**2.1 Transmission error**

Gears, by their inherent nature, cause vibrations due to the large pressure which occurs between the meshing teeth when gears transmit power. Meshing of gears involves changes in the magnitude, the position and the direction of large concentrated loads acting on the contacting gear teeth, which as a result causes vibrations. Extended period of exposure to noise and vibration are the common causes of operational fatigue, communication difficulties and health hazards. Reduction in noise and vibration of operating machines has been an important concern for safer and more efficient machine operations.

Design and development of quieter, more reliable and more efficient gears have been a popular research area for decades in the automotive and aerospace industries. Vibration of gears, which directly relates to noise and vibration of the geared machines, is typically dominated by the effects of the tooth meshing and shaft revolution frequencies, their harmonics and sidebands, caused by low (shaft) frequency modulation of the higher toothmesh frequency components. Typically, the contribution from the gear meshing components dominates the overall contents of the measured gearbox vibration spectrum (see Figure-2.1.1).

Transmission Error (TE) is one of the most important and fundamental concepts that forms the basis of understanding vibrations in gears. The name 'Transmission Error' was originally coined by Professor S. L. Harris from Lancaster University, UK and R.G. Munro,

Gearbox Simulation Models with Gears and Bearings Faults 19

What made the TE so interesting for gear engineers and researchers was its strong correlation to the gear noise and the vibrations. TE can be measured by different types of instruments. Some commonly used methods are: Magnetic signal methods, straingauge on the drive shaft, torsional vibration transducers, tachometers, tangential accelerometers and rotary encoders systems. According to Smith [2], TE results from three main sources: 1) Gear geometrical errors, 2) Elastic deformation of the gears and associated components and 3) Errors in mounting. Figure-2.1.3 illustrates the relationship between TE and its sources.

Transmission Error exists in three forms: 1) Geometric, 2) Static and 3) Dynamic. Geometric TE (GTE) is measured at low speeds and in the unloaded state. It is often used to examine the effect of manufacturing errors. Static TE (STE) is also measured under low speed conditions, but in a loaded state. STE includes the effect of elastic deflection of the gears as well as the geometrical errors. Dynamic TE (DTE) includes the effects of inertia on top of all the effects of the errors considered in GTE and in STE. The understanding of the TE and the behaviour of the machine elements in the geared transmission system leads to the

> **Transmission Error**

A typical GTE of a spur gear is shown in Figure-2.2.1. It shows a long periodic wave (gear shaft rotation) and short regular waves occurring at tooth-mesh frequency. The long wave is often known as: Long Term Component: LTC, while the short waves are known as: Short

The LTC is typically caused by the eccentricity of the gear about its rotational centre. An example is given in Figure-2.2.2 to illustrate how these eccentricities can be introduced into the gears by manufacturing errors; it shows the error due to a result of the difference

The effect of errors associated with gear teeth appears in the STC as a localized event. The parabolic-curve-like effect of tooth tip relief is shown in Figure-2.2.3(a). The STC is caused mainly by gear tooth profile errors and base pitch spacing error between the teeth. The effect

Thermal Distortions

Pinion and Gear Movement

Gear and Pinion Distortions

Pinion and Gear Tooth

Deflection

Gearcase Distortions

development of realistic gear rotor dynamics models.

Quality of Contact Surface Finish

**2.2 Effect of gear geometric error on transmission error** 

Gearcase Accuracy (Mounting Error)

Pinion and Gear Profile Accuracy

Pinion and Gear Helix Accuracy (Lead Error)

Fig. 2.1.3. Sources of Transmission Error.

between the hobbing and the shaving centres.

Pinion and Gear Pitch Accuracy (Tooth Spacing Error)

Term Component: STC.

his PhD student at the time. They came to the realization that the excitation forces causing the gears to vibrate were dependent on the tooth meshing errors caused by manufacturing and the bending of the teeth under load [1].

(Number of Teeth = 32)

Fig. 2.1.1. Typical spectrum composition of gear vibration signal.

TE is defined as the deviation of the angular position of the driven gear from its theoretical position calculated from the gearing ratio and the angular position of the pinion (Equation-2.1.1). The concept of TE is illustrated in Figure-2.1.2.

$$TE = \left(\Theta\_{gear} - \frac{R\_{pinnou}}{R\_{gear}} \Theta\_{pinnou}\right) \tag{2.1.1}$$

Fig. 2.1.2. Definition of Transmission Error.

his PhD student at the time. They came to the realization that the excitation forces causing the gears to vibrate were dependent on the tooth meshing errors caused by manufacturing

> harmonic of gear tooth meshing frequency (896Hz)

TE is defined as the deviation of the angular position of the driven gear from its theoretical position calculated from the gearing ratio and the angular position of the pinion (Equation-

> *pinion gear pinion gear*

> > *gear*

Theoretical rotation of gear Measured rotation of gear

*R*

3rd

 harmonic of gear tooth meshing frequency (1344Hz)

*R* (2.1.1)

**Transmission Error** 

and the bending of the teeth under load [1].

1st

Sideband

(Number of Teeth = 32)

*pinion*

Shaft rotational frequency (14Hz) and harmonics

2nd

Fig. 2.1.1. Typical spectrum composition of gear vibration signal.

