Geometry design of trapezoidal threaded variable pitch lead screws

Computers Math. Applic. Vol. 23, No. 1, pp. 65-73, 1992 Printed in Great Britain. All rights reserved

0097-4943/92 $5.00 + 0.00 Copyright(~ 1992 Pergamon Press plc

G E O M E T R Y D E S I G N OF T R A P E Z O I D A L T H R E A D E D VARIABLE PITCH LEAD SCREWS J E N - Y U LIU Department of Power Mechanical Engineering National Yuenlin Institute of Tedmo]ogy Huwei, Yuenlin, Taiwan 63208, R.O.C. HONG-SEN YAN

Department of MechanicalEngineering National Cheng Kung University Tainan, Talwan 70101, R.O.C. (Received Ms~ 1991)

A b s t r a c t - - O u r purpose is to derive equations for variable pitch helix and helicoids, based on sarew matrix transformations. Also mathematical models for the surface geometry of single, double, and Archlmedean double trapezoidal threaded variable pitch lead screws are presented. The result of this work is important in the computer-aided design and manufacturing of trapezoidal threaded variable pitch lead screws for industrial applications.

NOMENCLATURE Q

distancefrom originO! along the positiveX! axis

d

length of line AA1 or line A2A3 length of generating line X I , Yy , and Z / components of position vector ~ a typical point on variable pitch

r x , ry~ ~*z

P

~',3, 1~ ~i Fi,j l~i,j

F~

helicoid

o!x!r!z! s,1 s., s~, s .

fixed reference coordinate system displacement along Z / axis X ] , Y], and Z ! components of displacement g X ! , Y! , and Z ! components of I~" axis unit vectors of the cylindrical coordinate

[S z ! ,~ ] "~

unit vectors of the fixed reference coordinate position vector of point i generating llne ij surface of variable pitch helicoid i j final position vector of point i aftcsr variable pitch screw motion screw m~trix angle between generating line and zj axis parameter of generating llne rotation angle of variable pitch screw motion

I. I N T R O D U C T I O N Normally, the pitch of a lead screw is constant in industrial applications. However, the pitch of some lead screws are varied for special purposes. Then we have a variable pitch lead screw. W e have a trapezoidal threaded variable pitch lead screw when its thread profile is trapezoidal. Trapezoidal threaded variable pitch lead screws have several interesting engineering applications. They can be used as extruders for compressing rubbers and plastic [1-3], employed as a tool The authors are grateful to the National Science Council of the Republic of China for supporting this research under grant NSCS0-0422-E006-06. Typeset by ,A.A~TEX

65

66

J.-Y. Liu, H.-S. YAN

Figure 1. A typical trapezoidal threaded variable pitch lead screw.

for grinding coffee beans [4], and worked as an Archimedean screw for driving variable speed conveyors [5]. Figure 1 shows a typical trapezoidal threaded variable pitch lead screw [1]. In 1968, Litvin [6] pointed that curves (surfaces) applied in engineering mechanics can be defined as the locus of points (lines) generated by a point (line) of a moving body. And, the equations of such curves (surfaces) can be derived by the rules of coordinate transformation. As a result, he presented many surface equations of tooth profile for pinions and worms with constant pitches. In 1971, Chakraborty [7] proposed that a helicoid can be defined as the surface generated by the helical motion of a curve about the axis of the base cylinder, and he derived the expressions for a helical pinion with constant pitch helicoid. In 1975, Litvin [8] presented the expression for a constant pitch helicoid surface of the worm shaped cutter. In 1988, Tsay [9] set up a mathematical model for constant pitch helical gears. In 1990, Tsai [10] showed the geometric design of helicoids for a constant pitch worm gear set. However, the literature of geometric design for variable pitch lead screws is not generally available. Our purpose here is to derive mathematical models of trapezoidal threaded variable pitch lead screws based on the screw matrix method. 2. S C R E W MATRIX It is well known that the general three dimensional motion of a rigid body can be achieved by a rotation about an axis and a translation along the axis, simultaneously. Figure 2 shows that position vector ~" rotates about and slides along the l~ axis (through the origin O/ of fixed coordinate O/X/Y/Z/) with an angle ¢ and a displacement ~, respectively, to the final position ~'~. In homogeneous coordinates the position vector ~"can be represented by a column matrix as:

