Large Shaker Seismic Mass

A question that comes up whenever a laboratory is planning its first large electrodynamic shaker installation: the specification and the quotations keep referring to a seismic mass, or a reaction mass, or an inertia block. Is that a real, physically heavy object that has to be purchased and installed? Or is it a tuned-mass arrangement already built into the shaker housing? And if it is real, who supplies it, and how is it sized?

The question is a good one, because the answer is usually not in the shaker sales literature, the item is frequently missing from the capital request, and the decisions that depend on it — pit geometry, standoff from adjacent footings, placement of vibration-sensitive equipment elsewhere in the building — are extremely expensive to revisit once the concrete has been poured.

Yes, It Is a Real, Physically Heavy Mass

A seismic mass — reaction mass, inertia block, inertia base, all the same animal — is a large block of reinforced concrete, sometimes with a steel top plate, sometimes a fabricated steel weldment filled with concrete, to which the shaker trunnion base and the slip table base are both bolted. The block is then supported on soft isolators, usually air springs, occasionally steel coil or elastomeric mounts, so that the entire assembly of block + shaker + slip table + fixture + test article floats as a single rigid body on a low-frequency suspension. Typical vertical suspension frequencies are 1.5 to 3 Hz.

It is not hidden inside the shaker housing. On most installations the block sits in a pit poured below floor level, so all that is visible from the laboratory floor is the trunnion, the slip table, and a flush plate or grating around them. That is usually why an engineer who has spent time in test labs has never actually seen one.

If you have been in a lab where the shaker was clearly sitting on the slab with nothing underneath it, that was most likely a smaller air-cooled unit. In those machines the shaker body itself is the reaction mass and the built-in pneumatic isolators are the entire isolation system. Somewhere in the tens of kN of force rating that stops being sufficient, and by 100 kN it generally is not.

What the Block Actually Does

Newton’s third law does not negotiate. Whatever force the armature applies to the test article, it applies equally and oppositely to the shaker body. The body is not bolted rigidly to the floor — it rides on soft isolators — so it recoils. A momentum balance gives the recoil amplitude directly:

$$m_{\text{moving}} x_{\text{armature}} = m_{\text{body}} x_{\text{body}}$$

Consider a 100 kN class machine with a 100 kg armature carrying a 200 kg fixture and article, so 300 kg of moving mass. Run a low-frequency sine sweep, or a time waveform replication, down near the 5 Hz low corner where the armature is using most of a 51 mm continuous stroke, roughly $\pm 25 \text{ mm}$. Against a bare shaker body of 4,500 kg the body recoils about $\pm 1.7 \text{ mm}$. Bolt that same shaker to a block that brings the total reacting mass to 20,000 kg and the recoil falls to about $\pm 0.4 \text{ mm}$.

That recoil matters for three reasons:

1. Control fidelity. The control accelerometer measures absolute motion. Body recoil is not what you commanded, and at low frequency it is not small.
2. Hardware life. That motion is taken up by cooling hoses, cable runs, the head expander interface, and — the expensive one — the drive bar and the oil film bearings of the slip table.
3. Neighbors. It is the source of the motion that ends up in the foundations of everything else in the building.

One Subtlety Worth Knowing

At frequencies well above the suspension resonance, the force delivered into the building is approximately the isolator stiffness times the block displacement:

$$F_{\text{transmitted}} \approx k x_{\text{block}} = (2\pi f_n)^2 m_{\text{block}} x_{\text{block}} = (2\pi f_n)^2 m_{\text{moving}} x_{\text{armature}}$$

Notice what dropped out of the right-hand side: the block mass. Substituting the momentum balance eliminates it entirely.

For the numbers above, with $f_n = 2 \text{ Hz}$, 300 kg moving, and 25 mm of armature stroke, the transmitted force is on the order of

$$(2\pi \cdot 2)^2 (300)(0.025) \approx 1200 \text{ N}$$

independent of whether the block is 15,000 kg or 30,000 kg.

