The frequency below both checks

A block on the ground has three ways of moving that matter to the machine bolted to it, and the essay that computed their damping took them one at a time. Vertical, horizontal and rocking each had a spring from the soil, a dashpot for the waves it sends away into the ground, and a natural frequency of its own. It ended on a practical rule: a machine foundation is checked for rocking, because rocking radiates least and so resonates hardest.

The vertical mode can be taken alone, because a block pressed straight down into the ground neither slides nor tilts, provided the machine’s mass is centred over the footing. The other two cannot. A block whose centre of mass sits above its base cannot slide without its own inertia trying to tip it, and cannot tip without its mass moving sideways. Sway and rocking are one system with two modes, and the two separate checks describe neither.

Two checks, and a peak where neither of them is #

The separate checks are each correct for what they describe. A footing forced to slide without turning has a horizontal stiffness from the soil and a horizontal radiation damping, and the block on it has a natural frequency of 11.2 Hz with 29 per cent of critical damping. A footing forced to turn without sliding has a rocking stiffness and a much smaller radiation damping, and the block turning about its base has a frequency of 15.0 Hz with 13 per cent.

Neither of those constraints exists. The block is free to do both, and it does. Driven from its top, it reaches its largest amplification of 2.49 at 9.7 Hz, which is 0.86 of the sway frequency and 0.65 of the rocking frequency. At that frequency the sway check predicts an amplification of 1.80 and the rocking check 1.65.

The difference in height is modest, because this block is squat and heavily damped. The difference in place is what matters. A common rule keeps a machine’s running speed at least a fifth away from every natural frequency of its foundation. A machine running at 8.9 Hz, 534 revolutions a minute, satisfies that rule against both checks: it is 0.79 of the sway frequency and 0.59 of the rocking frequency. On the block it is within a few per cent of the true peak, amplifying 2.35 where the checks predict 1.71 and 1.51.

Why the mass above the base joins them #

The free body is the same one every period in this field comes from, and the question it asks is which mass is moving.

Describe the block by two numbers: the horizontal displacement of its base, $u$, and its rotation about the base, $\theta$. The soil resists the first with a horizontal spring and the second with a rocking spring, and for a footing on the surface those two springs are independent — sliding the base does not twist it and twisting it does not slide it.

The mass is where they meet. The centre of mass sits at a height $z_c$ above the base, so it moves by $u + z_c\theta$, and its inertia force acts at that height. That force appears in the horizontal equation, where it pulls on the rotation, and its moment about the base appears in the rocking equation, where it pulls on the sliding. Written about the base, the mass matrix carries $m z_c$ in both off-diagonal places while the stiffness matrix carries nothing there. The coupling is entirely inertial.

The two natural frequencies are then the roots of

$$\gamma\,\omega^4 - \left(\omega_x^2 + \omega_\theta^2\right)\omega^2 + \omega_x^2\,\omega_\theta^2 = 0,$$

where $\omega_x$ and $\omega_\theta$ are the two separate checks and $\gamma$ is the block’s moment of inertia about its centre of mass divided by its moment of inertia about its base — 0.655 for this block. If the mass were concentrated at base level $\gamma$ would be one, and the roots would be the checks.

With $\gamma$ below one, something definite happens, and it can be read off without solving. The left-hand side is positive at $\omega = 0$ and positive for very large $\omega$. At $\omega = \omega_x$ it equals $(\gamma - 1)\,\omega_x^4$, and at $\omega = \omega_\theta$ it equals $(\gamma - 1)\,\omega_\theta^4$, both negative. So it changes sign once below the lower check and once above the higher. One mode sits below both separate frequencies and one above both, whatever the block and whatever the soil, as long as its mass is above its base.

The same equation says how far apart they are pushed. The product of its two roots is the product of the two checks divided by $\gamma$, so the geometric mean of the coupled frequencies is the geometric mean of the separate ones divided by $\sqrt{\gamma}$. For this block, 9.9 Hz times 20.9 Hz is 207, and 11.2 Hz times 15.0 Hz divided by the square root of 0.655 is 207 as well. A block with more of its inertia high up has a smaller $\gamma$, and its modes are pushed further apart from both sides.

Neither mode is a sway or a rocking #

Every rigid motion in a plane is a rotation about some point, and a mode’s shape is fully described by where that point is. A pure sway is a rotation about a point infinitely far away; a pure rocking is a rotation about the middle of the base. The two coupled modes are neither.

