Gear Geometry and Applied Theory Episode 2 Part 9 pps

18.9 Geometry of Parabolic Rack-Cutters 525Figure 18.9.1: Illustration of rack-cutter profiles; b and c parabolic profiles of the shaper and pinion rack-cutters, i Two imaginary rigidly co

18.7 Pointing of Face-Gear Teeth Generated by Involute Shaper 523

Figure 18.7.2: Cross section profiles of

face-gear and shaper in plane 2

2 is designated by “A” (Fig 18.7.2) Point A has to be located on the addendum line of the face-gear, and therefore its location with respect to axis y a is determined by

r ps − 1/P d (Fig 18.7.2) The goal is determination of magnitude L2defined by distance

l between planes 1 and 2 (Fig 18.6.2) Figures 18.6.2 and 18.7.2 illustrate theprocedure of derivation of magnitude l and L2 The computation of L2is based onthe following procedure:

Step 1: Determination of pressure angleα of pointed teeth (Fig 18.7.2).

We use vector equation (Fig 18.7.2)

where P d is the diametral pitch; point M is the point of tangency of profiles of the shaper

and the face-gear in plane 2(Fig 18.7.2);|MA| = λ s;|NM| = r bs θ s

Vector equation (18.7.1) yields two scalar equations in two unknownsα and λ s:

r bs(cosα + θ ssinα) − λ scosα = N s− 2

2P d

(18.7.3)

r bs(sinα − θ scosα) − λ ssinα = 0. (18.7.4)

Here, r bs = (N s /(2P d)) cosα0;θ s = α − θ0s;θ 0s = π/(2N s)− inv α0 Eliminating λ s, weobtain the following equation for determination ofα:

Step 2: Determination of magnitude L2(Fig 18.6.2).

Two types of fillet surfaces might be provided: (i) those generated by the generatrix G

of the addendum cylinder [Fig 18.4.3(a)], and (ii) those generated by the rounded top

of the shaper (Fig 18.8.1)

Case 1: Generation of the fillet by edge G [Fig 18.4.3(a)].

Using Fig 18.5.1, we represent edge G [Fig 18.4.3(a)] in coordinate system S s by

Case 2: Generation of the fillet by the rounded top of the shaper.

The fillet is generated as the envelope to the family of circles of radiusρ (Fig 18.8.1).

The investigation of bending stresses shows that application of a shaper with a roundedtop reduces bending stresses on approximately 10% with respect to those obtained byapplication of an edged top shaper

Figure 18.8.1: Rounded top of the shaper tooth.

18.9 Geometry of Parabolic Rack-Cutters 525

Figure 18.9.1: Illustration of rack-cutter profiles; (b) and

(c) parabolic profiles of the shaper and pinion rack-cutters,

(i) Two imaginary rigidly connected rack-cutters designated as A1 and A s are applied

for generation of the pinion and the shaper, respectively Designation A0indicates

a reference rack-cutter with straight-line profiles (Fig 18.9.1)

(ii) Rack-cutters A1 and A s are provided with mismatched parabolic profiles that

de-viate from the straight-line profiles of reference rack-cutter A0 Figure 18.9.1(a) shows schematically an exaggerated deviation of A1 and A s from A0 The parabolic profiles of rack-cutters A1 and A s for one tooth side are shown schematically inFigs 18.9.1(b) and 18.9.1(c)

(iii) The tooth surfaces 1 and  s of the pinion and the shaper are determined as

envelopes to the tooth surfaces of rack-cutters A1 and A s, respectively

(iv) The tooth surfaces of the face-gear 2 are generated by the shaper and are termined by a sequence of two enveloping processes wherein (a) the parabolic

de-rack-cutter A s generates the shaper, and (b) the shaper generates the face-gear Theface-gear tooth surface2may also be ground (or cut) by a worm (hob) of a specialshape (see Section 18.14)

(v) The pinion and face-gear tooth surfaces are in point contact at every instant because:

