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In a gear pair the system vibration response is due to the deformation and elasticity of the

contacting components and not due to the force or the response time history. So a piece-wise

linear function was used rather than a rectangular or a half-sine function. This linear function

can be easily modified to include different effects including tooth breakage and tooth profile

69 To simplify the model, just the major factors, the mesh stiffness, k t( ) has been considered in this study The mesh stiffness is defined to be the ratio between the force acting

along the line of action, and the tooth displacement along the same line. In order to obtain a

mathematical model, the stiffness for a single pair of meshing teeth was first investigated.

The stiffness for the meshing of a gear pair is affected by several characteristics [91], firstly

the stiffness due to the Hertzian contact, the axial stiffness between both teeth and the

bending stiffness of the two teeth from the drive and the driven gear . All these factors

can lead to an increase in the stiffness amplitude in the middle phase of the meshing time.

Due to the complexity in precisely predicting the actual pattern of the variation in stiffness in

real cases, approximations are usually used. Therefore, this research study presumes that the

tooth mesh stiffness has the combined profile shown in Figure 4.5 (a).

As mentioned above, a piece-wise linear function is used to represent the transient value of

the stiffness during the gear mesh-in and gear mesh-out of a gear pair. The stiffness is

assumed to have a constant amplitude variation in the middle part of the engagement of the

tooth pair. Maximum stiffness value of a single gear pair is usually in the order of 107 N/m

[92].

In a single pair of teeth contact, the meshing stiffness stays mostly unchanged, since the

reduction of stiffness in one of tooth in the pair is then compensated by an increase in

stiffness of the other mating tooth. When the gears rotate the point of contact between a tooth

pair will be higher in one tooth and lower in the other tooth of the matching pair.

While the gears are rotating, the number of teeth in contact also varies, as a consequence the

70 spur gears, where the contact ratio is lower, such variations are mainly because of a load

transfer occurring over a single tooth in one double tooth pairing. However, in the case of a

helical gear pairs, the meshing change of the overall length in the line of contact line is

relatively much smaller since there is a higher contact ratio which is distributed between

three or four teeth at a time. Therefore, the mesh overall meshing stiffness variations in

helical gears are considerably smaller compared against that of spur gears. Thus models of

helical gear dynamics consider meshing stiffness as a constant value that uses its time

averaged result [93].

In order to simplify this research model, the meshing stiffness is assumed as a function of

angular displacement to depend upon two factors: overlap and contact ratio . The first

parameter measures the overlap of adjacent teeth in the axial or face-width direction and the

other parameter measures the contact between adjacent teeth in a diagonal section.

The overall profile of a single pair stiffness can be interpreted as linear piecewise, which is

shown in Figure 4.5 (a).

Figure 4.5(b) shows the x-axis denoting the number of base pitches, which needs be

multiplied by ( ) to provide the angular variation. The total meshing stiffness can then

be calculated by summing the overall stiffness of a single tooth pair, and incrementing by one

tooth at a time for the total number of teeth.

Further, in Figure 4.7 (b) illustrates the resulting meshing stiffness waveform for the

underlying gear pair of this research study. The minimum number of teeth in contact is three

and the maximum number is four. This variation between the minimum and maximum

71 fluctuation shows that even in a healthy condition (given that there are manufacturing faults

or faults) a gear pair will still produce vibrations. With the existence of manufacturing faults,

the variations in the stiffness will be non-uniform. The minimum and maximum number of

contacting teeth pairs will vary depending on the shaft angular position, as does the resulting

error’s amplitude, typically causing greater variations in the stiffness amplitude and resulting

in vibrations with higher amplitudes.

72 It is well known and mentioned in chapter three that tooth breakage (failure) is a common

fault in helical gears. This kind of fault begins with small damages to the structure of the

tooth, which then leads to a change in its meshing stiffness. The colours in Figure 4.7 (b),

blue, pink, red and black, represents the influence of a single tooth breakage on the overall

meshing stiffness. The tooth breakage are missing 25%, 50%, 75% and 100% of a complete

single tooth. The figure shows that even local faults can result in a sudden drop in the

stiffness amplitude. Additionally, at lower stiffness values, the width of the interval increases

along with the severity of the fault, however the low amplitude value remains the unchanged.

The sudden drops in the meshing stiffness value should then create an equivalent rise in the

overall amplitude of the system vibration.

4.6 Model for Tooth Wear