Kinematic Geometry of Surface Machinin Episode Episode 15 pps

In this method of surface machining, the work having axial profile 1 of variable curvature is rotating about its axis of rotation O P with a certain rotation ωP Figure 11.9.. Examples of

of the cutting edge of the cutter at the current CC-point. In this method of

surface machining, the work having axial profile 1 of variable curvature is

rotating about its axis of rotation O P with a certain rotation ωP (Figure 11.9)

The cutting edge of the cutter, an arc segment of a curve 2 having

perma-nently variable radius of curvature R T, is used for performing this

machin-ing operation

The cutter is traveling along the axial profile of the part surface in the

peripheral direction 3 On the lathe, this motion is obtained as the

superposi-tion of the axial mosuperposi-tion 4 of the cutter and its reciprocal mosuperposi-tion toward the

part axis of rotation 5 and in backward direction 6

In addition, the cutter is performing the orientational motion of the

sec-ond kind (see Chapter 2) This motion of the cutter is performed either in

the direction 7 or in the direction 8 The actual direction of the orientation

motion of the cutter depends upon the actual geometry of the axial profile

1 of the part surface at two neighboring CC-points K i and K i+1 Ultimately,

due to the cutter motion either in the direction 7 or in the direction 8, the

cutting edge is rolling with sliding over the axial profile 1 of the part surface

being machined

Implementation of the orientational motion of the cutter allows for better

fit of the radius of curvature R T to the part surface radius of curvature R P

at every CC-point K In this way (see Equation 11.10), the surface generation

output is increasing

It is important to note a possibility of machining form surfaces of

revolu-tion in compliance with the method (Figure 11.9), when not the cutter, but a

Int Cl B23B 1/00, Filed November 3, 1984.

2 7

4

1

8

K i+1

K i

3

О P

ωP

Figure 11.9

Utilization of the orientational motion of the second kind of the cutter in the method of turning

form surfaces of revolution (SU Pat No 1171210).

Examples of Implementation of the DG/K-Based Method 473

milling cutter or grinding wheel having a corresponding axial profile of the

generating surface of the tool is used instead

Similar to the utilization of the orientational motion of the second kind

(Figure 11.9), the orientational motion of the first kind of the cutter can be

utilized in the turning of surfaces of revolution as well

A method of turning of form surfaces of revolution (Figure 11.10) is

fea-turing in its kinematics the orientational motion of the first kind [8] This

method of surface machining is similar to the earlier discussed method of

surface machining shown in Figure 11.9 For convenience, designations of

the major elements in Figure 11.10 are identical to the designations of the

corresponding major elements in Figure 11.9 So, there is no reason to repeat

all the details of the method under consideration

In the method of surface machining (Figure 11.10), the orientational

motion of the first kind is utilized The orientational motion of this kind

allows for turning of the cutter about the axis O K i. along the unit normal

vector nP either in the direction 9 or in the direction 10 The actual

direc-tion of the orientadirec-tion modirec-tion depends upon parameters of geometry of the

surface P at the two neighboring CC-points K i and K i+1 In the neighboring

CC-point K i+1, the orientation motion is designated as 11/12

Int Cl B23B 1/00, Filed September 13, 1984.

8

10 9

3

О K.i+1

O P

O K.i

2

11

1

6

5

4

12

P

7

ω P

K i+1

K i

Figure 11.10

Utilization of the orientational motion of the first kind of the cutter in the method of turning of

form surfaces of revolution (SU Pat No 1232375).

Evidently, the parameters of the orientational motion of the first kind are

strongly constrained by the limit values of the geometrical parameters of the

cutting edge of the cutter to be used, first of all by the clearance angle of the

cutting edge

Implementation of the orientational motion of the cutter allows for better

fit of the radius of curvature R T to the part surface radius of curvature R P

at every CC-point K In this way (see Equation 11.10), the surface-generation

output is increasing

It is the right point to stress that it may also be possible to machine form

surfaces of revolution in compliance with the method (Figure 11.10) when

not the cutter, but a milling cutter or grinding wheel having a corresponding

axial profile of the generating surface of the tool is used instead Under such

a scenario, no constraints are imposed by the limit values of the geometrical

parameters of the cutting edge of the cutting tool to be used

11.2.2  Milling Operations

The earlier discussed methods of turning form surfaces of revolution (see

Figure 11.9 and Figure 11.10) allow substitution of the cutter with a

mill-ing cutter or with a grindmill-ing wheel havmill-ing a correspondmill-ing profile of axial

