GEAR GEOMETRY AND APPLIED THEORY Second Edition docx

1.2 Coordinate Transformation in Matrix Representation 21.4 Rotational and Translational 4× 4 Matrices 14 3 Centrodes, Axodes, and Operating Pitch Surfaces 44 3.4 Axodes in Rotation Betw

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GEAR GEOMETRY AND APPLIED THEORY

Second Edition

Revised and expanded, Gear Geometry and Applied Theory, 2nd edition,

cov-ers the theory, design, geometry, and manufacture of all types of gears and gear

drives Gear Geometry and Applied Theory is an invaluable reference for

de-signers, theoreticians, students, and manufacturers This new edition includesadvances in gear theory, gear manufacturing, and computer simulation Amongthe new topics are (1) new geometry for modified spur and helical gears, face-geardrives, and cycloidal pumps; (2) new design approaches for one-stage planetarygear trains and spiral bevel gear drives; (3) an enhanced approach for stressanalysis of gear drives with FEM; (4) new methods of grinding face-gear drives,generating double-crowned pinions, and generating new types of helical gears;(5) broad application of simulation of meshing and TCA; and (6) new theories onthe simulation of meshing for multi-body systems, detection of cases wherein thecontact lines on generating surfaces may have their own envelope, and detectionand avoidance of singularities of generated surfaces

Faydor L Litvin is Director of the Gear Research Center and DistinguishedProfessor Emeritus in the Department of Mechanical and Industrial Engineering,University of Illinois at Chicago He holds patents for twenty-five inventions, and

he was recognized as Inventor of the Year by the University of Illinois at Chicago

in 2001

Alfonso Fuentes is Associate Professor of Mechanical Engineering at thePolytechnic University of Cartagena

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ii

Gear Geometry and Applied Theory

First published in print format

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© Faydor L Litvin and Alfonso Fuentes 2004

2004

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This publication is in copyright Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press

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First published in print format

- ----

- ----

© Faydor L Litvin and Alfonso Fuentes 2004

2004

Information on this title: www.cambridge.org/9780521815178

This publication is in copyright Subject to statutory exception and to the provision ofrelevant collective licensing agreements, no reproduction of any part may take placewithout the written permission of Cambridge University Press

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Cambridge University Press has no responsibility for the persistence or accuracy of sfor external or third-party internet websites referred to in this publication, and does notguarantee that any content on such websites is, or will remain, accurate or appropriate

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1.2 Coordinate Transformation in Matrix Representation 2

1.4 Rotational and Translational 4× 4 Matrices 14

3 Centrodes, Axodes, and Operating Pitch Surfaces 44

3.4 Axodes in Rotation Between Intersected Axes 51

3.6 Operating Pitch Surfaces for Gears with Crossed Axes 56

5.4 Representation of a Surface by Implicit Function 82

6.1 Envelope to a Family of Surfaces: Necessary Conditions

6.6 Envelope to Family of Contact Lines on Generating

6.7 Formation of Branches of Envelope to Parametric

6.8 Wildhaber’s Concept of Limit Contact Normal 118

7.7 Gaussian Curvature; Three Types of Surface

8 Mating Surfaces: Curvature Relations, Contact Ellipse 202

Contents vii

9.5 Application of Finite Element Analysis for Design

10.5 Meshing of Involute Gear with Rack-Cutter 28010.6 Relations Between Tooth Thicknesses Measured

12.6 Conjugation of an Eccentric Circular Gear with

13.7 Overcentrode Cycloidal Gearing 367

15.3 Profile-Crowned Pinion and Gear Tooth Surfaces 41115.4 Tooth Contact Analysis (TCA) of Profile-Crowned

15.5 Longitudinal Crowning of Pinion by a Plunging Disk 41915.6 Grinding of Double-Crowned Pinion by a Worm 42415.7 TCA of Gear Drive with Double-Crowned Pinion 430

