An accelerometer is itself a tiny spring-mass-damper system. Adjust the natural frequency, damping ratio and the frequency being measured to see the response factor (indicated / true), the measurement error and the usable upper frequency in real time, and understand why it reads accurately only in the flat band well below resonance.
Parameters
Sensor natural frequency fₙ
Hz
Frequency at which the internal spring-mass system resonates
Damping ratio ζ
The response is flattest at ζ ≈ 0.7
Measured vibration frequency f
Hz
Frequency of the vibration you are trying to measure
True acceleration a
m/s²
Amplitude of the acceleration the target actually experiences
Results
—
Frequency ratio r
—
Response factor (ind./true)
—
Measurement error (%)
—
Usable upper freq. (Hz)
—
Resonance amplification (×)
—
Measurement verdict
—
Frequency response curve — measured-frequency sweep animation
The sensor response factor (indicated / true) is plotted against frequency: a flat band at low frequency, a resonance peak at the natural frequency and the roll-off above it. The usable band (under 5% error) is shaded. The small inset at top right is the seismic mass and spring inside the sensor.
Frequency response (FRF) — response factor vs frequency
Accelerometer response factor (indicated / true acceleration). r is the ratio of the measured frequency f to the sensor's natural frequency fₙ, and ζ is the damping ratio. The sensor is accurate only in the flat band where r is much smaller than 1, and ζ ≈ 0.7 gives the widest flat band.
Measurement error and indicated acceleration. A positive error means over-reading, a negative error means under-reading. The closer the response is to 1, the closer the indicated value is to the true value.
The response factor — the amplification — at resonance (r = 1). The smaller the damping ratio ζ, the higher it shoots; for a lightly-damped sensor it can exceed ten.
What is the Accelerometer Frequency Response Simulator?
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An accelerometer is just a handy part you stick on something and it reads the acceleration directly, right? Why would I need to worry about "frequency response"?
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Roughly speaking, an accelerometer is not a perfect sensor that instantly gives the right number. Look inside and you find a small "seismic mass" hanging on a spring. When the case shakes, the device infers acceleration from how much that mass lags behind. So the sensor itself is a spring-mass system with its own natural frequency, and how accurate it is depends on which frequency you are measuring.
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Wait, the sensor itself is a vibrating system? So when does it read accurately and when does it go off?
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The key is where the frequency you want to measure sits relative to the sensor's natural frequency. Well below it, the seismic mass moves faithfully with the case and the output is a constant multiple of the true acceleration — that flat band is where it is accurate. With the defaults above (fₙ = 30000 Hz, f = 1000 Hz) the frequency ratio r is only 0.0333 and the response factor is 1.0022 — a measurement error of just 0.22%.
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And what happens if I keep raising the frequency I want to measure?
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As the measured frequency climbs toward the natural frequency, the response factor shoots up — that is resonance. With the default light damping (ζ = 0.05) the peak is exactly ten times. The sensor "indicates" ten times the true acceleration. Push past the natural frequency and the seismic mass can no longer keep up, the response drops, and it under-reads. So slide the measured frequency f upward and watch the error explode.
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So how high can I safely go?
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The practical rule is simple. For a lightly-damped sensor the trustworthy band reaches only about one-fifth of the natural frequency. With the defaults, that is roughly one-fifth of 30000 Hz, about 6500 Hz, for a 5% amplitude error. So always pick a sensor whose natural frequency lies far above the highest frequency you mean to measure. To measure 5 kHz machine vibration, for example, you would choose a sensor with a natural frequency of 25 kHz or more.
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Other than choosing a sensor with a higher natural frequency, is there any way to widen the band?
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Yes — deliberately raise the damping ratio. Setting ζ near 0.7 (exactly 1/√2) almost removes the pre-resonance peak and keeps the response flat up to much higher frequencies. Slide ζ from 0.05 to 0.7 and you will see the usable upper limit grow from about one-fifth of the natural frequency to nearly half. Piezoelectric sensors have small internal damping and are lightly damped, but servo and MEMS designs tune the damping electrically to chase a flat, wide-band response.
Frequently Asked Questions
Inside an accelerometer there is a small 'seismic' (proof) mass held by a spring, and the device infers acceleration from how much that mass lags behind the moving case. So the accelerometer is itself a spring-mass system with its own natural frequency. When the vibration being measured is well below the sensor's natural frequency the response is flat and accurate, but as the measured frequency climbs toward resonance the response shoots up, and above resonance the mass can no longer keep up and the response rolls off. That is why the flat, trustworthy band always has an upper limit.
The accelerometer response factor comes from the seismic-instrument transfer function: response = 1 / sqrt((1-r^2)^2 + (2*zeta*r)^2). Here r = f/fn is the ratio of the measured frequency f to the sensor's natural frequency fn, and zeta is the damping ratio. In the flat band where r is much smaller than 1, response is approximately 1 and the indicated value matches the true acceleration. As r approaches 1 the response peaks, and for r > 1 it drops below 1. The measurement error is (response - 1)*100 [%]; a positive value means the sensor over-reads.
For a lightly-damped sensor with a damping ratio zeta around 0.05, the response factor response = 1/sqrt((1-r^2)^2+(2*zeta*r)^2) rises monotonically with the frequency ratio r, and |response-1| reaches 5% near r = 0.22. In other words, the '5% amplitude error' band only reaches about one-fifth of the natural frequency. That is why you must choose a sensor whose natural frequency is far above the highest frequency you intend to measure. Conversely, deliberately raising the damping ratio to zeta = 0.7 flattens the response over a wide band and extends the usable upper limit to nearly half the natural frequency.
