Rapid prototying and manufacturing technology applied to the forming of spherical gear sets with skew axes

Rapid prototyping and manufacturing technology appliedto the forming of spherical gear sets with skew axes Ying-Chien Tsaia,*, Wern-Kueir Jehngb,1 a Department of Mechanical Engineering,

Rapid prototyping and manufacturing technology applied

to the forming of spherical gear sets with skew axes

Ying-Chien Tsaia,*, Wern-Kueir Jehngb,1

a Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan, 80424, R.O.C.

b Department of Industrial Engineering and Management, National Kaohsiung Institute of Technology, Kaohsiung, Taiwan, R.O.C.

Received 5 May 1998

Abstract

This paper proposes a systematic method to design and synthesize spherical gear sets in space Based on the kinematics of spherical mechanisms, a parametric mathematical model for spherical gear sets with skew axes is presented and a computer solid-model for spherical gears is designed and formed Using the gear solid-model database, non-ambiguous data set descriptions of rapid prototyping and manufacturing (RP&M) machines have been generated A design example is presented to demonstrate RP&M procedures for the surface generating of a complex spherical gear set with skew axes The result of this work can be of crucial bene®t in research in new gear types and industrial development In this paper, important detail techniques are also proposed for the solid-models of CAD/CAM systems, describing how they can be converted successfully into the RP&M (SL) system, and built up gear set models # 1999 Elsevier Science S.A All rights reserved

Keywords: Rapid prototyping and manufacturing (RP&M); Solid-model; Gear set with skew axes; Three-dimensional printing (3DP); Instantaneous skew axis (ISA)

1 Introduction

In kinematics, spur and helical gears can be classi®ed as

plane mechanisms, whilst bevel gears are spherical

mechan-isms, and hypoid and worm gears can be deemed as spatial

mechanisms Therefore, the study of bevel gears should be

based on spherical mechanisms Bevel gears are essential to

power transmission between two intersecting axes Various

mathematical models of tooth geometry for bevel gears have

been presented by many authors; Huston and Coy [1]

initially presented a new approach to the surface geometry

of ideal spiral-bevel gears Tsai and Chin [2] further

devel-oped the surface geometry of straight and spiral-bevel gears

with parametric representations of the spherical involute and

the involute spiraloid to study the fundamental geometrical

characteristics Most of these models are immediately

avail-able for 3D computer solid-models to provide for the

analysis of their characteristics and performance Huston

et al [3] investigated spiral-bevel gears of circular-cut pro®les for their geometric characteristics, such as radius

of curvature, speci®c sliding and relative curvature, which are important to the analysis of meshing kinematics, contact stress, fatigue life and lubrication Existing bevel gears are mostly of spherical inviolate tooth pro®les or of tooth pro®les generated by tools with straight cutting edges In contrast, there are also many types of tooth geometry that may be used to give bevel gears conjugate tooth pro®les However, for simplicity of manufacturing, spherical tooth bodies are usually cut into conical shapes In this paper, using rapid prototyping and manufacturing technology, two meshing spherical gear sets with skew axes are designed and presented The RP primitives provide actual full-size phy-sical models that can be handled, analyzed, and used for further development

Rapid prototyping and manufacturing (RP&M) technol-ogy was developed in the late 1970s and early 1980s In the USA, Hebert of 3 M in Minneapolis, Hull of UVP (Ultra Violet Products Ins.) in California, and Kodama of the Nagoya Prefectural Research Institute in Japan [4] worked independently on rapid prototyping concepts based on selectively curing a surface layer of photopolymer and

*Corresponding author Tel.: +886-7-525-2100; fax: +886-7-525-2149

E-mail address: yctsai@mail.nsysu.edu.tw (Y.-C Tsai)

1 Ph.D Candidate of Mechanical Engineering Department, National Sun

Yat-Sen University, Kaohsiung, Taiwan.

0924-0136/99/$ ± see front matter # 1999 Elsevier Science S.A All rights reserved.