~.={r}=

[1:1 r~

,

where, r~, r~, and rz are the components of position vector ~' along X!, Y!, and Z!, respectively. And, the previous screw transformation can be achieved by screw matrix [11], [Sw,~], as follows. Let

w2v¢ +c¢

w~w~v¢ w~,wzv¢ -w~s¢ +wys¢ s.

w~%y¢

w]v¢

%w~v¢

+w,s¢

+c¢

-w~s¢

[Sw,~] =

w,w,v¢ wywzv¢ -wys¢

+w~s¢

Sy

(1)

wive +c¢

0 0 0 1 in which V ¢ = 1 - cos¢, C ¢ = cos¢, and S¢ = sine and W~, Wy, Wz, Sx, Sy and Sz are the components of I~ and ~ along X!, Y/, and ZI, respectively. Then:

~'= [Sw,~]{r).

(2)

If the relation between S and ¢ is nonlinear, then [Sw,~] is called a variable pitch screw matrix. W h e n I~ is coincident with the Z! axis, components W~, W~, W~, S~ and S~ become W = = 0,

Variable pitch lead screws

67

7; r'

Figure 2. Screw motion about l~ axis.

7,

•Sz y

/~-i/ Figure 3. Variable pitch helix.

Zf

~)f

~ Yf BII

////

I

I X~ Figure 4. Variable pitch helicoid.

68

J.-Y. LIU, H.-S. YAN

Wy = O, Wz = 1, S= = 0, and S v = 0, respectively. Substituting these quantities into Equation (1), the screw matrix for a screw motion about the Z! axis is:

0 0

1 0

"

3. VARIABLE P I T C H HELIX Let point P be located at a distance "a" from origin O] along the positive X! direction, as shown in Figure 3, and ~'p be the position vector of point P. Then, = a-f=

[a001] T.

The corresponding position of ~ after a screw transformation with respect to Z 1 axis, ~ , is:

(3) 0

0

When Sz is a linear function of ¢, a constant pitch helix is given by Equation (3). If Sz is a nonlinear function of ¢, Equation (3) represents a variable pitch helix. Equation (3) can also be expressed in cylindrical coordinate form as:

e$ = .~ + s.g,

(4)

where Er - cos ¢i" + sin ¢~, ~', = - sin ¢~" + cos Cj,

= ~.

(5)

Here, Er, E~ and /~ are the unit vectors of the cylindrical coordinate in r, ~b and z directions, respectively. A right-hand variable pitch helix is given by Equation (4). And, if S, is negative, Equation (4) becomes a left-hand variable pitch helix. 4. VARIABLE P I T C H H E L I C O I D As shown in Figure 4, line A B with length "£" is parallel to O / Y ! Z ! plane and intersects positive )(I axis at point A to yield a distance "a" from origin 01. Let 7 be the angle between line A B and the Z! axis. Then the equation of line A B is:

When line A B carries out a variable pitch screw motion about the Z! axis with constant angle 7, a variable pitch helicoid /~AB is generated. Then /~AB is given by:

= (~c~

-

~s~s¢)~ + (as~ + ~s~c¢)j + (~c~ + s.)~, O < y < e and O <_ S, <_ S,1.

(7)

Equation (7) can be expressed in cylindrical coordinates as: TlAB = aEr + ~SvE# + ( ~ C 7 + S , ) / ~ ,

O_<~
(7a)

Variable pitch lead screws

69

Zf

All Figure 5. Single trapezoidal threaded variable pitch lead screw. 5. SINGLE T R A P E Z O I D A L T H R E A D E D V A R I A B L E P I T C H LEAD S C R E W S Figure 5 shows a trapezoid ABB1A1 which intersects the positive X! axis at point A, and lines A B and B1A1 axe parallel to the O / Y / Z ! plane. The two angles 3' between line A B and the Z! axis and between line B1A1 and the Z! axis are equal. If angle ,/is constant during a variable pitch screw motion, variable pitch helicoids are generated by both generating lines A B and BIA1. Here the equation of line A B is:

(8) and that of line B1AI is:

(8a) Then, the helicoid generated by line A B is:

~AB = [S.,.,]{~AB} = (.C¢ -- ~S,/S¢)[+ (,S¢ + ,S,/C4')'] + (~C,/ + S.); = aE. + rlS,/E¢ + (r/C'/+ S.)['. 0 _< r/<_ e and 0 _< S. _< S.1

