Kinematic Geometry of Surface Machinin Episode 2 pdf

The major problem in the theory of surface generation is the absence of methods by use of which the challenging problem of optimal surface generation can be successfully solved.. Chapter

Preface xix

number of research and application papers and articles Commonly, isolated

theoretical and practical findings for a particular surface-generation process

are reported instead of methodology, so the question “What would happen if

the input parameters are altered?” remains unanswered Therefore, a

broad-based book on the theory of surface generation is needed

The purpose of this book is twofold:

To summarize the available information on surface generation with a

critical review of previous work, thus helping specialists and

prac-titioners to separate facts from myths The major problem in the

theory of surface generation is the absence of methods by use of

which the challenging problem of optimal surface generation can be

successfully solved Other known problems are just consequences of

the absence of the said methods of surface generation

To present, explain, and exemplify a novel principal concept in the

the-ory of surface generation, namely that the part surface is the primary

element of the part surface-machining operation The rest of the

elements are the secondary elements of the part surface-machining

operation; thus, all of them can be expressed in terms of the desired

design parameters of the part surface to be machined

The distinguishing feature of this book is that the practical ways of

synthe-sizing and optimizing the surface-generation process are considered using

just one set of parameters — the design parameters of the part surface to be

machined The desired design parameters of the part surface to be machined

are known in a research laboratory as well as in a shop floor environment

This makes this book not just another book on the subject For the first time,

the theory of surface generation is presented as a science that really works

This book is based on the my varied 30 years of experience in research,

practical application, and teaching in the theory of surface generation, applied

mathematics and mechanics, fundamentals of CAD/CAM, and engineering

systems theory Emphasis is placed on the practical application of the results

in everyday practice of part surface machining and cutting-tool design The

application of these recommendations will increase the competitive

posi-tion of the users through machining economy and productivity This helps

in designing better cutting tools and processes and in enhancing technical

expertise and levels of technical services

Intended Audience

Many readers will benefit from this book: mechanical and manufacturing

engineers involved in continuous process improvement, research workers

whoareactiveorintendtobecomeactiveinthefield,andseniorundergraduate

and graduate students of mechanical engineering and manufacturing

xx Preface

This book is intended to be used as a reference book as well as a textbook

Chapters that cover geometry of sculptured part surfaces and elementary

kinematics of surface generation, and some sections that pertain to design

of the form-cutting tools can be used for graduate study; I have used this

book for graduate study in my lectures at the National Technical University

of Ukraine “Kiev Polytechnic Institute” (Kiev, Ukraine) The design chapters

interest for mechanical and manufacturing engineers and for researchers

The Organization of This Book

The book is comprised of three parts entitled “Basics,” “Fundamentals,” and

“Application”:

Part I : Basics — This section of the book includes analytical description

of part surfaces, basics on differential geometry of sculptured

sur-faces, as well as principal elements of the theory of multiparametric

motion of a rigid body in E3space The applied coordinate systems

and linear transformations are briefly considered The selected

mate-rial focuses on the solution to the problem of synthesizing optimal

machining of sculptured part surfaces on a multi-axis NC machine

The chapters and their contents are as follows:

