Bearings and gearbox vibration are fundamental issues for rotating machines in many industrial applications. These are critical components and, as such, any failure can prove expensive in both repair cost and down-time. Because of this condition monitoring has become increasingly important over the years, usually centred around vibration measurement taken at critical locations, either continuously (online) or as part of a monitoring schedule. Vibration monitoring has become an integral part of most maintenance regimes and relies on the detection of various well-known frequency characteristics associated with this type of component. Detailed knowledge of design of the bearing or gearbox allows characteristic vibration frequencies to be calculated. However these frequencies are often masked by vibration from nearby components or by noise, sometimes making diagnosis difficult. Vibration time signatures are also often subject to both amplitude and frequency modulation which affect the resultant frequency spectra. Here we are going to look at the causes and effect of amplitude modulation in particular and how it is manifested in the frequency domain.

Rolling Contact Bearing Features

Rolling contact bearings, either rolling ball or rolling element are used extensively in all types of rotating machine. A rolling element bearing consists of a number of balls or rollers within an inner and outer bearing ring. When faults develop within such a bearing it is often due to pitting on the surface of the elements or on the inner or outer bearing face. Standard equations exist which allow the frequency of occurrence of these impacts to be estimated from the speed of rotation and the detailed geometry of the bearing which include the inner and outer race diameters and the diameter of the individual elements. Typically we have the most common defect frequencies:

Fundamental Train Frequency (FTF)

Ball Passing Frequency Outer Race (BPFO)

Ball Passing Frequency Inner Race (BPFI)

Ball Spin Frequency (BSF)

where

D = pitch diameter
d = element diameter
a = contact angle
n = number of elements
f = Revolution speed (revs/sec).

Note that f is the relative speed difference between the inner and outer races. In most cases the outer race is stationary and so f is the shaft rotational speed. See Figure 2 below.

Gearbox Features

There are a large number of different designs of gearbox. The simplest are known as Spur gears and are used to transfer power between two parallel shafts, usually at different speeds, as shown in Figure 3. The fundamental speeds of rotation of the shafts are defined by the ratio of the number of teeth on each gear-wheel.

f_i = Input Speed
f_o = Output Speed
N_i = Number of teeth on Input Gear
N_o = Number of teeth on Output Gear

Many more complicated gearboxes, including multiple compound gear trains, follow the same principles, but the internal gear speeds are more difficult to calculate.

Gear Mesh Frequency

The main frequencies seen within a vibration spectra for a gearbox are the rotational speeds of each gear, f_i and f_o and the gear-meshing or the tooth mesh frequency, f_m. The gear-mesh frequency defines the rate at which gear teeth mesh together. The Gear Mesh Frequency is given by:

or

.

Hunting Tooth Frequency (HTF)

Another frequency which is sometimes used is the Hunting Tooth Frequency (HTF). This is found when one tooth on each gear are damaged and represents the frequency at which the two teeth contact each other. The calculation of this frequency is also dependent on the number of teeth per gear and involves finding the highest common factor (CF) between the gear ratios. For example if the input gear has 9 teeth then its factors are 1×9 and 3×3. If the output gear has 15 teeth then its factors are 1×15 and 3×5 and so the highest common factor is 3. The Hunting Tooth Factor (HTF) is given by: where CF is the highest Common Factor.

Amplitude Modulation

As discussed above, any surface defects in either a bearing or a gearbox will result in a vibration signal that will contain individual impulses due to the impacts generated when the defect comes into contact with other elements. The frequency of the impacts is dependent on design features of the component. The ability to measure and identify these frequencies will allow us to better identify these problems. In practice the signals caused by these impacts will also be modulated. This modulation could be both in amplitude and in frequency, but for the purposes of this discussion we will concentrate on amplitude modulation. Amplitude modulation is caused when the load on a bearing of a gearbox varies, typically with rotational speed. This change in load will affect the strength of the impact seen. For example, imagine a gearbox which shows signs of wear and which also has a slight bend in the shaft. The worn teeth will cause peaks in vibration to occur at the gear meshing frequency. The bend in the shaft will cause the pressure on the gear teeth to increase and then decrease during a complete revolution of the shaft. The same might apply to a bearing which is mounted horizontally. Due to gravitational forces the pressure between the element and the bearing surfaces may be greater at the bottom of the bearing rather than the top. If there is a defect on one of the elements then the impact from this may be stronger when the element rotates to the bottom of the bearing than when it is at the top. [Note that if there is a defect in the outer race and this is stationary then the impulses from this defect will not generally be subject to load variations and so will not show amplitude modulation]. The following graphs show the effect of amplitude modulation. Assume that our defect causes a pure sinusoidal output at the defect frequency, in this example 75Hz. We then amplitude modulate this signal at a lower frequency of 10Hz.

Figure 5 shows the effects of applying amplitude modulation to this signal. The frequency of the modulation signal is 10Hz. The relative amplitude of the modulation signal, or the modulation factor, is 0.25.

This amplitude modulation, caused by the multiplication of the higher frequency signal (f_H) and the lower frequency modulation signal (f_L), results in a signal, a(t) that is described by the standard equation:

This expands to:

That is, the resultant frequency spectrum will contain peaks at frequencies f_H, (f_H-f_L ) and (f_H+f_L )Hz. The components (f_H-f_L) and (f_H+f_L) are known as ‘sidebands’ and are a common characteristic when performing frequency analysis on bearings and gearboxes. These are shown below in the frequency spectra of the amplitude modulated signal. Here we can clearly see the main component at 75Hz and the two side-bands at 65Hz and 85Hz.

In most situations the main vibration signal from a defect will not be sinusoidal but often a series of impulses, repeating at the defect frequency. In the frequency domain this signal will also contain a number of harmonics of the main defect frequency. Each of these harmonics will also have side-bands. In most cases the modulation frequency will be the shaft rotational speed. Therefore when analysing data from rolling element bearings or from gearboxes, the measurement and detection of sidebands in the frequency domain is very important. In gearboxes the gears rotate at different speeds and so there may be many sidebands present but with knowledge of the shaft speed and number of teeth per gear then detailed diagnostics are possible which can pin-point faults with specific gears and/or shafts.

Conclusion

In this discussion we have briefly looked at how we can use the detailed design information of bearings and gearboxes to look for specific fault conditions by collecting vibration information and analysing their frequency spectra. Amplitude modulation of the vibration signatures is common and we have seen how this causes side-bands to be present in the frequency domain.

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Don Davies

Applications Group Manager at Prosig

Don Davies graduated from the Institute of Sound and Vibration Research (ISVR) at Southampton University in 1979. Don specialises in the capture and analysis of vibration data from rotating machines such as power station turbine generators. He created and developed the PROTOR system and is Applications Group Manager at Prosig. Don is a member of the British Computer Society.

Latest posts by Don Davies (see all)

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• Bearing & Gearbox Vibration Analysis using Demodulation Techniques? (Part 2) - May 2, 2014