Gear Geometry and Applied Theory Episode 2 Part 5 pptx

iii Profile crowning provides localization of bearing contact, and the path of contact on the tooth surface of the pinion or the gear is oriented longitudinally see Section15.4.. 15.2 Axo

14.10 Nomenclature 403

parabolic function of transmission errors that is able to absorb the linear functions oftransmission errors caused by misalignments

14.10 NOMENCLATURE

α n rack profile angle in normal section (Fig 14.4.7)

α t rack profile angle in transverse section (Fig 14.4.7)

β k (k = p, ρ) helix angle on pitch cylinder (k = p), on cylinder of

radiusρ (k = ρ) (Figs 14.2.1 and 14.4.7)

λ i (i = p, b, ρ) lead angle on the pitch cylinder (i = p), on the base cylinder

(i = b), and on the cylinder of radius ρ (Figs 14.2.1, 14.4.5

φ, φ1, andφ2 angle of gear rotation (Figs 14.4.1 and 14.5.1)

η2 half of the angular tooth thickness on pitch circle of gear 2

F(12,n) normal component of contact force (Fig 14.8.2)

p n circular pitch measured perpendicular to the direction of skew

teeth of the rack [Fig 14.4.7(c)]

p t circular pitch in the cross section [Fig 14.4.7(c)]

P n and P t diametral pitches that correspond to p n and p t

q orientation angle of straight contact lines on rack tooth surface

(Fig 14.4.3)

r b radius of base cylinder (Fig 14.4.4)

r o radius of operating pitch cylinder, axode

r pi radius of pitch cylinder i (Figs 14.3.2 and 14.3.3)

s t tooth thickness on the pitch circle in the cross section

u surface parameter of a screw involute surface

w t space width measured on the pitch circle in cross section

X(12)f , Y(12)f , Z(12)f components of contact force (Figs 14.8.2 and 14.8.3)

15 Modified Involute Gears

15.1 INTRODUCTION

Involute gears, spur and helical ones, are widely used in reducers, planetary gear trains,transmissions, and many other industrial applications The level of sophistication in thedesign and manufacture of such gears (by hobbing, shaping, and grinding) is impressive.The geometry, design, and manufacture of helical gears was the subject of research

presented in the works of Litvin et al [1995, 1999, 2001a, 2003], Stosic [1998], and Feng et al [1999].

The advantage of involute gearing in comparison with cycloidal gearing is that thechange of center distance does not cause transmission errors However, the practice

of design and the test of bearing contact and transmission errors show the need formodification of involute gearing, particularly of helical gears Figure 15.1.1 shows a 3Dmodel of a modified involute helical gear drive

The existing design and manufacture of involute helical gears provide instantaneouscontact of tooth surfaces along a line The instantaneous line of contact of conjugated

tooth surfaces is a straight line L0that is the tangent to the helix on the base cylinder

(Fig 15.1.2) The normals to the tooth surface at any point of line L0are collinear andthey intersect in the process of meshing with the instantaneous axis of relative motionthat is the tangent to the pitch cylinders The concept of pitch cylinders is discussed inSection 15.2

The involute gearing is sensitive to the following errors of assembly and manufacture:(i) the changeγ of the shaft angle, and (ii) the variation of the screw parameter (of one

of the mating gears) Angleγ is formed by the axes of the gears when they are crossed,

but not parallel, due to misalignment (see Fig 15.4.4) Such errors cause discontinuouslinear functions of transmission errors which result in vibration and noise, and theseerrors may also cause edge contact wherein meshing of a curve and a surface occursinstead of surface-to-surface contact (see Section 15.9) In a misaligned gear drive, thetransmission function varies in each cycle of meshing (a cycle for each pair of meshingteeth) Therefore the function of transmission errors is interrupted at the transfer ofmeshing between two pairs of teeth [see Fig 15.4.6(a)]

This chapter covers (i) computerized design, (ii) methods for generation, (iii) lation of meshing, and (iv) enhanced stress analysis of modified involute helical gears

simu-404

Contact lines L 0

Base cylinder helix

Figure 15.1.2: Contact lines on an involute

helical tooth surface.