Chapter 1 Part Surfaces: Geometry — The basics of differential

geometry of sculptured part surfaces are explained The focus

here is on the difference between classical differential geometry

and engineering geometry of surfaces Numerous examples of the

computation of major surface elements are provided A feasibility

of classification of surfaces is discussed, and a scientific

classifica-tion of local patches of sculptured surfaces is proposed

Chapter 2 Kinematics of Surface Generation — The

general-ized analysis of kinematics of sculptured surface generation

is presented Here, a generalized kinematics of instant relative

motion of the cutting tool relative to the work is proposed For

the purposes of the profound investigation, novel kinds of

rela-tive motions of the cutting tool are discovered, including

gen-erating motion of the cutting tool, motions of orientation, and

relative motions that cause sliding of a surface over itself The

chapter concludes with a discussion on all feasible kinematic

schemes of surface generation Several particular issues of

kine-matics of surface generation are discussed as well

Chapter 3 Applied Coordinate Systems and Linear

Transforma-tions — The definitions and determinations of major applied

coordinate systems are introduced in this chapter The matrix

and practical implementation of the proposed theory (Part III) will be of

Preface xxi

approach for the coordinate system transformations is briefly

discussed Here, useful notations and practical equations are

provided Two issues of critical importance are introduced here

The first is chains of consequent linear transformations and a

closed loop of consequent coordinate systems transformations

The impact of the coordinate systems transformations on

funda-mental forms of the surfaces is the second

These tools, rust covered for many readers (the voice of experience), are

resharpened in an effort to make the book a self-sufficient unit suited for

self-study

Part II : Fundamentals — Fundamentals of the theory of surface

genera-tion are the core of the book This part of the book includes a novel

powerful method of analytical description of the geometry of contact

of two smooth, regular surfaces in the first order of tangency; a novel

kind of mapping of one surface onto another surface; a novel

analyti-cal method of investigation of the cutting-tool geometry; and a set of

analytically described conditions of proper part surface generation A

solution to the challenging problem of synthesizing optimal surface

machining begins here The consideration is based on the analytical

results presented in the first part of the book The following chapters

are included in this section

Chapter 4 The Geometry of Contact of Two Smooth Regular

Sur-faces — Local characteristics of contact of two smooth, regular

surfaces that make tangency of the first order are considered The

sculptured part surface is one of the contacting surfaces, and the

generating surface of the cutting tool is the second The performed

analysis includes local relative orientation of the contacting

sur-faces and the first- and second-order analyses The concept of

conformity of two smooth, regular surfaces in the first order of

tangency is introduced and explained in this chapter For the

pur-poses of analyses, properties of Plücker’s conoid are implemented

Ultimately, all feasible kinds of contact of the part and of the tool

surfaces are classified

Chapter 5 Profiling of the Form-Cutting Tools of Optimal Design

— A novel method of profiling the form-cutting tools for

sculp-tured surface machining is disclosed in this chapter The method

is based on the analytical description of the geometry of contact

of surfaces that is developed in the previous chapter Methods of

profiling form-cutting tools for machining part surfaces on

con-ventional machine tools are also considered These methods are

based on elements of the theory of enveloping surfaces

Numer-ous particular issues of profiling form-cutting tools are discussed

at the end of the chapter

xxii Preface

Chapter 6 Geometry of Active Part of a Cutting Tool — The

gen-erating body of the form-cutting tool is bounded by the

generat-ing surface of the cuttgenerat-ing tool Methods of transformation of the

generating body of the form-cutting tool into a workable cutting

tool are discussed In addition to two known methods, one novel

method for this purpose is proposed Results of the analytical

investigation of the geometry of the active part of cutting tools in

both the Tool-in-Hand system as well as the Tool-in-Use system

are represented Numerous practical examples of the

computa-tions are also presented

Chapter 7 Conditions of Proper Part Surface Generation — The

satisfactory conditions necessary and sufficient for proper part

surface machining are proposed and examined The conditions

include the optimal workpiece orientation on the worktable of a

multi-axis NC machine and the set of six analytically described

conditions of proper part surface generation The chapter

con-cludes with the global verification of satisfaction of the

condi-tions of proper part surface generation

Chapter 8 Accuracy of Surface Generation — Accuracy is an

impor-tant issue for the manufacturer of the machined part surfaces

Analytical methods for the analysis and computation of the

devia-tions of the machined part surface from the desired part surface are

discussed here Two principal kinds of deviations of the machined

surface from the nominal part surface are distinguished Methods

for the computation of the elementary surface deviations are

pro-posed The total displacements of the cutting tool with respect to

the part surface are analyzed Effective methods for the reduction

of the elementary surface deviations are proposed Conditions

under which the principle of superposition of elementary surface

deviations is applicable are established

Part III : Application — This section illustrates the capabilities of the

novel and powerful tool for the development of highly efficient

methods of part surface generation Numerous practical examples of

implementation of the theory are disclosed in this part of the

mono-graph This section of the book is organized as follows:

Chapter 9 Selection of the Criterion of Optimization — In order to

implement in practice the disclosed Differential

Geometry/Kine-matics (DG/K)-based method of surface generation, an

appropri-ate criterion of efficiency of part surface machining is necessary

This helps answer the question of what we want to obtain when

performing a certain machining operation Various criteria of

effi-ciency of machining operation are considered Tight connection

of the economical criteria of optimization with geometrical

ana-logues (as established inChapter 4) is illustrated The part surface

Preface xxiii

generation output is expressed in terms of functions of

confor-mity The last significantly simplifies the synthesizing of optimal

operations of part surface machining

Chapter 10 Synthesis of Optimal Surface Machining Operations

— The synthesizing of optimal operations of actual part

sur-face machining on both the multi-axis NC machine as well as

on a conventional machine tool are explained For this purpose,

three steps of analysis are distinguished: local analysis, regional

analysis, and global analysis A possibility of the development of

the DG/K-based CAD/CAM system for the optimal sculptured

surface machining is shown

Chapter 11 Examples of Implementation of the DG/K-Based

Method of Surface Generation — This chapter demonstrates

numerous novel methods of surface machining — those

devel-oped on the premises of implementation of the proposed

DG/K-based method surface generation Addressed are novel methods of

machining sculptured surfaces on a multi-axis NC machine, novel

methods of machining surfaces of revolution, and a novel method of

finishing involute gears

The proposed theory of surface generation is oriented on extensive

appli-cation of a multi-axis NC machine of modern design In particular cases,

implementation of the theory can be useful for machining parts on

conven-tional machine tools

Stephen P Radzevich

Sterling Heights, Michigan

Stephen P Radzevich, Ph.D.,is a professor of mechanical engineering and

manufacturing engineering He has received an M.Sc (1976), a Ph.D (1982),

and a Dr.(Eng)Sc (1991) in mechanical engineering Radzevich has

exten-sive industrial experience in gear design and manufacture He has

devel-oped numerous software packages dealing with computer-aided design

(CAD) and computer-aided manufacturing (CAM) of precise gear finishing

for a variety of industrial sponsors Dr Radzevich’s main research

inter-est is kinematic geometry of surface generation with a particular focus on

(a) precision gear design, (b) high torque density gear trains, (c) torque share

in multiflow gear trains, (d) design of special-purpose gear cutting and

fin-ishing tools, (e) design and machining (finfin-ishing) of precision gears for

low-noise/noiseless transmissions of cars, light trucks, and so forth He has spent

more than 30 years developing software, hardware, and other processes for

gear design and optimization In addition to his work for industry, he trains

engineering students at universities and gear engineers in companies He

has authored and coauthored 28 monographs, handbooks, and textbooks; he

authored and coauthored more than 250 scientific papers; and he holds more

than 150 patents in the field At the beginning of 2004, he joined EATON

Corp He is a member of several Academies of Sciences around the world

I would like to share the credit for any research success with my numerous

doctoral students with whom I have tested the proposed ideas and applied

them in the industry The contributions of many friends, colleagues, and

students in overwhelming numbers cannot be acknowledged individually,

and as much as our benefactors have contributed, even though their

kind-ness and help must go unrecorded

Basics

1

Part Surfaces: Geometry

The generation of part surfaces is one of the major purposes of

machin-ing operations An enormous variety of parts are manufactured in various

industries Every part to be machined is bounded with two or more

sur-faces. Each of the part surfaces is a smooth, regular surface, or it can be

composed with a certain number of patches of smooth, regular surfaces that

are properly linked to each other

In order to be machined on a numerical control (NC) machine, and for

com-puter-aided design (CAD) and comcom-puter-aided manufacturing (CAM)

appli-cations, a formal (analytical) representation of a part surface is the required

prerequisite Analytical representation of a part surface (the surface P) is

based on analytical representation of surfaces in geometry, specifically, (a) in

the differential geometry of surfaces and (b) in the engineering geometry of

surfaces The second is based on the first

For further consideration, it is convenient to briefly consider the principal

elements of differential geometry of surfaces that are widely used in this

text If experienced in differential geometry of surfaces, the following

sec-tion may be skipped Then, proceed directly to Secsec-tion 1.2

1.1  Elements of Differential Geometry of Surfaces

A surface could be uniquely determined by two independent variables

Therefore, we give a part surface P (Figure 1.1), in most cases, by expressing

its rectangular coordinates X P , Y P , and Z P, as functions of two Gaussian

coor-dinates U P and V P in a certain closed interval:

 The ball of a ball bearing is one of just a few examples of a part surface, which is bounded

with the only surface that is the sphere.

Part Surfaces: Geometry 

Significance of the vectors uP and vP becomes evident from the following

considerations First, tangent vectors uP and vP yield an equation of the

tan-gent plane to the surface P at M:

Tangent plane

t p P M P P

where rt .P is the position vector of a point of the tangent plane to the surface P

at M, and r P( )M is the position vector of the point M on the surface P.

Second, tangent vectors yield an equation of the perpendicular NP, and of

the unit normal vector nP to the surface P at M:

When the order of multipliers in Equation (1.) is chosen properly, then the

unit normal vector nP is pointed outward of the bodily side of the surface P.

