Modulation concept
In this article, which is the third in a series, This is the topic of the Concept of Modulation and its Physical Origin, which plays a fundamental role in diagnosis by vibration analysis in machines.
Articles on this topic already published were:
1 Introduction to vibration and envelope modulation
2 History of modulation in vibration analysis
3 Concept of modulation and its physical origin
3.1 Introduction
Modulation is one of the most important phenomena in the analysis of vibrations of rotating machines.. A palavra derives from a broad modulation, which means change or controlled variation of a certain quantity; in engineering, the term describes the process through which a characteristic of a signal is modified by the influence of another signal.
Although the concept was initially developed in the field of telecommunications, where modulation is used intentionally to transport information, It turns out that many mechanical systems present exactly the same physical behavior: A periodic vibration of high frequency has its amplitude, frequency or phase altered by another mechanical phenomenon of lower frequency. In rotating machines, although, modulation is not an intentional process, but rather a consequence of the dynamic interaction between components subject to variable loads, periodic impacts, elastic deformations and geometric imperfections.
This change produces a signal whose information is no longer concentrated solely on the original frequency, but distributed by new components called side bands (banda lateral). These bands carry extremely valuable information about the mechanical state of the machine and are often the first indication of a developing defect..
Figure 3.1 – Sinal sinusoidal pure vs. amplitude modulated signal
It is important to clarify a point that is often misunderstood. Modulation is not a passive process of redistributing carrier energy. For an amplitude modulated signal,
| s(t) = A · [1 + m·cos(2p·f_m·t)] · cos(2p·f_c·t) | (3.1) |
the total power is (A²/2)·(1 + m²/2), that is, grows with the modulation index, while the power associated with the carrier component remains at A²/2. The energy contained in the sidebands is additional energy, provided by the modulating mechanism itself: the impact of the rolling element on the defect, the cyclic variation of the gear force, the periodic fluctuation of electromagnetic force.
What modulation does, and in this lies its true interest for the diagnosis, is to locate this energy. Instead of remaining dispersed throughout the analysis range in the form of low-energy, broadband pulses, virtually indistinguishable from background noise, the energy associated with the defect is concentrated in a narrow region of the spectrum, around the carrier frequency, where the signal/noise ratio is favorable and where demodulation techniques can recover it.
Figure 3.2 – Physical origin of modulation in rotating machines.
3.2 Carrier and modulating signal
To understand modulation it is useful to distinguish two fundamental signals:
- Carrier signal (carrier): relatively high frequency vibration, normally associated with a structural resonance, the gearing frequency of a transmission or other dominant machine component.
Modulator signal (modulating signal): periodic phenomenon of lower frequency that changes the characteristics of the carrier. This sign is often associated with rotational speed, the characteristic frequency of a bearing or the frequency of passage of the teeth of a gear.
Figure 3.3 – Carrier and modulating signal
Mathematically, modulation results from the combination of these two signals. Physically, corresponds to the fact that a low-frequency mechanical phenomenon influences the higher-frequency vibrational response.
The modulating phenomenon
The modulating signal corresponds to the mechanical phenomenon that periodically changes the existing vibration.
In bearings, each passage of a defect through the loaded zone produces an impact.
In the gears, a cracked or worn tooth causes a periodic variation in the load transmitted.
In electrical machines, electromagnetic forces vary cyclically depending on the electrical supply and rotor slip.
All these phenomena produce a modulation of the carrier vibration.
Figure 3.4 – Excitation of a structural resonance by repetitive impacts on a bearing.
Why side bands appear?
Suppose a high frequency structural vibration whose amplitude varies slowly due to a periodic defect.
In the temporal domain, a fast wave is observed whose amplitude increases and decreases cyclically.
In the frequency domain, instead of there being just a peak in the carrier frequency, two new components appear:
- banda lateral inferior;
- banda lateral superior.
These components appear in:
| fc-fm fc fc+fm | (3.2) |
at where:
- fc is the carrier frequency;
- fm is the modulating frequency.
The greater the modulation intensity, the greater the amplitude of the side bands will be.
Figure 3.5 – Relationship between the modulated signal and the appearance of sidebands.
3.3 How a mechanical modulation is born
Physically, modulation arises whenever a periodic phenomenon alters the dynamic properties of another vibratory phenomenon.
There are three fundamental mechanisms:
- periodic variation of the exciting force;
- periodic variation of stiffness;
- periodic variation of damping.
They all cause periodic changes in the machine's dynamic response.
Consider a perfectly manufactured gear. The gearing frequency remains practically constant and the spectrum presents only one dominant peak..
