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Platform and floor
A soft isolator gives a low natural frequency. Floor vibration above the square root of two times that frequency is reduced, while vibration near the natural frequency is amplified unless the damping is high.
Model checks
The science
What is modelled
A rigid platform of mass m on a linear spring k and viscous damper c above a floor that moves sinusoidally. The page integrates the equation of motion in time and also evaluates the closed-form steady-state transmissibility, so the two can be compared.
Equations
m x'' + c (x' - y') + k (x - y) = 0,y = Y sin(2π f t).fn = √(k/m) / 2π,r = f / fn,c = 2 ζ √(k m).T = √(1 + (2ζr)²) / √((1 - r²)² + (2ζr)²); attenuation-20 log10 T.- T = 1 at
r = √2for every damping ratio.
Checks run in the page
- The natural frequency from the stiffness and from the period.
- The simulated resonance peak against the closed form.
- The crossover at root two times the natural frequency.
- Roll-off at four times the natural frequency.
Honest simplifications
One vertical degree of freedom, linear spring, viscous damping and a rigid floor motion. Real isolation stages have several degrees of freedom, rocking modes, structural resonances in the platform, air-spring nonlinearity and ground noise that is not a single sine.
What to try
- Press At resonance: the platform moves several times more than the floor.
- Press Soft isolator: the low natural frequency gives strong attenuation at 8 Hz.
- Press Heavily damped: the resonance peak is small but the high-frequency attenuation is worse.