PII: S 0 9 2 4 - 0 1 3 6 ( 9 9 ) 0 0 2 8 7 - 3

building three-dimensional objects with successive layers of

the polymer Both Herbert and Kodama had dif®culty

maintaining ongoing support from their research

organiza-tions, and each of them stopped his work before proceeding

to a commercial or product phase UVP continued to support

Hull, who worked through numerous problems of

imple-menting photopolymer part building until he developed a

complete system that could build detailed parts

automati-cally Hull coined the term stereolithography, or

three-dimensional printing [5] This system was patented in

1986, at which time Hull and Freed, jointly with the

stock-holders of UVP, formed 3D Systems, to develop commercial

applications in three-dimensional printing

A different RP&M technology, developed by DTM in

USA is a process of fusing or sintering called the selective

laser sintering (SLS) system [6] This process generates

three-dimensional parts by fusing powdered thermoplastic

materials with the heat from an infrared laser beam A thin

layer of powdered thermoplastic material is evenly spread,

by a roller, over the build region The pattern of the

corresponding part cross-section is then ``drawn'' by the

laser on the powder surface With amorphous materials, the

laser heat causes powder particles to soften and bind to one

another at their points of contact, forming a solid mass

The third RP&M technology was developed at the

Mas-sachusetts Institute of Technology (MIT) and is called

three-dimensional printing (3DP) [7] The 3DP process can use a

number of powder materials including those well-known to

the investment casting industry for the production of shells:

refractory powder, such as silica or alumina, combined with

a liquid colloidal silica binder Three-dimensional parts and

ceramic molds are fabricated by selectively applying binder

to thin layers of powder, causing the particles of powder to

stick together Each layer is formed by generating a thin

coating of powder and then applying binder to it with an

ink-jet-like mechanism Layers are formed sequentially and

adhere to one another, thus generating a 3D object in a

manner similar to that of the other RP&M systems The 3DP

apparatus includes a cylinder ®tted with a piston that can be

lowered in small increments under computer control The

piston is coated with a thin layer of powder supplied by a

hopper Above the powder is an ink-jet-like mechanism that

is supplied with binder and can move along both horizontal

(X±Y) axes Small droplets of liquid are ejected downwards

continuously from the nozzle toward the powder surface

Unwanted droplets are skimmed before reaching the powder

by electrically charging them at the nozzle and then

de¯ect-ing them from the stream by applyde¯ect-ing a voltage to electrodes

located below the nozzle The nozzle is moved across the

powder surface in a raster scan whilst the electrical signals

control the deposit of binder onto the powder in speci®c

locations After the molds have been fabricated from

cera-mic powder within the cylinder, they are placed in a furnace

to cure the binder and strengthen the mold Following this

step, excess unbound power is removed and the object is

then ready for use

In summary, RP&M technology is clearly a technology with enormous potential to form complicated solid proto-types The materials used include liquid resin, fusing pow-dered thermoplastic materials, PVC, polycarbonate, investment wax, nylon, ABS, powdered metals, ceramics, refractory powder (such as silica or alumina), etc The layer-additive methods have both laser and non-laser point-by-point fabrications The layer faceted attachment also uses layer-additive and layer-subtractive fabrications The tech-nology also uses many types of laser beams, such as ultra violet (UV) laser, infrared laser, UVargon-ion laser, UV He±

Cd radiation, CO2laser, etc Development work is continu-ing in USA, Japan, Germany, Sweden, Israel, and other European countries [8±10]

2 Parametric tooth surfaces of spherical gear sets with skew axes

Consider the moving coordinate systems S1(X1, Y1, Z1) and S2( X2, Y2, Z2) to be rigidly connected to gear 1 and gear

2 (shown in Fig 1) that transform rotation between two skew axes L1and L2, and angles of gear rotation 1and 2

for gear 1 and gear 2, respectively Fixed coordinate systems S(X, Y, Z) and S0(X0, Y0, Z0) coincide with S1(X1, Y1, Z1) and

S2(X2, Y2, Z2

twisting angle of these two rotating axes By PluÈcker line coordinate theory, the common perpendicular line of L1and

L2can be evaluated The X-axis and X1-axis coincide with the rotating axis L1, whilst the X0-axis and X2-axis coincide with the rotating axis L2 The Z-axis and Z0-axis coincide with the common perpendicular line of axes L1and L2 The speed ratio of gear 2 to gear 1 is m The notations of  and u are the position parameters of the instantaneous screw axes,

 is the twisting angle, whilst u is the Z-axis direction offset distance relative to L1and the coordinate system S(X, Y, Z) Fig 2 shows the kinematic vector relations for a two gear set with contact at point P The relative velocity Vrel of the conjugate tooth pro®les at contact point P is the rotating velocity of the L2axis relative to the L1axis

