Gear Geometry and Applied Theory Episode 2 Part 10 doc

19.3.1b]: Lead Angle on Worm Operating Pitch Cylinder The lead angles on the worm operating pitch cylinder and ordinary pitch cylinder are related as where ro is the chosen radius of the

19.3 Design Parameters and Their Relations 553

worm pitch diameter may be chosen as

dp= 2rp= q

The value of q depends on the number N1of threads of the worm and the number N2

of gear teeth and may be picked up from a recommended set (7 ≤ q ≤ 25).

Let us develop the pitch cylinder on a plane [Fig 19.3.1(b)] The helix for each

worm thread is represented by a straight line The distance pa xbetween the neighboring straight lines is

pa x= H

N1

(19.3.2)

where N1 is the number of worm threads, and H is the lead Considering as known

rp and Pa x, we can determine the lead angle on the pitch cylinder from the following equation [Fig 19.3.1(b)]:

Lead Angle on Worm Operating Pitch Cylinder

The lead angles on the worm operating pitch cylinder and ordinary pitch cylinder are related as

where ro is the chosen radius of the operating pitch cylinder The difference between

ro and rp affects the shape of contact lines between the surfaces of the worm and the worm-gear.

Relation Between Worm and Worm-Gear Pitches

We emphasize that we now consider the worm and worm-gear pitches on the operating pitch cylinder (Fig 19.3.2) The axial section of two neighboring teeth represents two

parallel curves Therefore, the worm axial pitch pa x is the same for the worm pitch

cylinder and the operating pitch cylinder The normal pitch pnis the same for the worm and the worm-gear and is represented by the equation

pn= pa xcos λ(o)

1 The worm-gear transverse pitch, pt, is represented by the equation (Fig 19.3.2)

90◦±  λ(o)

pa xcos λ(o)

1sin

γ − λ(o)

1

provided γ − λ(o)

= 0 .

(19.3.6)

Figure 19.3.2: Worm and worm-gear operating pitch cylinders.

Radius of Worm-Gear Operating Pitch Cylinder

We take into account that

Equations (19.3.6), (19.3.7), and (19.3.8) yield the following:

(i) Rois represented for the right-hand worm and worm-gear as

The upper sign corresponds to the case when γ > λ(o)

1 , and the lower sign sponds to the case when γ < λ(o)

corre-1 (ii) For the left-hand worm and worm-gear, we have

Ro= pa xN2cos λ(o)

1

19.3 Design Parameters and Their Relations 555

Representation of m21 in Terms of N1and N2

The gear ratio m21was represented for the right-hand and left-hand worms and gears by Eqs (19.2.8) and (19.2.9), respectively Equations (19.2.8), (19.2.9), (19.3.9), and (19.3.10) yield

1

+ N2

*

Relations Between Profile Angles in Axial, Normal,

and Transverse Sections

Consider the transverse, normal, and axial sections of the worm surface The transverse

section is obtained by cutting of the surface by plane z = 0 [Fig 19.3.3(a)] The axial

sec-tion is obtained by cutting of the surface by plane y = 0 [Fig 19.3.3(d)] Figure 19.3.3(b)

shows the unit tangent a to the helix on the pitch cylinder at point P of the helix The

normal section [Fig 19.3.3(c)] is obtained by cutting of the surface by plane that

passes through the x axis and is perpendicular to vector a [Fig 19.3.3(b)] The normal section is shown in Fig 19.3.3(c), and the unit tangent to the profile at point P is b The unit normal n to the worm surface at P is represented as

where

a = [0 cos λp sin λp]T

b = [cos αn sin αnsin λp − sin αncos λp]T, (19.3.16)

and λp is the lead angle of the helix at the pitch cylinder Equations (19.3.15) and (19.3.16) yield

n = [− sin αn cos αnsin λp − cos αncos λp]T. (19.3.17) Projections of the unit normal are shown in Fig 19.3.3 The orientations of the tangents to the profiles in the transverse, normal, and axial sections are represented by

Figure 19.3.3: Sections of worm face: (a) tooth cross section; (b) worm pitch cylinder in 3D-space; (c) section

sur-of pitch cylinder by plane ; (d) axial

section of worm tooth.

angles αt, αn, and αa x, respectively It is evident from Fig 19.3.3 that

tan αt = − nx

ny = tan αnsin λp, tan αa x= nx

nz = tan αncos λp.

