the geometrie modelling of mechanical elements with...

The Geometrie Modelling of Mechanical Elements with Complex Shapes

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Contributions to the Geometrie Modelling of Mechanical Elements with Complex

Shapes

Mahmoud J. AI-Daccak

Bachelor of Engineering-Electrical. (McGiII University). 1987

Department of Mechanical Engineering

McGili University

Montréal. Canada

A thesis submitted to the Faculty of Graduate Studies and Research

in partial fulfillment of the requirements for the degree of

Master of Engineering-Mechanical

March 17. 1989

© Mahmoud J. AI-Daccak

'" UNIVERSITF McGILL

"''' , FACULTE DES ETUPES AVAl.'lCEES ET T)F

~m1 DE L' AUTEU!\: ,

DEPARTEHE~T : GRADE:

, TITRE DE LA THESE:

1. Par la présente, l'auteur accorde à l'université McGill l'autorisation de mettre cette thèse à la disposition des lecteurs dans une biblio thèque de ~cGill ou une autre bibliothèque, soit sous sa forme actuelle, soit sous forme d'une réproduction. L'auteur détient: cependant les autres droits de publications. Il est entendu, par ailleurs, que ni la thèse, ni '.es longs extraits de cette thèse ne pourront être imprimés ou reproduits par d'autres ~oyens sans l'autorisation écrite de l'auteur.

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Adresse permanente:

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(English on reverse)

• Abstract

Geometrie modelling helps us produce optimal mechanical components more

quickly and accurately by enhancing the power of the CAO/CAM systems used for their

design and manufacture. The modelling of comp/ex shapes. I.e .. shapes that cannot be

produced from a combination of primitives-lines. planes. cylinders. cones. tori. etc.-.

presents a particular challe:lge. /n this context. the modelling of bevel gears is addressed

as a paradlgm of this type of shapes. Actually. various approaches have been proposed to

approximate the theoretical invo/ute-generated contact surface of beve/ gears. As a means

to accurately represent contact surfaces of these gears. the notion of the exact spherica/

invo/ute is introduced. The so/id models of straight and spiral bevel gears are obtained

by app/ying simple sweeping techniques to their tooth profiles which are described by the

exact spherical involute.

A key issue in the geometric analysis of mechanieal elements is the accu rate and

economic computation of their volumetrie propertles. na me/y. volume. centrold coordlnates

and inert:a tensor. Explicit. readlly implementable formulae are developed to eva/uate these

properties for a sol id. glven Its pieeewise-/inearly approxlmated boundary. The formulae

are based upon a repeated application of the Gauss Divergence Theorern. that reduces the

computation of the said propertles to Ime integratlon Moreover. a method is proposed for

the computation of the volumetrie properties for sweep-generated sollds. based only on thelr

2D generating contour and their sweeping parameters Thus. the direct 3D calculations of

the volumetrie properties of such solids are reduced to simpler 20 ca/cu/ations.

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Résumé

La modélisation géométrique a!de à produire rapidement des composantes méca

niques plus précises en améliorant la puissance des systèmes CAO /FAO utilisés pour leur

conception et leur fabrication La modélisation de formes complexes. comme celles qui

ne peuvent pas être produites par une combinaison de formes primaires-lignes. plans.

cylindres. cones. tores. etc -. représente un réel défi pour les ingénieurs de conception

les engrenages coniques sont représentatifs de cette classe de formes et c'est pourquoI

nous nous y intéressons Diverses approches ont été proposées pour approximer la surface

de contact théorique des engrenages coniques. qui est produite par une développante. Afin

de représenter avec précision ces surfaces de contact. la notion de la développante sphéflque

exacte est introdUite Les modèles géométriques des engrenages COntques à denture drOite

et à denture spirale sont obtenus en appliquant des méthodes simples de balayage sur les

profils des dents des engrenages. ceux-cI étant décrits par la développante sphérique exacte

Un élément essentiel de l'analyse géométrique des composantes mécaniques

est le calcul préCIS et efficace de leur propriétés volumiques. notamment le volume. les

coordonnées du barycentre. et le tenseur d'inertie. Nous développons des formules explicites

et faciles à mettre en œuvre qui évaluent ces propriétés pour un solrde. étant donnée une

approxImation polyhédrale. Les formules sont basées sur l'application répétée du théorème

de la divergence de Gauss. ce qUI permet de réduire le calcul des dites propriétés à une

intégration de ligne De plus. nous proposons une méthode pour le calcul des propriétés

volumiques des solides prodUIts par balayage. qUi est basée seulement sur leur contour

bi-dimensionnel et leurs paramètres de balayage AinSI. les calculs tn-dimensionnels directs

des propriétés volumiques de ces solides sont ramenés à des calculs bi-dimensionnels plus

simples.

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Acknowledgemcnts

1 would like to extend my deepest gratitude to my thesis supervisor. Prof. Jorge

Angeles. for his continuous advlce. enthuslastic guidance and constant support throughout

the course of thls research.

1 am grateful to Prof. P J Zsombor-Murray for hls suggestions and constructive

criticism. Additionally. 1 would Irke to thank Mr. J. Goldrich of The Gleason Works Company

for his remarks on the state-of-the-art of bevel-gear CAD /CAM technology 'Il industry and

my colleague Mr. Stephane Aubry for hls French translation of the abstract Moreover.

the computer and research facilrties furnished by McRCIM (McGIII Research Center for

Intelligent Machines) arP. duly appreclated

1 am eternally Indebted to my parents and two slsters for the Invaluable love.

constant encouragement and endless support they have provlded me through the long

distance between us Special thanks are due to my father for being an inspiration to

me. My profound thankfulness to my ':Atta ". MIss Zuhra Shukn. for her utter love and

persistent be!ref ln me Flnally. slnce 1 began my study in Canada. It has been my blessing

to know many best frrends who have been my family away from home and have dlrectly or

indirectly contributed to the accomplrshment of thls Vlork ln diverse ways

The research work reported here was possible under the following' a NSERC

(Natural Science and Engineering Research (ouncii. of Canada) Grant#A4532. FCAR

(Fonds pour la formation de chercheurs et l'aide à la recherche. of Quebec) Grant# 88-

AS-2517: and IRSST (Institut de recherche en santé et en sécurrté du travail. of Quebec)

Grant# RS8706.

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To my belo'tled parents.

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Contents

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ............................ IX

Chapter 1 Introduction.. . . . ... . . . . . . . . . . .... . . . .... . . . . .... . . . ... .. 1

1.1 Sol id Modelling and Evaluation of Volumetrie Properties of

Meehamcal Elements. .. ................. ..... .................. 2

1.2 3D Modelling of Bevel Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 3

1.3 Volumetrie Properties of Boundary-Represented Solids . . . . . . . . . . . . . . . . .. 5

1.4 Volumetrie Properties of Sweep-Generated Solids. . . . . . . . . . . . . . . . . . . . . .. 7

Chapter 2 Solid Modelling Schemes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1 If'~roduction . . . . . . .. ............................................. 9

2.2 Primitive Instancing .. . .. ..... . ......... . 10

2.3 Cel! Decomposition and Spatial Occupancy Enumeration .. . . . . . . . . . . . .. 11

2.4 Constructive Solid Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12

2.5 Boundary Representations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Sweep Representations ................ . 15

Chapter 3 3D Modelling of Bevel Gears ........ . . . . . . . . . .. . . . . . . .. 17

3.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 The Exact Sphericallnvolute. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . 18

3.3 Generating the Involute Bevel-Gear Profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

34 Sol id Modelling of Bevel Gears . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.5 Examples and Results ......... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

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Chapter 4 Volumetrie Properties of Boundary-Represented

Solids .................................................. " 35

4.1 Introduction. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 35

4.2 General Transformation Formulae . . . . . . . . . . . ... . . . . . .. . . . . . . . . . . . . .. 36

4.3 Two-Dimensional Regions. . . . . . . . . . . . . . . . . .. .. . . . . ... . . . . . . . . . . . .. 37

4.4 Three-Dimensional Regions ............................. . . ...... 40

4.5 Examples........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 46 4.5.1 Example 1: Computation of the Volume. Centroid Coordinates.

and Inertia Tensor of a Cam Disk. . . . . . . . . . . . . . .. . .. .......... 46 4.5.2 Example 2: Computation of Volume. Centroid Coordinates. and

Inertia Tensor of Spur and Helieal Gears. ... . . . . . . . . . . . . . . . . . . .. 48

Chapter 5 Volumetrie Properties of Sweep-Generated

Solids .................................................... 53

5.1 Introduction.... ............ ................................... 53

5.2 On Notation and Basic Definitions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 53 5.3 Reduced Formulae for the Volumetrie Properties of Sweep-Generated

Solids ......................................................... 54

5.4 Applications..................................................... 58

5.4.1 Straight Extrusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 58

5.4.2 Extrusion While Twisting . . . . . . . . . . . . . . .. . . . . . . . . . . . . . .. . . . .. 59

5.4.3 Extrusion While Scaling .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 61

5.5 Examples........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 62

Chapter 6 Conclusions and Remarks .... . . . . .. . . . . .. . . . . . . . .. . . . .. 64

References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 66

Appendix A. Gear Terminology ............ , ......................... " 69

A.t General Terminology . . . . . . .. . . . . .. . . . . . .. . . . . . . .. . . . . . . . . . . .. 69

A.2 Revel Gear Terminology and Geometry . . . . .. . . . . . .. . . . . . . . . . . . .. 70

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Appendix B. Sorne Useful Tensor Relations .. , . . . . . . . . . . . . .. . . . . . . . . . . . .. 73

B.1 Tensor Notation and its Relation to Multi-Linear Algebra ......... " 73

B.2 Divergence of a nth-rank Tensor ....... . . . . . . . . . . . . . . . . . . . . . . .. 74

B.3 2D-to-3D Mapping of Vectors and Second-Rank Tensors . . . . . . . . . .. 74

B.4 Proof of The Divergence Identities Used . . . . . . . . . . . . . . . . . . . . . . . .. 75

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List of Figures

2.1 Primitive Instancing. (adapted from Mortenson (1985). p. 448.

Fig. 10.15) .... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11

2.2 Cel! Decomposition. (adapted from Mortenson (1985). p. 451.

Fig. 10.19) .................................. . . . . . . . . . . . . . . . . .. .. 12

2.3 Constructive Solid Geometry. (adapted from Mortenson (1985).

p. 462. Fig. 10.28) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13

2.4 Boundary Representations. (adapted from Lee and Requicha

(1982)) ........ . . . . . . . .. ....................................... 14

2.5 Nonhomogeneous surface generated by translating a 2D curve.

(adapted from Mortenson (1985). p. 457. Fig. 10.23) . . . . . . . . . . . . . . . . . .. 15

2.6 Translational and rotational sweeping. (adapted from Mortenson

(1985). p. 456. Fig. 10.22) ......................................... 16

2.7 General sweeping along a ~D curve. (adapted from Mortenson

(1985). p. 456. Fig 10.26) ...................... .. ................ 16

3.1 Generating the exact spherica/ in volute 1 by unwrapping the arc of ,.....

great circle TP from the base circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 19

3.2 The exact spherica/ invo/ute 1 generated on a sphere of radius T. •••.••.• 20

3.3 The exaci spherica/ invo/ute function. ........................... . . . . 21

3.4 Projection of curves on the !. - ansverse sphere. ........................ 23

3.5 The béfse cone and the generated spherical involute. . . . . . . . . . . . . . . . . . . . 24

3.6 Projection of transverse sphere showing two opposing spherical

involutes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. ...... ?5

3.7 Spherical profiles of a pinion-and-gear train shown on a transverse

sphere. (adapted from Merritt (1946). p. 57. Fig. 5.18) ................. 26

3.8 Obtaining a spiral bevel gear from a straight one. (adapted from

Sioane (1966). p. 202. Fig. 216) .................................... 26

3.9 Comparison of results in the X axis. ...... . . . . . . . . . . . . . . . . . . . . . . . . . . 30

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• 3.10 Comparison of results in the Y axis. ............................... 31

3.11 Comparison of results in the Z axis. ............................... 31

3.12 The spherical profile used to model the gears. ....................... 32

3.13 The straight bevel gear model .................................... 33

3.14 The spiral bevel gear model. ...... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 34 4.1 Line segment representing one side of a polygon approximating a

planar cloSE'd curve. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 38 4.2 Polygon representing one face of a polyhedron approxjmating a

closed surface. .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 43

4.3 The ith polygon contained in plane ni .............................. 44

4.4 Profile of a cam disk ............................................. 47

4.5 Cam obtained by a straight extrusion of its profile. . . . . . . . . . . . . . . . . . . .. 48

4.6 Gear profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 50

4.7 Spur gear generated by a straight extrusion of its profile. . . . . . . . . . . . . . .. 51 4.8 Helical gear generated by rotating its profile while it is being

extruded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52

5.1 Sweeping region n2 to generate the 3D region fl3 " . . . . . . . . . • . . . . . . . • .. 55

A.l Spur gears. (adapted from Watson (1970). p. 13. Fig. 2.1) ............. 71

A.2 Helical gears. (adapted From Watson (1970). p. 17. Fig. 2.7).. . . . . . . . . .. 71

A.3 Straight bevel gears. (adapted froll1 Watson (1970). p. 20. Fig.

