Gear Geometry and Applied Theory Episode 3 Part 7 pptx

cur-Further derivations are based on the following equations: k n = kIcos2q + kIIsin2q= 1 2kI + kII+1 2kI − kII cos 2q 26.4.48 Here, kI and kII are the surface principal curvatures and

26.4 Generation of a Surface with Optimal Approximation 763

Figure 26.4.6: Determination of maximal

deviations along line L gk.

MINIMIZATION OF DEVIATIONS δ i, j. Consider that deviations δ i , j (i = 1, , n; j =

1, , m) of  g with respect to p have been determined at the (n , m) grid points The

minimization of deviations can be obtained by corrections of previously obtained tionβ(1)(θ p) The correction of angle β is equivalent to the correction of the angle that

func-is formed by the principal directions on surfaces t and g The correction of angleβ

can be achieved by turning of the tool about the common normal to surfaces  t and

 p at their instantaneous point of tangency Mk.

The minimization of deviationsδ i , j is based on the following procedure:

Step 1: Consider the characteristic L gk, the line of contact between surfaces  t and

 g , that passes through current point Mk of mean line Lmon surface p(Fig 26.4.6).Determine the deviationsδ kbetween t and p along line Lgkand find out the maximaldeviations designated asδ(1)

kmaxandδ(2)

kmax Points of Lgkwhere the deviations are maximal

are designated as N k(1)and N k(2) These points are determined in regions I and II of surface

 g with line Lmas the border The simultaneous consideration of maximal deviations

in both regions enables us to minimize the deviations for the whole surface g

Note The deviations of  t from p along Lgkare simultaneously the deviations of

 g from p along Lgk because Lgkis the line of tangency of t and g

Step 2: The minimization of deviations is accomplished by correction of angleβ kthat

is determined at point Mk(Fig 26.4.6) The minimization of deviations is performed

lo-cally, for a piece k of surface  g with the characteristic Lgk The process of minimization

is a computerized iterative process based on the following considerations:

(i) The objective function is represented as

(ii) The variable of the objective function isβ k Then, considering the angle

β(2)

k = β(1)

and using the equation of meshing withβ k, we can determine the new characteristic,

the piece of envelope (k)

g , and the new deviations The applied iterations provide

the sought-for objective function The final correction of angle β k we designate

asβ (opt)

Note 1 The new contact line L(2)gk (determined withβ(2)

k ) slightly differs from the real

contact line because the derivative d β(1)

k /ds but not dβ(2)

k /ds is used for determination

L(2)gk However, L(2)gk is very close to the real contact line

Step 3: The discussed procedure must be performed for the set of pieces of surfaces

 g with the characteristic Lgkfor each surface piece

Recall that the deviations for the whole surface must satisfy the inequality δ i , j ≥ 0.The procedure of optimization is illustrated with the flowchart in Fig 26.4.7

Curvatures of Ground Surface Σg

The direct determination of curvatures of g by using surface g equations is a plicated problem The solution to this problem can be substantially simplified using thefollowing conditions proposed by the authors: (i) the normal curvatures and surfacetorsions (geodesic torsions) of surfaces pand g are equal along line Lm, respectively;

com-and (ii) the normal curvatures com-and surface torsions of surfaces  t and  g are equal

along line Lg This enables us to derive four equations that represent the principal vatures of surface g in terms of normal curvatures and surface torsions of pand t.However, only three of these equations are independent (see below)

cur-Further derivations are based on the following equations:

k n = kIcos2q + kIIsin2q= 1

2(kI + kII)+1

2(kI − kII ) cos 2q (26.4.48)

Here, kI and kII are the surface principal curvatures and angle q is formed by unit

vectors eI and e and is measured counterclockwise from eI and e; eI is the principal

direction with principal curvature kI; e is the unit vector for the direction where the

normal curvature is considered; t is the surface torsion for the direction represented by e.