A damping ratio of zeta = 0.7 (exactly 1/sqrt(2) = 0.707) makes the accelerometer's frequency response as flat as possible. A lightly-damped sensor's response shoots up just before resonance, but raising zeta toward 0.7 almost eliminates the resonance peak and keeps the response factor close to 1 up to much higher frequencies. As a result, the 5% usable upper limit widens from about one-fifth of the natural frequency to nearly half. Piezoelectric sensors have small internal damping and are lightly damped, but servo and MEMS designs adjust the damping electrically and mechanically to obtain a flat, wide-band response.
Real-World Applications
Machine vibration measurement and vibration testing: When measuring the vibration of motors, pumps and gearboxes, you must accurately capture frequency components up to several orders of the running speed. If, for example, the gear-mesh frequency of a 10000 rpm motor reaches several kHz, you must check that this frequency falls within the sensor's flat band. The basic approach is to choose a small piezoelectric sensor with a high natural frequency and select a model so that the highest frequency of interest is below one-fifth of the sensor's natural frequency.
Shock and drop testing: A product drop or an automotive crash test produces an acceleration pulse with very high frequency content over a very short time. Because a shock signal contains components from several kHz to tens of kHz, a sensor with a low natural frequency has its resonance excited and records a peak acceleration larger than the real one. Shock measurement uses dedicated sensors with high natural frequencies, removing resonant content with a mechanical filter or a low-pass filter as needed.
Modal analysis and structural health monitoring: In experimental modal analysis, where a bridge, building or aircraft structure is excited to obtain an FRF, a flat accelerometer frequency response is a prerequisite. If the sensor's resonance falls inside the measurement band, the structure's natural frequency and the sensor's resonance get confused and the extracted modal parameters are wrong. Before measuring, check the sensor's frequency-response chart and make sure the highest mode frequency of interest lies within the flat band.
Sensor calibration and model selection: An accelerometer datasheet always lists a "frequency response (±5%, ±10%)" and a "resonant frequency". The response factor and usable upper limit that this tool shows are exactly what those specifications mean, made visible. When planning a new measurement, use it to work backward from the frequency range and required accuracy to decide which natural frequency and damping ratio the sensor should have.
Common Misconceptions and Pitfalls
The biggest misconception is "an accelerometer reads correctly at any frequency once it is mounted". An accelerometer is itself a spring-mass system with a natural frequency. The closer the measured frequency comes to the natural frequency, the further the response factor strays from 1, and at resonance it shoots up beyond ten for a lightly-damped sensor. The "usable frequency range" on a datasheet marks exactly the band in which this error stays within tolerance. Measuring high-frequency vibration without reading the spec risks believing a value that resonance has wildly amplified.
Next, mistaking "you can use it up to the resonant frequency". The resonant frequency is the frequency at which the sensor produces its largest error — it is not the usable upper limit. A lightly-damped sensor holds a "5% amplitude error" only up to about one-fifth of its natural frequency. Even if the datasheet says "resonant frequency 30 kHz", the practical upper limit should be considered to be one-fifth of that, around 6 kHz. Confusing the resonant frequency with the usable upper limit overestimates the measurement band by a factor of five.
Finally, assuming "the damping ratio is fixed and cannot be changed". It is true that piezoelectric sensors have small internal damping and are lightly damped, but servo (force-balance) and MEMS designs let you engineer the damping electrically and mechanically. Bringing the damping ratio close to ζ ≈ 0.7 removes the resonance peak, flattens the response over a wide band and greatly extends the usable upper limit. On the other hand, too much damping introduces phase lag and reduced response at low frequencies, so there is an optimum point that depends on the application. The right mindset is not "just pick a sensor" but to design the whole measurement chain, damping characteristics included.
How to Use
Set the accelerometer's natural frequency (fn) in Hz using naturalFreqNum or naturalFreqRange slider—typical MEMS sensors range 1–50 kHz, piezoelectric accelerometers 10–100 kHz.
Adjust damping ratio (ζ) from 0.1 to 2.0; industrial accelerometers typically use ζ = 0.7 for flat response, higher values reduce resonance peak.
Input the measured vibration frequency (f_meas) and true acceleration amplitude; the simulator calculates frequency ratio r = f_meas/fn, response magnification factor, measurement error percentage, and safe operating bandwidth where error remains below 5%.
Worked Example
A 10 kHz piezoelectric accelerometer with ζ = 0.7 measures a bearing vibration at 2 kHz with stated amplitude 5 g. Frequency ratio r = 2/10 = 0.2, response factor ≈ 1.02 (well below resonance), measurement error ≈ 2%, usable upper frequency approximately 7 kHz (where 5% error threshold intersects). If the same sensor reads a 9.5 kHz fault signature with ζ = 0.3, resonance amplification reaches 1.67×, introducing 67% measurement error—unacceptable for condition monitoring.
Practical Notes
Select damping ζ ≥ 0.6 for machinery diagnostics; underdamped sensors (ζ < 0.4) show sharp resonance peaks that mask true acceleration and introduce phase lag errors exceeding 40°.
Usable bandwidth typically extends to fn/3; a 30 kHz accelerometer safely covers DC to ~10 kHz vibration analysis without amplification exceeding 1.15×.
Cross-check measurement verdict against ISO 20816 velocity thresholds; if simulator flags high error, verify sensor mounting (loose mounting reduces effective fn by up to 30%) or switch to lower natural frequency model.