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:
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• 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
5
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
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(
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
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2 Solid Modelling Schemes
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
2 Solid Modelling Schemes
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
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L... _____ _
2 Solid Modelling Schemes
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
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3. 3D Modelling of Bevel Gears
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
3. 3D Modelling of Bevel Gears
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
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- -- - -------~----------------
3. 3D Modelling of Bevel Gears
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
3. 3D Modelling of Bevel Gears
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
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where
-- --------------~-----
3. 3D Modelling of Bevel Gears
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
-: . .....
3. 3D Modelling of Bevel Gears
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
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3. 3D Modelling of Bevel Gears
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
3. 3D Modelling of Bevel Gears
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 (
3. 3D Modelling of Bevel Gears
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
3. 3D Modelling of Bevel Gears
~----------------------------------------------------------------~
Figure 3.12 The sphericill profIle us~d to mode! the gears
32
-------------- ---
3. 3D Modelling of Bevel Gears
c
(
'------------------------ --- ----------'
Figure 3.13 The sltaight bevel geat model
33
3. 3D Modelling of Bevel Gears
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
..
(
4. Volumetrie Properties of Boundary-Represented Solids
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. Volumetrie Properties of Boundary-Represented Solids
(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
4. Volumetrie Properties of Boundary-Represented Solids
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
4. Volumetrie Properties of Boundary-Represented Solids
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
(
4. Volumetrie Properties of Boundary-Represented Solids
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
-
4. Volumetrie Properties of Boundary-Represented Solids
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
(
4. Volumetrie Properties of Boundary-Represented Solids
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
4 Volumetrie Properties of Boundary-Represented Solids
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
(
(
4 Volumetrie Properties of Boundary-Represented Solids
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):
Products of inertia (mm5):
Helical gear.
Volume(mm3):
Centroid coordinates(mm):
Moments of inertia (mm5):
Products 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
4 Volumetrie Properties of Boundary-Represented Solids
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
(
5. Volumetrie Properties of Sweep-Generated Solids
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
(
(
5. Volumetrie Properties of Sweep-Generated Solids
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
-.....
5. Volumetrie Properties of Sweep-Generated Solids
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
5. Volumetrie Properties of Sweep-Generated Solids
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):
Products of inertia (mm5):
Helkal gear:
Volume(mm 3).
Centroid coordinates (mm)'
Moments of inertla (mm5):
Products of mertla (mm5)
----- ------------------
5. Volumetrie Properties of Sweep-Generated Solids
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
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68
(
Appendix A. Gear Terminology
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
....
Appendix A. Gear Terminology
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
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 •
Appendix B. Sorne Useful Tensor Relations
(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