Thus,

tan αn= tan αtsin λp= tan αa xcos λp. (19.3.18) Equation (19.3.18) relates the profile angles in normal, transverse, and axial sections Let us now consider a particular case, an involute worm We may express the radius

rbof the base cylinder of an involute worm in terms of the screw parameter p, the lead

angle on the pitch cylinder λp, and the axial profile angle αa x The derivations are based

19.4 Generation and Geometry of ZA Worms 557

on the following considerations:

19.4 GENERATION AND GEOMETRY OF ZA WORMS

The worm is generated by a straight-lined blade (Fig 19.4.1) The cutting edges of the blade are installed in the axial section of the worm.

Henceforth we consider two generating lines, I and II, that generate the surface sides I and II of the worm space, respectively (Fig 19.4.2) The generating lines are represented

in coordinate system Sbthat is rigidly connected to the blade The respective surfaces of

both sides of the worm thread are generated while coordinate system Sb performs the screw motion about the worm axis (Fig 19.4.3) The generated surface is represented

in coordinate system S1by the matrix equation

r1(u , θ) = M1b( θ) rb(u) (19.4.1)

Here, the coordinate system S1 is rigidly connected to the worm; θ is the angle of rotation in the screw motion; parameter u determines the location of a current point

on the generating line and is measured from the point of intersection of the generating

line with the zbaxis Thus u = |B B| for the current point Bof the left generating line

II Similarly, u = |AA| for the current point Aof the right generating line I.

Figure 19.4.1: Installation of blade for generation of an Archimedes worm.

Figure 19.4.2: Geometry of straight-lined blade.

The unit surface normal is represented in coordinate system S1by the equations

n1(u , θ) = ± k N1= ± k

 ∂r1

where k = 1/|N1| The upper or lower sign must be chosen with the condition that the

surface unit normal will be directed toward the worm thread.

Matrix M1bis represented by the equation (Fig 19.4.3)

19.4 Generation and Geometry of ZA Worms 559

Here, p is the screw parameter that is considered as an arithmetic value ( p > 0) The upper and lower signs for p θ correspond to the cases when a right-hand worm and

a left-hand worm are generated, respectively Figure 19.4.3 shows the generation of a

right-hand worm The surface sides I and II for right-hand and left-hand worms are generated by generating line I and generating line II, respectively.

Using Eqs (19.4.1) and (19.4.2) we may represent the surface equations and the

surface unit normals for both sides of the worm thread in S1as follows:

(i) Surface side I, right-hand worm:

n1= −k[(p sin θ + u sin α cos θ) i1− (p cos θ − u sin α sin θ) j1+ u cos α k1]

where Pa xis the axial diametral pitch.

(ii) Surface side II, right-hand worm:

The surface unit normal is

n1= k[(p sin θ − u sin α cos θ) i1− (p cos θ + u sin α sin θ) j1+ u cos α k1]

The surface unit normal is

n1= −k[(−p sin θ + u sin α cos θ) i1+ (p cos θ + u sin α sin θ) j1+ u cos α k1]

where k = 1/(p2+ u2)0.5.

(iv) Surface side II, left-hand worm:

The surface unit normal is

n1= k[−(p sin θ + u sin α cos θ) i1+ (p cos θ − u sin α sin θ) j1+ u cos α k1]

where k = 1/(p2+ u2)0.5.

Problem 19.4.1

The worm surface 1is represented by Eqs (19.4.7) Consider the axial section of 1

as the intersection of 1by plane y1= 0 Equations (19.4.7) with y1= 0 provide two solutions:

(i) Derive the equations of two axial sections as x1= x1(u), and z1= z1(u).

(ii) Determine coordinates x1and z1for the point of intersection of the respective axial

section with the pitch cylinder of radius rp.

x1= −u cos α, y1= 0, z1= u sin α − $ rptan α − sp

θ = π, x1= −rp, z1= sp

2 + pπ.

Problem 19.4.2

The worm surface 1is represented by Eqs (19.4.7) Consider the cross section of 1

by plane z1= 0 Investigate the equation r1= r1( θ), where r1= (x2

1+ y2

1)0.5, and verify that it represents the Archimedes spiral.

19.5 Generation and Geometry of ZN Worms 561

Figure 19.4.4: Cross section of an Archimedes

worm.