(9)

and the helicoid generated by line BIA1 is: RS,A, = [ S . , , , ] { . B , A , } = ( . C ¢ -- ,S,/S¢)[ + (.S¢ + ,S,/C¢)'j + (d - ,C,/ + S.)E

= a¢. + ~S,/¢~ + (d - ~C-/ + S.)~,

0< ~__.eand0

(lO)

Equations (9) and (10) represent variable pitch helicoids generated by lines A B and B1A1. These surfaces form a single trapezoidal threaded variable pitch lead screw. EXAMPLE 1. Design a single trapezoidal threaded variable pitch lead screw for Sz - +260ram and ~ - 520 °. The relation between ¢ and S~ is a modified trapezoidal curve [12]. Let the equal lengths of both generating lines A B and B1A1, angle ,/, and d be 15nun, 60°, and 21mm, respectively. Substituting these quantities into Equations (9) and (10), we obtain two variable pitch helicoids. Figure 6 shows the solid modeling, based on PATRAN Plus [13], of this single trapezoidal threaded variable pitch lead screw.

J.-Y. Lm, H.-S. YAN

70

Figure 6. Solid modeling of a single trapezoidal threaded variable pitch lead screw.

Zf !

jA3 / B2

"q l

A1

"
/ Xf Figure 7. Double trapezoidal threaded variable pitch lead screw.

6. D O U B L E T R A P E Z O I D A L T H R E A D E D V A R I A B L E P I T C H LEAD S C R E W S Figure 7 shows two trapezoids ABB1A1 and A2B2BaA3 located symmetrically at the same distance from O I Y I Z ] plane. As stated previously, all generating lines are at constant angle 7 with the Z! axis during a variable pitch screw motion. If these two trapezoids have a variable pitch screw motion about the Z! axis simultaneously, a double trapezoidal threaded variable pitch lead screw is obtained. The equations of generating lines AB, B1A1, A2B2, and BsAs, respectively, are:

{BtA,

=

a[ + ~$7~ + (d - ~C7)/~,

FA~B~ = -a-~ - ~STj + ~CTk,

0 _< ~ _< t, 0 < ~ < f,

(11)

Variable pitch lead screws

71

The corresponding variable pitch helicoids are:

,fiA,~ = [S.,,¢]{.A~} = (.C¢ -~S~S¢)~

+ ( . S ¢ + ~S~C¢)j + (~C~ + S.)f:

= aft + r l S 7 f 4, + (r/C7 + S,)k,

(12)

0 _< T/< t and 0 _< S, _< S,~.

EB, A, = [S,,.¢]{,B,A,} = (aCe -- r/STS¢)i'+ (aSfb -I- r/$70¢)3' + (d - riG7 -t- Sz)f¢

(13)

= aft + ~}S7E+ + (d - ~1C7 + Sz)g, 0 < rI <_ f and 0 _< Sz <_ S,1.

EA.8, = [S.,,¢]{~..B:} = (-.C¢

+ ~S~S¢)[ + ( - ~ S ¢ - nS~C¢)'] + (~C~ + S.)g

(14)

= - a t , - ~ s T ~ + (~c-: + s.)~, o < ,7 < t and 0 <_ s. < s.~. fiB.A. = [S.,,,]{~B.A.} = ( - - . C ¢ + ~S~S¢)'f + ( - - . S ¢ -- ~S~C¢)j + (d - ~C~ + S , ) ~

(i5)

= - a E r - rlS7f¢ + (d - riG 7 + S,)k, 0 <_ rl <_ t and 0 <_ Sz <_ S,~.

EXAMPLE 2. Design a double trapezoidal threaded variable pitch lead screw to meet the requirements of Example 1. Let the equal lengths of all four generating lines, angle 7, and d be 15mm, 60°, and 21mm, respectively. Substituting these quantities into Equations (12-15), we obtain four variable pitch helicoids. Figure 8 shows the solid modeling, based on PATRAN Plus, of the corresponding double trapezoidal threaded variable pitch lead screw.

Figure 8. Solidmodding of a double trapezoidal threaded variable pitch lead screw. 7. A R C H I M E D E A N DOUBLE T R A P E Z O I D A L T H R E A D E D VARIABLE P I T C H LEAD SCREWS If trapezoids ABB1A1 and A2B2B3Aa, which are located symmetrically at the same distance from the Z! axis on O/XtZ ! plane as shown in Figure 9, carry out a variable pitch screw motion, the Archimedean double trapezoidal threaded variable pitch lead screws can be generated.