Appendix 16.C: Relations Between Parametersα ptandα pn 473

Appendix 16.E: Derivation of Additional Relations Between

17 New Version of Novikov–Wildhaber Helical Gears 475

18.6 Conditions of Nonundercutting of Face-Gear Tooth

18.7 Pointing of Face-Gear Teeth Generated by Involute

18.10 Second Version of Geometry: Derivation of Tooth

18.11 Second Version of Geometry: Derivation of Face-Gear

19.6 Generation and Geometry of ZI (Involute) Worms 574

19.8 Geometry and Generation of F-I Worms (Version I) 59019.9 Geometry and Generation of F-II Worms (Version II) 597

19.11 Equation of Meshing of Worm and Worm-Gear

20.2 Generation of Worm and Worm-Gear Surfaces 614

20.5 Contact Lines 622

21.5 Local Synthesis and Determination of Pinion

21.6 Relationships Between Principal Curvatures and

21.8 Application of Finite Element Analysis for the Design

21.9 Example of Design and Optimization of a Spiral Bevel

21.10 Compensation of the Shift of the Bearing Contact 676

22.6 Generation of Face-Milled Hypoid Gear Drives 690

23.8 Illustration of the Effect of Regulation of Backlash 716

26.4 Generation of a Surface with Optimal Approximation 752

27.3 Measurement of Involute Worms, Involute Helical

27.4 Measurement of Asymmetric Archimedes Screw 779

28 Minimization of Deviations of Gear Real Tooth Surfaces 782

28.2 Overview of Measurement and Modeling Method 78328.3 Equations of Theoretical Tooth Surface t 78428.4 Coordinate Systems Used for Coordinate

New ideas of gear design presented in the book include:

(1) Development of gear drives with improved bearing contact, reduced sensitivity

to misalignment, and reduced transmission errors and vibration These goals areachieved by (i) simultaneous application of local synthesis of gear drives and com-puterized simulation of meshing and contact and (ii) application of a predesignedparabolic function of transmission errors that is able to absorb linear functions oftransmission errors caused by misalignments

(2) Development of enhanced finite element analysis of stresses with the followingfeatures: (i) the contacting model of teeth is developed automatically, on the basis

of analytical representation of equations of tooth surfaces; (ii) the formation ofbearing contact is investigated for several pairs of teeth in order to detect andavoid areas of severe contact stresses

(3) Improved conditions of load distribution in planetary gear trains by modification

of the applied geometry and regulation of installment of planet gears on the carrier.New approaches are presented for gear manufacture that enable (i) grinding of face-gear drives by application of a grinding worm of a special shape and (ii) design andmanufacture of new types of helical gears with double-crowned pinions for obtaininglocalization of bearing contact and reduction of transmission errors

The developed theory of gearing presented in the book will make the authors theexperts in this area The book includes the solution to the following important complexproblems:

(i) development of new approaches for determination of an envelope to the family ofsurfaces including the formation of the envelope by two branches;

(ii) avoidance of singularities of tooth surfaces and undercutting in the process ofgeneration; and

xii

Foreword xiii

(iii) simplification of the contacting problem by a new approach for the determination

of principal curvatures and directions of an envelope

The developed ideas have been applied to the design of gear drives, including a newversion of Wildhaber–Novikov helical gear drives, spiral bevel gears, and worm-geardrives Computerized simulation of meshing and contact and testing of prototypes ofgear drives have confirmed the effectiveness of the ideas presented in the book Threepatents for new manufacturing approaches have been obtained by Professor Faydor L.Litvin and representatives of gear companies

The main ideas in the book have been developed by the authors and their associates atthe Gear Research Center of the University of Illinois at Chicago They have also been thesubject of a great number of international publications of permanent interest Thanks tothe wonderful leadership of Professor Faydor L Litvin, who is universally well known

in the field of gears, this Center has involved representatives of various universities inthe United States, Italy, Spain, and Japan in gear research The publication of this bookwill certainly enhance the education and training of engineers in the area of gear theoryand design of gear transmissions

Prof Eng Graziano Curti Politecnico di Torino, Italy

The contents of the second edition of the book have been thoroughly revised and stantially augmented in comparison with the first edition of 1994

sub-New topics in the second edition include the following new developments:

(1) A new geometry of modified spur gears, helical gears with parallel and crossedaxes, a new version of Novikov–Wildhaber helical gears, a new geometry of face-gear drives, geometry of cycloidal pumps, a new approach for design of one-stageplanetary gear trains with improved conditions of load distribution, and a newapproach for design of spiral bevel gear drives with a reduced level of noise andvibration and improved bearing contact

(2) Development of an enhanced approach for stress analysis of gear drives by tion of the finite element method The advantage of the developed approach is theanalytical design of the contacting model based on the analytical representation ofthe gear tooth surfaces

applica-(3) Development of a new method of grinding of face-gear drives, new methods ofgeneration of double-crowned pinions for localization of the bearing contact andreduction of transmission errors, and application of modified roll for reduction oftransmission errors

