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Technology / Technical guide
Vibration Envelopes: Modulation, Sidebands & Interpretation
An envelope describes slower amplitude variation around a faster oscillation under an appropriate model. It differs from the carrier waveform and overall RMS. A visible envelope does not uniquely identify a bearing condition: different signals and mechanisms can produce similar amplitude patterns.
A carrier and its envelope occupy different time scales.
The interpretation depends on spectral separation, the sign of the modulation and the mathematical definition used for the envelope.
01 / Two time scales
Amplitude modulation multiplies a faster waveform.
Use the illustrative expression x(t) = A0[1 + m cos(ωm t)] cos(ωc t), where A0 has signal units, m is dimensionless and the modulation frequency is much lower than the carrier frequency.
Use the illustrative expression x(t) = A0[1 + m cos(ωm t)] cos(ωc t), where A0 has signal units, m is dimensionless and the modulation frequency is much lower than the carrier frequency. Choose m between zero and one for a nonnegative varying amplitude.
The bracketed term describes the slowly varying amplitude factor. The cosine at ωc supplies the rapid oscillation. This separation is a model assumption, rather than an explanation automatically applicable to every vibration waveform with changing peaks.
02 / Sidebands
Multiplication creates components on either side of the carrier.
The product-to-sum identity expands the model into a carrier at ωc and two sidebands at ωc − ωm and ωc + ωm.
The product-to-sum identity expands the model into a carrier at ωc and two sidebands at ωc − ωm and ωc + ωm. Each sideband has amplitude A0m/2. The amplitude pattern and the spectral components are two descriptions of the same constructed signal.
For hypothetical A0 = 2 units, m = 0.25, fc = 20 Hz and fm = 1 Hz, the carrier amplitude is 2 units. Sidebands at 19 and 21 Hz each have amplitude 0.25 unit. These frequencies represent an invented mathematical example, not a component fault frequency.
03 / Envelope magnitude
An analytic signal defines a particular envelope.
MathWorks describes an envelope based on the magnitude of the analytic signal.
MathWorks describes an envelope based on the magnitude of the analytic signal. Write z(t) = x(t) + jH{x(t)}, with H denoting the Hilbert transform. Its magnitude |z| is a mathematically defined quantity, while its physical interpretation depends on the represented signal.
For the separated, nonnegative modulation model, that magnitude follows A0[1 + m cos(ωm t)]. The toy envelope therefore ranges from 1.5 to 2.5 units. The result relies on the carrier–modulation separation; analytic magnitude is not a universal recovery of an unknown physical modulator.
04 / Sign and mixture
An amplitude-looking curve can hide phase or beating.
If the signed modulation factor crosses zero, its magnitude no longer preserves the sign.
If the signed modulation factor crosses zero, its magnitude no longer preserves the sign. A phase reversal can accompany that crossing. Treating absolute amplitude as the original signed factor would erase information even when the mathematical envelope remains defined.
Two close tones can also produce beating without a single physical source undergoing the assumed amplitude modulation. A similar envelope pattern can therefore arise from different constructions. The pattern alone does not distinguish a varying force from several superposed frequency components.
05 / Other measures
Peak, RMS and analytic envelopes answer different questions.
A moving RMS describes mean-square content within a selected interval; a curve joining local peaks describes a different feature.
A moving RMS describes mean-square content within a selected interval; a curve joining local peaks describes a different feature. The analytic magnitude uses a quadrature representation. Their values need not coincide for an arbitrary waveform or a changing amplitude.
Window duration, component content and the definition of “envelope” affect interpretation. Reporting only an envelope amplitude omits the transformation that produced it. Likewise, an overall RMS can hide the timing of amplitude variation even when it represents the same raw signal.
06 / Scope
A modulation feature is evidence about a signal model first.
Interpreting sidebands or envelopes as a physical condition requires a relationship between the feature and the mechanism.
Interpreting sidebands or envelopes as a physical condition requires a relationship between the feature and the mechanism. Multiple sources, changing speed and measurement effects can alter that relationship. A signal pattern by itself does not provide a unique diagnosis or severity classification.
No MARSTON envelope algorithm, bearing diagnosis or monitoring feature is asserted. This explanation supplies no band-selection, test or fault-identification procedure. It separates carrier mathematics, amplitude representation and physical cause without assigning an actual equipment result.
Related technical guides: Vibration Displacement, Velocity & Acceleration · Notch Filters: Selective Attenuation & Phase Consequences · Model Residuals: Prediction Gaps & Ambiguous Causes.
General background: MathWorks: analytic, RMS and peak envelope concepts. These references explain industry concepts; they are separate from the applicable MARSTON product and project documents.
Key concepts and conditions.
Read each quantity with its reference, units and stated conditions. A general relation does not establish a model-specific value.
| Concept / quantity | Meaning / relation | Conditions / limits |
|---|---|---|
| Carrier fc | Faster oscillation in hertz | Specified component and waveform model |
| Modulation fm | Slower amplitude variation | Suitable separation from the carrier |
| Index m | Dimensionless modulation depth | The example keeps the amplitude factor nonnegative |
| Analytic magnitude | |x + jH{x}| in signal units | Physical envelope meaning depends on component separation |
| Sideband amplitude | A0m/2 in the toy model | Product form and frequency convention stated |
Engineering questions.
Is an envelope the same as overall RMS?
No. They describe different features of the represented signal.
Where do the toy sidebands occur?
At the carrier frequency minus and plus the modulation frequency.
Why keep the example modulation nonnegative?
It avoids losing the sign of the amplitude factor in its magnitude.
Does analytic magnitude identify a physical modulator universally?
No. That interpretation needs suitable signal separation and model conditions.
Can two tones produce an envelope-like beat?
Yes. Similar amplitude patterns can have different constructions.
Does a visible envelope prove bearing damage?
No. A pattern alone does not establish a unique mechanism.
Technical enquiry
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An envelope describes slower amplitude variation around a faster oscillation under an appropriate model. Share the model and applicable operating or design information for an engineering discussion.