THE ANNALS OF DUNAREA DE JOS UNIVERSITY OF GALATI FASCICLE X APPLIED MECHANICS, ISSN 1221-4612

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1. Introduction The contact stresses are responsible about surface

contact fatigue. The surface contact fatigue leads to pitting on teeth's surface. The Hertzian contact stresses theory is used to determine the allowable contact stresses and allowable loading which is re-sponsible about them. Because of simplifying pre-sumptions which are made it should to introduce a number of coefficients to render an account of spe-cial features of gear wheel teeths contact. Neverthe-less it is no sufficiently to give an account all of them.

Therefore the use of possibilities of CAD (com-puter assisted design) technology to generating three dimensional models of gear wheels and CAE (com-puter assisted engineering) technology to obtaining stresses is very important to get a more realistic pic-ture of loading's influence and of distributions of contact stresses.

This work is a report of efforts to simulate the contact stresses at loading between one tooth of dif-ferent gear wheels and one tooth of gear rack by means of possibilities of software products Solid-Works2005 and CosmosWorks2004.

2. Modeling of Gear Wheel and Gear Rack

The software package SolidWorks2005 now have an advanced tool Equations by means of which it is possible to assign mathematically rela-tion between elements of geometry, and the possi-bility to introduce global variables. This allow the models to be maximally close to real and the possi-bility to reshape the models according new values of parameters number of teeth z, module of teeth m, pressure angle , profile shift coefficient x (also known as the addendum modification coefficient), etc.

The technique of modeling begins with genera-tion of basic sketch. In conditions of this generation and after the activation of Equations (under the menu Tools) are introduced global variables - num-ber of teeth z, module of teeth m, pressure angle , profile shift coefficient x and as global constants the coefficients - of addendum h*a = 1,0 , of dedendum h*f = 1,25 , of fillet radius *a = 0,38. Then are drawn the standard pitch circle, the base circle, the addendum circle and the dedendum cir-

Some Results from Simulating of Contact Stresses in the Case of Meshing of Gear Wheel and Gear Rack

Dimitar Petrov

Technical University Sofia branch Plovdiv Bulgaria

ABSTRACT Technique of modeling in the environment of CAD software SolidWorks'2005 about gear gears with different number of teeth z, module of teeth m, pressure angle , profile shift coefficient k, about gear rack and about its assembly are presented in detail. Then three-dimensional CAD models of the assembly are subjected by means of simulation loading to reaching allowable limit contact stresses. The investigation's results derived by means of finite elements calculation method about the influence profile shift coefficient x, number of teeth z, fillet R of gears wheels tooth on its limit capacity of loading are presented. These results and the results obtained by means of Hertzian contact theory are compared. An analysis about its capacity of loading is made and some conclusions are formulated .

KEY WORDS: contact stresses; CAD, CAE simulation.

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cle. Its diameters are determined by means of the tool Equations as follows [1], [2]:

d = z.m db = d.cos() da= m.( z + 2.h*a + 2.x) df = m.( z 2.(h*a + c*) + 2.x)

Fig. 1 2-D sketch with standard pitch circle,

base circle, addendum circle and dedendum cir-cle

Then is generated the involute as a spline line through n points, beginning from most upper point of base circle (fig.2). In general it is done in such way: - it is determined the profile angle a which corresponds to the addendum circle, this angle is divided into n-1 equal intervals for determining of the step of increasing a, this step is used in a cy-cle FOR NEXT i in the calculating of polar coor-dinates (inv i , ri) of consecutive i-points of the involute by the formulas:

a = arccos(db da) a = a (n1) FOR i =1 TO n invI = [tg(i.a) (i.a).180].180 ri = db (2.cos(i.a)) NEXT i

In such way the first point of the involute curve

will be on the base circle, n3 points will lie be-tween the addendum circle and dedendum circle, the next to last point will lie on the addendum circle, and the last point will be outside the addendum cir-cle. Through these points is stretched the involute. Then the fillet is drawn between the involute and dedendum circle with radius:

=*a .m. After that is made a mirror image of the both, the

involute and its fillet, about a line that divides

equally intertooth's space. The mirror line (Fig.2) is a line which passes through the center of base circle at an angle = psi from first point of the involute. The angle = psi is defined by means of sequence of equations:

p = m. pb = p.cos() s = m. [ 2+2.x.tan()] inv() = [tg() .180] sb = db.[s d + inv()] eb = pb sb = [(eb 2 ) (db 2 ) ] .(180 )

,where p is the standard pitch, pb the base pitch, s the tooths width on standard pitch circle, inv() the involutes function of profile angle which corresponds to standard pitch circle, sb the tooths width on the base circle, eb the intertooth's on the base circle.