There can be described two trihedrons, one being coincident with point P, and the other with point Q The trihedron at P has unit vectors nsa, n3 and n4, whilst the trihedron at Q has unit vectors n2, n3and n5 The vector

nsa is the unit direction vector of the ISA (de®ned pitch line) The unit vectors n1 that lies on the plane zˆu is perpendicular to the unit vector nsa The unit vectors n3 and n5construct a datum plane called the N-plane [11] that is

a plane necessary to depict the complex motion of the contacting point P The unit vector n2 is the unit normal vector of the N-plane The vector n3 is the unit direction vector of the line QP (de®ned ) Fig 2 shows the para-meters `, and û, which are the position parapara-meters of the N-plane Tsai and Sung [13] presented the ruled surface of the tooth pro®le of gear 1 in coordinate system S1(X1, Y1, Z1)

as follows:

The ruled surface of the tooth pro®les of gear 2 in

coordinate system S2(X2, Y2, Z2) is described as:

The constraint for the parametric conjugate tooth pro®les of

a gear set with skew axes was proposed by Tsai and Sung [8] as

InEq (3),andarefunctionof`and1,i.e.,ˆ(`,1)and

ˆ(`,1) Tsai and Sung [11], however, only directly

simpli-®ed the constraints as point contact types to suit spur and bevel gears According to Litvin [12], ``Mating gear surfaces will differentiate two cases of tangency: (i) the interacting surfaces P

1andP2are in line contact at every instant, andP2is the envelopetothefamilyofsurfacesthatisgeneratedbyP1inthe coordinate system S2, and (ii) surfacesP1andP2are in point contactateveryinstant(thecontactofP1andP2islocalized).'' Therefore, it is inconsequential to simplify Eq (3) to point contact only, whereas the meshing constraint equation of the spurandbevelgearshasalreadybeende®nedbyChangandTsai [13,14] In this paper, a special analytical solution of Eq (3) is

Fig 1 Spatial rotation between two crossed skew axes, the derivation of coordinates, the instantaneous screw axes, and the pitch vertical.

r1…; ; `; 1† ˆ …` sin  ‡  cos cos † cos ` cos  ÿ  cos sin 1‡ …u ‡  sin † sin 1

ÿ…` sin  ‡  cos cos † sin 1‡ …u ‡  sin † cos 1

2 4

3

r1…a; ; `; 1† ˆ 2‡ …u ÿ E ‡  sin †sin 2

2‡ …u ÿ E ‡  sin †cos 2

2

4

3

 sin @@

1

 

@

@`

 

ÿ @@

1

 

@

@`

 

ÿ cos @@

1

 

‡ …` sin  cos sin ‡  cos  sin ‡  sin cos sin 

‡ u sin sin cos  ‡ u cos sin † @@`

 

‡ …ÿ`  sin  sin sin ‡ u  cos sin cos † @@`

 

‡ ` sin sin  cos ÿ u cos cos  cos ˆ 0: (3)

proposed to apply for general situations and it is logically

deducedfromspatialmechanismtosphericalmechanism.This

solutioncanbeusedtodesignanytypeofsphericalgearsetwith

skew axes

After extensive use of the trial and modi®cation methods,

a special analytic solution for Eq (3) has been found, as

follows:

@

@1ˆ



u

…sin  cot cot ‡ tan cos cot2 cos 

ÿ sin tan tan cos  ÿ cot sin  tan

ÿ sin tan cos † ‡`…sin  sin tan2 tan

‡ sin  cos cot ÿ sin  cos cot tan †



 tan ‡ cot

cot

‡ …cos  ‡ sin cot sin †; (4)

@

@1ˆ ‰u…sin  cot cot ‡ tan cos cot2 cos 

ÿ sin tan tan cos  ÿ cot sin  tan

ÿ sin tan cos † ‡ `…sin  sin tan2 tan

‡ sin  cos cot ÿ sin  cos cot tan †Š; (5)

@

@`ˆ

1

…cot2 ÿ tan2†; (6)

@

@`ˆ cot …cot ÿ tan †: (7) Consequently, by use of Eqs (1), (2)±(7), any types of skew gear sets in a spatial mechanism can be logically constructed

2.1 Deduction of general spherical gear sets with skew axes

Since the spherical mechanism is a degeneration of the spatial mechanism, by using the theoretical kinematics of spherical mechanisms as proposed by Chiang [15], the spherical curvatures and torsion relations can be de®ned

in Frenet±Senet form as:

t0

n0

b0

2 4

3

5 ˆ ÿkgn ÿ gt ‡ ngbb

nt ÿ gn

2 4

3

5 ˆ gÿn ÿ bgt

t

2 4

3

where unit vectors t, n, and b are taken at the current points

of the spherical curve, as shown in Fig 3, forming a

Fig 2 N-plane, unit vectors, relative velocity, pitch line and pitch vertical.

trihedron which indicates the curve direction and curvatures.