(4) Broad application of simulation of meshing and tooth contact analysis (TCA) fordetermination of the influence of errors of alignment on transmission errors andshift of the bearing contact This approach has been applied for almost all types ofgear drives discussed in the book

(5) The authors have contributed to the development of the modern theory of gearing

In particular, they have developed in this new edition of the book (i) formation of anenvelope by two branches, (ii) an extension of simulation of meshing for multi-bodysystems, (iii) detection of cases wherein the contact lines on the generating surfacemay have their own envelope, and (iv) detection and avoidance of singularities ofgenerated surfaces (for avoidance of undercutting during the process of generation).The authors are grateful to the companies and institutions that have supported theirresearch and to the members of the Gear Research Center of the University of Illinois

at Chicago who tested their ideas as co-authors of joint papers (see Acknowledgments)

xiv

The authors express their deep gratitude to the institutions and companies that havesupported their research and to colleagues and co-authors of accomplished researchprojects The following list of names only partially covers those to whom the authorsare obliged for their valuable help and inspiration:

(1) Dr John J Coy, Assistant Director, NASA Airspace System Program, NASAAmes Research Center; formerly Manager, Mechanical Components Branch, NASAGlenn Research Center

(2) Dr Robert Bill, Director; James Zakrajsak, Transmission Chief; Dr Robert F schuh, Senior Researcher; NASA John Glenn Research Center and Army ResearchLaboratory

Hand-(3) Dr Gary L Anderson, U.S Army Research Office

(4) James S Gleason, Chairman, The Gleason Corporation; Gary J Kimmet, President, Worldwide Sales & Marketing, The Gleason Corporation; Ralph E.Harper, Secretary and Treasurer, The Gleason Corporation; John V Thomas,Director of Gear Technology, The Gleason Works

Vice-(5) Ryuichi Yamashita, Vice-President; Kenichi Hayasaka, Manager, Gear Researchand Development Group; Yamaha Motor Co., Japan

(6) Terrel W Hansen, Manager; Robert J King, former Manager; Gregory F Heath,Project Engineer; The Boeing Company – McDonnell Douglas Helicopter Systems(7) Dr Robert B Mullins, Director of Engineering; Ron Woods, Technical ResourceSpecialist; Bell Helicopter Textron

(8) Edward Karedes, Chief – Transmissions; Bruce Hansen, Manager, Research andDevelopment; Sikorsky Aircraft Corporation

(9) Daniel V Sagady P.E., Vice-President, Engineering & Product Development;Theresa M Barrett, Executive Engineer; Dr Mauro De Donno, Area Manager –Gears (Guanajuato Gear & Axle); Dr Jui S Chen, Senior Engineer Gear Design;American Axle & Manufacturing

(10) Tom M Sep, Senior Technical Fellow; Visteon Corporation

xv

(11) Matt Hawkins, Gear Specialist; Rolls-Royce Corporation (in Indiana)(12) Daniele Vecchiato, Gear Research Center for UIC

(13) Ignacio Gonz ´alez-P´erez, Gear Research Center for UICAnd the following scholars formerly associated with the Gear Research Center of UIC:(14) Dr C.B Patrick Tsay

(15) Dr Wei-Jiung Tsung(16) Dr Sergei A Lagutin(17) Dr Wei-Shing Chaing(18) Dr Ningxin Chen(19) Dr Andy Feng(20) Dr Yi Zhang(21) Dr Chinping Kuan(22) Dr Yyh-Chiang Wang(23) Dr Jian Lu

(24) Dr Hong-Tao Lee(25) Dr Chun-Liang Hsiao(26) Dr Vadim Kin

(27) Dr Inwan Seol(28) Dr David Kim(29) Dr Shawn Zhao(30) Dr Anngwo Wang(31) Giuseppe Argentieri(32) Alberto Demenego(33) Dr Kazumasa Kawasaki(34) Dr Qi Fan

(35) Claudio Zanzi(36) Matteo Pontiggia(37) Alessandro Nava(38) Luca Carnevali(39) Alessandro Piscopo(40) Paolo Ruzziconi

The subscript “m” indicates that the position vector is represented in coordinate system

S m (x m , y m , z m) To save space while designating a vector, we will also represent theposition vector by the row matrix,