In the lower mode the block turns about a point 3.69 m below its base. That is far enough down for the motion to look mostly like sliding, with the top moving further than the base in the same direction. The mass moves further for the same stretch of the two soil springs than it would in either pure motion, and a system in which the mass moves more for the same spring force is a softer system: hence a frequency below both. This mode carries more than nine tenths of the block’s mass in horizontal motion, which is why most of the mass moving together is also the mode a horizontal force finds.

In the upper mode the block turns about a point 1.40 m up, inside the block and close to its centre of mass. The base slides one way while the mass barely moves the other, so both springs are worked hard for very little motion of the mass — a stiff system, and a frequency above both. It carries the remaining 8 per cent of the mass, and the two shares add up to the whole block, as the modal masses of any structure must. Like every structure with more than one period, the block now has a second mode that the first check never mentioned. This one is at twice the frequency of the first and heavily damped, and for most machines it is a mode that can safely be left out.

The height of the machine cannot avoid the lower mode #

A natural response to a rocking problem is to lower the machine, on the reasoning that a force applied near the base produces less moment about it. It does produce less moment. It does not avoid the resonance.

The reason is a general property of modes. A force puts energy into a mode in proportion to how far its point of application moves in that mode. A force applied at the point a mode turns about does no work on it and cannot excite it at all.

For the lower mode that point is 3.69 m under the ground. No position on the block is anywhere near it, so every position drives the mode, and moving the machine from the top to the base reduces the displacement from 5.94 to 2.51 µm per kN while leaving the peak within a few per cent of the same frequency. For the upper mode the point is 1.40 m up, on the block, and a machine placed there would not excite the upper mode at all. That is the one mode that did not need avoiding.

A tall block, and the check that was supposed to be safe #

The squat block is the forgiving case: its mass sits only a metre above its base and its rocking damping is 13 per cent. Put the same 150 tonnes on a narrower, deeper block — 3.2 m across and 3.2 m deep, the proportions of a pedestal carrying a machine up to a working level — and both of those protections go.

The rocking check here predicts an amplification of 76.7 at 6.5 Hz. That is an alarming number, and it is the reason the rule to check rocking exists. But the block does not peak at 6.5 Hz. It peaks at 5.5 Hz, and at 5.5 Hz the rocking check predicts an amplification of 3.6.

So the check is conservative about the height of the peak and wrong about where it is. A compressor running at 330 revolutions a minute, 5.5 Hz, would be read off the rocking check as sitting 15 per cent below resonance with a modest amplification. On the block it sits on the peak. Its top displaces 191 µm for every kilonewton of unbalanced force, against a static 12.3 µm per kN.

The danger is not in the peak’s height, which the check overstates. It is in the peak’s frequency, which the check places too high. Every coupled lower mode is below both separate checks, so an operating speed chosen to sit safely below the checks moves towards the true resonance rather than away from it.

The lower mode borrows its damping from sway #

The tall block’s lower mode turns about a point only 0.97 m under its base, so it is mostly a rocking. But the base still slides in it, and a sliding base radiates shear waves at the sway rate. The mode’s damping is therefore a blend of the two separate values: 3.2 per cent, between rocking’s 0.7 and sway’s 15, and weighted towards rocking because that is mostly what the mode is.

That borrowing is exactly why the peak is 15.7 rather than 76.7. The peak amplification of a lightly damped mode is close to one over twice its damping ratio, inversely proportional to whatever the damping really is, and 3.2 per cent gives about 16. So the coupling does two things at once, and they go in opposite directions. It lowers the frequency into the machine’s running range, and it lends the rocking some of sway’s radiation, so the resonance it creates is milder than the check’s.

The tall block’s lower mode still carries about four fifths of the block’s mass in horizontal motion. That is less than the squat block’s, because more of this mode is rotation.

Deeper blocks pull the modes further from both checks #

The two blocks so far differ in two ways at once. The effect of depth alone shows when the footprint and the mass are held and only the depth of the block carrying the mass is changed.

The sway check does not move at all, because nothing about sliding a block across the ground knows how high its mass is. The rocking check falls steadily, because a deeper block has a larger moment of inertia about its base. The two coupled modes straddle both at every depth.

