fol

Ou

hon

g, Q

13

In this paper, an approach for mesh generation of gear drives at any meshing position is presented and an automatic modeling pro-gram for tooth mesh analysis is developed. Based on the derivation of a exibility matrix equation in the contact region, a nite elementmethod for 3D contact/impact problem is proposed. Using this method, tooth load distribution and mesh stiness results are derived

Mostly in the form of involute proles, gear tooth mesh is a

ing gears. The mesh impact is also present when the geardrive is under the conditions of the sudden loading and

system.

to many other forms of face gears, which result in signi-cant weight reduction used in helicopter transmissions [36]. Similar approaches have been used to carry out toothcontact analysis and stress analysis of gear drives by Braucr[7], Zanzi and Pedrero [8] and Guingand et al. [9]. Pimsarnand Kazerounian [10] presented a new pseudo-interference

* Corresponding author. Tel.: +44 28 9097 4102; fax: +44 28 9066 1729.E-mail address: h.ou@qub.ac.uk (H. Ou).

Comput. Methods Appl. Mech. Engrcomplex process involving, e.g. multi-tooth engagement,multi-point contact and varying load conditions. Toachieve improved static and dynamic characteristics of geardrives and enhanced load carrying capacity and reliability,accurate determination of the tooth load distribution, meshstiness as well as the deformation and stress of tooth faceis an important part in gear drive design. Owing to themanufacturing and assembly errors and elastic deforma-tion of loaded gears, vibration and noise are generated par-ticularly during the approach and recess of the toothmeshing. Backlash causes intermittent impact to the mesh-

In recent years the research on the static and dynamiccontact problems of gear drives has been reported by manyresearchers. Gosselin et al. [1] analyzed the contact stressand displacement of the line contact condition in spur gearsand the point contact condition in bevel gears by nite ele-ment method in comparison with the results from analyti-cal formulations. Litvin et al. [2] developed computerprograms that integrate computerized design, tooth con-tact analysis and automatic modeling and nite elementsimulation of a new type of helical gear drives. The sameapproach has been applied by Litvin and his colleaguesunder static loading during meshing process. This method is also used to simulate the gear behavior under dynamic loading conditions.The dynamic responses of the gear drives are obtained under the conditions of both the initial speed and the sudden load being applied.The inuence of the backlash on impact characteristics of the meshing teeth is analyzed. 2006 Elsevier B.V. All rights reserved.

Keywords: Gear drive; Impact; Dynamic response; Contact problem; Finite element method

1. Introduction

Gear drives transmit motion and power by tooth mesh.

change of speed. Such mesh impact caused by the backlashand due to the approach and recess of tooth meshing has adetrimental eect on dynamic characteristics of a gearA nite element methodcontact/impact ana

Tengjiao Lin a, H.a State Key Lab of Mechanical Transmission, C

b School of Mechanical and Aerospace Engineerin

Received 11 May 2005; received in revised form

Abstract0045-7825/$ - see front matter 2006 Elsevier B.V. All rights reserved.doi:10.1016/j.cma.2006.09.014r 3D static and dynamicysis of gear drives

b,*, Runfang Li a

gqing University, Chongqing 400044, PR China

ueens University Belfast, Belfast BT9 5AH, UK

September 2006; accepted 20 September 2006

www.elsevier.com/locate/cma

g. 196 (2007) 17161728

c1 t

a arctgN N r tga =r ; 1

be divided into two parts. The discrete nodes along toothwidth in the non-contact region are uniformly distributed,and the x-coordinates of discrete nodes in the contactregion may be determined by the coordinates of the toothend face. The involute ank of helical gears and its inter-secting line are illustrated in Fig. 2. Point C is the intersect-ing point of contact line KK

0and certain cross-section S

along the tooth width.Let ak be the pressure angle at the start point of the con-

tact line. ac is the pressure angle at discrete node C and bb isthe helix angle of the base circle. The x-coordinate of nodeC can be given as follows:

xc AN 2CB KN 2=KN 2 rb2tgac tgak=tgbb; 3

where KN2 = rb2tgak, CB = rb2tgac, AN2 = rb2tgak/tgbb.

l. Mstiness estimation method to evaluate the equivalent meshstiness and the mesh load in a gear system. The resultsshowed a good agreement between the proposed mesh sti-ness method and nite element contact analysis.

There has been ongoing interest on nite element basedcomputational methods for dynamic contact problems [1117]. Dierent formulations have been developed to solvethe dierential equations of motion of frictional dynamiccontact/impact problems and corresponding proceduresare veried to a number of benchmark problems such asthe contact of two elastic rods or wheels. Little researchhas been reported on nite element based methods to inves-tigate the dynamic contact/impact problems of gear drives.Bajer and Demkowicz [18] developed a method to simulatea general class of dynamic contact/impact problems forsystems such as gear drives with the consideration ofmomentum and total energy. Based on the proposed for-mulation, they developed a 2D parallel nite element sim-ulator to calculate the dynamic stresses of a planetary geartrain. Whilst this is a new step for dynamic modeling ofcomplex mechanical systems such as gear drives, it isworthwhile to investigate the detailed dynamic behaviorof gear drives under the conditions such as impact loading,change of speed and the eect of backlash.