The superscript “T” means that rT

mis a transpose matrix with respect to rm

A point – the end of the position vector – is determined in Cartesian coordinates with

three numbers: x, y, z Generally, coordinate transformation in matrix operations

needs mixed matrix operations where both multiplication and addition of matricesmust be used However, only multiplication of matrices is needed if position vectors arerepresented with homogeneous coordinates Application of such coordinates forcoordinate transformation in theory of mechanisms has been proposed by Denavit &Hartenberg [1955] and by Litvin [1955] Homogeneous coordinates of a point in a three-

dimensional space are determined by four numbers (x∗, y∗, z∗, t∗) which are not equal

to zero simultaneously and of which only three are independent Assuming that t∗= 0,ordinary coordinates and homogeneous coordinates may be related as follows:

Figure 1.1.1: Position vector in Cartesian nate system.

coordi-With t∗= 1, a point may be specified by homogeneous coordinates such as (x, y, z, 1),

and a position vector may be represented by

1.2 COORDINATE TRANSFORMATION IN MATRIX REPRESENTATION

Consider two coordinate systems S m (x m , y m , z m ) and S n (x n , y n , z n) (Fig 1.2.1) Point

M is represented in coordinate system S mby the position vector

1.2 Coordinate Transformation in Matrix Representation 3

Figure 1.2.1: Derivation of coordinate

Here, (in , j n , k n) are the unit vectors of the axes of the “new” coordinate system;

(im , j m , k m ) are the unit vectors of the axes of the “old” coordinate system; O n and

O mare the origins of the “new” and “old” coordinate systems; subscript “nm” in the

designation Mnm indicates that the coordinate transformation is performed from S mto

S n The determination of elements a l k (k = 1, 2, 3; l = 1, 2, 3) of matrix M nmis based

on the following rules:

(i) Elements of the 3× 3 submatrix

represent the direction cosines of the “old” unit vectors (im , j m , k m) in the “new”

coordinate systems S n For instance, a21= cos( y n , x m ), a32= cos( z n , y m), and so

on The subscripts of elements a kl in matrix (1.2.5) indicate the number l of the

“old” coordinate axis and the number k of the “new” coordinate axis Axes x , y, z

are given numbers 1, 2, and 3, respectively

(ii) Elements a14, a24, and a34represent the “new” coordinates x (O m)

n , y (O m)

n , z (O m)

n of

the “old” origin O m

Recall that nine elements of matrix Lnmare related by six equations that express thefollowing:

(1) Elements of each row (or column) are direction cosines of a unit vector Thus,

To determine the new coordinates (x n , y n , z n , 1) of point M, we have to use the rule

of multiplication of a square matrix (4× 4) and a column matrix (4 × 1) (The number

of rows in the column matrix is equal to the number of columns in matrix M nm )

The inverse matrix Mmnindeed exists if the determinant of matrix Mnm differs fromzero

1.2 Coordinate Transformation in Matrix Representation 5

There is a simple rule that allows the elements of the inverse matrix to be determined

in terms of elements of the direct matrix Consider that matrix Mnmis given by

The submatrix Lmnof the order (3× 3) is determined as follows:

The columns to be multiplied are marked

To perform successive coordinate transformation, we need only to follow the productrule of matrix algebra For instance, the matrix equation

represents successive coordinate transformation from S1to S2, from S2to S3, , from

S p−1to S p

To perform transformation of components of free vectors, we need only to apply

3× 3 submatrices L, which may be obtained by eliminating the last row and the last column of the corresponding matrix M This results from the fact that the free-vector

components (projections on coordinate axes) do not depend on the location of the origin

of the coordinate system

The transformation of vector components of a free vector A from system S m to S nisrepresented by the matrix equation

surface point where the surface normal is considered will be transferred simultaneously

1.3 ROTATION ABOUT AN AXIS Two Main Problems

We consider a general case in which the rotation is performed about an axis that doesnot coincide with any axis of the employed coordinate system We designate the unit

vector of the axis of rotation by c (Fig 1.3.1) and assume that the rotation about c may

be performed either counterclockwise or clockwise

Henceforth we consider two coordinate systems: (i) the fixed one, S a; and (ii) the

movable one, S b There are two typical problems related to rotation about c The first

one can be formulated as follows

Consider that a position vector is rigidly connected to the movable body The initial

position of the position vector is designated by OA = ρ (Fig 1.3.1) After rotation

through an angleφ about c, vector ρ will take a new position designated by OA∗= ρ∗.Both vectors,ρ and ρ∗(Fig 1.3.1), are considered to be in the same coordinate system, say S a Our goal is to develop an equation that relates components of vectors ρ aandρ∗

describes the relation between the components of vectorsρ and ρ∗that are represented

in the same coordinate system S a The other problem concerns representation of the same position vector in different

coordinate systems Our goal is to derive matrix Lba in matrix equation

1.3 Rotation About an Axis 7

Figure 1.3.1: Rigid body rotation.