2.12) ........... . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 71

A.4 Spiral bevel gears. (adapted from Watson (1970). p. 20. Fig.

2.13) ...... . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. 72

A.5 Bevel gear geometry. (adapted From Wilson. Sadler and Michels

(1983). p. 439. Fig. 7.24) .. . .. .. .. .. .. .... . . .. . .. . . . .. .. .. . .. .. ... 72

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Chapter 1 Introduction

The production of a mechanical element or component from its conceptual

ization to its final packaging involves many steps such as product specification. design.

engineering analysis. planning for manufacture. material procurement. and production con

trol. These processes are not independent: they ail interact to produce the final good.

For example. analysis may reveal the need for redefining product specification. Evidently.

it is more efficient and effective to have a central description of the object or product for

the purpost. of reference. rnodificatio". manipulation. and further processmg. Obtalnmg

such a central object description underlines the basic motivation for shape description and

modelling with computers. Computer-based geometric models of mechanical elements that

can be edited. modified and updated throughout the production process is the obvious

alternative to engineering drawings obtained through drafting-a process with inherent de

scriptive limitations (Baer. Eastman and Henrion 1979) ln recent years. Computer-Aided

Design (CAD) and Computer-Aided Manufacturing (CAM) of mechanical elements have

been fields of intensive research and development and have emerged as essential tools in

the production process. The trend has evolved From the design and modelling of simple

mechanical components with limited complexity to the design and modelling of sculptured

and highly complex ones. Advances in the CAD and CAM technologies have been the driv

ing force behind the rapid development in the geometric modelling field. This relatively new

field blends geometry with the computer and forms the backbone of CAD/CAM systems

(Mortenson 1985).

1. Introduction

This thesis focuses on two areas of geometric modelling. The first is solid

modelling. which can be defined as the process of obtaining an unambiguous and infor

mationdlly complete mathematical representation of the shape of a physical object in a

(orm that a computer can process (Mortenson 1985). The second is extraeting the global

geometric properties. i.e .. the volumetrie properties. of the modelled physical object based

on its representation. Attention is given to the modelling of objects with complex shapes

whose boundaries are defined by sculptured surfaces. i.e .. obJects with shapes that cannot

be described by boolean combinations of blocks. cylinders. wedges. etc. The problem of

modelling bevel gears is addressed as a paradigm of mechanical elements with complex

shapes. Moreover. a methodology is developed for the calculatlon of volumetric properties

of ûbjects. induding volume. centroid location and inertia tensor. modelled using certain

representation schemes capable of descnbrng mechanical elements with comJ::lex shapes.

It is shown how simple mode"rng techniques can be used to model different gear

types. including bevel gears. The 3D modelhng of bevel gears has received little attention

in spite of their practical importance as mechanical elements to transmit rotation oetween

nonpara"el shafts The evaluation of the volumetrlc propertles of mechanlcal elements

is also of similar importance The moment of inertia. for example. is a very Important

property in the motion of rigid bodies. Wlth objects of complex shapes. its calculation

requires special attention. The enhancement and optimization of volumetrie calculations is

fundamental in the field of CAO/CAM of mechanical elements.

1.1 Solid Modelling and Evaluation of Volumetrie Properties of

Meehanieal Elements

Accordil.g to Reqt!icha (1980). there are five major categories of unambiguous

sofid-modelling scneme.:-: primitive instancing. cell decomposition and spatial occupancy

enumeration. sweep representations. constructive solid geometry (CSG). and boundary

representations (B-reps). The last three are the most important to contemporary mode"ing

systems Aigorithms for the computation of volumetric properties of models produced by

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1 Introduction

a certain scheme are directly related to the properties of that scheme (Lee and Requicha

1982a). This natural association. along with a brief description of each modelling scheme.

will be the subject of Chapter 2

The subject of modellmg mechanical elements or components has been ad

dressed by several researchers. An incomplete list includes: 8raid and Lang (1973): Voel

cker and Requicha (1977): Requicha and Voelcker (1979): Wesley et al (1980): Dewhirst

and Hillyard (1981); and Chen and Perny (1983). The purpose of the work presented here

is to apply general sweepmg techniques to model mechanical elements and to illustrate

how these techniques can lead to very powerful and intuitive schemes for the modelling of

such elements with either constant or varying cross-sectIon. In addition. comprehensive

and efficient algorithms for the volumetrie calculations of boundary-represented and sweep

generated obJects that take into consideration the special properties of these representation

schemes are derived.

The following three sections attempt to summarize the prcvious research re

ported concerning the problems addressed here and highllght the contributions of the thesis

in solving these problems.

1.2 3D Modelling of Bevel Gears

The geometric characteristics and design parameters of bevel gears have been

studied extensively in the specialized literature (Dudley 1954: Dudley 1962: Sioane 1966).

The same cannot be said about the 3D modelling of these gears. In facto representing

the surface geometry of these widely used gears in a way suitable for computer modelling

and analysis is still a major research challenge. which has been addressed by very few

researchers (Huston and Coy 1981. 1982. Tsai and Chin 1987). In most of the current

work. the tooth-surface geometry of bevel gears is analyzed by expandlng on the planar

involute geometry which is widely used in spur-gear analysis and design. Huston and Coy

(1981) used planar involute geometry and introduced the idea of spindling a disk into a cone

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1 1. Introduction

to describe the tooth-surface geometry of logarithmic spiral bevel gears. They approximated

the tooth profile by describing it on the surface of a cone rather than on the surface of a

sphere The same researchers later expanded their analysls to describe the geometry of

circular-cut spiral bevel gears (Huston and Coy 1982). but limited thelr discussion to crown

gears. Tsal and Chin (1987) descnbed the tooth-surface geometry of bevel gears based on

a spherical involute generated by "unwrapping" the developable surface of a right cone. A

point on the unwrapped edge of the surface. describing a taut chord. traces out a trajectory

that Iles on a spherical surface. Their analysis uses a family of spherical involute curves

initiatmg from a straight hne or a spiral curve on the cone to describe straight or spiral

bevel gears. respectlvely.

The work presented hele introduces the exact spherical inllolute. derived from

the fundamental involute geometry. to describe the bevel-gear tooth profile on the surface of

a sphere. The objective is to use the generated profile to model the bevel-gear tooth surface.

The solid models of such gears are obtained by a simple extrusion of thelr descnbed tooth

profiles. This consists of radial extrusion for stralght bevel gears and further tWlstmg for

the case of spiral bevel gears The general approach can be adapted to produce the sohd

models of difTerent types of spiral bevel gears mcJudlng logarithmic and clrcular-cut ones.

Although the theoretical approach presented for the modelling of bevel-gear tooth surface

may not exactly match the actual manufactunng of bevel gears with current technology. it

provldes the foundations for the geometnc modellmg and computer analysis of this type of

gear.

During the course of this work. a representative of Gleason Works Company-a

world leader in the design and manufacturing of bevel gears-. was privately contacted

to acquire information about the state-of-the-art of bevel-gear CAD /CAM technology m

industry. The following remarks were issued:

- Unlike the spur- and helical-gear surface geometry. the bevel-gear surface geometry

is not weil understood and the industry is credited with most of the research and

literature made available in this field.

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1. Introduction

- Software packages exist for the design and analysis of different types of bevel gears

given their design parameters and the machine setting required for their manufactur

mg.

- The CAM of such gears is still in its infancy. although currently. it has been gaining

ground with prototypes of new packages a,>pearing in recent exhibits.

- Software packages for the 3D modelling and rendering of bevel gears are not currently

available.

Chapter 3 includes a detailed discussion on the terms and concepts introduced

in this section. along with some illustrative examples.

1.3 Volumetrie Properties of Boundary- Represented Solids

As mentloned above. the accurate and fast computation of the first three

moments-volume. vector flrst moment. and rnertia tensor-of regions bounded by 2D

contours and 3D surfaces emerged as an Important issue given the current advances of and

increased interest in the computer representatlon of solids. The calclliation of the above

mentioned volumetrie propertles for geometrically complex objects IS a key issue in the field

of CAD / CAM of mechanrcal elements A large domam of mechanical elements with com

plex shapes can be modelled usrng boundary representatlons schemes. Curved surfaces of

such shapes are approximated either by parameterized surface patehes or polygonal faces.

The focus here is on the technIques for calculating the volumetrie properties for 2D and

3D regions given the piecewise-linear approximation of their boundary. For a 2D region.

the boundary is approximated by a polygon: for a 3D region. by a polyhedron.

Several approaches have been reported to l-alculate the volumetrie properties of

solids given their boundary ~!1resentation. They ail evaluate these properties by surface

integration. using either direct integration or the Gauss Divergence Theorem (GDT) (Lee

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1 Introduction

and Requicha 1982-a.b). Methods resorting to the GOT hav<! been introduced (Messner and

Taylor 1980. Timmer and Stern 1980: Lien and Kajiya 1984. Ota. Aral and Tokumasu 1985.

Angeles et al 1988) whlch reduce the integrals of mterest to surface integrals Formulae for

the transformation of integrals deflning the volume and the tirst moment of sohd regions

bounded by closed surfaces. into surface integrais have been available for sorne tlme (Brand

1965) However. computer-onented algonthms maklng use of such formulae and extendmg

them to include integrals defming the mertla tensor of solid regions are rather recent. In facto

Messner and Taylor (1980) were among the f"st to apply the GDT to compute the moments

of solids polyhedra through surface integration. Surface integrals are evaluated numerically

through a quadrature rule whlch is exact only if applied to trlangular elements of the surface

polygons. It seems that Lien and Kajiya (1984) were the flrst to propose Simple formulae

based on the GDT that allow ::ln exact evaluation of the moments of arbltrary nonconvex

polyhedra. Although the formulae they propose are general enough, as to compute moments

of any order. they elre hmlted to polyhedra comprised of a single type of polygon Slnce no

expliclt general formulae are glven for evaluatlng surface integrations of arbitrary polygonal

faces. the method is not applicable to solids represented by a mixture of dlfferent, posslbly

nonconvex boundlng polygons. Ota. Aral and Tokumasu (1985) proposed more exphclt

formulae for sohd polyhedra employlng the GDT as weil These formulae. however, do not

supply ail components of the mertla tensor. but only ItS projection onto one glven aXIS,

excludlng products of mertla

Timmer and Stern (1980) computed the moments of solids bounded by surfaces

which are apprmtlmated by parametenzed patches rather than polygons. Usmg the GDT and

Green's Theorem. they reduced the problem to evaluating line integrais over the contours of

the surface patches The hne integrals are computed by approxlmate integration formulae.

These formulae provide the required moments, excludmg off-diagonal entries of the inertla

tensor-the products of inertia. Angeles et al (1988) proposed a method. based on the

GDT, for the computation of moments of planar and axially symmetric reglons glven the"

spline approximation of the boundary The method is general enough as to supply ail

components of the inertia tensor.

6

•

(

1. Introduction

Presented here is the praetieal implementation of general formulae. derived by

Angeles (1983). aimed at computing the first three moments of boundeé two- and three

dimensional regions The general formulae permit the computation of a volume mtegral.

defmed over a reglon Imbedded ln a Euchdean space of arbltrary dlmensic 1. via a reduced

Îlltegration on the boundary of the glven region For 3D regions. the method adopted here

reduces the resulting surface integrais to simple line mtegrals over the edges representing

the region. The desired line mtegrals are obtained by a repeated appltcation of the GDT

in the planes defmed by the surfaces forming the 3D boundary. Une integratlon IS then

introdueed to derive practlcal. simple formulae that provide the volumetrie properties for

the pleeewise-imearly approximated regiol1s of mterest. The devised formulae supply ail

component~ of the inertia tensor. includmg moments and prodL!ets of inertia. and they are

directly applicable to sol Ids represented by difTerent arbltrary polygonal faces.

Chapter 4 contains the detailed derivation of these formulae. along with some

numerieal examples.

1.4 Volumetrie Properties of Sweep-Generated Solids

Although eommereially available geometrie-modelling software includes the fa

cilities to ealculate the volumetrie properties for general sohds. they do not seem to exploit

the fact that many sohds are sweep generated. and hence. thelf volumetrie properties can

be derived from information on the 20 generatmg contour and the sweeping parameters.

General sweeping is a very powerful modelling seheme due to the vast number of solids

with complex shapes that can be model/ed using this intuitive and easy-to-visualize tech

nique (Mortenson 1985: Casale and Stanton 1985: Coquillart 1987). In its simplest form.

sweeping can be used to model many interesting shapes by just sweepmg a c.onstant cross

section of the shape along a prescribed line normal to the eross section Some variations of

this basic form. such as allowing the cross section to be scaled or twisted whde bemg swept

along a curve" can be used to model a large domain of objects. including many complex

meehanical elements.