Normal curvature nis equal to 1/ in spherical mechanism

ˆr, where r is the radius of the assumed unit length, and

nˆ1gis geodesic curvature From the characteristics of

spherical curvatures, curves of spherical bodies have

uni-form normal curvature, and spherical gear sets with skew

axes must have their two rotating axes intersecting at the spherical original point, so that it can be deduced that the offset distance of the two rotating axes is zero This means that uˆ0 (refer Eqs (1) and (2)), and Fig 4 shows the locational relationships of spherical gear sets with skew axes, where coordinates S(X, Y, Z) and S0(X0, Y0, Z0) coincide

at the center point O of the spherical ball

For analyzing the loci of the contact point, it is necessary

to create a datum for the X±Y plane (as shown in Fig 4) The connecting line of PP1is the action line along the normal direction, and point P is the intersection point of the two pitch spherical cones The notation  is the radius expanding angle from contact point P to point P1, therefore, OP1is the instantaneous screw axis (ISA) of the two rotating axes Then PP1is de®ned as p, OP1is de®ned as `, OP is the radius

of the spherical ball, and the triangle OPP1relationships are deduced:

d

d`ˆ tan  ‡ `sec2

d

Substituting uˆ0 and Eqs (9) and (10) into Eq (1), the tooth pro®le of gear 1 in the coordinate system S1(X1, Y1, Z1) degenerates to

Fig 4 Spherical gear sets rotating between intersected skew axes.

Fig 3 Unit vectors of a moving trihedron at the current point of the

spherical curve.

Similarly the tooth pro®les of gear 2 in coordinate S2(X2,

Y2, Z2) are

From Fig 1, the position vector rLof the ISA in

coordi-nate S(X, Y, Z) and the relative velocity VLof the points on rL

only have components along the ISA, and therefore, they can

be represented as:

rL…`† ˆ `…cos  i ‡ sin  j† ‡ uk (15)

VLˆ !2 ‰rL…`† ÿ EkŠ ÿ !1 rL…`† (16)

where !1ˆ ÿ!1i; !2ˆ !2 2

relative velocity VL, and the position vector rLof the ISA

have the same direction, i.e.,

drL…`†

By setting m2to be the speed ratio of gear 2 to gear 1, i.e.,

letting m ˆ ÿ…!2=!1†, Eq (17) can be deduced as [16]:

 ˆ tanÿ1 

u ˆ mE…m ÿ cos r†

m2ÿ 2m cos r ‡ 1: (20)

Amongst Eqs (18)±(20), only two equations are

inde-pendent, i.e., Eqs (18) and (19) are the same equation

Fig 2 shows unit vectors nsa, n1, n2, n3, n4 and n5 as

follows:

nsaˆ cos  i ‡ sin  j; (21)

n1ˆ R =2;jnsaˆ ÿsin  i ‡ cos  j; (22)

n2ˆ R;n sa

R;n 1

 

k ˆ cos  sin ‡ sin  sin cos sin  sin ÿ cos  sin cos

cos cos

2 4

3 5;

(23)

n3ˆ R;n sa

n1ˆ ÿsin  cos cos  cos

sin

2 4

3

n4ˆ …sin  sin †i ÿ …cos  sin †j ‡ cos k;

n5 ˆ …sin  sin sin ÿ cos  cos †i

ÿ …cos  sin sin ‡ sin  cos †j ‡ …cos sin †k:

The relative velocity of gear 2 with respect to gear 1, as well as the position vector of contact point rp, can be deduced as:

Vrelˆ !2 …rpÿ Ek† ÿ !1 rp; (25)

rpˆ uk ‡ `nsa‡ n3ˆ …` cos  ÿ  sin  cos †i

‡ …` sin  ‡  cos  cos †j ‡ …u ‡  sin †k: (26)

