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Variable pitch screw drives
Mechanism and Machine TheoryVol. 13, PC,.523-531 © Pergamon Press Ltd., 1978. Printed in Great Bdtain
0094-114XI7811001-05231$02.0010
Variable Pitch Screw Drives R. F. Gilllet Received 20 April 1977
Abstract The propulsion of vehicles by a variable pitch screw parallel to the track has been proved in the past to be safe and efficient, but entailed a large and expensive screw shaft. Distributing the contact between the vehicle and the shaft amongst a series of roller followers mounted on a development of a Sarrus mechanism, reduces the problems causing the size and expense. In particular lateral reactions on the vehicle and the shaft can be balanced and gaps in the screw shaft, required for support bearings, can be crossed.
Introduction A MECHANICALmeans of propelling a stream of vehicles over short distances has been sought for many years. Several techniques are suitable for applications in mechanical handling and passenger transport where constant speeds are maintained: it is the provision for variation of speed along a route that poses the greater problems. Two passenger transport systems now under development are based on driven roller propulsion[I,2]. The vehicles move slowly through stations as a continuous rank, but speed up and space out between them. The safety, efficiency and reliability of roller propulsion has always been in doubt; so developers in the past have experimented with a continuous screw of varying pitch, parallel with the track to propel roller followere projecting from the vehicles (Fig. 1). This method has a high mechanical efficiency and provides absolute vehicle spacing. The most notable system of this type was the Adkins-Lewis Never-Stop railway built for the Wembley Empire Exhibition of 192413]. It was popular and reliable. Running costs were very low, but the construction cost was high, mainly on account of the large size of the screw (600 mm dia.) installed along the track. A screw drive of reasonable size and cost would be superior to existing forms of mechanical propulsion. This paper describes a type of mechanism for linking vehicles to a variable pitch screw drive that retains the advantages mentioned above, but enables the screw to be very much smaller.
Design The bulkiness of the Adkins-Lewis screw drive stemmed from its dependence on one roller follower per vehicle engaging with the propulsion screw. This dictated a coarse pitch screw which in turn complicated the problem of crossing gaps in the screw shaft required for support bearings. Figure 2 shows the constraints on maximum and minimum pitch. A large diameter screw must be used unless there are several roller followers per vehicle. If a number of followers are attached to each vehicle and each engages the screw, they can share the propulsion forces and so be smaller in size. They can also mesh with a screw of finer pitch. This screw can then spin faster with a lower torque than a coarse one. However, as the pitch of the screw will change with changes in vehicle speed, the roller followers on the vehicle must mounted in an expanding mechanism that can accommodate the changes in pitch. Where there is a redundant number of followers, and one is disengaged from the screw in a gap, the tDepartmentof Engineering,Universityof Warwick,CoventryCV4 7AL, England. 523
524
High Sl~¢d
Velocity
Low Speed Distance
E][3DD
71
R
OOOOOOO0000 OOOOOO OO OOOOOO000000OOOOO OOO OOO
Rgure 1. Driven roller propulsion and screw propulsion can both provide variation in vehicle velocity according to position on the track. Low=prod
Rotational velocity =ira
"
Lowspeed Low ipGcd pitch
determinn minimum pitch,
Normal
force
-
Thrust cos j~
Thc maximumpitch angle detcmlnes the maximum contact forct.
Rgure 2. The influence of pitch on rotational velocity and pitch angle on reaction forces. others can maintain propulsion and retain the disengaged follower in correct relation to the screw. In addition, followers distributed around the shaft can restrain all but axial movement of the follower mechanism. Roller followers, arranged in opposed pairs either side of the shaft on a two start thread incur equal and opposite tangential forces when driven by the shaft. Thus lateral forces are balanced, except for an unbalanced couple (Fig. 3). The follower array can act as a separate tractor riding on the screw shaft, that needs only to be attached to the main vehicle by a drag link and a torque arm. The vehicle could even depend on the tractor for its own steering guidance (Fig. 4). The mechanism holding the followers needs to space opposed follower pairs equally, but allow that spacing to change. In effect the mechanism must restrain a series of parallel plates perpendicular to the drive shaft with equal but varying spacings. This is most easily achieved with a development of a Sarrus mechanism to retain the plates parallel combined with a pantograph mechanism to keep them equi-spaced (Fig. 5).
525 ltQl't 1.