Screw Involutesurface

Profile-crowned pinion

tooth surface

Screw Involutesurface

Double-crowned pinion

tooth surface(a)

(b)

Figure 15.1.3: Crowning of pinion tooth surface.

(iii) Profile crowning provides localization of bearing contact, and the path of contact

on the tooth surface of the pinion or the gear is oriented longitudinally (see Section15.4)

(iv) Longitudinal crowning enables us to provide a parabolic function of transmissionerrors of the gear drive Such a function absorbs discontinuous linear functions

of transmission errors caused by misalignment and therefore reduces noise andvibration (see Section 15.7) Figures 15.1.3(a) and 15.1.3(b) illustrate the profile-crowned and double-crowned pinion tooth surface

(v) Profile crowning of the pinion tooth surface is achieved by deviation of the ating tool surface in the profile direction (see Section 15.2) Longitudinal crown-ing of the pinion tooth surface can be achieved by: (i) plunging of the tool, or(ii) application of modified roll (see Sections 15.5 and 15.6)

gener-(vi) The effectiveness of the procedure of stress analysis is enhanced by automatization

of development of the contacting model of several pairs of teeth The derivation

of the model is based on application of the equations of the tooth surfaces; CADcodes for building the model are not required Details of application of the proposedapproaches are presented in Section 15.9

15.2 Axodes of Helical Gears and Rack-Cutters 407

-Figure 15.2.1: Axodes of pinion, gear, and rack-cutter: (a) axodes; (b) tooth surfaces of two skew rack-cutters.

15.2 AXODES OF HELICAL GEARS AND RACK-CUTTERS

The concept of generation of pinion and gear tooth surfaces is based on application ofrack-cutters The idea of the rack-cutters is the basis for design of such generating tools

as disks and worms The concept of axodes is applied when the meshing and generation

of helical gears are considered

Figure 15.2.1(a) shows the case wherein gears 1 and 2 perform rotation about parallelaxes with angular velocitiesω(1)andω(2) with the ratioω(1)/ω(2)= m12 where m12is

the constant gear ratio The axodes of the gears are two cylinders of radii r p1 and r p2,

and the line of tangency of the cylinders designated as P1–P2is the instantaneous axis

of rotation (see Chapter 3) The axodes roll over each other without sliding

The rack-cutter and the gear being generated perform related motions:

(i) translational motion with velocity

v= ω(1)× O1P = ω(2)× O2P (15.2.1)

where P belongs to P1–P2(ii) rotation with angular velocityω (i ) (i = 1, 2) about the axis of the gear.

The axode of the rack-cutter that is meshing with gear i is plane that is tangent to

the gear axodes

In the existing design, one rack-cutter with a straight-line profile is applied for eration of pinion and gear tooth surfaces Then, the tooth surfaces contact each other

gen-along a line and edge contact in a misaligned gear drive is inevitable.

Point contact in the proposed design (instead of line contact) is provided by tion of two mismatched rack-cutters, as shown in Fig 15.2.1(b), one of a straight-lineprofile for generation of the gear and the other of a parabolic profile for generation ofthe pinion This method of generation provides a profile-crowned pinion

applica-It is shown below (see Sections 15.5 and 15.6) that the pinion in the proposed new sign is double-crowned (longitudinal crowning is provided in addition to profile crown-

de-ing) Double-crowning of the pinion (proposed in Litvin et al [2001c]) allows edge

contact to be avoided and provides a favorable function of transmission errors

Normal and Transverse Sections

The normal section a −a of the rack-cutter is obtained by a plane that is perpendicular to

plane and whose orientation is determined by angle β [Fig 15.2.1(b)] The transverse

section of the rack-cutter is determined as a section by a plane that has the orientation

case of design, we choose s12= 1

The rack-cutter for gear generation is a conventional one and has a straight-lineprofile in the normal section The rack-cutter for pinion generation is provided with

a parabolic profile The profiles of the rack-cutters are in tangency at points Q and