In spherical gear sets with skew axes, the two rotating axes intersect at the center point of the spherical ball, and therefore, the length of the common perpendicular line of the axes is zero, meaning that Eˆ0 Substituting Eˆ0 into

Eq (19), therefore uˆ0 also Then Eq (25) and Eq (26) can be deduced as:

Vrelˆ !2 rpÿ !1 rp; (27)

rpˆ …` cos  ÿ  sin  cos †i ‡ …` sin  ‡  cos  cos †j

As n2is the unit normal vector of the N-plane, and Vrelis normal to the line of , Vrelis also normal to the N-plane Therefore, the vectors n2 and Vrel are mutually parallel vectors on the N-plane and may have different directions The mathematical relationship between n2and Vrelcan be represented as:

Substituting Eqs (23), (27) and (28) into Eq (29), and taking the k terms to be simpli®ed, Eq (29) can be deduced as:

2

‡  sin cos  sin ‡  sin2 sin  cos ˆ 0: (30) Then substituting Eq (17) into Eq (30) gives:

However, only if sin ˆ0, Eq (31) is equal to zero That means that in spherical gear sets with skew axes, the twisting angle of the N-plane (shown Fig 2) is 08 or 1808, and cos ˆ1 or ÿ1 As mentioned above, in spherical gear sets with skew axes it can be assumed that ˆ08, uˆ0 and Eˆ0

2 sin  ‡  sin cos  sin ‡  sin2 sin  cos ˆ 0 (31)

r2…r; 1† ˆ r 2‡ sin  sin sin 2

2‡ sin  sin cos 2

2

4

3

r1…r; 1† ˆ r …cos  sin  ‡ sin  cos cos †cos cos  cos  ÿ sin  cos sin 1‡ sin  sin sin 1

ÿ…cos  sin  ‡ sin  cos cos †sin 1‡ sin  sin cos 1

2

4

3

Substituting these relations into Eq (3), the constraint

gov-erning equation degenerates to

ÿ @@

1

‡ … sin sin † @`

 

‡ ` sin sin  ˆ 0: (32)

Eq (32) is the constraint governing equation of spherical

gear sets with skew axes From the geometric characteristics,

substituting Eqs (10)±(12) into Eq (32) yields

d

d1ˆ sin sin  1 ‡ tan  tan  ‡ ` sec2

d

d`

: (33)

Eq (33) is the constraint condition for gear 1: similarly it

can be developed analogously for gear 2 Therefore, the

constraint condition for gear 2 is

d

d2

2d

d`

: (34) Eqs (33) and (34) are two quasi-linear partial differential

equations of order unity, where the parameter may be a

constant, or may be a function of 1 In spur gear sets if is

constant, the pro®les of the gears are involutes, whereas if

ˆ/2‡3.51, the pro®les are cycloid In addition, when

ˆ1.6581‡0.51, there is a circular-arc dedendum; when

ˆ1.6581ÿ0.51, there is a full addendum circular-arc [13]

In this paper, it is assumed that the parameter is a function

of 1(or 2) i.e., ˆ(1) (or ˆ(2)) The solution of

Eqs (33) and (34) may be expressed as:

 ˆ sin 

Z

sin …1† d1‡ f1…r†; (35)

Z

sin …2† d2‡ f2…r†: (36) When the parameter  is separately formulated as a

function of r and 1(or 2), i.e., ˆ(r, 1), ˆ(r, 2),

then Eqs (35) and (36) can be derived as:

d

d1ˆ sin  sin …1†; (37)

d

dr ˆ

d

d

d

dr ˆ

d

where …d=dr†‰ f1…r†Š ˆ …d=dr†‰ f2…r†Š Eqs (37) and (38) are

the constraint equations for gear 1 Eqs (39) and (40) are the

constraint equations for gear 2

Assuming that the contact type is point contact, and that

the parameters of  and ` are mutually orthogonal, they are

always independent at any contacting instant so that

…@=@`† ˆ 0 Eq (32) then degenerates to

ÿ @

@1

‡ ` sin sin  ˆ 0: (41)

Substituting Eq (12) into Eq (41), the constraint

Eq (41) for gear 1 becomes

@

Here,

 ˆ

Z sin sin  d1‡ f1…r†; and ˆ …1†: (43) Analogously for gear 2, yields:

 ˆ

Z

2‡ f2…r†; and ˆ …2†:

(44) Furthermore, assuming that the pro®les of the gear sets are involutes, then the parameter is a constant if the initial value of 1and 2are equal to zero Thus Eqs (43) and (44) yield:

 ˆ 1sin sin  ‡ f1…r†; (45)

 ˆ 2sin sin … ÿ r† ‡ f2…r†: (46) Then Eqs (45) and (46) can be used as the constraint governing equations to develop the spherical gear set with skew axes

3 Technology of RP&M applied in forming spherical gear sets

The RP&M machines for three-dimensional duplicating machines are highly dependent on the electronic database input These systems take an electronic description of a three-dimensional object and reproduce that description into

a solid object If the description is inadequate, the part generated will also be inadequate To avoid the phenomenon

of ``garbage in and garbage out'', the geometric descriptions required for current RP&M equipment are provided by computer-aided design (CAD) systems Therefore, it is necessary to incorporate CAD into the RP&M environment, RP&M input ®les, data representation, part con®guration, etc [11]

3.1 CAD system integration within an RP&M environment RP&M model data requirements that are proposed cur-rently are non-ambiguous data descriptions of the part geometry to be generated Non-ambiguous data sets result

in unique possible interpretations [17], so the model data must facilitate the generation of closed paths and differenti-ate between the ``inside'' and ``outside'' of the part Using cross-hatching algorithms can create vectors that solidify the area between the part boundaries and borders Problems will occur if the part geometry is not completely closed because either adjacent surface vertices may not connect or whole surfaces may be missing Therefore, the RP&M system software must have the ability to close gaps, otherwise

the system could leave an opening that would negatively

affect part building because gaps in the borders will cause

the hatching vectors to be incomplete or escape outside the

part

The boundary data of a part must also convey the

orienta-tion of the solid area Surface normal informaorienta-tion pertaining

to the object's boundary is used to indicate the orientation of

the object's mass If this information is incorrect, walls with

zero thickness, or twisted surfaces with con¯icting,

impos-sible orientations that create a part called a ``MoÈbius Strip''

can occur [18] Therefore, an unambiguous geometrical

description is necessary for the model In this paper, the

authors use two kinds of solid-model CAD systems: one is

Parametric Technologies' PRO/ENGINEER, and the other

is SDRC I-DEAS

A solid-model can be de®ned as a geometric

representa-tion of a bounded volume This volume is represented

graphically via curves and surfaces, as well as

non-graphi-cally through a topological tree structure which provides a

logical relationship that is inherent in solid-models [19] The

topological data de®nes and maintains the connective

rela-tionships between the various faces and surfaces of

geome-try Each face of the object, including its normal orientation

with respect to the objects mass, is maintained Therefore,

solid-models, by de®nition, satisfy the requirements for

RP&M input

Most popular 3D solid-model CAD systems require

translated processors to create the STL ®le for an RP&M

system In PRO/E system, the users are allowed to create

STL ®les Firstly, the user must specify the degree of

resolution for the model curved surfaces by entering a

quality value in the CAD system's interface The value

can range from 1 to 10, with 10 being the highest resolution,

although higher resolution values generate larger STL ®le

sizes Using a triangle range density higher than eight seems

to give little improvement to the ®nal part A CAD system's

internal accuracy also affects STL accuracy [20] The

I-DEAS system calls this accuracy the point coincidence value

[19] This value is often determined in the system's default

start-up ®le Some systems allow users to interactively

change the model's accuracy With the PRO/E system,

one accuracy value applies to all features within the part

Therefore, geometries with relatively large and small radii

with bene®t from using a tighter accuracy value PRO/E will

regenerate all previous features using the new accuracy

value [21]

After creating STL part model ®les, the next step is to

con®gure these ®les located in the space of the vat used to

construct the part STL part ®les must always build supports

to hold the part in place whilst it is being generated For the

SL (stereolithography) system, supports are required for

every part because SL is an additive process using a

layer-by-layer approach occurring on the liquid resin

sur-face., and the resin is ®lled in a large vat To properly

constrain a given layer, it must be attached to the previous

layer The initial layer is attached to the platform that always

provides support Liquid-based RP&M systems must also be concerned with trapped volumes Trapped volumes are de®ned as spaces that hold liquid separate from the liquid

in the vat These regions may require special recoating parameters that slow the build rate A change in orientation can often eliminate the trapped volumes For example, a cup right side up contains a trapped volume, but when turned upside down, it does not If simple reorientation is imprac-tical, the CAD designer can drill (or cut) drain holes The holes can reduce the resin in the trapped volumes These holes can be plugged later, during the post-processing stage Using the above-mentioned detail technologies, perfect STL