cross-section of the generating surface T These methods of surface

machin-ing indicate that efficient methods of millmachin-ing of form surfaces of revolution

can be developed

A method of milling of form surfaces of revolution on NC machine tools was

developed by the author [26] In compliance with the method (Figure 11.11), a

form surface of revolution P having axial profile 1 is machined with the

mill-ing cutter havmill-ing a curved axial profile of the generatmill-ing surface T The work

is rotating about its axis of rotation OP with a certain rotation ωP The axis

6 4

2

1

View А (Turned)

7 8

K A

3 5

K

O P

O T

R o.P

R o.T

d P

d T

d P

d T

ω T

ω P

Figure 11.11

G V., Mashinostroitel’, No 5, 17–19, 1987.)

Examples of Implementation of the DG/K-Based Method 475

of rotation of the work OP and the axis of rotation of the milling cutter OT

are crossing at a right angle

The milling cutter is traveling in the direction 3 along the axial profile 1 of

the part with a certain peripheral feed rate This motion is a superposition

of the milling cutter motion in the axial direction 4 of the work, and of its

motion 5 toward the work axis of rotation and in backward direction 6 In

addition to the mentioned motions, the milling cutter is also performing the

motion of orientation of the second kind While traveling in the axial

direc-tion 4 of the work, the milling cutter simultaneously performs linear modirec-tion

along its axis of rotation OT This motion is performing either in the

direc-tion 7 or in the opposite direcdirec-tion 8 depending upon the geometry of the

surfaces P and T at the current CC-point K.

The orientational motion of the milling cutter provides a possibility for

increasing the rate of conformity of the generating surface T of the milling

cutter to the form surface of revolution P at every CC-point K In this way, the

surface generation output is increased

Grinding of form surfaces of revolution can be performed in the same way

as shown in Figure 11.11

11.2.3  Machining of Cylinder Surfaces

Orientation motions of the cutting tool are also used for the improvement

of machining of general cylinder surfaces Such a possibility is illustrated

below by the method of machining of a camshaft.

The method of machining of a camshaft [1] is targeting the maximal

pos-sible material removal rate

In compliance with the method, a grinding wheel having conical

generat-ing surface T is used for the machingenerat-ing of the surface P of a cam (Figure 11.12)

The grinding wheel is rotating about its axis of rotation OT with a certain

rotation ωT.

The surface generation motions are performed by the work The set of these

motions includes the rotation ωP of the work about the axis of rotation OP

and the reciprocal motion 1 in the direction of the common perpendicular

to the axes OP and OT The rotation ωP of the grinding wheel can be either

uniform or nonuniform

The grinding wheel is performing an auxiliary straight motion 2

Direc-tion of the moDirec-tion 2 is parallel to the axis of rotaDirec-tion of the work OP The

straight motion 2 is timed with work rotation ωP in the way under which the

material removal rate is constant and equal to its greatest feasible value:

Q crmax=0 5, [ ( )]⋅ Lϕ 2⋅v T( )ϕ ⋅ =b Const (11.11)

 SU Pat No 1703291, A Method of Machining of Form Surfaces./S.I Chukhno and S.P Radzevich,

Int Cl B23C 3/16, Filed August 2, 1989.

Reinforcement of form surfaces of revolution can be performed with the tool

having a conical generating surface T [21,22] In this method (Figure 11.13),

the work is rotating about its axis O P with a certain rotation ωP The axis

O T of the conical indenter 1 (conical tool) is crossing the work axis of rotation