_ . . %// _ Pitch start 2.
ppcsed follower= haw hb~,,IQncedIQt.ral thrusts but an unbalanced couple.
I
Figure 3. Opposed followers running on two thread starts.
v
-'~ Follower
_
/
~- ....... ........ I~_
.
~
J
~
3 Main drag link J
I
. ...
m¢clmism I " ~ ' - - - - vCncle (or tractor) I
r ..... ~'~ L . . . . . . .k..~lr~
Rgure 4. A method of attaching the propulsion mechanism to the vehicle.
Shaft
s
I
~
I/
/
s ~ i . g , r , . u ,qua~ but can vary b size
Sarrus °".mcchanism
l~ntograph anism
I
Rgure 6. One form of mechanism (a development of a Sarrus mechanism) that can retain and space three sets of opposed followers.
526 Errors between Thread and Follower Spacing On a constant velocity section of screw, threads are equi-spaced, but on an accelerating section the thread pitch increases and cannot be matched by equi-spaced followers. The resulting errors must be accommodated by the followers and their suspensions, but they can be very small. Consider two successive time intervals of length t during which a follower meshed with a screw accelerates at constant rate A from an initial velocity u. If D1 is the distance travelled by the follower in time t and/92 is the distance travelled by time 2t. Dt = ut + ~ A t 2
(!)
1)2 = 2ut + 2 A t z.
(2)
If the difference between the distances travelled in successive time intervals is AD AD = DE - 2Dr.
(3)
Substituting eqns (1) and (2) into (3) A D = A t 2.
(4)
If t is expressed as a fraction, q, of the time for the screw to rotate once Aq z
(5) where Z is the rotational velocity of the screw. AD also represents the difference in spacing of adjacent pairs of rollers in a follower mechanism if they are to mesh with a thread that provides an acceleration A. AD increases with the square of the axial span of a follower pair and (for a given acceleration) falls with the square of shaft speed. If A is 1.3 m/s 2 and Z is 7.85/s, when q is 1 AD is 21 ram, when q is ~ AD is 1.3 mm. The roller followers might be 25 mm dia. and thus the error when q is ~ is substantially smaller than a roller diameter.
Gap Crossing The manner in which followers are spaced about the shaft influences the length of carrier mechanism required to hold them, the gap width that they can jump and the circumferential space left at bearings for their support brackets. If adjacent pairs of opposed followers are set at 90 ° to each other about the shaft axis, (i.e. q is ~), 3 pairs occupy ½ a pitch. The mechanism can cross a gap of ~ of a pitch and a 90° arc is left for the bearing support bracket. Other follower layouts are possible as shown in Figs. 6 and 7. When a pair of followers disengages from the screw in a gap, it not only ceases to propel its vehicle, it also ceases to restrain its carrier. The remaining pairs continue to provide sufficient restraint, but the disengaged followers can still be guided by a stationary spline surrounding a support bearing (Figs. 8 and 9). A long carrier with more followers can jump large gaps, but the mechanism becomes more expensive and the errors to be tolerated increase. It appears that 3 pairs of followers are best, but it can be seen that initial experiments were done with 4 pairs that can cross gaps of twice the width (Figs. 7 and 9). There are many other arrangements of followers together with many mechanisms to space them.
Two-statl Threads and Variations in Pitch Opposed pairs of followers running on two start threads have been shown above to contribute to the feasibility of this type of mechanism. However, if the pitch of a screw varies, there can be three forms of a two-start thread. (1) Fixed Axial Displacement. A second thread is a fixed axial length from the first so that on a length of screw of coarse pitch there are two closely spaced threads then a long space before the next pitch. This type of thread was used by Wilson[4].
527 j/j/Opposed
follo~r pair
Acklitio~]l fourth pair of roller= optional, but doubles the size of gap that can b¢ jumped.
with Q ~honism length of of a pitch.
Figure 6. Alternative layouts for the roller followers around the shaft.
Figure 7. A model of a carrier mechanism with 4 sets of opposed followers.
(2) Fixed Angular Displacement. The second thread is a fixed angle from the first, usually 180°, in which case it is directly opposed to the first in a section through the shaft. (3) Proportional Axial Displacement. The second thread is at the mid point in axial distance (or any other proportion) between two pitches of the first thread. It does not necessarily lie directly opposed to the first in a section through the shaft. Fixed angular displacement threads can be used to propel opposed followers. Proportional axial displacement threads can provide three points of contact parallel to the screw that remain equi-spaced when the pitch changes and may prove useful in other applications.