*TE*

*pinion gear R R*

*pinion*

Rgear Rpinion

Pinion Gear

Fig. 2.1.2. Definition of Transmission Error.

2.1.1). The concept of TE is illustrated in Figure-2.1.2.

 harmonic of gear tooth meshing frequency (448Hz) What made the TE so interesting for gear engineers and researchers was its strong correlation to the gear noise and the vibrations. TE can be measured by different types of instruments. Some commonly used methods are: Magnetic signal methods, straingauge on the drive shaft, torsional vibration transducers, tachometers, tangential accelerometers and rotary encoders systems. According to Smith [2], TE results from three main sources: 1) Gear geometrical errors, 2) Elastic deformation of the gears and associated components and 3) Errors in mounting. Figure-2.1.3 illustrates the relationship between TE and its sources.

Transmission Error exists in three forms: 1) Geometric, 2) Static and 3) Dynamic. Geometric TE (GTE) is measured at low speeds and in the unloaded state. It is often used to examine the effect of manufacturing errors. Static TE (STE) is also measured under low speed conditions, but in a loaded state. STE includes the effect of elastic deflection of the gears as well as the geometrical errors. Dynamic TE (DTE) includes the effects of inertia on top of all the effects of the errors considered in GTE and in STE. The understanding of the TE and the behaviour of the machine elements in the geared transmission system leads to the development of realistic gear rotor dynamics models.

Fig. 2.1.3. Sources of Transmission Error.

#### **2.2 Effect of gear geometric error on transmission error**

A typical GTE of a spur gear is shown in Figure-2.2.1. It shows a long periodic wave (gear shaft rotation) and short regular waves occurring at tooth-mesh frequency. The long wave is often known as: Long Term Component: LTC, while the short waves are known as: Short Term Component: STC.

The LTC is typically caused by the eccentricity of the gear about its rotational centre. An example is given in Figure-2.2.2 to illustrate how these eccentricities can be introduced into the gears by manufacturing errors; it shows the error due to a result of the difference between the hobbing and the shaving centres.

The effect of errors associated with gear teeth appears in the STC as a localized event. The parabolic-curve-like effect of tooth tip relief is shown in Figure-2.2.3(a). The STC is caused mainly by gear tooth profile errors and base pitch spacing error between the teeth. The effect

Gearbox Simulation Models with Gears and Bearings Faults 21

Fig. 2.2.3. (a) GTE of a meshing tooth pair (b) Resulting Short Term Component of GTE.

Elastic deflections occurring in gears are another cause of TE. Although gears are usually stiff and designed to carry very large loads, their deflection under load is not negligible. Typical deflection of gear teeth occurs in the order of microns (μm). Although it depends on the amount of load gears carry, the effect of the deflection on TE may become more

A useful load-deflection measure is that 14N of load per 1mm of tooth face width results in 1μm of deflection for a steel gear: i.e. stiffness = 14E109 N/m/m for a tooth pair meshing at the pitch line. It is interesting to note here that the stiffness of a tooth pair is independent of its size (or tooth module) [3]. Deflection of gear teeth moves the gear teeth from their theoretical positions and in effect results in a continuous tooth pitch error: see Figure-2.3.1 (a). The effect

of the gear deflection appears in the TE (STE) as a shifting of the GTE: Figure-2.3.1 (b).

Fig. 2.2.4. Effect of spacing error appearing in short term component of GTE.

**2.3 Effect of load on transmission error** 

significant than the contribution from the gear geometry.

of individual tooth profile errors on the GTE is illustrated in Figure-2.2.3(a). The GTE of a meshing tooth pair is obtained by adding their individual profile errors. The STC of gear GTE is synthesised by superposing the tooth pair GTEs separated by tooth base pitch angles (Figure-2.2.3 (b)).

Another common gear geometric error is tooth spacing or pitch errors, shown in Figure-2.2.4. The tooth spacing error appears in GTE as vertical raise or fall in the magnitude of a tooth profile error.

Fig. 2.2.1. A typical Geometrical Transmission Error.

Fig. 2.2.2. (a) Eccentricity in a gear caused by manufacturing errors, (b) Resulting errors in gear geometry.

of individual tooth profile errors on the GTE is illustrated in Figure-2.2.3(a). The GTE of a meshing tooth pair is obtained by adding their individual profile errors. The STC of gear GTE is synthesised by superposing the tooth pair GTEs separated by tooth base pitch angles

Another common gear geometric error is tooth spacing or pitch errors, shown in Figure-2.2.4. The tooth spacing error appears in GTE as vertical raise or fall in the magnitude of a

Fig. 2.2.2. (a) Eccentricity in a gear caused by manufacturing errors, (b) Resulting errors in

(Figure-2.2.3 (b)).

tooth profile error.

gear geometry.

Fig. 2.2.1. A typical Geometrical Transmission Error.

Fig. 2.2.3. (a) GTE of a meshing tooth pair (b) Resulting Short Term Component of GTE.

Fig. 2.2.4. Effect of spacing error appearing in short term component of GTE.
