Kinematic Geometry of Surface Machinin Episode 1 doc

The shortest shaving time, and the highest accuracy of the involute gear-tooth surface could be obtained if and only if the feed rate per gear-tooth Fi of the shaving cutter remains equa

Examples of Implementation of the DG/K-Based Method 483

for the coefficients L g , M g , and N g for the screw involute surface P g

ψψψ

for the unit normal vector ng to the gear-tooth surface P g

Computations similar to those above must be performed for the generating

surface T sh of the shaving cutter

Vector VΣ of the resultant relative motion of the surfaces P g and T sh passes

through the point K, and it is located in a common tangent plane to the

sur-faces P g and T sh Consider a plane through the unit normal vector ng that is

orthogonal to the direction of VΣ Radii of curvature of the surfaces P g and

T sh in this cross-section differ The width of the tool-path over the lateral

tooth surface P g depends upon the direction of the vector VΣ By varying

the direction of feed Fdiag, say timing in various manner angular velocities

w g and w sh with feed Fdiag, tool-paths of various width F(i could be obtained

The shortest shaving time, and the highest accuracy of the involute

gear-tooth surface could be obtained if and only if the feed rate per gear-tooth F(i of the

shaving cutter remains equal to its maximal value — that is, if F(i =F(cnf(max) In

order to make the equality F(i=F(cnf(max) valid, it is necessary to remain at the

highest possible rate of conformity of the surface T sh to the surface P g

In general, the rate of conformity of an involute gear-tooth surface T sh to

the involute tooth surface P g at the point K varies, as the normal plane

sec-tion rotates around the common unit vector ng The direction of the major

axis of the spot of contact aligns with the direction at which the highest rate

of conformity of the involute tooth surfaces P g and T sh is observed

The tangent plane to the gear-tooth surface P g at the point K is the plane

through two unit tangent vectors ug and vg These yield the equation for the

tangent plane through point K (i.e., through the point rK) on the gear-tooth

surface P g:

(rg tang−rK)×u vg⋅ g=0,

where rg.tang is the position vector of a point of the tangent plane

The angle of the gear and of the local relative orientation of the shaving

cutter tooth surfaces (see Equation 4.1 through Equation 4.3) is equal:

where φn is the normal pressure angle, ψg is the gear helix angle, ψsh is

the shaving cutter helix angle, and Σ is the gear and shaving cutter crossed

axes angle

484 Kinematic Geometry of Surface Machining

For the case under consideration, the equation of the indicatrix of

confor-mity Cnf P T R( /g sh) can be derived from the general form of equation of this

characteristic curve (see Equation 4.59)

The first f1.g, and the second f2.g, f2.sh fundamental forms are initially

com-puted in the coordinate systems X Y Z g g g and X Y Z sh sh sh, correspondingly

(see Figure 11.18) It is necessary to convert these expressions to the common

local coordinate system x y z g g g Such a transformation can be performed by

means of the formula of quadratic form transformation (see Equation 3.37

K

n g µ

µ

x g

Figure 11.18

The major coordinate systems (From Radzevich, S.P., International Journal of Advanced

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

Examples of Implementation of the DG/K-Based Method 485

where [φ1 2, g sh g sh( )]( ) and [φ1 2, g sh k( )] are the fundamental forms of the

sur-faces P T g( sh ), initially represented in the coordinate systems X Y Z g g g and

X Y Z sh sh sh , and finally in the common coordinate system x y z g g g

In the local coordinate system x y z g g g , the equation for the Cnf P T R( /g sh)

where r cnf(R1.g,R1.sh, , )µ ϕ is the position vector of a point of the characteristic curve

Cnf P T R( /g sh ) — for finishing of a given gear, the function r R R cnf( 1.g, 1.sh, , )µ ϕ

reduces to r cnf(R1.sh, , )µ ϕ ; and j is the polar angle (further the argument ϕ is

employed for determining the optimal direction of resultant relative motion VΣ

of the surfaces P g and T sh)

The characteristic curve Cnf P T R( /g sh) is depicted in Figure 11.19 The rate

of conformity of the surfaces P g and T sh in the normal cross-section through

the minimal diameter d cnf(min) (or, the same, through the direction t(max)cnf of the

maximal rate of conformity of the surfaces P g and T sh) is the highest

pos-sible (Figure 11.20) This plane section of the surfaces P g and T sh is referred

to as the optimal normal cross-section.

