Deformation overlap in the design of spur and helical gear pair

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Finite Elements in Analysis and Design 40 (2004) 1361 – 1378 www.elsevier.com/locate/"nel

Deformation overlap in the design of spur and helical gear pair Su-Jin Parka;∗ , Wan-Suk Yoob a

Department of Mechanical Design Engineering, Pusan National University, 30 Jangjeon-dong, Geumjeong-gu, Busan 609-735, South Korea b School of Mechanical Engineering, Pusan National University, 30 Jangjeon-dong, Geumjeong-gu, Busan 609-735, South Korea Received 25 March 2003; accepted 15 October 2003

Abstract The elastic de6ection of gear teeth is analyzed to investigate the deformation overlap. The deformation overlap, which is a numerically calculated quantity through displacement analysis at the initial contact, is de"ned as the piled region of a contact tooth pair due to the elastic deformation. The deformation overlap is suggested for an e9ective indicator to represent the whole deformation of a meshing gear pair. The elastic contact theory and "nite element method are used to compute the contact force and teeth de6ection. The contact problem is de"ned as a QP problem, and the contact forces between teeth are calculated from the transmitted torque. Then the deformation overlap is calculated with the contact forces as boundary conditions. For a spur gear pair, the calculated deformation overlap is used for the basis of the tooth tip relief, analysis of deformation characteristics for a pro"le shifted gear pair, and the selection of pro"le shift coe;cient considering teeth de6ection. Deformation overlap is extended to a three-dimensional problem, and implemented to a helical gear pair. ? 2003 Elsevier B.V. All rights reserved. Keywords: Teeth de6ection; Deformation overlap; Contact problem; Finite element analysis; Helical gear pair; Spur gear pair

1. Introduction A gear system, which transmits power by means of meshing gear teeth, is conceptually simple and e9ective in power transmission. The knowledge of gear teeth de6ection can be used to calculate ∗

Corresponding author. Tel.: +82-51-510-1457; fax: +82-51-514-7640. E-mail addresses: [email protected] (S.-J. Park), [email protected] (W.-S. Yoo).

0168-874X/$ - see front matter ? 2003 Elsevier B.V. All rights reserved. doi:10.1016/j."nel.2003.10.003

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tooth load sharing, pro"le relief, deformation characteristics, and pro"le shift coe;cient [1]. Thus, the de6ection analysis of a elastic teeth pair is important for the high performance of geared system [1–4]. Gear tooth de6ections were investigated by many authors. Umezawa and Ishikawa [5,6] presented experimental studies for the de6ections and moments of a gear tooth. In their approaches, de6ections and moments of rack-shaped cantilever with "nite width were taken into consideration. Among the computer-aided approaches to predict the displacement "eld on a loaded gear tooth, the dynamic stress, deformation, and fracture of gear teeth are simulated by Wallace and Seireg [7]. Chabert et al. [8] also presented valuable results using "nite element method (FEM) on the stresses and de6ection of spur gear teeth induced by a static load. The complex potential method was used to calculate spur gear tooth 6exibility by Cardou and Tordian [9]. Stegemiller and Houser [10] developed some models to compute the base 6exibility of wide facewidth gear teeth. Fukunaga [11] presented a study of the tooth de6ection and strength analysis on the spur gears including ratio of contact. The analysis was performed based on the measurement of root stress and FEM. The Rayleigh-Ritz approach had been applied by Yau et al. [2] to study the bending and shear de6ections of gear teeth. Gagnon et al. [1,3] analyzed gear de6ection by the thick plate "nite strip formulation. They used Mindlin’s theory to extend the application to thick plates. The de6ection of gear teeth is mainly caused by the load sharing between meshing gear teeth, and gear tooth height and thickness a9ect on the gear tooth de6ections. In the analysis and design of a geared system, it is necessary to consider the deformations of two gears, a driving gear and a driven gear. This paper presents an approach of studying the de6ection of gear teeth using the deformation overlap. Deformation overlap, which is a calculated quantity through displacement analysis, is de"ned as the piled region of the contact tooth pair at the initial contact. Tooth load sharing was obtained by the FEM and elastic contact theory, and it was used as the force boundary conditions for the displacement analysis. Deformation overlap is "rst adopted for the tooth pro"le modi"cation and the design of pro"le shifted gears, and then implemented to three-dimensional helical gear pairs. 2. Denition of deformation overlap The tooth pairs that make contact before and immediately after are referred to as the leading and the trailing tooth pairs, respectively. The deformation overlap is de"ned when contact begins in the trailing tooth pair. In ideal conditions, the contact is a smooth one at the instant when the trailing tooth pair makes contact. In other words, the relative angular positions of the two gears are such, that they allow the tooth pair to come smoothly into contact. However, if the tooth spacing on each gear is not exactly constant, the discrepancy of the velocities of contact points along the common normal may be produced, and there may be an impact at the initial contact of some tooth pairs [12] Also, when the gear is heavily loaded, the tooth 6exibility allows the driven gear to lag behind its ideal angular position, and thus the trailing tooth pair contacts before the ideal time of contact. Just prior to the initial contact of trailing tooth pair, load sharing between tooth pairs should be calculated, and used as force boundary conditions for the displacement analysis. When the deformation analyses of driving and driven gear are performed, there is a region that overlaps by the deformation at the tooth tip of the driven gear as presented in Fig. 1. The deformation overlap q in Fig. 1 is de"ned as the maximum depth of the piled region of a contact tooth pair. The deformation

