Gear Geometry and Applied Theory Episode 3 Part 3 ppsx

right-Table 21.3.2: Machine-tool settings of a formate-cut gear21.4 DERIVATION OF PINION TOOTH SURFACE We limit the discussion to the generation of the pinion by straight-line blades of

Parameters V2, H2, XB2, XD2, and γm2 (Fig 21.3.6) are the gear machine-tool

settings The upper and lower signs in front of V2 correspond to right-hand and hand gears, respectively The whole set of machine-tool settings for a formate-cut gear

left-is presented in Table 21.3.2.

Figure 21.3.6: Coordinate systems applied for cutting or grinding of a formate-cut gear: (a) for hand gear; (b) for left-hand gear.

right-Table 21.3.2: Machine-tool settings of a formate-cut gear

21.4 DERIVATION OF PINION TOOTH SURFACE

We limit the discussion to the generation of the pinion by straight-line blades of the head-cutter However, application of blades of parabolic profile for pinion generation

is beneficial in some cases, for instance for design of a gear ratio close to 1.

Applied Coordinate Systems

Coordinate systems applied for generation of the pinion are shown in Fig 21.4.1

Co-ordinate systems Sm1, Sa1, Sb1are the fixed ones and they are rigidly connected to the

cutting machine The movable coordinate systems S1and Sc1 are rigidly connected to

the pinion and the cradle, respectively They are rotated about the zb1axis and the zm1

axis, respectively, and their rotations are related with a polynomial function ψ1 ( ψc1) wherein modified roll is applied (see below) The ratio of instantaneous angular ve-

locities of the pinion and the cradle is defined as m1c( ψ1 ( ψc1)) = ω(1)( ψc1) /ω(c) The

magnitude m1c( ψ1 ) at ψc1 = 0 is called ratio of roll or velocity ratio Parameters XD1,

XB1, Em1, and γm1are the basic machine-tool settings for pinion generation.

Coordinate system Sp [Figs 21.4.1(a) and 21.4.1(b)] is applied for illustration of installment of the head-cutter on the cradle and corresponds to generation of the right- hand and left-hand pinion, respectively.

Head-Cutter Surfaces

The pinion generating surfaces are formed by surface (a)

p and (b)

p generated by line and circular arc parts of the blades Surface (a)

Figure 21.4.1: Coordinate systems applied for pinion generation: (a) and (b) illustration of tool stallment for generation of right- and left-hand pinions; (c) illustration of installment of machine-tool settings.

in-where spand θpare the surface coordinates, αpis the blade angle, and Rpis the cutter point radius (Fig 21.4.2) The upper and lower signs in Eq (21.4.1) correspond to the convex and concave sides of the pinion tooth, which are in mesh with the concave and convex sides of the gear, respectively.

The unit normal to the pinion generating surface (a)

p is represented by the equations

n(a) p ( θp) = Np

 Np, Np= ∂r

(a) p

∂sp × ∂r

(a) p

Figure 21.4.2: Blades and generating cones for pinion generating tool with straight blades: (a) vex side blade; (b) convex side generating cone; (c) concave side blade; (d) concave side generating cone.

The unit normal to the pinion generating surface of part (b) is represented by the

equations

n(b) p ( θp) = N

(b) p

 N(b)

p , Np= ∂r

(b) p

∂λf × ∂r

(b) p

Families of Pinion Tooth Surfaces

Such families are represented as

where b1, b2, and b3 are the modified roll parameters and C and D are the modified

roll coefficients The derivative of function ψ1 ( ψc1) taken at ψc1= 0 determines the

so-called ratio of roll or velocity ratio, determined in Eq (21.4.9) by b1or m1c.

where n(a) m1 is the unit normal to the surface, and v( p1) m1 is the velocity in relative motion.

The vectors are represented in the fixed coordinate system Sm1as follows:

elimination of the last row and column of Mm1c1 and Mc1p, respectively Elements of

matrices Lm1c1and Lc1prepresent the direction cosines formed by the respective axes of

coordinate systems Sm1 and Sc1 for Lm1c1 and coordinate systems Sc1 and Sp for Lc1p

(see Chapter 1).

Position vector rm1 in Eq (21.4.12) is determined as

m1c − 2b2ψc1− 3b3ψ 2

c1

(21.4.13)

where C and D are the modified roll coefficients.

