**2.3 Effect of load on transmission error**

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 significant than the contribution from the gear geometry.

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).

Gearbox Simulation Models with Gears and Bearings Faults 23

line indicating the ideal involute profile of the tooth: Figure-2.3.2 (a). The effect of mesh stiffness variation due to the change in the number of meshing tooth pairs appears as steps in the STE plot: Figure-2.3.2 (b). The amount of deflection increases when a single pair of teeth is carrying load and decreases when the load is shared by another pair. The share of

A paper published jointly by S.L. Harris, R. Wylie Gregory and R.G. Munro in 1963 showed how transmission error can be reduced by applying appropriate correction to the involute gear profile [4, 5]. The Harris map in Figure-2.3.3 shows that any gear can be designed to have STE with zero variation (i.e. a flat STE with constant offset value) for a particular load. The basic idea behind this technique is that the profile of gear teeth can be designed to

Additionally, variation of TE can be reduced by increasing the contact ratio of the gear pair. In other words, design the gears so that the load is carried by a greater number of tooth

It is a standardized design procedure to perform STE analysis to ensure smoothly meshing gears in the loaded condition. It was explained in this section how the strong correlation between the TE and the gear vibration makes the TE a useful parameter to predict the quietness of the gear drives. However, a more realistic picture of the gear's dynamic properties can not be captured without modelling the dynamics of the assembled gear drive system. Solution of engineering problems often requires mathematical modelling of a physical system. A well validated model facilitates a better understanding of the problem and provides useful information for engineers to make intelligent and well informed

A comprehensive summary of the history of gear dynamic model development is given by Ozguven and Houser [6]. They have reviewed 188 items of literature related to gear dynamic simulation existing up to 1988. In Table-2.4.1, different types of gear dynamics models were classified into five groups according to their objectives and

Long relief Short relief No relief

force carried by a tooth through the meshing cycle is shown in Figure-2.3.2 (c).

cancel the effect of tooth deflection occurring at the given load.

**Optimum STE for load 2** 

Fig. 2.3.3. Optimum tooth profile modification of a spur gear.

**2.4 Modelling gear dynamics** 

decisions.

pairs.

Fig. 2.3.1. (a) Deflection of gear tooth pair under load, (b) Effect of load on transmission error (TE).

Consider the more general situation where the deflection in loaded gears affects the TE significantly. Note that the following discussion uses typical spur gears (contact ratio = 1.5) with little profile modification to illustrate the effect of load on TE. Figure-2.3.2 illustrates the STE caused by the deflection of meshing gear teeth. The tooth profile chart shows a flat

Fig. 2.3.2. Effect of Load on TE, (a) Tooth Profile Chart, (b) Static Transmission Error, (c) Loading acting on a tooth

(b)

Fig. 2.3.1. (a) Deflection of gear tooth pair under load, (b) Effect of load on transmission

Consider the more general situation where the deflection in loaded gears affects the TE significantly. Note that the following discussion uses typical spur gears (contact ratio = 1.5) with little profile modification to illustrate the effect of load on TE. Figure-2.3.2 illustrates the STE caused by the deflection of meshing gear teeth. The tooth profile chart shows a flat

Double Pair Contact Single Pair Contact Double Pair Contact

Line of action

Rotation

Fig. 2.3.2. Effect of Load on TE, (a) Tooth Profile Chart, (b) Static Transmission Error, (c)

Loaded TE

No load

Perfect involute tooth profile

**F/2**

**F F/2**

error (TE).

(a)

**(a)** 

Profile Chart

**F/2**

**F/2**

Unloaded teeth positions

Deflection

Driver

0 μm

STE (Deflection)

Load Share

Loading acting on a tooth

**(b)** 

**(c)** 

F/2

F

line indicating the ideal involute profile of the tooth: Figure-2.3.2 (a). The effect of mesh stiffness variation due to the change in the number of meshing tooth pairs appears as steps in the STE plot: Figure-2.3.2 (b). The amount of deflection increases when a single pair of teeth is carrying load and decreases when the load is shared by another pair. The share of force carried by a tooth through the meshing cycle is shown in Figure-2.3.2 (c).

A paper published jointly by S.L. Harris, R. Wylie Gregory and R.G. Munro in 1963 showed how transmission error can be reduced by applying appropriate correction to the involute gear profile [4, 5]. The Harris map in Figure-2.3.3 shows that any gear can be designed to have STE with zero variation (i.e. a flat STE with constant offset value) for a particular load. The basic idea behind this technique is that the profile of gear teeth can be designed to cancel the effect of tooth deflection occurring at the given load.

Additionally, variation of TE can be reduced by increasing the contact ratio of the gear pair. In other words, design the gears so that the load is carried by a greater number of tooth pairs.

Fig. 2.3.3. Optimum tooth profile modification of a spur gear.