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Fig. 1. Deformation overlap in (a) ideal contact (deformation overlap q = 0) and (b) actual contact (deformation overlap q = 0).

overlap q is zero in ideal contact, and it is not zero in actual situation. Gear teeth contact may be already occurred by deformation before their ideal time of contact, and the deformation overlap is proportional to the de6ections of gear teeth. The tooth load sharing can be obtained by the solution of the elastic contact problem for the geared system. 3. The elastic contact problem for the geared system The contact problems involve unknown boundary conditions. Speci"cally, the actual contacting surface, stresses, and displacements on the contacting surface are all unknown prior to the solution of the problem [13,14]. The two bodies that come into contact are named as body 1 and body 2, respectively. In the contact problem of gear pair, driving gear (body 1) is a contactor and driven gear (body 2) is a target [15], and the two bodies occupy domains 1 and 2 in a "xed global coordinate system as presented in Fig. 2. Let the vector x be the position of a generic material particle with respect to an inertial coordinate frame and x0 be the position vector of the same particle in the undeformed state. The displacement vector u of the same particle is given by u = x − x0 :

(1)

The Cauchy stress ij and the strain tensor ij are written as ij = Cijkl kl ;

(2)

ij = (ui; j + uj; i )=2;

(3)

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Fig. 2. Schematic representation of contact between two bodies.

where Cijkl denotes the elastic constitutive coe;cient. The notation (); j denotes di9erentiation with respect to the coordinate xj , and the summation convention of summing on all repeated indices is used. The equilibrium equation is given by ij; j + bi = 0;

(4)

where bi is the body force, which will be neglected for the static analysis of a geared system. The boundary condition set of each body is divided into three disjoint parts: U on which displacements are prescribed, F on which tractions are prescribed, and the potential contact surface C . The basic assumption along the contact surface is that contact force is compressive only when the contact surface is assumed unbonded. No friction is included such that tangential components of the contact force are zero. Then the condition on the normal traction S is S = −ijk nki nkj ¿ 0;

(5)

where the superscript k denotes the kth body and n = {ni } is the outward unit normal vector on the boundary surface. The geometry of the potential contact surface of the kth undeformed body shown in Fig. 2 is described by Hu and Kwak [13,16] gk (x0k ) = 0;

(6)

then the impenetration compatibility condition states as g1 (x02 + u2 − u1 ) 6 0

on C :

(7)