(i) rack-cutters A1 and A are mismatched [Fig 18.9.1(a)] due to application of two

different parabola coefficients, and (ii) the pinion and the shaper are provided with

a different number of teeth Figures 18.9.1(b) and 18.9.1(c) show schematically theprofiles of the rack-cutters of the pinion and the shaper, respectively Application

of both items, (i) and (ii), provides more freedom for observation of the desireddimensions of the instantaneous contact ellipse and for the predesign of a parabolicfunction of transmission errors

(vi) An alternative method of generation of face-gears is based on application of a worm

of a special shape, which might be applied for grinding or cutting (Fig 18.1.3).Grinding enables us to harden the tooth surfaces and to increase the permissiblecontact stresses It is shown below that the derivation of the worm thread surface

is based on simultaneous meshing of the shaper with the face-gear and the worm(see Section 18.14)

Reference and Parabolic Rack-Cutters

Reference rack-cutter A0has straight-line profiles [Fig 18.9.1(a)] Parabolic rack-cutters

designated as A s and A1are in mesh with the shaper and the pinion Parabolic profiles

of A s and A1 deviate from straight-line profiles of A0.

Coordinate systems S q and S r are applied for derivation of equations of shaper

rack-cutter A s Parameters u r and parabola coefficient a r determine the parabolic profile of

rack-cutter A s [Fig 18.9.1(b)] Respectively, coordinate systems S k and S e are applied

for derivation of equations of rack-cutter A1 Parameters u e and parabola coefficient a e determine the parabolic profile of rack-cutter A1 [Fig 18.9.1(c)] Origins O q and O k

of coordinate systems S q and S k, respectively [Figs 18.9.1(b) and 18.9.1(c)], coincide,

and their location is determined by parameter f d The profiles of the rack-cutter areconsidered for the side with profile angleα d [Fig 18.9.1(a)]

The design parameters of reference rack-cutter A0 [Fig 18.9.1(a)] are w0, s0, and α d.Taking into account that

Here,λ = w0/s0, and p and P are the circular and diametral pitches, respectively.

The tooth surface of rack-cutter A s is represented in coordinate system S r [Fig.18.9.1(a)] as

18.10 Derivation of Tooth Surfaces of Shaper and Pinion 527

Parameterθ r is measured along the z r axis Parameter l d is shown in Fig 18.9.1(a)

Normal Nr to the shaper rack-cutter is represented as

18.10 SECOND VERSION OF GEOMETRY: DERIVATION OF TOOTH

SURFACES OF SHAPER AND PINION

Shaper Tooth Surface

We apply for derivation of shaper tooth surface  s: (i) movable coordinate

sys-tems S r and S s that are rigidly connected to the shaper rack-cutter and the shaper,

and (ii) fixed coordinate system S n [Fig 18.10.1(a)] Rack-cutter A s and the shaper

perform related motions of translation and rotation determined by (r ps ψ r) and ψ r

[Fig 18.10.1 (a)]

Figure 18.10.1: For generation of shaper of

pinion by rack-cutters: (a) generation of the

shaper, (b) installation of pinion rack-cutter,

and (c) generation of the pinion.

The shaper tooth surface sis determined as the envelope to the family of rack-cutter

surfaces A s considering simultaneously the following equations:

rs(u r , θ r , ψ r)= Msr(ψ r)rr (u r , θ r) (18.10.1)

Nr(u r)· v(s b)

Here, vector function rs (u r , θ r , ψ r ) represents in S s the family of rack-cutter A s tooth

surfaces; matrix Msr(ψ r ) describes coordinate transformation from S r to S s; vector

function Nr (u r ) represents the normal to the rack-cutter A s [see Eq (18.9.4)]; vr (s b)isthe relative (sliding) velocity

Equation (18.10.2) (the equation of meshing) yields

Pinion Tooth Surface

Movable coordinate systems S e and S1are rigidly connected to the pinion rack-cutter

and the pinion, respectively [Figs 18.10.1(b) and 18.10.1(c)]; S n∗is the fixed coordinatesystem The installation angleβ [Fig 18.10.1(b)] is provided for the improvement of

the bearing contact between the pinion and the face-gear (see Section 18.13) Derivations

of pinion tooth surfaces are similar to those applied for derivation of shaper toothsurfaces and are based on the following procedure:

Step 1: We obtain the family of pinion rack-cutters represented in coordinate system

18.11 Derivation of Face-Gear Tooth Surface 529

18.11 SECOND VERSION OF GEOMETRY: DERIVATION

OF FACE-GEAR TOOTH SURFACE

Preliminary Considerations

The face-gear tooth surface is determined as the result of two enveloping processeswherein (i) a parabolic rack-cutter generates the shaper tooth surface (see Section 18.10),and (ii) the shaper generates the face-gear tooth surface The second enveloping process

is based on the algorithm presented in Section 18.5 wherein an involute shaper generatesthe face-gear tooth surface of the first version of geometry Recall that the shaper toothsurface of the second version of geometry is represented in two-parameter form by vector

function Rs(ψ r , θ r) [see Eq (18.10.4)] The normal to the surface mentioned above isrepresented by vector function (18.10.5) Investigation of undercutting of surface2(of the second version of geometry) is based on the algorithm discussed in Section 18.6

The type of a surface may be defined by the Gaussian curvature that represents theproduct of principal surface curvatures at the chosen surface point Thus, the Gaussian

curvature K at a surface point M is defined as

Investigation shows that surface 2 has elliptical (K > 0) and hyperbolic (K < 0)

points (Fig 18.11.1) The common line of both sub-areas is the line of parabolic points.The dimensions of the area of surface elliptical points depend on the magnitude of

the parabola coefficient a r of the shaper rack-cutter Surface2of the first version ofgeometry contains only hyperbolic points

18.12 DESIGN RECOMMENDATIONS

The bending stresses in a face-gear drive depend on the unitless coefficient

c = P d l = P d (L2 − L1) (18.12.1)

(See the designations of L2 and L1 in Fig 18.6.2.) Usually, the coefficient c is chosen as

c = 10 for high-power transmissions The coefficient c can be increased for face-gear

drives by choosing a higher gear ratio and increasing the tooth number This statementcan be confirmed by the graphs shown in Fig 18.12.1 for face-gear drives of the firsttype of geometry

Figure 18.11.1: Areas of elliptical and hyperbolic points of face-gear tooth surface2 for rack-cutter

parabola coefficients (a) a r = 0.01 1/mm, (b) a r = 0.02 1/mm, and (c) a r = 0.03 1/mm.

The investigation of the influence of coefficient c on the structure of the face-gear teeth is based on the following considerations: Assume that the outer radius L2is known(it has been determined from the conditions of avoidance of pointing) We are able toeliminate the portion of the tooth where the fillet exists (Figs 18.6.2 and 18.11.1) by

increasing the inner radius L1 This means that the coefficient c will be decreased [see

Figure 18.12.1: Graphs of coefficient c for

face-gears of the first type of geometry.

18.13 Tooth Contact Analysis (TCA) 531

Figure 18.12.2: Illustration of influence of parabola coefficient a r and gear ratio m 2s on coefficient c.

Eq (18.12.1)] However, observing a sufficient value of c enables us to obtain a more

uniform structure, eliminating the weaker part of the face-gear tooth

Figure 18.12.2 shows the influence of the parabola coefficient a r of the parabolicprofile of the rack-cutter and the gear ratio on the possible tooth length of the face-gear

of the second type of geometry Results of the investigation of undercutting and pointing

are shown in Fig 18.12.2, which represents the influence of gear ratio m2sand parabola

coefficient a r on the coefficient c represented in Eq (18.12.1).

18.13 TOOTH CONTACT ANALYSIS (TCA)

Tooth contact analysis is directed at simulation of meshing and contact of surfaces1and2and enables investigation of the influence of errors of alignment on transmissionerrors and the shift of bearing contact The algorithm for simulation of meshing isbased on equations that describe the continuous tangency of surfaces1and2and ispresented in Section 9.4

