---
title: "Inertial Seismometer Interactive Lab | Sandbox Physics"
description: "Does a seismometer record the ground, or the motion inside its case? A seismometer first measures motion inside its case."
canonical_url: "https://sandboxphysics.com/labs/inertial-seismometer/"
language: "en"
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  zh-CN: "https://sandboxphysics.com/zh/labs/inertial-seismometer/"
last_modified: "2026-10-28"
---

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.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

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.

- [USGS · seismometers and inertia](https://www.usgs.gov/faqs/seismometers-seismographs-seismograms-whats-difference-how-do-they-work)
- [Güralp 3T · MAN-030-0001 Issue V (2026) · pp. 24–27, 45–47](https://www.guralp.com/documents/MAN-030-0001.pdf)
- [Sercel · Geophones Specifications 01/26 · pp. 1–4](https://www.sercel.com/sites/default/files/2026-01/geophones_specifications_sercel_en.pdf)

Physics tutorial

## Start with the gap, then interpret the voltage

**Background** The case moves with the ground. A spring and damper connect it to a mass, whose inertia resists acceleration. The difference between mass and case is the first physical signal.

**Why it matters** A voltage trace does not have the same units or frequency response as ground displacement. Each sensor and feedback loop changes the conversion.

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 misconception** The mass is an absolute fixed point at every frequency.

**Better mental model** At 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

1. 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.

2. 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.

3. 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.

Physical conclusion

Use independent calibration and a stated frequency band before interpreting recovered ground motion.

## Related experiments and articles

How physics instruments measure · Reading path

### How physics instruments measure

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a seismometer reads the ground through motion inside its case**](https://sandboxphysics.com/stories/instruments/seismometer/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)