In recent years, we have been working on nite elementbased methods for dynamic contact/impact problems andparticularly their applications to gear drives [1921]. Thefocus of the work has been on the development of ecientapproaches for mesh generation of dierent tooth prolesand the prediction of dynamic contact/impact behaviorsof gear drives under certain loading and initial conditions.This paper outlines our recent work on this subject. In thefollowing sections, the basic theory used for mesh genera-tion of gear drives is rst presented with an example given.This is followed by the nite element formulation of staticand dynamic contact/impact problems for gear drives. Toverify the method and the program developed, a bench-mark problem is evaluated. Finally detailed results andevaluation of the dynamic contact/impact behaviors ofboth a spur and a helical gear drives are presented. Theconclusions are given at the end of the paper.

2. Mesh generation of gear drives

Fig. 1 shows the global coordinate system yoz and therotational coordinate system y1o1z1, on which the centersof shafts for the pinion and the gear are at points O1 andO, respectively. The tooth position shown in the yoz coor-dinate system is the original position for generating toothface proles. The mesh generation method for each toothpair at certain rotational positions is presented as follows[20].

2.1. Mesh generation of tooth end face

T. Lin et al. / Comput. Methods AppThe mesh position of the pitch point is assumed to bethe initial position of the meshing gears. When the pinionc2 1 2 b1 c1 b2

where a0t is the meshing angle. N1N2 is the length of mesh-ing line. rb1 and rb2 are the base radii of the pinion and thegear, respectively.

In the geometry of two meshing gears, the involuteequation on the end face of teeth can be given by

xi0 0;yi0 rbk cos ki= cos aizi0 rbk sin ki= cos ai;

8>: k 1; 2; 2where xi0, yi0, zi0 are the coordinates of the discrete node ion the tooth surface. ai and ki are the pressure angle andthe central semiangle of node i on the tooth surface,respectively.

2.2. Mesh generation of tooth width

The mesh discretization process on the tooth width mayrotates an angle of w, the pressure angles of contact pointsfor the pinion and the gear can be expressed as

a arctgtga0 w;

1a

1ar

1br

1r

1N

2N

1O

O2r

2br

a2r

21, cc y

1y

1zz

T

Fig. 1. Global and rotational coordinate systems of a gear train.

ech. Engrg. 196 (2007) 17161728 1717Hence the coordinates of discrete nodes of each cross-section on the tooth width can be obtained

2.3. Mesh generation of tooth root llet

For the rack cutter with only a single round edge, themachining parameters of the tool can be dened by

a hatmt ct mt ro;b pmt=2;ro pmt 4hatmttgat=4 cos at;

8>: 6where ro is the radius of the tool. h

at and c

t are the adden-

dum coecient and radial clearance coecient. xt is a mod-ication coecient. mt and at are the transverse moduleand pressure angle of the gears, respectively.

The equation for the root curve on the end face can bewritten as

i0 i sin a0titi i 7

For the wheel cutter with two round edges, the machin-

O C r h m cm r ; 8

2N

K

B

1o1N

2N

1N

CK

b

2o

involute flank

base cylinder B1

base cylinder B2

A

x

y

z

1718 T. Lin et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 17161728Rack cutter and wheel cutter are common gear cuttingtools. If a rack cutter such as hobbing cutter is adopted,the tooth root curve is the equidistant line of the prolateinvolute as shown in Fig. 3.

For the rack cutter with two round edges, the machiningparameters of the tool are dened by

a h mt cmt ro;8>xi xc;yi yi0 cos hk zi0 sin hk;zi yi0 sin hk zi0 cos hk;

8>: 4where hk is the turn angle between the cross-section S andthe tooth end face

hk xitgbb=rbk k 1; 2:

Fig. 2. Contact line of meshing gears.at t

b pmt=4 hatmttgat ro cos at;ro ct mt=1 sin at:

: 5

r

ttmx

t 0r 0

c

P

1a b

pitch circle of gear

z

yo

pr

n

= t

a

pitch line of tool

n

Fig. 3. Tooth root llet generated by rack cutter.

P n= t t

0r

0c pitch circle of tool

n

y

t

pr

o

c o c at t t t o

b0 p2zc

invaac invat sin1 ro cos aacOcCo

:

>>>>:For the wheel cutter with a single round edge, the machin-ing parameters of tool can be dened by

pcr

co

z

t

ing parameters of the tool are dened by

ro ct mt

1 sin aac ;8>>>>>>>>:where rp is the pitch radius of gear; a0ti is a parameter be-tween at and 90.

a1 a xtmt; ui a1ctga0ti b=r:

If a wheel cutter such as slotting cutter is adopted, thetooth root curve will be the equidistant line of the prolateepicycloid as shown in Fig. 4.xi0 0;y rp cosu a1 ro

sina0 u ;

8>>>>>

;zi0 rp sinui cosa0tibi ro cosati ui;:

h1 3p=2 p=2 2x1tgat=z1 tgat at tgac1 a0t;h2 p=2 p=2 2x2tgat=z2 tgat at tgac2 a0t:

2.5. Boundary conditions

For the static contact analysis of gears, the xed dis-placement constraint is applied on the boundary surface

Fig. 5. Boundary constraint and load condition of gears.