Finally, we obtain the equations for pinion tooth surface part (a) as

r(a)1 (sp, θp, ψc1) = M1 p( ψc1) r(a) p (sp, θp) (21.4.14)

f1 p (sp, θp, ψc1) = 0. (21.4.15) Using similar derivations, the fillet surface may be represented as

r(b)1 ( λf, θp, ψc1) = M1 p( ψc1) r(b)

p ( λf, θp) (21.4.16)

f1 p ( λf, θp, ψc1) = 0. (21.4.17)

21.5 LOCAL SYNTHESIS AND DETERMINATION OF

PINION MACHINE-TOOL SETTINGS

The purpose of local synthesis is to obtain favorable conditions of meshing and contact

at the chosen mean contact point M Such conditions at M are defined by η2 , a, and m21

(Fig 21.2.1) The gear machine-tool settings are considered as known and they may be adapted, for instance, from the manufacturing summary.

The procedure of local synthesis is a part of the proposed integrated approach for the design of spiral bevel gears with localized bearing contact and reduced levels of vibration and noise based on application of a longitudinally directed path of contact and application of parabolic blades for generation of the gear to avoid hidden areas of severe contact.

The procedure of local synthesis is represented as a sequence of three stages that

provide: (i) the tangency at M of gear tooth surface 2 and gear head-cutter surface

g, (ii) the tangency at M of gear and pinion tooth surfaces 2 and 1 , and (iii) the

tangency at M of pinion tooth surface 1 and pinion head-cutter surface p Finally,

we obtain that all four surfaces 2 , g, 1 , and pare in tangency at M At all stages,

the relationships between the principal curvatures and directions of meshing surfaces are applied (provided in Section 21.6) Then, it becomes possible to obtain the sought-for pinion machine-tool settings.

The procedure of local synthesis is applied for both cases of design of spiral bevel gear drives: (i) face-milled generated gears, and (ii) formate-cut spiral bevel gears The procedure for the case of face-milled generated spiral bevel gear drives is represented below The procedure of local synthesis for formate-cut spiral bevel gear drives can be considered as a particular case of the one applied for face-milled generated spiral bevel gear drives and is discussed below.

Local Synthesis of Face-Milled Generated Spiral Bevel Gear Drives

The procedure of local synthesis of face-milled generated spiral bevel gear drives is illustrated by the following three stages:

2 is chosen as a candidate for the mean contact point M of pinion–gear tooth surfaces.

Step 1: The meshing of surfaces 2 and g is represented in coordinate system Sm2(Fig 21.3.2) by the following equations:

rm2(sg, θg, ψ2 ) = Mm2g( ψ2 ) rg(sg, θg) (21.5.1)

Equation (21.5.1) represents in Sm2 the family of surfaces g, and Eq (21.5.2) is the equation of meshing The generated surface 2 is represented in S2by the matrix equa- tion

r2 (sg, θg, ψ2 ) = M2g( ψ2 ) rg(sg, θg) (21.5.3)

and the equation of meshing (21.5.2).

Step 2: Mean point A on surface 2 is chosen by designation of parameters LAand

RA(Fig 21.5.1), where A is the candidate for the mean contact point M of surfaces 2

Figure 21.5.1: Representation of point A

in coordinate system S2.

and 1 Then we obtain the following system of two equations in three unknowns:

Step 3: Equations (21.5.2), (21.5.3), and (21.5.4) considered simultaneously allow

the determination of parameters (sg∗, θ∗

g, ψ∗

2) for point A Vector functions rg(sg, θg)

and ng( θg) determine the position vector and surface unit normal for a current point of surface g Taking in these vector functions sg = s∗

g and θg = θ∗

g, we can determine the

position vector r(A) g of point A and the surface unit normal at A.

Step 4: Parameters s∗g and θ∗

g and the unit vectors eg and eu of principal directions

on surface g are considered as known For a head-cutter with blades of straight-line profiles:

eg = ∂r

(a) g

∂sg ÷ 





∂r(a) g

∂θg ÷ 





∂r(a) g

The approach discussed in Section 21.6 enables the determination at point A of

(i) the principal curvatures ks and kq of 2 , and (ii) the unit vectors es and eq of principal directions on surface 2 The unit vectors es and eq are represented in Sm2 The general procedure presented in Section 8.4 can be applied for determination of

principal curvatures kg and kuof the surfaces of the blades of parabolic profile.

Step 1: The derivations accomplished at Stage 1 enable the determination of the position vector r(A)2 and the surface unit normal n(A)2 of point A of tangency of surfaces

2 and g The goal now is to determine such a point M in the fixed coordinate system

Figure 21.5.2: Coordinate systems S2, S , and

S (Fig 21.5.2) where three surfaces, 2 , g, and 1 , will be in tangency with each other.