If localized wear appears on a tooth, each passage of this tooth produces a slight increase in the force transmitted. The gear frequency continues to exist, but its amplitude now starts to oscillate periodically.
This oscillation constitutes precisely an amplitude modulation.
Figure 3.6 – Formation of side bands by amplitude modulation. Comparison between an unmodulated signal and a modulated signal.
3.4 Physical example: outer race defect
Consider a bearing with a small defect in the outer race.
Whenever a rolling element crosses this imperfection, an impact occurs.. This impact excites a natural frequency of the machine structure, for example, a resonance located in the 5 kHz.
If the characteristic frequency of the defect (BPFO) It is therefore important to choose a suitable test mass to obtain a good result. 104 Hz, the amplitude of the vibration of 5 kHz stops being constant and starts to vary 104 times per second.
The result is not just a vibration 5 kHz, but a vibration whose amplitude periodically oscillates at the frequency BPFO.
In the frequency domain,:
- the carrier frequency (5 kHz);
- a lower side band (5000 − 104 = 4896 Hz);
- a superior side band (5000 + 104 = 5104 Hz).
It is important to distinguish two effects that are often confused.
Figure 3.7 – Formation of lateral bands (BPFO example)
The same can be seen in the figure below..
Figure 3.8– Complete example of modulation in a bearing with a defect on the outer race.
3.5 Fundamental types of modulation
In rotating machines there are three main mechanisms.
Amplitude modulation (AM)
The amplitude of the vibration varies periodically while the frequency remains practically constant.
It is the most common form of modulation in:
- bearings;
- gears;
- straps;
- impacts;
- mechanical clearances.
It is also the basis of analysis by environment.
Frequency modulation (FM)
In this case the amplitude remains approximately constant, but the instantaneous frequency fluctuates over time.
It is often observed in:
- eccentric gears;
- variable speed motors;
- systems subject to periodic speed oscillations.
Frequency modulation produces multiple sidebands whose distribution depends on the modulation index.
Phase modulation (PM)
In phase modulation, the instantaneous phase of vibration undergoes periodic variations.
Although less evident in conventional FFT analyses,, the PM assumes importance in:
- synchronous analysis by orders (Analysis by orders);
- servo drives;
- turbomachines;
- high precision systems.
Figure 3.9 – Comparison between amplitude modulation (AM), frequency modulation (FM) and phase modulation (PM).
The table below presents the same comparison.
Table 3.1 Comparison between amplitude modulation (AM), frequency modulation (FM) and phase modulation (PM).
| 1. Amplitude Modulation (AM): | 2. Frequency Modulation (FM): | 3. Phase Modulation (PM): |
| Example: Eccentric gear. Gearing is stronger at certain points in the rotation, modulating to amplitude (Note: gears can also generate FM). | Example: Localized sliding bearing. Sliding causes instantaneous variations in the relative speed of elements, changing the vibration frequency. | Example: misalignment, or periodic variation of the gear angular position. |
| Weather: The amplitude of the high frequency signal (the gear) varies sinusoidally with rotation frequency ( ). The surrounding (dashed red line) it is clear. | Weather: The amplitude is constant, but the signal frequency varies. Notice how the oscillations are “stretched” e “pills” over time. | Weather: the amplitude and average frequency remain, but zero crossings and peaks periodically appear early or late. |
| Spectrum: Features a strong central carrier ( ) and exactly two symmetrical sidebands at e . Note that pure AM produces exactly two symmetrical sidebands — this is its distinguishing feature. | Spectrum: Presents a carrier ( ) and multiple side bands on . The amplitude of these sidebands depends on the modulation index, and the spectrum is more complex than that of AM — the presence of several side bands can be noted, not just two. | Spectrum: carrier accompanied by multiple sidebands of symmetrical amplitudes, given by first-species Bessel functions. It is spectrally very similar to FM, which is only distinguished by the way in which the modulation index depends on the modulating frequency. |
Note on sideband asymmetry: I do not produce FM or PM, alone, asymmetrical side bands: in both cases, the amplitudes follow Bessel functions of the first kind and are distributed symmetrically around the carrier. The asymmetry effectively observed in the spectrums of gears and reducers results from the coexistence of amplitude modulation with frequency or phase modulation, with a certain phase relationship between both: on one side of the carrier the contributions are added, from the other they are subtracted. The presence of clearly asymmetrical lateral bands constitutes, therefore, an indicator of the simultaneous action of the two mechanisms, and not a signature of phase modulation.
3.6 Representation in time and frequency
In the time domain, an unmodulated signal has an approximately constant envelope.
When amplitude modulation occurs, an envelope is observed that increases and decreases periodically. This variation corresponds precisely to the modulating signal.
