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Load Sharing in Metallic and Non-Metallic Gears D Walton, A A Tessema, C J Hooke and J M Shippen Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 1994 208: 81 DOI: 10.1243/PIME_PROC_1994_208_104_02 The online version of this article can be found at: http://pic.sagepub.com/content/208/2/81

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81

Load sharing in~etallic)and~on-metallic gears D Walton, BTech, PhD, CEng, MIMechE, A A Tessema, BSc, PhD, C J Hooke, BSc, PhD, DEng and J M Shippen, BSc, PhD, CEng, MIMechE School of Manufacturing and Mechanical Engineering, The University of Birmingham

A review of work o d o o t h deformatiohnd load sharing in non-metallic gears is presented. Ahnite element analys;&employing the flexibility method for contacting bodies is used to model tooth deflections and contact patterns ’Betweenheshing g e a r y For metallic gears the change in contact ratio between the theoretical and running values is shown to be small. However, for low modulus, nonY t a l l i c gears the change in contact ratio is large and can give cause for concern. The benefits and disadvantages of this increase in +.Qperating contact r a t i a s stated. Finally, the possibility of characterizing the change in operating contact ratio in a non-dimensional form is discussed. NOTATION

elastic modulus force tangential transmitted force final gap between contacting nodes initial gap between contacting nodes ideal contact ratio application factor module number of teeth rotational speed base pitch rigid body displacement base radius radius real contact ratio flexibility matrix torque face width contact ratio factor for bending stress tooth form factor for bending stress actual length of the line of contact roll angle deformation viscosity rigid body rotation coefficient of friction Poisson’s ratio non-dimensional elasticity parameter pressure angle root bending stress Subscripts and superscripts i, j p T w

arbitrary node labels pinion transpose wheel

being approximately 100 times less than for steel gears. As a result, the real contact ratio in non-metallic gears is greater than the ideal value. The term real contact ratio was first used by Yelle and Burns (1) to differentiate from the ideal contact ratio which assumes the gear teeth to be perfectly rigid. In this paper the term has the same meaning and is also used synonymously with the expression ‘operating contact ratio’. The increase in load sharing is due to the extended duration of contact between the gear teeth (edge contact). The gears mesh prematurely and go out of mesh beyond the theoretical point of last contact, resulting in an increase in the real contact ratio. The increase in real contact ratio is most significant for low module (fine pitch) gears and high loads. Due to this, the allowable bending fatigue strength of polymer gears was found to be dependent upon the module. Yelle and Burns (1, 2) sought to eliminate this dependei,ce on the module by formulating their own rating procedure which called for the redefinition of the Lewis form factor to allow for load sharing. The method had been applied to rate the bending fatigue strength of lubricated acetal/acetal and acetal/steel gear pairs and was found to be successful. However, it should be noted that it is not clear whether their work considered size effects in their fatigue and analysis. They were also less concerned about the effect of premature contact on the wear behaviour of polymer gears. When running polymer gears in dry unlubricated conditions, premature contact can lead to severe tooth interference, which can cause progressive flank wear and eventual fracture (3), and is a case when the increase in real contact ratio does not assist gear performance. In this paper, the increase in real contact ratio from the ideal is presented in non-dimensional form, enabling comparisons to be drawn between real contact ratios for metallic as well as non-metallic gears. A finite element procedure is given for determining the operatL.4 contact ratio.

1 INTRODUCTION

2 TOOTH DEFORMATION AND LOAD SHARING

Non-metallic gears are more flexible than their metallic counterparts owing to their low modulus of elasticity,

An early study on gear tooth stiffness was conducted by Timoshenko and Baud in 1926 (4). They measured tooth deflection and put forward an analytical method for its approximation. In their theoretical analysis, they divided gear tooth deflection into three parts: that due

The M S wus recriued on 23 Murch 1993 and w0s accepted for publication on

I5 February 1994. C02193 Q IMechE 1994

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D WALTON, A A TESSEMA, C J HOOKE AND I M SHIPPEN

