# Frequency modulation of industrial gears

Frequency modulation (also known as phase modulation) is the frequency conversion signal formed when the carrier signal is modulated by the modulation signal.

Some factors, such as gear speed fluctuation, pitch error caused by indexing error in machining and so on, will cause the change of meshing speed and produce the phenomenon of frequency modulation. The load fluctuation of gear surface caused by gear fault will not only produce amplitude modulation, but also cause torque fluctuation, resulting in angular velocity change and frequency modulation.

If the carrier signal is asin (2 π FMT + φ) and the modulation signal is β sin (2 π FRT + φ), the frequency modulation can be expressed as X (T) = asin [2 π FMT + β sin (2 π FRT) + φ]

Where, FM ~ carrier frequency (meshing frequency); fr ~ modulation frequency (rotational speed frequency of gear);

β – frequency modulation index, i.e. the maximum angular displacement (phase shift) generated by modulation, β = △ FM / FR;

△ FM ~ maximum frequency deviation value.

With the help of Bessel function, the above formula can be expanded into infinite series in the form of:

x(t)=(A/2){J0(β)sin[2πfmt+φ]

+J1(β)sin[2π(fm－fr )t+φ]+J1(β)sin[2π(fm+fr )t+φ]

+J2(β)sin[2π(fm－2fr )t+φ]+J2(β)sin[2π(fm+2fr )t+φ]+… A kind of

Where, jn (β) – is the nth order Bessel coefficient of independent variable, n = 0,1,2,3.

According to the above formula, there are infinite frequency components in the frequency modulation vibration signal, among which the first term J0 (β) is carrier component, the second term J1 (β) has the first-order upper and lower sideband components, the third term J2 (β) has the second-order upper and lower sideband components. They take carrier frequency FM as the center and modulation frequency fr as the interval to form an infinite pair of sidebands with symmetrical distribution.

There are three common points in amplitude modulation and frequency modulation of gear vibration signal:

① Carrier frequency is equal (meshing frequency);

② The frequency interval of sideband is equal (the rotation frequency of gear);

③ The sideband is symmetrically distributed on both sides of the carrier frequency.

The difference between amplitude modulation and frequency modulation is that the envelope of the signal remains unchanged after frequency modulation, that is, the total energy remains unchanged after modulation, which is equivalent to dispersing the energy on the carrier frequency to the side frequency.

It must be noted that the gear modulation sidebands on the actual spectrum are not symmetrically distributed, as shown in the right figure. This is because the phenomenon of AM and FM always exist at the same time. The side frequency components in the actual spectrum are the superposition of the side frequency components generated when the two kinds of modulation act on each other. Because the phase of the side frequency components is not the same, after the vector superposition, the amplitude of some side frequencies increases, some of them decrease, so the original symmetry is destroyed. 