It can be imagined that surface g is rigidly attached to 2 at point A and that both

surfaces, g and 2 , perform motion as a rigid body turning about the gear axis on a certain angle φ(0)

2 Using the coordinate transformation from S2to S(Fig 21.5.2), we

may obtain r(A)  and n(A)  The new position of point A in Swill be the point of tangency

of 2 and 1 (designated as M), if the following equation of meshing between 2 and

1 is observed:

n(A)  

φ(0) 2

Here, n(A)  ≡ n(M)

 and v(21 ,A)≡ v(21,M)

 ; v(21 ,A) is the relative velocity at point A

deter-mined with the ideal gear ratio

m(0)21 = ω(2)

The solution of Eq (21.5.8) for φ(0)

2 provides the value of the turning angle φ(0)

2 It is evident that three surfaces, 2 , g, and 1 , are now in tangency with each other at

point M We emphasize that the procedure in Step 1 enables us to avoid the tilt of the

head-cutter for generation of the pinion.

Step 2: We consider as known at point M the principal curvatures ksand kqof surface

2 , and the unit vectors es and eqof principal directions on 2 The unit vectors esand

eq are represented in S The goal is to determine at point M the principal curvatures

kf and khof surface 1 , and the unit vectors ef and ehof principal directions on 1 This goal can be achieved by application of the procedure described in Section 21.6 It

is shown in Section 21.6 that the determination of kf, kh, ef, and ehbecomes possible

if parameters m21, η2 (or η1 ), and a /δ are assumed to be known or are used as input

data.

consider in this stage two sub-stages: (a) derivation of basic equations of surface gency, and (b) determination of pinion machine-tool settings that satisfy the equations

tan-of surface tangency Tangency tan-of 2 , g, and 1 at mean contact point M has already

been provided in the previous stages The position vector r(M)  of point M and the surface

unit normal n(M)  at point M were determined in coordinate system S Let us imagine

now that coordinate system S1that coincides with S(Fig 21.5.2) and surface 1 are

installed in coordinate system Sm1(Fig 21.4.1) Angle ψ(0)

1 shown in Fig 21.4.1 is the

installment angle of the pinion Using coordinate transformation from S1 to Sm1, we

may determine in Sm1position vector r(M) m1 of point M and the surface unit normal n(M) m1 .

In Section 21.6 the conditions of improved meshing and contact of pinion and gear tooth surfaces 1 and 2 are considered, and the relationships between the principal curvatures and directions of surfaces for such conditions of meshing and contact are de- termined [see Eqs (21.6.27)] The point of tangency of surfaces 1 and pis designated

in Section 21.6 as point B The pinion generating surface pis installed in Sm1, taking that the cradle angle ψc1is equal to zero The position vector of point B of surface pand the surface unit normal at B are represented in Sm1 as r(B) m1 and n(B) m1 The tangency

of 1 and pat the mean contact point M is satisfied, if the following vector equations

(ii) Design parameter Rpof the head-cutter (Fig 21.4.2).

(iii) Parameters Sr1 and q1 that determine the installment of the head-cutter on the cradle (Fig 21.4.1).

(iv) Parameter ψ(0)

1 that determines the initial installment of coordinate system S1

with respect to Sb1 (Fig 21.4.1), and surface parameter θpof the head-cutter face p.

sur-After completion of the first sub-stage (the derivation of equations of tangency of surfaces 2 , g, 1 , and p, we may start the next sub-stage, the derivation of pin- ion machine-tool settings that provide the tangency of surfaces mentioned above The procedure for computation is as follows:

Step 1: Calculate the values of θp and ψ(0)

1 (two unknowns) Equation (21.5.10) is used for determination of θpand ψ(0)

1 , taking into account that

Equations (21.5.10) and (21.5.14)–(21.5.17) yield the following expressions for θp

where αpis the given value of the profile angle of the head-cutter and nx, ny, nzare the

three components of vector n(M)  The great advantage of the approach developed is that the requirement of the coincidence of the normals does not require a nonstandard profile angle αp or the tilt of the head-cutter with respect to the cradle Using θp, it becomes

possible to determine the unit vectors epand etof principal directions on surface pat

point M.

Step 2: Determination of machine-tool settings XB1( XD1), Em1, m1 p, and the

design parameter Rpof the head-cutter (five unknowns).