For example, A high-frequency vibration generated by a defective bearing may present an envelope that oscillates at the frequency of the rolling elements.
This characteristic is the physical basis of the Analysis of Engaging, developed to recover the information contained in the signal’s surroundings.
Frequency domain representation
One of the most intuitive ways to identify modulation is to observe the espectro FFT.
A pure component produces a single spectral peak.
When this component is modulated, new components appear distributed symmetrically around the main frequency..
These components are called side bands.
The distance between consecutive sidebands exactly corresponds to the frequency of the modulating phenomenon.
Like this, the spacing between sidebands has a direct physical meaning and can be used to identify the origin of the modulation.
3.7 Physical interpretation of sidebands
Sidebands do not constitute independent frequencies produced by the machine. They represent the energy that the modulating phenomenon introduces into the signal, distributed symmetrically around the carrier frequency.
It is important to distinguish two effects that are often confused.
The intensity of modulation, translated by the modulation index m, determines the amplitude of the sidebands. In amplitude modulation with a sinusoidal modulator, each sideband has amplitude mA/2 and grows, therefore, proportionally to m. The amplitude of the carrier component remains equal to A and is not affected by modulation.
The number of side bands, to be, it doesn't depend on me: depends on the harmonic content of the modulating signal. A purely sinusoidal modulator produces exactly two sidebands, one lower and one upper. A modulator with harmonics — like the train of periodic pulses generated by a bearing defect, or the non-sinusoidal variation of the coupling force in a transmission — produces a pair of sidebands for each harmonic of the modulator, giving rise to the families of equally spaced stripes that are observed in industrial practice.
It is for this reason that, as a defect evolves, two distinct phenomena with the same cause are observed simultaneously: the existing side bands gain width, because the modulation index increases; and higher order side bands appear, because the impacts become clearer and their harmonic content richer.
This behavior allows the relative amplitude of the side bands to be used as an indicator of the severity of the defect..
3.8 Physical origin of the envelope signal
When an impact occurs in a bearing, this has an extremely short duration and contains energy distributed over a wide range of frequencies.
Much of this energy excites a structural resonance of the machine.
The observed response then ceases to be an isolated impulse, becoming a damped high-frequency oscillation.
As impacts recur periodically, the amplitude of this oscillation varies depending on the frequency of repetition of impacts.
This envelope corresponds to the envelope signal.
When applying a demodulation technique, the high resonance frequency is eliminated and only the surrounding is preserved, allowing to directly identify the characteristic frequencies of the defect.
Figure 3.10 – Formation of the envelope signal.
3.9 Importance for diagnosis
A mechanical defect in the initial phase usually generates very small impacts.
These impacts can remain completely hidden in the background noise of the machine.
However, when they excite a structural resonance, start to modulate a high amplitude vibration.
The information associated with the defect thus becomes much easier to extract through demodulation techniques..
This principle explains why envelope analysis can often identify defects in bearings many months before a significant increase in the overall level of vibration occurs..
The impacts produced by defects normally have reduced energy.
In conventional FFT this energy is distributed across thousands of spectral lines, can be completely masked by noise.
Demodulation changes this situation. By removing the carrier frequency, Practically all the energy related to the defect starts to focus on the frequency of repetition of impacts, significantly increasing the signal-to-noise ratio.
Figure 3.11 – Comparison between conventional FFT and envelope FFT. On the left an FFT spectrum is shown where the defect is barely visible. On the right is the corresponding envelope spectrum, clearly highlighting the characteristic frequencies of the bearing.
Most mechanical defects begin in the form of small local changes.
These changes produce little additional energy, but they profoundly modify the way in which vibrational energy is distributed across the spectrum.
Consequently, The most relevant information is often found in the sidebands and surrounding signal, and not in the simple increase in the global level of vibration.
Understanding the physical origin of modulation constitutes, therefore, an essential skill for any diagnostic engineer, allowing to correctly interpret the observed phenomena and select the most appropriate analysis technique for each type of machine and failure mechanism.
3.10 Synthesis
Modulation constitutes a physical mechanism through which a periodic phenomenon changes the amplitude, the frequency or phase of an existing vibration.
In rotating machines, This phenomenon results from the interaction between mechanical components subject to impacts, variable loads or geometric imperfections.
The presence of side bands in the spectrum, as well as information retrieved through environment analysis, provides one of the most important diagnostic signatures of condition monitoring. Understanding these principles forms the basis for the following chapters, where mathematical modulation models and demodulation techniques applied to bearing diagnosis will be developed, gears and other rotating machines.
Figure 3.12 – Synthesis: relationship between mechanical defect, modulation and diagnosis