7

l

1 Introduction

The volumetrie properties of solids represented by simple lranslational or rota

tional sweeping can be calculated by exploiting dimensional separability to convert volume

integrais lOto surface integrals (ReqUlcha 1982-a) Sohds generated by a more general

sweeping have not been accommodated and are consldered here The work introdueed in

this thesis addresses the problem of caleulating the volumetrlc propertles of objeets mod

elled by sweeplng a planar cross section along a normal or obhque line. Whllè allowing the

section to be transformed as It IS being swept We dertve relattons between the volumetric

properties of sweep-generated solids and those of thelr generatlng 20 cross section. to

gether with the sweeping parameters. whieh determines the transformation of this cross

section. From these general relations. we denve the special relations applicable to the most

common types of sweeplng. and i"ustrate the results with some numertcal examples

The method presented here for the caleulatlon of the volumetrie properties of

sweep-generated solids reduces drastlcally the amount of required computations by uSlOg

20 caleulatlons instead of direct 3 D calculations Additlonally. the results obtatned âre far

more accurate than those obtamed by direct caleulatlons applied on an approxlmate 30

model of a sweep-generated soltd

Chapter 5 deals wlth the Items presented in thls section and contams the detailed

derlvation of developed formulae.

Chapter 6 includes some concludlllg remarks. along with a discussion on the

limitations of the work presented here and suggestions for future research.

8

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Chapter 2 Solid Modelling Schemes

2.1 Introduction

Solid modelling is a relatively new field that has been developing since the early

1970·s. The need for powerful and practical CAO/CAM systems has been the driving

force behind the advancement in the field. As mentioned in Chapter 1. there are five

major unambiguous solld-modelling sehemes (Requicha 1980)-a representation is said

to be unambiguous If it defines a unique sohd. Algorithrr.s for calculating the volumetnc

properties of modelled objects are dependent on the charactenstlcs of the model/mg scheme.

ln general. computmg the volumetrie properties of solids requires the evaluation of volume

integrals of the form

Ik = ln Ik(r)dO (2.1)

where f is a kth-rank homogeneous tensor of order k m the position vedor r.

The vanous modelling schemes are mtroduced in this chapter. along with a brief

discussion on their adequacy to the modelling of mec.hanical elements and the calculation

of volumetrie properties. The reader is referred to Mortenson (1985). Requicha (1980) and

Lee and Requicha (1982-a.b) for a comprehensive and formaI treatment.

ln discussing the various modelhng schemes. certain properties are taken int(\

consideration. Requicha (1980) classifies these propert1es as formaI and informai proper

ties. The formaI properties are: domain. validity. completeness. and uniqueness. Domain

n -

2 Solid Modelling Schemes

is the set of objects representable by a scheme. Validity describes the ability of a scheme to

produce valid representation of objects. i.e. to avoid representatlons that do not correspond

to any solid. Completeness measures the abllity of a scheme to produce informative rep

resentations that can answer geometnc questions about the mode lied object Uniqueness

is the abllity to produce a unique representatlon of an object mdependent of the obJect' s

substructure or orientation. The less formai but equally important propertles illclude. con

ciseness. user frl€'1dliness and efficacy. Conciseness of a scheme represents the amount of

data required to represent an object. while user {riendlines5 measures the ease of creating

valid models using that scheme Efhcacy measures the adaptabliity of a scheme to different

applications and geometric analyses. Table 2 1 provides a summary the mentloned formai

and informai properties for each of the modelling schemes discused next.

Propertle<

Primitive Instancmg r 1 GIF ! G G G F

Cel! DecompûsltlOn (. P G 1 P P F P

Sp3tlal Enumeratlür F (. 1 G G P 1 P F

C, GIG P Gle p l------------f

Boundary R ep' f, P G F ~P F l----~--'----- -- ~ ---~--1 --j---

CSG

Simple Sweepmg f' (. C; P G: G P ------------r-~--~, General Sw", .. pm~ (, P G pp! P,F

Table 2.1 Properttes of the major modelllng schemes Key G=Good. F=Fair. P=Poor

2.2 Primitive Instancing

This representation scheme relies on the concept of grouping objects that can

be de'icribed by a few parameters !nto families of similar shapes called generic primitives. A

range of objects. called primitive Instances. within a iamily can be mo.delled by varying the

defining parameters of that family. Figure 2.1 illustrates the concept of modelling different

10

t .

c

f

(


instances of a Z section. Pure primitive instancmg does not provide the ability to combine

difTerent instances to present new and more complex objects. The scheme. however. is

unamblguous. unique. easy to validate. concise. and user frrendly.

Figure 2.1 Primitive Instancing ladapted from Morte:lson (1985). p. 448 Fig 1015)

The domain of thls scheme IS limited to the number of familles it represents.

The notion of familles. though. is best suited for the design and mode"mg of standardized

mechanrcal elements that can be described by a finite number of parameters. To calculate

the volumetrre properties of objects represented by this scheme. special formulae must be

developed for each primitive. no uniform general treatment IS possible. This task becomes

more difflcult as the number and complexity of primitives Increases. Consequently. only a

small number of parametrized familles of relatlvely simple objects can be accommodated.

2.3 Cell Decomposition and Spatial Occupancy Enumeration

An object can be represented by decomposing It mto cells (see Fig. 2.2). Each

cell is considered to be easler to represent than the original object. In cell decompositions

schemes. a solid is represented as a union of disjoint cells

5 = U Cellt (2.2)

11


these cells need not be identical. Such representation schemes are unambiguous but not

unique and their validity is hard to establish. For artifacts such as mechanical elements.

these schemes are not concise and representmg elements wlth curved surfaces is not easy.

Figure 2.2 Cel! Decomposition (adapted from lv1ortenson (1985). p 451 Fig 1019)

Spatial occupancy enumeration is a special case of cell decompositions where

cells are cubical and he in a flxed spatial grid. This scheme 15 unambiguous and unique. An

array of cublcal cells is easy to valldate but is not concise. It is best suited fvr representing

objects with highly irregular shapes such as those occumng ln nature.

The volumetrie properties of obJects represented by cell decomposition or spatial

occupancy enumeratlon can be calculated by evaluatmg volume integrais over the disjoint

cells and summmg up the results. ThiS can be done directly for cells wlth Simple shapes but

numerical integration methods must be used in the case of more complex on es (O'Leary

1980).

2.4 Constructive Solid Geometry

.. Constructive solid geometry (CSG) schemes represent comp'lex solids as boolean

combinations of solid primitives of simple shapes. Fig. 2.3. A solid is represented with a

12

(

2 Solid Modelllng Schemes

binary tree that has boolean operations such as union. intersection or difference at its

nonterminal nodes and sol id prim;tives or 3D transformations al its leaves. CSG schemes

are unambiguous but not unique. Th~;r domam depends on the set of primitives available

for the scheme along witl! the possible transformations and combinations applicable to

these primitives. These schemes can be very concise if thelr primitives reflect the domain

of objects to be represented. They are easy to use for unsculptured mechanical elements.

CSG's main drawback is that obtaining the geometric data of objects represented with this

scheme is hlghly mefficlent.

1 1

~----------------------------------------~

Figure 2.3 Constructive Solid Geometry. (adapted from Mortenson (1985). p 462. Fig 1028)

A natural way to calculate the volumetric properties of an object represented by

a CSG scheme is addmg and subtracting volume integrals over the primitives. making up the

object. and the intersectIons of these primitives. The number of integrals to be evaluated

grows exponentially. m the worst case. as the number of primitives and their mteractions

increases. Moreover. numerieal caneellations occurring rn the case (,If greatly overlapped

primitives lead to inaccurate calculations To overcome these drawbacks for such widely

used representatlon schemes. other techniques for the volumetrie ealculations are eonsidered

(Lee and Requicha 1982-b). This includes conversion algorithms to .other representation

schemes such as boundary representations for which the volumetrie properties can be

13

2. Solid Modelling Schemes

computed more accurately.

2.5 Boundary Representations

A solid is represented by segmenting its boundary into nonoverlapping faces.

each in turn is represented by a set of bounding edges (see Fig. 2.4). The faces mak

ing up the surface of a solid must satisfy certain conditions. these are described in Re

quicha (1980). Polyhedral objects can be represented directly with boundary representa

tions schemes where the faces are made ur of polygons Objects with sculptured surfaces

are represented by using surface patehes or polygona! ; pproximation of the surface. These

schemes have a vast domain and are generally unamblguous. They are complete with a

wealth of geometric data readily aVdilable about the represented obJects. This data is es

sential for su ch areas as graphie display and interac:tion. and volumetrie calculations. The

validity of boundary representations schemes is not a trivial Issue and It IS the subJect of

interesting research Approximatmg curved surfaces with parametnc surface patches or

polygonal faces enables this scheme to represent a wide range of mechanical eiements with

complex. sculptured shapes. This. however. can be quite verbose and computer assistance

must be provided to construct such surfaces.

Figure 2.4 Boundary Representations (adapted from Lee and Requicha (1982))

14

(

(

L... _____ _


The volumetrie properties of objects modelled using boundary representations

schemes are evaluated by surface integration, using direct integration or thp GDT. The

more attractive approach is to use the GDT since direct integratlon is not always possible.

For polyhedral objects, this method is readily applicable Curved obJects are accommodated

b) approxlmatmg their surfaces wlth polygonal fdces or by approximate integratlon over

surface patches. The thesis deals with the former approach and reduces the problem to

line integration via a repeated application of the GDT This has been introduced in Chapter

1 and will be the subJect of Chapter 4.

2.6 Sweep Representations

Sweeping schemes are based on the simple notion of moving a surface along

a path to generate a solid ln general sweeping, the surface is allowed to be deformed as

it moves along a 3D curve. This concept of modelling solids IS easy to use and visuallze

yet it is the least understood, mathematically. of ail the other schemes Conditions for

creating valid representations are unknown, invalid obJects can be easily created as shown

in Fig. 2.5

Figure 2.5 Nonhomogeneous surface generated by translatlng a 2D curve (adapted trom Mortenson (1985). p 457. Fig 1023)

Translational and rotational sweepl'lg are weil known and are widely used for

modelling of constant cross-section and axially symmetnc turned mechanical elements. re

spectively. The domain of such simple sweeping techniques is limited although general

sweeping techniques can be devised to accommodate a '.vide range of objects. The volu

metrie properties .of objects generated by translational and rotationa~ sweeping (Fig 2.6)

can be calculated by reducing the volume integrals to surface integrals over the generating

15

r---~~~------------------

2 Solid Modelling Sc hem es

planar cross section. The surface integrals introduced can be further reduced to line inte

grals over the edges representing the crOS5 section. Methcds for calculating the volumetrie

properties of objects represented by more general sweeping (Fig. 2.7) are not known.

SwtPI \01,d

Ge".r.l0r .. ,flCe

, 1 AIIII of rrvo1utlon

Figure 2.€ Translational and rotational sweeping (adapted from Mortenson (1985). p 456 Fig 1022)

Constant crOS$-lKtton (gener,tOf CUry.)

Figure 2.7 General sweeping along a 3D curve (adapted from Mortenson (1985). P 456. Fig 10.26)

As mentioned in Chapter 1. the evaluation of volumetric properties of obJects

represented with sweeping a planar cross section along a line while allowmg It to be trans

formed as it moves is addressed in this thesis. This will be the subject of Chapter 5

The sweepmg technique represented here is general enough to repre~ent many interesting

mechanical elemcnts wlth sculptured surfaces.

16

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Chapter 3 3D Modelling of Bevel Gears

3.1 Introduction

The {jrst mathematical investigation into possible gear

tooth curves is credited to de la Hire in France who published a

treatise on gear design in 1694. As a result of his researc·'7 he

concluded that the involute curve had considerable advantages over

other shapes and his pioneering work was confirmed by the great

Swiss mathematician Leonard Euler in the eighteenth century. Euler

studied involute gear geometry extensively and demonstrated math

ematically the superiority of invo/ute tooth form. These fmdings.

however. were not put into practice for more than a century.

Watson. H.J.. 1970. Modern Gear Production.

As mentioned in Chapter 1. the 3D modelling of bevel gears has received little

attention. The work presented here relies on the fundamental involute geometry to define

the exact profile of such gears on the surface of a sphere. Once a profile is obtained. the 3D

model of its corresponding bevel gear is generated by applying special sweepmg techniques

on this profile. The purpose here is to introduce a methodology for the 3D modelling of

be"l'I gears rather than the development of a comprehensive design package.

• 3. 3D Modelling of Bevel Gears

The reader is referred to Appendix A for a quick reference on the bevel-gear

terminology and geometry. For a thorough reference. the reader is referred to Dudly (1962).