Geometric parameters Pinion Gear

Normal module, mn 3.5 mm 3.5 mmNumber of teeth, Z 44 88

l. Mech. Engrg. 196 (2007) 17161728 1719where ap is the engagement angle of the slotting cutter.a0ti is a parameter between ap and 90.

ui zczbi b0;

bi cos1rpc

r2pc r2pc sec2 a0ti sec2 a0titg2a0tiOcCo2

qOcCo sec2 a0ti

:

2.4. Mesh adjustment

Using the above method, the discrete coordinates of thepinion and the gear are generated at their original positionsas shown in Fig. 1. It is necessary to transform the discretecoordinates of the pinion and gear to their meshing posi-tions. The transformation equations for the pinion coordi-nates may be given by

xiyizi

8>:9>=>;

1 0 0

0 cos h1 sin h10 sin h1 cos h1

264375 xi1yi1

zi1

8>:9>=>;

0

0

a1

8>:9>=>;: 11

The transformation equations for gear coordinates can bewritten as

xiyizi

8>:9>=>;

1 0 0

0 cos h2 sin h20 sin h2 cos h2

264375 xi2yi2

zi2

8>:9>=>;; 12

where a1 is the center distance between the meshing gears.xi1, yi1 and zi1 are the coordinates of the pinion at its origi-ro ractghacos aac sin aactgha ;

ct mt ro ractghatg aac ha rac cos aac1= cos ha 1

cosaac haOcCo rc hatmt ct mt ro;b0 p

zc;

8>>>>>>>>>>>>>>>>>:9

where ro, zc and rc are the radius, number of teeth and ref-erence radius of the tool, respectively. rac and aac are the ra-dius and the pressure angle of the addendum circle of thetool.

ha p=2zc invaac inva:The equation of the root curve on the end face can be writ-ten as

xi0 0;yi0 rp cosui rpc sin bicosa0tibi ro

sina0ti ui;

rpc sin bi

0

8>>>>> 10

T. Lin et al. / Comput. Methods Appnal position, and xi2, yi2 and zi2 are the coordinates of thegear at its original position.Normal pressure angle, an 20 20Helix angle, b 13.5 13.5Face width, b 50 mm 50 mmof the gear and the radial constraint is dened on theboundary surface of the pinion. A tangential distributedload is applied uniformly on the pinion surface as shownin Fig. 5. The distributed forces applied on the pinioncan be obtained by

q Trslsb

; 13

Table 1Geometric parameters of the helical gear driveFig. 6. Computation model of the helical gear drive. (a) Solid model and(b) nite element mesh.

where T is the torque applied on the surface of the pinion.rs and ls denote the radius and the arc length of the loadedsurface of the pinion, respectively. b is the tooth width.

2.6. Numerical example

An automatic mesh generation procedure for gear drivesis developed. As an example, the geometric parameters of ahelical gear drive are shown in Table 1. Fig. 6a shows the

1720 T. Lin et al. / Comput. Methods Appl. Msolid model of the helical gear drive and the nite elementmesh for dynamic contact/impact analysis is illustrated inFig. 6b.

3. Governing equations of contact/impact problems

3.1. Flexibility matrix equation of contact region

The dynamic contact model of two bodies is shown inFig. 7, in which O-xyz is the global coordinate systemand o-nts is the local coordinate system dened only onthe local contact surface, where subscript n refers to thenormal direction to the contact surface and subscripts t, srefer to the two tangential directions on the contactsurface. Assume that N is the matrix of shape functionsand ue(t), ve(t) and ae(t) are the vectors of nodal displace-ment, velocity and acceleration at time (t), respectively.The displacement, velocity and acceleration elds can bedescribed as

ut Nuet;vt Nvet N _uet;at Naet Nuet;

8>: 14where _uet; uet are the rst and second derivatives ofthe nodal displacement vector.

Employing the principle of virtual work incorporatingthe inertia and damping forces as a part of the total exter-nal forces, the dAlembert principle of the dynamic contactproblem of two bodies may be derived as follows [22]:

miait civit kiuit pit Rit i 2 X1;X2; 15where mi, ci, ki are the mass, damping and stiness matricesof the two bodies, respectively. pi(t), Ri(t) are the appliedload vector and the contact force vector of the two bodies.ui(t), vi(t), ai(t) are the displacement, velocity and accelera-tion vectors of the two bodies, respectively.

)(tp

1)(tR s

nt 2

z

O yx

)(tR

oFig. 7. Dynamic contact model of two bodies.The Newmark direct integration method is adopted tosolve Eq. (15). In this method, it is assumed that

vtDt vt 1 dat datDtDt; 16

utDt ut vtDt 12 a

at aatDt

Dt2; 17

where a and d are adjustable parameters depending onthe integration accuracy and stability. When aP0.25(0.5 + d)2 and dP 0.5, it is conrmed that the New-mark method has unconditional stability [11,21]. a = 0.25and d = 0.5 are used in this research. Substituting Eqs.(16) and (17) into Eq. (15) forms the eective stiness ma-trix equation of the dynamic contact/impact bodies, inwhich only displacement vector ui(t+Dt) are present in theequation and needs to be solved, i.e.

