From acceleration to vibration velocity without errors

In brief

• For a sine wave, the velocity amplitude is A/(2πf). This shortcut does not work for any arbitrary impulse by substituting one “dominant” frequency.
• A constant offset of 0.005 m/s² over 60 s creates a false 0.30 m/s if it is integrated directly.
• A low-frequency filter is not cosmetic. It can remove drift together with the structure's real response.
• PPV requires an explicit definition of direction, band, filtering, and maximum. Changing the label from m/s² to mm/s does not compute anything.
• The current Inclify dynamic analysis works on acceleration waveforms and does not have a general integrator to velocity or ready-made PPV. It shows the waveform, FFT, and RMS/MAX in third-octave bands.

It is easy to make velocity on a screen. You select an acceleration channel, add an integration block, and get mm/s. The plot is smooth, and the number has the right unit. That still does not mean it describes the structure's motion.

Every small constant error in acceleration grows linearly with time after integration. Slow temperature change, sensor tilt, a gravity component, transducer offset, or asymmetric saturation can turn into major velocity drift. A filter removes part of the problem, but it does so at the expense of information. The boundary between error and real motion does not define itself.

That is why the question “does the system calculate velocity?” is too weak in an RFQ. You need to ask what it calculates from, how, in which band, with what uncertainty, and whether the result can be reproduced.

Three quantities describe the same motion, but they weight frequency differently

Displacement x(t), velocity v(t), and acceleration a(t) are linked by derivatives:

v(t) = dx(t)/dt

a(t) = dv(t)/dt = d²x(t)/dt²

In the other direction, velocity is the integral of acceleration:

v(t) = v(t₀) + ∫[t₀,t] a(τ) dτ

The initial condition v(t₀) appears. For vibration, the average velocity over a suitably chosen window is often assumed to be zero, but that assumption must fit the phenomenon. It must not be applied carelessly to motion that actually contains a low-frequency component or permanent displacement.

For a pure sine wave:

a(t) = A sin(2πft)

the velocity amplitude is:

V = A/(2πf)

the displacement amplitude is:

X = A/(2πf)²

For the same acceleration, the lower the frequency, the larger the velocity and displacement. An accelerometer naturally emphasizes higher frequencies, and integration increases the importance of the low end of the band. That difference explains why the same acceleration value can lead to a different assessment after conversion.

The metric comes from the assessment document and the purpose. Before you design integration, separate the questions answered by PPV, RMS, FFT and third-octave bands. In machine analysis, ISO 20816-1 allows different quantities and methods, provided they are defined and their limitations are described. For structures, ISO 4866 ties measurement to frequency, duration, and response amplitude. Do not transfer a machine procedure to a building just because both display mm/s.

Numeric example: the same acceleration amplitude, different velocity

Assume an acceleration amplitude of A = 0.20 m/s².

For a 10 Hz sine wave:

V = 0.20/(2π × 10) = 0.00318 m/s ≈ 3.18 mm/s

For a 2 Hz sine wave:

V = 0.20/(2π × 2) = 0.0159 m/s ≈ 15.9 mm/s

The acceleration is the same, yet the velocity amplitude at 2 Hz is five times higher. The conclusion only makes sense for pure sine waves. A short impulse contains many frequencies and phases. You cannot take its peak acceleration, divide it by 2π and one frequency from the FFT, and then call the result PPV.

For a multi-frequency signal, you integrate the full calibrated waveform or each component in the frequency domain with correct handling of phase, zero, and record edges. Then you calculate the maximum according to the required definition.

Where drift comes from

Assume a constant acceleration offset of b = 0.005 m/s². This is a small number compared with a short impulse. After simple integration over 60 s, it gives:

Δv = b × t = 0.005 × 60 = 0.30 m/s = 300 mm/s

The resulting value does not describe vibration. It is a mathematical consequence of a constant error. In a real signal chain, the offset may come from electronics, calibration, temperature change, or a gravity component.

A DC-capable accelerometer also records gravity. If the axis rotates by a small angle, the projection of g on that axis changes. A small-angle approximation shows the scale of the problem: a rotation of 0.1° gives a change on the order of g × sin(0.1°), which is about 0.017 m/s². That is more than three times the offset in our example. The exact impact depends on the sensor type, orientation, and the method used to remove gravity.