®les are transferred from the CAD system to the RP&M's slice computer There, the RP&M operators can complete and form the prototypes of the CAD design parts

4 An example of spherical gear sets with skew axes formed by RP&M

Fig 4 shows a spherical gear set with two twisting intersecting axes The set is con®gurated in a shell spherical ball, with an inner spherical radius of 105 mm and an outer spherical radius of 115.5 mm The radii are conjugated contacting in the spherical space The common perpendi-cular distance is zero, and the twisting angle of the N-plane

as shown in Fig 2 relative to the pitch line (the ISA) is zero This indicates that sin ˆ0 and cos ˆ1 The twisting angle

of these two rotating axes is 16.2988, the speed ratio is ÿ1.33 and the constant pressure angle is 27.4858 From Eqs (13), (14) and (42), the pro®le surfaces of the couple gear are presented in Figs 5 and 6 Fig 5 shows the surface functions of Eq (13) rotating from 08 to 3608 along axis

L1 whilst Fig 6 shows the surface functions of Eq (14) rotating from 08 to 3608 along axis L2 Firstly, from Figs 5 and 6, the useful gear surfaces must be searched and picked

in rotating spherical space Obviously gear 1 is assumed

Fig 5 Ruled curve contours of a spherical gear.

as rotating counter clockwise 308 per tooth, whilst gear

2 rotates clockwise 408 per tooth By the gear law of

transmission:

N1…!1†averageˆ N2…!2†average: (47) The speed ratio mˆÿ1.33 is given, so that !1/!2ˆÿ40/ 30ˆÿ1.33 From Eq (47) if the number of teeth on gear 1 is

16, then the number of teeth on the pinion is 12

From Fig 5, and the above-mentioned angle, the parameter is a constant and if the initial values of 1 and 2are equal to zero, then the pro®le of gear 1 takes the rotating angle from 08 to 308, and the pinion from ÿ58 to ÿ458 to avoid serious undercut when they are meshing The synthesis of one tooth of the gearing contour pro®les of the gear and pinion are shown in Fig 7 The contour pro®les can

be rotation copied by the PRO/E CAD system to complete the circular-teeth in the spherical ball shell space, as shown

in Fig 8

As mentioned in the above section on RP&M technolo-gies, the gears couple may need to have some drain holes drilled for the inner trapped volumes and have supports built

if they are formed by the SL type machines Fig 9 shows the completed solid-models as built by PRO/E CAD systems

By the postprocessing of the PRO/E systems, these models have proposed to the SLA translator to translate the data base

of the PRO/E solid-model to STL ®les for RP&M machines

Fig 7 Two gearing teeth synthesized on the spherical gear shell blank.

Fig 6 Ruled curve contours of a spherical pinion.

Fig 8 A completed gear computer solid-model by the moving copy method.

Fig 9 Supports and drain holes of computer sold gear models for RP and M machines.

Entering the PRO/E SLA module, ®rstly the system must

specify the resolutions of the solid-models and the supports

To avoid excessively large STL ®les, as mentioned above,

the solid-models of parts can have higher resolution, whilst the supports do not The completed STL model ®les with triangle meshes are shown in Fig 10 These ®les can be transferred to RP&M machines to construct prototypes by

SL machines as shown in Fig 11

5 Conclusion

In this paper, the complicated meshing contour surfaces of spherical gear set with skew axes have been derived, their constraint governing equations have been systematically investigated, and the kinematic relationships between the relative velocity and the instantaneous screw axis have been presented This information is very useful for spherical gear design and manufacture The new technology of RP&M with the stereolithography method has been used to verify the completeness and correctness of the derived mathema-tical models for spherical gear sets

Acknowledgements The authors are grateful for research support from the National Science Council of the ROC through contract no NSC-87-2212-E-110-010, as well as assistance in the form-ing of rapid prototypform-ing models from the Aero Industry Development Company

References

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Fig 10 The SLA models using triangle density created by PRO/E

systems.

Fig 11 The RP and M prototypes for spherical gear sets with skew axes.