The designations ρ a and ρ b indicate that the same position vector ρ is represented

in coordinate systems S a and S b , respectively Although the same position vector is

considered, the components ofρ in coordinate systems S a and S bare different and wedesignate them by

ρ a = a1ia + a2ja + a3ka (1.3.3)and

ρ b = b1ib + b2jb + b3kb (1.3.4)

Matrix Lba is an operator that transforms the components [a1 a2 a3]T into

[b1 b2 b3]T It will be shown below that operators La and Lba are related

Problem 1. Relations between components of vectorsρ a andρ∗

a.Recall thatρ aandρ∗

a are two position vectors that are represented in the same coordinate system S a Vectorρ represents the initial position of the position vector, before rotation,

andρ∗represents the position vector after rotation about c The following derivations

are based on the assumption that rotation about c is performed counterclockwise The

procedure of derivations (see also Suh & Radcliffe, 1978, Shabana, 1989, and others)

is as follows

Step 1: We representρ∗

a by the equation (Fig 1.3.1)

OM= (ca · ρ a)ca = (ca · ρ∗

and ca is the unit vector of the axis of rotation that is represented in S a

Step 2: Vectorρ a is represented by the equation

ρ a = OM + MA = (c a · ρ a)ca + MA (1.3.7)that yields

We emphasize that a vector being rotated about c generates a cone with an apex

angleα Thus, both vectors, ρ and ρ∗, are the generatrices of the same cone, as shown inFig 1.3.1

Step 3: Vector MN has the same direction as MA and this yields

|MN| = |MA∗| cos φ = |MA| cos φ = ρ sin α cos φ (1.3.9)whereα is the apex angle of the generated cone, |MA| = ρ sin α, and ρ is the magnitude

|NA∗| = |MA∗| sin φ = ρ sin α sin φ, |c a × ρ a | = ρ sin α.

Step 5: Equations (1.3.5), (1.3.6), (1.3.10), and (1.3.11) yield

ρ∗

a = ρ acosφ + (1 − cos φ)(c a · ρ a)ca + sin φ(c a × ρ a). (1.3.12)

Step 6: It is easy to prove that

(ca · ρ a)ca = ca× (ca × ρ a)+ ρ a (1.3.13)because

ca× (ca × ρ a)= (ca · ρ a)ca − ρ a(ca· ca).

Step 7: Equations (1.3.12) and (1.3.13) yield

ρ∗

a = ρ a + (1 − cos φ)[c a × (ca × ρ a)]+ sin φ(c a × ρ a). (1.3.14)Equation (1.3.14) is known as the Rodrigues formula According to the investigation

by Cheng & Gupta [1989], this equation deserves to be called the Euler–Rodrigues,formula

1.3 Rotation About an Axis 9

Step 8: Additional derivations are directed at representation of the Euler–Rodrigues

formula in matrix form

The cross product (ca × ρ a) may be represented in matrix form by

Step 9: Equations (1.3.14), (1.3.15), and (1.3.16) yield the following matrix

repre-sentation of the Euler–Rodrigues formula:

ρ∗

a = I+ (1 − cos φ)(C s)2+ sin φC s ρ a = La ρ a (1.3.18)

where I is the 3× 3 identity matrix While deriving Eqs (1.3.14) and (1.3.18), weassumed that the rotation is performed counterclockwise For the case of clockwiserotation, it is necessary to change the sign preceding sinφ to its opposite The expression

for matrix La that will cover two directions of rotation is

La = I + (1 − cos φ)(C s)2± sin φC s (1.3.19)The upper sign preceding sinφ corresponds to counterclockwise rotation and the lower

sign corresponds to rotation in a clockwise direction In both cases the unit vector c must be expressed by the same Eq (1.3.17) that determines the orientation of c but

not the direction of rotation The direction of rotation is identified with the proper signpreceding sinφ in Eq (1.3.19).