Since the normal traction must vanish whenever the gap opens, or vice versa, the following condition should be satis"ed: S · g1 = 0 on C :

(8)

Another geometric compatibility condition using g2 for the potential contact surface of body 2 can be expressed as follows: g2 (x01 + u1 − u2 ) 6 0

on C :

(9)

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Since this condition is the same as condition g1 of Eq. (7), the simpler one of these two equations can be used. Geometric boundary conditions for each body can be written as ui = uR i on U ;

(10)

where uR i is the prescribed displacement. Force boundary conditions are written as ij nj = FR i on F ; where FR i is the prescribed traction. The governing equation of contact problem is given by J (u1 ; u2 )

min

(12)

s:t: g2 (x01 + u1 − u2 ) 6 0 where


J=






1 2

(11)

ij ij d −

F

on C ;

FR 1i ui1 dF −


F

(13)

FR 2i ui2 dF :

(14)

The "nite element equation and linearization of constraint g2 are as follows [13,17]: T

T

T

T

J = 12 U1 K1 U1 + 12 U2 K2 U2 − R1 U1 − R2 U2 ;

(15)

g2 ≈ M 1 U 1 + M 2 U 2 − G 6 0

(16)

on C ;

where U, K, R are the nodal displacement vector, the sti9ness matrix, and nodal force vector, respectively. M is the gradient matrix of geometric compatibility condition g(x) and G is the initial gap size. For the e;cient calculation of Eq. (12), a dual problem is set as follows [13,16]: max min P

U1 ;U2

s:t:

[J + PT (M1 U1 + M2 U2 − G)] P ¿ 0:

(17) (18)

Necessary conditions for the minimization problem of Eq. (17) are T

(19)

T

(20)

K1 U1 − R1 + M1 P = 0; K2 U2 − R2 + M2 P = 0;

where P is the Lagrange multipliers as dual variables. Finally, dual QP (quadratic programming) equations are written as follows [13,16]: max P

[ − 12 PT QP − LT P]

(21)

s:t:

(22)

P ¿ 0;

where T

T

Q = M1 (K1 )−1 M1 + M2 (K2 )−1 M2 ;

(23)

L = G − M1 (K1 )−1 R1 − M2 (K2 )−1 R2 :

(24)

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In this study, the Lemke [18] method is used for the solution of the above QP problem. The gear teeth contact problem can be treated as an elastic contact problem, and the above elastic contact problem is adopted to obtain gear contact forces by using Fortran language. 4. Investigation of deformation overlap 4.1. Finite element model The gear pro"les generated by rack cutters are used in this study. The involute curve of each gear tooth is de"ned by the straight part of the rack cutter tooth, while the shape of the "llet is de"ned by the curved section at the tip of the rack cutter tooth, i.e. tip circle [12,19]. When pressure angle, module, number of teeth, and pro"le shift coe;cient are given, the involute curve is computed by the involute function. The "llet curve is obtained by tracking the trace of the cutting point of tip circle. The "nite mesh is developed with four-node plane strain quadrilateral elements. In Fig. 3, the FE model of standard driving gear and driven gear is generated with module m = 2:25, number of driving gear teeth Z1 = 42, number of driven gear teeth Z2 = 49, pressure angle  = 17:5◦ . Tables 1 and 2 show the detailed data of the standard gear system and its "nite element model with boundary conditions, respectively. The isoparametric four-node quadrilateral element [15] was used and 40 elements were inscribed at each potential contact tooth surface.

Fig. 3. FE model of a standard spur gear system.