(ii) The cross section is represented by equations

x1= (a − pθ) cot α cos θ, y1= (a − pθ) cot α sin θ.

(iii) Equation

r1=  x2

1+ y2 1

0.5

yields

r1= (a − pθ) cot α.

The magnitude of the initial position vector for θ = 0 is r1= a cot α The increment

and decrement of the magnitude of the position vector is proportional to θ, and this is

the proof that the cross section is an Archimedes spiral Figure 19.4.4 shows the cross section of the ZA worm with three threads.

19.5 GENERATION AND GEOMETRY OF ZN WORMS

thread Straight-lined shapes are provided in the normal section of the space with the

second version of installation [Fig 19.5.1(b)] The surfaces of the worm will be generated

by the blade performing a screw motion with respect to the worm.

To describe the installation of the blade with respect to the worm, we use coordinate

systems Sa and Sb that are rigidly connected to the blade and the worm We start the

discussion with the generation of the worm space (Fig 19.5.2) Axis z coincides with

Figure 19.5.1: Blade installation for tion of ZN worm: (a) for thread generation; (b) for space generation.

genera-Figure 19.5.2: Coordinate systems applied for blade stallation.

in-19.5 Generation and Geometry of ZN Worms 563

Figure 19.5.3: Representation of generating lines in

coordinate system S a.

the worm axis; axes za and zb form angle λp that is the lead angle on the worm pitch

cylinder; the origins Oa and Oblie on the worm axis.

The straight-lined shapes of the blades are shown in Fig 19.5.3 The extended straight lines are in tangency with the cylinder of the to-be-determined radius ρ The intersection

of plane ya = 0 of coordinate system Sa with the cylinder represents an ellipse with axes

2 ρ and 2ρ/sin λp The coordinate transformation from Sa to Sb is represented by the

Figure 19.5.4: Interpretation of ellipse equations.

Representation of Generating Lines in Coordinate Systems Sa

Henceforth we consider the generating lines I and II (Fig 19.5.3) Each generating line

is tangent to the ellipse whose equations are represented in Sa in parametric form as

Figure 19.5.4 illustrates the determination of coordinates of current point C of the

ellipse; µ is the variable parameter.

The unit tangent τa to the ellipse is represented by the equation

sin λp

Ta = dRadµ

It is evident that at the point of tangency of the generating line with the ellipse (point

M and respectively M), we have b(I ) a = τ(I )

a and b(II ) a = −τ(II )

a Equations (19.5.3), (19.5.4), and (19.5.5) yield (see additional explanations in Notes 1 and 2, which

19.5 Generation and Geometry of ZN Worms 565

follow this section)

ρ

|Ta| = cos δ, cos µ

(I )= cos α cos δ

sin µ(I )= sin α sin λp

cos δ = tan δ tan λp

cos µ(II ) = − cos α

cos δ , sin µ

(II )= sin α sin λp

cos δ = tan δ tan λp.

(19.5.6)

Here,

cos δ = (cos2α + sin2α sin2λp)0.5, sin δ = sin α cos λp. (19.5.7)

The designations “I” and “II” indicate the generating lines I and II.

The generating lines are represented in Sa by the equations

xa = ρ sin µ ± u cos δ cos µ

ya = 0

za = ρ cos µ sin λp ∓ u cos δ sin µ

Note 1: Determination of Expressions for cos δ and sin δ

Using the equality b(I ) a = τ(I )

a , and Eqs (19.5.3) and (19.5.4), we obtain

 ρ

|Ta|

−1sin α sin λp. (19.5.10) Using Eqs (19.5.10), we obtain that

ρ

|Ta| = (cos2α + sin

2α sin2λp)0.5. (19.5.11) Using for the purpose of simplification the designation

ρ

we obtain the following expressions for cos δ and sin δ:

cos δ = (cos2α + sin2α sin2λp)0.5, sin δ = (1 − cos2δ)0.5= sin α cos λp.

Equations (19.5.7) are confirmed.

Note 2: Derivation of Expressions for cos µ and sin µ

Determination of ρ

Equations (19.5.8) represent generating lines that are tangents to the ellipse shown in

Fig 19.5.3 The points of tangency are M and M, respectively Equations (19.5.8) for

point N of the generating lines (Fig 19.5.3) are represented as

ρ sin µ ± u∗cos δ cos µ = d, ρ cos µ

sin λp ∓ u∗cos δ sin µ

sin λp = 0 (19.5.15)

where u∗= |MN| = |MN |, d = OaN = rp− (sp/2) cot α.