Unit tangent vectors uP and vP to a surface at a point are of critical

impor-tance when solving practical problems in the field of surface generation

Numerous examples, as shown below, prove this statement

Consider two other important issues concerning part surface geometry —

both relate to intrinsic geometry in differential vicinity of a surface point

The first issue is the first fundamental form of a surface P The first

funda-mental form f 1.P of a smooth, regular surface describes the metric properties

of the surface P Usually, it is represented as the quadratic form:

φ1.P⇒ds P2 =E dU P 2P+2F dU dV P P P+G dV P P2 (1.)

where s P is the linear element of the surface P (s P is equal to the length of a

segment of a certain curve line on the surface P), and E P , F P , G Pare

funda-mental magnitudes of the first order

Equation (1.) is known from many advanced sources In the theory of

sur-face generation, another form of analytical representation of the first

funda-mental form f 1.P is proven to be useful:

P P

00

(1.6)

6 Kinematic Geometry of Surface Machining

This kind of analytical representation of the first fundamental form f 1.P

is proposed by Radzevich [10] The practical advantage of Equation (1.6)

is that it can easily be incorporated into computer programs using

mul-tiple coordinate system transformations, which is vital for CAD/CAM

applications

For computation of the fundamental magnitudes of the first order, the

fol-lowing equations can be used:

E P=U UP⋅ P, F P=U VP⋅ P, G P=V VP⋅ P (1.7)

Fundamental magnitudes E P , F P , and G P of the first order are functions of

U P and V P parameters of the surface P In general form, these relationships

can be represented as E P = E P (U P , V P ), F P = F P (U P , V P ), and G P = G P (U P , V P)

Fundamental magnitudes E P and G P are always positive (E P > 0, G P > 0),

and the fundamental magnitude F P can equal zero (F P ≥ 0) This results in the

first fundamental form always being nonnegative (f 1.P ≥ 0).

The first fundamental form f 1.P yields computation of the following major

parameters of geometry of the surface P: (a) length of a curve-line segment

on the surface P, (b) square of the surface P portion that is bounded by a

closed curve on the surface, and (c) angle between any two directions on the

surface P.

The first fundamental form represents the length of a curve-line

seg-ment, and thus it is always nonnegative — that is, the inequality f 1.P ≥ 0 is

always observed

The discriminant H P of the first fundamental form f 1.P can be computed

from the following equation:

It is assumed that the discriminant H P is always nonnegative — that is, H P=+

E G P P−F P2

The fundamental form f 1.P remains the same while the surface is

band-ing This is another important feature of the first fundamental form f 1.P The

feature can be employed for designing three-dimensional cam for finishing

a turbine blade with an abrasive strip as a cutting tool

The second fundamental form of the surface P is another of the two

above-mentioned important issues The second fundamental form f 2.P describes

the curvature of a smooth, regular surface P Usually, it is represented as the

quadratic form

φ2.P⇒ −drP⋅dnP=L dU P 2P+2M dU dV P P P+N dV P P2 (1.9)

Equation (1.9) is known from many advanced sources

Part Surfaces: Geometry 7

In the theory of surface generation, another analytical representation of

the second fundamental form f 2.P is proven useful:

P P

00

(1.10)

This analytical representation of the second fundamental form f 2.P is

pro-posed by Radzevich [10] Similar to Equation (1.6), the practical advantage of

Equation (1.10) is that it can be easily incorporated into computer programs

using multiple coordinate system transformations, which is vital for CAD/

CAM applications

In Equation (1.10), the parameters L P , M P , N P designate fundamental

mag-nitudes of the second order Fundamental magmag-nitudes of the second order

can be computed from the following equations:

Fundamental magnitudes L P , M P , N P of the second order are also functions

of U P and V P parameters of the surface P These relationships in general form

can be represented as L P = L P (U P , V P ), M P = M P (U P , V P ), and N P = N P (U P , V P)

Discriminant T P of the second fundamental form f 2.P can be computed

from the following equation:

For computation of the principal directions T1.P and T2.P through a given point

on the surface P, the fundamental magnitudes of the second order L P , M P , N P,

together with the fundamental magnitudes of the first order E P , F P , G P, are used

Principal directions T1.P and T2.P can be computed as roots of the equation

The first principal plane section C 1.P is orthogonal to P at M and passes

through the first principal direction T1.P The second principal plane section

8 Kinematic Geometry of Surface Machining

C 2.P is orthogonal to P at M and passes through the second principal

direc-tion T2.P

In the theory of surface generation, it is often preferred to use not the

vec-tors T1.P and T2.P of the principal directions, but instead to use the unit vectors

t1.P and t2.P of the principal directions The unit vectors t1.P and t2.P are

com-puted from equations t1.P = T1.P/|T1.P| and t2.P = T2.P/|T2.P|, respectively

The first R 1.P and the second R 2.P principal radii of curvature of the surface

P are measured in the first and in the second principal plane sections C 1.P

and C 2.P, correspondingly For computation of values of the principal radii of

curvature, use the following equation:

H T

P P

Another two important parameters of local topology of a surface P are (a) mean

curvature MP, and intrinsic (Gaussian or full) curvature GP These

param-eters can be computed from the following equations:

The formulae for MP= k1P+k2P

2 and GP=k1 P⋅k2 P yield a quadratic equation:

with respect to principal curvatures k 1.P and k 2.P The expressions

k1 P=MP+ MP2−GP and k2 P=MP− MP2−GP (1.18)

are the solutions to Equation (1.17)

Here, k 1.P designates the first principal curvature of the surface P, and k 2.P

des-ignates the second principal curvature of the surface P at that same point The

principal curvatures k 1.P and k 2.P can be computed from k1.P=R1−.P and k 2.P= k2.P=

The first principal curvature k 1.P always exceeds the second principal curvature

k 2.P — that is, the inequality k 1.P> k 2.P is always observed

This brief consideration of elements of surface geometry allow for the

intro-duction of two definitions that are of critical importance for further discussion

Definition 1.1: Sculptured surface P is a smooth, regular surface with

major parameters of local topology that differ when in differential

vicin-ity of any two infinitely closed points.

 Remember that algebraic values of the radii of principal curvature R 1.P and R 2.P relate to each

other as R 2.P > R 1.P.

Part Surfaces: Geometry 9

It is instructive to point out here that sculptured surface P does not allow

slid-ing “over itself.”

While machining a sculptured surface, the cutting tool rotates about its axis

and moves relative to the sculptured surface P While rotating with a certain

angular velocity ωT or while performing relative motion of another kind, the

cutting edges of the cutting tool generate a certain surface We refer to that

face represented by consecutive positions of cutting edges as the generating

sur-face of the cutting tool [11, 1, 1]:

Definition 1.2: The generating surface of a cutting tool can be represented

as the set of consecutive positions of the cutting edges in their motion

rela-tive to the stationary coordinate system, embedded to the cutting tool itself.

In most practical cases, the generating surface T allows sliding over itself The

enveloping surface to consecutive positions of the surface T that performs

such a motion is congruent to the surface T When machining a part, the

surface T is conjugate to the sculptured surface P.

Bonnet [1] proved that the specification of the first and second

fundamen-tal forms determines a unique surface if the Gauss’ characteristic equation

and the Codazzi-Mainardi’s relationships of compatibility are satisfied, and

those two surfaces that have identical first and second fundamental forms

are congruent. Six fundamental magnitudes determine a surface uniquely,

except as to position and orientation in space

Specification of a surface in terms of the first and the second fundamental

forms is usually called the natural kind of surface parameterization In

gen-eral form, it can be represented by a set of two equations:

The natural form

Equation (1.19) specify that same surface P In further consideration, the

nat-ural parameterization of the surface P plays an important role.

Illustrative Example

Consider an example of how an analytical representation of a surface in a

Cartesian coordinate system can be converted into the natural

parameteriza-tion of that same surface [1]

A gear tooth surface G is analytically described in a Cartesian coordinate

system X g Y g Z g (Figure 1.2)

 Two surfaces with the identical first and second fundamental forms might also be

symmetri-cal Refer to the literature— Koenderink, J.J., Solid Shape, The MIT Press, Cambridge, MA, 1990,

p 699—on differential geometry of surfaces for details about this specific issue.

Part Surfaces: Geometry 11

Substituting the computed vectors Ug and Vg into Equation (1.7), one can

come up with formulae for computation of the fundamental magnitudes of

the first order:

cos

2

The computed values of the fundamental magnitudes E g , F g, and Gg can be

substituted to Equation (1.6) for f 1.g In this way, matrix representation of the

first fundamental form f 1.g can be computed The interested reader may wish

to complete this formulae transformation on his or her own

The discriminant H g of the first fundamental form of the surface G can be

computed from the formula H g = U g cosf b.g

In order to derive an equation for the second fundamental form f 2.g of the

gear-tooth surface G, the second derivatives of r g (U g , Vg ) with respect to U g

and V g parameters are necessary The above derived equations for the vectors

Ug and Vg yield the following computation:

0001

U

V V

.

ττ

01

Further, substitute these derivatives (see Equation 1.2 and Equation 1.8

into Equation 1.11) After the necessary formulae transformations are

com-plete, then Equation (1.11) casts into the set of formulae for computation of

the second fundamental magnitudes of the surface G is as follows:

L g=0 M g=0 and N g= −U gsinτb g. cosτb g. (1.2)

After substituting Equation (1.2) into Equation (1.9), an equation for the

computation of the second fundamental form of the surface G can be obtained:

φ2 g⇒ −drg⋅dNg = −U gsinτb g. cosτb g. dV g2 (1.26)