Table 1 Data of a standard spur gear system Gear

Driving gear

Teeth Pressure angle (deg) Module Transmitted torque per unit facewidth (kgf mm) Young’s modulus (kgf =mm2 ) Poisson’s ratio

42

Driven gear 17.5 2.25 1432 21 000 0.3

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Table 2 Data of FE model for the standard spur gear system Gear

Driving gear

Driven gear

Node Element

4666 4360

4697 4400

Boundary conditions

Geometric B. C. on the rim (mm) Radial Circumferential

0.0 0.0158

Radial Circumferential

0.0 0.0

Table 3 Tooth load sharing of the standard spur gear system Radial distance (mm) Contact force (kgf) Transmitted torque (kgf mm)

47.53 0.41 1432.9

47.43 16.40

47.34 12.15

47.25 2.88

4.2. Contact forces on the contact tooth pair The contact forces are obtained by solving of contact problem and the "nite element analysis at the contact points of contact tooth pairs. According to the de"nition of deformation overlap, just prior to the initial contact of fourth tooth pair (from left to right) in Fig. 3, the "xed (zero) and prescribed geometric (displacement) boundary conditions are imposed on the rim of driven and driving gear, respectively. The prescribed displacement, which is known after some iterations until the transmitted torque is achieved, is imposed in the circumferential direction of gear rotation. After three iterations, the "nally modi"ed displacement is 0:0158 mm, which is shown in Table 2. In Table 3, the computed contact force due to the transmitted torque of 1432 kgf mm at each contact node, and the radial distance of each contact node from the center of driving gear are presented. 4.3. Computation of deformation overlap The contact nodal forces, which satisfy the transmitted torque, are utilized as the force boundary conditions for the displacement analysis of a gear pair. The two displacement analyses are executed for driving and driven gears respectively. As a result of displacement analysis, the deformed position of each node is obtained. Then, in the deformed state, the deformation overlap is computed as the maximum depth of the piled region at the trailing tooth pair as presented in Fig. 1. Since the deformations of two gears, driving gear and driven gear, are considered in this computation, the deformation overlap indicates the deformation of a gear pair. For the standard spur gear system shown in Fig. 3, the overlap depths are shown in Table 4. In Table 4, ‘+’ sign and ‘−’ sign denote separation and overlap, respectively. The deformation overlap q, which is de"ned as the maximum

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Table 4 Overlap depths for the standard spur gear system Gear

Node

Radial distance (mm)

Overlap depth (m)

Driven gear

4018 4091 4093 4092 4094 4095

57.375 57.281 57.187 57.094 57.000 56.906

−16.885 −15.985 −13.369 −9.134 −3.353 +3.912

overlap depth, is appeared as q = 16:885 m at the tooth tip of driven gear. Radial distances in Table 4 are measured from center of driven gear to each node in the undeformed state. 5. Application of deformation overlap 5.1. Calculation of tooth tip relief The deformation overlap is applicable to a pro"le modi"cation. Di9erences in the angular positions of the gears due to the tooth 6exibility sometimes cause an impact at the initial contact of each tooth pair. In order to avoid these problems, some parts should be removed from the tooth face of each gear near the tooth tip [12,20]. In this procedure, the deformation overlap presents the amount of material that should be removed. The simple trial scheme of linear tooth tip relief for the standard gear system is shown in Fig. 4. The extra material, which is located outside the line from node 4095 to the modi"ed tip, is suggested to be removed. The modi"ed tip value () of 21:8 m is obtained after three iterations of deformation overlap computation. The initial tip value is calculated from the "rst deformation overlap q = 16:885 m. Then, the tip value is modi"ed after each iteration. Although all the overlaped material can be

Fig. 4. A simple scheme for linear tooth tip relief.

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Table 5 The comparison of overlap depths before and after linear tooth tip relief Gear

Node

Driven gear

4018 4091 4093 4092 4094 4095

Overlap depth (m) Before modi"cation

After modi"cation

−16.885 −15.985 −13.369 −9.134 −3.353 +3.912

+3.262 +0.962 −0.050 +0.143 +1.489 +3.912

removed at one stroke, the suitable tip value is found after some iterations to prevent the excessive drop of contact ratio. After modi"cation, the deformation overlap of 0:050 m at node 4093 can be decreased to zero. As stated above, the deformation overlap serves as a good guide to the tip relief to avoid the problems due to the teeth de6ection. 5.2. Design of a optimal pro9le shift coe:cient For the case when the center distance is equal to the standard center distance, the design method for a pro"le shifted gear pair is known as the “long and short addendum system” [12,21]. In this situation, unequal tooth thickness, increment in driving gear tooth and decrement in driven gear tooth, should be designed. The amount of changes in the tooth thicknesses are suggested as follows [12,20]: VT1 = 2e tan  = 2xm tan ;