Q∗ [Fig 15.2.2(a)] that belong to the normal profiles of the driving and coast sides of

the teeth, respectively The common normal to the profiles passes through point P that belongs to the instantaneous axis of rotation P1–P2[Fig 15.2.1(a)]

Pinion Parabolic Rack-Cutter

The parabolic profile of the pinion rack-cutter is represented in parametric form in an

auxiliary coordinate system s a (x a , y a) as (Fig 15.2.3)

where a is the parabola coefficient The origin of s coincides with Q.

15.2 Axodes of Helical Gears and Rack-Cutters 409

Figure 15.2.2: Normal sections of pinion and gear rack-cutters: (a) mismatched profiles; (b) profiles of

pinion rack-cutter in coordinate systems s a and S b; (c) profiles of gear rack-cutter in coordinate systems

S e and S k.

The surface of the rack-cutter is denoted by c and is derived as follows:

(i) The mismatched profiles of pinion and gear rack-cutters are represented in Fig.15.2.2(a) The pressure angles are α d for the driving profile andα c for the coast

profile The locations of points Q and Q∗are denoted by|QP | = l dand|Q∗P | = l c where l d and l care defined as

(ii) Coordinate systems s a (x a , y a ) and S b (x b , y b) are located in the plane of the normal

section of the rack-cutter [Fig 15.2.2(b)] The normal profile is represented in S b

by the matrix equation

rb (u c)= Mbara (u c)= Mba [a c u2 u c 0 1]T (15.2.7)

Figure 15.2.3: Parabolic profile of pinion rack-cutter in normal section.

(iii) The rack-cutter surface  c is represented in coordinate system S c (Fig 15.2.4)

wherein the normal profile performs translational motion along c–c Then we

obtain that surface cis determined by vector function

15.3 Profile-Crowned Pinion and Gear Tooth Surfaces 411

form in coordinate system S e (x e , y e) as:

The coordinate transformation from S k to S t is similar to the transformation from S bto

S c (Fig 15.2.4), and the gear rack-cutter surface is represented by the following matrixequation:

rt (u t , θ t)= Mtk(θ t)Mkere (u t). (15.2.10)

15.3 PROFILE-CROWNED PINION AND GEAR TOOTH SURFACES

Profile-crowned pinion and gear tooth surfaces are designated as σand2, respectively,wherein1indicates the pinion double-crowned surface

Generation of Σσ

Profile-crowned pinion tooth surface  σ is generated as the envelope to the pinionrack-cutter surface  c The derivation of  σ is based on the following considera-tions:

(i) Movable coordinate systems S c (x c , y c ) and S σ (x σ , y σ) are rigidly connected to thepinion rack-cutter and the pinion, respectively (Fig 15.3.1(a)) The fixed coordinate

system S mis rigidly connected to the cutting machine

(ii) The rack-cutter and the pinion perform related motions, as shown in Fig 15.3.1(a),

Figure 15.3.1: Generation of profile-crowned tooth surfaces by application of rack-cutters: (a) for pinion generation by rack-cutter; (b) for gear generation by rack-cutter.

where s c = r p1 ψ σ is the displacement of the rack-cutter in its translational motion,andψ σ is the angle of rotation of the pinion.

(iii) Using coordinate transformation from coordinate system S c to coordinate system

S σwe obtain a family of generating surfaces σ represented in S σby the followingmatrix equation:

rσ (u c , θ c , ψ σ)= Mσ c(ψ σ)rc (u c , θ c). (15.3.1)(iv) The pinion tooth surface  σ is determined as the envelope to the family of sur-

faces rσ (u c , θ c , ψ σ) and requires simultaneous application of vector function rσ (u c ,

θ c , ψ σ) and the equation of meshing represented as follows (see Zalgaller [1975],

Litvin [1994], and Litvin et al [1995]):

coordinates and 4x4 matrices (Chapter 1).

