Mathematical foundations of vibration modulation
In this article, which is the fourth in a series, This is the topic of Mathematical Foundations of Vibration Modulation, 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
4.1 Introduction
The previous section demonstrated that modulation is a direct consequence of the interaction between different mechanical phenomena present in a rotating machine.. A localized defect, a periodic variation in load or a change in stiffness does not just generate new vibrations; They also change the characteristics of existing vibrations, redistributing energy in the spectrum and generating sidebands.
Although the origin of modulation is physical, its interpretation requires a mathematical description. It is this description that allows us to understand why certain frequencies appear in the spectrum, predict the position of side bands and develop algorithms capable of automatically identifying defects in bearings, gears, electric motors and other mechanical components.
The mathematical formulation of modulation constitutes, like this, based on surround analysis, of cepstral analysis, cyclostationary analysis and many modern diagnostic techniques.
Figure 4.1 – Relationship between the physical phenomenon and the mathematical representation of modulation.
4.2 Representation of a harmonic signal
Consider initially a purely sinusoidal vibration produced by a perfectly balanced rotor.
The vibration can be described by:
at where:
- A is the amplitude;
- fc is the vibration frequency (carrier frequency);
- t represents time.
In the frequency domain, this signal corresponds to a single spectral line located at fc.
In practice, this situation is rare, since practically all machines present small load variations, velocity, stiffness or contact between surfaces.
Figure 4.2 – Relationship between a pure sinusoidal signal and its FFT spectrum.
4.3 Amplitude modulation (AM)
Suppose now that the amplitude of this vibration varies slowly due to the periodic passage of a defect.
The signal is now described by:
at where:
- m is the modulation index;
- fm is the modulating frequency.
Developing this expression we obtain:
This equation demonstrates that amplitude modulation generates three main components:
- fc carrier frequency;
- banda lateral inferior fc-fm;
- banda lateral superior fc+fm.
This result constitutes one of the fundamental relationships of vibration analysis.
Figure 4.3 – Mathematical formation of sidebands by amplitude modulation.
4.4 Modulation Index
The parameter m is called modulation index.
Physically represents the intensity with which the mechanical phenomenon changes the carrier vibration.
When:
- m=0, there is no modulation;
- 0<m<1, there is partial modulation;
- m approximately equal to 1, the modulation is intense.
As the defect evolves, the modulation index tends to increase, causing a progressive increase in the amplitude of the side bands.
For this reason, Many algorithms use the evolution of sidebands as an indicator of defect severity.
Figure 4.4 – Influence of the modulation index on the amplitude of the sidebands.
4.5 Modulation produced by impacts
Bearing defects produce practically impulsive impacts.
Mathematically, an impact can be approximated by a Dirac impulse function.
The structural response of the machine can be represented by:
at where:
- fr represents the resonance frequency;
- alpha represents structural damping.
Each impact generates a damped response.
The periodic repetition of these impacts naturally leads to the formation of a modulated signal.
This description constitutes the mathematical foundation of the envelope analysis.
Figure 4.5 – Damped response of the structure after an impact and formation of the envelope signal.
4.6 Frequency modulation (FM)
On certain machines, the defect mainly changes the instantaneous frequency of the vibration.
The signal can be represented by:
where β represents the frequency modulation index.
Unlike amplitude modulation, FM produces multiple sidebands distributed symmetrically around the carrier.
This behavior is particularly common in:
- gears;
- transmissions;
- systems subject to periodic variations in speed.
Figure 4.6 – Comparison between AM and FM modulation and respective spectrums.
Frequency modulation theoretically produces an infinite number of sidebands, located at f_c ± k·f_m with k = 1, 2, 3, …, whose amplitudes are given by Bessel functions of the first kind J_k(b), where β is the frequency modulation index. In practice only bands with k less than about β + 1 present significant amplitude, so the number of observable sidebands increases with the modulation index — unlike what happens in amplitude modulation, where this number depends exclusively on the harmonic content of the modulator.
4.7 Phase modulation (PM)
When the defect causes small periodic variations in the angular position, modulation occurs in the signal phase.
The mathematical expression is similar to that of FM:
where Qsi (t) represents the temporal variation of the phase.
In industrial practice, AM, FM and PM often coexist, making the spectrum significantly more complex.
In effect, FM and PM constitute two descriptions of the same phenomenon — the modulation of the cosine function argument — differing only in the relationship between the modulation index and the modulating frequency: in FM the index is inversely proportional to f_m, in the PM it is independent of it. From a single record, the two mechanisms are practically indistinguishable; their separation requires observation of the behavior of the side bands depending on the rotation speed.
4.8 Relationship between modulation and envelope analysis
Envelope analysis can be interpreted mathematically as a process composed of three steps:
- selection of a frequency band where modulation exists;
- signal demodulation (for example, through the Hilbert transform or rectification followed by filtering);
- envelope FFT calculation.
The result is a spectrum where the carrier frequency disappears and the characteristic frequencies of the defect remain..
This technique significantly increases sensitivity in early fault detection.
Figure 4.7 – Mathematical steps of the analysis by envelope: filtering, envelope demodulation and FFT.
4.9 Limitations of the mathematical model
The models presented assume ideal conditions, such as:
- constant rotation speed;
- structural linearity;
- absence of noise;
- a single carrier frequency.
Real industrial machines often have:
- multiple resonances;
- variable speeds;
- non-linear phenomena;
- several modulation mechanisms simultaneously.
Consequently, interpretation of spectra generally requires more advanced techniques, how order-tracked analysis (order analysis), cyclostationary analysis and methods based on artificial intelligence.
4.10 Mathematical Foundations of Vibration Modulation – Synthesis
The mathematical formulation demonstrates that sidebands are not artifacts of digital processing, but rather an inevitable consequence of the interaction between periodic phenomena present in the machine.
Understanding these relationships allows you to predict the position of the side bands, correctly interpret vibration spectra and select the most suitable processing technique for each application.
The following chapters will develop practical methods for extracting modulated information, covering in detail the analysis by environment, to Hilbert transform, synchronous demodulation and other advanced signal processing techniques applied to the diagnosis of rotating machines.
