Technical Note · Vibration Fundamentals
Shaker Performance Limits: Force, Velocity and Displacement
How to tell whether a shaker can run your test before you mount the article: the three limits every electrodynamic shaker has, the formulas that connect them, and worked examples for sine and random.
Key takeaways
- Every electrodynamic shaker has three limits: force, velocity and displacement (stroke). A test fits only if it respects all three at every frequency.
- Force is set by the total moving mass: armature, head expander or slip table, fixture, test article and hardware. The test article is often the smallest part of it.
- For a sine, velocity falls as 1/f and displacement as 1/f². That gives every shaker a sine performance envelope: stroke-limited at low frequency, velocity-limited in the middle, force-limited above.
- For random, force is checked against the rms rating, and stroke against the peak displacement, which is dominated by the lowest octave of the profile.
- A resonant test article can demand far more force than its weight suggests. Check force at the article's modes, not just against the static mass budget.
Why check shaker limits before the test
A vibration controller will try to deliver whatever profile it is given. If the profile asks for more than the shaker can physically produce, the result is at best an abort and at worst an armature hitting its stops, an amplifier fault, or a test article exposed to an uncontrolled transient. All of these cost lab time, and some cost hardware.
The good news is that the check takes a few minutes with a calculator. You need the shaker's datasheet, an honest estimate of the moving mass, and the test profile. This note walks through the three limits, the formulas that connect them, and worked examples for sine and random using an illustrative 2,000 lbf shaker.
| Limit | What sets it | Typically stated as | Where it bites |
|---|---|---|---|
| Force | Current in the armature coil (amplifier current, coil heating) | Peak force for sine, rms force for random, peak force for shock | Mid and high frequencies, heavy payloads, resonant articles |
| Velocity | Back-EMF: the faster the armature moves, the more voltage the amplifier must supply | Peak velocity, in/s or m/s | The middle of the range, between the two crossovers |
| Displacement | Mechanical stroke between the armature's stops | Peak-to-peak displacement, in or mm | Low frequencies; the lowest octave of random profiles; shock compensation |
The moving mass
Newton's second law sets the force limit: F = M·a. In US units the arithmetic is convenient, because force in lbf equals weight in lb multiplied by acceleration in g. The trap is M. It is not the test article's weight. It is everything that moves with the armature.
| Item | Weight (lb) |
|---|---|
| Armature (moving element) | 25 |
| Head expander | 30 |
| Fixture | 28 |
| Test article | 15 |
| Bolts, accelerometers, cable clamps | 2 |
| Total moving mass | 100 |
With 100 lb moving, the 2,000 lbf sine rating supports 2,000 / 100 = 20 g, not the 130 g the article's weight alone would suggest. Head expanders and slip tables are often the heaviest items after the armature, and fixtures designed for stiffness can grow quickly. Weigh them rather than estimating, and remember the article's own mounting hardware.
Rated force is not usable force. Many labs keep a margin of 10 to 20% below the rated force for sine and random. It covers the extra force a test article demands at its resonances, equalization overshoot and the uncertainty in the mass estimate.
From acceleration to velocity and displacement
For a sine at frequency f, acceleration, velocity and displacement are tied together by factors of 2πf:
The practical consequence is easy to remember. Hold the acceleration constant and lower the frequency: velocity rises in proportion and displacement rises with the square. A 1 g sine needs only 0.002 in pk-pk at 100 Hz, but 0.2 in at 10 Hz and 4.9 in at 2 Hz. That is why low-frequency sine profiles are almost always specified as constant displacement, and why the low end of every shaker is limited by stroke.
| Quantity | US units | Metric units |
|---|---|---|
| Peak velocity from peak acceleration | v (in/s) = 61.45 · a (g) / f | v (m/s) = 1.561 · a (g) / f |
| Peak-to-peak displacement from peak acceleration | dpk-pk (in) = 19.56 · a (g) / f² | dpk-pk (mm) = 496.8 · a (g) / f² |
| Acceleration from displacement | a (g) = 0.0511 · dpk-pk (in) · f² | a (g) = 0.00201 · dpk-pk (mm) · f² |
| Force | F (lbf) = W (lb) · a (g) | F (N) = m (kg) · a (g) · 9.807 |
| Displacement–velocity crossover | f = 2v / (2π · dpk-pk) | same, consistent units |
| Velocity–force crossover | f = 61.45 · (F/W) / v | f = 1.561 · (F / (m · 9.807)) / v |
The sine performance envelope
Put the three limits on one log-log plot of acceleration against frequency and they form the shaker's sine performance envelope. Stroke gives a line rising at 40 dB per decade, velocity a line rising at 20 dB per decade, and force a flat line at F/W. The usable region is under all three.