Applied Coordinate Systems

The following coordinate systems are applied for TCA: (a) coordinate system S f, rigidlyconnected to the frame of the face-gear drive [Fig 18.13.1(a)]; (b) coordinate sys-

tems S1 [Fig 18.13.1(a)] and S2[Fig 18.13.2(b)], rigidly connected to the pinion and

the face-gear respectively; and (c) auxiliary coordinate systems S d , S e , and S q, plied for simulation of errors of alignment of the face-gear drive [Figs 18.13.2(a) and18.13.2(b)]

ap-All misalignments are referred to the gear ParametersE, B, and B cot γ determine

the location of origin O q with respect to O f [Fig 18.13.1(b)] Here,E is the shortest

distance between the pinion and the face-gear axes when the axes are crossed but not

intersected The location and orientation of coordinate systems S d and S ewith respect

to S q are shown in Fig 18.13.2(a) The misaligned face-gear performs rotation about

the z axis [Fig 18.13.2(b)]

Figure 18.13.1: Coordinate systems applied for lation of meshing, I.

simu-Figure 18.13.2: Coordinate systems plied for simulation of meshing, II.

ap-18.13 Tooth Contact Analysis (TCA) 533

Computational Procedure

The algorithm of TCA is based on simulation of continuous tangency of surfaces1and2accomplished as follows (see Section 9.4):

(1) Surfaces1and2and their unit normals are represented in the fixed coordinate

system S f by vector functions

r(i ) f (u i , θ i , φ i) (i = 1, 2) (18.13.1)

n(i ) f (u i , θ i , φ i) (i = 1, 2) (18.13.2)(2) Continuous tangency of1and2is represented by vector equations

∂ ( f1, f2, f3, f4, f5)

∂ (u1, θ1, u2, θ2, φ2) = 0. (18.13.6)Then the solution of system of equations (18.13.5) may be represented by functions

{u1(φ1), θ1(φ1), u2(φ1), θ2(φ1), φ2(φ1)} ∈ C1. (18.13.7)

The solution of system of equations (18.13.3) and (18.13.4) by functions (18.13.7)

is an iterative process and requires as a first guess the set of parameters

(18.13.8)that satisfies system of equations (18.13.3) and (18.13.4)

(4) The solution by functions (18.13.7) enables us to obtain:

(a) transmission functionφ2(φ1) and function of transmission errors

φ2(φ1)= φ2( φ1)− N1

(b) the paths of contact on surfaces1and2that are represented, respectively,as

bearing contact may cause an edge contact wherein the formation of the bearing contact

is considered (in addition to stress analysis)

The sensitivity of face-gear drives of the first version of geometry to the changeγ

of the shaft angle may be compensated by the axial correctionq of the face-gear in

the process of assembly [Fig 18.13.3(c)] The advantage of the first version of geometry

is that the transmission errors of the gear drive are equal to zero This is the result ofapplication of an involute shaper for generation that has equidistant profiles

The results of TCA of the second version of geometry are represented in Figs 18.13.4and 18.13.5 The main advantages of the mentioned type of geometry are as follows:(i) Longitudinal orientation of bearing contact that enables us to avoid the edgecontact

(ii) Reduction of stresses (see Section 18.15)

The sensitivity of the gear drive of the second type of geometry to error γ may be

compensated as well by correctionq.

For face-gear drives of the second type of geometry, the misalignment of the geardrive is accompanied with transmission errors However, application of a predesignedparabolic function of transmission errors provides a favorable shape of the function

of errors of the drive and reduces the magnitude of maximal transmission errors (see

Figure 18.13.3: Path of contact, bearing contact, and major axis of contact ellipses for the following examples: (a) no errors

of alignment, (b)|γ | = 3 arcmin, and (c)

adjustment of path of contact by applying the axial displacementq of the face-gear

with respect to the pinion (|γ | = 3 arcmin,

|q| = 550 µm).

18.14 Application of Generating Worm 535

Figure 18.13.4: Path of contact, bearing contact, and major axis of contact ellipses for the following examples: (a) no errors of alignment, (b)|γ | = 2 arcmin, and (c) adjustment of path of contact by

application of correctionq: |γ | = 2 arcmin, |q| = 350 µm.