~kiuitDt ~pitDt RitDt i 2 X1;X2; 18where ~ki and ~pitDt are the eective stiness matrix andequivalent load vector, respectively.

~ki ki 1aDt2mi daDt

ci; 19

~pitDt pitDt mi1

aDt2uit 1aDt vit

1

2a 1

ait

ci daDt uit

da 1

vit d

2a 1

aitDt

:

20Due to the fact that the contact problem of gear drivesinvolves only a very small region in contact between thepinion and the gear during the meshing process, the com-putational iterations are far from ecient if the iterationsare carried out by solving Eq. (18). Instead a condensationof Eq. (18) in the contact region of the meshing teeth isconducted using a modied Cholesky factorization ap-proach. Hence the eective exibility equation on the con-tact surface of the two contact bodies can be derived asfollows:

~f Rt eS pt e0; 21where ~f is the eective exibility matrix; eS pt is the relativedisplacement vector caused by equivalent applied load ~piton the contact surface; e0 is the vector of the initial gap be-tween contact node pairs.

To solve the dynamic contact/impact problems, the con-tact force vector R(t) is obtained by Eq. (21) in each step oftime integration. Then the displacements, velocities andaccelerations of the two bodies are computed using Eqs.(18), (16) and (17), accordingly.

For static contact problems of two bodies, one of thecontact bodies, e.g. X1 in Fig. 7, may not have sucientconstraints. Thus the rigid body displacement vector ur1of body X1 is considered to be unknown and the rigid bodymotion and force equilibrium equations are used to solve

ech. Engrg. 196 (2007) 17161728the rigid body displacement vector. The rigid body motionequation of body X1 is given as

ui1 u01 x r; 22where ui1 is the displacement at any nodal position of bodyX1 due to rigid body motion; x and r are angular displace-ment and position vectors of body X1, respectively. Theforce and moment equilibrium equations of body X1 canbe written asX

i2X1pi Ri 0;X

i2X1M01pi M01Ri 0;

8>>>: 23where M01pi and M01Ri are moment vectors of theexternal and contact force components of body X1,respectively.

Thus the expanded form of the eective exibility matrix

condition for the three contact conditions may be devel-

T. Lin et al. / Comput. Methods Appl. MEq. (21) is given as follows:

~f f df e 0

" #R

ur1

eS p e0pr1

( ); 24

where fd and fe are the transformation matrices due to therigid body motion and force equilibrium of body X1,respectively; pr1 is a force component vector of body X1.Depending upon the constraint condition of body X1 ofthe contact problem, displacement components of the re-quired degrees of freedom at certain positions of body X1are used to form the rigid body displacement vector ur1.For the gear drive case that the gear is fully constrainedand the pinion has only one rotational degree of freedomunder torque T as shown in Fig. 1, one additional angulardisplacement about O1 and one force equilibrium equationare added in Eq. (24). Therefore the same procedure forsolving the exibility matrix Eq. (21) in the contact regionand the global stiness matrix equation may be applied forthe static contact problems.

3.2. Treatment of impenetrability condition

To implement the impenetrability condition in contactregion, three possible contact conditions are dened as

Table 2Impenetrability conditions of contact nodal pairs

Contactcondition

Impenetrability conditions

Stick deRk1t deRk2t deRkt; d~uk2t d~uk1t ~ek 0k n; t; s

Sliding deRn1t deRn2t deRnt; d~un2t d~un1t ~en 0deRt1t deRt2t l sin hsjdeRntj; d~ut2t d~ut1t d~lttdeRs1t deRs2t l cos hsjdeRntj; d~us2t d~us1t d~lst

Separation deRk1t deRk2t 0; d~uk2t d~uk1t d~lktk n; s; t

deRk1t;deRk2t;d~uk1t and d~uk2t (k = n, s, t) are the increment of thecontact force and displacement of two bodies in the k direction of thelocal coordinate system (o-nts); ~ekk n; s; t is the initial gap of the con-tact node pairs in the k direction; l is the friction coecient; hs is theangle between the sliding direction and the coordinate direction s;

d~lkt k n; s; t is the gap increment of contact node pairs in the coor-dinate direction k.oped as given in Table 2.To ensure the correct impenetrability condition in solv-

ing Eq. (21) and to accommodate possible changes of con-tact conditions in time integration, a set of criteria areintroduced as given in Table 3. As shown in the table foran existing contact condition, a new contact conditionmay be reached by evaluating the relative displacementand contact force components in a contact nodal pair inthe local coordinate system (o-nts). Thus the convergedsolution on the contact surface may be obtained quicklyby alternating and solving the exibility matrix equationof the contact region, i.e. Eq. (21). As the eective exibilitymatrix ~f is derived by the assumption of the stick conditionfor all the contact pairs of a contact/impact problem, thealternation of Eq. (21) is straightforward as given by

25where i, j, k are contact node pairs corresponding to thecontact conditions of stick, sliding and separation,respectively.