to the flattening of high spots, deflections due to depression at the contact surface and the deflection of the tooth when considered as a cantilevered beam giving shear and bending deflections. They suggested how the flattening of high spots could be neglected on the grounds that they mainly depended on surface finish. In any case their magnitude was considered not appreciable compared to the other deflections. They adopted Hertz’s equation for two contacting bodies to calculate the deflection due to depressions at the contact surfaces, while bending and shear deflections were calculated assuming the gear tooth could be represented as a beam of variable width. The most comprehensive experimental analysis of tooth deflections was carried out by Walker (5) as long ago as 1938. Walker, after a long series of tests, was able to develop empirical equations for gear tooth deflection and was able to demonstrate that tooth deflection is proportional to load and is independent of the module. Based on his experimental investigation of tooth deflections, Walker was also able to suggest a method of profile modification to avoid premature tooth contact and challenged the then British Standard recommcndations for tipping or tip easing. There have been a number of attempts to use the actual profile of a gear tooth to calculate its deflection. Weber (6) gave gear tooth deflection formulae which considered contact, bending, shear and rim deflections. Attia (7,8) developed equations based on the work of Weber but added components to account for the deformation of the gear body and the effect of neighbouring teeth. Cornell (9) gave an equation in summation and integral form for three different forms of gears, indicating standard, under-cut and high contact ratio forms. All these equations relied upon computers for solutions. Conformal mapping function methods have been used to determine gear tooth deflection (1&13). In this method the actual gear tooth profile is transformed into a semi-infinite boundary so that equations developed for a semi-infinite body can be used to evaluate its deflection. The finite element method has also been used to evaluate gear tooth deflection by a number of researchers. Coy and Chao (14) looked at the selection of a suitable grid size to account for Hertzian deformations. By applying a concentrated load at a single point for different grid sizes to Hertzian contact width ratios, they reached an optimum grid size to give the Hertzian deformation accurately. Their models consisted of two contacting cylinders, as their deformation could be accurately calculated using Hertz’s equation. This work emphasized the significance of Hertzian deformation, which was quoted as reaching as much as 25 per cent of the total gear tooth deflection. Tanaka and Kohida (15) used the finite element method to simulate spur gcar tooth meshing in order to examine tooth stresses, while also taking into effect load sharing. They were able to derive a stress concentration factor that took into account load sharing, but ignored interference. Steward (16) presented a three-dimensional finite element analysis coupled with two-dimensional Hertzian contact theory to evaluate the deflection of solid, wide-faced spur gears. His finite element analysis results agreed well with his experimental investigation. Determination of tooth deflection is an important step in determining load sharing between teeth. In Part C Journal of Mechanical Engineenng Science

general, for two ideal pairs of teeth in contact, the load fractionsf,, for pair 1, andf, ,for pair 2, are given by K2

*’ = K , + K 2

Kl

and

f2=K,+Kz

where K , = combined mesh stiffness for pair 1 K,

= combined

mesh stiffness for pair 2

In these formulae neither the stiffness nor the load fraction is known at a particular meshing point. Hence the load fractions are determined by an iteration procedure that changes the load fractions from an initial guess of, say, 50 per cent each (for two pairs of teeth in contact) until the load fractions are found that cause the mesh deflections to be equal. The problem in the above formulae is that they fail to predict load sharing in gears when there is an edge contact, manufacturing error or both. To calculate load sharing in real gears, Harris and co-workers (17-19) have developed techniques that take into account both mesh deflection and manufacturing error. The technique relies on the determination of mesh deflection by any one of the above-mentioned analytical methods and has found a wider application in determining transmission errors in gears. The finite element method of determining load sharing presented in this paper simulates the mcshing of real gears by taking into account only mesh deflection. The method has been successfully applied to predict edge contact in highly flexible non-metallic gears. An efficient and economical use of commercially available finite element analysis software is also demonstrated. Since the method simulates the actual meshing of gears, it can be applied to determine load sharing with manufacturing errors. 3 MODELLING OF TOOTH DEFLECTION AND LOAD SHARING

3.1 General

The first step in evaluating tooth deflection and hence load sharing is to model the gear tooth contact. This is shown in Fig. 1. To simplify matters, the pinion tooth is taken to be rigid. In this simulation of gear tooth contact, two problems are encountered. The first is

Fig. 1 Tooth contact and deformation @ IMechE 1994

83

LOAD SHARING IN METALLIC A N D NON-METALLIC GEARS

finding successive contact points as the gears rotate and the second is determining the nature of the forces at every contact point, both in magnitude and direction. Due to bending, shear and Hertzian deflection, there will not be conjugate action. Thus the second contact point will no longer lie on the ideal contact line f-1 and the direction of the normal force will no longer be tangent to the base circles. As a result the torque acting on the gears will fluctuate. To simplify the simulation process, it is necessary to make certain assumptions. One is letting the torque remain constant, allowing contact between the wheel and pinion to take place along their whole range of active tooth profiles. Thus, by dividing their active tooth profiles into equal intervals, predetermined meshing contact points can be obtained. However, this by itself does not guarantee correct evaluation of mesh stiffness. Factors such as friction, load sharing and the geometry of contact (point or line contact in two dimensions) have to be considered. In most cases sliding friction can be neglected as its effect on deformation is small. The contact area at a particular meshing point can be modelled using finite element methods. As found in the literature, there are methods available to model such contact problems (20-23). In this work the flexibility method of contact problem analysis was adopted.