Equation (26.4.48) is known as the Euler equation Equation (26.4.49) is known indifferential geometry as the Bonnet–Germain equation (see Chapter 7)

The determination of the principal curvatures and principal directions for g is based

on the following computational procedure (see Section 7.9):

Step 1: Determination of k n(1)and t(1)for surface g at the direction determined by

the tangent to Lm The determination is based on Eqs (26.4.48) and (26.4.49) applied

to surface  p Recall that  p and  g have the same values of k(1)n and t(1)along thepreviously mentioned direction

Step 2: Determination of k n(2) and t(2) The designations k n(2) and t(2) indicate thenormal curvatures of g and the surface torsion along the tangent to Lg Recall that

k n(2)and t(2)are the same for t and g along Lg We determine k(2)n and t(2)for surface

 t using Eqs (26.4.48) and (26.4.49), respectively

Step 3: We consider at this stage of computation that for surface  g the following

are known: k n(1)and t(1), and k n(2)and t(2)for two directions with tangentsτ1 andτ2that form the known angle µ (Fig 26.4.8) Our goal is to determine angle q1(or q2)

26.4 Generation of a Surface with Optimal Approximation 765

Figure 26.4.7: Flowchart for optimization.

for the principal direction e(g) I and the principal curvatures k (g) I and k (g) II (Fig 26.4.8)

Using Eqs (26.4.48) and (26.4.49), we can prove that k n (i ) and t (i ) (i = 1, 2) given for

two directions represented byτ1andτ2are related with the following equation:

t(1)+ t(2)

k(2)n − k(1)

n

Figure 26.4.8: For determination of cipal directions of generated surface g.

prin-Step 4: Using Eqs (26.4.48) and (26.4.49), we can derive the following three

equa-tions for determination of q1, k (g) I , and k (g) II :

Equation (26.4.51) provides two solutions for q1(q1(2)= q(1)

1 + 90◦) and both are

cor-rect We choose the solution with the smaller value of q1

Numerical Example: Grinding of an Archimedes Worm Surface

The worm surface shown in Fig 26.4.9 is a ruled undeveloped surface formed by the

screw motion of straight line K N (|K N| = up) The screw motion is performed in coordinate system S p[Fig 26.4.9(b)] The to-be-ground surface p is represented in Sp

as

r p = u pcosα cos θ pip + u pcosα sin θ pjp + (pθp − u psinα) k p (26.4.54)

where u pandθ pare the surface parameters

The surface unit normal is

p sin θ p + u psinα cos θ p

−p cos θ p + u psinα sin θ p

26.4 Generation of a Surface with Optimal Approximation 767

Figure 26.4.9: Surface of an Archimedes worm.

The design data are: number of threads N1= 2; axial diametral pitch Pa x = 8/in;

α = 20◦; the radius of the pitch cylinder is 1.25 in The remaining design parametersare determined from the following equations:

(i) The screw parameter is

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0

2 3 4 5 6 7 8 9

Figure 26.4.10: Deviations of the ground surface from ideal surface of an Archimedes worm.

(ii) The lead angle is

The optimal angle β (opt) = −94.6788◦ has been determined by the developed mization method The deviations of the ground surface g from p with the optimal

opti-β (opt)are positive and the maximal deviation has been reduced to 0.35µm (Fig 26.4.10).

27 Overwire (Ball) Measurement

27.1 INTRODUCTION

Indirect determination of gear tooth thickness by overwire (ball) measurement has foundbroad application This topic has been the subject of research by many scientists Theearliest publications dealing with such measurement of worms and spatial gears areLitvin’s papers and books Detailed references regarding the history of the performedresearch are given in Litvin [1968]

The application of computers and subroutines for the solution of systems of linear equations is a significant step forward in this area that was accomplished by

non-Litvin et al [1998b] This chapter covers the following topics:

(i) Algorithms for determination of location of a wire or a ball placed into the space

of a workpiece with symmetric and nonsymmetric location of tooth surfaces(ii) Relation between the tooth thickness and overwire measurement – this relationenables us to use the developed algorithms for a workpiece with various tolerances.The developed theory was applied for measurement of tooth thickness of worms,screws, and involute helical gears Computer programs for this purpose have been de-veloped

27.2 PROBLEM DESCRIPTION

Consider that a ball (a wire) is placed into the space of a workpiece (a worm, screw,

or gear) The surfaces of the ball and the workpiece are in tangency at two pointswhose location depends on the geometry of the surfaces of the workpiece, the width

of the space, and the diameter of the ball The surface geometry of the workpiece isrepresented analytically and the width of the space and the ball (wire) diameter aregiven Then, we can determine analytically (i) the distance of the center of the ball fromthe axis of the workpiece, or (ii) the shortest distance between the axes of the wire andthe workpiece The measurement over the ball (the wire) and the comparison of theobtained data with the analytically determined data enable us to find out if the spacewidth satisfies the requirements

769

The following is a description of the analytical approach that has been developedfor the determination of points of tangency of a ball (wire) with the surfaces of thespace of a workpiece In the most general case there is no symmetry in the location andorientation of the two surfaces that form the space of the workpiece This is typical, forinstance, wherein the workpiece is a spiral bevel pinion or a hypoid pinion Thus, wehave to consider in such a case the simultaneous tangency of the ball or the wire withboth surfaces of the space.