Two regions of the figure read differently. At shallow depths the lower mode is nearly the sway: at 0.5 m it is 11.1 Hz against the sway check’s 11.2, because the mass is so close to the base that tipping it hardly moves it. At large depths the lower mode approaches the rocking check from below. The coupling matters most where the two checks nearly agree. At a depth of 3 m the sway check is 11.2 Hz and the rocking check 11.9, and a designer reading them sees one comfortable frequency near 11.5 Hz. The block’s lower mode is at 8.9 Hz, more than a fifth below both.

The peak the check predicts, and the one the block reaches #

Across the whole range the rocking check overstates the block’s own peak, by a factor of 1.4 on the shallowest block and 2.1 on the deepest. That is the half of the rule that works: a foundation designed against the rocking check’s peak has margin on amplitude.

The other half is the lower mode’s damping, which falls from 28 per cent to 5 as the block deepens and the mode turns from mostly sway into mostly rocking. A deep block therefore loses on both counts at once. Its resonance moves lower, towards the running speeds of heavy reciprocating machines, and it carries less of the sway’s radiation with it. The amplification is lower than the check’s, but the frequency is not where the check put it.

Widening the footing is the repair, and it moves the peak #

If a pedestal must be deep, the variable left is its footprint. The rocking stiffness of a footing grows as the cube of its size, and a wider footing radiates more in both modes.

For the compressor at 5.5 Hz the change is large. The top of the widened block moves 6.3 µm per kilonewton at that speed, against 191 µm per kilonewton on the narrow one, a thirtieth of it. The peak amplification falls from 15.7 to 3.6, because the lower mode now carries far more of the sway’s damping: about 14 per cent of critical, against 3.2 on the narrow block. Part of that is the wider footing radiating more in rocking, 8 per cent against 0.7, and part is sway’s own radiation, which doubles with the wider footing and which the lower mode borrows.

But the peak has not gone. It has moved to 8.6 Hz, again below both separate checks, which now nearly coincide at 11.2 and 11.4 Hz. That is exactly the case the depth figure warned about. A widened pedestal that happens to carry a machine running at 8.6 Hz has been moved onto its own resonance by the repair. Widening is the right move for the compressor, and it is right only after the coupled frequencies have been computed again, not the separate ones.

Where the block cannot be changed at all, the remedy moves elsewhere — to a mass tuned to the lower mode, which works best on exactly the lightly damped mode the tall block has, or back to springs under the machine that keep its force off the block in the first place.

What the two-spring model leaves out #

The footing sits on the surface. A block set into the ground picks up resistance from the soil against its sides, and that resistance acts at mid-depth rather than at the base. It adds coupling terms to the stiffness matrix as well as the mass matrix, and depending on where the side resistance acts relative to the centre of mass they can reinforce the inertial coupling or partly cancel it.

The springs and dashpots are constants. The analogues are fitted to the exact elastic solution over the range of frequencies machines usually run at, and outside that range the soil’s stiffness falls and its radiation changes with frequency. The coupled frequencies inherit that approximation, and they inherit the uncertainty in the soil’s shear modulus as well, which enters every frequency as a square root.

The modes are not classical. Sway radiates strongly and rocking weakly, so the damping is not proportional to the mass or stiffness, and strictly the modes are complex, with parts of the block moving out of phase. The responses drawn above are computed exactly, frequency by frequency. The damping ratios quoted beside the mode shapes are estimates from the real shapes, and they reproduce the exact peaks to within a few per cent.

The machine’s force acts in one plane and its mass is centred. An eccentric machine couples the vertical mode into the rocking too, and a machine whose unbalanced force rotates in plan drives sway and rocking in two directions at once, with a torsion about the vertical axis on top.

Still open: how much of the coupling survives a block set into the ground #

Every number in this essay belongs to a block on the surface, where the only coupling between sway and rocking is its own inertia. Most real machine blocks are embedded, and the side soil introduces a second coupling through the stiffness, acting at a height the designer partly chooses by choosing the depth of embedment. Placed below the centre of mass, the side resistance works with the inertial coupling and spreads the modes further; placed near it, it could in principle remove the coupling altogether, leaving two nearly independent modes and a pair of separate checks that were right after all. Whether an embedment depth exists that decouples a given block, and whether it survives the uncertainty in how much the side soil really contributes after backfill and settlement, is a question about the stiffness matrix — and it is the one that decides whether the separate checks are an approximation or a choice.