respective rigid body rotations as

3.2 The flexibility method for contact problem analyses

Equation (5) shows that for those nodes in contact, their relative displacement due to rigid body displacement and deformation is equal to the initial gap between them. Combining equations (3) and (6) gives

Two bodies, 1 and 2, are shown in Fig. 2a with their boundaries subdivided into elements connected at grid points; hi represents the initial separation distance or gap between the two potentially contacting nodes subjected to a torque TI on body 1 and T2 on body 2. Figure 2b represents the rigid body movement of the two bodies under the action of the applied torques. The rigid body translation can be given in terms of their

rli

=

Rlidl

and

rZi = R2,02

where, in general,

Torques and forces can be related as T,

= RliFi

and

T, = R z i F i

where, in general,

(3) If the two bodies are constrained in some way, their relative displacement due to deformation (Fig. 2c), is given as

hi = h I i + 82i

and

hi = [ q F i

(4)

Then the final gap, g i , between the two contacting nodes is given by gi = hi

+ rZi8,

-

rli ei + hi

(5)

To assemble the system equation n numbers of nodes in contact after the application of a torque T can be assumed. For these nodes the following equation can be written

(7) Solving equation (7) for the values for F and 8, it is possible to check that the assumed number of nodes, n, really are in contact. This is done by finding the direction of the contact force 10. If the contact force is in tension for a particular pair of nodes on solution of an iteration, they are released for the next iteration; otherwise they remain in contact. Along with this a check is made as to whether there is interference between the boundaries of the two bodies by using equation (5) and the initial gap value, hi.

3.3 A particular case Due to the special properties of the involute profile, the direction of all loads, F , , F , , F,, ..., F, must be tangent to the base circle as shown in Fig. 3. Hence all radii in equation (7) are equal to the base radius rb, that is r1 = 1, = r3 = ... = r n = rb

(b)

(C)

Fig. 2 Contacting bodies and their displacements 0 IMechE 1994

These properties of the involute profile enable contact to be modelled outside the theoretical line of action, as shown by points A and B. Knowing the contact force, from equation (7), for each pair of nodes and then for each pair of teeth, enables the load sharing to be determined. For a unity gear ratio the rigid body rotation 8 Proc Instn Mech Engrs Vol 208

D WALTON, A A TESSEMA, C J HOOKE AND J M SHIPPEN

84

Theoretical contact line

V Fig. 3 Meshing of involute spur gear teeth

is equal for both driver and driven gear; otherwise the division will be dependent on the gear ratio.

3.4 General procedure adopted for the evaluation of deflection and load sharing The procedure adopted in using the flexibility method for contact problem analysis is given in Fig. 4. A preprocessor routine using I-DEAS (integrated design engineering analysis software) was used to prepare the finite element mesh at a particular meshing point (roll angle), as shown in Fig. 5a. Linear, triangular and plain strain elements were used for the finite element mesh, as these enabled the transition from fine to coarse mesh to be modelled relatively easily. A fine mesh was used at

Model preparation using I-DEAS

Flexibility matrix extraction using MSciNastran I

I

I

(h)

Fig. 5 Finite element mesh

each contact position to model the contact zone accurately (Fig. 5b). MSC/Nastran (finite element analysis software) was used to generate the flexibility matrix for the gear teeth models. The necessary reduction of the system flexibility matrix to selected potentially contacting nodes at the boundary of the gear teeth was also carried out in MSC/Nastran. This reduction greatly reduced the computing time required for the iterations to arrive at a contact solution. A program was written in FORTRAN which solved the contact problem and directly calculated load sharing. In this program, the flexibility method of contact analysis was used, as described in Sections 3.2 and 3.3.