3.3. Velocity and acceleration correction of intermittent

contact/impact

Since the Newmark direct integration method is pro-posed for the continua of bodies of the contact/impactproblem, the velocities and accelerations of the contactbodies are computed with the step-forward time integra-tion using Eqs. (16) and (17). However, at the instance ofinitial impact or release of contact, the direct integrationscheme for the velocity and acceleration results is nostick, sliding and separation. In the stick condition, a con-tact nodal pair has no relative motion in both the normaland tangential directions in the local coordinate system(o-nts) on the contact surface, i.e. the contact nodal pairsticks together. In the sliding condition, relative motion isallowed only in the tangential direction between the pairof contact nodes while the stick condition remains in thenormal direction. The separation condition indicates thatthe pair of contact nodes is separate. Using the Coulombsfriction model for the sliding condition, the impenetrability

ech. Engrg. 196 (2007) 17161728 1721longer valid due to the discontinuity of the contact pressureand velocity as demonstrated in [11,14]. Hughes et al.

3.4. Numerical example of dynamic contact/impact problem

Criteria for the change of contact conditions

eRn1t > 0eRn1t 6 0; eR2t1t eR2s1tq 6 ljeRn1tjeRn1t 6 0; eR2t1t eR2s1tq > ljeRn1tjeRn1t > 0eRn1t 6 0; deRt1td~ltt < 0; deRs1td~lst < 0eRn1t 6 0; deRt1td~ltt P 0 or deRs1td~lst P 0d~un2t d~un1t ~en > 0

l. Mech. Engrg. 196 (2007) 17161728developed a correction approach using a local wavepropagation analysis [11] and Taylor and Papadopoulosdeveloped a method for a priori satisfaction of the impene-trability constraint and its rate forms by means of a Larg-range multiplier formulation [14], whilst Chen and Yehproposed a set of correction formulas for velocity and accel-eration based on the concept of equivalent nodal forces andthe area weighting factor of the contact element [13].

In this research, a similar approach to [13] is employedfor the correction of the velocity and acceleration at theinstance of initial impact and release of contact in the con-tact region. The only simplication for the gear drive caseis that the density and the depth of the contact element arethe same for both the pinion and the gear. Assume vitDtand aitDt i 1; 2 are the velocity and acceleration ofthe two bodies after correction. The superscripts n and tdenote the normal and tangential directions in the localcoordinate system (o-nts) for the nodes in the contactregion. If the current contact condition is stick, the correc-tions of velocity and acceleration at the instant of contactbetween the two bodies may be given by

vitDt 12 v1tDt v2tDt;aitDt 12 a1tDt a2tDt:

(26

When the current contact condition is sliding, the following

Table 3Criteria for the change of contact conditions

Contact conditions

Present Updated

Stick SeparationStick

Sliding

Sliding SeparationStick

Sliding

Separation SeparationStick

1722 T. Lin et al. / Comput. Methods Appcorrections may be used:

vnitDt 12 vn1tDt vn2tDt;vtitDt vtitDt l vn1tDt vn1tDt;anitDt 12 an1tDt an2tDt;atitDt atitDt l an1tDt an1tDt:

8>>>>>>>: 27At the instant of separation or release of contact betweentwo bodies, the formulas of the correction are given by

vitDt vitDt 1i

2v1tDt v2tDt;

aitDt aitDt 1i

2a1tDt a2tDt:

8>>: 29Table 3. Necessary changes of the eective exibility matrixEq. (21) are made according to the criteria of impenetrabil-ity conditions as given in Table 2. As the iterative compu-tation of Eq. (21) in the contact region is also part of thedirect integration, in each time step the criteria of contactconditions in Table 3 are used to check whether initial im-pact or release of contact takes place in each contact nodalpair. If such a condition of initial impact or release of con-tact is detected the above correction formulas for the veloc-ity and acceleration are used accordingly. Otherwise thenormal direct integration is performed for the rest of thecontact bodies.

d~un2t d~un1t ~en 6 0where l is the length of the bars. cw is the velocity of elasticwave, cw

E=q

p. Using Eq. (29) the impact time and con-

m/s0.10 =V

m1.0 m01.0

m1.0

m01.0

Poissons ratio Youngs Modulus

Density0=

211 N/m101.2 =E

33 kg/m108.7 =

Fig. 8. Finite element model of two bars with initial velocity.

300)

Tooth 3

l. MNumerical solution

00

5

10

15

20

25

12010080604020 140Time t (s)

Cont

act p

ress

ure

(M

Pa) Numerical solution

Analytical solution

Fig. 9. Time history of dynamic contact pressure.

T. Lin et al. / Comput. Methods Apptact pressure may be calculated to be t = 38.54 ls andr = 20.24 MPa.

A comparison of the analytical and numerical solutionto the time history of the normal contact pressure andvelocity on the contact surface are shown in Figs. 9 and10, where the solid lines represent the numerical resultsand the dashed lines represent the analytical solution. Itcan be seen that a good agreement of the two solutions isobtained. Fig. 11 shows the time history of the normal dis-placement on the contact surface.

