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The tilting couple inherent to power screw-nut systems
Wear, 39 (1976) 277 - 264 0 Elsevier Sequoia S. A., Lausanne - Printed in the Netherlands
277
THE TILTING COUPLE INHERENT TO POWER SCREW-NUT SYSTEMS
H. R. EL-SAYED
and H. A. KHATAAN
Production Engineering Alexandria (U.A.R.) (Received
February
Department,
Faculty of Engineering,
7, 1975; in final form February
Alexandria
University,
13, 1976)
Summary The load-carrying capacity of the externally pressurized power screwnut system represents the axial component of its normal load-carrying capacity. The tangential component is responsible for the restoring torque applied to the nut to prevent its possible rotation. The radial component, which seems to be balanced within the engaged length of the nut, results in a tilting couple which tends to misalign the working axes of the power screw and nut. An analytical study dealing with the generation and magnitude of the tilting couple is presented. Possible methods of eliminating its undesirable effects are considered.
1. Introduction The externally induced pressure in the lubricant film, which separates the working flank surfaces of the threads of the externally pressurized power screw-nut system, gives rise to the appearance of a set of normal load elements acting on a definite helix of action [l - 41. When dealing with the functions of the different components of the normal load elements, several authors [5 - 91 disregarded the effects of the radial components and considered them to be balanced within the engaged length of the nut. This conclusion may be drawn from a study of the distribution of these components only along the axial projection of the system. The radial components of the normal load elements tend to misalign the power screw and nut axes and this has been investigated. 2. Concept and magnitude of the tilting couple To illustrate the concept of the tilting couple, it is necessary to explain the method by which the power screw thread flank surface may be generated.
Fig, 1. Generator line of the screw thread flank surface.
Fig. 2. Generation
of the screw flank surface.
2.1, Generation of the screw thread flank surface The thread flank surface of the power screw can be generated by rotating a specified line (a surface generator) about an axis in a definite manner (Fig. 1). The surface generator is specified by the following. (a) It should be of the same length as the radial length I of the thread flank to be generated. (b) It should intersect the plane perpendicular to the axis at a constant angle equal to the thread angle #. (c) It should remain a certain distance from the axis during rotation, which will be the 2 axis of the surface generated.
279
, ,’ ; 8 5
‘. ‘.
Fig. 3. Mathematical representation of the thread flank surface.
The surface generator should be rotated in such a manner that, when its projection on a plane perpendicular to the axis rotates through an angle 8, the point of intersection 0 of the generator extension with that axis moves linearly along it for a distance equal to (0/2n) p, where p is the pitch of the thread to be generated. In this way, the paths of the different points lying on the surface generator, and/or its extension, form a set of concentric helixes intersecting the generator line at right angles (Fig. 2). Each helix has a helix angle X, which depends upon its distance r from the axis. Therefore, the path of any point lying on the screw thread flank surface (Fig. 3) may be expressed by 2=(8/2n)p--tan4
(1)
Accordingly, the surface formed between any two neighbouring helixes represents a screw flank surface. Thus the screw flank surface may be defined as that surface bounded by the intersection of the prescribed generated surface with two coaxial cylinders which are symmetrically set about the 2 axis, and whose radii are ri and rO. 2.2. The tilting couple Due to the absence of dry friction, the tilting couple occurs in both the stationary and running conditions of externally pressurized power screw-nut systems. The stationary working condition of the system is considered first.
280
Fig. 4. Projection of the stationary helix of action on three perpendicular planes.
Figure 4 shows the stationary helix of action [2] generated as outlined. The projection of the stationary helix of action on plane I is a circle whose radius rr is that of the helix itself. The projections of the helix on planes II and III are sine and cosine curves. They may be represented mathematically by the following parametric equations: sine
r..II
=r,
zii
=
(e/277)P
riii
=
r, cos 0
3ii
= (~/2n)
t
(2)
and P
t
(3)
Figure 5 shows the projections of the stationary helix of action, for one thread, on planes I, II and III. The distribution of the corresponding acting components is also shown on each projection. Figure 5 clearly shows that a tilting couple acting on the power screw-nut system will occur. For an integral number of threads m, which may be involved within the engaged length of the nut, the total tilting couple acting on the system is given by T, = m L roftc where ft, = tan c)tan A,,, L has been defined previously [ 21 and A, is the outer helix angle in degrees. The magnitude of the total tilting couple acting on the system during running conditions may be derived similarly:
(4)
281
) Fm’
I-+I
0 Ill
Fig. 5. Projections of the stationary helix of action combined with their radial component elements.
T, = m where F,,
W roFtc
(5)
= (F,/F,)tan A, and W, F, and F, have been defined previously
r31. 3. Elimination of the tilting couple effect Figure 6 shows two possible methods for eliminating the undesirable effect of the tilting couple: (a) by unthreading a length p/2 inside the nut (this length may be bored to a diameter slightly greater than the screw outer diameter, so that an integral number of complete threads are left on both sides); (b) by using even thread-ways, so that the tilting couple produced by one thread-way will be counteracted by that produced by the other.
