IN08 · The instrument filters the ground

Inertial Seismometer

Does a seismometer record the ground, or the motion inside its case? A seismometer first measures motion inside its case. A spring-mounted mass follows slow ground motion but lags behind rapid motion; a moving coil turns their relative velocity into voltage. Calibration describes this frequency-dependent response. Dividing it out can recover ground motion within a useful band, while weak response and noise limit what can be recovered.

Shake an inertial seismometer, compare displacement and moving-coil voltage, then recover a declared frequency band from saved calibrated signals.

Live relative gap · ideal model $-0.570\,\mathrm{mm}$

Live raw coil voltage $40.1\,\mathrm{mV}$

Live calibrated actual force $0\,\mathrm{N}$

Saved recovered ground RMS · band limited $565\,\mu\mathrm{m}$

A moving case, an inertial mass

Drag the gold shaker-stroke knob. Change the instrument, then save a new calibrated voltage record.

InstrumentBase motionShaker stroke

0.8

A/\mathrm{mm}

Shaker frequency

2

f/\mathrm{Hz}

Natural frequency

1

f_0/\mathrm{Hz}

Damping ratio

0.7

\zeta

Default electrical record and independent calibration are saved.

Readout, recovery and zero

Saved channel for recoveryVoltage noise bound

30

\epsilon/\mu\mathrm{V}

Regularization fraction

0.005

\lambda/|H(1\,\mathrm{Hz})|

Band lower edge

0.15

f_L/\mathrm{Hz}

Band upper edge

5

f_H/\mathrm{Hz}

Controller zero offset

0

z_0/\mu\mathrm{m}

Force per mass limit

0.35

u_{\max}/(\mathrm{m}\,\mathrm{s}^{-2})

Detector seed

181

s_0

Sine acquisition preserves the requested frequency and changes the sample clock to retain whole cycles. The exact clock and duration are saved; pulse edges must settle.

Model assumptions

One horizontal degree of freedom, linear spring and viscous damping. Positive displacement points right; relative displacement is mass minus base. Position pickup, high-impedance velocity coil and force actuator are separate. Feedback is a bounded proportional/derivative controller with a finite actuator time constant, not a commercial broadband instrument. The acquisition filter is a first-order 8 Hz low pass, not an ideal anti-alias filter; the 18-bit ADC spans minus five to plus five volts. Calibration uses independent ideal channel fixtures, a high-bandwidth loop-disabled free release, and a fixed-base zero. Reconstruction reads saved voltages and calibration, removes DC and uses a declared cosine-edged band with Tikhonov regularization. Sine records contain whole cycles by visibly adapting the sample clock; pulse edges must settle. The periodic finite-record assumption, residual noise, finite bandwidth and saturation limit recovery. Scene displacement is enlarged twenty times; hardware geometry and material values are teaching choices.

Saved electric signal

Default simulated detector record

$t/\mathrm{s},\qquad V/\mathrm{V}$

Copper: raw selected voltage, including offset, noise and digitization.

Recovered ground motion has a frequency band

$t/\mathrm{s},\qquad y/\mathrm{mm}$

Teal: recovery from saved voltages and calibration. Dashed white: separate ideal base displacement, never used by recovery. The shaded outer tenth at each end is excluded from RMS; this trim does not cure an invalid record boundary.

Saved response available.

$f_s=64\,\mathrm{Hz},\quad T=32\,\mathrm{s}$

Response, calibration, raw samples and limits

$f/\mathrm{Hz},\qquad |H|/(\mathrm{V}\,\mathrm{m}^{-1})$

$f/\mathrm{Hz},\qquad |H_{\mathrm{inv}}|/(\mathrm{m}\,\mathrm{V}^{-1})$

$t/\mathrm{s},\qquad V_q/\mathrm{V}$

$f_0=1\,\mathrm{Hz},\quad\zeta=0.7$

Ideal independent fixtures supply known displacement, relative velocity and actuator force per mass. A separate loop-disabled, unfiltered free-release tap identifies mechanical poles; a stationary-base observation measures zero. These teaching calibrations have no assigned traceable uncertainty.

$t/\mathrm{s}$	$V_q/\mathrm{V}$	$V_v/\mathrm{V}$	$V_u/\mathrm{V}$

The velocity pickup and force coil are separate. The former is read at high impedance; coil loading is omitted and damping is prescribed. A first-order 8 Hz acquisition filter attenuates high frequencies but cannot remove arbitrary out-of-band interference. The selected inverse removes DC, tapers its band and regularizes weak response. Its force-feedback model is valid only without clipping. Periodic sine records use a synchronous clock; pulse edges are checked for settling. No absolute ground offset is recovered. Scene displacement is enlarged twenty times; capacitor overlap, supports and coils are schematic apparatus geometry.

Start with the essentials

Focus question: Does a seismometer record the ground, or the motion inside its case?
One-sentence intuition: Relative displacement, relative velocity and feedback force are three different outputs of one dynamical system.

Core mathematical model

The base drives relative motion

\ddot q+2\zeta\omega_0\dot q+\omega_0^2q=-\ddot y+u,\qquad q=x-y

Positive points right. $x$ is the mass position, $y$ the base position, and $u$ the actual feedback force divided by mass. The mechanical mode has zero feedback force.

Ideal pickup signals before acquisition

V_q=G_q q,\qquad V_v=G_v\dot q,\qquad V_u=G_u u

These are the ideal voltages before acquisition filtering, electrical offset and noise. Separate position, high-impedance velocity and actuator-monitor channels then pass through the declared filter and digitizer. Offset and gain come from saved calibration.

A finite loop cannot hold perfect zero

\tau_a\dot u+u=\operatorname{clip}[-K_p(q+z_0)-K_d\dot q]

The actuator has lag and a force limit. Its actual output approaches base acceleration only where loop gain is large and the linear model is unsaturated.

Restore only a declared band

\widehat Y= W\,\frac{H^*}{|H|^2+\lambda^2}\widehat V

$H$ is the independently calibrated voltage response per metre of ground displacement. $W$ tapers the frequency band; regularization suppresses weak-response noise. DC is removed, and finite-record boundary assumptions are checked.

Common difficulties

A nearly still mass is a limiting case

Typical misconceptionThe mass is an absolute fixed point at every frequency.

Better mental modelAt low frequency the spring carries the mass with the case. Its apparent stillness is useful only above the mechanical natural frequency, within other instrument limits.

Run the experiment

01
Slow the shaker
Choose Mass follows the case, then compare it with Mass stays behind.
What to observe: Slow motion moves the base and mass together; fast motion leaves a much larger relative gap.
02
Read a different channel
Switch between relative position, coil voltage and actual feedback force. Save a new record.
What to observe: Their amplitudes and phases differ. A coil voltage follows relative velocity, not displacement.
03
Restore a noisy trace
Choose Noise after inversion, then raise regularization and save a new record.
What to observe: The saved raw samples stay intact. Suppressing noise also suppresses some weak-frequency signal.

Inertial Seismometer

Saved electric signal

Recovered ground motion has a frequency band

Start with the gap, then interpret the voltage

Core mathematical model

The base drives relative motion

Ideal pickup signals before acquisition

A finite loop cannot hold perfect zero

Restore only a declared band

Common difficulties

A nearly still mass is a limiting case

Run the experiment

Slow the shaker

Read a different channel

Restore a noisy trace

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