Sources of drift also include:

• asymmetry between positive and negative samples;
• initial condition error;
• a gap or duplicate in time;
• a non-zero mean caused by a clipped record;
• saturation of one polarity of the impulse;
• very low-frequency noise;
• an offset step after overload;
• motion of the base or cable.

Removing the mean from the full record helps only in some cases. If the offset changes over time or the record contains an asymmetric impulse, one subtraction does not repair the waveform. Polynomial detrending can, in turn, remove the true low-frequency response. Each choice is part of the measurement model.

A high-pass filter solves the problem and creates a new one

Dewesoft documentation for dynamic waveform integration states that offset increases drift, so a high-pass filter is used before or within integration. The cutoff frequency defines the compromise. A higher cutoff stabilizes the result faster, but removes more of the real low-frequency content. A lower cutoff preserves more signal, but requires a longer settling time and a sensor capable of measuring that range.

A filter has a type, order, cutoff frequency, and amplitude and phase response. The label “HP 1 Hz” is not enough. Near the cutoff frequency, the amplitude is already changed, and a causal filter shifts phase. Forward and reverse filtering can remove phase shift offline, but it uses future data and behaves differently at the edges. It must not be presented as the same thing as an online alarm.

In the frequency domain, integration is equivalent to division by j2πf:

V(f) = A(f)/(j2πf)

For f = 0, the expression is undefined. For very small f, even small noise is strongly amplified. You need to state explicitly what happens to the DC component and the lower band limit.

Zeroing the DC bin does not solve leakage from a slow trend. A short record and discontinuity at its ends spread energy into neighboring frequencies. A window function reduces leakage, but it also changes amplitude. The parameters must be stored together with the result.

A calculation procedure that can be accepted

1. Define the result

Decide whether you need a velocity waveform, RMS, the maximum of one axis, the largest component, the vector sum, or PPV according to a specific method. Record the band and the measurement point. Without that, there is no way to judge the filter.

2. Confirm the input unit

A raw number from a converter may mean volts, g, m/s², or a device unit. Apply the calibration factor, sensitivity, and orientation before integration. If the input is in g, convert it to m/s² first, and only then choose the result scale in mm/s. The card from raw value to engineering unit helps you record the coefficients, version, and responsibility for this stage.

3. Check time and completeness

Integration assumes known intervals between samples. A gap, duplicate, or jitter changes the area under the curve. Do not interpolate a missing peak without marking it. An incomplete record should not automatically generate a final PPV. Time selection, trigger, and buffer selection must be accepted according to the rules described in the guide on sampling and pre-trigger.

4. Assess saturation and offset

Check flattened peaks, mean shift, behavior before the event, and return to background. Pre-trigger provides material for assessing offset and setting filters. Without it, the algorithm may impose an incorrect initial condition.

5. Set the filtering

Record the chain's anti-aliasing filter, the integration high-pass filter, any low-pass filter, their type, order, and frequencies. Justify the lower band on the basis of the phenomenon and the assessment document, not on the basis of where the plot stops drifting.

6. Integrate numerically

The trapezoidal method, a digital integrator, or a frequency-domain operation can give similar results within the band, but they differ at the edges and for slow components. Name the implementation, version, and precision. Do not treat it as a neutral programming detail.

7. Verify with reference signals

Use sine waves at several frequencies, a multi-tone signal, and an impulse. Compare amplitude and phase with an independent chain or calculation. Add a controlled offset to check whether the procedure produces artificial velocity.

8. Keep the data needed for reproduction

Archive the acceleration waveform, time, calibration, orientation, filters, algorithm, result, and completeness status. ISO 21289 indicates the parameters that should be specified for acquisition, analysis, reporting, and archiving of vibration data. This is a good requirements framework, even when a different document defines the detailed metric.

Filter before or after integration

The order of operations matters for a finite signal and for implementation. Ideal linear operations on an infinite, correctly represented signal can be described in a simple frequency-domain way. A real record has a start, an end, offset, noise, and saturation.

Filtering acceleration before integration limits the low end of the band before division by frequency amplifies noise. An integrator with a built-in filter performs both actions in a single chain. Filtering the final velocity can produce a different initial state and edge transition. Do not choose the order based on which plot looks calmer.