We consider Eqs (19.5.15) as a system of two linear equations in the unknowns u∗

and ρ and represent them as

a11ρ + a12u∗= d, a21ρ + a22u∗= 0. (19.5.16) The solution for the unknown ρ is

19.5 Generation and Geometry of ZN Worms 567

Figure 19.5.5: Worm thread generation [Fig.

19.5.1(a)]: representation of generating lines in

Here, wpis the distance between the blades measured as shown in Fig 19.5.5.

Equations of Surfaces of Worm Thread

The surface of the worm thread is generated by the edge of the blade (the generating line) that performs a screw motion about the worm axis The vector equation of the

surface is represented in S1by the following matrix equation:

r1( θ, u) = M1b( θ)Mbara (u) (19.5.22)

Here, ra(u) is the vector equation of the generating line that is represented in coordinate system Sa; matrix Mbais represented by (19.5.1); matrix M1bis represented by (19.4.3) The surface unit normal is represented as follows:

Choosing the proper sign in Eqs (19.5.23), we may obtain that the surface normal will

be directed toward the worm thread.

Surfaces and surface unit normals of ZN worms are represented as follows:

(i) Surface side I, right-hand worm:

x1= ρ sin(θ + µ) + u cos δ cos(θ + µ)

y1= −ρ cos(θ + µ) + u cos δ sin(θ + µ)

z1= ρ cos α cot λp

cos δ − u sin δ + pθ.

(19.5.24)

(ii) Surface side II, right-hand worm:

x1= ρ sin(θ + µ) − u cos δ cos(θ + µ)

y1= −ρ cos(θ + µ) − u cos δ sin(θ + µ)

z1= −ρ cos α cot λp

cos δ + u sin δ + pθ.

(19.5.27)

Here, cos µ = −cos α/cos δ, sin µ = sin α sin λp/cos δ; the expressions for cos δ

and sin δ are the same as those in Eqs (19.5.25).

Surface unit normal components:

(iii) Surface side I, left-hand worm:

x1= −ρ sin(θ − µ) + u cos δ cos(θ − µ)

y1= ρ cos(θ − µ) + u cos δ sin(θ − µ)

19.5 Generation and Geometry of ZN Worms 569

Surface unit normal components:

(iv) Surface side II, left-hand worm:

x1= −ρ sin(θ − µ) − u cos δ cos(θ − µ)

y1= ρ cos(θ − µ) − u cos δ sin(θ − µ)

cos δ = tan δ tan λp. (19.5.33)

Surface unit normal components:

Kinematic Interpretation of Surface Generation

The visualization of generation of the worm surface is based on the following erations:

consid-(i) The generating line L( j )( j = I, II) may be represented in plane ( j )that is tangent

to the cylinder of radius ρ (superscripts I and II indicate the generating lines I and

II, respectively).

(ii) L( j ) and the worm axis represent two crossed straight lines Thus, L( j ) may be

represented in a coordinate system Sτ ( j ) whose unit vectors we designate as e( j )1 ,

e( j )2 , and e( j )3 The unit vector e( j )3 is directed along the worm axis and e( j )3 = kb.

Unit vector e( j )1 is directed along the shortest distance between the unit vectors of

the generating line and the worm axis Unit vector e( j )2 is determined as the cross

product of e( j )1 and e( j )3 (see below).

(iii) Coordinate systems Sτ ( j ) ( j = I, II) and Sb (Figs 19.5.6 and 19.5.7) are rigidly

connected to each other and perform a screw motion with the screw parameter p about the worm axis Point M( j ) of the intersection of L( j ) with e( j )1 generates in screw motion a helix on the cylinder of radius ρ The unit tangent to the helix at point M( j )and the unit vector b( j )of L( j )do not coincide in the case of ZN worms and form a certain angle.

The gear ratio m21 was represented for the right-hand and left-hand worms and gears by Eqs (19 .2. 8) and (19 .2. 9),...

(19.5.6)

Here,

cos δ = (cos2< /small>α + sin2< /small>α sin2< /small>λp)0.5,... δ and sin δ:

cos δ = (cos2< /small>α + sin2< /small>α sin2< /small>λp)0.5,