3.2 . The Exact Spherical Involute

The spherical involute is the 3D counterpart of the familiar planar involute of a

circle. The well-known planar Involute of a wcle (Sloane 1966) can be deflned as the curve

traced by a point on a taut chord which unwraps from the circle. The circle is called the

base circle of the involute. The sphencal involute can be defmed simllarly wlth the base

circle now Iying on a sphere Contlary to Tsal and Chin (1987). we deflne the spherical

involute as the curve. on a sphere. traced by a point on an arc of a great Clrcle. which

unwraps from the base circle. This is illustrated ln Fig 3 1. where P 15 a typlcal pOint on

the sphencal involute and T P IS the mentioned arc which unwraps From the base Clrcle. r--

Note that. the plane defmed by the great circle containing T P is perpendicular to the pla'le

deflned by the great circle containing OT. The above definition. based on the fundamental

involute geometry. glves the exact spherical ;nvolute traced on a spher::... the term exact

meant to distinguish It from previously defined spherical involutes

Similar to the planar Involute. the parametnc equatlons deflnlng the spherical

involute can be found by consldenng the ~phencal right triangle DT P in Fig 3~. along

with the fact that the arc of great wcle T P IS equal to the arc of the base circle TQ.

Figure 3.2 is slmllar to Fig 3 1 showing only the sphencal triangle OT P for clarity. The

base circle IS defined as the intersection of a sphere. of radius r. with a right cone of half

angle "t having its apex at the center of the sphere. Slnce the lengths of the arcs of great ,....., ,.....,

circles are usually designated as angles. arcs T P. OT. and OP are represented by angles

0:. "t. and 6. respectively.

Referring to Fig. 3.2. angle {3 is defined as (J

represented as ,..... TQ = r{Jsin')'

,...., 4> + t/;. and arc TQ can be

(3.1)

18

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(

3. 3D Modelling of Bcvcl Gears

o

Figure 3.1 Generating the exact spherica/ invo/ute 1 by unwrapping the arc of great "...,

circle T P from the base circle

slnce the radius of the base circle is equal to r sin ,. Equating arc TQ to the arc of great '"""' ,.....

circle T P (given as TP= Ta). we derive an expression for angle Q.

Q = .Bsin, (3.2)

By applying the laws of cosines and sines (Gasson 1983) to the right spherical triangle

OTP. we can write the following relations:

COS Q = cos ,cos h + sin 1 sin {) cos t/>

cos {) = cos a cos "Y + sin Ct sin 1 cos 90°

= cos Ct cos "Y . . . c _ Sin a . 900 _ sm Ct

sm v - . '" sm -. Â.. sln~ sln~

Substitutmg for cos fJ and sin fJ in eq. (3.3) we derive

cos 0: = COS21 cos Ct + sin 1 sin 0: cot t/>

(3.3)

(3.4)

(3.5)

(3.6)

19

3. 3D Modelling of Bevel Gears

a

Figure 3.2 The exact spherical involute 1 generated on a sphere of radius r

By rearranging eq. (3.6) and collecting terms we obtain an expression for angle Ct in terms

of angles 4> and ï as follows.

tan a = sin Î tan 4> (3.7)

Using eq. (3.2) to substitute for angle 0:' in eq. (3 7) yields

tan(;1sinï) =sinïtan4> (3.8)

Equation (3.8) is the basic equation describing the spherical involute.

The two parametric equations defining the spherical involute can be derived

from eqs. (3.4. 3.5 and 3.8). keeping in mind that (J == V; + 4> and a = (3 sin 1. namely.

V; = tan-l(t~nfbsinl) - 4> sm r

1: tanl( V; + 4» sin ï] tan v = --,-,--:--:---..:.--:--;--=-

sin(4)) cosh)

(3.9)

(3.10)

The right-hand side of eq.(3.9) is defined as the exact spherical invo/ute of 4>. denoted by

20

J (

c

-


12r-----r---r---,...----r--r-~-____,

ID

8

î 6

4

2

ID 20 JO 40 50 60

~ (deg)

Figure 3.3 The exact spherical in volute function

tf; = spinv( 4». Figure 3.3 shows plots of tf; versus cP. for various values of "'Y. Note that

eq. (3.10) defines angle {; uniquely. where 0 ::; 6 :S 7r.

Any point on the spherical involute is uniquely defined by specifying an angle

tf; > O. To determine the coordinate of such a point. angle b must be found for the given

angle t/J. This 15 do ne by first solving for rp in terms of t/J and then substitut;ng both angles

in eq. 3.10 to find 8. To obtain the angie 4> corresponding to a given angle t/J. one should

find the inverse relation, 4> = spinv-1(tf;). This is achleved by finding the root of the

function /(cP). defined below. for given t/J and 8.

f (4)) = tan[( t/J + 4» sin ,] - tan 4> sin "'Y (3.11)

Finding the root 4>. for given t/J and " is done numerically using the Newton-Rhapson

method as follows

(3.12)

21


where

(3.13)

3.3 Generating the Involute Bevel-Gear Profile

The involute bevel-gear profile is generated on the surface of a sphere from

the exact spherical involute described in Section 3.2. This sphere is called the transverse

sphere in analogy to the transl/'erse plane in which the profile of a spur gear lies when it is

generated from the planar involute of a circle.

Figure 3.4 IS a view of the transverse sphere looking along the pitch element

of a pmion-and-gear train, lines in the figure being understood to be arcs of great circ/es.

The pitch element is the intersection of the two pitch cones of the set. which have their

common apex at the center of the sphere. The figure shows the intersection circ/es of the

pitch and base cones with the transverse sphere. these wcles bemg cal/ed the pltch and

base circ/es. respectively. Points 01 and 02 are the traces of the axes of the gear and

plnlon. respectively. The spherical mvolute IS generated. on the transverse sphere. between

a point Q on the base circle and a pOint on the intersection wcle of the face cone with the

transverse sphere For a glven angle 1/J. angle </J IS flrst solved for as outlined in Section 3.2.

and then angle 8 IS obtamed uSlng eq.(3 10). Hence. by incrementing angle '!/J. successive

values for angle b. representing the arc of great circle 0lP, are obtained. with b varying

from the base cone angle lb to the face cone angle 10' '1 he successive values of o. along

with their corresponding values of t/J. define the exact spherical involute.

Figure 3.5 shows a coordinate system defined on the base cone of a gear. The

(x, y, z) coordinates of any point on the tooth profile defined by the spherical involute

shown are given in terms of t/J and 6 as.

x = rsinocost/J

y = r sin 0 sin t/J (3.14)

z = rcoso

22

•

(

- -- - -------~----------------


Figure 3.4 Projection of curves on the transverse sphere.

where.

The above formulae give the involute profile at a cone distance T. i.e .. on the

transverse sphere of radius r. Taking r = Ao. the outer cone distance of a pinion-and

gear train. one can oLtain the exact spherical in volute describing the tooth profile on the

outer transverse sphere of radius Ao. Mirror-imaging the spherical involute obtained for

a tooth about the tooth centerline will give the symmetrical opposed s~herical involute

that completes the tooth profile. To achieve that. the angle 0 subtended by the circular

thickness of the tooth at any point P of the calculated spherica! involute is found from the

angular thickness Op and the involute angle tbp of the tooth profile at the pitch circle. The

circular thickness t and the pressure angle <Pp at the pitch circle on the transverse sphere of

radius Ao are given as design parameters for a specifie gear train. The angular thickness

23


y

Figure 3.5 The base co ne and the generated spherical involute

Op is then calculated for a given t as

o = t p Ao sin "1

(3.15)

1 being the pitch angle. while the involute angle t/;p is calculated from eq.(3.9) by substi

tuting the specified pressure angle cPp for angle 4>.

Referring to Fig. 3.6. angle 0 is given as.

0= Op - 2(t/I - tPp) (3.16)

Consequently. the (x, y, z) coardinates of the opposed symmetrical spherical involute com

pleting the tooth profile are given by

x = Ao sin {) cos(t/; + 0)

y = Ao sin {) sin(tb + 0)

z = Aocosh

(3.17)

24

c

('

(

where

-- --------------~-----


Outer circle

Figure 3.6 Projection of transverse sphere showing two opposing spherical involutes

lb $ 6 $ 10

The tooth profiles for the whole gear are then generated by a series of rotations

in the xy plane of the previously found tooth profile. The angle of rotation a which maps

the kth tooth profile into the k + 1st tooth profile is found using the gear train circular

pitch p. namely.

a= p Ao sin ,

The rotation is repeated for N number of teeth to geilerate the whole profile.

[

Xk+l] [cosa Yk+l = sma Zk+l 0

- sin (J

cosa o

3.4 Solid Modelling of Bevel Gears

k=l, ... ,N-l

(3.18)

(3.19)

The bevel-gear solid model can be obtained once the tooth profiles describing

the gear on a traverse spnere are produced (such profiles. for a pinion-and-gear train are

shown in Fig. 3.7). To illustra te that. consider several concentric transverse spheres. each

located at a difTerent radius from the common apex of the base and pitch cones of a

25

1

-: . .....


bevel-gear train. The tooth profiles on each concentric sphere are constructed as spherical

involute curves. The shape of the loci of points connecting a point on a tooth from one

sphere to the other determines the shape of the surface of that tooth. For a straight bevel

gear. the loci of points connecting two spheres is a straight line. whereas for a spiral bevel

gear it is curved as a spiral line originating at the apex of the cones (see Fig. 3.8).

Figlne 3.7 Spherical profiles of a pinion-and-gear train shown on a transverse sphere (adapted from Merritt (1946). p 57. Fig 5.18)

Figure 3.8 Obtaining a spiral bevel gear from a straight one (adapted trom Sioane (1966). p_ 202. Fig. 216)

Straight bevel gears can be thought of as the bevel gear counterparts of spur

gears. The latter are constructed by considering several parallel transverse planes. each

located at a different distance on the axis ofthe base or pitch cylinders of a parallel-axis gear

train. The tooth profiles on each parallel plane are constructed as planar circular involute

curves. Connecting a point on a tooth from one plane to the other by a straight line provides

26

f

(


the shape of a spur-gear tooth surface. Rotating the transverse planar sections forming

a spur gear relative to one another about the axis of rotation of the gear will produce a

corresponding hellcal gear. The same idea is used to produce the spiral bevel gears which

are the bevel gear counterparts of the he!ical gears. Rotating the transverse spheri<.al

sections of the straight bevel gear relative to one another about the axis of rotation of the

gear will produce the corresponding spiral bevel gear. By controlling the angle of rotation

of one spherical section relative to another. one can control the shape of the spiral bevel

gear tooth.

The ideas discussed above are used to obtain the solid models of straight and

spiral bevel gears. For a given gear. with an outer cone distance Ao. th.a tooth profiles

are generated on the transverse sphere surface of radius A 1. as described in Section 3.3.

A simple radial extrusion of the profiles from the outer radius Ao to the inner radius At

produces the solid model of the corresponding straight bevel gear. For such a gear. the

tooth centerline on the pitch plane is a radial stralght line. Rotating the profile while it is

being extruded produces the solid model of the corresponding spiral bevel geai. The type

of spiral bevel g~ar produced depends on the change in the angle of rotation w as a functlon

of the radial distance r while the profile is being extruded. i.e ..

w = F(r) (3.20)

where

The tooth centerline on the pitch plane will follow a specifie spiral depending on the function

F. For example. for a logarithmic spiral. in which the spiral angle is fixed at ail points along

the centerline. function F( r) will have the form (T sai and Chin 1987)

w = F(r) = tan("') In(T / Am} ~in(Qp)

(3.21)

where '" is the median spiral angle. Am is the median cone distance and O:p is the pitch

cone angle.

27


A software package. BEVEL. hi)S been developed for the 3D modelling of bevel

gears. The gears are modelled in pairs of pinion-and-gear trains. given the following design

parameters:

Number of pinion teeth: Np

Number of gear teeth: Na

Diametral pitch: Pd

Face width: F

Working depth: hk

Whole depth: ht

Gear addendum: aa

Pressure angle: <Pp

The shaft angle E is taken to be 90° which is the case for most commonly used

bevel gears. The bevel-gear nomenclature is adopted from (Dudley 1962). a book to which

we refer the reader for a description of the design parameters.

The type of bevel gear deslred can be specifled as stralght. logarithmic spiral.

circular-cut spiral or involute spiral. The design parameters are used to produce the profile

of the gear or pinion on the outer transverse sphere. As described before. the gear solid

model is obtained by sweeping that spherieal profile. The model is displayed and its

volumetrie properties ar«; calculated using its piecewise-linear approximation.

3.5 Examples and Results

The approach proposed here is used to model straight and logarithmic !.pirtll

bevel gears. Table 3.1 contains the design parameters for a pinion-and-gear train used to

procfuce the gear models of the spccified train. Note that English units are used because

the diametral pitch Pd is specified as a standard design parameter1. in in- l .

1 The equivalent parameters in SI units are F = 38.10mm. hk = 20.32mm. kt = 21.59mm and

aa = 10.16mm

28

•

J (


Np Ne Pd F hk ht aC ljJp \II

(in) (in) (in) (in)

16 20 2 1.50 0.80 0.85 0.40 20° 35°

Table 3.1 Pinion-and-gear train design parameters.