Generation of Gear Tooth Surface Σ 2

The schematic of generation of2is shown in Fig 15.3.1(b) Surface2is represented

by the following two equations considered simultaneously:

Necessary and Sufficient Conditions of Existence of an Envelope

to a Parametric Family of Surfaces

Such conditions in the case of profile-crowned pinion tooth surface σ are formulated

as follows (see Zalgaller [1975] and Litvin [1989, 1994]):

(i) Vector function rσ (u c , θ c , ψ σ ) of class C2is considered

(ii) We designate by point M(u(0)c , θ(0)

c , ψ(0)

σ ) the set of parameters that satisfies the

equation of meshing (15.3.2) at M and satisfies as well the following conditions

[see items (iii)–(v)]

15.3 Profile-Crowned Pinion and Gear Tooth Surfaces 413

(iii) Generating surface  c of the rack-cutter is a regular one, and we have at M

to surface cdiffers from zero The designations of N(c)

σ indicate that the normal

to c is represented in coordinate system S σ

(iv) Partial derivatives of the equation of meshing (15.3.2) satisfy at M the

By observation of conditions (i)–(v), the envelope  σ is a regular surface, it

con-tacts the generating surface  c along a line, and the normal to  σ is collinear tothe normal of  c Vector function rσ (u c , θ c , ψ σ) and Eq (15.3.2) considered simul-taneously represent surface  σ in three-parameter form, by three related parameters

(u c , θ c , ψ σ)

Representation of Envelope Σσin Two-Parameter Form

The profile-crowned surface σmay also be represented in two-parameter form, takinginto account the following considerations:

(i) Assume that inequality (15.3.7) is observed, say, because

Rσ(θ c , ψ σ)= rσ (u c(θ c , ψ σ), θ c , ψ σ). (15.3.11)

Base circles

Common normal

Centrodes

Figure 15.4.1: Illustration of cross-profiles of profile-crowned helicoids.

15.4 TOOTH CONTACT ANALYSIS (TCA) OF PROFILE-CROWNED

PINION AND GEAR TOOTH SURFACES

Meshing of Profile-Crowned Helicoids: Conceptual Considerations

Two profile-crowned helicoids are considered The concept of the meshing is based onthe following considerations discussed in Litvin [1962, 1989] and Litvin & Tsay [1985]:(1) The helicoids transform rotation between parallel axes

(2) The helicoid tooth surfaces are in point contact and this is achieved by the cation of the cross-profile of the pinion tooth surface This statement is illustratedfor the example in Fig 15.4.1 in which an involute helicoid of the gear and pinionmodified helicoid are shown Profile crowning of the pinion is provided becausethe cross-profile deviates from the involute profile The gear and the pinion toothsurfaces are in point contact provided by mismatched crossed profiles

modifi-(3) The formation of each of the mating helicoids may be represented as the result ofscrew motion of the cross-profile Figure 15.4.2 shows the formation of a helicoid

by a family of planar curves that perform a screw motion about the axis of thehelicoid

(4) The screw parameters p1and p2of the profile-crowned helicoids have to be relatedas

15.4 Tooth Contact Analysis (TCA) 415

(6) It is easy to verify that during the process of meshing, point M of tangency of

cross-profiles performs in the fixed coordinate system a translational motion along

a straight line that passes through M and is parallel to the axes of aligned gears The motion of a contact point along line M–M may be represented by two com-

ponents:

(i) transfer motion with gear i (i = 1, 2) that is performed as rotation about the

gear axis(ii) relative motion with respect to the helicoid surface that is a screw motion with

parameter p i.The screw motion by its nature represents a combination of rotation about thegear axis with angular velocity designated as (i ) and translational motion with

the velocity p i  (i ) The resulting motion of the contact point in the fixed coordinate

system is a translational motion with the velocity p i  (i ) along line M–M because

rotations in transfer and relative motions are performed withΩ(i ) = −ω (i ).(7) It is easy to verify that the contact point moves over the helicoid surface along

a helix that is generated by point M while it performs a screw motion over the

surface of the helicoid The path of contact on the surface of the helicoid is a helix

in which radiusρ i and the lead angleλ i are related by p i = ρ itanλ i (i = 1, 2).