The same caution applies to third-octave bands. RMS acceleration in a band, integrated through a simple coefficient for the center frequency, is not exactly the same as the RMS of a velocity waveform after filtering. A band contains a frequency range, and the amplitude does not have to lie at the center. If the criterion requires band-limited velocity, calculate it according to that method.

Three purchasing strategies

A direct transducer does not remove all problems. It has its own low and high band, damping, transverse sensitivity, and mounting requirements. An accelerometer is not automatically better, but it gives flexibility if you preserve the full signal. An edge device can be the best choice in the field, provided its calculation is transparent and qualified.

Questions for the supplier before purchase

• What physical quantity is measured directly?
• Is the full input waveform retained and exportable?
• What sensor, band, range, noise, and phase response does the chain have?
• How are gravity, offset, and the post-overload transient removed?
• What filter acts before sampling, and what filter acts before integration?
• What is the type, order, and cutoff of each filter?
• How is the velocity initial condition calculated?
• Does the algorithm run online, offline, causally, or bidirectionally?
• How does the system handle gaps and an incomplete record?
• How does it define PPV for multiple axes, and which band does it use?
• What signals and tolerances are covered by FAT and SAT?
• Can the history be recalculated after an algorithm change without overwriting the evidence?
• Does the result include the profile version and configuration?

An answer of “according to the standard” without the number, edition, and parameters does not satisfy any of these questions. The required signals, tolerances, and evidence must be entered into FAT and SAT for the vibration monitoring system.

How Inclify supports this and where the boundary is

In the current dynamic implementation of Inclify, the full waveform is treated as acceleration. The platform displays it in time, calculates FFT, and evaluates 21 third-octave bands from 1 to 100 Hz. The profile with 4th-order Butterworth filters calculates RMS and MAX acceleration in bands. Calibration is set separately for each axis, and the profile used for the event remains stored with the result.

Inclify does not have a general acceleration-to-velocity integrator and does not expose ready-made PPV. You should therefore not describe an acceleration plot as velocity or compare acceleration third-octave bands with a threshold expressed in mm/s. If the project requires PPV, the calculation must be produced in a qualified device chain or outside the platform's current analysis, with explicit filters and a validation procedure.

Preserving the full record, time, axis coefficients, completeness, and analysis profile makes independent recalculation easier. A specific dynamic signal can be exported to XLSX. That is audit material, not an automatic confirmation of a velocity metric.

Inclify's dynamic alarm operates on available RMS/MAX results in third-octave bands against a reference envelope. It does not turn them into PPV. That is the current boundary of vibration analysis in the platform; velocity requirements must be added separately to the integration project.

Limitations that must not be smoothed over

You cannot recover true velocity from acceleration after sensor saturation. A missing fragment may also contain the maximum. A filter or AI will not reconstruct information that the chain did not record.

There is also no single “good” cutoff. For a machine, 10 Hz may be reasonable, while for a building it may remove a significant part of the response. Dewesoft documentation gives typical ranges as application examples, not universal settings for structural standards.

Velocity calculated by two software tools may differ even from the same file if they remove the mean, filter, handle edges, and scale the FFT differently. For that reason, a result without configuration is not suitable for supplier comparison.

Sources and further reading

• ISO 4866:2010, Vibration of fixed structures - principles for measuring and processing structural vibration with respect to frequency, duration, and response amplitude.
• ISO 21289:2008, parameters for acquisition of vibration data - parameters for specifying measurement, analysis, reporting, and archiving of vibration waveforms.
• ISO 20816-1:2016, measurement and evaluation of machine vibration - an example framework in which the measured quantity and method limitations must be defined.
• Dewesoft X Manual, Integration and Double Integration - the influence of offset, high-pass filtering, filter order, and unit scale.
• PCB Piezotronics, Accelerometer Selection Based on Applications - sensor selection and the influence of a small offset on the post-integration error.
• NIST Technical Note 334, sampled data and switching logic - sampling, aliasing, and limits of signal reconstruction.

Decision before purchase

Ask the supplier for one acceleration file and the velocity result together with filters, version code, and a reference calculation. Then change the lower cutoff and check how much PPV shifts. If it cannot be reproduced, you are buying a number, not a measurement method. Talk to the Inclify team if you want to assess which data from an existing chain can be retained in the platform and what is missing for a qualified velocity result.