The gear profile on the outer transverse sphere is first obtained as outlined in

Section 3.3. Figures 3.9. 3.10 and 3.11 show a comparison between the tooth profile

described by the exact spherical involute proposed here and the model proposed in Tsai

and Chin (1987). The spherical profile. shown in Fig. 3.12. is then swept to generate the

solid models of both the straight and spiral bevel gears. Figura 3.13 shows the solid model

obtained for the straight bevel gear. while Fig. 3.14 shows the model for the spiral bevel

gear. For the straight bevel gea~. the profile of Fig. 3.12 is extruded radially for a distance

equal te the gear face wldth F For the spiral bevel gear. the profile is rotated while it is

being extruded for the same distance. The angle of rotation is controlled so that the tooth

center line forms a logarithmic spiral with a 35.0° medlan spiral angle.

The volumetrie properties for the model/ed straight2 and spiral3 bevel gears are

readily computed using their piecewise-linear approximation (Refer to Chapter 4).

Straight bevel gear:

Volume(m3):

Centroid eoordinates(in):

Moments of inertia (in5):

Products of inertia (in5):

73.527

x = 0.0. fi = 0.0. z = 4.0569

Ixx = 1.6556 x 103.

Iyy = 1.6556 x 103 .

/zz = 8.8013 x 102

/xy = 14.809. I xz = 0.0. Iyz = 0.0

2 The equivalent volumetrie properties in SI units are V = 1.2049 x 106mm 3. Z = 1.0305 x

102mm. Ixx = Iyy = 1 7503 x 10100101

5. /zz = 9 3050 x 109mm 5. Ix" = 1 4801 x 108mm5

3 The equivalent volumetrie properties in SI units are. V = 1 2057 x 106mm 3. Z = 1.0304 x

102mm. Ixx = Iyy = 1.7515 x 1010mm5. I zz = 9.3153 x 109mm5. lxy = 1.4801 x 108mm5

29

~ ,!~ , 'f)} t' '~

Spiral bevel gear:

Volume(in3):

Centroid coordinates(in):

Moments of inertia (in5):

Products of inertia (in5):

5.3

5.25

5.2

5.15

..§. 5.1

1l 50s .. c .,;;

8 5 u ><

4.95

4.9

485

4.8 15

73.577

x = 0.0. fi = 0.0. z = 4.0566

lxx = 1.6567 x 103.

lyy = 1.6567 x 103.

lzz = 8.8111 x 102

. 3 3D Modelling of Bevel Gears

lxy = 14.809. lxz = 0.0. lyz = 0.0

~ _______ Tsal and Chm model

_- Proposed model

20 25 30 )5 40 45

{J (deg)

Figure 3.9 Comparison of results in the X axis

The volume and centroidal location of the straight and spiral gears should be

identical. since the former is obtained by subjecting the latter to an isochoric mapping.

However. the results obtained show a relative error of the order of 10-3. This is due to the

error introduced by the linear approximation of the twisted surface of the spiral gear mode!.

Morl:over. the gears are found to have identical inertia tensors within a relative error of the

order of 10-3. This result was expected. since the generating tooth profile of the gears

has an inertial axial symmetry.

30

c

l

! .. c ~ 8 v ;...

;!

~ c

1 u ~

3 3D Modelling of Bevel Gears

O.l5r-----,,----"-T----r----r·----r---.,

0.3

0.2~

0.2

O.I~

01

0.05

.------_. Tsai and Chin model _--- Proposed model

O~-~--~--~--~--~--~ 15 20 2S JO lS 40

(1 (deg)

Figure 3.10 Comparison of results in the Y axis

4.3

4.2

4.1

4

39

3.8

3.7

3.6 IS 20 25

.-------- Tsai ilnd Chin model ---- Proposed model

JO 35 40

Il (deg)

Figure 3.11 Comparison of results in the Z axis

45

45

31


~----------------------------------------------------------------~

Figure 3.12 The sphericill profIle us~d to mode! the gears

32

-------------- ---


c

(

'------------------------ --- ----------'

Figure 3.13 The sltaight bevel geat model

33


Figure 3.14 The spiral bevel gear model

34

•

{

Chapter 4 Volumetrie Properties of Boundary-Represented Solids

4.1 Introduction

Through measurements made upon the horizontal mo

tion of bodies moving with ve/ocities acquired by falling down an

inclined plane. Galileo notieed a phenomena which he described in

his "Diseourses".

"but ln the hOflzontal plane GH Its [the mov

ing body's] equable motion. according to its

velocity as aequired in the deseent from A to

B. will be eontinued ad tnfmztum"

Galileo (1564-1642)

ln this statement we find the nue/eus of the concept inertia.

Edwards. H.W .. 1933. Ana/ytie and Veetor Mechanics.

As mentioned in Chapter 1. the GOT is applied to reduee the problem of eal

culating the volumetrie propertles of solids to hne integration. The solids are represented

by their closed surfaces which are approximated by polygonal faces Reducing the problem

to li ne integration results in readily implementable formulae that are general enough as to

calculate moments of any order for solids represented by different arbitrary polygonal faces.

•

l

4. Volumetrie Properties of Boundary-Represented Solids

4.2 General Transformation Formulae

let [V be a lI-dimen~ional Euclidean space. in which a bounded region il is

imbedded. The formulae presented here are valid for Euclidean spaces of arbitrary (finite)

dimension. A few defmitions are introduced first.

The kth moment of il is defined as the following integral:

(4.1 )

where fk(r) is a homogeneous function of kth degree of the position vector r. It is.

moreover. a tensor of the kth rank. and hence 50 is h. The most familiar moments are the

first three defined for k = 0,1 and 2. The zeroth moment ]0 is slmply the volume of n. the first moment Il is the flrst-rank tensor. I.e .. the vector producing the position vector i

of the centroid of n by

i = ]t!]o (4.2)

The second moment of n. h. a second-rank tensor. is the inertia tensor of il.

Now let 4»m(r) be an mth rank tensor functlon of r. div4»m denoting its diver

gence. Furthermore. let (.) denote the inner product of the ten50r quantlties operands. The

GDT states the following relationship:

r div4»mdn = j' 4»m' ndan Jn an (4.3)

where an and n den ote the boundary of il and the outward unit normal of thi~ boundary.

respectively. If the function /k(r). whose integral is to be computed. is the divergence of

the function 4»m(r). then the integral reduces to an integral on an by application of the

GDT. The relation between k and mis. clearly. m = k + 1. i.e .. 4»m(r) is a tensor of

a rank hlgher than that of fk. the difference between both ranks being Unit Y However.

finding a function 4»m(r) whose divergence be a given function Idr) can be. m general.

a more difficult task than computing the volume integral of Idr) directly. Nevertheless.

36

..

(


the computation of the moments of regions. particularly the first three ones. involves the

derivation of cJ)m(r) functions that can be readily obtained. as described next. In tact. let

VII. qe and le denote the volume. the (vector) first moment and the (second-rank tensor)

second moment of n. the last two on es being taken with respect to a given point O. The

computation of these quantities can be reduced to that of integrals on an by application

of the GDT. as follows (Angeles 1983):

VII =! f r.ndan v Jan

q~ =-21 f (r. r)ndan = _1_ f {(r· n)d8n

Jan 1 + v" Gn

10 3 1

I~ = r· rI ( ) 1(r· n) - -2r ® nJdan an 2 v + 2

(4.4a)

(4.4b)

(4.4c)

where 1 denotes the identlty second-rank tensor and the symbol ® the tensor product of

Its two tensor operands. Included ln Appendix B is a section on tensor notation. In the

next sections. the formulae (4 4a-c) are applied to two- and three-dimensional regions.

under a piecewise-linear approximation of their boundaries. It is pointed out that qe can

be computed by two alternative formulae. as ln eq.(4 4b). Moreover. the frrst is dimension

invariant. whereas the second IS more sUltable for applications involving piecewise-linear

approximation. due to the simple forms which the r . n term produces in such cases. Both

formulae will prove to be useful in derrving practical simple formulae for our purposes.

4.3 Two-Dimensional Regions

For planar regions l/ = 2. the formulae (4.4a-c) reduce to

V2 =~ lan r· ndan

q? =-21 f (r. r)ndan = ~ r r(r· n)dan

Jan 3 Jan o i 3 1 12 = r· r[-l(r· n) - -r ® nJd8n

an 8 2

(4.5a)

(4.5b)

(4.5c)

Since 1I2 reduces ta the area of a planar region n. eq (4 Sa) is a restatement

of Green's formula. Explicit formulae are now derived which apply to a piecewise-linear

approximation of the boundary.

37

a 4. Volumetrie Properties of Boundary-Represented Solids

Figure 4.1 Une segment representing one side of a polygon approximating a planar closed curve

If an in eqs.(4.5a-c) is approximated by a closed n-sided polygon. then

(4.6)

an1 denoting the ith side of the polygon. Thus. the above formulae can be approximated

by

(4.7a)

(4.7b)

(4_7c)

the ith side of the approximating polygon.

Furthermore. let V21' Q2, and 121 be the contributions of the ith side an" of

the polygon to the corresponding integral. and let St and ft denoting the length and the

position vector of its centroid. as shown in Fig. 4.1. The following relations are readily

obtained:

38

•

(

- -------------------~----------------


(4.8)

Using each of the two formulae of eq.(4.7b). two alternate expressions for q~

are obtained:

(4.9a)

(4.9b)

Subtracting twice both sides of eq (4.9a) from thrice both sides of eq.(4.9b)

yields

(4.10)

The right-hand side of eq.(4.10) is readily recognized to be the projection onto

-nt of the second moment of segment an, with respect to O. represented here as I~.

Thus.

(4.11)

Furthermore.

o i 3 1 3 1 12 = r· r[-(r· n )1- -r ® n ]dan = -l(n·. y) - -y' ® n t 8 t 2 ' , 8" 2' , an,

(4.12a)

with y, defined as

(4.12b)

Now express vector r which appears in the integrand of eq.(4.12b) as

(4.12c)

where r, and rHl denote the position vectors of the end points of ani' Thus. the foregoing

vectors are the position vectors Of the ith and the (i + 1 )st vertices of the approximating

39


polygon. which are assumed to be numbered in counterdockwise order. Moreover. since

the polygon is closed. for t = n. i = imodn in eq.(4.12c) and in what follows. Substitution

of eq.(4.12c) into eq.(4.12b) yields. for i = 1, ... , n:

2 St [( 1 St) ( 1 3 2) ]

V t =3 '1' r,+1 + 2r1 . rt + "4 rt + 2rt' rHl - 2rt . rt + 4"S' 'Hl (4.12d)

Formulae (4.8). (4.11) and (4.12a-d) are the desired relations.

4.4 Three-DimensÎonal Regions

Similar formulae for solid regions are now presented. By setting v = 3 in the

general relations (4.4a-c). they reduce to:

V3 =~ hn r . ndan (4.13a)

qr =-21

{ (r· r)ndan =: { r(r· n)d8n Jan 4 Jan

(4.13b)

o ( 3 1 13 = Jan r· r[10 1(r. n) - 2r ® n]dan (4.13c)

Next. explicit formulae are presented that are applicable to piecewise-linear ap

proximations of boundaries of solids of arbitrary shapes.

One simple approximation of the boundary can be obtained by means of a

polyhedron formed by polygonal faces. The integrals appeanng in eqs.(4.13a -cl thus can

be expressed as sums of integrals over the polyhedral faces. the whole boundary an thus

being approximated as

(4.14) 1

which is similar to the approximation appearing in eq.(6). except that now each part an,

is a polygonal portion of a plane The integral formulae can thus be approximated as:

(4.15a)

40

,(

(

.

- -- ---~------------_.-------- - --------.--._-------------------.--- -- ----------------------

4 Volumetrie Properties of Boundary-P .ented Solids

(4.15b)

(4.15c)

Now. let V3l' q~ and I~ be the contribution of ith face. an,. of the polyhedron.

to the corresponding integral. Â%. f 1 and Ig being the area. the position vector of the t

centroid and the inertia tensor of the polygon an! respectively. the two last quantities

being taken with respect to O. Then.

(4.16)

The polygon area fJ.z and its centroid f l are calculated as outlined for 20 regions

in a plane defined by the polygon.

By subtracting twice the second part of (4.15b) from the first one. the following

relation I~ readily obtained:

(4.17)

The second integral of eq. (4.17) is readily identified as I~ -the second moment l

of polygon an,. Hence.

(4.18)

which is a relation similar to that represented by eq.(4.11). i.e .. the contribution of an, to

the first moment of n is recognized to be the projection onto - ~n! of the second moment

of an,. both moments being taken. of course. with respect to the same point O. The

centroidal inertia tensOi of the polygon al hand is caJculated using formulae developed for

41

-

-, 1


2D regions in a plane defined by the polygon. Using the parallel axes theorem and a rotation

of axes. I~ is then found from the calculated centroidal inertia tensor. ,

Additionally. one has. for the contribution of the ith polygonal face to the second

moment of the 3D region under study.

