Training Multiphysics Differential Equations Piezoelectric & Electromagnetic Actuators
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Piezoelectric & Electromagnetic Actuators

30 min Multiphysics Differential Equations

Piezoelectric & Electromagnetic Actuators

Actuators turn electrical energy into mechanical displacement — the quintessential multiphysics problem. A piezoelectric stack strains when a voltage is applied; a solenoid or voice-coil pulls a plunger when current flows. The governing equations couple Kirchhoff's voltage law with Newton's second law through a constitutive law that links charge (or flux) to displacement.

A lumped piezoelectric actuator can be modelled by $m\ddot{x} + c\dot{x} + kx = n\,v(t)$ and $C_b\dot{v} = i - n\dot{x}$, where $n$ is the electromechanical transformer ratio. The same structure applies to voice-coils, moving-iron relays, and some MEMS devices.

Modelling these as two coupled ODEs lets us predict resonance peaks, blocked-force behavior, and the free-displacement limit — information essential for control design.

Piezo Lumped Model

Mechanical: $m\ddot{x} + c\dot{x} + kx = n\,v$.

Electrical: $i = C_b\dot v + n\dot x$.

Two states: $x$ (tip displacement) and $v$ (terminal voltage) or equivalently $q = C_b v + n x$.

Blocked-Force and Free-Displacement

Blocked ($x = 0$): force $F_{\text{blk}} = n v$.  Free (no load, static): displacement $x_{\text{free}} = n v / k$.

Plotting $F$ vs $x$ at fixed $v$ gives a straight line from $(0, F_{\text{blk}})$ to $(x_{\text{free}}, 0)$.

Example 1 — Free displacement

A piezo has $n = 10\,\mu\text{m}/\text{V}\cdot k$ (i.e., effective $d_{33}$ is known via $n/k$). If $k = 10^7\,\text{N/m}$ and $n = 100\,\text{N/V}$, find free displacement at $v = 50\,\text{V}$.

$x_{\text{free}} = n v / k = (100)(50)/10^7 = 5\times 10^{-4}\,\text{m} = 0.5\,\text{mm}$.

Example 2 — Blocked force

Using the same $n = 100\,\text{N/V}$, find $F_{\text{blk}}$ at $v = 50\,\text{V}$.

$F_{\text{blk}} = n v = (100)(50) = 5000\,\text{N}$. (5 kN — piezo stacks are remarkably stiff actuators.)

Example 3 — Electrical resonance frequency

For $m = 0.05\,\text{kg}$, $k = 10^7\,\text{N/m}$ and $c = 400\,\text{N·s/m}$, find the undamped natural frequency and damping ratio.

$\omega_n = \sqrt{k/m} = \sqrt{10^7/0.05} = \sqrt{2\times 10^{8}} \approx 14{,}142\,\text{rad/s} \Rightarrow f_n \approx 2.25\,\text{kHz}$.

$\zeta = c/(2\sqrt{km}) = 400/(2\sqrt{10^7\cdot 0.05}) = 400/(2\cdot 707) \approx 0.283$ (under-damped).

Interactive Demo: Piezo Actuator Line
x_free =500µm
F_blk =5000N
f_n (with m=50 g) =2251Hz

Practice Problems

1. What physical quantity does $n$ (the piezo transformer ratio) convert between?
2. For $n=200$ N/V, $k=10^7$ N/m, $V=20$ V, find $x_{free}$ and $F_{blk}$.
3. If $C_b = 1$ µF and $n=200$ N/V, compute the electrical time constant with $R = 100\,\Omega$ (ignore mechanics).
4. Why is the $F$-vs-$x$ actuator line linear at fixed voltage?
5. How does the blocked-force curve change if the stiffness $k$ doubles but $n$ is unchanged?
6. Name three real devices modelled with this two-port structure.
Show Answer Key

1. Voltage $v$ ↔ mechanical force/displacement; electrically, also current ↔ velocity.

2. $x_{free} = (200)(20)/10^7 = 4\times 10^{-4}\,\text{m} = 400\,\mu\text{m}$; $F_{blk} = 4000$ N.

3. $\tau = RC_b = 100 \cdot 10^{-6} = 100\,\mu\text{s}$.

4. Because the governing equation $kx = nV - F$ is linear in $x$ and $F$ when $V$ is held fixed.

5. Same $F_{blk} = nV$ but $x_{free}$ halves; the slope (stiffness) steepens.

6. Piezo stack actuator, voice-coil (loudspeaker), moving-iron relay, MEMS mirror, proof-mass accelerometer.

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