Consider that the two surfaces of the space and the unit normals to the surfaces arerepresented by vector equations,

is the position vector of center C of the ball, or the point of intersection of both normals

with the wire axis It is evident from the conditions of force transmissions by the

mea-surement, that both normals to the wire intersect the wire axis at the same point We consider that z is the axis of the workpiece, and Z is chosen to determine the location

of point C in a plane that is perpendicular to the axis of the workpiece.

The tangency of the wire (ball) with the surfaces of the space is represented by theequation (Fig 27.2.1)

27.2 Problem Description 771

Figure 27.2.1: Measurement by a single ball.

Our goal is to determine the distance

◦N



where N is the number of gear teeth There is a special approach for determination

of M when a gear with an odd number of teeth is measured by two wires (see Section

27.3)

Procedure of Computation

The procedure of computation for overwire (ball) measurement may be represented asfollows:

Step 1: Equations (27.2.5) represent a system of six equations in six unknowns: X,

Y, u (i ), θ (i ) (i = 1, 2) We may represent this system of equations by a subsystem of

four non-linear equations, and a system of two linear equations The subsystem of four

Figure 27.2.2: Measurement by two balls.

four unknowns: u(1),θ(1), u(2), andθ(2) A subroutine for the solution of the above system

of four nonlinear equations is required [for instance, we can use the one contained inthe IMSL library [Visual Numerics, Inc., 1998]]

The unknowns X and Y may be determined from the following two linear equations:

X = x (i )

u (i ) , θ (i )

− ρn (i ) x

u (i ) , θ (i )

(i = 1 or 2). (27.2.14)

Step 2: We have considered the radiusρ of the ball (the wire) as known In reality,

we have to determine a value ofρ that satisfies the equation

Here, ra is the radius of the addendum circle in the plane Z = d, and δ is the desired difference between (R + ρ) and ra The proper value ofρ can be determined by variation

27.3 Measurement of Involute Worms, Involute Helical Gears, and Spur Gears 773

ofρ in Eqs (27.2.9) to (27.2.15) until the desired value of δ is obtained Then, an even

value ofρ may finally be chosen, and we can start the computations using Eqs (27.2.9)

to (27.2.14)

Step 3: The width w t of the space on the reference (pitch circle) rp may vary in

accordance to the prescribed tolerance dwt The nominal value of M is obtained for the nominal value of wt The determination of the ratio d M /dw t in addition to the nominal

value of M enables us to determine the real value of the space width.

Particular Case 1

The surfaces of the space have a plane of symmetry, say Y= 0 Then, we may consider

conditions of tangency of the wire (ball) with one side surface only Equations (27.2.5)

applied for this case yield

where Z is considered as chosen.

Equations (27.2.16) represent a system of three equations in three unknowns The lution of a subsystem of two nonlinear equations enables us to determine the unknowns

so-(u , θ) Then, we may determine R from the remaining equation that is the linear one with respect to the unknown R.

Particular Case 2

A screw with asymmetric space surfaces is considered Both surfaces of the space arehelicoids A cross section of the screw will coincide with another one after rotating

through a certain angle about the axis of the screw For this reason, any value of Z can

be chosen in Eqs (27.2.9) and (27.2.10), for instance, Z= 0

Particular Case 3

The surfaces of the space are symmetric and they are helicoids In this case, we can use

Eqs (27.2.16) and take Z= 0

27.3 MEASUREMENT OF INVOLUTE WORMS, INVOLUTE HELICAL GEARS,

AND SPUR GEARS

Basic Equations

Equations of involute worms are represented in Section 19.6 Equation (27.2.16) with

Z= 0 yields the following computational procedure:

These equations work for the right-hand and left-hand involute worms and helical gears.