(4.19a)

with w t defined as:

(4.19b)

The integral appearing in eq.(4.19b) is evaluated next. To this end. ris expressed

as:

(4.20)

where p is a vector Iying in the plane of the polygon ant and is originating from its centroid.

as shown in Fig. 4.2. Now w t becomes

where an exponent k over a vector quantity indicates the kth power of the magnitude of

the said vector (r2 == r· r = IlrI12). a notation that will be used in the following discussion

Three surface integrals over ant need to be evaluated in the expression for w,. namely:

(4.22a)

(4.22b)

(4.22c)

42

c

(


Figure 4.2 Polygon representing one face of a polyhedron approximating a closed surface

Since p is a vector Iying entirely ln the plane I1t defined by the polygon ant • It

can be represented uniquely in the 2D subspace as a 20 vector of that plane. Conc;equently.

the GOT can be applied in this 20 subspace to reduce the surface integrals (4.22a-c) to

the following hne integrals (refer to the Appendix for a proof of these relations):

At = ~ 1 div(p 0 p @ p)dan1 = :. ! p ® p(p. "l)drl (4.23a) an1 4 ft

at = ~ 1 dIV(p2 p)danl = ~ l p2(p. "t)drt (4.23b) a[Jt 4 r?

bl = ~ 1 div(p2 p ® p)danl = ~ i p2p(p. iil)d~ (4.23c) anl 5 fI

where rI denotes the polygonal boundary of an?. nt denoting the outward unit normal

vector of this boundary that is contained in plane nt. Note that. similar to applying

the GOT in 3D to reduce volume integrals to surface integrals. the GOT is used in the

mentioned 20 subspace to reduce surface integrals to li ne mtegrals.

Now. let fI k denote the kth side of the m sided polygon an! that comprises !

43

1

-


the kth and the (k + 1)st vertices which are numbered counterclockwise when the face of

interest is viewed from outside of the polyhedron. Moreover. a sum over subscript k is to

be understood. henceforth. as being modulo m. Furthermore. the position vector p of any

point of Ft k is defined in plane nt as follows (refer to Fig 4 3): ,

O~s:Sl, (4.24)

where mk and hk are the veclors (rt k - ft) and (rt k+l - It k)' respectlvely. rt k being the , " J

position vector of the kth vertex of polygon ant . Similar to vector p. vectors mk and hk

lay solely in the plane nt. Consequently. their representation as 2D vectors in that plane

is used in order to apply the GDT to reduce the surface integrals defined over ailt to line

integrals defined over Ft k. ,

v

1"'·L 10---1-" •• 1:---1

hA:

u

Figure 4.3 The tth polygon contained in plane nt

Let n, k be unit normal vector to ft k-pointing outwards of anl -. and St k 1 1 1

be the length of the kth side of rt. Thus. quantities At. al' and bt can be evaluated as

indicated next. keeping in mindthat St k = Ilhkll and p·nt k = mk·nt k since hk·n t k = 0 . "" ,

(4.25a)

44

C

(


where

p ® P =(mk ® mk) + (mk ® hk + hk ® mk)s + (hk ® hk)s2 (4.25b)

and hence.

101

1 1 o p ® pds =(mk ® mk) + 2(mk ® hk + hk ® mk) + 3(hk ® hk) (4.25c)

Aiso.

1 m 101 a, =4" L sz,k(mk . ""k) 0 p2ds (4.26a)

k=l

where

p2 =ml + 2mk . hks + hls2 (4.26b)

and hence.

10 1 2 2 1 2 o P ds =mk + mk . hk + 3hk (4.26c)

Furthermore.

(4.27a)

where

(4.27b)

and hence.

(4.27c)

Now that the three surface integrals (4.22a.b.c) have been reduced to line inte

grals and evaluated in plane n,. the results obtained in this 2D subspace should be mapped

to produce the needed results necessary for calculating w, in the 3D space. eq.(4.21). The

45

l

-

4 Volumetrie Properties of Boundary-Represented Solids

scalar quantity at poses no problems and it is readily multiplied by the 3D identity tensor 1

in the expression for w t . The second rank tensor At and the vector bt in the ?D subspace

are transformed to their counterpart second-rank tensor and vector. respectively. in the 3D

space. before being substituted in the expression for w t • The detailed transformation is

explained in Appendix B.

This completes the calculation of wt . in terms of whlch the second moment

tensor of the piecewise-linear approximation of [} is determined -see eqs.(4.19a.b).

4.5 Examples

4.5.1 Example 1: Computation of the Volume, Centroid Coordinates, and Inertia

Tensor of a Cam Disk.

Shown ln Fig 4 4 is the profile of a cam disk which was synthesized usmg

periodic parametric splmes. as discussed in deta i ! in (Angeles and L6pez-Cajun 1988). The

purpose of the cam IS to produce a dwel/-rise-dwel/-return motion of its oscillating flat-face

follower The angle of rotation of the cam for each of the foregolng phases IS' tu;'(dwell).

2l:lt/J (rise). 4~tP(dwell). and 3l:ltf;(return). wlth ~tf; = 36° The amplitude of the follower

oscillations are prescnbed to be 30°.

The cam profile is obtained through an optlmlzation procedure which minimizes

the area enclosed by the profile in order to produce a minimum-welght cam while keeping

the eccentricity of the contact pOint below 50% of the base clrcle (Angeles and Lapez-Cajun

1988).

The method presented in Section 4.3 was applied to the cam profile and the

computation of the geometrical properties listed below yielded

Arel(mm2): 1.1901 x 104

Centloid coordinates(mm). x = -1.6560 x 10. fi = -8.4425

Moments of lnertla(mm4):

Product of inertia(mm4):

lxx = 1.3371 x 107, lyy = 1.3637 x 107

Ixy = 3.2208 x 105

46

(

f


y (mm) 75.0

~--=f-7---=-t-=---+---t--"""+'_-...... X (mm) 7 .0

'-----__ J

Figure 4.4 Profile of a cam disk

Now. if the cam disk is given a thickness of 40mm. the cam sol id model is

obtained by sweeping the cam profile in a direction normal to its surface for a length equal

to the prescribed thickness-see Fig. 4.5. The method presented in Section 4.4 was applied

to this cam model and the following volumetrie properties were obtained:

Volume(mm3):

CentrOld coordinates(mm):

Moments of inertia (mm 5):

Products of inertia (mm5):

4.7605 x 105

x = -1.6560 x 10. fi = -8.4425, z = 20.0

Ixx = 7.8872 x 108,

lyy = 7.9938 x 108.

Izz = 10.8031 x 108

Ixy = 1.2883 x 107•

I xz = -1.5767 x 107•

Iyz = -8.0381 x 107

The results obtained are aceurate within the computer's floating-point precision

and the linear approximation of the cam profile. The cam profile was approximated by a

polygon of 100 vertices. More accu rate results cano of course, be obtained if the profile is

47

L __ ~ ______ _

4 Volumctrie Propertics of Boundary·Reprcsented Solids

Figure 4.5 Cam obtained by a straight extrusion of its profile

approximated by a polygon with a higher number of verllces.

4.5.2 Example 2: Computation of Volume, Centroid Coordinates, and Illertia

Tensor of Spur and Helical Gears.

Spur and helical gears can be modelled by sweeping their profile for a di st 31lC(!

equal ta the gear face width. The tooth profile of such gears. in turn. is desuibcd by

planLlr involute and ils mirror image about the axis of symmelry of the loolh ((nlbour gc

1981). The whole gear profile is obtained by applymg successive rotations on tl13l lootll

profile. of amount 21'1 D. where l' is the circ ular pitch and J) IS the pit ch diallleter. A spur

gear can be modelled by sweeping ils planar gear profile in a dlrecllon normal to the plane

48

c

(

(


containing that profile. Similarly. a helical gear can be modeiled by additionally rotating its

gear profile as it is being swept. In this example. the following gear parameters describe the

tooth profile We refer the reader to Appendlx A for a description of the design parameters. Pressure angle 4>p : 20°

Number of teeth N : 16

Module m : 8mm

The resulting gear profile. with a pitch-circ/e radius of 64mm. is shown in Fig. 4.6. Given

a face width of 50mm. the profile was swept. as outllned above. for a length equal to

the prescribed face width to generate the corresponding 3D spur gear model. Moreover.

given a hellx angle of 23.0°. the corresponding 3D heltcal gear model was similarly obtained

by rotating the profile. as It is being swept. to produce the prescribed helix angle. Bath

generated models are shown ln Figs. 4.7 and 4.8

The method presented in Section 4.4 was applied to the two gear models de

scribed above. The following volumetrlc properties were obtained:

Spur gear

Volume(mm 3):

Centroid coordtnates(mm):

Moments of mertia (mm5):


Helical gear.

Volume(mm3):

Centroid coordinates(mm):

Moments of inertia (mm5):


6.7137 x 105

x = 0.0. fi = 0.0. 'Ji = 25.0

Ixx = 1.2976 x 109.

Iyy = 1.2976 x 109.

Izz = 1.4763 x 109

Ixy = 0.0. lxz = 0.0. Iyz = 0.0

6.7090 x 105

x = 0.0. fi = 0.0. 'Ji = 25.0

lxx = 1.2962 x 109.

lyy = 1.2962 '> 109 .

lzz = 1.4743 x 109

Ixy = 0.0. lxz = 0.0. Iyz = 0.0

49


25.0

t-:-::.,..L-=f-=o-::T:::---+---:T~~r.:-r~ X (mm) 7 .0

Figure 4.6 Gear profile

The volume of the helical gear and of the spur gear should be identical since the

former is obtained by subjecting the latter to an isochoric mappmg. However. the results

obtained show a relative error of the order of 10-3. This is due to the error Introduced by

the Imear approximation of the twisted surface of the helical gear model Moreover. the

results show that the two gears have identical centroldal locations. whlch was as expected.

due to the particular type of isochoric transformation mvolved ln addition. the gears are

found to have Identlcal mertia tensors wlthin a relative error of the order of 10-3. This

result was 1150 expected. since the generatmg tooth profile of the gears has an inertlal axial

symmetry. i e .. the two principal moments of inertla of the profile are identical. and hence

Üle inertia tensor of the profile' 5 2D reglon remains unchanged under rotations about its

centroid.

It should be noted that the method presented here for the calculation of volu

metrie properties of modelled objects was first tested on objects of simple shapes whose

momer.ts can be found from simple formulae. The results obtamed by applying the pre

sented method were identical to those obtained with the formulae.

50

c 4. Volumetrie Properties of Boundary-Represented Soiids

(

Figure 4.7 Spur gear generated I>y a straight extrusion of its profile.

(

51

4. Volumetrie Properties of Boundaly-Represented SoUds

Figure 4.8 Helital gear generated by rotating its profile while it is bcing cxtrudcd

il -52

c

(

(

- -------- ------------ -------

Chapter 5 Volumetrie Properties of Sweep-Generated Solids

5.1 Introduction

As discused in Chapters 1 and 2. general SWl~ep representations schemes pro

vide a powerful medelling technique for a vast domain of objects. In this chapter • we

derive formulae for the calculation of volumetrie properties of sweep-generated solids. The

sweeping technique considered here is based on the notion of extruding a deformable 2D

cross section along a hne Dimensional separability is exploited to reduce the volumetrie

computations to calculatioils on the untransformed 2D cross section of objects gen'erated

by this sweeping technique. The derived formulae express the volumetrie properties of

solids in terms of those of their generating cross section.

It IS shown that sorne tr; 'sformations applied on the 2D cross section as it is

being swept preserve the volumetrie properties of the solid obtained by just sweeping the

untransformed cross section. For example. the volumetrie proper~ies of a solid generated

by twisting its cross section about an axis passing through the centroid of this 2D cross

section while it is bemg swept is identical to those of a solid generated by sweeping without

twisting the cross section.

5.2 On Notation and Basic Definitions

For the purpose of this chapter. we will use [}/I to denote a bounded region in

-

.....

5. Volumetrie Properties of Sweep-Generated Solids

ê /.1. the v-dimensional EucJidean space. where /1 is assumed to be either 2-representing

2D planar regions. or 3-representing solids. The volume. first and second moments of nI) are defined in terms of the position vector r as

(S.1a)

(S.1b)

(S.1e)

The second expression for le. eq.(5.1c). is given in matrix notation. whereas

the fi, .t one. in lensor notation. Since we will be dealing with the components of the above

quanti.ies. the matrix notation Will be used in the rest of the chapter.

5.3 Reduced Formulae for the Volumetrie Properties of

Sweep-Generated Solids

Since the shape of sweep-generated solids is determined by information on the

generating 20 cross section and the sweeping parameters. the volumetrie properties of such

solids can be calculated using solely that information. In the following section a general

approach is described for the calculation of the mentioned properties for such solids based

on the information determiniog their shape. Let the generating 20 contour be given ln ô

plane II. the solid being obtained by extruding this contour along a curve Iying outside n and passing through a point 0 of this plane. Let el and e2 be two orthogonal unit vectors

Iying in the plane il and e3 be a unit vector normai to the plane. where e3 = el x :!2' A

coordinate frame is now defined with origin at 0 and its X. Y alld Z axes parallel to el.

e2 and e3. respectively.