As a reminder, XB1 and XD1 are related by Eq (21.5.13) The determination

of the machine-tool settings mentioned above is based on application of the system

of equations (21.6.27) and Eq (21.5.12) that represent a system of four nonlinear equations with four unknowns: XD1, Em1, m1 p, and Rp Also, the design parameters mentioned above provide improved conditions of meshing and contact at the mean

contact point M.

Step 3: Determination of machine-tool settings Sr1and q1(Fig 21.4.1) and the pinion

surface parameter sp(three unknowns).

Determination of the three parameters is based on application of Eq (21.5.11), sidering that generating surface pis a cone The final equations are as follows:

con-Sr1cos q1+ (Rp∓ spsin αp) cos θp= X(M)

We can summarize all the stages as follows:

(i) It is necessary to determine ten unknowns: six machine-tool settings ( XB1, Em1,

XD1, q1, Sr, m1 p), two surface parameters ( θp, sp), one cutter parameter Rp, and one position parameter ψ(0)

1 which defines the pinion turn (Fig 21.4.1).

(ii) The equation system for determination of the unknowns is formed as follows:

In addition, the three following curvature equations are applied:

d33

(21.5.28)

Equation (21.5.24) is equivalent to two independent scalar equations; Eq (21.5.25) is equivalent to three scalar equations; and Eqs (21.5.26), (21.5.27), and (21.5.28) represent five scalar equations Thus, the system of equations pro- vides ten scalar equations for determination of ten unknowns The solution for the unknowns requires: (1) a solution of a subsystem of four nonlinear equations (see Step 2), and solution of six remaining equations represented in echelon form (each

of the six equations contains one unknown to be determined).

Local Synthesis of Formate-Cut Spiral Bevel Gear Drives

The local synthesis procedure for formate-cut spiral bevel gear drives is based on the same four stages previously represented for generated spiral bevel gear drives The only modification of the procedure of local synthesis for generated spiral bevel gears affects Stage 1 Stages 2, 3, and 4 are applied without modification for the case of formate-cut spiral bevel gears.

The formate-cut gear tooth surface is the copy of the surface of the generating tool The cradle and the gear being cut or ground are held at rest Only the head-cutter is rotating around its own axis of rotation with the desired velocity of cutting or grinding.

Therefore, principal curvatures ksand kqof 2 and the unit vectors esand eqof principal directions on surface 2 coincide with the principal curvatures kg and ku and the unit

vectors eg and eu, respectively, of principal directions on surface g of the surface

of revolution of the generating tool The procedure represented in Section 21.6 for

determination of (i) the principal curvatures ks and kq of 2 , and (ii) the unit vectors

es and eqof principal directions on surface 2 at point A is not applied for the case of formate-cut spiral bevel gear drives.

21.6 RELATIONSHIPS BETWEEN PRINCIPAL CURVATURES AND

DIRECTIONS OF MATING SURFACES The relationships represented below are used for the procedure of local synthesis for the determination of the pinion machine-tool settings Henceforth, two types of instan- taneous contact of meshing surfaces are considered: (i) those along a line, and (ii) those

at a point Line contact is provided in meshing of the surface being generated with the tool surface Point contact is provided for the generated pinion and gear tooth surfaces The determination of the required relationships is based on the approach proposed in

Chapter 8 The basic equations in the approach developed are as follows:

Equations (21.6.1) and (21.6.2) relate the velocities of the contact point and the tip

of the unit normal in their motions over the contacting surfaces Equation (21.6.3) represents the differentiated equation of meshing Equations (21.6.1) and (21.6.2) yield

a skew-symmetric system of three linear equations in two unknowns x1and x2of the following structure:

The tool surface g generates the gear tooth surface 2 Surfaces g and 2 are in

line contact and their meshing is considered in coordinate system Sm2 (Fig 21.3.2) Equations (21.6.1) to (21.6.3) yield a system of three linear equations,

ci1vg(2)+ ci2vu(2)= ci3 (i = 1, 2, 3) (21.6.5) where

v(2)g = v(2)

r · eg, v(2)

u = v(2)

and eg and euare the unit vectors of principal directions on g.

The following are considered known: point A of tangency of gand 2 , the common

surface unit normal and the relative velocity v(g2) g , and the principal directions and

curvatures kg and ku of surface g at A The goal is to determine (i) the principal curvatures ks and kq of surface 2 , and (ii) angle σ(g2) that is formed between the

unit vectors eg and es that represent the first principal directions on g and 2 The solution is based on the property that the rank of the system matrix (21.6.5) is 1 and is

The formate-cut gear tooth surface is... gand 2 , the common

surface unit normal and the relative velocity v(g2) g , and the principal directions and< /h2>