528
~
Oplxlmfol dlov~rpair, Support bearing
¢lbeari~l CrOll
I~lr. folower
cldjac4ml,
Support bl~chrt Rgure 8. A cross section through a shaft support bracket that includes a guide spline for the roller followers.
Figure 9. The model of a follower mechanism straddling a screw support bracket,
529
Thread Cross-Section and Roller Followers The simplest follower is a cylindrical roller running in a square section groove. However, a pair of opposed bevelled rollers loaded against one another and riding in triangular grooves can absorb axial or radial errors in the thread by moving in their own axis. A single bevelled roger in a triangular groove jams, but two circumferentially adjacent rollers occupy a narrow groove when the pitch is fine and a fatter one at coarse pitch and do not jam (Fig. 10). One roller propels during acceleration the other brakes the vehicle during deceleration. Bevelled rollers can also maintain line contact between the thread and themselves with little scuffing.
C"=--------..-
- - -
..
Rgure 10. An adjacent pair of followers spaced circumferentially. They occupy a narrow groove when the pitch is fine which is also the stage at which it is desirable to keep the groove as narrow as possible.
Efficiency The high efficiency of roller screw drives comes as a surprise to those accustomed to the low efficiency of a plain screw. It can be seen from Fig. 11 that the efficiency of a roller screw may not drop below 95% until the pitch angle reaches about 70*. It can also be seen that the efficiency of balanced pairs of followers is better than an unbalanced follower with an external means of reaction. The efficiency of an unbalanced follower with external reaction surface can be found as follows: assume that a single follower propels a vehicle and the lateral force upon it is reacted by a roller running on a wall parallel to the screw shaft (Fig. 12). If the screw pitch
I.O O.9 ~ O.8 Efficient/ 0.7 O-6 0-5 O4
Balancedopposedfollowers. Unbalancedfollowersusingan ¢xternollateralrestraint.
~%~
,4J.- O.O1
O'3I
0.2 0.1 0
I
go
I
70
I
60
I
I
so
40
!
30
I
I
lo
Io
0
PitchAn~le Rgure 11. The efficiency of balanced and unbalanced followers plotted against pitch angle.
530 i
DimKtiu of s=mw
f
IsF
Direction of movement of follower mechanism
Direction of movement of follower R c h a n i m
Rgure 12. Forces on unbalanced and balanced followers. angle is ~b, the thread reaction P, the wall reaction Q and both rollers have coefficients of friction/z, then resolving perpendicularly to the shaft F = P sin ~b+ ~P cos 4~.
(6)
If the resultant force along the shaft is R R = P cos $ - / z P sin ~b-/~F.
(7)
Substituting equation (6) in (7) R = P cos $ - 2/~P sin $ -/~2p cos $.
(8)
The efficiency of the device, while accelerating, r/, can be expressed as
~/=
R tan $ F
(9)
Substituting from eqns (6) and (8), this can be expressed as (1 - / z tan ok) (tan ~ +/z) n - (1 +/~ cotS) /~ (1 +/z cot 4ff _
(10)
Similarly for a pair of opposed followers, assume that each runs on a different start of a two start thread. Their reactions laterally to the shaft will be equal and opposite. The torque that the reactions apply to the vehicle need not increase resistance to motion. Using similar notation to the unbalanced example above F = P s i n s +/~P cos $
(11)
531
R = 2(P cos 4' - t~P sin 4,)
(12)
(1 - / z tan 4,) *7 - (1 +/z cot 4,)"
(13)
_
The efficiency during deceleration can be found in a similar manner.
Curving The maximum curvature of the track is limited by: the length of shaft elements between couplings, the type of coupling, the matching errors created between the followers and the thread and --if a passenger vehicle--the tolerance of passengers to lateral acceleration. Standard gear couplings can tolerate 1° of misalignment and this gives smaller matching errors between followers and thread than those experienced during typical rates of acceleration (1.0-1.5 m/s2). If it is assumed that 1° is the maximum tolerable misalignment, the number of couplings used in a given change of direction will not be influenced by the radius of the curve.