(25)

VT2 = −2e tan  = −2xm tan ;

(26)

where e is the pro"le shift value, x(=e=m) is the pro"le shift coe;cient. And VT1 and VT2 are changes of tooth thickness in the driving gear tooth and driven gear tooth, respectively (Table 5). Again, the deformation overlap can be used to analyze the deformation characteristics of this type of gear design, and to select the optimal pro"le shift coe;cient with respect to the minimum teeth de6ection. The FE model of a gear pair with pro"le shift coe;cient x = 0:5 for the previous gear system is presented in Fig. 5. Fig. 5 shows the change of tooth thicknesses due to the pro"le shift, and the data of FE model and contact forces are shown in Table 6. The computation follows exactly the same procedure described earlier. The changes of deformation overlap according to the pro"le shift coe;cient are presented in Fig. 6. This results can be divided into three sections according to their characteristics. In the "rst section of shift coe;cient 0.0 – 0.125, the deformation overlap is decreasing. In this section, the dominant tooth for the gear pair’s deformation is the driving gear tooth. In other words, the decreased de6ection of driving gear tooth has more e9ect on the gear pair’s deformation than the increased de6ection of driven gear tooth. As a result, the deformation overlap is decreasing.

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Fig. 5. FE model of a pro"le shifted gear system with x = 0:5. Table 6 Data of FE model and tooth load sharing for a pro"le shifted gear system with x = 0:5 Gear

Driving gear

Driven gear

Node Element

4875 4590

4155 3860

Boundary conditions

Geometric B. C. on the rim (mm) Radial Circumferential

Radial distance (mm) Contact force (kgf)

0.0 0.0170

Radial Circumferential

0.0 0.0

48.53

48.41

48.29

48.18

2.22

15.57

11.71

2.24

Transmitted torque (kgf mm)

1432.05

The second section ranges from shift coe;cient 0.125 – 0.25. In this section, the deformation overlap is stagnated because the de6ections of both gear teeth have an equal footing on the gear pair’s deformation. Therefore, the deformation overlap has little change in this range of coe;cient. In the third section, where the pro"le shift coe;cient is larger than 0.25, the deformation overlap is increasing. The situation is reversed to the "rst section. This is the deformation characteristic analysis of long and short addendum system by using deformation overlap, and the optimum pro"le shift coe;cient considering deformational aspect of gear pair can be chosen in the second section. 5.3. Shift coe:cient for a low velocity ratio gear system To calculate the optimum pro"le shift coe;cient of another system, the same procedure is applied for a system where the velocity ratio is very low. In Figs. 7 and 8, the FE model of a low velocity ratio gear system is shown with module m=2:0, driving gear teeth Z1 =19, driven gear teeth Z2 =48, pressure angle  = 23:5◦ . Velocity ratio Z1 =Z2 of this system is much lower than that of the previous

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Deformation Overlap (µm)

18.0 17.8 17.6 17.4 17.2 17.0 16.8 16.6 16.4 0.0

0.1

0.2

0.3

0.4

0.5

0.6

Profile Shift Coefficient

Fig. 6. Deformation overlap of a pro"le shifted gear system.

Fig. 7. Standard FE model for the low velocity ratio system.

Fig. 8. Pro"le shifted FE model for the low velocity ratio system with x = 0:56.