Fig. 6. Elimination of the tilting couple, using even thread ways.
(a) by unthreading
half a pitch length, (b) by
4. Discussion and conclusions Equations (4) and (5) are general equations expressing the magnitude of the tilting couple inherent to externally pressurized power screw-nut systems under stationary and running conditions. The tilting couple factor is shown for the suggested new profile externally pressurized power screws in Fig. 7, together with the resulting tilting couple factor for the ACME power screw also. The performance of the square power screws in the stationary and running conditions is free from tilting couple effects. This is due to the zero value of its thread angle. From the results presented, the following can be concluded. (1) When the externally pressurized power screw-nut systems are loaded, a tilting couple takes place. It tends to misalign the coincidence of the working axes of the power screw and nut.
283
0.07
) STATIONARY
((----)
RUNNING, I
X = 0.010
I
I
(A*),
DEGREES
I
I
0.01
0
HELIX
ANGLE
Fig. 7. Tilting couple factors us. helix angle.
(2) The performance of the externally pressurized power screw-nut systems having a square power screw is free from the tilting couple effect, in both the stationary and running conditions. (3) The alignment of the power screw and nut axes of the externally pressurized power screw-nut systems could be secured by simple production precautions. References H. R. El-Sayed and H. A. Khataan, The exact performance of externally pressurized power screws, Wear, 30 (2) (1974) 237 - 247.. H. R. El-Sayed and H. A. Khataan, A suggested new profile for externally pressurized power screws, Wear, 31 (1) (1975) 141- 156. H. R. El-Sayed and H. A. Khataan, The running performance of externally pressurized power screws, Wear, 39 (1976) 285 - 306. H. R. El-Sayed and H. A. Khataan, Study of performance of power screw-nut systems, Wear, 39 (1976) 15 - 23.
284 5 G. Porsch, Ausfiihrung und Untersuchung einer Hydrostatischen Spindlemutter, Ind.Anz., 17 (5) (1969) 37. 6 J. H. Rumberger and G. Wertwijn, Hydrostatic lead screw, Mach. Des., (April 11,1968) 218 - 224. 7 Hydrostatic lead screw transmission system, Hydraul. Pneum. Power, 10 (118) (1964) 604. 8 Hydrostatic lead screws make their bow, Met. Work. Prod., 108 (10) (1964) 62. 9 G. Wertwijn and S. P. Kolarich, Selecting a hydrostatic lead screw, Power Transm. Des., 9 (11) (1967) 61.
277
THE TILTING COUPLE INHERENT TO POWER SCREW-NUT SYSTEMS
H. R. EL-SAYED
and H. A. KHATAAN
Production Engineering Alexandria (U.A.R.) (Received
February
Department,
Faculty of Engineering,
7, 1975; in final form February
Alexandria
University,
13, 1976)
Summary The load-carrying capacity of the externally pressurized power screwnut system represents the axial component of its normal load-carrying capacity. The tangential component is responsible for the restoring torque applied to the nut to prevent its possible rotation. The radial component, which seems to be balanced within the engaged length of the nut, results in a tilting couple which tends to misalign the working axes of the power screw and nut. An analytical study dealing with the generation and magnitude of the tilting couple is presented. Possible methods of eliminating its undesirable effects are considered.
1. Introduction The externally induced pressure in the lubricant film, which separates the working flank surfaces of the threads of the externally pressurized power screw-nut system, gives rise to the appearance of a set of normal load elements acting on a definite helix of action [l - 41. When dealing with the functions of the different components of the normal load elements, several authors [5 - 91 disregarded the effects of the radial components and considered them to be balanced within the engaged length of the nut. This conclusion may be drawn from a study of the distribution of these components only along the axial projection of the system. The radial components of the normal load elements tend to misalign the power screw and nut axes and this has been investigated. 2. Concept and magnitude of the tilting couple To illustrate the concept of the tilting couple, it is necessary to explain the method by which the power screw thread flank surface may be generated.
Fig, 1. Generator line of the screw thread flank surface.
Fig. 2. Generation
of the screw flank surface.
2.1, Generation of the screw thread flank surface The thread flank surface of the power screw can be generated by rotating a specified line (a surface generator) about an axis in a definite manner (Fig. 1). The surface generator is specified by the following. (a) It should be of the same length as the radial length I of the thread flank to be generated. (b) It should intersect the plane perpendicular to the axis at a constant angle equal to the thread angle #. (c) It should remain a certain distance from the axis during rotation, which will be the 2 axis of the surface generated.
279
, ,’ ; 8 5
‘. ‘.
Fig. 3. Mathematical representation of the thread flank surface.