Contact problem solution FORTRAN program

4 CHANGE IN OPERATING CONTACT RATIO Solution (displacement and stress) MSciNastran

Graphical output using I-DEAS

Fig. 4 Procedure for the analysis of deflection and load sharing Part C: Journal of Mechanical Engineering Science

4.1 Formulation of a non-dimensional gear elasticity parameter T Changes in operating contact ratio from the ideal value indicate a function of the gear tooth mesh stiffness. The stiffness of gears is dependent on geometry, materials and operating conditions. A standard gear tooth form can be represented by the module, number of teeth and pressure angle. Gear material properties are given by the elastic modulus, Poisson’s ratio and coefficient of friction, while the operating conditions can be charac0 IMechE 1994

LOAD SHARING IN METALLIC AND NON-METALLIC GEARS

terized by the amount of backlash, force per unit face width, rotational speed, lubricant viscosity and temperature. Thus, for a solid gear, the difference in operating contact ratio, R,, ,from the ideal value, I,, ,can be expressed as

85

no matter which parameter was varied, the change in contact ratio was found to be the same.

4.2 Gear specification

Five gears were used in the analyses, designated as types I to V, details of which are given in Table 1. However, not all of the parameters listed above are independent. For example, the elastic modulus for polymer gears is dependent on temperature. The effect of some of the parameters are less important and can be neglected, such as lubricant viscosity, friction and rotational speed, or can be kept constant, such as the backlash. The effect of friction can be ignored as it has only a minimal effect on load sharing. In the same way the effect of Poisson's ratio on tooth deformation is secondary and is generally about the same for most gear materials. It is possible to group pressure angle and number of teeth together, resulting in the parameter 'ideal contact ratio'. For standard tooth forms the ideal contact ratio can be defined by specifying only the number of teeth in the wheel and pinion and the pressure angle. Hence, equation (8) reduces to

4.3 Variation of load sharing with roll angle

Load sharing between teeth can be shown as the load fraction against roll angle (Fig. 6). The broken line represents the variation in ideal load sharing with roll angle. First and last contact points take place at 9.9" roll angle from the pitch point and are designated as f and 1. At one base pitch, 12" roll angle, from the first contact point there is another pair at position 1 and one base pitch from the last contact point there is another pair of teeth in contact at position 2. Hence points 1 and 2 designate the transition from a single to a double pair of teeth contact and from a double to a single pair of tooth contact respectively. Between the first point of contact and position 2 and from position 1 to the last contact point, there are two pairs of teeth in contact, each carrying half the total load. Between positions 1 and 2 there is only one pair of teeth in contact, which carries the whole load. As shown in Fig. 6, the variation of real load sharing for small transmissible loads is not significantly different from the ideal. For 0.1 N m applied load, premature contact occurs near the first contact point and the tooth pair also comes out of contact near the theoretical last

(9)

According to the theorem of dimensional analysis, as there are five variables and two reference dimensions (N, mm), three non-dimensional groups can be created, T , I,, and R,, (24). T is defined as

8l

Therefore, for standard tooth forms, it is possible to characterize the variation of operating contact ratio from the ideal value using the ideal contact ratio and the non-dimensional parameter T. Equation (10) relates the meshing of gears made of the same material. For polymer gears of different pinion/wheel material combinations or for polymer gears meshing with a steel gear an effective value of E might be adopted, that is

120

Theoretical 0.1 N m x x x 6.0Nm A A A 40.0 N m E = 3.0 GPa +++

=

40

0

To test that the non-dimensional parameter was linear with the parameters T , r b , W , E and m,a series of analysis was made by varying one of the parameters while keeping the rest constant. For the same value of I-r;

~

- 6 3

15

3

0

6

9'

12

Roll angle deg

Fig. 6 Variation of load sharing with roll angle

Table 1 Specification of gear models [all models are to BS 436: Part 2: 1970 (25)]

@ IMechE 1994

TYPe

I

I1

111

1v

Module (mm) Number of teeth Pressure angle (deg) Profile modification (mm) Ideal contact ratio for 1:1 gear ratio Face width (mm)

2.0 30 20.0 0.0 1.65 17.0

1.o 60 20.0 0.0 1.78 17.0

2.0 18 20.0 0.0 1.53 17.0

3.0 40 20.0 0.0 1.71 17.0

V 4.0 100

20.0 0.0 1.85 17.0 Proc lnstn Mech Engrs Vol208

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D WALTON, A A .TESSEMA, C J HOOKE A N D J M SHIPPEN

