Engineering · Simulator 096
Sensitive equipment sits on springy mounts to keep floor vibration away, and noisy machines sit on mounts to keep their shaking out of the building. Change the mass, the mount stiffness and damping, and how fast the floor shakes. See when the mounts help, when they make things worse, and why damping is a trade-off.
A mass m on a spring of stiffness k with a damper c has a natural frequency
f_n = (1/2π) √(k/m), damping ratio ζ = c / (2√(km))
When the floor moves as y = Y sin ωt, the mass follows with steady amplitude X = T·Y, where the transmissibility depends on the frequency ratio r = f / f_n:
T = √[ (1 + (2ζr)²) / ((1 − r²)² + (2ζr)²) ]
Near r = 1 the system resonates and T can be much larger than 1. Every curve crosses T = 1 at r = √2; above that, the mounts isolate, and the higher the ratio, the better. Soft mounts (low f_n) therefore isolate best. Damping limits the peak at resonance but lets more vibration through at high frequencies. The same T also gives the share of a machine's shaking force passed to the floor.
The simulation integrates m x'' + c(x' − y') + k(x − y) = 0 from rest, so you also see the start-up transient before the motion settles.
One degree of freedom (vertical only), linear springs and viscous damping, a rigid machine and a sinusoidal floor motion. Real mounts also rock, and their stiffness and damping change with frequency.
A 200 kg pump runs at 1,800 rpm. You want at most 10% of its vibration to reach the floor, with light damping (ζ ≈ 0.05). What natural frequency, and so what total mount stiffness, do you need?
The forcing frequency is 1800/60 = 30 Hz. With little damping T ≈ 1/(r² − 1) ≤ 0.1 gives r² ≥ 11, so r ≥ 3.3 (damping raises this slightly, to about 3.4). So f_n ≤ 30/3.4 ≈ 8.8 Hz and k = m(2πf_n)² ≤ 200 × (55.3)² ≈ 6.1×10⁵ N/m in total. Softer mounts isolate better, provided they also cope with the static sag and the start-up pass through resonance.