The indicatrix of conformity Cnf P T R( /g sh ) of the tooth flanks P g and T sh (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 487

The same angle i opt makes vector VΣ with the perpendicular to the cutting

edge (Figure 11.21)

At every point of the tooth flank P g , the first principal curvature k1.g is

uniquely determined by the topology of the surface P g The second

princi-pal curvature k2.g of the screw involute surface P g is always equal to zero

(k .g≡0) Similarly, the second principal curvature k2.sh of the screw

invo-lute surface T sh is also always equal to zero ( k2 sh ≡ 0) At this point, the rest

of the parameters of the indicatrix of conformity Cnf P T R( /g sh) of the

sur-faces P g and T sh (that is, the parameters R1.sh, j, and m) can be considered

as the variable parameters It is necessary to determine the optimal

combina-tion of values of the parameters R1 opt sh =R1 opt sh(U V g, g), ϕopt =ϕopt(U V g, g), and

µopt =µopt(U V g, g) If the proper combination of the parameters Ropt1.sh

, ϕopt, and µopt is determined, then computation of the optimal design parameters of

the shaving cutter and of the optimal parameters of kinematics of the diagonal

shaving operation turns to the routing engineering calculations

The indicatrix of conformity Cnf P T R( /g sh) reveals how close the tooth

sur-face T sh of the shaving cutter is to the gear-tooth surface P g in every

cross-section of the surfaces P g and T sh by normal plane through K It enables

specification of an orientation of the normal plane section, at which the

sur-faces P g and T sh are extremely close to each other — that is, the normal plane

section through the unit tangent vector tcnf(max) in the direction of the maximal

rate of conformity of the surfaces P g and T sh This normal plane section

sat-isfies the following conditions:

cnf sh

90°

pt

Figure 11.21

Elements of local topology of the tooth surfaces P g and T sh referred to the lateral plane of the

auxiliary phantom rack R (From Radzevich, S.P., International Journal of Advanced

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

488 Kinematic Geometry of Surface Machining

Equation (11.20) of the indicatrix of conformity Cnf P T R( /g sh) yields the

fol-lowing necessary conditions of the maximal rate of conformity of the

shav-ing cutter tooth surface T sh to the involute gear-tooth surface P g:

The Necessary Conditions

for the Minimal Shavving

Time and the Maximal

Accuracy of the Shaaved

The sufficient conditions for the maximum of the function r cnf(R1.sh, , )µ ϕ of

three variables are also satisfied

The first equality in Equation (11.22) consists in condensed form all the

necessary information on the optimal design parameters of the shaving

cut-ter Analysis of this equality reveals that it could be satisfied when R 1.sh→∞

Thus, for a conventional diagonal shaving operation when the gear and the

shaving cutter are in external mesh, it is recommended to finish the gear

with the shaving cutter of the maximal possible pitch diameter In the ideal

case, the gear can be shaved with a rack-type shaving cutter Application of

the shaving cutter of larger pitch diameter increases the difference between

pitch diameters of the gear and of the shaving cutter This yields a larger rate

of conformity of the surfaces P g and T sh Actually, the pitch diameter of the

shaving cutter to be applied for a rotary shaving operation is restricted by

the design of a shaving machine

Analysis of the function R1.sh=R1.sh( ,φ ψn sh)reveals that the rate of

confor-mity of the surfaces P g and T sh increases when both normal pressure angle

φn and helix angle ψshare smaller — that is, φn →0o and ψsh→0o The

interested reader may wish to refer to [20] for details of the analysis

The second and the third equalities in Equation (11.22) together enable one to

give an answer to the question on the optimal relative orientation of the surfaces

P g and T sh (µ →0o , however, the inequality Σ ≠0o is required) and on the

optimal parameters of instant kinematics of diagonal shaving (ϕ ϕ= opt)

The resultant relative motion VΣ of the surfaces P g and T sh is

decom-posed on its projections onto directions of the motions to be performed on

the gear-shaving machine

Vector Vsl of the velocity of relative sliding of the surfaces P g and T sh is

located in the common tangent plane It is convenient to decompose the vector

Vsl at the point K onto two components V sl =Vφ +Vψ The first component

Examples of Implementation of the DG/K-Based Method 489

Vφ represents sliding along the tooth profile, and the second component Vψ

represents sliding in the longitudinal tooth direction

The feed Fdiag is directed parallel to the plane surface that is tangent to

pitch cylinders of the gear and of the shaving cutter It also affects the

resul-tant speed VΣ of cutting (VΣ =Vsl+Fdiag) Varying parameters of the

diago-nal shaving operation and of design parameters of the shaving cutter enable

one to control the resultant speed VΣ =Vsl+Fdiag of cutting For this purpose,

the speed and direction of the shaving machine reciprocation and shaving

cutter rotation have to be timed with each other

In the local coordinate system x y z g g g (Figure 11.22), the vector VΣ of the

resultant motion makes a certain angle ϕΣ with the y g axis Thus,

V

V V

01

Timing of the feed F diag with rotations of the involute gear and of the shaving cutter (From

Radzevich, S.P., International Journal of Advanced Manufacturing Technology, 32 (11–12), 1170–1187,

