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Mechanical · worked example · 3 pages

Machine vibration and isolation

A fan on resilient mounts, modelled as a single degree of freedom: one mass on the summed mount stiffness with light damping. The question is how much of the rotating unbalance force reaches the floor — the force transmissibility — and it hangs on the ratio of forcing to natural frequency. Note the three speed quantities: rotational speed N (rev/min), frequency f (Hz = cycles/s), and angular frequency ω = 2πf (rad/s).

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What this calculation covers

Machine and mounts

  • Fan mass on the mounts
  • Operating speed
  • Number of mounts
  • Stiffness of each soft mount
  • Damping ratio of the mounts

Speed, three ways

  • Forcing frequency (same quantity in Hz)
  • Angular forcing frequency

The one-degree-of-freedom model

  • Total stiffness under the fan
  • Natural angular frequency
  • Natural frequency
  • Standard gravity (from the constants sheet)
  • Static deflection under the fan's weight
  • Frequency ratio r
  • Isolation only exists above r = √2

Force transmissibility

  • Transmissibility for ratio r, damping ζ
  • Chosen soft mounts
  • Isolation achieved

Mounting options compared

  • Frequency ratio for a mount stiffness k
  • Ten-times-stiffer mounts
  • Running AT resonance (r = 1)

Results summary

  • The 1450 rev/min fan forces at 24.2 Hz. On soft mounts (180 kN/m total) the system sits at 5.5 Hz — r = 4.4 — and only 6% of the unbalance force reaches the floor: 94% isolation with an 8 mm static sag. Mounts ten times stiffer put r at 1.39, below √2's benefit line, and actually AMPLIFY the force by 8%; passing through r = 1 at this damping would multiply it tenfold. Soft is the point.

Try changing…

  • Run the fan at 960 rev/min — r drops to 2.9 and the transmitted share more than doubles.
  • Raise ζ to 0.2 — the resonance peak collapses to 2.6×, at the price of a little isolation up high.
  • Bolt the fan straight down (huge k) — r → 0 and every newton of unbalance goes into the slab.

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