For the example shaker the stroke limit gives way to the velocity limit at 11.1 Hz, and velocity gives way to force at 17.6 Hz. Above that, the shaker can deliver 20 g at any frequency up to the top of its rated range, as long as the mass stays at 100 lb. Add a heavier fixture and the force line drops; the velocity and stroke lines do not move.
Checking a sine profile
A sine profile fits if it stays inside the envelope at every frequency. Because a constant-displacement segment has velocity rising with frequency, and a constant-acceleration segment has velocity falling with frequency, the highest velocity in a typical profile occurs at its own crossover frequency. Check the low end for stroke, the crossover for velocity, and the high-g segment for force.
| Profile point | Freq (Hz) | Accel (g pk) | Disp (in pk-pk) | Vel (in/s pk) | Force (lbf) | Result |
|---|---|---|---|---|---|---|
| A: low end | 5 | 0.64 | 0.500 | 7.9 | 64 | Fits |
| A: crossover | 19.8 | 10.00 | 0.500 | 31.1 | 1,000 | Fits |
| A: top end | 2,000 | 10.00 | < 0.001 | 0.3 | 1,000 | Fits |
| B: crossover | 31.3 | 25.00 | 0.500 | 49.1 | 2,500 | Exceeds force |
| C: low end | 5 | 3.20 | 2.500 | 39.3 | 320 | Exceeds stroke |
Profile A fits comfortably. Profile B asks for 25 g, which needs 2,500 lbf from a 2,000 lbf shaker. Profile C asks for 2.5 in pk-pk at 5 Hz, more stroke than the shaker has. Neither problem can be fixed by the controller; they need a lighter fixture, a bigger shaker or a profile change.
Random tests: rms force and peak displacement
Random profiles are specified as an acceleration spectral density in g²/Hz, and the overall level is the Grms: the square root of the area under the profile (see Power Spectral Density (PSD) and Grms). The force check uses the shaker's random (rms) force rating: Frms = W × Grms.
Velocity and displacement are found by dividing the spectrum by (2πf)² and (2πf)⁴ before integrating:
Random signals have peaks. A common convention is to assume displacement peaks of about 3 × drms in each direction, so the stroke needed is about 6 × drms peak-to-peak. Clipping the drive at 3σ does not strictly cap the displacement, which is a filtered version of the drive, so some labs allow 4 to 5σ for long tests or when kurtosis control raises the peaks.
| Quantity | 20–2000 Hz | 10–2000 Hz | Limit |
|---|---|---|---|
| Acceleration (Grms) | 8.90 | 8.92 | — |
| Force (lbf rms) = 100 lb × Grms | 890 | 892 | 1,800 |
| Velocity (in/s rms) | 2.73 | 3.88 | — |
| Velocity, 3σ peak (in/s) | 8.2 | 11.6 | 70 |
| Displacement (in rms) | 0.0126 | 0.0357 | — |
| Displacement, 3σ peak-to-peak (in) | 0.076 | 0.214 | 2 |
The flat 20–2000 Hz profile uses 890 lbf rms of a 1,800 lbf rms rating and only 0.08 in of a 2 in stroke. Extend the same level down to 10 Hz and the Grms barely changes, but the stroke needed rises to 0.21 in: nearly three times as much, from one extra octave. This is why most random specifications slope up at the low end instead of starting flat.
For the flat profile, 87.5% of the displacement variance comes from below 40 Hz. When a random test runs out of stroke, the fix is almost always at the bottom of the profile.
When the test article fights back: resonance and apparent mass
The mass budget assumes everything moves together as a rigid body. A real test article has resonances, and at a resonance part of it moves much more than the table. The force needed to drive it rises accordingly: at the resonant frequency the article can behave like a mass up to about Q times heavier than it weighs, though because that motion is out of phase with the table it adds to the rest of the moving mass only in part (see Resonance).
