Presets
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Apparatus
Model checks
The science
What is modelled
The ground moves as b(t) = A sin(2π f t). A carriage of mass m1 is held by a spring k1 and damper c1; a small absorber of mass m2 = μ m1 hangs on k2, c2 from the carriage.
Equations
m1 x1'' = -k1(x1 - b) - c1(x1' - b') + k2(x2 - x1) + c2(x2' - x1'),m2 x2'' = -k2(x2 - x1) - c2(x2' - x1'), integrated with RK4 at 240 Hz.- Steady-state transmissibility from the complex solve
X1/b = Z1 / [Z1 + Z2 - m1ω2 - Z22/(Z2 - m2ω2)], Z = k + iωc. - Den Hartog tuning
f2 = f1/(1+μ),ζ2 = √(3μ / (8(1+μ)3)).
Checks run in the page
- The time simulation after the transient against the analytic steady-state amplitude.
- Zero-frequency limit: transmissibility tends to 1.
- Energy conserved by the integrator for the undamped free system.
- A Den Hartog-tuned absorber lowers the peak below the no-absorber peak.
Honest simplifications
Linear springs and dampers, one horizontal direction, rigid masses and a single sinusoidal ground input. Real buildings have many modes and earthquakes are broadband.
What to try
- Pick Tuned, then Detuned: the same drive shakes the carriage far more when the absorber frequency is wrong.
- Look at the transmissibility graph: the single tall peak splits into two lower peaks.
- Press Kick: the absorber moves out of phase with the carriage and the ringing dies quickly.