Conclusions Mechanical forms of variable speed linear drive are appropriate when a series of vehicles must pass through several operations conducted at different speeds. This occurs on production lines and where loading and unloading is conducted at slow speed. A variable pitch screw can provide a linear drive that is compact, mechanically efficient and gives exact vehicle control. Unlike other drives, vehicles that slow down do not dissipate their kinetic energy as heat in brakes, but feed it via the screw to those increasing in speed. Most of the components are standard. The screw shaft, the major novel item, can be made by known methods. The drive technique rests on distributing the thrust amongst several roller followers. This is an established method for fixed pitch screws, but carries additional advantages for variable pitch ones. Correct distribution and freedom for followers provides (1) gap jumping; (2) balanced lateral forces; (3) self guidance of the drive mechanism, (4) finer pitches, smaller forces and lighter components. Speed ratios of 1 : 20 are possible. Higher ratios require a transfer to a shaft of higher speed with a finer pitch. Maximum rates of acceleration are determined by the provisions made for matching the followers to the thread, 6 mls z should be possible.
References 1. Poma 2000 s.a. Grenoble, Manufacturers Literature (1975). 2. SAVEC. Systeme de Transport Urbain. Versailles, Manufacturers Literature (1975). 3. W. Y. Lewis, Proposal to Ville de Paris Competition for the Establishment of a Mechanical System of Continuous Transport for passenger service in the city of Paris. London (1922). 4. D. G. Wilson, A. M. Valaas and L M. Malarkey, Variable.Pitch Screw Accelerators for Synchronous Transportation Systems. Massachusetts Institute of Technology (1972).
SCHRAUB~ANTRIEBE
MIT VER/4NDERLICHER STEIGUNG
R. F. Gillie Kurzfassun~ - Es werden gleitkontaktlose Schraubenantriebe mit ver~nderlicher Stelgung beschrleben, die als Linearantriebe geeignet sind und wesentliche Vor~eile gegenGber bisherigen Ausf~hrungen besitzem. Sie slnd kompakt und leistungsf~hig. Das haupts~chlich neue Element ist die Schraubenspindel! die meisten der Gbrigem Elemente sind Standardteile. Der Wirkungsgrad liegt bis zu einem Steigungswinkel von 70 ° bel 90 ... 95 %. Die Anordnung der Rollen, Lager und Verstellelemente wird behandelt, desgleichen das Geschwindigkeitsverh~ltnis. Mit eimem Spimdeldurchmesser yon 150 mm (blsher 600 mm) kSnnen Geschwindigkeitsver~nderungen von q : 20 ezTelcht werden.
0094-114XI7811001-05231$02.0010
Variable Pitch Screw Drives R. F. Gilllet Received 20 April 1977
Abstract The propulsion of vehicles by a variable pitch screw parallel to the track has been proved in the past to be safe and efficient, but entailed a large and expensive screw shaft. Distributing the contact between the vehicle and the shaft amongst a series of roller followers mounted on a development of a Sarrus mechanism, reduces the problems causing the size and expense. In particular lateral reactions on the vehicle and the shaft can be balanced and gaps in the screw shaft, required for support bearings, can be crossed.
Introduction A MECHANICALmeans of propelling a stream of vehicles over short distances has been sought for many years. Several techniques are suitable for applications in mechanical handling and passenger transport where constant speeds are maintained: it is the provision for variation of speed along a route that poses the greater problems. Two passenger transport systems now under development are based on driven roller propulsion[I,2]. The vehicles move slowly through stations as a continuous rank, but speed up and space out between them. The safety, efficiency and reliability of roller propulsion has always been in doubt; so developers in the past have experimented with a continuous screw of varying pitch, parallel with the track to propel roller followere projecting from the vehicles (Fig. 1). This method has a high mechanical efficiency and provides absolute vehicle spacing. The most notable system of this type was the Adkins-Lewis Never-Stop railway built for the Wembley Empire Exhibition of 192413]. It was popular and reliable. Running costs were very low, but the construction cost was high, mainly on account of the large size of the screw (600 mm dia.) installed along the track. A screw drive of reasonable size and cost would be superior to existing forms of mechanical propulsion. This paper describes a type of mechanism for linking vehicles to a variable pitch screw drive that retains the advantages mentioned above, but enables the screw to be very much smaller.