points of contact. Hence the increase in the length of the line of action and change in contact ratio is not large. On the other hand, at higher loads, the variation from the ideal load sharing diagram is significant. First and last contact occurs well before the theoretical point which results in an increase in the length of the line of action and contact ratio. As shown in the figure, the maximum load carried by a single pair of teeth, for an applied load of 40 N m, is about 65 per cent of the total load. This implies a large decrease in the load carried by a single pair of teeth and a potential improvement in the bending strength of the gears. While a torque of 40 N m might be carried by some composite polymer gears of the size modelled here, the maximum load that can be applied to dry running acetal gears is about 10 N m. For loads below 6 N m, a single tooth pair just carries the whole load at the pitch point. Thus no appreciable improvement in the bending strength can be expected for module 2 mm polymer gears with the properties described in Table 1 and running at normal rated loads. However, the increase in real contact ratio and hence improvement in bending fatigue strength depends on the pitch, number of teeth and pressure angle of the gear. This is shown in the next section. 4.4 Variation of real load sharing with

The variation of real load sharing with 'T is given in Fig. 7 for the gears described in Table 1 and for a gear ratio of 1:l. Real load sharing here is defined in the

I I 400

04

0

I

800

!

1200

7 = T/(rbWmE) x lo-' (a) In linear scale

i: = T/(r,WmE) x

---I J

same way as by Yelle and Burns (I), that is R,, = Z,,/Pb. Figure 7a shows, on a linear scale, the variation of real load sharing with T for a range of ideal contact ratios. When the non-dimensional parameter T is zero the real contact ratio is the operating contact ratio. As T is increased, real load sharing increases at a variable rate. Figure 7a can be seen either as the measure of increase in load sharing or as a measure of interference. The increase in load sharing is due to contact outside the theoretical line of action, but contact outside this line also indicates tooth interference. Interference should be avoided in metallic gears since it can lead to catastrophic tooth failure. However, for some polymer gears a small amount of interference may cause no problems, For acetal gears, in which wear is the main cause of failure (3), interference may result in the tooth flank wearing at premature points of contact, although experimental evidence suggests that they wear to provide a conformal pattern. Nylon gear teeth have been shown to deform viscoplastically at points of premature contact without sign of gross wear. Figure 7b shows the curve in Fig. 7a for a contact ratio of 1.65 where the abcissa has been redrawn to a logarithmic scale. On this figure, the operating regimes for metallic and non-metallic gears can be shown and are designated by zones 1 and 2 respectively. A typical operating load for a type I (see Table 1) dry running polymer gear is 5-10 N m while the same sized gear made of steel could, depending on speed, carry a load from 20 to 70 N m [as rated by BS 434: 1970 (25)]. For typical elastic modulii of 209 and 3 GPa for steel and polymers respectively, T varies from around 10 to 34 x for steel gears and from around 170 to 350 x for polymer gears. This figure thus shows that the difference in real contact ratios from the ideal for metallic gears tends to be small compared to nonmetallic gears.

5 CONCLUSIONS

A finite element method of determining load sharing between teeth with edge contact is given. Analyses of the change in operating contact ratio from the ideal were carried out using a non-dimensional parameter T = T/(r, WmE). The parameter has been found to be useful for indicating the differences in operating conditions between metallic and non-metallic gears. It shows that the change in operating contact ratio for nonmetallic gears is large and can give cause for concern. The method has reduced the amount of computing time that might be required to analyse the effect of different parameters, such as materials, tooth size, torque, etc. It provides design information concerning load sharing and interference without any particular reference to material properties, gear tooth dimensions or operating conditions. 'T might also be used in the rating procedure of nonmetallic gears. For example, in BS 6168 (26) the tooth root bending stress is given by

(b) In semi-logarithmic scale

Fig. 7

Variation of real contact ratio with

Part C : Journal of Mechanical Engineering h e n c e

@ IMechE 1994

LOAD SHARING IN METALLIC AND NON-METALLIC GEARS

where Y, is a factor which allows for load sharing and is given by

r, = 0.2 + 0.8 7 '

Cf

The factor allows for the ideal contact ratio. However, this work and others (1, 2) suggest that the increase in real contact ratio for non-metallic gears is significant and might affect their bending fatigue strength, requiring a redefinition of Ye using the real contact ratio. It should also be noted that it is not yet clearly known whether the increase in fatigue strength for fine pitch non-metallic gears is due to increases in real load sharing, size effects or both. A further development of this work would be to determine the amount of tip easing and backlash required in gears as a function of T , an area of study currently under investigation. ACKNOWLEDGEMENTS

The authors would like to thank the ORS (Overseas Research Students) awards scheme, the School of Manufacturing and Mechanical Engineering, the University of Birmingham and the Higher Education Main Department of Ethiopia for their financial support which enabled this project to be carried out. REFERENCES 1 Yelle, H. and Burns, D. J. Calculation of contact ratios for plastic/ plastic or plastic/steel spur gear pairs. Trans. A S M E , J . Me&

