Presets
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Ground motion and sensor
The relative displacement x obeys x'' + 2 zeta w0 x' + w0^2 x = -a_base. Below resonance the mass follows the frame, near resonance it amplifies by about 1/(2 zeta), and far above it stays still so x equals the ground displacement.
Model checks
The science
What is modelled
Three uncoupled single-degree-of-freedom sensors. Each proof mass hangs on a spring and damper inside a frame that moves with the ground, so the quantity measured is the displacement of the mass relative to the frame. The excitation is a sine, a swept sine or a pulse train, and the integration uses a fourth-order Runge-Kutta step with a mechanical end-stop at 14 mm.
Equations
x'' + 2ζω0 x' + ω0² x = -abase(t).- Steady response:
X ω0² / A = 1 / √((1 - r²)² + (2ζr)²), withr = f / f0. - Log decrement:
ln(xn/xn+1) = 2πζ / √(1 - ζ²).
Checks run in the page
- Amplitude at resonance against A / (w0² 2 zeta).
- Stiff, low-frequency following.
- High-frequency ground-motion limit A / w².
- Free-decay peak ratio against the log decrement.
Honest simplifications
Linear single-axis oscillators without cross-coupling, noise, feedback force balancing or electronics. It teaches the physics of an inertial sensor; it is not an earthquake predictor or a calibrated instrument model.
What to try
- Press At resonance and compare the amplitude with the other presets.
- Press Heavy damping and watch the resonance peak flatten.
- Press Frequency sweep and follow the red dot along the resonance curve.