Problem 2 Recall that our goal is to derive the operator Lbain matrix equation (1.3.2)

that transforms components of the same vector (see Eqs (1.3.3) and (1.3.4)) It will be

shown below that the sought-for operator is represented as

Lba = LT

a = I + (1 − cos φ)(C s)2∓ sin φC s (1.3.20)

Operator Lba can be obtained from operator La given by Eq (1.3.19) by changing thesign of the angle of rotation,φ The upper and lower signs preceding sin φ in Eq (1.3.20) correspond to the cases where S a will coincide with S bby rotation counterclockwiseand clockwise, respectively The proof is based on the determination of components of

the same vector, say vector OA shown in Fig 1.3.1, in coordinate systems S a and S b

Step 1: We consider initially that vector OA is represented in S a as

Step 2: To determine components of vector OA in S bwe consider first that coordinate

system S and the previously mentioned position vector are rotated as one rigid body

about c After rotation through angleφ, position vector OA will take the position OA∗and can be represented in S bas

OA∗ = a1ib + a2jb + a3kb (1.3.22)

It is obvious that vector OA∗ has in S b the same components as vector OA has in S a

Step 3: We consider now in S b two vectors OA∗ and OA Vector OA∗ will coincide

with OA after clockwise rotation about c The components of vectors OA∗and OA in

S b are related by an equation that is similar to Eq (1.3.19) The difference is that we

now have to consider that the rotation from OA∗to OA is performed clockwise Then

we obtain

(OA) b= Lb (OA∗)b= I+ (1 − cos φ)(C s)2− sin φC s (OA∗)b (1.3.23)

Designating components of (OA) b by [b1 b2 b3]T, we receive

[b1 b2 b3]T= [I + (1 − cos φ)(C s)2− sin φC s ][a1 a2 a3]T. (1.3.24)

Step 4: We have now obtained components of the same vector OA in coordinate

systems S a and S b, respectively The matrix equation that describes transformation of

In our identification of coordinate systems S a and S bwe do not use the terms fixed

and movable We just consider that S a is the previous coordinate system and S b is

the new one, and we take into account how the rotation from S a to S b is performed:counterclockwise or clockwise

Using Eqs (1.3.26) and (1.3.27), we may represent elements of matrix Lba in terms of

components of unit vector c of the axis of rotation and the angle of rotationφ Thus,

1.3 Rotation About an Axis 11

Figure 1.3.2: Derivation of coordinate

When the axis of rotation coincides with a coordinate axis of S a, we have to make

two components of unit vector ca equal to zero in Eqs (1.3.30) For instance, in the

case in which rotation is performed about the z a axis (Fig 1.3.2), we have

We emphasize again that in all cases of coordinate transformation only elements (1.3.30)

of matrix Lba, and not the components of ca, depend on the direction of rotation The

unit vector c can be represented in either of the two coordinate systems, S a and S b, bythe equations

c= c1ia + c2ja + c3ka = c1ib + c2jb + c3kb (1.3.32)

This means that the unit vector c of the axis of rotation has the s ame components in

both coordinate systems, S a and S b It is easily verified that

[c1 c2 c3]T= Lba [c1 c2 c3]T. (1.3.33)

Although the significance of this observation has not been recognized in the literature,

it has been found to be advantageous in obtaining coordinate transformations in thisbook

The proof of Eq (1.3.32) is based on the following considerations The unit vector c

of the axis C of rotation is directed along the axis that is common to the two coordinate systems, S a and S b Thus, the orientation of c is not changed when one of the two

coordinate systems rotates with respect to the other about C For instance, assume that

a unit vector of S a, say ia , is directed along OA, and the unit vector i b of S bis directed

along OA∗(see Fig 1.3.1) Both unit vectors, ia and ib, are the generatrices of the samecone and therefore

c · ia = c · ib = c1.

Similarly, we may prove that

c · ja = c · jb = c2, c · k a = c · kb = c3.

Thus, Eq (1.3.32) is confirmed

Employment of Additional Coordinate Systems

Generally, the axis of rotation does not coincide with any coordinate axis of S a The

movable coordinate system S b coincides with S a in the beginning of rotation Thus,

there is no coordinate axis of S bthat coincides with the axis of rotation Our goal is to

employ two additional coordinate systems S m and S nthat will enable us to make one

of their coordinate axes coincide with the axis of rotation The auxiliary coordinate

system S m is rigidly connected to S a , and the auxiliary coordinate system S nis rigidly

connected to S b

The determination of the structure of matrix Lma is based on the following

consider-ations Let us represent Lma as

A respective axis of S m will coincide with c if one of three unit vectors (a, b, and d)

coincides with c We limit this discussion to the case in which the z maxis coincides with

c Two other cases can be discussed similarly For the previously mentioned case we

1.3 Rotation About an Axis 13

It is obvious that

because a· c = 0, b · c = 0, and c · c = 1 While choosing one of the two unit vectors (a

and b), say b, we have to take into account the following relations: (i)

because b is a unit vector Equations (1.3.39) and (1.3.40) relate two of three components

of b, and only one of them can be chosen.