1371

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Table 7 Data of the low velocity ratio system Gear

Driving gear

Teeth Pressure angle (deg) Module Transmitted torque per unit facewidth (kgf mm) Young’s modulus (kgf =mm2 ) Poisson’s ratio

19

Driven gear 23.5 2.0 1432 21 000 0.3

48

Table 8 Data of FE model for the low velocity ratio system Gear

Driving gear

Driven gear

x = 0:0 Node Element

6553 6230

6383 6060

Boundary conditions

Geometric B. C. on the rim (mm) Radial Circumferential

0.0 0.0115

Radial Circumferential

Gear

Driving gear

Driven gear

x = 0:56 Node Element

6523 6230

5913 5600

Boundary conditions

0.0 0.0

Geometric B. C. on the rim (mm) Radial Circumferential

0.0 0.0119

Radial Circumferential

0.0 0.0

spur gear system of 0.857. The pro"le shift coe;cient in Fig. 8 is 0.56. The data of gear system and FE model for this gear system are presented in Tables 7 and 8, respectively. Following the same computational procedure with the previous spur gear system, the contact forces and the deformation overlap are obtained as presented in Table 9 and Fig. 9, respectively. It is shown that the deformation overlap represents the deformation characteristics of this low velocity ratio system as in the case of the previous example. Again, the results are divided into three sections by the two boundary points of shift coe;cient 0.156 and 0.325. As shown in Fig. 9, the change of deformation overlap is minimized when the pro"le shift coe;cient is selected within 0.156 and 0.325. Also, the deformation of this gear system will be minimized in this section.

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Table 9 Tooth load sharing of the low velocity ratio system x = 0:0 Radial distance (mm) Contact force (kgf) Transmitted torque (kgf mm) x = 0:56 Radial distance (mm) Contact force (kgf) Transmitted torque (kgf mm)

54.98 4.34 1431.9

54.89 10.35

54.80 12.37

54.72 1.67

56.07 2.56

55.99 9.17

55.92 10.24

55.85 6.69

1431.6

14.2

Deformation Overlap (µm)

14.1 14.0 13.9 13.8 13.7 13.6 13.5 13.4 -0.1

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

Profile Shift Coefficient

Fig. 9. Deformation overlap of the low velocity ratio system.

5.4. Implementation to three-dimensional helical gear system In a three-dimensional helical gear problem, driving gear and driven gear are divided into four-node tetrahedra for the e;cient search of contact compatibility condition and deformation overlap. In this case, points(nodes) and triangular surfaces are used for the point-to-surface contact search. The gap size between point and surface should be investigated for the contact search, and then the gap size calculated from displacement analysis will be the three-dimensional deformation overlap as stated in earlier chapters. Contact search point of contactor body is denoted by p1 , and the points constituting the boundary surface of the target body are represented by p2 , p3 , and p4 as shown in Fig. 10. The outward unit normal vector n on the boundary surface is given by −−−−→ −−−−→ p 2 p 3 × p 2 p4 n = −−−−→ −−−−→ ; | p 2 p3 × p 2 p4 |

(27)

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Fig. 10. Nodes and element’s surface in contact boundary.

Fig. 11. Compatibility conditions for contact.

where → means the vectorial quantity. If the end point of a perpendicular line from point p1 is denoted by p5 , the gap size G is written as −−−−→ G = | p5 p1 |; (28) −−−−→ = p 2 p1 · n

(29)

= (r2 − r1 ) · n;

(30)

where r1 and r2 are de"ned from the origin O of the global coordinate system to the points P1 and P2 , respectively. The contacting element’s surface can be determined by area coordinates. Point p5 in a triangle divides its area A into three areas, A1 , A2 , and A3 , as shown in Fig. 11. These areas are given by 2A = 2(A1 + A2 + A3 )

(31)

−−−−→ −−−−→ = | p2 p3 × p2 p4 |;

(32)

−−−−→ −−−−→ 2A1 = ( p3 p4 × p3 p5 ) · n;

(33)

−−−−→ −−−−→ 2A2 = ( p4 p2 × p4 p5 ) · n;

(34)

−−−−→ −−−−→ 2A3 = ( p2 p3 × p2 p5 ) · n:

(35)

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Fig. 12. FE model of helical driving gear.

