# Amplitude modulation of industrial gears

All kinds of faults of gears are embodied in a transmission error in operation, that is, the difference between the actual angular position of the output shaft and the ideal angular position of the output shaft without error or deformation when the gears transmit constant torque, which constitutes the main source of vibration and noise of gears. When the transmission error is large, the collision between the gear entering and disengaging will be intensified, which will produce a high vibration peak value and a short-term amplitude change and phase change. Amplitude modulation is generated by amplitude change and frequency modulation is generated by phase change. Different faults produce different modulation forms, which will be explained in detail below.

The amplitude modulation of gear is caused by load fluctuation on the tooth surface, gear machining error (such as uneven pitch), gear eccentricity, and local and uniform defects caused by gear failure.

From the point of view of signal processing, amplitude modulation is equivalent to the multiplication of two signals in the time domain, and the conversion to the frequency domain is equivalent to the convolution of the corresponding two spectra, as shown in the figure below. The relatively high frequency is called carrier frequency, and the low frequency is called modulation frequency. For gear signal, the meshing frequency is carrier frequency, and the rotation frequency of gear is modulation frequency.

The amplitude modulation principle of simple harmonic vibration can be explained as follows.

Let the carrier signal representing gear engagement frequency be g (T) = asin (2 π FMT + φ)

The modulation signal representing the gear rotation frequency is e (T) = 1 + bcos2 π FET

The vibration signal after amplitude modulation is x (T) = g (T) • e (T) = a (1 + bcos2 π FET) sin (2 π FMT + φ)

In the formula, a-carrier signal amplitude; b-modulation index; fm-carrier frequency (meshing frequency);

Fe ~ modulation frequency (rotation frequency); φ ~ initial phase angle.

After the above formula is expanded, X (T) = asin (2 π FMT + φ) + (1 / 2) absin [2 π (FM + Fe) t + φ]

+(1/2)ABsin[2π(fm－fe)t+φ]

In this way, in addition to the original meshing frequency component FM, a pair of sum frequency (FM + Fe) and difference frequency (FM FE) of meshing frequency and rotation frequency are added to the modulated signal. On the spectrum diagram, they are symmetrically distributed on both sides of FM with meshing frequency FM as the center, rotation frequency Fe as the interval, and (1 / 2) AB as the amplitude. They are called sideband, which is short for sideband, as shown in (c) on the right.

If the modulation signal e (T) is not a simple harmonic, but a periodic signal composed of multiple frequency components, each frequency component of E (T) will generate a sideband, forming a sideband family, as shown in the right figure.

Due to the influence of system transmission characteristics and the existence of frequency modulation, the actual side frequency components on the spectrum diagram will not be so symmetrical as shown in the right figure. However, the distribution shape of the sideband mainly depends on the modulation signal, and it is the modulation signal that reflects various transmission errors and fault conditions of the gear.

According to the shape of the sideband, it can be distinguished whether the gear has local defects or distributed defects.

If there are local defects such as broken teeth or large peeling off, when the meshing point enters the defect, the vibration of the gear is modulated by a short pulse, and the length of the pulse is equal to the meshing period TM = 1 / FM of the gear. The pulse repeats every time the gear rotates. Since the pulse can be divided into many sinusoidal components, the carrier frequency FM, 2fm, 3FM A series of side frequencies centered. It is characterized by a large number of side frequencies, a wide range, a small number, a uniform and relatively flat distribution, and the interval between each side frequency is equal to the rotation frequency of the gear. See (a) above.

If there are some defects such as pitting, scratches (gluing) and so on, the components of modulation frequency are more, but it is an envelope with smaller amplitude change and longer pulsation period in the time domain. Therefore, the edge frequency band on the spectrum is characterized by high and narrow distribution and large amplitude fluctuation. See (b) above.

In a word, the more concentrated the gear defects are, the lower, wider and flatter the sideband will be; the more evenly distributed the defects are, the higher, narrower and more undulating the sideband will be. 