Section 9.2) The predesigned parabolic function of transmission errors is obtained by(i) mismatch of parabolic rack-cutters for the shaper and the pinion of the gear drive,

and (ii) application of a shaper with tooth number N s > N p , where N p is the toothnumber of the pinion of the gear drive

18.14 APPLICATION OF GENERATING WORM

Concept of Generating Worm

The conventional method for generation of a face-gear is based on (i) application of aninvolute shaper, and (ii) manufacturing of the face-gear performed as the simulation ofmeshing of the shaper and the face-gear being generated

Figure 18.13.5: Parabolic function of transmission errors for proposed geometry.

Figure 18.14.1: Illustration of simultaneous meshing of shaper, worm, and face-gear.

Edward W Miller proposed in 1942 the generation of the face-gear by a hob [Miller,

1942] The next step was done by the patent proposed by Litvin et al [Litvin et al.,

2000a] that has formulated the exact determination of the thread surface of a generatingworm that provides the necessary conditions of conjugation of the tooth surfaces of thehob, the shaper, and the face-gear; the concept of worm design; and avoidance of wormsingularities The worm design as proposed above may be applied for grinding and

cutting of face-gears [Litvin et al., 2002a].

Designations  s, w, and 2 indicate surfaces of the shaper, worm, and face-gear,respectively Simultaneous meshing of  s,  w, and 2 is illustrated by Fig 18.14.1.Shaper surface s is considered as the envelope to the family of rack-cutter A s surfaces

and is represented by vector function Rs(ψ r , θ r) [see Eq (18.10.4)] Surfaces w and

2are generated as the envelopes to the family of shaper surfaces s.Recall that with the second type of geometry, the shaper is provided with non-involuteprofiles (see Section 18.10) We discuss in this section application of the worm forgeneration of a face-gear of the second type of geometry However, the discussed ideamay be applied as well for the generation of face-gears of the first type of geometry

Crossing Angle Between Axes of Shaper and Worm

Figure 18.14.2 shows fixed coordinate systems S a , S b , and S c applied for illustration

of installation of the worm with respect to the shaper Movable coordinate systems S s and S w are rigidly connected to the shaper and the worm Axis z s (it coincides with z a)

is the axis of rotation of the shaper Axis z w (it coincides with z c) is the axis of rotation

of the worm Axes z s and z w are crossed and form a crossing angle of 90o± λ w Theupper (and lower) sign corresponds to application of a right-hand (left-hand) worm

The shortest distance between axes z s and z w is designated as E ws.The crossing angleλ w is

λ w= arcsin r ps

18.14 Application of Generating Worm 537

Figure 18.14.2: Coordinate systems S s , S w, and worm installation.

Here, r ps is the pitch radius of the shaper, and E ws(Fig 18.14.2) is the shortest distance

between the axes of the shaper and the worm The magnitude of E wsaffects the sions of the grinding worm and the conditions of avoidance of surface singularities ofthe worm (see below)

The worm surface  w is determined in coordinate system S w (Fig 18.14.2) by thefollowing equations:

Here, relative velocity v(s w) s is determined by differentiation and transformation of

ma-trix Mwsthat are similar to derivations in Section 2.2; vector function rw(ψ r , θ r , ψ w) isthe family of shaper surfaces s represented in S w; matrix Mws(ψ w) describes coordinate

transformation from S s to S w; Eq (18.14.3) is the equation of meshing between sand

 w Parameters (ψ r , θ r) in vector function Rs(ψ r , θ r) represent the surface parameters

of the shaper; parameter ψ w is the generalized parameter of motion in the process ofgeneration of the worm by the shaper Recall that during generation of the worm, the

shaper and the worm perform rotations about crossed axes z and z (Fig 18.14.2)

l between planes 1 and 2< /small> (Fig 18.6 .2) Figures 18.6 .2 and 18.7 .2 illustrate... face -gear

of the second type of geometry Results of the investigation of undercutting and pointing

are shown in Fig 18. 12. 2, which represents the influence of gear ratio m2sand. .. face -gear drive [Fig 18.13.1(a)]; (b) coordinate sys-

tems S1 [Fig 18.13.1(a)] and S2[Fig 18.13 .2( b)], rigidly connected to the pinion and

the face -gear respectively; and