Structural Engineering & Statics
where the load goes
Concept

Transmissibility — where it appears

The fraction of a machine's force that reaches its support, or of a support's motion that reaches what stands on it. It is above one below the resonance, exactly one at √2 times it, and falls with frequency beyond that — which is why isolation means making a mounting soft rather than stiff.

Named by 4 essays across one field — each of them below, with the objects they name alongside it.

How much of a force a mount lets through, at three damping ratios. The force transmitted to the support divided by the force applied, against the ratio of the forcing frequency to the structure's own, at 2%, 5%, 20% of critical damping. Every curve passes through exactly 1 at a frequency ratio of root two, whatever the damping: below that ratio a mount amplifies what it was installed to isolate, and above it more damping lets more through.

The machine that shakes the building

Put a machine on springs to keep its vibration out of the floor, and below a frequency ratio of root two the springs make things worse. Every transmissibility curve ever drawn passes through exactly one at that ratio, whatever the damping — so a soft mount either works well or fails badly, with nothing in between.

dynamics · Vibration isolation Damping that is computed rather than measured. Radiation damping of a 2.5 m block on soil with a shear wave speed of 178 m/s, against how heavy the block is made — the horizontal axis is a multiple of the 150 tonne block drawn. At that mass the three modes are at 47, 29 and 13 per cent of critical. None of this is dissipation: the energy leaves as waves travelling away into the half-space, so the quantity is a geometrical coupling and it can be computed from the size of the footing, the density of the soil and its shear modulus. Every curve falls as the block gets heavier, because a heavier block moves less for the same wave field — which is the one counter-intuitive thing here: mass buys frequency and costs damping.

The damping that is radiated

Every response in this field is quoted with a damping assumption attached, because damping is measured rather than designed and the measurement has a factor of two in it. A machine block on the ground is the exception: its damping is not dissipation at all, and it can be computed from three numbers none of which is a material property of anything that dissipates.

dynamics · Vibration isolation Two separate checks, and the block that is neither. Amplification of the steady displacement at a machine 2.0 m above the base of a 150 tonne block 5.0 m across and 2.0 m deep, against forcing frequency. Checked separately, the block sways at 11.2 Hz with 29 per cent of critical damping and rocks at 15.0 Hz with 13 per cent, and the rocking check's peak is 3.99. Because its mass sits above its base the two are one system, with modes at 9.9 Hz and 20.9 Hz — one below both checks and one above — and the block's own peak is 2.49 at 9.7 Hz.

The frequency below both checks

A machine block is checked for sway and for rocking as if they were two oscillators, each with its own spring, its own radiation damping and its own natural frequency. A block whose mass sits above its base is one oscillator with two modes, and neither of them is a sway or a rocking — one sits below both checks, one above both, and the lower one is where the machine resonates.

dynamics · Vibration isolation Two blows on three foundations. The movement of a 150 tonne block struck by a hammer delivering 18 kN·s, twice, 0.75 s apart. On soil at 12.5 Hz with the 47 per cent of damping a half-space radiates, the block moves 0.86 mm and is still before the second blow. On the same soil over shallow rock, with 5 per cent, it moves 1.42 mm and rings for the whole interval. On springs at 4.0 Hz it moves 4.42 mm, and its ringing has not died when the second blow arrives. The peak force passed to the ground is 1.34, 1.32 and 0.42 MN: the heavily damped block and the lightly damped one pass almost the same force.

The blow that has no frequency

A forge hammer does not shake its foundation; it strikes it. The transmissibility curve every isolation design is drawn on has nothing to say about a blow, and the rules it teaches mislead. The force a struck block passes to the ground is least at a quarter of critical damping, not the most; five per cent and forty-seven pass almost the same; and the springs that soften every blow can bring the block down to the hammer's own rate and make a train of blows ring four times as high as one.

dynamics · Vibration isolation

Named alongside it

The objects these essays reach for when they reach for this one.

Vibration isolationDampingMachine foundationNatural frequencyResonanceRadiation dampingRockingServiceabilityDamping ratioDynamic amplificationFrequency ratioHalf-space

All concepts