Fig. 2 An involute and a mirror image of it.

After that new sketches are generated. The geo-

metrical elements from the first (basic) sketch are converted in these new sketches. Then they are used to draw out the three-dimensional Features - it is illustrated from Fig. 3 to Fig.6.

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Fig.3 - Drawing out of a cylindrical body of a future gear wheel.

Fig.4. Cutting through of a cavity between teeth.

Fig.5 - Multiplying of the cavitys cut-through.

Fig.6 -Three-dimensional model of gear wheel

having z = 20, m = 2 , x = 0.

The diameter of the gear wheels model is cho-sen to be:

din = da - 12 . m

Then only three cogs (Fig.7) of this model will be used (for reducing of the number of finite elements and therefore for reducing the volume of calculations) in the static analysis of loading by means of software package CosmosWorks'2004.

Fig.7 -Three-dimensional model of gear wheel

having z = 20, m = 2 , x = 0. The used technique of gear wheels modeling

gives opportunity quickly and easily to generate the models of these gear wheels which can have: dif-ferent profile shift coefficient x; different number of teeth z; different module of teeth m; different pressure angle , etc.

Analogically but much more simply is created model of one tooth of gear rack (Fig.8).

Fig.8 - First stage of the creating of a tooth of

gear racks model.

3. Assembly, constraints and loading Flank pitting is caused by alternating normal

pressure on the contact surfaces of the teeth. It is found to occur most frequently at the pitch circle - where relative sliding of the teeth is zero and the hydrodynamic lubricant film tends to break down - so attention is focused on this location.

For this reason it is needed to set some reference planes which will guarantee the contact line to be on pitch cylinder of gear wheel. Therefore is set

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reference plane Dopiratelna which touch tangen-tially the gear wheel tooths surface at the pitch cyl-inder (Fig.9).

Fig. 9 The reference plane Dopiratelna which touch tangentially the gear wheel tooths surface

at the pitch cylinder.

Because of this, the touching (between the gear wheel's tooth and gear rack's tooth) appears different at different profile shift coefficients x = 0,3 (Fig.10); x = + 0,3 (Fig.11) and x = + 0,9 (Fig.12). The touching occurs always at a point on pitch cir-cle, but at profile shift coefficients x = 0,3 this point is near to the addendum circle, whereas at pro-file shift coefficients x = + 0,9 this point is near to the dedendum circle.

Fig.10 -Three-dimensional assemblys model with gear wheel having z = 20, m = 2 , x = 0,3.

Fig.11 -Three-dimensional assemblys model

with gear wheel having z = 20, m = 2 , x = + 0,3.

Fig.12 -Three-dimensional assemblys model

with gear wheel having z = 20, m = 2 , x = + 0,9. The simulation of the loading is made in the Cos-

mosWork'2004's environment. The gear wheel is fixed at the hole's cylindrical face (Fig.13). The gear rack's tooth is restrained at flat faces on which it can slide (Fig.13).

The loading force is normal to the face (showed on Fig. 13) of the gear rack. This force is equal to useful tangential component Ft of the contact force at the pitch point [3].

Contact pair is of type surface touching. The program CosmosWorks2004 [4] creates node-to-

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area pairs automatically. Each node from one of the faces is associated with an area defined by nodes on the other face. The faces may move away from each other but will preserve the physical requirement that they may not penetrate each other. The meshes on both faces may or may not be identical. Infinitesimal sliding is considered.

Fig.13 Scheme of loading - imposing of re-

straints and applying of force.

The different investigated models of gear wheels was having: standard module m = 2 [mm], number of teeth z = 20, 40 or 60; profile shift coefficient x from 0,6 to + 0,9 trough step of increasing 0,3; width of gear-wheel crown b=10.m.

The material for both, the gear wheel and the gear rack, was chosen to be plain carbon steel. The op-tions of mesh were: Mesher Used: Standard; Auto-matic Transition: Off; Smooth Surface: On; Jaco-bian Check: 4; Element size: 1 mm; Tolerance: 0.05 mm; Mesh quality: High; Total nodes: 45000; Total elements: 30000. Used solver method was FFEPlus iterative method [4].

