Capstone · Vibrations & Dynamics
A design brief, not a demo. A pump on the roof is rattling the office below. You choose the mounts it sits on, how stiff and how damped, and three specs pull in different directions. Everything underneath is the forced spring–mass–damper you already know. What's new is finding a design that satisfies all three at once.
A 120 kg pump runs at 1,800 rpm (30 Hz). A small rotor imbalance shakes it at exactly that frequency, and today the whole force goes straight into the roof slab. Pick a set of mounts, with total stiffness k and damping ratio ζ, that meets all three specs:
Transmissibility T vs. frequency ratio r = f / fₙlog scale
Design space · tap to try a designx: natural frequency fₙ · y: damping ζ
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A design capstone built on the forced response from the Spring–Mass–Damper and Frequency Response labs. Transmissibility is the standard single-degree-of-freedom force-transmission result; the specs and design-space map are original to this exercise.
The force reaching the floor, as a fraction of the shaking force, is the transmissibility: T = √(1 + (2ζr)²) / √((1 − r²)² + (2ζr)²), with r = f / fₙ. Every curve, whatever the damping, passes through T = 1 at r = √2. Below that, mounts make things worse: a stiff rubber pad tuned near the running speed amplifies the shaking. Isolation only begins once the natural frequency sits well below the forcing frequency, so the first instinct is to go soft.
A machine that runs at 30 Hz still has to get to 30 Hz, and on the way up it passes straight through its own natural frequency. With almost no damping the startup peak is enormous, roughly 1/(2ζ), so spec ② wants damping. But past √2, damping makes isolation worse. The (2ζr)² term in the numerator means damped mounts pass more force at high r. Slide ζ up and watch the curve's tail lift off the floor of the chart. You need just enough damping to survive startup, and no more.
For a mass on a spring, static deflection and natural frequency are locked together: δ = g / ωₙ², or about fₙ ≈ 15.8 / √δ with δ in millimetres and fₙ in hertz. Notice what's missing: the mass. A 15 mm sag limit means fₙ can't go below about 4.1 Hz, whatever the machine weighs. Heavier machines need stiffer springs to reach the same fₙ, but the sag comes out the same. That's why the vertical edge on the design map doesn't move.
Switch to variable speed. The pump can now run as slow as 900 rpm (15 Hz), so ① has to hold at the slowest speed, where r is smallest. With 15 mm of allowed sag the green region vanishes: there's no answer, and that's the correct engineering conclusion, not a bug. A real designer would renegotiate. One option is to allow more sag (spring mounts with travel limiters; try 25 mm). Another is to tell the drive to skip the lowest speeds. A third is to point out that imbalance force grows with speed squared (F = m·e·ω²), so at half speed the shaking force is only a quarter as large, and a looser spec at low speed may be perfectly fine. Knowing which spec to push back on is part of the design.
EngineeringCandy · Capstone · forced-response physics from the Spring–Mass–Damper lab, specs original to this exercise · design it, test it, beat it