(8) The meshing of the mating helicoids is not sensitive to the change of the centerdistance Using Fig 15.4.3, it is easy to verify that the change of the center distancedoes not cause transmission errors We may assume that the crossing profiles form

a center distance E∗ = E This involves that the point of tangency will be M∗

instead of M and the pressure angle will be α∗ instead of α The new radii of

centrodes will be r∗(i = 1, 2) However, the line of action in the fixed coordinate

Figure 15.4.3: Operating circles in an aligned gear drive: (a) change of center distanceE = 0 when

no errors are applied; (b)E = 0.

system is again a straight line but now passes through point M∗ instead of M.

The line of action is the set of points of tangency of meshing surfaces in a fixedcoordinate system

(9) Considering the contact of helicoid surfaces in the 3D space, we find out that thesurfaces have a common normal and common position vectors at any point ofsurface tangency The normal does not change its orientation during the process

of meshing in a fixed coordinate system

(10) Although profile-crowned helicoids are not sensitive to the change of center tance and have localized surface contact, this type of gearing should not be appliedbecause the change of the shaft angle and the difference of lead angles will cause adiscontinuous linear function of transmission errors (see below) Then, vibrationand noise become inevitable This is the reason why a double-crowned pinion has

dis-to be applied instead of a profile-crowned one Application of a double-crownedpinion provides a predesigned parabolic function of transmission errors, and thelinear function of transmission errors caused by errors of assembly and manufac-ture is absorbed (see Section 15.7)

(11) The conceptual considerations for meshing for profile-crowned helicoids are truefor all types of Novikov–Wildhaber gears, including the meshing of profile-crowned involute helical gears

(12) The analytical investigation of profile-crowned modified helical gears is plished by application of TCA (Tooth Contact Analysis) (see below)

accom-Algorithm of Analytical Simulation

Simulation of meshing and contact have been performed for two cases of designwherein: (i) the pinion of the gear drive is profile-crowned, and (ii) the pinion is double-crowned (see Sections 15.5, 15.6, and 15.7) Comparison of the output for both cases

15.4 Tooth Contact Analysis (TCA) 417

Figure 15.4.4: Illustration of installment of coordinate systems for simulation of misalignment.

(Sections 15.4 and 15.7) shows that double-crowning of the pinion reduces transmissionerrors and noise and vibration of the gear drive

The algorithm of simulation of meshing and contact is based on conditions of tinuous tangency of contacting tooth surfaces of the pinion and the gear (see Section9.4) The algorithm for profile-crowned involute gears is applied as follows Know-ing the representation of tooth surfaces  σ and2 in coordinate systems S σ and S2that are rigidly connected to the pinion and the gear, we may represent surfaces  σ

con-and2 in fixed coordinate system S f taking into account the errors of alignment (see

Fig 15.4.4) We use for this purpose the coordinate transformation from S σ and S2to

S f (Fig 15.4.4)

We recall that tooth surfaces σ and2are profile-crowned and therefore they are

in point tangency Tangency of σ and2at common point M means that they have at

M the same position vector and the surface normals are collinear Then we obtain the

following system of vector equations:

15 .2 Axodes of Helical Gears and Rack-Cutters 409

Figure 15 .2. 2: Normal sections of pinion and gear rack-cutters: (a) mismatched... ( 15 .2. 10)

15. 3 PROFILE-CROWNED PINION AND GEAR TOOTH SURFACES

Profile-crowned pinion and gear tooth surfaces are designated as σand< i>2< /small>,... meshing and contact have been performed for two cases of designwherein: (i) the pinion of the gear drive is profile-crowned, and (ii) the pinion is double-crowned (see Sections 15. 5, 15. 6, and 15. 7)