Gear Geometry and Applied Theory Episode 3 Part 1 potx

Figure 19.7.5: Line of contact between generating cone and K worm surface: representation in plane of parameters... Figure 19.7.6: Contact lines between generating cone and wormon worm s

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19.7 Geometry and Generation of K Worms 583

Figure 19.7.2: Coordinate systems applied for generation of K worms.

1is represented as the family of lines of contact of surfacescand1by the followingequations:

Equation (19.7.2) is the equation of meshing Vectors Nc and v(c1) c are represented

in S c and indicate the normal to c and the relative (sliding) velocity, respectively It

is proven below [see Eq (19.7.8)] that Eq (19.7.2) does not contain parameter ψ.

Equations (19.7.1) and (19.7.2) considered simultaneously represent the surface of the

worm in terms of three related parameters (u c, θc, ψ).

For further derivations we will consider that the surface side I of a right-hand worm

is generated The cone surface is represented by the equations (Fig 19.7.3)

rc = u ccosαc(cosθcic + sin θ cjc)+ (u csinαc − a) k c. (19.7.3)

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Figure 19.7.3: Generating cone surface.

Here, u c determines the location of a current point on the cone generatrix; “a”

deter-mines the location of the cone apex

The unit normal to the cone surface is determined as

where u c > 0 Equation (19.7.8) with the given value of ucprovides two solutions for

θc and determines two curves, I and II in the plane (u c, θc) (Fig 19.7.5) Only curve I is

the real contact line in the space of parameters (u c, θc)

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Figure 19.7.4: Installment of grinding cone: (a)

illustration of installment parameter E c; (b)

illust-ration of installment parameterγ c.

Figure 19.7.5: Line of contact between generating cone and K worm surface: representation in plane

of parameters.

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Figure 19.7.6: Contact lines between generating cone and worm

on worm surface.

Equations (19.7.3) and (19.7.8) considered simultaneously represent in S c the line

of contact betweencand1 The line of contact is not changed in the screw motion

of the worm because equation of meshing (19.7.8) does not contain parameter of tionψ The worm surface 1 is represented by Eqs (19.7.1) and (19.7.8) consideredsimultaneously

mo-Figure 19.7.6 shows the contact lines on1between1andc The design eters of the worm surface are related with the equations

param-tanαc = tan α a xcosλp (19.7.9)

whereαa x is the proﬁle angle of the worm in its axial section, andλpis the lead angle

on the worm pitch cylinder, and

sc ≈ w a xcosλp (19.7.10)

where w a x is the width of worm space in the axial section, and w a xis measured on the

pitch cylinder The exact value of required s c can be determined using the equations ofthe axial section of the generated worm

The design parameters r c and a are represented as

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The ﬁnal expressions for both sides of the right-hand and left-hand worms and thesurface unit normals are represented by the following equations:

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

x1= u c(cosαccosθccosψ + cos αccosγcsinθcsinψ

− sin α csinγcsinψ) + a sin γcsinψ + Eccosψ

y1= u c(− cos α ccosθcsinψ + cos αccosγcsinθccosψ

− sin α csinγccosψ) + a sin γccosψ − Ecsinψ

z1= u c(sinαccosγc + cos α csinγcsinθc)− pψ − a cos γ c

(19.7.13)

nx1 = cos ψ sin α ccosθc + sin ψ(cos γ csinαcsinθc + sin γ ccosαc)

ny1 = − sin ψ sin α ccosθc + cos ψ(cos γ csinαcsinθc + sin γ ccosαc)

nz1 = sin γ csinαcsinθc − cos γ ccosαc

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

+ sin α csinγcsinψ) − a sin γcsinψ + Eccosψ

y1= u c(− cos αccosθcsinψ + cos αccosγcsinθccosψ

+ sin α csinγccosψ) − a sin γccosψ − Ecsinψ

z1= u c(− sin αccosγc + cos α csinγcsinθc)− pψ + a cos γ c

(19.7.16)

nx1 = cos ψ sin α ccosθc + sin ψ(cos γ csinαcsinθc − sin γ ccosαc)

ny1 = − sin ψ sin α ccosθc + cos ψ(cos γ csinαcsinθc − sin γ ccosαc)

nz1 = sin γ csinαcsinθc + cos γ ccosαc

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

+ sin α csinγcsinψ) − a sin γcsinψ + Eccosψ

+ sin α csinγccosψ) − a sin γccosψ − Ecsinψ

z1= u c(sinαccosγc − cos α csinγcsinθc)+ pψ − a cos γ c

(19.7.19)

nx1= cos ψ sin α ccosθc + sin ψ(cos γ csinαcsinθc − sin γ ccosαc)

ny1= − sin ψ sin α ccosθc + cos ψ(cos γ csinαcsinθc − sin γ ccosαc)