Fig. 13. FE model of helical driven gear.

If the following four conditions are satis"ed simultaneously, then point p5 locates on the contacting element’s surface. 0 6 A1 =A 6 1;

(36)

0 6 A2 =A 6 1;

(37)

0 6 A3 =A 6 1;

(38)

A1 =A + A2 =A + A3 =A = 1:

(39)

Finite element models of an involute helical gear pair are shown in Figs. 12 and 13. Four-node tetrahedral elements are used, and the "nal model is completed by the in-house mesh generation computer program. In Figs. 12 and 13, FE model is generated with normal module mn = 2:25, normal pressure angle n = 17:5◦ , helix angle = 31:33◦ , and four potential contact tooth pairs are denoted by , , , and . At the initial contact of tooth pair , tooth pairs and are already in contact, and the deformation overlap is investigated at the driven gear tooth tip of tooth pair . Other data of this helical gear pair and its material constants are shown in Table 10. Tooth load sharing is obtained through the contact force calculation as stated in the previous cases of spur gear systems. The nodes on the rim of driven gear are "xed, and prescribed boundary conditions to cause the contact force which accomplish the transmitted torque are imposed on the rim of driving gear. Several quantities of FE model and boundary conditions are summarized in Table 11.

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Table 10 Data of helical gear pair and material constants Driving gear teeth Driven gear teeth Normal module Normal pressure angle (deg) Helix angle (deg) Face-width (mm) Contact ratio Transmitted torque (kgf mm) Young’s modulus (kgf =mm2 ) Poisson’s ratio

26 30 2.25 17.5 31.33 17.0 2.87 24 340.0 21 000.0 0.3

Table 11 Data of FE model and the material constants Gear

Driving gear

Driven gear

Node Element

4594 17 640

4594 17 640

Boundary conditions

Geometric B. C. on the rim (mm) Radial Circumferential

0.0 0.0354

Radial Circumferential

0.0 0.0

Table 12 Tooth load sharing of helical gear pair Contact tooth pair

Contact position in Fig. 14

Contact force (kgf)

a b c d

3.67 245.16 144.30 139.83

e f g

130.03 79.87 116.67

As the results of contact force analysis, tooth load sharing is obtained as shown in Table 12 and Fig. 14. The contact forces are produced at seven places; four of them are on the contact tooth pair , and the others are on the contact tooth pair as presented in Fig. 14. Pair begins its contact earlier than pair , and is contacting at the latter part of gear facewidth. Pair begins its contact later than pair , and the contact is in progress at the fore part of gear facewidth (see Fig. 14).

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Fig. 14. Seven contact positions and deformation overlap search point.

Through these results of contact force analysis, it is well known that pair has more progress in its teeth contact than pair . In addition, pair has the smaller number of contact points and tooth load sharing because the state of contact is close to its end. The deformation overlap is calculated through the displacement analysis as stated in the previous spur gear systems. The computed contact forces are imposed as the force boundary conditions, and displacement analyses are conducted for driving and driven gears, respectively. The deformation overlap is investigated at the tooth tip of driven gear in contact tooth pair as presented in Fig. 14, and its maximum magnitude was 21:25 m. Since this deformation overlap is resulted from the deformation of both gears, it can be utilized as the guideline for the improved design of a helical gear pair. 6. Conclusions The deformation overlap was de"ned for numerical investigation of elastic de6ection in the design of a gear pair. In the state of initial contact, the deformation overlap is calculated by the de6ection of both contact teeth. The contact forces and displacements were computed by the solution of elastic contact problem and FEM. The application "elds of deformation overlap are demonstrated for the tooth tip relief calculation and the design of a pro"le shifted gear pair. The standard amount of removable material for the tip relief was proposed, and deformation characteristics and optimal shift for long and short addendum system are obtained by using deformation overlap. The deformation overlap is extended to a three-dimensional problem and implemented to a helical gear pair, and the results showed a good extension to three-dimensional problems. The deformation overlap is regarded as a useful item for consideration in the design of a gear pair, and may be further extended to describe three-dimensional problems arising in other types of gearing. Acknowledgements The authors would like to thank to Ministry of Science and Technology of Korea for the "nancial support by a Grant (M1-0203-00-0017-02J0000-00910) under the NRL (National Research Laboratory).