Design The bulkiness of the Adkins-Lewis screw drive stemmed from its dependence on one roller follower per vehicle engaging with the propulsion screw. This dictated a coarse pitch screw which in turn complicated the problem of crossing gaps in the screw shaft required for support bearings. Figure 2 shows the constraints on maximum and minimum pitch. A large diameter screw must be used unless there are several roller followers per vehicle. If a number of followers are attached to each vehicle and each engages the screw, they can share the propulsion forces and so be smaller in size. They can also mesh with a screw of finer pitch. This screw can then spin faster with a lower torque than a coarse one. However, as the pitch of the screw will change with changes in vehicle speed, the roller followers on the vehicle must mounted in an expanding mechanism that can accommodate the changes in pitch. Where there is a redundant number of followers, and one is disengaged from the screw in a gap, the tDepartmentof Engineering,Universityof Warwick,CoventryCV4 7AL, England. 523
524
High Sl~¢d
Velocity
Low Speed Distance
E][3DD
71
R
OOOOOOO0000 OOOOOO OO OOOOOO000000OOOOO OOO OOO
Rgure 1. Driven roller propulsion and screw propulsion can both provide variation in vehicle velocity according to position on the track. Low=prod
Rotational velocity =ira
"
Lowspeed Low ipGcd pitch
determinn minimum pitch,
Normal
force
-
Thrust cos j~
Thc maximumpitch angle detcmlnes the maximum contact forct.
Rgure 2. The influence of pitch on rotational velocity and pitch angle on reaction forces. others can maintain propulsion and retain the disengaged follower in correct relation to the screw. In addition, followers distributed around the shaft can restrain all but axial movement of the follower mechanism. Roller followers, arranged in opposed pairs either side of the shaft on a two start thread incur equal and opposite tangential forces when driven by the shaft. Thus lateral forces are balanced, except for an unbalanced couple (Fig. 3). The follower array can act as a separate tractor riding on the screw shaft, that needs only to be attached to the main vehicle by a drag link and a torque arm. The vehicle could even depend on the tractor for its own steering guidance (Fig. 4). The mechanism holding the followers needs to space opposed follower pairs equally, but allow that spacing to change. In effect the mechanism must restrain a series of parallel plates perpendicular to the drive shaft with equal but varying spacings. This is most easily achieved with a development of a Sarrus mechanism to retain the plates parallel combined with a pantograph mechanism to keep them equi-spaced (Fig. 5).
525 ltQl't 1.
_ . . %// _ Pitch start 2.
ppcsed follower= haw hb~,,IQncedIQt.ral thrusts but an unbalanced couple.
I
Figure 3. Opposed followers running on two thread starts.
v
-'~ Follower
_
/
~- ....... ........ I~_
.
~
J
~
3 Main drag link J
I
. ...
m¢clmism I " ~ ' - - - - vCncle (or tractor) I
r ..... ~'~ L . . . . . . .k..~lr~
Rgure 4. A method of attaching the propulsion mechanism to the vehicle.
Shaft
s
I
~
I/
/
s ~ i . g , r , . u ,qua~ but can vary b size
Sarrus °".mcchanism
l~ntograph anism
I
Rgure 6. One form of mechanism (a development of a Sarrus mechanism) that can retain and space three sets of opposed followers.
526 Errors between Thread and Follower Spacing On a constant velocity section of screw, threads are equi-spaced, but on an accelerating section the thread pitch increases and cannot be matched by equi-spaced followers. The resulting errors must be accommodated by the followers and their suspensions, but they can be very small. Consider two successive time intervals of length t during which a follower meshed with a screw accelerates at constant rate A from an initial velocity u. If D1 is the distance travelled by the follower in time t and/92 is the distance travelled by time 2t. Dt = ut + ~ A t 2
(!)
1)2 = 2ut + 2 A t z.
(2)
If the difference between the distances travelled in successive time intervals is AD AD = DE - 2Dr.
(3)
Substituting eqns (1) and (2) into (3) A D = A t 2.
(4)
If t is expressed as a fraction, q, of the time for the screw to rotate once Aq z
(5) where Z is the rotational velocity of the screw. AD also represents the difference in spacing of adjacent pairs of rollers in a follower mechanism if they are to mesh with a thread that provides an acceleration A. AD increases with the square of the axial span of a follower pair and (for a given acceleration) falls with the square of shaft speed. If A is 1.3 m/s 2 and Z is 7.85/s, when q is 1 AD is 21 ram, when q is ~ AD is 1.3 mm. The roller followers might be 25 mm dia. and thus the error when q is ~ is substantially smaller than a roller diameter.