2007 With permission.)

490 Kinematic Geometry of Surface Machining

To represent the vector VΣ in the global coordinate system X Y Z k k k (Figure 11.22),

the operator Rs(g→k) of the resultant coordinate system transformation is

used:

V V V

Σ Σ Σ

where Vg, Vsh are the linear velocities of the rotations g and sh,

respec-tively; and R w g., R w sh. are radii of pitch cylinders of the gear and the shaving

cutter And,

| |Vsl= ⋅2| |ωg⋅R w g. ⋅cos( 0 5⋅ = ⋅Σ) 2| |ωωsh ⋅R w sh. ⋅ccos( 0 5⋅ Σ) (11.27)

In the coordinate plane Y Z k k, the resultant motion Vyz of the gear and the

shaving cutter can be represented as follows:

Thus, reciprocation is equal to

This is the way the values of the shaving cutter rotation and its reciprocation

are timed with each other

The synthesized method of diagonal shaving of involute gears is disclosed

in detail in [4,20,29,30]

492 Kinematic Geometry of Surface Machining

[21] Radzevich, S.P., Differential-Geometric Method of Surface Generation, DrSci thesis,

Tula, Polytechnic Institute, 1991.

[22] Radzevich, S.P., Fundamentals of Surface Generation, Rastan, Kiev, 2001.

[23] Radzevich, S.P., Methods of Milling of Sculptured Surfaces, VNIITEMR, Moscow,

1989.

[24] Radzevich, S.P., R-Mapping Based Method for Designing of Form Cutting Tool

for Sculptured Surface Machining, Mathematical and Computer Modeling, 36 (7–8),

921–938, 2002.

[25] Radzevich, S.P., Sculptured Surface Machining on Multi-Axis NC Machine, Vishcha

Shkola, Kiev, 1991.

[26] Radzevich, S.P., and Dmitrenko, G.V., Machining of Form Surfaces of

Revolu-tion on NC Machine Tool, Mashinostroitel’, No 5, 17–19, 1987.

[27] Radzevich, S.P., Goodman, E.D., and Palaguta, V.A., Tooth Surface

Fundamen-tal Forms in Gear Technology, University of Niš, the Scientific Journal Facta

Universitatis , Series: Mechanical Engineering, 1 (5), 515–525, 1998.

[28] Radzevich, S.P., and Palaguta, V.A., Advanced Methods in Gear Finishing,

VNI-ITEMR, Moscow, 1988.

[29] Radzevich, S.P., and Palaguta, V.A., CAD/CAM System for Finishing of

Cylin-drical Gears, Mekhanizaciya i Avtomatizaciya Proizvodstva, No 10, 13–15, 1988.

[30] Radzevich, S.P., and Palaguta, V.A., Synthesis of Optimal Gear Shaving

Opera-tions, Vestink Mashinostroyeniya, No 8, 36–41, 1997.

[31] Radzevich, S.P et al., On the Optimization of Parameters of Sculptured Surface

Machining on Multi-Axis NC Machine, In Investigation into the Surface

Genera-tion, UkrNIINTI, Kiev, No 65-Uk89, pp 57–72, 1988.

A novel method of surface generation for the purposes of surface machining

on a multi-axis numerical control machine, as well as on a machine tool of

conventional design is disclosed in this monograph The method is

devel-oped on the premises of wide use of Differential Geometry of surfaces, and

of elements of Kinematics of multiparametric motion of rigid body in

Euclid-ian space Due to this, the proposed method is referred to as the DG/K-based

method of surface generation

The DG/K-based method is targeting synthesizing of optimal methods of

part surface machining, and of optimal form-cutting tools for machining

of surfaces

A minimal amount of input information is required for the

implementa-tion of the method Potentially, the method is capable of synthesizing optimal

surface machining processes on the premises of just the geometry of the part

surface to be machined However, any additional information on the surface

machining process, if any, can be incorporated as well Ultimately, the use of

the DG/K-based method of surface generation enables one to get a maximal

amount of output information on the surface machining process while using

for this purpose a minimal amount of input information The last illustrates

the significant capacity of the disclosed method of surface generation

The developed DG/K-based method of surface generation is a cornerstone

of the subject theoretical machining/production technology to study by

uni-versity students

496 Kinematic Geometry of Surface Machining

Rt(j x , X) Operator of rotation through an angle j x about the X axis

Rt(j y , Y) Operator of rotation through an angle axis j y about the Y

Rt(j z , Z) Operator of rotation through an angle j z about the Z axis

Rt(j A , A) Operator of rotation through an angle j A about the A 

Tr(a x , X) Operator of translation at a distance a x along the X axis

Tr(a y , Y) Operator of translation at a distance a y along the Y axis

498 Kinematic Geometry of Surface Machining

KINEMATIC GEOMETRY

OF SURFACE MACHINING

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