The surface generator should be rotated in such a manner that, when its projection on a plane perpendicular to the axis rotates through an angle 8, the point of intersection 0 of the generator extension with that axis moves linearly along it for a distance equal to (0/2n) p, where p is the pitch of the thread to be generated. In this way, the paths of the different points lying on the surface generator, and/or its extension, form a set of concentric helixes intersecting the generator line at right angles (Fig. 2). Each helix has a helix angle X, which depends upon its distance r from the axis. Therefore, the path of any point lying on the screw thread flank surface (Fig. 3) may be expressed by 2=(8/2n)p--tan4
(1)
Accordingly, the surface formed between any two neighbouring helixes represents a screw flank surface. Thus the screw flank surface may be defined as that surface bounded by the intersection of the prescribed generated surface with two coaxial cylinders which are symmetrically set about the 2 axis, and whose radii are ri and rO. 2.2. The tilting couple Due to the absence of dry friction, the tilting couple occurs in both the stationary and running conditions of externally pressurized power screw-nut systems. The stationary working condition of the system is considered first.
280
Fig. 4. Projection of the stationary helix of action on three perpendicular planes.
Figure 4 shows the stationary helix of action [2] generated as outlined. The projection of the stationary helix of action on plane I is a circle whose radius rr is that of the helix itself. The projections of the helix on planes II and III are sine and cosine curves. They may be represented mathematically by the following parametric equations: sine
r..II
=r,
zii
=
(e/277)P
riii
=
r, cos 0
3ii
= (~/2n)
t
(2)
and P
t
(3)
Figure 5 shows the projections of the stationary helix of action, for one thread, on planes I, II and III. The distribution of the corresponding acting components is also shown on each projection. Figure 5 clearly shows that a tilting couple acting on the power screw-nut system will occur. For an integral number of threads m, which may be involved within the engaged length of the nut, the total tilting couple acting on the system is given by T, = m L roftc where ft, = tan c)tan A,,, L has been defined previously [ 21 and A, is the outer helix angle in degrees. The magnitude of the total tilting couple acting on the system during running conditions may be derived similarly:
(4)
281
) Fm’
I-+I
0 Ill
Fig. 5. Projections of the stationary helix of action combined with their radial component elements.
T, = m where F,,
W roFtc
(5)
= (F,/F,)tan A, and W, F, and F, have been defined previously
r31. 3. Elimination of the tilting couple effect Figure 6 shows two possible methods for eliminating the undesirable effect of the tilting couple: (a) by unthreading a length p/2 inside the nut (this length may be bored to a diameter slightly greater than the screw outer diameter, so that an integral number of complete threads are left on both sides); (b) by using even thread-ways, so that the tilting couple produced by one thread-way will be counteracted by that produced by the other.
Fig. 6. Elimination of the tilting couple, using even thread ways.
(a) by unthreading
half a pitch length, (b) by
4. Discussion and conclusions Equations (4) and (5) are general equations expressing the magnitude of the tilting couple inherent to externally pressurized power screw-nut systems under stationary and running conditions. The tilting couple factor is shown for the suggested new profile externally pressurized power screws in Fig. 7, together with the resulting tilting couple factor for the ACME power screw also. The performance of the square power screws in the stationary and running conditions is free from tilting couple effects. This is due to the zero value of its thread angle. From the results presented, the following can be concluded. (1) When the externally pressurized power screw-nut systems are loaded, a tilting couple takes place. It tends to misalign the coincidence of the working axes of the power screw and nut.
283
0.07
) STATIONARY
((----)
RUNNING, I
X = 0.010
I
I
(A*),
DEGREES
I
I
0.01
0
HELIX
ANGLE
Fig. 7. Tilting couple factors us. helix angle.
(2) The performance of the externally pressurized power screw-nut systems having a square power screw is free from the tilting couple effect, in both the stationary and running conditions. (3) The alignment of the power screw and nut axes of the externally pressurized power screw-nut systems could be secured by simple production precautions. References H. R. El-Sayed and H. A. Khataan, The exact performance of externally pressurized power screws, Wear, 30 (2) (1974) 237 - 247.. H. R. El-Sayed and H. A. Khataan, A suggested new profile for externally pressurized power screws, Wear, 31 (1) (1975) 141- 156. H. R. El-Sayed and H. A. Khataan, The running performance of externally pressurized power screws, Wear, 39 (1976) 285 - 306. H. R. El-Sayed and H. A. Khataan, Study of performance of power screw-nut systems, Wear, 39 (1976) 15 - 23.
284 5 G. Porsch, Ausfiihrung und Untersuchung einer Hydrostatischen Spindlemutter, Ind.Anz., 17 (5) (1969) 37. 6 J. H. Rumberger and G. Wertwijn, Hydrostatic lead screw, Mach. Des., (April 11,1968) 218 - 224. 7 Hydrostatic lead screw transmission system, Hydraul. Pneum. Power, 10 (118) (1964) 604. 8 Hydrostatic lead screws make their bow, Met. Work. Prod., 108 (10) (1964) 62. 9 G. Wertwijn and S. P. Kolarich, Selecting a hydrostatic lead screw, Power Transm. Des., 9 (11) (1967) 61.