Gap Crossing The manner in which followers are spaced about the shaft influences the length of carrier mechanism required to hold them, the gap width that they can jump and the circumferential space left at bearings for their support brackets. If adjacent pairs of opposed followers are set at 90 ° to each other about the shaft axis, (i.e. q is ~), 3 pairs occupy ½ a pitch. The mechanism can cross a gap of ~ of a pitch and a 90° arc is left for the bearing support bracket. Other follower layouts are possible as shown in Figs. 6 and 7. When a pair of followers disengages from the screw in a gap, it not only ceases to propel its vehicle, it also ceases to restrain its carrier. The remaining pairs continue to provide sufficient restraint, but the disengaged followers can still be guided by a stationary spline surrounding a support bearing (Figs. 8 and 9). A long carrier with more followers can jump large gaps, but the mechanism becomes more expensive and the errors to be tolerated increase. It appears that 3 pairs of followers are best, but it can be seen that initial experiments were done with 4 pairs that can cross gaps of twice the width (Figs. 7 and 9). There are many other arrangements of followers together with many mechanisms to space them.
Two-statl Threads and Variations in Pitch Opposed pairs of followers running on two start threads have been shown above to contribute to the feasibility of this type of mechanism. However, if the pitch of a screw varies, there can be three forms of a two-start thread. (1) Fixed Axial Displacement. A second thread is a fixed axial length from the first so that on a length of screw of coarse pitch there are two closely spaced threads then a long space before the next pitch. This type of thread was used by Wilson[4].
527 j/j/Opposed
follo~r pair
Acklitio~]l fourth pair of roller= optional, but doubles the size of gap that can b¢ jumped.
with Q ~honism length of of a pitch.
Figure 6. Alternative layouts for the roller followers around the shaft.
Figure 7. A model of a carrier mechanism with 4 sets of opposed followers.
(2) Fixed Angular Displacement. The second thread is a fixed angle from the first, usually 180°, in which case it is directly opposed to the first in a section through the shaft. (3) Proportional Axial Displacement. The second thread is at the mid point in axial distance (or any other proportion) between two pitches of the first thread. It does not necessarily lie directly opposed to the first in a section through the shaft. Fixed angular displacement threads can be used to propel opposed followers. Proportional axial displacement threads can provide three points of contact parallel to the screw that remain equi-spaced when the pitch changes and may prove useful in other applications.
528
~
Oplxlmfol dlov~rpair, Support bearing
¢lbeari~l CrOll
I~lr. folower
cldjac4ml,
Support bl~chrt Rgure 8. A cross section through a shaft support bracket that includes a guide spline for the roller followers.
Figure 9. The model of a follower mechanism straddling a screw support bracket,
529
Thread Cross-Section and Roller Followers The simplest follower is a cylindrical roller running in a square section groove. However, a pair of opposed bevelled rollers loaded against one another and riding in triangular grooves can absorb axial or radial errors in the thread by moving in their own axis. A single bevelled roger in a triangular groove jams, but two circumferentially adjacent rollers occupy a narrow groove when the pitch is fine and a fatter one at coarse pitch and do not jam (Fig. 10). One roller propels during acceleration the other brakes the vehicle during deceleration. Bevelled rollers can also maintain line contact between the thread and themselves with little scuffing.
C"=--------..-
- - -
..
Rgure 10. An adjacent pair of followers spaced circumferentially. They occupy a narrow groove when the pitch is fine which is also the stage at which it is desirable to keep the groove as narrow as possible.
Efficiency The high efficiency of roller screw drives comes as a surprise to those accustomed to the low efficiency of a plain screw. It can be seen from Fig. 11 that the efficiency of a roller screw may not drop below 95% until the pitch angle reaches about 70*. It can also be seen that the efficiency of balanced pairs of followers is better than an unbalanced follower with an external means of reaction. The efficiency of an unbalanced follower with external reaction surface can be found as follows: assume that a single follower propels a vehicle and the lateral force upon it is reacted by a roller running on a wall parallel to the screw shaft (Fig. 12). If the screw pitch
I.O O.9 ~ O.8 Efficient/ 0.7 O-6 0-5 O4
Balancedopposedfollowers. Unbalancedfollowersusingan ¢xternollateralrestraint.
~%~
,4J.- O.O1
O'3I
0.2 0.1 0
I
go
I
70
I
60
I
I
so
40
!
30
I
I
lo
Io
0
PitchAn~le Rgure 11. The efficiency of balanced and unbalanced followers plotted against pitch angle.