After determination of c and b we can define the unit vector a using the cross product

The motion of movable coordinate system S n with respect to S m and S a (S m and S a are

rigidly connected) is rotation about the z maxis through angleφ Matrix L nmcan be termined in accordance with Eqs (1.3.29) and (1.3.30) The coordinate transformation

de-from S a to S nis based on the matrix equation

We have discussed above the coordinate transformation from S a to S bthat is represented

by matrix equation (1.3.30)

At the start of motion, coordinate system S b coincides with S a , coordinate system S n

(which is rigidly connected to S b ) coincides with S m (which is rigidly connected to S a).With these considerations we can develop the following matrix equations:

LnmLma [c1 c2 c3]T= LmaLba [c1 c2 c3]T= m = [m1 m2 m3]T (1.3.47)

Here, m is the unit vector of the axis of S mthat is the axis of rotation (two components

of m are equal to zero and the third is equal to one) Matrix equations (1.3.44) to

(1.3.47) are illustrated in Problem 1.5.4

1.4 ROTATIONAL AND TRANSLATIONAL 4× 4 MATRICES

Generally, the origins of coordinate systems do not coincide and the orientations of thesystems are different In such a case the coordinate transformation may be based on theapplication of homogeneous coordinates and 4× 4 matrices that describe separatelyrotation about a fixed axis and displacement of one coordinate system with respect tothe other

Consider that the same point must be represented in coordinate systems S p and S q

(Fig 1.4.1) The origins of S p and S qdo not coincide and the orientations of coordinateaxes in these systems is also different It is useful in such a case to apply an auxiliary

coordinate system, S n, and a matrix, Mnp , that describes translation from S p to S n

Coordinate systems S p and S qhave a common origin and the coordinate transformation

from S p to S q is based on the Euler–Rodrigues equation

The coordinate transformation from S p to S qis represented by the matrix equation

rq = MqnMn prp= Mqprp (1.4.1)The 4× 4 matrix Mnp describes translation from S p to S nand is represented by

sys-1.5 Examples of Coordinate Transformation 15

The 4× 4 matrix Mqndescribes rotation about a fixed axis with unit vector c and is

1.5 EXAMPLES OF COORDINATE TRANSFORMATION

The examples of coordinate transformation presented in this section are based on plication of 4× 4 rotational and translational matrices The study of these problemswill give the reader experience in coordinate transformation as it relates to the theory

(i) Determine the position vector r2of the same point in coordinate system S2

(ii) Express the inverse matrix M12= M−1

21 in terms of elements of matrix M21 and

then determine the position vector r1considering that r2is given

The rotation from S1to S2is performed clockwise and the lower sign in Eq (1.3.30)

must be chosen Taking into account that c1= c2= 0, c3= 1, we obtain the

fol-lowing expression for the rotational matrix Mf 1:

Figure 1.5.1: Centrodes in translation– rotation motions.

The drawings of Fig 1.5.1 yield that

1.5 Examples of Coordinate Transformation 17

(ii) Matrix M21is not singular and the inverse coordinate transformation is possible

To determine the inverse matrix M12= M−1

21, we use equations (1.2.10) to (1.2.13),which yield

cosφ sinφ 0 ρ(sin φ − φ cos φ)

− sin φ cos φ 0 ρ(cos φ + φ sin φ)

x1= x2cosφ + y2sinφ + ρ(sin φ − φ cos φ)

y1= −x2sinφ + y2cosφ + ρ(cos φ + φ sin φ)

Figure 1.5.2: Centrodes in rotational

mo-tions of opposite direction.