In this example the static budget says 1,000 lbf at 10 g. Near the article's 300 Hz mode the demand reaches about 2,600 lbf, well above the shaker's rating. Just above the mode the article moves against the table and the demand briefly falls well below the rigid-mass estimate. The controller sees all of this as a drive that must climb steeply, then fall, over a narrow band. Depending on the specification, the answer may be response limiting (notching) on the article, force limiting, or a larger shaker.
Run a low-level survey first. A low-level sine or random survey shows where the article's resonances are and how strongly they load the shaker, before the full-level test asks for the force.
Other limits worth checking
- Amplifier limits. Amplifier current sets force and amplifier voltage sets velocity. A shaker paired with a smaller amplifier has a smaller envelope than its datasheet.
- Static payload and overturning moment. The armature suspension carries the static weight of the payload, and an off-center center of gravity loads it with a moment. Both have limits on the datasheet, especially for vertical tests with tall fixtures.
- Slip tables. Horizontal tests add the slip table's moving plate and drive bar to the moving mass, and the bearing system has its own load and moment ratings.
- Upper frequency. The armature's first axial resonance and the fixture's own modes set a practical upper limit. Above them, the table no longer moves as a rigid body (see Fixture Design for Vibration Testing).
- Duty and cooling. Ratings assume the shaker's cooling is working. Long tests near full force need the cooling system sized for continuous duty.
Common misconceptions
| Misconception | Reality |
|---|---|
| "The article weighs 15 lb, so a 2,000 lbf shaker can do 130 g." | The shaker also moves its own armature, the head expander or slip table, the fixture and the hardware. Here that is 100 lb, so the force limit is 20 g. |
| "If the force budget passes, the test will run." | Force is only one of three limits. Below about 18 Hz this shaker is limited by velocity or stroke, not force. |
| "Random displacement is small because the acceleration is small." | Displacement weights the spectrum by 1/f⁴. Extending a flat profile down one octave nearly triples the stroke needed (Table 5). |
| "Static mass is all that matters." | At the article's resonances its apparent mass multiplies by up to Q. Force demand can exceed the rating even though the static budget passes (Figure 5). |
| "Sine, random and shock force ratings are interchangeable." | They are different numbers measured differently. Check sine against peak, random against rms, and shock against the shock rating. |
Quick reference
| Check | Formula or rule | Example result |
|---|---|---|
| Moving mass | Armature + head expander/slip table + fixture + article + hardware | 100 lb |
| Force-limited acceleration | a = F / W | 2,000 / 100 = 20 g |
| Stroke check (sine) | dpk-pk = 19.56 · a / f² | 0.64 g at 5 Hz → 0.50 in |
| Velocity check (sine) | v = 61.45 · a / f | 10 g at 19.8 Hz → 31 in/s |
| Random force | Frms = W · Grms | 100 lb × 8.9 = 890 lbf rms |
| Random stroke | ≈ 6 × drms for 3σ peaks | 0.08 in pk-pk |
| Resonant article | Apparent mass up to Q × article mass at resonance | Check force at the article's modes |
How Panther helps
- Shaker limits in every application: Panther's Shaker Limits page sets positive and negative limits for acceleration, velocity, displacement and drive voltage. If a test reaches any of them, it is aborted. The settings can be saved as a shaker file and reused across tests.
- Force check before the test starts: enter the test equipment weights and a maximum force, and Panther checks the demanded force against it. It can display a warning or prevent the test from starting.
- Protecting the article at resonance: limit channels reduce the drive when a response exceeds its limit profile and return control to the control channels when it falls back below; abort channels stop the test.
- Know your shaker's real envelope: ShakerScope maps the stroke-, velocity- and force-limited regions of the sine performance envelope from a measured sweep of the rig.
- Quick calculations: the free Spectral Dynamics vibration calculator converts between acceleration, velocity and displacement for sine and random.
References and further reading
- C. M. Harris and A. G. Piersol (eds.), Harris' Shock and Vibration Handbook, 5th ed., McGraw-Hill.
- D. S. Steinberg, Vibration Analysis for Electronic Equipment, 3rd ed., Wiley.
- ISO 5344, Electrodynamic vibration generating systems — Performance characteristics.
- MIL-STD-810H, Method 514.8, Vibration.
- NASA-HDBK-7004, Force Limited Vibration Testing.
- Spectral Dynamics Technical Library: Power Spectral Density (PSD) and Grms; Resonance; Fixture Design for Vibration Testing.