Now. let p denote the position vector of a point P of region n2 Iying in II. as

shown in Fig 5.1. Moreover. let 173 d~rtote the 30 region produced by sweeping region n2 •

r being the position vector of any point R of [}3' The general sweeping is then defined as a

54

(~

5 Volumetrie Properties of Sweep-Generated Solids

mapping S carrying region n2 into [J3' Upon this mapping. veetor p is carried into vector

r. namely.

z

r = S(p)

--T 1

, i .1--1-. 1

1

1

1 1

Figure 5.1 Sweeping region D2 to generate the 3D region D3

(5.2)

For coneiseness. we will first foeus on parallel orthogonal sweeping along a li ne

perpendicular to JI. skewed sweeping being discussed later. In this eontext. region n2 is

mapped into its planar parallellmage n~ under a transformation M which carries vector p

of n2 into vector p' of n~. namely.

p' = Mp (5.3a)

where

(5.3b)

55

-

5. Volumetrie Propcrties of Sweep-Generated Solids

Hence. in generating region []3' region n2 is allowed to be transformed as it is being swept.

M being the transformation matrix that maps n2 into its image n~. Now. referring to

Fig. 5.1. vector r of the generated region n3 can be wntten as

(5.40)

where

(5.4b)

ln the foregoing relations. x' and y' are iinear functions of x and y and. in general. nonlinear

functions of ç. i.e ..

x' = xI!(d + yg1(d

y' = xh(ç) + YQ2(Ç)

The transformation matrix M is then defined as

g1 (ç) 0] g2(d 0

o 1

and. hence. the position vector r of n3 is given as

r = Çe3 + Mp

(5.5a)

(5.5b)

(5.6)

Eqs.(5.5a.b) describe an affine transformation (Gans 1969) in a plane parallel

to n2 in which M is constant. The transformation matrix M varies as a function of ç

only. i.e .. it varies as a function of the sweeping direction. Functions ft and gl' which

need not be linear. control the shape of region n3

. This sweepmg operation can be 'Iery

general and a vast number of obJects can be modcled by varying ft and gt which control

the transformation of fl2 Slnce we are deahng with ail affine transformation. the Jacobian

matrix J of x' and yi with respect to x and y. as given by eqs.(5.5a.b). is equal to M and

hence. its determinant is also constant for a given ç.

56

c

(


Next. let l:l == det(J) = det(M). which allows us to write

(5.7)

and hence. any volume integral over the 3D region n3

occupied by the solid generated by

the sweeping defined above. with the prescribed Ime of sweeping is written as

r f(r) dD3 = !i r f( Çe3 + s)dn~)dç la ( la' 3 . 2

= ll!n2 f(Çe3 + Mp)dil2].1dç

(5.8)

The volume integral over il3 is evaluated as a function of the surface integral over [}2' the

untransformed generating 2D region.

Now. define the components of q~ and I~ as

[Qi] [I~l I~2 001] qq = ~ If = I~2 I~2 (5.9)

This allows us to write the general formulae. eqs. (5.1a-c). for the volumetrie properties of

the sweep-generated solid. namely. V3. q? and I? in terms of \T2. qf and If. the area.

first and second moments of [}2' respe\ . ·ely. as follows

V3 = V2hlldÇ (5.10a)

q? = V2e3h çl:ldç + Cl Mlldç)q~ (5.10b)

I? = V2 (1 - e3ef) h ç2 11dç + A + tr(A)e3ef - B - BT (5.10c)

Matrices A and B are given as

A = l GI2GT 6dç (5.11a)

B = (l çM6dç)qfef (5.11b)

matrix G being obtained by interchanging the tirst two rows and columns of M. namely.

G = [~ ~ ~] 57

-

5 Volumetrie Properties of Sweep-Generated Solids

ln deriving eq.(5.10c). the sealar quantity IIrl1 2 and the matrix rrT were ex

panded using eq. (5.6) as follows:

IIrl1 2 = ç2 + pTMTMp

rrT = ç2e3ej + ~Mpej + ~e3pTMT + MppTM T

= ç2e3ej + çMpej + dMpef)T + MppTMT

(5.12a)

(5.12b)

The general formulae for the volumetrie propertles of solids generated by skewed

sweeping or sweepmg along a 20 curve can be readlly obtained be generalizing the mapping

of region D2 into n~ Consequently. the transformation of vector p of fl 2 into vector p' of

n~ will be given as

X' =xft (ç) + ygdç} + hdç)

y' =xh(ç) + Y92(ç) + h2{ç)

(5.13a)

(5.13b)

where hl (ç). for 1 = 1,2. determlne the li ne or curve along which the 20 reglon IS swept

As before. n~ remalns parallel to n2 .

5.4 Applications

The foregomg general relations are now applied to some of the most commonly

used types of solid sweep generation

5.4.1 Straight Extrusion

For straight extrusion. the transformation of the generating 2D region n2 is

given as

x'=x

y' =y

(5.14a)

(5.14b)

58

L _________ ~ __ ~_~~~ ~~ ___ ~

c

(

(


I.e .•

M =1, Il = 1 (5.14c)

A~.3U1ning that the 2D region is extruded for a length L. i.e .. from ~ = 0 to ç = L. the

volume. first moment. and second moment of the generated sol id are next derived. 12) being

the components of the inertla tensor If.

(5.15a)

(5.15b)

L3 1

112 = LIb· L2

III =TV2+ L1ll' 113 = --q~ (5.15c) 2

L 3 1 L 2

1 h3 = (J~ 1 + 1h) L (5.15d) 122 ="3 V2 + L 122 , 123 = --Q2'

2

5.4.2 Extrusion While Twisting

Twistlng the 2D reglon while it is being extruded gives the following transfor

mation of the generating 2D reglon n2 •

i.e ..

X' =x cos(aç) - y sin(aç)

y' =x sin(aç) + ycos(aç)

[

cos(aç)

M = sin~aç) - sln(aç) 0] co~aç, ~ :

(5.16a)

(5.16b)

(5.16c)

59

-.....


where a is a constant defining the twisting angle per unit length of extrusion. If the 2D

region is extruded from ç = 0 to ç = L. the volume. first moment. and second moment of

the generated solid are given as

1 [ sin (aL) + - 1 - cos(aL)

a 0

cos(aL) - 1 sin(aL)

o

L3

1 (' ') 1 . ( , , 111 = 3l'2 + 2 L 111 + ln + 4a Sin 2aL) (111 - 122 )

(5.17a)

(5.17b)

+ 21alcos(2aL) - IJ1b (5.I7e)

112 = 41a[1 -- cos(2aL)](l~1 -lh) + L sin(2aL)lb (5.17d)

113 = - ! [! cos(aL) + Lsin(aL) - !J-' q~ + ! [! sin(aL) - LCOS(aL)] q;(5.17e) a a a a a

L3

1 1 ') 1 " ln =3V2 + 2L (111 + 122 + 4a sln(2aL)(l11 - 122 )

1 [ 1 - 2a cos(2aL) - 1 )I12 (5.17f)

1 [1 ], 1 [1 . ( 1], ) h3 = - - - s,"(aL) - Lcos(aL) qi - - - cos(aL) + L sm aL) - - q2(5.17g a a a a a

h3 =(I~l + Ih) L (5.17h)

Note that the transformation defmed ln eqs.(5 16a-c) IS an eqUiaffine transfor

mation. slnce II = 1. and also an Isometnc one. slnce 11

2 + g; = 1. for l = 1,2 (Gans

1969) Several remarks can be made about transformations wlth such properties. First.

since the transformation is an equiafflne one. the area of n2 is preserved as it IS being

transformed. Hence. the volume of the resultant sweep-generated obJect IS identical to an

object obtamed by Just straight extrusion. cf eqs.(5.15a) and (5 17 a). Second. If twi5ting

is done along an axis passlng through the centrold of [}2' je .. ql = q2 = O. the centroid

is fixed under the Isometric transformation Consequently. the centroldal location of the

resultant obJect IS identlcal to the case of straight extrusion Wlth Il -::: 1. the centroidal

location along the extrusion aXIs is always preserved. Third. if the generatlng region n2 has

60

c

/ (

ft

5. Volumetrie Propert;es of Sweep-Generated Solids

an axial inertial symmetry. i.e .. if the two principal moments of inertia of D2 are identical.

I~l = Ih and Ih = O. the inertia tensor of D2 remains unchanged under an isometric

transformation. As a result. the inertia tensor of the sweep-generated object is identical

to the one obtained by Just straight extrusion. The inertia tensor is preserved under such

equiaffine and Îsometnc transformation. this 15 conflrmed by ta king qi = q2 = O. I~l = Ih and Ih = 0 and comparing the mertia tensors defined by eqs. (5.1Se.d) with the one defined

by eqs.(5.17 e-h).

5.4.3 Extrusion While Scaling

If the generating 20 regiùn is scaled linearly in both the x and y directions.

while it is being eKtruded. then the transformation of the 2D region D2 will be

1 e .

[

a lç + 1 M= 0

o

x' =x(al ç + 1)

y' =y(a2ç + 1)

(S.18a)

(5.18b)

(5.18c)

where al and a2 are constants defining the amount of scaling in each direction. Extruding

the 20 region from ç = 0 to ç = L gives the following volumetrie properties

L 3 L 2 V3 = V2 lala2'3 + (al + aÛT + L] (S.19a)

q~ = V2 [n + [~2 ~3 ~] [~] (5.19b)

(5.19c)

61


where the constants CI' i = 1,"',9 are as follows

L 4 L 3 L2 Cl = ala2

4 + (al + a2)T + T (5.20a)

2 L 4 L3 L2 C 2 = ala2

4 + al (al + 2a2)T + (2al + a2) T + L (5.20b)

L 4 L 3 L2 C3 = 01a~4 + a2(2a l + a2)T + (al + 2a2) T + L (5.20c)

L 5 L 4 L3 C4 = ala2 5 + (al + a2)T + '3 ~5.20d)

L 5 L4 L3 L2 C5 = ata~5 + a~(3al + a2)4 + 3a2(al + a2)T + (al + 3a2)2 + L (5.20e)

L 5 L4 L3 L2

C6 = ata25

+ ai(at + 3a2)4 + 3al(al + a2)T + (3at + a2)2 + L (5.20J)

2 2 L 5 L4 2 2 L3 C7 = 01025 + 2ala2(ot + a2)4 + (al + 4a1a2 + a2)T

L2

+ 2(a1 + a2)T + L (5.20g)

L 5 L4 L3 L 2 Ca = afa2"5 + al (al + 2a2)4 + (2al + a2) 3 + "2 (5.20h)

L 5 L4 L3 L 2 Cg = ala~5 +a2(20 1 +a2)4 + (al +2a 2)-3 +"2 (5.20t)

ln the foregoing relations. fl 2 was scaled Imearly as a function of ç. By intro

ducing other sealmg functions. possibly nonlmear in ç. several familles of sealing trans

formations can be similarly aecommodated. Moreover. twisting and scaling can be easily

combined into one transformation to form a more general sweepmg

5.5 Examples

The volumetrie properties of both the spur and helical gears. specified ln Ex

ample 2 of Section 4.5. are calculated using the formulae presented in Sections 5.4.1 and

5.4.2. respectively. The following volumetrie propertles were obtained.

62

c

(

(

Spur gear:

Volume(mm3):

Centroid coordinates (mm):

Moments of inertia (mm5):


Helkal gear:

Volume(mm 3).

Centroid coordinates (mm)'

Moments of inertla (mm5):

Products of mertla (mm5)

----- ------------------


6.7137 x 105

x = 0.0. fj = 0.0. z = 25.0

Ixx = 1.2976 x 109•

Iyy = 1.2976 x 109.

Izz = 1.4763 x 109

Ixy = 0.0. I xz = 0.0. Iyz = 0.0

6.7137 x 105

x = 0 O. fj = 0.0. z = 25.0

Ixx = 1.2976 x 109.

Iyy = 1.2976 x 109.

Izz = 1.4763 x 109

Ixy = 0.0. I xz = 0.0. I yz = 0.0

As expected. the volumetric propertles are identlcal for both gears smce the latter is ob

tained from the former by twisting the axially symmetfle gear profile about ItS centroid.

Companng the results obtained here with the on es obtamed in Section 4 5.

Example 2. we notice that they are identleal in the ease of the spur gear However. the

results obtamed for the heheal gear show that the error introduced by calculating the

volumetrie propertles dlrectly from the gear's approxlmate 3D modells aVOIded here. This

illustrates that the direct ealculation of the volumetrie properties from the 2D profile and

the sweeping parameters not only reduees the amount of ealeulations required but also

avoids the error introdueed by approxlmating the 3D model of sweep-generated objeets.