nz1 = − sin γ csinαcsinθc − cos γ ccosαc

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

− sin α csinγcsinψ) + a sin γcsinψ + Eccosψ

− sin α csinγccosψ) + a sin γccosψ − Ecsinψ

z1= u c(− sin αccosγc − cos α csinγcsinθc)+ pψ + a cos γ c

(19.7.22)

nx1= cos ψ sin α ccosθc + sin ψ(cos γ csinαcsinθc + sin γ ccosαc)

ny1= − sin ψ sin α ccosθc + cos ψ(cos γ csinαcsinθc + sin γ ccosαc)

nz1 = − sin γ csinαcsinθc + cos γ ccosαc

by Eqs (19.7.13), (19.7.16), (19.7.19), and (19.7.22), respectively

The proof is based on the following considerations:

(i) The equation of meshing (19.7.15) provides that

sinθc = p cot αc

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This means thatθcis constant andccontacts1along a straight line, the atrix of the cone

gener-(ii) The worm surface is generated by a straight line, that is, it is a ruled surface It is adeveloped surface as well because the surface normal does not depend on surface

coordinate u c Recall that u c determines the location of a current point on thegenerating line

(iii) Considering the equations of the worm surface and the unit normal to the surface,

we may represent a current point of the surface normal by the equation

R1(u c , ψ, m) = r1(u c, ψ) + mn1(ψ) (19.7.26)

where the variable parameter m determines the location of the current point on

the surface normal Function R1(u c, ψ, m) represents the one-parameter family of

curves that are traced out in S1by a current point of the surface normal

(iv) The envelope to the family of curves is determined with Eq (19.7.26) and theequation (see Section 6.1)

(v) Equations (19.7.26) and (19.7.27) yield that the normals to the worm surface are

tangents to the cylinder of radius r band form the angle of (90◦− λ b) with the wormaxis

Here,

rb = E csinθc = p cot α c, λb = α c. (19.7.28)

Problem 19.7.1

Consider that the worm surface represented by Eqs (19.7.13) is cut by the plane y1= 0

Axis x1is the axis of symmetry of the space in axial section The point of intersection

of the axial proﬁle with the pitch cylinder is determined with the coordinates

x1= r p, y1= 0, z1= −wa x

2 = −pa x

4 = − π

4P a x

Here, w a xis the nominal value of the space width in axial section that is measured along

the generatrix of the pitch cylinder; p a xis the distance between two neighboring threads

along the generatrix of the pitch cylinder, and P a x = π/p a xis the worm diametral pitch

in axial section Derive the system of equations to be applied to determine s c(Fig 19.7.1)

considering r p , r c , E c,αc , p, and w a x as given

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a = r ctanαc+sc

2.

The derived equation system contains four equations in four unknowns:θc,ψ, uc, and

a The solution of the system for the unknowns provides the sought-for value of sc

19.8 GEOMETRY AND GENERATION OF F-I WORMS (VERSION I)

F worms with concave–convex surfaces have been proposed by Niemann and Heyer(1953) and applied in practice by the Flender Co., Germany The great advantage of the

F worm-gear drives is the improvement of conditions of lubrication that is achieved due

to the favorable shape of contact lines between the worm and the worm-gear surfaces

We consider two versions of F worms: (i) the original one, F-I, and (ii) the modiﬁedone, F-II, proposed by Litvin (1968) Both versions of worm-gear drives are designed

as nonstandard ones: the radius r p (o)of the worm operating pitch cylinder differs from

the radius r p of the worm pitch cylinder, and r p (o) − r p ≈ 1.3/P a x To avoid pointing of

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19.8 Geometry and Generation of F-I Worms (Version I) 591

Figure 19.8.1: Installation of grinding wheel

generating worm F-I: (a) illustration of

instal-lation parameterγ c; (b) illustration of

installa-tion parameter E c.

teeth of worm-gears, the tooth thickness of the worm on the pitch cylinder is designed

as t p = 0.4p a x = 0.4π/P a x

Installment of the Grinding Wheel for F-I

The surface of the grinding wheel is a torus The axial section of the grinding wheel

is the arcα–α of radius ρ [Fig 19.8.1(b)] In the following discussion we consider the

generation of the surface side II of the right-hand worm.

The radius ρ is chosen as approximately equal to the radius rp of the worm pitchcylinder The installation of the grinding wheel with respect to the worm is shown inFig 19.8.1(a) The axes of the grinding wheel and the worm form the angleγc = λ p,whereλpis the lead angle on the worm pitch cylinder, and the shortest distance between

these axes is E c Figure 19.8.2(a) shows the section of the grinding wheel and the

worm by a plane that is drawn through the z caxis, which is the axis of rotation of the

grinding wheel, and the shortest distance O c O1[Fig 19.8.1(b)] It is assumed that the

line of shortest distance passes through the mean point M of the worm proﬁle; a and b determine the location of center O bof the circular arcα–α with respect to Oc.Here,

whereρ is the radius of arc α–α.