4. Contact stress and mesh stiness of helical gear drive

Determination of load distribution along the toothwidth during meshing process is a basis for gear drivedesign. Due to the assumptions and simplications made,

Time t (s)

Velo

city

v(m

m/s)

Analytical solution

-500-250

0250500750

100012501500

0 12010080604020 140

Fig. 10. Time history of normal velocity.

Time t (s)

Disp

lace

men

t (

m)

0

20

40

60

80

100

120

0 12010080604020 140

Fig. 11. Time history of normal displacement.it is dicult to accurately predict the load distributionand contact stresses using conventional methods. On thecontrary, the load distribution and contact stresses can be

0306090

120150180210240270

0 5 10 15 20 25 30 35 40 45 50

Cont

act f

orce

F(N

/mm

Face width b (mm)

Tooth 2Tooth 1

Fig. 12. Distribution curve of tooth contact force while three teeth ismeshing.

Fig. 13. The von Mises stress of the gear.ech. Engrg. 196 (2007) 17161728 1723accurately predicted using the nite element method forcontact problems with proper denition of gear geometry,loading and boundary conditions.

In this paper, a helical gear drive is considered with a1500 N m torque applied on the pinion. For the static con-tact analysis of the gear drive, xed displacement con-straints are applied to the bottom surface of the gear.The nite element mesh is shown in Fig. 6. As can be seenfrom the gure, three pairs of teeth are engaged at the meshposition. Fig. 12 shows the contact force distributionsalong the face width. It is apparent that the middle tooth(tooth 2) takes most of loading and the maximum contactforce appears close to one end of the gear. However, themaximum von Mises stress as shown in Fig. 13 is at thetip of tooth 3 on the right. This suggests that tooth prolemodication is needed in order to reduce the peak stresslevel.

The mesh stiness is dened as the value of the appliedload that can generate 1 lm deformation on 1 mm facewidth while a pair of teeth or several pairs of teeth areengaged. Assume that the number of teeth is n and thedeformations of the pinion and gear teeth are dpi(i = 1, . . . ,n) and dgi (i = 1, . . . ,n), respectively. The meshstiness can be given as follows:

C Xni1

F idpi dgi : 30

q 0:04723 0:15551ZV 1

0:25791ZV 2

; 31

C0 1qcos an cos b cos 20

cos 13:5

0:05317; 32

where ZV1 and ZV2 are the equivalent number of teeth ofthe pinion and the gear, respectively. With the given valuesof ZV1 = 47.854 and ZV2 = 95.717 in this gear train, themesh stiness is calculated to be C 0 = 17.2 (N/mm lm).Therefore the mesh stiness results are in good agreementusing the conventional and the nite element methods.

0

2

4

6

8

10

12

14

0 3 6 9 12 15 18 21 24

Tooth 1

Tooth 2 Tooth 3 Tooth 4

Tooth 5

Angle of rotation (o)

Cont

act f

orce

F(kN

)

Fig. 14. Contact force curve of each tooth pair during meshing process.

1724 T. Lin et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 171617280

36

912

1518

21

0 3 6 9 12 15 18 21 24Mes

hing

stiff

ness

C(N

/mm

)

Angle of rotation (o)

m

.

Fig. 15. Mesh stiness curve of gear during meshing process.Therefore the variations of contact pressure and mesh sti-ness during the whole meshing process can be calculatedusing the nite element method developed in this research.For the helical gear drive as shown in Table 1, there aretwo possible meshing conditions of either two or threepairs of teeth engaged. Each tooth pair rotates an angleof 22.7 from the tooth root to tip and the contact forcesof each tooth pair during the meshing process is shownin Fig. 14.

The mesh stiness curve of the gear drive is shown inFig. 15. The mesh stiness varies in the range from 17 N/mm lm to 19 N/mm lm dependent upon whether two orthree tooth pairs are engaged.

The mesh stiness C 0 can be approximately determinedby the conventional formula as given by [24]

ion, which has initial rotating speeds of n1 = 500 rpm and

0

2040

6080

100120

140

r/min10002 =n

r/min5001 =n

Time t (s)

Tota

l con

tact

forc

e F n

(kN)

0 2 4 6 8 10 12 14 16 18 20 22

Fig. 16. Total contact forces under initial speed with displ0

20

40

60

80

100

120

140

0 3 6 9 12 15 18 21 24 27

r/min10002 =n

r/min5001 =n

Tota

l con

tact

forc

e F n

( kN)

n2 = 1000 rpm, respectively. Using the developed nite ele-ment method for contact/impact problems, the dynamicresponses of both the spur gear and the helical gear drivesare obtained. Fig. 16a and b compare the results of thetotal contact forces under dierent initial rotating speedsof the pinion and between the two gear types.

It is apparent that the impact time is independent of theinitial speed, and the total contact force is directly propor-tional to the initial speeds of the pinion. Due to the gradualengagement of the tooth pairs and longer distance on theline of contact in the helical gear drive, smaller values ofthe total contact forces are predicted when the same rotat-5. Impact characteristics of gear drives

The phenomenon of initial speed impact occurs whenthere is a sudden change of the rotating speed in a geardrive and another important consideration in gear designand dynamics is the situation of sudden loading to a gearsystem. To investigate the dynamic behavior of a gear sys-tem under the both conditions, a spur gear and a helicalgear drives are investigated in this research. The parame-ters of the helical gear drive are given in Table 1. For thespur gear case, all gear parameters are the same with anexception of a zero helix angle being dened, i.e. b = 0.