Des., 1981,103, 528-542. 2 Yelle, H. and Burns, D. J. Root bending fatigue strength of acetal spur gears-a design approach to allow for load sharing. Presented at the AGMA Fall Technical Meeting, 1981. 3 Walton, D., Hooke, C. J., Mao, K., Breeds, A. and Kukureka, S. A new look at testing and rating non-metallic gears. Proceedings of Third World Congress on Gearing and power transmission, Paris, 1992, pp. 683-693. 4 Timoshenko, S. and Baud, R V. Strength of gear teeth is greatly affected by fillet radius. Automotiue Industries, 1926. 5 Walker, H. Gear tooth deflection and profile modification. The Engineer, 1938, pp. 319-412 and 434-436. 6 Weher, C. The deformation of loaded gears and the effect of their load carrying capacity. British Scientific and Industrial Research Sponsored Research (Germany), Report 3, 1949.

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7 Attia, A. Y. Dynamic loading of spur gear teeth. Trans. A S M E , J. Engng Industry, 1959. 8 Attia, A. Y. Deflection of spur gear teeth cut in thin rims. T r a m . A S M E , J. Engng Industry, 1964. 9 Cornell, R. W. Compliance and stress sensitivity of spur gear teeth. Trans. A S M E , J. Mech. Des., 1981,103-447. 10 Baronet, C. N. and Tordion, G. V. Exact stress distribution in standard gear teeth and geometry factors. Trans. A S M E , J . Engng Industry, 1973,1159-1163. 11 Terauchi, Y. and Nagnmura, K. Study on deflection of spur gear teeth (first report, calculation of tooth deflection by two dimensional elastic theory). Bull. J. Soc. Mech. Engrs, 1980, 231843, 1682-1688. 12 Aida, T. and Terauchi, Y. On the bending stress of a spur gear (first report, stress at the fillet curve and inner stress at the root of gear tooth). Bull. J . Soc. Mech. Engrs, 1962, q17), 161-170. 13 Chakraborty, J. and Hanoshikatti, H. G. Determination of the combined stiffness of a spur gear pair under load. Trans. A S M E , 1974,74-DET. 39. 14 Coy, J. J. and Chao, C. H. A method of selecting grid size to account for Hertz deformation in finite element analysis of spur gears. Trans. A S M E , J . Mech. Des., 1982,104,759-766. 15 Tanaka, M. and Kohida, H. Stress analysis of spur gears in consideration of the ratio of contact. Bull. J . SOC.Mech. Engrs, 1985, 28(243),2 128-2135. 16 Steward, J. H. The compliance of solid, wide faced spur gears. Proceedings of the 1989 International Conference on Power transmission and gearing, 1989, Vol. 1, pp. 181-188. 17 Harris, S. L. Dynamic loads on the teeth of spur gears. Proc. Instn Mech. Engrs, 1958, 172,87-112. 18 Gregory, R. W., Harris, S. L. and Munro, R. G. Dynamic behaviour of spur gears. Proc. Instn Mech. Engrs, 1963,178,207-226. 19 Munro, R. G. A review of the theory and measurement of gear transmission error. IMechE Conference on Gearbox noise and vibration, Cambridge, 9-11 April 1990, paper C404/032, pp. 3-10 (Mechanical Engineering Publications, London). 20 Bath, K. and Chaudhary, A. A solution method for planar and axisymmetric contact problems. Int. J. Numer. Methods Engny, 1985,21.65-68. 21 Simo, J. C., Wriggers, P. and Taylor, R. L. A perturbed Lagrangian formulation for the finite element solution of contact problems. Computer Methods in Appl. Mechanics and Engng, 1985,163-180. 22 Francavilla, A. and Zienkiewicz, 0. C. A note on numerical computation of elastic contact problems. Int. J . Numer. Methods Engng, 1975,9,913-924. 23 Chandrasekaran, N., Haisler, W. E. and Goforth, R. E. A finite element solution method for contact problems with friction. Int. J . Numer. Methods Engng, 1987,24,477-495. 24 Huntley, H. E. Dimensional analysis, 1951 (Macdonald, London). 25 BS 436: Part 2: 1970 British standard specijcation for spur and helical gears (British Standards Institution, London). 26 BS 6168: 1987 British standard specification f o r non-metallic gears (British Standards Institution, London).

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