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(a) (b)

Figure 19.8.2: Generation of grinding wheel with torus surface: (a) section of the grinding wheel and (b) applied coordinate systems.

We set up coordinate systems S c and S p that are rigidly connected to the grinding

wheel; coordinate systems S b and S aare rigidly connected to the circular arc of radiusρ

(Fig 19.8.2) The circular arcα–α is represented in Sbby the equation

rb = ρ[− sin θ 0 cos θ 1]T. (19.8.2)

Figure 19.8.2(a) shows coordinate systems S a and S b in the initial position The

surface of the grinding wheel is generated in S c while the circular arc with coordinate

systems S a and S b is rotated about the z paxis [Fig 19.8.2(b)]

The coordinate transformation is based on the following matrix equation:

rc(θ, ν) = Mc pMpaMabrb= Mcbrb. (19.8.3)Here,

cosν sinν 0 −d cos ν

− sin ν cos ν 0 d sin ν

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nc = [sin θ cos ν − sin θ sin ν − cos θ ]T. (19.8.8)

Equations of Meshing of Grinding Wheel and Worm

The unit normal nc is directed toward the generating surface and outward to the worm

surface The worm surface is generated as the envelope to the family of surfaces that

is generated in S1 by c in its relative motion with respect to the worm surface1

Coordinate system S1is rigidly connected to the worm

The equation of meshing is

nc· v(c1)

where v(c1) c is the velocity in relative motion of the grinding wheel with respect to the

worm Vectors in Eq (19.8.9) are represented in S c

We consider that the worm performs the screw motion with the screw parameter p

(Fig 19.7.4) with respect to the grinding wheel, and v(c1) c is represented by Eqs (19.7.7).After transformations, the equation of meshing of the grinding wheel surface with theworm surface is represented by

f (θ, ν) = tan θ − Ec − p cot γ c − d cos ν

b cos ν + (Eccotγc + p) sin ν = 0. (19.8.10)

The equation of meshing does not contain parameter ψ in screw motion because the

relative motion is the screw one Equation (19.8.10) with the given value ofθ provides

two solutions for ν, but only the solution for 0 < ν < 90◦ should be used for further

derivations Recall that Eq (19.8.10) is derived for the case when the surface side II of

a right-hand worm is generated

Lines of Contact on Worm Surface

The line of contact between c and1 is a single line onc and is represented in S c

by Eqs (19.8.7) and (19.8.10) considered simultaneously Figure 19.8.3 shows the line

of contact in the space of parametersθ, ν; the dashed line represents the line of contact

that is out of the working part of the grinding wheel

The worm surface is represented in S1as the set of contact lines between surfaces

c and1 Using this approach, we have derived the equations of the worm surfaces

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Figure 19.8.3: Line of contact between grinding wheel and worm F-I surfaces.

for both sides, considering the right-hand and left-hand worms Axis x1in the derived

equations is the axis of symmetry for any section of the worm space by a plane that is drawn through the x1axis An axial section of the worm space is obtained by intersecting

the space by the plane y1= 0 To provide the above-mentioned location of the x1axis, asthe axis of symmetry of the axial section of the space, we have to consider the following:

(i) The initially applied coordinate system S1∗ is substituted by a parallel coordinate

system S1 whose origin is displaced along the z∗1 axis at the distance a o (Fig.19.8.4)

(ii) The coordinates of the point of intersection of the axial section of the worm spacewith the pitch cylinder must be

x1= r p, y1= 0, z1=wa x

where w a x is the space width on the pitch cylinder

Figure 19.8.4: Derivation of axial section of worm F-I.

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The results of derivations of the worm surface and the surface unit normal are asfollows:

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

x1= (ρ sin θ c + d)(− cos ν cos ψ + sin ν sin ψ cos γ c)

+ (ρ cos θ c − b) sin ψ sin γ c + E ccosψ

y1= (ρ sin θ c + d)(cos ν sin ψ + sin ν cos ψ cos γ c)

+ (ρ cos θ c − b) cos ψ sin γ c − E csinψ

z1= (ρ sin θ c + d) sin ν sin γ c + (b − ρ cos θ c) cosγc − pψ + a o

(19.8.12)

where

ao= −wa x

2 − (ρ sin θ c + d) sin ν sin γ c − (b − ρ cos θ c) cosγc + pψ (19.8.13)

nx1 = sin θ c(− cos ν cos ψ + sin ν sin ψ cos γc)+ cos θ csinψ sin γc ny1 = sin θ c(cosν sin ψ + sin ν cos ψ cos γc)+ cos θ ccosψ sin γc nz1 = sin θ csinν sin γc − cos θ ccosγc.

