Seismograph Frequency Response

What is modelled

An inertial seismograph holds a heavy mass on a spring in a frame fixed to the ground. When the ground moves the mass lags, so the motion of the mass relative to the frame is the record. How faithfully it follows the ground depends on the frequency ratio r, the ground frequency divided by the natural frequency of the pendulum, and on the damping ratio.

Equations

• Relative displacement z″ + 2 ζ ω0 z′ + ω0² z = − ag(t).
• Steady response z / ug = r² / √((1 − r²)² + (2 ζ r)²), which tends to 1 for ground motion much faster than the pendulum.
• Damped period T0 / √(1 − ζ²) and log decrement 2 π ζ / √(1 − ζ²).
• Resonance peak 1 / (2 ζ √(1 − ζ²)) at r = 1 / √(1 − 2 ζ²) for ζ below 0.707.

Checks run in the page

• Runge-Kutta steady amplitude against the response formula.
• Damped period and log decrement (or, above critical damping, no overshoot and the decay rate).
• Scan of the response for the resonance peak.

Honest simplifications

The pendulum is a linear single-degree-of-freedom oscillator along one axis, with viscous damping and a synthetic earthquake of three wave packets. A real instrument has a pen lever, friction, tilt sensitivity and an electronic feedback loop, and real records come from a three-component network.

• Press Resonance and see a lightly damped pendulum exaggerate the ground motion many times.
• Press Slow ground waves and see the record shrink when the ground moves more slowly than the pendulum swings.
• Press Critically damped and watch the free decay return without a single swing.