So what does the block buy? Motion control and stability. Much smaller recoil displacement and velocity. A low center of gravity. The rotational inertia needed to resist the overturning couple that appears in the horizontal axis, where the drive force acts well above the isolator plane. And a rigid, flat, dimensionally stable datum to which the shaker and slip table can be aligned and kept aligned. Those are not small benefits. They are simply different benefits from the ones people usually assume.

Sizing and “Tuning”

The classical machinery rule of thumb is an inertia block of five to ten times the reciprocating mass. For shakers, practice generally lands at 10 to 20 times the total moving mass, or 3 to 5 times the shaker body mass. For a 100 kN class machine that puts the block somewhere in the 15,000 to 30,000 kg range — a pit on the order of $3 \text{ m} \times 2 \text{ m} \times 1.2 \text{ m}$ of reinforced concrete.

Do not size the block from a rule of thumb. Size it from the OEM foundation drawing, then check it. The rule of thumb is for sanity, and for the capital request.

“Tuning” in this context does not mean tuning in the vibration-absorber sense. There is no auxiliary mass-spring being tuned to a troublesome frequency. What is actually being chosen is the following.

Who Supplies It

Almost never the shaker OEM. The typical division of responsibility is:

• Shaker OEM. Supplies the loading data and an installation/foundation drawing: force levels, body mass, moving mass, bolt pattern, pit envelope, static and dynamic loads, and whether the trunnion base includes integral pneumatic isolators.
• Local structural or foundation contractor. Designs and pours the block to that drawing, and coordinates with the building structure.
• Isolation specialist. Supplies the air springs or coil mounts and performs the six-DOF rigid-body analysis. Fabreeka, Bilz, GERB, Kinetics Noise Control, Vibro/Dynamics and similar firms do this work routinely.

Budget the isolation package as a separate line item. It is frequently the item nobody put in the capital request.

Get the foundation drawing early — before the placement of other equipment in the bay is finalized. It will state the required standoff from adjacent footings, and it will reveal whether an isolation joint or relief cut in the slab is even meaningful given the pit geometry.

On the Granite Base of the Slip Table

A common instinct is to view the granite base of a slip table as the reaction mass, by analogy with a coordinate measuring machine. That instinct is half right.

On a slip table the granite base serves several purposes at once. It provides the flat, stiff, dimensionally stable reference surface for the oil film or T-film bearing beneath the slip plate. It provides stiffness, so the plate does not have a low bending mode. And it provides local mass to react the horizontal drive force. In that last sense it is indeed a reaction mass for the lateral axis.

But the granite is normally tied rigidly to the shaker body, or to a common base, precisely so that the drive-bar reaction and the shaker body recoil react against each other through a common structure rather than through the floor. The granite is a structural element of the load path, not an isolation element. The isolation lives entirely in what is underneath the whole assembly.

So the CMM analogy is right about why granite and wrong about why it is heavy. On a CMM one is buying flatness, thermal stability, and damping. On a slip table one is buying flatness, stiffness, damping, and mass, in roughly that order of importance.

A related item to settle early: whether the installation is a mono-base configuration, in which the shaker and granite share one integrated base, or separate bases. That decision affects the pit layout and the path by which the horizontal reaction enters the block, and it is far easier to sort out on paper than after the concrete is in.

Floor Vibration Criteria

The question “how much vibration is acceptable at the other end of the bay?” has a standard answer set. The generic vibration criteria, usually drawn as the VC curves, are a family of horizontal lines of constant one-third octave band rms velocity. The upper four lines are derived from the base curve of ISO 2631-2 multiplied by scaling factors, and the VC-A through VC-G lines below them were developed by Ungar, Gordon and co-workers for semiconductor and research facilities. They are reproduced in IEST-RP-CC012 and are what nearly every equipment OEM cites when a site vibration limit appears in a specification.