(19.8.14)

Parametersθc andν in Eqs (19.8.12) and (19.8.14) are related to the equation of

meshing,

tanθc = Ec − p cot γ c − d cos ν

b cos ν − (Eccotγc + p) sin ν . (19.8.15)

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

− (ρ cos θ c − b) sin ψ sin γ c + E ccosψ

− (ρ cos θ c − b) cos ψ sin γ c − E csinψ

z1= (ρ sin θ c + d) sin ν sin γ c − (b − ρ cos θ c) cosγc − pψ + a o

(19.8.16)

where

ao=wa x

2 − (ρ sin θ c + d) sin ν sin γ c + (b − ρ cos θ c) cosγc + pψ (19.8.17)

nx1 = sin θ c(− cos ν cos ψ + sin ν sin ψ cos γc)− cos θ csinψ sin γc ny1 = sin θ c(cosν sin ψ + sin ν cos ψ cos γc)− cos θ ccosψ sin γc nz1 = sin θ csinν sin γc + cos θ ccosγc

(19.8.18)

Parametersθcandν in Eqs (19.8.16) and (19.8.18) are related with the equation

of meshing,

b cos ν + (Eccotγc + p) sin ν . (19.8.19)

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

− (ρ cos θ c − b) sin ψ sin γ c + E ccosψ

− (ρ cos θ c − b) cos ψ sin γ c − E csinψ

z1= − (ρ sin θ c + d) sin ν sin γ c + (b − ρ cos θ c) cosγc + pψ + a o

(19.8.20)

where

ao= −wa x

2 + (ρ sin θ c + d) sin ν sin γ c − (b − ρ cos θ c) cosγc − pψ (19.8.21)

nx1 = sin θ c(− cos ν cos ψ + sin ν sin ψ cos γc)− cos θ csinψ sin γc ny1 = sin θ c(cosν sin ψ + sin ν cos ψ cos γc)− cos θ ccosψ sin γc nz1 = − sin θ csinν sin γc − cos θ ccosγc.

(19.8.22)

Parametersθc andν in Eqs (19.8.20) and (19.8.22) are related to the equation of

meshing,

b cos ν + (Eccotγc + p) sin ν . (19.8.23)

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

+ (ρ cos θ c − b) sin ψ sin γ c + E ccosψ

+ (ρ cos θ c − b) cos ψ sin γ c − E csinψ

z1= − (ρ sin θ c + d) sin ν sin γ c − (b − ρ cos θ c) cosγc + pψ + a o

(19.8.24)

where

ao=wa x

2 + (ρ sin θ c + d) sin ν sin γ c + (b − ρ cos θ c) cosγc − pψ (19.8.25)

nx1 = sin θ c(− cos ν cos ψ + sin ν sin ψ cos γc)+ cos θ csinψ sin γc ny1 = sin θ c(cosν sin ψ + sin ν cos ψ cos γc)+ cos θ ccosψ sin γc nz1 = − sin θ csinν sin γc + cos θ ccosγc.

(19.8.26)

Parametersθcandν in Eqs (19.8.24) and (19.8.26) are related with the equation

of meshing,

b cos ν − (Eccotγc + p) sin ν . (19.8.27)

Figure 19.8.5 shows the cross section and axial section of the F-I worm that have

been obtained for the following input parameters: N1= 3; N2= 31; r p= 46 mm; axial

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19.9 Geometry and Generation of F-II Worms (Version II) 597

Figure 19.8.5: Cross section and axial section of worm F-I.

module m a x = 8 mm The radius of the operating pitch cylinder is r po = r p + 1.25m a x =

56 mm;ρ = 46 mm; γc = λ p= 14◦3715; αn= 20◦; a = r p + ρ sin α n = 61.733 mm;

b = ρ cos α n = 43.226 mm.

19.9 GEOMETRY AND GENERATION OF F-II WORMS (VERSION II)

Method for Grinding

The grinding of F worms of version II can be performed by the same tool that is usedfor generation of worms of version I The difference is in the application of specialsetting parameters The geometry of F worms of version II has certain advantages incomparison with the worms of version I: (i) the line of contact between the grindingsurfacecand the worm surface is a planar curve, the circular arc of the axial section

of the torus; and (ii) the shape of the line of contact does not depend on the diameter

of the grinding wheel and the shortest center distance E c

The main idea of the proposed method for grinding is based on application of axes

of meshing There are two axes of meshing when a helicoid is generated by a peripheral

tool with a surface of revolution One of the axes of meshing, I –I , coincides with the

axis of rotation of the tool (Fig 19.9.1); the location and orientation of the other axis of

Tiêu đề	Geometry and Generation of K Worms
Trường học	Unknown University
Chuyên ngành	Mechanical Engineering
Thể loại	Lecture Note
Năm xuất bản	2004
Thành phố	Unknown City

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