5.1. Initial speed impact

For the initial speed impact problem, it is assumed thatthe gear is fully constrained and it is meshed with the pin-Time t ( s)

acement constraint. (a) Spur gear and (b) helical gear.

ing speed of the pinion is assumed as shown in Fig. 16a andb. It is also noticeable that the impact time and the contactforce vary corresponding to the tooth meshing positions;i.e. the more tooth members are in engaged, the less vari-ances of the contact forces and thus a better loadingcondition.

Fig. 17 presents the normal velocities of the contact sur-faces for the spur gear drive when the initial speed of the

reduces considerably and the maximum contact forcesbecome much smaller. It is also noted that there is a signif-icant change of the contact force distributions as a functionof time. Further evaluation of the dynamic behavior maybe carried out for a gear drive when more detailed opera-tion conditions such as the loading conditions of the systemare known.

5.2. Sudden load impact

For the sudden load impact case, torques applied onthe pinion surface are dened to be T1 = 1500 N m andT2 = 2500 N m, respectively. The total contact forces ofthe spur and helical gear drives are shown in Fig. 20 whentwo tooth pairs are engaged.

The impact time is independent of the magnitude of thecontact forces. But the total contact forces are directly pro-portional to the loading and are depended upon the type ofthe gear forms. Because the loads are applied to the pinionboundary surface, the stresses take a short period of timeto propagate from the pinion to the meshing positionand then the gear. This process is shown in Fig. 21 in

Time t (s)

Velo

city

v(m

m/s)

-3000-2000-1000

0100020003000400050006000

0 10 20 30 40 50 60

Fig. 17. Time history of normal velocity of gears.

T. Lin et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 17161728 1725pinion is 500 r/min. It can be seen that the velocities oscil-late around certain average values. Fig. 18 shows selectedconsecutive still frames of the stress contours of the spurgear drive. These stress variations in dierent time framesshow the propagation of elastic waves in the meshing gears.

When the gear is stationary without tangential con-straints, the gear can rotate around its centre if a force isapplied to it from the pinion. Under this boundary condi-tion, the time histories of the total contact forces are shownin Fig. 19a and b. Comparing to Fig. 16, the impact timeFig. 18. Propagation of elastic waves induced by the initial speed impact and(c) t = 3.0 ls, (d) t = 10.0 ls, (e) t = 15.0 ls and (f) t = 20.0 ls.selected consecutive still frames of von Mises stresscontours.

5.3. Approach impact

An inherent vibration and noise source of a gearsystem is the approach impact, in which the meshingprocess changes from a single pair to double pairs oftooth meshing for a spur gear drive. To evaluate thedynamic response of the spur gear drive, the niterepresented by von Mises stress distribution. (a) t = 0.6 ls, (b) t = 1.0 ls,

l. M7080 r/min10002 =n

n(kN

)

1726 T. Lin et al. / Comput. Methods Appelement analysis starts at t = 3 ls before the single pairmeshing changes to the double pair tooth meshing.Fig. 22 shows the total impact force versus time at dier-ent rotational speeds and dierent loads applied to thespur gear drive.

0102030405060

0 2 4 6 8 10 12 14

r/min5001 =n

Time t (s)

Tota

l con

tact

forc

e F

Fig. 19. Total contact forces under initial speed without dis

01020304050607080

0 20 40 60 80 100 120

Time t (s)

Tota

l con

tact

forc

e F n

(kN)

mN25002 =T mN15001 =T

Fig. 20. Total contact forces with sudden appli

Fig. 21. Propagation of elastic waves induced by the sudden load impact and(c) t = 4.0 ls, (d) t = 10.0 ls, (e) t = 20.0 ls and (f) t = 40.0 ls.7080 r/min10002 =n

n(kN

)

ech. Engrg. 196 (2007) 17161728From the result shown in Fig. 22a, the total approachimpact force is directly proportional to the rotating speed,i.e. the higher the rotating speed, the larger the total con-tact force. The total contact force before the transitionfrom the single to double pairs of tooth meshing denotes

0102030405060

0 2 4 6 8 10 12 14 16 18

r/min5001 =n

Time t (s)

Tota

l con

tact

forc

e F

placement constraint. (a) Spur gear and (b) helical gear.

01020304050607080

0 20 40 60 80 100 120

Time t ( s)

Tota

l con

tact

forc

e F n

( kN)

mN25002 =T mN15001 =T

ed load. (a) Spur gear and (b) helical gear.

represented by von Mises stress distribution. (a) t = 1.5 ls, (b) t = 3.0 ls,

matically and the peak contact force reaches twice as high

It can be seen from Fig. 23 that the impact process is

020406080

100120140160

0 2 4 6 8 10 12 14 16 18 20 22

n1=500 r/min n2=1000 r/min

Time t (s)

Tota

l con

tact

forc

e F n

(kN)

00 4 8 12 16 20 24 28 32 36 40 44

1020304050607080 T1=1500 Nm T2=2500 Nm

Time t (s)

Tota

l con

tact

forc

e F n

/ kN

r d

T. Lin et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 17161728 1727as the average value of the contact forces. This is under-stood to be a main source for vibration and noise of a geardrive. As can be seen in Fig. 22b, the total contact forcesare directly proportional to the applied torques. A peakcontact force can be seen at the instant of transition oftooth meshing condition but the magnitude of the contactforces and the degree of oscillation are not as severe as thecase of dierent rotating speeds.