φ1andφ2are related with the equation

φ2

φ1 =ρ1

ρ2

(1.5.11)whereρ1andρ2are the radii of gear centrodes (see Section 3.2) E is the shortest distance between the axes of rotation The fixed coordinate system S f is rigidly connected to the

gear housing S pis an auxiliary coordinate system that is also rigidly connected to thegear housing

(i) Derive equations for coordinate transformation from S2to S1

(ii) Derive equations for coordinate transformation from S1to S2

cos(φ1+ φ2) sin(φ1+ φ2) 0 E sin φ1

− sin(φ1+ φ2) cos(φ1+ φ2) 0 E cos φ1

Using equations (1.5.12) and (1.5.14), we obtain

x1= x2cos(φ1+ φ2)+ y2sin(φ1+ φ2)+ E sin φ1

y1= −x2sin(φ1+ φ2)+ y2cos(φ1+ φ2)+ E cos φ1

z1= z2.

(1.5.15)

1.5 Examples of Coordinate Transformation 19 (ii) The inverse matrix M21= M−1

12 can be expressed in terms of elements of M12asfollows [see Eqs (1.2.10) to (1.2.14)]:

cos(φ1+ φ2) − sin(φ1+ φ2) 0 E sin φ2

sin(φ1+ φ2) cos(φ1+ φ2) 0 −E cos φ2

x2= x1cos(φ1+ φ2)− y1sin(φ1+ φ2)+ E sin φ2

y2= x1sin(φ1+ φ2)+ y1cos(φ1+ φ2)− E cos φ2

z2= z1.

(1.5.18)

Problem 1.5.3

Consider that two gears transform rotation about parallel axes in the same direction

(Fig 1.5.3) Coordinate systems S1 and S2are rigidly connected to gears 1 and 2; S f

and S p are fixed coordinate systems; E is the shortest distance; ρ1andρ2are the radii

of gear centrodes (see Section 3.2) (i) Determine matrices M21and M12= M−1

21 and (ii)

perform the coordinate transformation in transition from S1to S2and from S2to S1

Figure 1.5.3: Centrodes in rotational

mo-tions of the same direction.

cos(φ1− φ2) sin(φ1− φ2) 0 −E sin φ2

− sin(φ1− φ2) cos(φ1− φ2) 0 E cos φ2

cos(φ1− φ2) − sin(φ1− φ2) 0 E sin φ1

sin(φ1− φ2) cos(φ1− φ2) 0 −E cos φ1

x2= x1cos(φ1− φ2)+ y1sin(φ1− φ2)− E sin φ2

y2= −x1sin(φ1− φ2)+ y1cos(φ1− φ2)+ E cos φ2

z2= z1

(1.5.21)

x1= x2cos(φ1− φ2)− y2sin(φ1− φ2)+ E sin φ1

y1= x2sin(φ1− φ2)+ y2cos(φ1− φ2)− E cos φ1

z1= z2.

(1.5.22)

Problem 1.5.4

The purpose of this problem is to illustrate the verification of Eqs (1.3.44) to (1.3.47)

Figure 1.5.4(a) shows two coordinate systems S a and S bthat coincide with each other

initially Coordinate system S b is rotated counterclockwise about the axis with unitvector

− sin φ cos γ cosφ cos2γ + sin2γ (1− cos φ) sin γ cos γ

sinφ sin γ (1− cos φ) sin γ cos γ cosφ sin2γ + cos2γ





 (1.5.24)

Figure 1.5.4(b) shows an auxiliary coordinate system S mthat is rigidly connected to

S a The coordinate axis z mcoincides with ca whose components represent the elements

of the third row in matrix L represented by Eq (1.3.36) Coordinate axis x coincides

1.5 Examples of Coordinate Transformation 21

Figure 1.5.4: Auxiliary coordinate systems: (a)

illus-tration of coordinate systems S a and S b; (b)

illustra-tion of coordinate systems S a , S m , and S n.

with x a According to Eq (1.3.30), matrix Lma is

Coordinate systems S n and S bwhich are rigidly connected to each other are rotated

about cm= kmthrough angleφ Matrix L nm, in accordance with Eqs (1.3.30), is resented by

cosφ sinφ cos γ − sin φ sin γ

− sin φ cos φ cos γ − cos φ sin γ





Figure 1.5.5: General case of coordinate formation.

trans-Using Eqs (1.5.24) to (1.5.27) we may be certain that matrix equations (1.3.43)and (1.3.45) are indeed observed Equations (1.3.46) and (1.3.47) in the discussed caseyield

transition from S1to S2and from S2to S1

matrix equation

r2= M21r1= M2 pMpmMmfMf 1r1. (1.5.30)