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ADVANCED ENGINEERING 3(2009)2, ISSN 1846-5900

INFLUENCE OF TIP RELIEF PROFILE MODIFICATION OF SPUR INVOLUTE GEARS ON STRESSES

Buljanović, K. & Obsieger, B.

Abstract: In this paper the linear tip relief profile modification has been observed. The amount of tip relief profile modification depends on elastic gear tooth deflection that needs to be compensated. The standard gear model without linear tip relief profile modification and also modified one have been developed and analyzed using FEM analysis to compare gear tooth root stress, influenced by mentioned profile modification. Keywords: Spur involute gears, profile modification, tip relief, FEM 1 INTRODUCTION

During the meshing of gear pair, there appears so-called contact shock due to the contact of two new teeth. This impact produces noise and amplifies inaccuracies in the pitch and cause deformation of the teeth under load. In order to reduce the impact influence, the involute in the tip region is modified through a relief curve. This process is called profile modification at the tip and depends on elastic gear tooth deflection that needs to be compensated [1].

2 LINEAR TIP RELIEF PROFILE MODIFICATION

Tip relief profile modification can be designed in few different ways. In this paper linear tip relief modification has been considered. This type of correction is shown in Fig. 1.

Fig. 1. Linear tip relief profile modification

Ca

da

dk

d

df

Δs(d)

Δs(d)

Linear tip relief:

150

Tip relief profile modification is defined as thickness Δs(d) of the material removed along the tooth flank with reference to the nominal involute profile. To define changes in tooth thickness Δs(d), tooth tip diameter da, profile relief at tooth tip Ca and diameter at the beginning of correction dk have to be calculated by (1).

ka

a k

( ) d ds d Cd d−

Δ =−

. (1)

3 TIP RELIEF CALCULATION FOR NOMINAL LOAD

Profile relief at tooth tip Ca has been obtained as a sum of elastic deflection of the spur gear caused by distributed load and Hertz contact deformation. Diameter at the beginning of correction dk has been found at characteristic point B of the tooth flank.

3.1 Elastic tooth deflection of the spur gear Elastic tooth deflection caused by nominal transverse load in plane of action is shown in Fig. 2.

Fig. 2. Elastic tooth deflection of the spur gear

Elastic tooth deflection caused by nominal transverse load has been calculated

using simplified expressions [2, 3]:

( ) ( )P

2

btib1,2

1C yF

A Be Db E

νδ ⋅

−= + + . (2)

Where: ( )1,75 1,68,11, 05 153 xxA e z− −−= − + , (3) ( ) 10, 63 7,35 0,924B x z−= + − , (4)

( ) 11, 28 2,88 3, 68C x z−= − + , (5) ( )n1, 06 0, 638lnD m z= − + , (6)

δb

Fbti

αFY

yP

df

y

x

151

( )p b b fp

Pn n

cosr ryy

m m

α ω− −= =

⎡ ⎤⎣ ⎦ , (7)

bb

arctanrρ

α =⎛ ⎞⎜ ⎟⎝ ⎠

, (8)

bp

bcosr

rα

= , (9)

( )b 1, 252

mzr m x= − − , (10)

bbrρ

ω φ= − , (11)

( )n

n

4 taninv

2x

zπ α

φ α+

= + , (12)

nf ncos

2m z

r α= , (13)

n n ninv tanα α α= − . (14) 3.2 Deformation caused by Hertzian contact stress Hertzian contact stress refers to the localized stresses that develop as two curved surfaces come in contact and deform slightly under the imposed loads. This deformation is dependent on the elasticity of the material in contact. Deformation caused by Hertzian contact stress is shown in Fig. 3.

Fig. 3. Deformation caused by Hertzian contact stress

The expression that has been used for calculation of deformation caused by

Hertzian contact stress is:

2

bti nH1,2

H

2 (1 )1, 27 0, 781ln

F mb E b

νδ

π−

= +⎛ ⎞⎜ ⎟⎝ ⎠

. (15)

ρ2

T2

ρ1

bH

bH

T1 2bH

δH2

δH1 δH1,2

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3.3 Relief at tooth tip Profile modification should be calculated for each tooth flank of the mating gears. The maximum values of the profile relief at the tooth tip of each gear are equal to the sum of elastic tooth deflection and deformation caused by Hertzian contact stress [1, 2, 3], so stands:

1,2 b1,2 H1,2a1,212

C δ δ δ== + . (16)

4 GEAR TOOTH ROOT STRESS The mashed gears teeth are subjected to bending, compression and shear. One side of the tooth root is strained by tension and the other side by compression. The stresses are analyzed on the tooth side loaded in tension because the first cracks are expected to appear there. ISO 6336 standard [4] specifies the fundamental formulae for bending stress calculations for spur gears.

The critical cross-section of the tooth is determined by defining tangents on the profile root fillet under the 30° angle to its axis symmetry, as shown in Fig. 4. Bending stress σbn has been calculated depending on tangential force component Ft.

