Understanding Integration in Vibration Analysis
Integration in vibration analysis is the mathematical process of converting a vibration signal from one parameter to another — performing integration in the time domain, or, equivalently, dividing by frequency in the frequency domain. Most often it turns acceleration (the quantity an accelerometer actually senses) into velocity, or velocity into displacement. Because the three are linked through calculus (velocity = ∫ acceleration dt; displacement = ∫ velocity dt), integration lets an analyst express the same vibration in whichever parameter best suits the machine, the fault, and the frequency range — and it is the mathematical inverse of differentiation.
1. Definition: One Sensor, Three Parameters
Integration matters because no single parameter is best for everything. Acceleration emphasises high frequencies and excels at early bearing-defect detection; velocity is the balanced general-purpose metric used by the international machine-vibration standards; displacement emphasises low frequencies and suits slow machines and clearance work. Rather than carry three kinds of sensor, an engineer measures acceleration once and integrates to reach the other two. This is why a modern analyser can show a single measurement as acceleration, velocity, and displacement at the flick of a setting.
2. The Mathematical Relationships
Time-domain integration
- Velocity from acceleration: v(t) = ∫ a(t) dt
- Displacement from velocity: d(t) = ∫ v(t) dt
- Displacement from acceleration: d(t) = ∫∫ a(t) dt dt (double integration)
Frequency-domain integration
The operation is far simpler once the signal is in the spectrum, where each frequency line is just scaled:
- Velocity from acceleration: V(f) = A(f) / (2πf)
- Displacement from velocity: D(f) = V(f) / (2πf)
- Consequence: dividing by frequency amplifies low frequencies and suppresses high ones — the single most important fact to remember about integration.
Integration is a 1/f operation. It boosts the low-frequency end of the signal and attenuates the high-frequency end — which is exactly why a velocity spectrum looks “tilted” toward the low end compared with the acceleration spectrum it came from.
3. Why Integration Is Needed
Sensor economics
Accelerometers are the most versatile and most common vibration sensors, but acceleration is not always the most informative parameter. Integration lets one rugged accelerometer serve every parameter need, which is far more economical than fitting separate velocity and displacement sensors.
Parameter selection by frequency
- High frequency (above ~1000 Hz): acceleration is best — it highlights bearing impacts and gear-mesh energy.
- Mid frequency (10–1000 Hz): velocity is best, and is the parameter used for general machinery condition.
- Low frequency (below ~10 Hz): displacement is best, for slow machines and clearance assessment.
- Integration is what lets you move into the optimal parameter for whichever range a fault lives in.
Standard requirements
The dominant machine-vibration standard, ISO 20816 (which superseded ISO 10816), specifies RMS velocity. If you measure acceleration, you must integrate to velocity to compare against the limits; if you measure displacement with a proximity probe, it must also be converted before any velocity comparison is valid.
4. The Challenges of Integration
Integration is mathematically simple but practically treacherous, because the same 1/f behaviour that is useful also magnifies errors at the low-frequency end.
Low-frequency drift
This is the primary problem. Any DC offset or very-low-frequency component gets divided by a tiny number, producing an enormous error that makes the integrated signal “drift” off scale. The fix is a high-pass filter applied before integration, typically with a 2–10 Hz cutoff.
Noise amplification
Because integration is a 1/f operation, low-frequency noise is amplified more strongly than the signal of interest, degrading the signal-to-noise ratio. Filtering the noise out before integrating is the remedy.
Double integration compounds the problem
Going from acceleration all the way to displacement requires integrating twice, so any DC offset or low-frequency noise is amplified twice and the errors multiply. Aggressive high-pass filtering — often 10–20 Hz — is essential to keep the result usable.
5. Doing It Properly
Single integration (acceleration → velocity)
- Acquire the acceleration signal at an adequate sample rate.
- Remove DC offset.
- High-pass filter at 2–10 Hz to kill drift.
- Integrate (divide by 2πf in the frequency domain).
- Verify the result is sensible and free of drift.
Double integration (acceleration → displacement)
- Apply an aggressive high-pass filter — a higher cutoff (10–20 Hz) than for single integration.
- First integration: acceleration → velocity.
- Check the intermediate velocity result.
- Second integration: velocity → displacement.
- Final verification: confirm the displacement is physically reasonable.
6. Frequency Domain vs. Time Domain
There are two ways to implement integration, and modern instruments overwhelmingly favour the first.
- Frequency-domain integration (preferred): take the FFT, divide each line by 2πf, and inverse-transform. It is straightforward, introduces no cumulative error, makes filtering trivial, and is the standard method in modern analysers — giving a clean, accurate result.
- Time-domain integration: numerical integration by the trapezoidal or Simpson’s rule. It suffers cumulative error and drift and needs more careful filtering, so it is reserved for cases where a frequency-domain approach is not practical.
7. Practical Applications and Field Use
In day-to-day work, integration shows up whenever measurements from different sensors must be compared on equal terms: converting accelerometer data to velocity for an ISO 20816 check, or converting proximity-probe displacement to velocity so the two can sit on the same chart. On slow machines (below ~500 RPM) acceleration and velocity both become small, so analysts integrate to displacement to get a meaningful number, and multi-parameter analysis — viewing one signal as acceleration, velocity, and displacement — gives the most complete picture because each parameter emphasises a different part of the frequency range.
This is exactly how a portable instrument behaves on a real job. A two-channel analyser such as the ಬ್ಯಾಲೆನ್ಸೆಟ್-1ಎ samples acceleration at the bearing housings and integrates internally to display velocity for an ISO 20816 severity check or the 1× amplitude and phase needed for field balancing — the high-pass filtering and integration happening transparently so the engineer simply selects the parameter that fits the task.
8. Common Mistakes
- Integrating without filtering: guarantees drift and unusable displacement values — always high-pass filter first.
- Wrong cutoff frequency: set too low and drift returns; set too high and valid low-frequency content is stripped away. The cutoff is always a balance between drift prevention and signal preservation.
- Comparing mixed parameters: never compare an acceleration value directly with a velocity value — convert both to the same parameter first, because frequency content alone changes which parameter reads higher.
Integration is a fundamental signal-processing operation that ties acceleration, velocity, and displacement together into one coherent description of a machine. Used with proper high-pass filtering and a frequency-domain implementation, it underpins standards compliance, sensor economy, and the multi-parameter analysis that lets an engineer see a fault clearly in whichever parameter shows it best.