63

1

Chapter 6 Conclusions and Remarks

The 3D modelling of bevel gears was addressed as a paradigm of modellrng

mechanical elements with complex shapes. In thls context. the exact spherical Involute

was derived from the fundamental Involute geometry and was used to describe the bevel

gear profile on the surface of a sphere This profile IS used to produce the sohd model of

the correspondmg gear by simple radiai extrusion and tWlstlng ooeratlons Radiai extrusion

produces a stralght bevel gear Radiai extrusion whlle tWlstlng will produce dlfferent types

of spiral bevel gears This modelhng methodology can be adopted to deswbe bevel gears

with different tooth profiles Tooth profiles that devlate from the theoretlcallnvolute profile

but are used ln Industry should be accommodated ln future work Moreover. m order to

evaluate the accuracy of the modellmg technique. the bevel-gear models thus obtamed

should be compared wlth actual manufactured gears.

ln the context of the evaluatlon of volumetnc propertles. the Gauss Divergence

Theorem was successfully applied to the computation of the flrst three moments of planar

and solid reglons. employing a plecewise-Imear approximation of their boundaries The

method adopted reduces the problem to evaluatlng Ime integrais over the edges definlllg

the boundaries. Areas for planar regions. as weil as volumes for solid reglons. together

with c.entrOld coordinates and inertia tensors. were computed. Practlcal simple formulae

were derived for planar regions whose boundary IS approxlmated by polygons. and for sohds

whose boundary IS approximated by polygonal faces Computer-oriented algorithms were

implemented based on these formulae. whlch compute the properties sought for planar and

c

(

6. Conclusions and Remarks

solid regions, given their boundary representation. The formulae derived are exact but the

accuracy attained depends upon both the piecewise-linear approximati.)n of the boundaries

and the eomputer's floatmg-point precision. Errors in the ealculation of the volumetrie

properties tntrodueed by the linear approximation of the boundaries should be analysed as

an extension of the work presented here. In this context. the volumetrie properties obtained

through ealculatlons on model'ed objects should be compared with the aetual properties of

these objects measured expenmentall l '

The volumetrie propertles of sweep-generated objects were also successfully

ealculated from those of the generating 2D cross section and information on the sweeping

parameters. This not only reduees signifieantly the amount of computations required to

ealeulate these propertles, but also avoids any numerical instabilities tntrodueed by the

direct calculatlon of these propertles from the approximate 3D models of objects. Numerieal

results have been presented to show that the volumetrie properties of sweep-generated

objects can be calculated aceurately and efflclently usmg the proposed method. A natural

extension to the approaeh presented here is to allow the generating 2D region D2 to oe

mapped mto a nonparallel reglon n~ ThiS tntroduces a more general sweepmg technique

in which the 2D cross section IS swept along a 3D curve while allowing it to be transformed

as it moves It should be noted that the generatlon of nonhomogeneous or invalid solids

through sweeping has not been aceounted for here and should be considered in future work.

65

-.......

References

References

Angeles. J .. 1983. "Calculo de cantidades ffsieas globales asoeiadas a volumenes

aeotados por superficies eerradas mediante integraci6n en la frontera. Ingenieria.

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Angeles. J .. et al. 1988. "The Evaluation of Moments of Bounded Regions Using Spline Approximation of the Boundary". Advances in Design Automation. Vol.

14. pp. 49-54.

Angeles. J . and Lapez-Cajun. c.. 1988. "Optimal Synthesis of Cam Meehanlsms with Oseillating Flat-face Followers". Mechanism and Machine Theory.

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Baer. A .. Eastman. c.. and Henrion. M .. 1979. "Geometrie Modelling a survey".

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Braid. 1. C .. and Lang. c.. 1973. "The design of Mechanical Components with

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North-Holland. Amsterdam. Holland.

Brand. L .. 1965. Advanced Caleulus. John Wlley & Sons. Inc .. New York. p. 390.

Brown. C. M .. Requieha. A. A. G .. and Voelcker. H. B .. 1978. "GeometricModelling Systems for Meehanieal Design and Manufacturing". Proceedings o(

the 1978 Annual Conference of A CM. Washington. D.C.

Casale. M.S. and Stanton. E. L.. 1985. "An Overview of Analytic Solid Model

ing". IEEE Computer Graphies and Applications. Vol. 5. No. 2. pp. 45-56.

Chen. Z .. and Pemy. D. B. 1983. "Computer-Assisted Methods for Defining 3D Geometrie Structures of Mechanieal Parts". In Computer-Based Automation.

Tao. J. T .. Plenum Press. New York.

Colbourge. J.R .. 1987. The Geometry of Involute Gears. Springer-Verlag. New

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Coquillart. S .. 1987. "A Control-Point-Based Sweeping Technique". IEEE Com

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(

------- ------~~---------------

References

Dewhirst. D. L.. and Hillyard. R. c.. 1981. "Application of Volumetrie Modelling tp Mechanical Design and Analysis". Proceedings of the Design Automation

Conference. 18th. Nashville. TN.

Dudley. D.W .. 1962. Gear Handbook. MacMillan. New York.

Dudley. D.W .. 1954. Practical Gear Design. MacMillan. New York.

Gans. D .. 1969. Transformations and Geometries. Appleton-Century-Crofts. New York.

Gasson. P.c.. 1983. Geometry of Spatial Forms. Halsted Press. New York.

Huston. R.L. and Coy. J.J.. 1981. "Ideal Spiral Bevel Gears - A New Approach to Surface Geometry" .ASME Journal of Mechanical Design. Vol. 103. No. 1. pp. 127-133.

Huston. R.L. and Coy. J.J .. 1982. "Surface Geometry of Circular Cut Spiral Bevel Gears" .ASME Journal of Mechanical Design. Vol 104. No. 4. pp. 743-748.

Krishnamurty. K .. 1967. Vector Analysis and Cartesian Tensors. Holden-Day.

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Lee. Y. T. and Requicha, A. A. G .. 1982-a. "Algorithms for computing the

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Lee. Y. T. and Requicha. A. A. G .. 1982-1I. "Algorithms for computing the volume and other integral properties of solids. Il. A family of algorithms based on representation conversion and cellular approximation". Communications of

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Lien. S. and Kajiya. J. T .. 1984. "A symbolic method for calculating the integral

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Mortenson. M.E .. 1985. Geometrie Modeling. John Wiley. New York.

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O·Leary. J. R .. 1980. "Evaluation of Mass Properties by Finite Elements". Jour

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68

(

Appendix A. Gear Terminology


The types of gear encountered in this thesis belong to two main classes. The

first class represents gears that connect parallel shafts. and comprises spur and helical

gears (see Figs. A.t. and A.2). The second class represents gears that connect intersecting

shafts. and comprises straight and spiral bevel gears (see Figs. A 3. and A.4).

A.t General Terminology

The following IS a definition4 of some general gearmg terms encountered in the

thesis

Pltch circle'

Pltch diameter.

Diametral pitch:

Module:

Circular pitch:

Pressure angle:

A ddendum:

Dedendum:

Working depth:

Whole depth:

the imaginary circle that rolls without slipping with a pitch

circle of a mating gear

the dlameter of the pitch circle

the ratio of the number of teeth to the pitch diameter. the

latter belng measured in inch

the ratio of the pltch dlameter. in millimeters. to the number

of teeth

the distance along the pltch drcle between corresponding

profiles of adjacent teeth

the angle between a tooth profile and the Ime normal to a

pltch surface. usually at the pitch pOint of the profile.

the height by whlCh a tooth projects beyond the pitch line.

the depth of a tooth space below the pitch line.

the depth of engagement of two gears. that is. the sum of

their addendums.

the total depth of a tooth space equal to addendum plus

dedendum

4 The definitions are adapted from Dudly (1962) for a quick reference.

69

~ -

---

Appendix A Gear Terminology

A.2 Bevel Gear Terminology and Geometry

Refer to Fig. A.5 for the followmg definitions4 related to bevel gears.

Pitch cone: the imaginary cone ln a bevel gear that rolls wlthout slippmg

on a pitch cone of another gear

Apex of pitch cone: the intersection of the elements making up the pitch cone.

Cone distance: the length of a pitch-cone element

Outer cone distance. the distance from the apex of the pltch cone to the outer

ends of the teeth.

Inner Ct. 7e distance. the distance from the apex of the pltch cone to the Inner

Face cone.

Root (base) cone.

Face angle:

Pitch angle:

Root (base) angle:

Face width:

Addendum angle:

Oedendum angle:

Spiral angle:

Shaft angle'

ends of the teeth

the cone formed by the elements passing through the top of

the teeth and the apex

the cone formed by the elements passmg through the boUom

of the teeth and the apex.

the angle between an element of the face cone and the axis

of the gear

the angle between an element of the pitch cone and the aXIs

of the gear

the angle between an element of the root cone and the axis

of the gear

the width of a tooth.

the angle between an element of the pitch cone and an ele

ment on the face cone

the angle between an element of the pitch cone and an ele-

ment on the root cone.

the angle between the tooth center line and an element of

the pltch cone.

the angle between the two gear shafts.

70

Appendix A Gear Terminology

•

Figure A.1 Spur gears (adapted trom Watson (1970), p 13, Fig. 2.1)

« Figure A.2 Helical gears (adapted from Watson (1970). p 17, Fig 27)

Figure A.3 Straight bevel gears (adapted from Watson (1970), p 20, Fig 212)

71

....


Figure A.4 Spiral bevel gears (adapted from Watson (1970). p 20. Fig 213)

,.-------_._--------------------,

Root

/ 1

1-------- Oullule dl4lM'~r ~-,\,----'" /

L, ____ __ Figure A.5 Bevel gear geometry (adapted from Wilson. Sadler and Michels (1983).

p 439. Fig 7.24)

72

{

Appendix B. Sorne Useful Tensor Relations


B.l Tensor Notation and its Relation to Multi-li ... ear Aigebra

let 8. b. and c be three 3D vectors (first-rank tensors) and A be a second-rank

tensor. The tensor product of two vectors a and b gives a second-rank tensor denoted by

ab or a ® b. defmed. in ter ms of the components of the vectors involved. in a 3D frame J

as follows (Krishnamurty 1967):

(B.l)

The tensor product of a vector with a second-rank tensor gives a third-rank tensor a ® A.

The dot product of a vector by a second-rank tensor is a vector defined as

follows (Krishnamurty 1967).

= [A]y(b]; (B.2)

(B.3)

For the three vectors a. b. and c. the following relations hold:

c . (a ® b) = (c . a)b (B.4a)

c x (a ® b) = (c x a)b (B.4b)

73

----------------------------

1

--

Appendix B Sorne Useful Tensor Relations

B.2 Divergence of 8 nth-rank Tensor

Following the summation convention (Krishnamurty 1967). the components of

the diverge:'1ce of an nth· rank tensor «1». whose components in 1 are represented as 4JtJ k n

with respect to index n. is an (n -- 1)st-rank tensor given as follows (Krishnamurty 1967):

( ) _ (') _ 8<i>t)k n V1 . «1» tJk n - dlv«l» lJk n - -a-- (B.5) In

Ali relations are derived taking the divergence with respect to the rightmost index of a

nth-rank tensor. The gradient of a ntn-rank tensor is an (n + 1)st-rank tensor defined as

follows:

(B.6)

ln the l/-dimensional Euclidean space. let p den ote the position vector of a point

of this space. Then the following relations hold:

V'p = 1

V'P=l/

where «1»1 and «1»2 are two nth-rank tensors.

B.3 20-to-30 Mapping of Vectors and Second-Rank Tensors

(B.7a)

(B.7b)

(B.7e)

ln a 3D frame 1. let us define a plane n This plane can be considered as a

2D subspace of the 3D space. A vector a of this 2D subspace can be represented in terms

the two unit vectors ê u and êv as:

[aJn = [:~l (B.8)

The two unit vectors êu and êv can be represented. in turn. in the 3D space in terms of

the J -frame unit vectors i, j, k as

(B.9a)

74

1 / {

(

__ 44 &CUlA l &JAtU 2 4 •


(B.9b)

The same vector a can be represented in the 3D space by substituting eqs.(B.9a.b)

in eq (B.8) to give'

(B.10a)

[a); = au[êul; + a,[ê,l; = [:: J (B.10b)

Where ax. a". and az are the components of a in the frame 1.

By the sa me token. a second-rank tensor A can be defined in the 2D subspace

(il plane) as follows:

(B.11a)

(B.11b)

This second-rank tensor has ils equlvalent second-rank tensor in the 3D space (in the Y

frame) whlch can be obtained by substltuting eqs.(B.9a.b) in eq.(B.l1a). keeping in mind

that êu ® êv = lêu11[êtl]~

8.4 Proof of The Divergence Identities Used

Proof of the three divergence identities that were used in deriving eqs. (4.23a-c)

is given next. the proof is not readlly available in the literature on tensors.

div(p2 p) = V . (p2p)

= (V p2) . P + p2(V'. p)

= 2p· p + p211 = (2+ 1I)p2 (8.13)

75

-

-.~

Appendix B Some Useful Tensor Relations

div(p2p ® p) = V . (p2p ® p)

= V p2 . (p ® p) + p2V' . (p ® p)

= 2p· (p ® p) + p2((V'p). P + p(V' . p))

::: 2(p. p)p + p2(1 . P + IIp)

= (3 + lI)p(p . p) = (3 + 1I)p2 P

div(p ® p ® p) = V . (p 0 P ® p)

= (Vp) . (p ~ p) + p(V' . (p ® pl)

= 1 . (p 0 p) + p((V'p) . p + p(V'. p))

= p ® p + p(l . p + vp)

(B.14)

= p 0 p + p ® P + vp 0 P = (2 + lI)p ® p (B.15)

Where v is the dimension of the space under study.

76

the geometrie modelling of mechanical elements with...

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