Fig. 4. Critical cross-section of tooth

Nominal tooth root stress for the ith point of contact on tooth flank can be

determined by B-method [4].

tF0-B F S β

n

F Y Y Ybm

σ = . (17)

The helix angle factor Yβ equals 1 for spur gears. Although the tooth form factor YF and stress correction factor YS are defined in [4] at the critical cross-section, the MAX method presumes the calculation of maximum stresses in tooth root. That means critical position for the stress analysis is for the (YFYS)max. 5 GEAR PAIR MODEL Gear pair with following geometrical parameters has been analyzed: - number of teeth z1,2 = 58/67, - profile shift correction x1,2 = 0 mm,

Fbti

αFY

yP

Fr Ft

sFn

ρF 30° 30°

153

- normal module mn = 12 mm, - normal pressure angle αn = 20°, - gear facewidth b1,2 = 330 mm, - tool addendum factors ha

*01,2 = 1,25mn,

- bottom clearance factors ca*

1,2 = 0,25mn, - tool tip radius factors ρa

*01,2 = 0,25mn,

- transverse contact ratio εα = 1,79. Material assigned to both gears has been steel with following material parameters:

- Modulus of elasticity E = 210000 N/mm2, - Poisson’s ratio ν = 0,3.

According to theoretical background for tip relief profile modification Ca and dk have been calculated: - relief at tooth tip Ca1,2= 0,061/0,061 mm, - diameter at the beginning of correction dk1,2= 349,223/403,358 mm. 6 NUMERICAL ANALYSIS

Finite element nonlinear contact analysis was chosen for modelling and simulation of gear pair in mesh.

The analysis has been carried out by using software package ANSYS 10.0. [5]. Newton-Rapson’s method [6] has been used for the convergence of the results for this non-linear analysis.

The load has been applied by putting in contact pinions’ and wheels’ teeth and applying the torsion moment on the pinion.

The gear models have been discretized by 2D finite elements that are adequate for the contact analysis. The stress state has been considered to be a plane stress and the friction has been neglected. 6.1 Geometrical model of gears Modelling of entire gears in mesh would significantly increase the complexity and size of geometric and numerical model which would, in turn, result in prolonged calculation time. Thus, already in modelling phase certain simplifications have been made. Only parts of the rims of the wheel and the pinion have been modelled (Fig. 5.), both with two whole teeth and two teeth segments [7].

Rim thickness has been set to 100 mm that is approximately 5mn in order to avoid the influence of too thin rim on the results.

6.2 Meshing of gear model Three types of finite elements have been used for meshing of gear models.

Gear models have been divided in areas and they have been mashed with elements PLANE183 [5]. These elements are defined by 8 nodes, having two degrees of freedom at each node: translations in the nodal x and y directions and are well suited to modelling irregular meshes. These elements may be used as plane elements (plane stress, plane strain and generalized plane strain) or as axisymmetric elements. These elements have plasticity, hyperelasticity, creep, stress stiffening, large deflection, and large strain capabilities (Fig. 6.).

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Fig. 5. Geometry of gears in mesh

Fig. 6. PLANE183 finite element

Due to contact problem analysis the contact elements usage have been necessary. Parts of teeth flanks in contact have been meshed with contact elements TARGE169 and CONTA172 [5]. These parabolic elements (Fig. 7.) with two nodes on end and one midside node each with two degrees of freedom (translations in the nodal x and y directions) are very suitable for analysis of problems with states of plane stress and plane strain. As they can’t be used as standalone elements, they must be overlaid over existing 2D solid elements – in this case PLANE183 Contact occurs when the element surface (CONTA172) penetrates one of the target segment elements (TARGE169) on a specified target surface.

Fig. 7. TARGE169 and CONTA172 contact finite elements

x

y

TARGE169

CONTA172

x

y

PINION

WHEEL

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In order to further decrease calculation time, finite element mesh has been adapted as well. Areas around contacting surfaces have been meshed with larger density of finite elements mesh because these areas are crucial for results accuracy. Coarser finite elements have been used in areas of less significance such as gear rim and parts of gear teeth that are not in the contact.

Meshed gear model is shown in Fig. 8.

Fig. 8. Meshed gear model

6.3 Boundary conditions The gears have been loaded by positioning mating teeth i.e. their flanks into contact due to inadequacy of other loading models [8]. Namely, concentrated force couldn’t be applied due to high local deformation of the material which takes place near point of force action and significant influence on the results.

After positioning the mating teeth in desired position the boundary conditions have been applied.

The wheels’ nodes placed on inner rim radius and on the ends of rim have been constrained in global Cartesian coordinate system (x, y) in all directions i.e. the movements in directions of both axis have been disabled (Δx=0, Δy=0). The pinions’ nodes placed on inner rim radius have been constrained in global cylindrical coordinate system (r, φ) in a way: Δr=0. Centre of both mentioned coordinate systems have been centre of rotation of the pinion.

Rotation of the pinions’ nodes placed on inner rim radius around the centre of the global cylindrical coordinate system has been enabled. Angle of rotation Δφ of these nodes has been increasing in stepwise fashion until it resulted with momentum which has been higher then nominal torque at the pinion. Final value Δφ has been determined from two closest rotation steps by the interpolation method.