Four points about how to apply the table:

• The two velocity columns are soft conversions, not exact equivalents. The VC series was originally defined in microinches per second as successive halvings of 2000, and the metric column is the rounded conversion. VC-A at 2000 microinch/s is really 50.8 µm/s, not 50. The columns disagree by about 1.6 percent throughout, which matters to nobody except the person checking the arithmetic.
• The values are one-third octave band rms velocity, not overall rms and not peak. Comparing an overall broadband rms number against a VC line is a very common error, and it is conservative by roughly the square root of the number of contributing bands.
• The ISO-derived curves at the top of the table are constant velocity from 8 to 80 Hz and follow a constant acceleration line below 8 Hz, which means the allowable velocity rises as frequency falls. VC-C and below are commonly drawn as constant velocity from 1 to 80 Hz in modern practice. Check which convention the equipment OEM intends before comparing survey data against a low-frequency band.
• VC-F and VC-G are not realistically achieved by a building slab. Equipment with those requirements sits on its own active or pneumatic isolation system, and the slab criterion is then set by what that isolation system can reject.

Do the Site Survey While the Bay Is Empty

The best time to characterize a laboratory bay is before anything is installed in it. Two measurements are worth the effort: ambient floor vibration, and transfer mobility between the planned shaker location and the planned locations of vibration-sensitive equipment.

Instrumentation

One caution that catches people out: for ambient floor noise, a standard 100 mV/g ICP accelerometer will not get you there. The motion of interest is in the tens of micrometers per second, and general-purpose units have a broadband noise floor an order of magnitude or more above that. Use low-noise seismic accelerometers — a 10 V/g unit such as the PCB 393B31, or the 393B12 / 393B05 depending on availability — and confirm that the low-frequency corner covers 1 Hz. Keep the general-purpose accelerometers for the hammer or drop-weight source measurements, where the levels are high.

Transfer Mobility

Place the reference at the planned shaker location and the receivers at the candidate positions for sensitive equipment, then reverse the pair as a reciprocity check. Process to one-third octave velocity rms, so that the results can be laid directly against the generic vibration criteria curves (VC-A through VC-E) and against whatever the equipment OEM specifies. Take enough averages to trust the coherence, and log what was running elsewhere in the building at the time.

Save the raw time histories, not merely the processed spectra. You will want to reprocess them a year from now to answer a question nobody has asked yet.

Sample Calculations

The following worked examples use a consistent hypothetical installation, so that the numbers carry from one example to the next.

Example 1. Body Recoil, Bare Shaker vs. Block

The armature is driven to its stroke limit at the 5 Hz low corner. Momentum balance gives

$$x_{\text{body}} = \frac{m_{\text{moving}}}{m_{\text{body}}} x_{\text{armature}}$$

Bare shaker body, 4,500 kg:

$$x_{\text{body}} = \frac{300}{4500} (25 \text{ mm}) = 1.67 \text{ mm peak}$$

Shaker bolted to the block, total reacting mass 22,000 kg:

$$x_{\text{body}} = \frac{300}{22000} (25 \text{ mm}) = 0.34 \text{ mm peak}$$

A factor of five reduction in recoil displacement, and the same factor in recoil velocity. This is the benefit the block genuinely delivers.

Example 2. Force Transmitted Into the Slab

Well above the suspension resonance, the isolators are stiffness-controlled and the force delivered to the building is

$$F_{\text{transmitted}} \approx k x_{\text{block}} = (2\pi f_n)^2 m_{\text{iso}} x_{\text{block}} = (2\pi f_n)^2 m_{\text{moving}} x_{\text{armature}}$$

Take $f_n = 2 \text{ Hz}$. At the 5 Hz stroke-limited condition,

$$F_{\text{transmitted}} = (2\pi \cdot 2)^2 (300)(0.025) = (157.9)(300)(0.025) = 1184 \text{ N}$$

Now repeat with a 30,000 kg block instead of 15,000 kg, holding $f_n = 2 \text{ Hz}$. The isolator stiffness must nearly double to keep the same suspension frequency, and the transmitted force is unchanged at 1184 N. Doubling the concrete bought nothing at all in transmitted force.

Repeat instead with the block unchanged and $f_n$ lowered from 2 Hz to 1.5 Hz:

$$F_{\text{transmitted}} = (2\pi \cdot 1.5)^2 (300)(0.025) = 666 \text{ N}$$

Example 3. Isolator Sizing

Required total isolator stiffness for a target suspension frequency:

$$k = m_{\text{iso}} (2\pi f_n)^2 \qquad\qquad \delta_{\text{static}} = \frac{g}{(2\pi f_n)^2}$$