530 i
DimKtiu of s=mw
f
IsF
Direction of movement of follower mechanism
Direction of movement of follower R c h a n i m
Rgure 12. Forces on unbalanced and balanced followers. angle is ~b, the thread reaction P, the wall reaction Q and both rollers have coefficients of friction/z, then resolving perpendicularly to the shaft F = P sin ~b+ ~P cos 4~.
(6)
If the resultant force along the shaft is R R = P cos $ - / z P sin ~b-/~F.
(7)
Substituting equation (6) in (7) R = P cos $ - 2/~P sin $ -/~2p cos $.
(8)
The efficiency of the device, while accelerating, r/, can be expressed as
~/=
R tan $ F
(9)
Substituting from eqns (6) and (8), this can be expressed as (1 - / z tan ok) (tan ~ +/z) n - (1 +/~ cotS) /~ (1 +/z cot 4ff _
(10)
Similarly for a pair of opposed followers, assume that each runs on a different start of a two start thread. Their reactions laterally to the shaft will be equal and opposite. The torque that the reactions apply to the vehicle need not increase resistance to motion. Using similar notation to the unbalanced example above F = P s i n s +/~P cos $
(11)
531
R = 2(P cos 4' - t~P sin 4,)
(12)
(1 - / z tan 4,) *7 - (1 +/z cot 4,)"
(13)
_
The efficiency during deceleration can be found in a similar manner.
Curving The maximum curvature of the track is limited by: the length of shaft elements between couplings, the type of coupling, the matching errors created between the followers and the thread and --if a passenger vehicle--the tolerance of passengers to lateral acceleration. Standard gear couplings can tolerate 1° of misalignment and this gives smaller matching errors between followers and thread than those experienced during typical rates of acceleration (1.0-1.5 m/s2). If it is assumed that 1° is the maximum tolerable misalignment, the number of couplings used in a given change of direction will not be influenced by the radius of the curve.
Conclusions Mechanical forms of variable speed linear drive are appropriate when a series of vehicles must pass through several operations conducted at different speeds. This occurs on production lines and where loading and unloading is conducted at slow speed. A variable pitch screw can provide a linear drive that is compact, mechanically efficient and gives exact vehicle control. Unlike other drives, vehicles that slow down do not dissipate their kinetic energy as heat in brakes, but feed it via the screw to those increasing in speed. Most of the components are standard. The screw shaft, the major novel item, can be made by known methods. The drive technique rests on distributing the thrust amongst several roller followers. This is an established method for fixed pitch screws, but carries additional advantages for variable pitch ones. Correct distribution and freedom for followers provides (1) gap jumping; (2) balanced lateral forces; (3) self guidance of the drive mechanism, (4) finer pitches, smaller forces and lighter components. Speed ratios of 1 : 20 are possible. Higher ratios require a transfer to a shaft of higher speed with a finer pitch. Maximum rates of acceleration are determined by the provisions made for matching the followers to the thread, 6 mls z should be possible.
References 1. Poma 2000 s.a. Grenoble, Manufacturers Literature (1975). 2. SAVEC. Systeme de Transport Urbain. Versailles, Manufacturers Literature (1975). 3. W. Y. Lewis, Proposal to Ville de Paris Competition for the Establishment of a Mechanical System of Continuous Transport for passenger service in the city of Paris. London (1922). 4. D. G. Wilson, A. M. Valaas and L M. Malarkey, Variable.Pitch Screw Accelerators for Synchronous Transportation Systems. Massachusetts Institute of Technology (1972).
SCHRAUB~ANTRIEBE
MIT VER/4NDERLICHER STEIGUNG
R. F. Gillie Kurzfassun~ - Es werden gleitkontaktlose Schraubenantriebe mit ver~nderlicher Stelgung beschrleben, die als Linearantriebe geeignet sind und wesentliche Vor~eile gegenGber bisherigen Ausf~hrungen besitzem. Sie slnd kompakt und leistungsf~hig. Das haupts~chlich neue Element ist die Schraubenspindel! die meisten der Gbrigem Elemente sind Standardteile. Der Wirkungsgrad liegt bis zu einem Steigungswinkel von 70 ° bel 90 ... 95 %. Die Anordnung der Rollen, Lager und Verstellelemente wird behandelt, desgleichen das Geschwindigkeitsverh~ltnis. Mit eimem Spimdeldurchmesser yon 150 mm (blsher 600 mm) kSnnen Geschwindigkeitsver~nderungen von q : 20 ezTelcht werden.