Piezoelectric Harvester — Two-Mode Beam Profile

Base Vibration

Drive frequency 15.0 Hz

Base acceleration 0.30 g

Load resistance 100 kΩ

Mechanical damping ζ 0.020

Live Readouts

Tip deflection (pk)

0.0 mm

Output voltage (RMS)

0.00 V

Mode 1 share

Mode 2 share

Average power delivered to load

0.0 µW

Frequency Response

Steady-state power vs. drive frequency — two peaks: f₁ ≈ 15.0 Hz, f₂ ≈ 94.0 Hz.

How it works

Unlike a single lumped mass-spring model, this simulator treats the cantilever as an actual Euler-Bernoulli beam and keeps its first two bending mode shapes, each coupled to a shared piezoelectric electrode:

mᵣ·φ̈ᵣ + cᵣ·φ̇ᵣ + kᵣ·φᵣ + θᵣ·V = mᵣ·a₀·sin(ωt)   (r = 1, 2)
Cp·V̇ + V/R = θ₁·φ̇₁ + θ₂·φ̇₂
w(x,t) = φ₁(t)·ψ₁(x) + φ₂(t)·ψ₂(x)

ψᵣ(x) are the standard clamped-free (cantilever) eigenfunctions, ψᵣ(x) = cosh(βᵣx) − cos(βᵣx) − σᵣ(sinh(βᵣx) − sin(βᵣx)), with root β₁L = 1.8751 and β₂L = 4.6941 fixed by the clamped-free boundary conditions — giving a second-mode natural frequency f₂ = f₁·(β₂L/β₁L)² ≈ 6.267·f₁, a textbook Euler-Bernoulli result, not a tunable parameter.

Two real resonances — sweep the drive frequency across both f₁ and f₂ and watch the beam bend as mode 1 (simple bow toward the tip) near the first peak, then reshape into an S-curve with an internal node — a point along the beam that stays still while both sides move in opposite directions — near the second peak. A single lumped mass-spring-damper (one degree of freedom) cannot produce this: it has no spatial extent to bend, so it can only ever show one peak and a beam that bows in one direction.
Load resistance — sets the RC corner 1/(2πR·Cp) and adds electrical damping to both modes through their shared electrode voltage.
Mechanical damping ζ — applied per mode; lower ζ gives taller, narrower peaks.
Mode share readouts — show how much of the instantaneous tip motion comes from each mode; near f₁ mode 1 dominates almost completely, and mode 2 only becomes significant near its own resonance, since off-resonance response falls off like 1/(k − mω²).

Both the beam profile (numerically integrated, semi-implicit Euler, 500 kHz sub-stepping) and the frequency-response chart (closed-form steady-state phasor solution of the same coupled two-mode equations) are computed independently of the 3D single-mode version of this simulator, and agree with each other to within numerical integration error.

Clamp / base shaker PZT patch (strain zone) Output LED (∝ power) Mode-2 internal node