Why Your Accelerometer Lies Above Its Natural Frequency

Why this calculation matters

Every accelerometer, whether it is a low-cost MEMS chip on a hobby drone or a piezoelectric sensor bolted to a turbine casing, is fundamentally a spring-mass-damper system in disguise. A small seismic mass hangs on a spring inside the housing, and when the housing accelerates, the mass lags behind just enough that its relative displacement (or the strain it induces) becomes the signal you read out. That internal mechanism has its own natural frequency and its own damping, and both properties silently distort every measurement you take.

Most datasheets quote a single number, something like "frequency response: 5 Hz to 10 kHz +/-5%," and engineers treat it as a hard boundary, as if the sensor works perfectly up to that point and then simply stops working. The truth is more continuous and more interesting: the error grows smoothly as you approach the sensor's own resonance, and if you ever cross it, the output amplifies instead of attenuates. Knowing where you sit relative to that resonance is the difference between trusting your vibration data and quietly logging garbage.

The physics: a second-order system between you and the truth

Treat the accelerometer's internal mass as a classic single-degree-of-freedom oscillator. It has a natural frequency fₙ set by the seismic mass and spring stiffness, and a damping ratio ζ that determines how sharply it resonates. When the true acceleration a oscillates at some measured vibration frequency f, the sensor's indicated output relates to the true input through a frequency-response function that depends only on the ratio of the two frequencies.

Define the frequency ratio:

r = f / fₙ

The magnitude of the response factor, the ratio of indicated to true acceleration, follows the standard second-order transfer function:

Response factor = 1 / sqrt[ (1 - r^2)^2 + (2 * zeta * r)^2 ]

At r approaching 0 (measured frequency far below the sensor's natural frequency), the response factor approaches 1.0: the sensor tracks the input almost perfectly. As r approaches 1.0, the first term in the denominator vanishes and the response is controlled entirely by damping — this is resonance, and the response factor can spike well above 1.0 if ζ is small. Beyond r = 1, the response factor falls off and can eventually attenuate the signal.

The measurement error is simply how far the response factor sits from unity:

Measurement error (%) = (Response factor - 1) x 100

A well-designed measurement stays only in the region where this error is small and predictable, conventionally under about 5%, which happens at frequency ratios below roughly 0.2 to 0.3 depending on damping.

Working through a real sensor

Suppose you're instrumenting a gearbox housing and picked a general-purpose piezoresistive accelerometer with a quoted natural frequency fₙ of 2,000 Hz and a damping ratio ζ of 0.65 (a fairly heavily damped design, typical of shock-rated sensors).

You expect the dominant gear-mesh vibration to sit around f = 400 Hz, with a true acceleration a of 50 m/s^2 at that frequency.

First, the frequency ratio:

r = f / fₙ = 400 / 2000 = 0.20

Plug into the response factor formula:

(1 - r^2)^2 = (1 - 0.04)^2 = 0.9216
(2 zeta r)^2  = (2 x 0.65 x 0.20)^2 = 0.0676
sum = 0.9892
Response factor = 1 / sqrt(0.9892) ~= 1.0055

That's a measurement error of about +0.55%, negligible for almost any industrial application. The sensor sits comfortably inside its usable range at this frequency.

Now push the scenario: imagine a resonance inside a rotating shaft excites a component at 1,600 Hz instead. The ratio becomes r = 0.80, and the same formula gives:

(1 - 0.64)^2 = 0.1296
(2 x 0.65 x 0.80)^2 = 1.0816
sum = 1.2112
Response factor = 1 / sqrt(1.2112) ~= 0.9087

Now you're under-reading true acceleration by about 9%, and if ζ were lower, say 0.1, typical of a lightly damped MEMS part, the same 1,600 Hz signal would sit close enough to resonance to produce substantial amplification instead. At r = 1 exactly, the response factor reduces to 1/(2ζ) — the resonance amplification figure — which for ζ = 0.1 works out to a 5x overshoot. This is exactly why a good simulator reports resonance amplification (x) as its own number: it tells you the worst case you'd see if your excitation frequency ever lands precisely on fₙ, the scenario every accelerometer datasheet is implicitly warning you about.

From these numbers you can also back out a usable upper frequency: the highest f at which the error stays under whatever tolerance your application demands, given the sensor's fₙ and ζ. For lightly damped sensors this ceiling is considerably lower than the resonant frequency itself, which surprises people who assume "usable range" simply means "up to fₙ."

Where engineers get this wrong

The most common mistake is selecting an accelerometer purely by full-scale range and price, checking fₙ against the top of the expected frequency band without leaving any margin. A common rule of thumb, that the usable frequency should not exceed roughly one-fifth of fₙ for lightly damped sensors, gets ignored under budget or schedule pressure, and the resulting data looks plausible — no clipping, no obvious noise — while being systematically biased.

A second mistake is ignoring damping ratio entirely. Two sensors with identical fₙ can behave completely differently near resonance: one with ζ = 0.7 barely deviates from unity gain even close to r = 1, while one with ζ = 0.05 produces a spike large enough to saturate downstream electronics. If a sensor's spec sheet doesn't quote ζ, treat that as a gap worth chasing down rather than an unimportant omission — assume worst-case behavior until you can measure it directly.

Finally, remember this entire analysis assumes a linear, single-degree-of-freedom sensor model. Real accelerometers have secondary resonances from mounting, cable whip, and cross-axis coupling that this idealized model doesn't capture. Treat the frequency-ratio calculation as a necessary first check on the sensor itself, not a complete validation of the whole measurement chain.

Reading a real datasheet with this model in hand

Once you understand the underlying second-order behavior, a manufacturer's frequency response plot stops looking like a black box and starts looking like a direct plot of the response factor curve above, just without the equation printed on it. A gentle rise near the top of the quoted range is the early stage of resonance amplification; a sharp peak followed by a steep drop is a lightly-damped sensor whose ζ is well below 0.3. If the datasheet only gives you a flat percentage tolerance and a single cutoff frequency, you're being handed a summary of this curve rather than the curve itself, and it's worth asking the manufacturer for fₙ and ζ directly if the application is at all frequency-critical — condition monitoring on rotating machinery, seismic instrumentation, or anything where a 5-10% systematic bias in reported acceleration would change a pass/fail decision.

This also explains why accelerometers meant for different jobs look so different mechanically. A shock accelerometer for crash testing needs a very high fₙ, often tens of kHz, specifically so that r stays small even for the fast transients in a crash pulse; the trade-off is lower sensitivity, since a stiffer spring produces less relative displacement for a given acceleration. A seismic accelerometer used for building monitoring does almost the opposite: a low fₙ and high sensitivity, because the frequencies of interest are only a few Hz and there's no need to chase a wide bandwidth at the cost of resolution. Neither design is "better" in the abstract; each is matched to keep the actual measurement inside the flat, low-error region of its own response factor curve.

Try it yourself

If you're specifying or troubleshooting a vibration measurement, it's worth running your actual fₙ, ζ, and expected frequency band through the numbers rather than eyeballing a datasheet plot. You can try the accelerometer frequency response simulator here — it walks through exactly the frequency ratio, response factor, and error calculation above and returns a direct measurement verdict for your sensor and target frequency. If your vibration is transmitted through a mounting structure before it ever reaches the sensor, the base excitation transmissibility tool covers the companion problem of how much vibration actually arrives at your sensor in the first place.