# The frequency modulation of gears

In addition to the influence on the vibration amplitude, the load fluctuation caused by gear load nonuniformity, gear pitch nonuniformity and fault will inevitably produce torque fluctuation, which will cause gear speed fluctuation. This kind of fluctuation is frequency modulation (or phase modulation) in vibration. For gear drive, any factor that causes amplitude modulation will also cause frequency modulation. Two kinds of modulation always exist at the same time. For the gear pair with small mass, the frequency modulation is particularly prominent.

Even if the carrier signal and modulation signal are single frequency components, many side frequency components will be formed. If the carrier signal is asin (2 π FCT + φ) and the modulation signal is β sin (2 π FZT), the frequency modulated signal is

f(t)=Asin［2πfct+βsin(2πfZt)+φ］ （1-11）

Where a – amplitude;

FC – carrier vibration rate;

FZ — modulation frequency;

β – modulation index, equal to the maximum phase shift generated by modulation;

φ – initial phase angle.

The above formula can be expanded by Besser function to obtain the characteristics of FM signal: the frequency modulated vibration signal contains infinite frequency components, and forms infinite pairs of modulation sidebands with the meshing frequency fc as the center and the modulation frequency FZ as the interval (Fig. 7).

Figure 7 frequency modulation and its sideband

Phase modulation has the same effect as frequency modulation. In fact, all phase modulation can also be regarded as frequency modulation, and vice versa.

For the gear vibration signal, the main reason of frequency modulation is the phase change of gear mesh stiffness function due to the influence of gear machining error and fault, which will be periodic due to the rotation of gear. Therefore, in the gear signal frequency modulation, the carrier function and the modulation function are both general periodic functions, which contain the basic frequency and its multiple frequency components. The result of modulation is to form a series of sidebands on both sides of meshing frequency. The interval of the side frequency is the rotation frequency FZ of the gear shaft, and the shape of the side frequency family mainly depends on the modulation index β.

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