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Model checks
The science
One degree of freedom
The bench of mass m rides on a spring of stiffness k and a damper with coefficient c = 2 ζ √(k m). If the ground moves as y(t), the bench position z obeys m z'' = -k (z - y) - c (z' - y'). The natural frequency is fn = (1/2π) √(k/m). For a sine ground motion the steady-state ratio of bench to ground amplitude (the transmissibility) is T = √[(1 + (2ζr)²) / ((1 - r²)² + (2ζr)²)] with r = f / fn. It is 1 at r = 0, peaks near r = 1 (resonance), crosses 1 at r = √2 and falls toward zero above it; that is the isolation region.
Why damping is a trade-off
More damping lowers the resonance peak, which matters when the ground sweeps through fn or an earthquake contains that frequency, but it raises the transmitted motion above √2 fn (the damper conducts vibration straight through). The model integrates the equation with a fourth-order Runge-Kutta step and the checks below run separate numerical experiments.
Honest simplifications: a single vertical degree of freedom with linear spring and viscous damper; no rocking, no payload flexibility; the earthquake signal is a synthetic sum of 14 sinusoids with a smooth envelope, not a real record; the picture exaggerates motion (the factor is printed on it).
What to try
- Choose At resonance: the bench moves several times more than the ground and the transmissibility dot sits on the peak.
- Choose Sine, well isolated and read the attenuation in the readout.
- Choose Sweep 1-20 Hz and watch the yellow dot run along the transmissibility curve while the bench swells and then calms.
- Raise damping with Shock pulse and see the ringing decay faster in the time history.