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References [1] P. Gagnon, C. Gosselin, L. Cloutier, Analysis of spur and straight bevel gear teeth de6ection by the "nite strip method, ASME J. Mech. Des. 119 (1997) 421–426. [2] E. Yau, H.R. Busby, D.R. Houser, A Rayleigh–Ritz approach to modeling bending and shear deformation of gear teeth, Comput. Struct. 50 (5) (1994) 705–713. [3] C. Gosselin, P. Gagnon, L. Cloutier, Accurate tooth sti9ness of spiral bevel gear teeth by the "nite strip method, ASME J. Mech. Des. 120 (1998) 599–605. [4] V.G. Sfakiotakis, N.K. Anifantis, Finite element modeling of spur gearing fracture, Finite Elem. Anal. Des. 39 (2002) 79–92. [5] K. Umezawa, De6ections and moments due to a concentrated load on a rack-shaped cantilever plate with "nite width for gears, Bull. JSME 15 (79) (1972) 116–130. [6] K. Umezawa, J. Ishikawa, De6ection due to contact between gear teeth with "nite width, Bull. JSME 16 (97) (1973) 1085–1093. [7] D.B. Wallace, A. Seireg, Computer simulation of dynamic stress, deformation, and fracture of gear teeth, ASME J. Eng. Ind. 95 (4) (1973) 1108–1114. [8] G. Chabert, T. Dang Tran, R. Mathis, An evaluation of stresses and de6ections of spur gear teeth under strain, ASME J. Eng. Ind. 96 (1) (1974) 85–93. [9] A. Cardou, G.V. Tordion, Calculation of spur gear tooth 6exibility by the complex potential method, ASME J. Mech. Transmissions Automat. Des. 107 (1985) 38–42. [10] M.E. Stegemiller, D.R. Houser, A three-dimensional analysis of the base 6exibility of gear teeth, ASME J. Mech. Des. 115 (1993) 186–192. [11] K. Fukunaga, A strength analysis of spur gears including ratio of contact, JSME Trans. Ser. C 53 (489) (1987) 1071–1076. [12] J.R. Colbourne, The Geometry of Involute Gears, Springer, Berlin, 1987. [13] G.J. Huh, B.M. Kwak, Constrained variational approach for dynamic analysis of elastic contact problems, Finite Elem. Anal. Des. 10 (1991) 125–136. [14] Z.H. Zhong, Finite Element Procedures for Contact-impact Problems, Oxford University Press, Oxford, 1993. [15] K.J. Bathe, Finite Element Procedures, Prentice-Hall, Englewood Cli9s, NJ, 1996. [16] B.M. Kwak, Complementarity problem formulation of three-dimensional frictional contact, ASME J. Appl. Mech. 58 (1991) 134–140. [17] G.B. Lee, B.M. Kwak, Formulation and implementation of beam contact problems under large displacement by a mathematical programming, Comput. Struct. 31 (3) (1989) 365–376. [18] Van de Panne, A. Whinstone, The simplex and the dual method for quadratic programming, Oper. Res. Quart. 15 (4) (1964) 355–388. [19] D.W. South, J.R. Mancuso, Mechanical Power Transmission Components, Marcel-Dekker, New York, 1994. [20] A.H. Burr, J.B. Cheatham, Mechanical Analysis and Design, Prentice-Hall, Englewood Cli9s, NJ, 1995. [21] J.E. Shigley, J.J. Uicker Jr., Theory of Machines and Mechanisms, McGraw-Hill, New York, 1995.