72

J.-Y. L~u, H.-S. VAN

7~

~ B 3

A1

J

Figure 9. Archimedeandouble trapezoidal threaded variablepitch lead screw. The equations of the generating lines AB, B1A1 , A2B2 and B3A3, respectively, are:

~B,A, = (~ + ~ST)~"+ (a -- ~CT)f~, ~A,~, = - ( a + ~S~)~" + ~ C ~ ,

0 _<~ _
~B.~. = - ( . + ,s~)~ + (d

o _<~ _< ~.

-

~c~)g,

(16)

Then the corresponding Archimedean variable pitch helicoids are:

-~A8 = [S,,,,]{rAB} = (. + ~ s ~ ) c ~ + (~ + ~s~)s~I + (~c~ + s,)~

(17)

= (a + TIST)E, + (T/C7 + S,)/~', 0 _< T/_< f and 0 _< S, _< S,1.

E~,.,

=

[s.,,,]{r~,~,}

= (. + ~s~)c¢~+ (. + ~s~)s¢] + (d

-

~c7 + s,)~

(18)

= (a + ,/ST)E, + (d - r/C7 + S,)/~, 0 < r/< t and 0 < S, <_ S,,.

= - ( . + ~ s ~ ) c ¢ / - (. + ~s~)s¢-] + (~c~ +

s.)~

(19)

= - ( a + rlST)Er + (r/C7 + S,)/[', 0 _< ~/_< f and 0 <_ S, <_ S,~.

E~.A. = [S.,,~]{.~.~.} = - ( a + ~/$7)C¢i'- (a + FIST)Sd~ + (d - ~1C7 + S,)k = - ( a + ,/S7)Er + (d - r/C7 + S,)k, 0 _< r/_< t and 0 _< S, _< S,,.

(2o)

Variable pitch lead screws

73

EXAMPLE 3. Design an Archimedean double trapezoidal threaded variable pitch lead screw to meet the requirements of Example 1. Let the equal lengths of all four generating lines, angle 7, and d be 15mm, 60 °, and 21mm, respectively. Substituting these quantities into Equations (17-20), we have four variable pitch helicoids. Figure 10 shows the corresponding solid modeling.

Figure 1O. Solid modeling of Archimedean double trapezoidal threaded variable pitch lead screw.

8. CONCLUSION In conclusion, based on screw matrix transformation we mathematically derive the surface geometry of single, double, and Archimedean trapezoidal threaded variable pitch lead screws. The technique presented here can also be applied to other types of variable pitch lead screws with trapezoidal threads if the relationship ¢ = ¢ ( S z ) is specified. The result of this work is important in the computer-aided design and manufacturing of trapezoidal threaded variable pitch lead screws for various industrial applications. REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

G.B. Nichols, Ez4ruder, United States Patent No. 3310836, (1967). J. Blomet, Machine for The Extrusion of Paste Products, United States Patent No. 3327346, (1967). J.D. Weir, Screw Plasticiser with Variable Pitch Screw, United States Patent No. 3449793, (1967). S.L.L. Cailliot, Continuously Operable CoSec Making Machine, United States Patent No. 4074621, (1978). R.U. Ayres, R.P. Mckenna and S.G. Keahey, Pleated, Variable Speed Conveyor and Conveyor Driving Means, United States Patent No. 3903806, (1975). F.L. Litvin, Theory of Gearing, NASA, Washington, D.C., (1989). J. Chakraborty and B.S. Bhadoria, Design parameters for face gears, Journal of Mechanisms 6 (4), 435-445 (1971). F.L. Litvin and N.N. Krylov, Generation of tooth surfaces by two-parameter enveloping, Mechanism and Machine Theory l0 (3), 365-373 (1975). C.B. Tsay, Computer simulation and theoretical analysis of involute helical gears, Journal of the Chinese Society of Mechanical Engineers 9 (1), 35-42 (1988). Y.C. Tsai, L.M. Sulmg and H.L. Chang, A study on torque parameters and mathematical models of worm gear sets, Journal of the Chinese Society of Mechanical Engineers 11 (2), 135-145 (1990). C.H. Sub and C.W. Radcliffe, Kinematic and Mechanisms Design, Wiley, New York, (1978). R.A. Rothbart, Cams, Wiley, New York, (1956). PDA Engineering, Patran Plus, Cosla Mesa, (1987).