Finite element analyses are also carried out for theapproach impact problem of the helical gear drive at dier-ent rotating speeds and torques. The results suggest a pro-portional reduction of the total contact forces in both thecomputational cases of varying rotating speeds and tor-ques. This is due to the fact that in the case of the helicalgear drive the transition of tooth engagement is from dou-ble pairs of tooth meshing to triple pairs of tooth meshing.

5.4. Inuence of backlash under sudden loading

The eect of backlash is also evaluated by deningdierent backlash values of each pair of teeth frome1 = 0.1 mm, e2 = 0.3 mm and e3 = 0.5 mm, respectively,only the contact forces of the rst tooth pair and the curveis smooth. At the instant of t = 3 ls, the second tooth pairstarts to be engaged and this causes an interference betweenthe meshing tooth pairs because of the elastic deformationof the rst tooth pair. Thus the impact forces increase dra-

Fig. 22. Total approach contact forces of the spur geawhen the impact load applied to the pinion isT = 1500 N m. The total contact forces under a sudden

0

50

100

150

200

250

0 10 20 30 40 50 60 70 80Time t (s)

Tota

l con

tact

forc

e F n

(kN)

1 =0.1 mm 2=0.3 mm 3 =0.5 mm

Fig. 23. Total contact forces of the spur gear train with dierentdelayed because of the tooth backlash and the delay timeis dependent upon the value of the backlash. When the sud-den load is applied, the engaged teeth bear not only theapplied load but also the impact loading caused by the ini-tial speed and acceleration due to the backlash. Comparedwith the results without backlash as shown in Fig. 22b, theimpact force increases signicantly and the impact timereduces at the mean time. In line with the increased valuesof the backlash, the total contact forces increase and thecontact time reduces accordingly. It is clear that the toothbacklash has a signicant eect on the dynamic character-istics of the gear drives. It is demonstrated through thenumerical results obtained that the nite element methodoutlined in the paper can be used not only to predict theoverall dynamic behavior of a gear drive but also to quan-tify the detailed eect of various factors such as the back-lash, approach impact and initial speed/load impact.Such an analysis would be very useful to improve thedesign and manufacturing processes of gear drives for theimproved performance and reduced noise.

6. Conclusions

In this paper we propose a nite element method for 3Dload applied at the positions of single pair of tooth meshingand double pairs tooth meshing for the spur gear drive areshown in Fig. 23.

rive. (a) At dierent speeds and (b) at dierent loads.dynamic contact/impact problems. This method is basedon the derivation of the eective exibility matrix equation,

0

50

100

150

200

250

0 10 20 4030 50 60 70 80 90 100

Tota

l con

tact

forc

e F n

(kN)

Time t (s)

1=0.1 mm 2=0.3 mm 3 =0.5 mm

gaps. (a) Single tooth meshing and (b) double teeth meshing.

which is condensed from the global motion equations to

under specically dened gear operation conditions. Anumber of conclusions may be drawn as follows:

1728 T. Lin et al. / Comput. Methods Appl. Mech. Engrg. 196 (2007) 17161728(1) The mesh stiness results during the operation pro-cess obtained from this research agree well with theresults calculated from the conventional method.

(2) When the case of initial speed impact is considered,the contact time is independent from the initial speedbut the total contact force is directly proportional tothe initial speed. The more the tooth pairs areengaged, the less variance of the impact forces is.The results also quantify the dierences of the totalcontact forces between the spur and helical geardrives.

(3) The approach impact time is mainly prescribed by thegeometric parameters of the gear drive and the totalapproach contact force is directly proportional tothe initial speed and the sudden load.

(4) The tooth engagement is delayed because of the toothbacklash. For the sudden load impact problem, thedelay time is proportional to the value of the back-lash. The total contact force increases signicantlywith the increase of the tooth backlash but the con-tact time reduces slightly.

Acknowledgements

This research has been supported by National NaturalScience Foundation of China under contracts No.50075088 and 50675232. The rst author is grateful forhis sabbatical leave as Visiting Senior Research Fellow atQueens University Belfast.

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A finite element method for 3D static and dynamic contact/impact analysis of gear drivesIntroductionMesh generation of gear drivesMesh generation of tooth end faceMesh generation of tooth widthMesh generation of tooth root filletMesh adjustmentBoundary conditionsNumerical example

Governing equations of contact/impact problemsFlexibility matrix equation of contact regionTreatment of impenetrability conditionVelocity and acceleration correction of intermittent contact/impactNumerical example of dynamic contact/impact problem

Contact stress and mesh stiffness of helical gear driveImpact characteristics of gear drivesInitial speed impactSudden load impactApproach impactInfluence of backlash under sudden loading

ConclusionsAcknowledgementsReferences