In case of spur gears, we have to takeλ b= 90o

Determination of M

In the case of three-wire measurement, M is determined from Eq (27.2.7) The same

equation is applied for measurement by two balls (wires) of gears with an even number

of teeth Equation (27.2.8) is applied for the measurement by two balls of helical andspur gears with an odd number of teeth and for the measurement of spur gears with anodd number of teeth by two wires Measurement of helical gears with an odd number

of teeth by two wires is based on the approach developed by Litvin [1968]

Representation of the Unit Vectors of Wire Axes and the Shortest Distance Between the Axes of Two Wires

Consider that wire 1 is installed into gear space(1)

2 Vectors a(1)and r(1)(Fig 27.3.1)

represent the unit vector of the wire axis in S2, and the shortest distance between a(1)and the gear axis, respectively Here,

cotβ R = p

where p is the screw parameter of the helicoid.

Consider now that wire 2 is installed into the space(2)

2 that forms angleγ with the

first space Angleγ is measured in the plane that is perpendicular to the gear axis.

where N is the teeth number.

Figure 27.3.1: Location and orientation of

wire 1 in S2

27.3 Measurement of Involute Worms, Involute Helical Gears, and Spur Gears 775

We set up two coordinate systems Sa and Sb that are rigidly connected to wire 2

Coordinate system Sa initially coincides with S2 The installment of wire 2 in Sb is

represented in Sbby equations similar to (27.3.4) and (27.3.5):

2 Vector of shortest distance r(2)2 between a(2)2 and the gear axis, and unit vector a(2)2

are represented in S2by the equations

r(2)2 = M2aMabr(2)b , a(2)

2 = L2aLaba(2)b (27.3.9)

Matrices M2aand Mabmay be derived using Fig 27.3.2 Figure 27.3.2(a) shows the

orientation of coordinate systems Sa and Sb, one with respect to the other Coordinate systems Sa and Sb are rigidly connected, and initially Sa coincides with S2 Figure

27.3.2(b) shows the orientation and location of Sa with respect to S2after the rotationthrough angleφ and the displacement of pφ in the screw motion.

Figure 27.3.2: For derivation of location and

ori-entation of wire 2 in S2

Figure 27.3.3: For derivation of shortest distance between two wires.

Equations (27.3.9) yield

r(2)2 = R[cos(γ − φ) i2− sin(γ − φ) j2+ φ cot βRk2] (27.3.10)

a(2)2 = sin βRsin(γ − φ) i2+ sin βRcos(γ − φ) j2+ cos βRk2. (27.3.11)The variable parameterφ represents the rotation of wire 2 in its screw motion about

the gear axis

Determination of the Overwire Measurement M

We may now derive the equation of the unit vector c2of the shortest distance between

vectors a(1)2 and a(2)2 as follows (Fig 27.3.3):

c2= a

(1)

2 × a(2) 2



− tan βRcos

γ − φ2

27.3 Measurement of Involute Worms, Involute Helical Gears, and Spur Gears 777

(See derivations below.) The overwire measurement is performed when function C( φ)

reaches its extreme value, that is,

The derivation of equation

r(1)2 − r(2) 2

is based on the following considerations:

(i) Unit vectors a(1)2 and a(2)2 of the axes of two wires are crossed and lie in parallelplanes 1and 2(Fig 27.3.3) The shortest distance between a(1)2 and a(2)2 is

C= B A = C a

(1)

2 × a(2) 2

a(1)

2 × a(2)

2 .

(ii) Position vectors r(1)2 = O2M, and r(2)2 = O2N are drawn from origin O2of

coor-dinate system S2to current points M and N of the axes of the wires.

(iii) Figure 27.3.3 yields

27 .3 Measurement of Involute Worms, Involute Helical Gears, and Spur Gears 7 73< /b>

ofρ in Eqs ( 27. 2.9) to ( 27. 2.15) until the desired value of δ is... class="text_page_counter">Trang 13< /span>

27 .3 Measurement of Involute Worms, Involute Helical Gears, and Spur Gears 77 5

We set up two coordinate systems Sa and. .. similar to ( 27 .3. 4) and ( 27 .3. 5):

2 Vector of shortest distance r(2)2 between a(2)2 and the gear axis, and unit