Target surface (TARGE169)

Contact surface (CONTA172)

PLANE183

PLANE183

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7 RESULTS Gear tooth root stresses along the path of contact in standard model have been calculated and then compared to the stresses in modified one to present the influence of determined profile modification on gear tooth root stresses. The results of FEM analysis for pinion and wheel are shown in Fig. 9.

Fig. 9. Tooth root stress for pinion (σF01) and wheel (σF02) for the i th point of contact

For standard unmodified model stands that when double contact exceeds into single contact (point B on path of contact) and reverse (point D on path of contact) gear

157

tooth root stress changes rapidly i.e., the wheel speed changes at two shifting points, and causes the additional dynamic load as visible in Fig. 9.

Instead of the first contact between meshing gears with linear tip relief profile modification on the pinion tooth tip (point A on path of contact), it occurs lower on tooth flank (point A’ on path of contact). The same situation appears at point E. Gear tooth root stress increment between points A’ and B’ (double contact) and decrement between points D’ and E’ (double contact) are almost linear. There aren’t rapid stress changes at the shifting points so gears run smoother then standard gear pair without additional dynamic load.

The analysis also showed that the highest values of the tooth root stresses appear in point B on path of contact for standard and in point B’ for modified model.

8 CONCLUSION The standard gear model and also modified one have been developed and analyzed by using finite element method. Nonlinear analysis has been used because it gives the most accurate results. Numerical calculation methods, such as finite element method, provides easier stress calculations on teeth with no limits in gears’ geometrical specifications and also allows determination of stress distribution on whole path of contact.

Obtained results show that in case of standard unmodified model when double contact exceeds into single contact and reverse gear tooth root stress changes rapidly i.e., the wheel speed changes at two shifting points, and causes the additional dynamic load, unlike, in case of modified model wheel speed don’t change rapidly so there aren’t rapid stress changes at the shifting points. Also, instead of the first contact between meshing gears with linear tip relief profile modification on the wheel tooth tip it occurs lower on tooth flank. The same situation appears at the end of contact between meshing gears with linear tip relief profile modification. This phenomenon results in a way that gear tooth root stress increment and decrement on double contact zones are almost linear so gear pair with linear tip relief profile modification runs smoother then standard gear pair. NOTATION

A,B,C,D auxiliary factors for calculating tooth deflection, - A,A’,B,B’,D,D’,E,E’ characteristic points on path of contact, - b facewidth, mm bH half of the Hertzian contact width between the meshing teeth, mm c* bottom clearance factor, - da tip diameter, mm dk diameter at the beginning of correction, mm df root diameter, mm Ca profile relief at tooth tip, mm E modulus of elasticity, N/mm2

Fbt transverse load in plane of action (base tangent plane), N Fr radial force, N Ft tangential force, N

* 0ah tool addendum factor, -

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mn normal module, mm rb base radius, mm rP distance between point of application of the force and centre of gear, mm x addendum modification coefficient, - Φ auxiliary angle, rad YF tooth form factor, - yP bending arm, mm YS stress correction factor, - Yβ helix angle factor, - z number of teeth, - αb auxiliary angle, ° αFY angle of action of nominal transverse load, ° αn normal pressure angle, ° Δs removed material, mm δ deflection, mm δb bending deflection, mm δH Hertzian contact deformation, mm εα transverse contact ratio, - ν Poisson’s ratio, - ρ roll distance, mm

* a0ρ tip radius of the tool factor, -

σF0 nominal tooth root stress, N/mm2 ωb auxiliary angle, ° Indexes 1 pinion 2 wheel i i th point of contact

References: [1] Obsieger, J. (1989). Some considerations to the choice of profile correction of involute

gears, STROJARSTVO 31(1989)1, pp. 17-23, ISBN 0562-1887 [2] Terauchi, Y. & Nagamura, J. (1981). On tooth deflection calculation and profile

modification of spur gear teeth, Intern. Symp. Gearing and Power Transmission, Proc. Vol. II, pp. C-27 (159-164), Tokyo, 1981

[3] Franulović, M. (2003.) Influence of base pitch deviation on stresses in involute gearing, Masters Thesis, University or Rijeka, Faculty of Engineering, Rijeka, 2003

[4] ISO 6336 (1996.), Calculation of load capacity of spur and helical gears, International standard, 1996

Part 1: Basic principles, introduction and general influence factors Part 2: Calculation of surface durability (pitting) Part 3: Calculation of tooth bending strength [5] ANSYS Structual analysis Guide // Canonsburg: ANSYS Inc. 2004 [6] Zienkewich, O.C. (1997). The Finite Element Method, Mc Graw-Hill, London, 1977 [7] Basan, R.; Franulović, M. & Križan, B. (2008.). Numerical model and procedure for

determination of stresses in spur gears teeth flanks, Proceedings of XII International conference on mechanical engineering, Starek, L. & Hučko, B. (Ed.), Bratislava, 2008

[8] Franulović, M.; Križan, B. & Basan, R. (2006.) Calculation methods of load carrying capacity of spur gears, Advanced Engineering Design AED 2006, Musilek, L. (Ed.), Prag, 2006

Received: 2009-07-15