O P at the right angle The tool is moving along the axial profile of the part

surface P with a certain peripheral feed rate The indenter 1 is pressed into

the part surface P by normal force P rnf In the relative motion, the CC-point

K traces the trajectory 2 on the machined part surface

Two configurations of the indenter 1 are possible The first configuration

is shown in Figure 11.13 In such a tool configuration, its bigger diameter is

below the smaller diameter The inverse configuration of the tool, when the

smaller diameter is below the bigger diameter, is feasible as well

When machining a form surface of revolution, the portion of the surface

P having bigger diameter is machined with the portion of the tool having

smaller diameter, and vice versa In this way, it is possible to maintain that

same pressure when machining portions of the part with different

geome-try of the surface P For this purpose, the indenter is performing an auxiliary

straight motion either downward 3 or upward 4, depending on the geometry

of the surface P being machined The auxiliary motion requires in

corre-sponding compensation of center distance between the axes O P and O T A

component of the auxiliary straight motion creates the orientational motion

of the second kind of the tool

Reinforcement of the part surfaces under the optimal pressure that is of

the same value at every CC-point K enables an increase of the quality of the

surface finish

For reinforcement of form surfaces of revolution, not only a conical tool but

a cylindrical tool can be used as well In the method of reinforcement of a

 SU Pat No 1463454, A Method of Reinforcement of Surfaces./S.P Radzevich, Int Cl B24B

39/00, 39/04, Filed May 5, 1987.

O P

2

1

P

4

3

K

O T

ω P

Figure 11.13

A method of finishing of a form surface of revolution with a conical indenter.

478 Kinematic Geometry of Surface Machining

form surface of revolution [13], finishing of the part surface is performed with

the cylindrical indenter When machining the surface P, the work is rotating

about its axis O P with a certain rotation ωP (Figure 11.14) The cylindrical

indenter 1 is pressed into the part surface P by normal force P rnf The tool 1 is

traveling along the axial profile of the part surface P with a certain peripheral

feed rate Simultaneously, the tool 1 is performing the orientational motion

of the first kind w n about unit normal vector nP to the part surface P The

ori-entational motion of the tool is timed with part diameter dP( )i at the current

CC-point K Due to the orientational motion of the tool, the angle that the

axis O P of the part makes with the axis O i at the current ith point is under

the control of the user At every CC-point K, the angle of crossing αi is of its

optimal value When the diameter d P( )i is bigger, then the cross-axis angle αi

is also bigger, and vice versa In this way, the optimal pressure that is of the

same value at every CC-point K is maintained.

In particular cases, two paths of the indenter 1 are required to be

per-formed On the second tool-path, the angle that the axis O P of the part

makes with the axis O i at the current ith point is reduced to a value αi On

the second tool-path, angle αi at the current CC-point K is always smaller

than that angle αi on the first tool-path (αi <αi)

Reinforcement of the part surfaces under the optimal pressure that is of

the same value at every CC-point K enables for an increase in the quality of

the surface finish

Similarly, reinforcement of part surfaces of revolution can be performed

with a form roller For example, a method of reinforcement of a surface of

revolution is featuring the implementation of a form tool [15]

The method of reinforcement of form surfaces of revolution is illustrated

with an example of finishing of a cylindrical part surface P (Figure 11.15)

However, the method of surface finishing can be implemented for the

rein-forcement form surfaces of revolution as well

P

2

α i**

α*i

O i*

O i

O P

**

d P (i)

1

ω P

ω n K

Figure 11.14

A method of reinforcement of a form surface of revolution with a cylindrical tool (SU Pat No

1463454).

11.3 Finishing of Involute Gears

Various methods of shaving are widely used for finishing spur and helical

involute gears [28] Most gear shaving operations are not optimized

Com-putation of the optimal parameters of a diagonal shaving operation

pro-vides a perfect example of implementation of the DG/K-based method of

surface generation In compliance with the method, it is possible to compute

the desired design parameters of the shaving cutter best suited for finishing

the given involute gear It is also possible to compute the optimal parameters

of the relative motions of the shaving cutter with respect to the gear to be

finished For this purpose, the indicatrix of conformity Cnf P T R( /g sh) of the

generating surface T sh of the shaving cutter to the screw involute tooth

sur-face Pg of the gear is commonly employed

In diagonal shaving (Figure 11.16), the work-gear rotates about its axis

O g with a certain angular velocity ωg The shaving cutter rotates about its

axis O sh with an angular velocity ωsh that is timed with the ωg— that is,

ωsh = ⋅u ωg , where u is the tooth ratio ( u N N= g sh ; here N g is the number

of the gear teeth, and N sh is the number of the shaving cutter teeth) Axes

of rotation O g of the gear and O sh of the shaving cutter are at a

center-distance C, and cross each other at an angle Σ The angle Σ is as follows:

Σ =ψg+ψsh Here ψp is the gear helix angle It is positive (+) to the

right-hand gear and negative (−) to the left-right-hand gear to be machined The same is

observed with respect to the shaving cutter helix angle ψsh In addition, the

Work Gear

Work Gear

Fdiag

Fdiag

C

L

Σ

Shaving Cutter

K2

K1 C

Shaving Cutter

ωsh

ωsh

O sh

O sh

O g

O g

B g

ωg

ωg

θ θ

Figure 11.16

Schematic of a diagonal shaving method (From Radzevich, S.P., International Journal of Advanced

Manufacturing Technology, 32 (11–12), 1170–1187, 2007 With permission.)

Examples of Implementation of the DG/K-Based Method 481

shaving machine table reciprocates relative to the shaving cutter with feed

Fdiag The axis of rotation O g of the gear and direction of the feed Fdiag make

a certain angle q.

The traverse path of the feed Fdiag is at a certain angle q to the gear axis

of rotation O g (Figure 11.16) The relationship between the face width of the

gear B g and the shaving cutter B sh is an important consideration It defines

the value of the diagonal traverse angle

The surface of tolerance P[ ]h is at a distance of the tolerance [ ]h to the

gear-tooth surface P g After tooth surface P g of a gear and tooth surface T sh of a

shaving cutter are put into contact at point K, then the surface T sh intersects

the surface P[ ] The line of intersection is a certain closed three-dimensional h

curve Cpt shown in Figure 11.17 It bounds the spot of contact of the gear and

the shaving cutter tooth It is recommended that the area of the spot of

con-tact Cpt be kept as small as possible (Figure 11.17)

Due to the tooth surfaces P g and T sh making contact at a distinct point

K, only discrete generation of the gear flank is feasible In order to increase

productivity of the gear finishing operation, it is required to maintain the

tool-paths on the gear-tooth flank P g as wide as possible For this purpose,

the major axis of the spot of contact Cpt has to be as long as possible, and

relative motion VΣ of the surfaces P g and T sh has to be directed

orthogo-nally to the major axis of the spot of contact Cpt

Tooth of the Shaving Cutter

Tooth of the Work-Gear

K

K

sh

g

pt

χ ≠ 90

Figure 11.17

The problem at hand (From Radzevich, S.P., International Journal of Advanced Manufacturing

Technology, 32 (11–12), 1170–1187, 2007 With permission.)

Fortunately, it is possible to control the shape, size, and orientation of the

spot of contact Cpt For this purpose, an optimal combination of the design

parameters of the shaving cutter, of direction and speed of the feed Fdiag,

of rotation of the gear ωg, and of rotation of the shaving cutter ωsh must be

computed This also makes possible the control of the direction of relative

motion of the surfaces Pg and T sh, and in such a way as to increase the gear

accuracy and to cut the shaving time

For the analysis below, equations of the tooth flank surfaces P g and T sh are

necessary The equation of the gear-tooth surface P g can be represented in

the form of the column matrix (see Equation 1.20):

rg

=

+

−

.

ψ

b g b g p b g

V

.

sin tanψ − sinψ

























1

(11.12)

where the gear base cylinder diameter d b g. =2r b g. can be computed from

g

.

.

cos cos sin

φ

1

25 4

φφ

n

P ⋅ 1−cos2 sin2 . (11.13)

where m is the gear modulus, N g is the number of gear teeth, φn is the

nor-mal pressure angle, λb g. is the gear base lead angle (λb g. =90o−ψb g. ), ψb g. is

the gear base lead angle, and P g is the diametral pitch

The Ug parameter in Equation (11.12) can be expressed in terms of

param-eters of the gear design [22,27]:

b g

y g b g g

.

(11.14)

where the diameter of a cylinder that is coaxial to the gear is designated as

d y g. , and ψg is the gear pitch helix angle

Equation (1.7) yields computation of the fundamental magnitudes of the

first order

b g

b g g

g b g b g

cos co

.

ψ

ψ

ss2ψb g. (11.15)

for the screw involute surface P g

For the fundamental magnitudes of the second order, use of Equation (1.11)

returns expressions

L g =0, M g =0, N g= −U g⋅sinτb g. ⋅cosτb g. (11.16)