sin(nθ)·cos(ωt) + sin(φ)·cos(nθ)·sin(ωt) ]

The lower-left panel is a genuinely distinct 2D representation: a phase-space portrait. Differentiating w with respect to θ gives the tangential displacement u = −(h/R)·∂w/∂θ. Evaluated at the fixed point θ = 0, algebra collapses the two fields to

w₀(t) = A·sin(φ)·sin(ωt)
u₀(t) = −u_amp·cos(ωt),  u_amp = h·(n/R)·A  (constant, independent of φ)

Plotting u₀ against w₀ traces the closed curve a real surface point rides through — a circle-to-ellipse-to-line family whose shape is set purely by φ: full ellipse at φ=±90° (pure traveling wave), and a flat horizontal segment at φ=0 (pure standing wave — the point at θ=0 sits on a vibration node, so w₀≡0, but the tangential motion u₀ never vanishes, confirming the node still ratchets against the rotor).

f, A — set ω = 2πf and the wave amplitude; together they set the tangential tip speed u_amp·ω plotted as the traveling dot's orbital speed.
φ — sets the ellipse's eccentricity and, through sin(φ), both direction and fraction of the wave that is genuinely traveling.
μ — friction/preload coupling μ/(μ+μ₀) between stator tip speed and rotor rim speed.

The lower-right panel is a plain top-view of the rotor disc, spinning at the true signed angular velocity computed from the same formulas the 3D model uses — no camera, no perspective, just rotorOmega·dt integrated directly.

Unrolled wave profile w(θ) Phase portrait (u₀,w₀) Rotor top view

Wave strip animated ~5000× slower than the real 20–60 kHz drive for visibility. Phase portrait traces the exact analytic (u₀,w₀) curve. Rotor spins at its true computed rpm.

Traveling-Wave Ultrasonic Motor — Unrolled Wave View