The loading force was increasing until the contact stresses do achieve, accepted here, the allowable contact stress HP = 480[MPa]. The force at which the maximal contact stresses become equal to 480[MPa] are named here limit force Flimit or Capac-ity of Loading. Comparison between different cases of gear wheels are made by means of the value of the Flimit

4. Results from loading simulation The first result was irregular distribution of contact

stresses (Fig.14) - the maximal contact stresses are concentrated in the two ends of the contact line.

For reducing the stress concentration was been used fillet on the surface's edges of gear wheel's tooth's (Fig.15). The tested values of fillets was from

R = 0,1 [mm] to R = 1[mm] with step of increasing 0,1 [mm] and for different number of gear wheels teeth z. (z =20, z=40 and z=60).

Fig.14 Picture of stresses distribution.

Fig.15 Applying of fillet.

The obtained results about influence of applying different fillets on the Capacity of Loading Flimit were contradictory (Fig.16).

Capacity of Loading

0500

10001500200025003000350040004500

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1R [mm]

Flimit [N]

z = 20 z = 40 z = 60

Fig.16 Simulation results about influence of R on Flimit.

The influence of different profile shift coefficient

x on loading capacity is illustrated in fig. 17 (at

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fillet radius R = 0[mm]) and in Fig.18 (at fillet ra-dius R = 0,3[mm]).

Capacity of Loading (at R = 0[mm])

0500

100015002000250030003500400045005000

-0.6 -0.3 0 0.3 0.6 0.9Profile shift coefficient x

Flimit [N]

z=20 z=40 z=60

Fig.17 - Simulation results about influence

of x on Flimit.

Capacity of Loading (at R = 0.3 [mm])

0500

100015002000250030003500400045005000

-0.6 -0.3 0 0.3 0.6 0.9Profile Shift Coefficient x

Flimit [N]

z=20 z=40 z=60

Fig.18 - Simulation results about influence

of x on Flimit.

If we calculate the tangential limit force FHlimit at the allowable contact stresses HP by means of Hertzian contact formula [1] (where: E* - equivalent modulus; * - equivalent radius; b - width of gear wheel, Ft - loading force; - standard pressure an-gle) :

HP = 0,418.[(Ftcos b).(E* *)]1/2 ,we will receive the values as is shown in Table 1. These forces are about 2 times less in comparison with these limits forces, which are received from simulation.

Table 1 Contact Capacity of Loading Flimit

z = 20 z = 40 z = 60 FHlimit[N] from Hertzian contact theory 807.25 1614.5 2421

FHlimit[N] from simulation (at x=0 and R=0) 2363.4 1945.7 4036.22 An another special feature from simulation was the

fact, that in some cases, the maximal stresses in the zone of contact were not only on the surface, but in the internal volume of gear wheel's tooth (Fig. 19) under the surface.

Fig.19 Maximal stress here is in the volume.

5. Remarks

The simulations of contact stresses presented: 1. The distribution of contact stresses in the zone

of contact is not constant. The maximal contact stresses are concentrated in the two ends of the contact line.

2. The Capacity of Loading reduces with the in-creasing of profile shift coefficient x.. We should refer to Fig.10, Fig.11 and Fig.12 to interpret this fact - when profile shift coefficient is bigger, the hardness of gear wheel's tooth is bigger too, and in contrast of that the hardness of gear rack's tooth declines. Thus the gear rack's tooth deforms and presses much more in the ends, not in the middle of contact line. 3. The explanation of less Capacity of Loading by Hertzian contact theory (particularly at less coeffi-cient x - Fig. 17) should search in the pliability of the gear wheel's tooth of the model in comparison with theoretical cylinder at the Hertzian contact theory. So in the model the stresses redistribute in the volume of tooth. 4. The increasing of fillet R does not increase Ca-pacity of Loading. One reason for this is the fact that with increasing of R, the length of contact line reduces.

REFERENCES [1] ., ., ., . , , , 1980. [2] ., ., ., ., ., . , ISBN 954-03-0030-4, , , 1992. [3] http://www.mech.uwa.edu.au/DANotes/gears/failure/ failure.html. [4] COSMOS Works 2004 Online Users Guide.