The sel f - regulated hydrostatic screw and nut

*R. Bassani

The principle of self-regulating flow control, previously applied to hydrostatic bearings, has been applied to hydrostatic screws and nuts with trapezoidal threads. There is a continuous and automatic distribution of the total f low in the two recesses. The results of tests carried out on the self-regulated screw and nut confirm theoretical predictions

In a paper published in 19741, the author described a new hydrostatic opposed-pad rotary bearing. This simple shaped self-regulating flow bearing, although fed by only one pump, allows a continuous and automatic distribution of the total flow rate in two half flow rates in the two recesses.

This principle of self-regulating flow has been applied 2 to hydrostatic screws and nuts with double trapezoidal threads, (Fig 1-A (Appendix 1)). The total volume flow rate, Q, supplied in the helicoidal cavity, C, is divided into two partial volume flow rates one of which, on crossing the thread faces, passes through the hydraulic resistance, R s, and the other through Ri. Rs and Ri are the sum of the hydraulic resistances due to the gaps between the screw and nut thread faces.

R s = R s , 1 + R s , l l , R i = Ri , I + R i , i i (1)

For zero load (P = O) the gap thicknesses h and 2h o- h are equal to ho, then

R s = Ri, (2)

and the two partial volume flow rates are equal, ie

Qs = Qi = Q/2 (3)

If equation (6-A) in Appendix 1 is to be satisfied, then

Rs,i =Rs,l l , Ri,I = Ri,II (4.)

and therefore

Rs,l = Ri,ii ' Ri, I = Rs,ll (5)

If the load is not zero, the screw moves relatively to the nut, but equation (5) continues to be valid; the equality between the right hand side of equation (1) follows and similarly equations (2) and (3). Thus there is a continuous and

i

*Institute of Applied Mechanics and Machine Design, University of Pisa. l t al y

automatic regulation of the flow rate into the two recesses. Constant flow control gives hi~Jaer stiffness than valve, capillary or orifice control.

To verify the theoretical predictions, an experimental rig for testing the self-regulating flow lead screw and nut was designed and built.

Screw and nut

Since ordinary hydrostatic screws and nuts are generally made with a standard trapezoidal thread 3 (Italian TpG 2ill., Tp = trapezoidal thread with thread angle of 30 °, G = large: 2fil = 2 threads, or American double Acme), these were amongst the self-regulated hydrostatic screws and nuts tested previously. A medium sized screw and nut based.on the 80 (80 = maximum diameter in ram) TpG 2fil. was chosen as an experimental prototype.

Screw

The screw thread was non-standard in that the major diameter was 81 mm and it was made to more precise tolerances.

It was deemed necessary for testing to have only the central section threaded for a length of 475 ram, while the side parts, a little shorter than the central one, were smooth (Figs 3 and 4).

Nut

The nut consists (Fig 1) of the central part (1) and the two sideparts (2) and (3).

The central part, based on the theoretically designed nut (Fig l-A), is the effective part of the self-regulated hydro- static system, and contains the recesses, supply holes and drain holes, which are hardly visible in the figure.

Nomenclature

h lubricant fihn thickness, mm 1l 0 lubricant film thickness for P = O, mm n number of engaged turns of a thread p differential pressure, kg/m 2 p supply differential pressure, kg/m 2 Po supply differential pressure for h = ho, mm Pi differential pressure in the 'inferior' recess, kg/m 2 Ps differential pressure in the "superior' recess, kg/m 2 P' P/Po Pl Pi/Po l)'s l)s/Po

Pc lead, mm P load capacity, kg /°7 load capacity, kg, see equation (8-A)

Q volumetric flow rate, m 3/s t temperature, °C

ct helix angle at radius r e ( h o - h ) / h o : eccentricity 0 half thread angle, o /J lubricant viscosity, kg s/m 2 ¢ rotation angle of the screw, rad co rotational speed of the screw, rad/s

0301-679X/79/040185-06502.00© 1979 IPC Business Press TR1BOLOGY international August 1979 185

3:he functional diameters, ¢2 ,Ca ,¢4 ,¢s, are in such ratios as to minimise pumping power.

The lateral parts act as seaJs, so they have a more preciseiy toleranced thread that differs from the standardised one for the smaller minor diameter, ¢! ; the radial tolerances are such that, combined with the corresponding tolerances of the screw, they allow a radial play of 0.2ram.

The axial tolerances of the screw and the nut, in particular those on the tend Pc, have been designed to allow an axial play, ga, so that the film thickness, at zero load, is h o = Vzg a cosc~ cos0 = 0.05 mm. Fig 2 shows the three parts of the nut.

Test equipment

The test rig was designed to allow axial movement of the screw while the nut is at rest (Fig 3), which simplified construction of the equipment and the nut feed.

© © @

Fig 1 Experimental self-re~lated hydrostatic nut

Fig 2 The three parts o f the nut

The screw and nut support (2) were attached to tlqe carrying frame (1); the capillary compensated hydrostat ic bearings, guides for the screw (V), and the stanchion (2~) of the dynamometer (5) were fLxed te (2); the nut (N) was engaged in the dynamometer ann was fed through holes made in the dynamometer itself, described in Appendix 2.

A variable speed motor (6) moves, by means of the pulley (7), the cogged belt (8) and the put]ey (9), connected to ( I ) by the stiffener (10), the slip joint ( t 1) which moves the screw through a universal joh~t so that it can rotate a~d advance. The screw is connected at the other end through a spherical joint to a hydrauIic load jack ( i 2) connected through a universal joint to the stiffener (13) of the structure (1).

The various joints eliminate the effects of errors in the construction and assembly of the eqmpment and those result~Jag from strains on the structure under load. Fig 3 shows the feed circuits for the nut, the screw bearings and hydraulic jack. Fig 4 shows the central part of the equipment, the screw and the dynamometer containirig t~e n u t ,

The nut was fed by a variable delivery pump, wittz an oil having the following properties at 20 °C:

viscosity, ~ = 6.28 x 1 0 . 3 kgs/m :~ density, p = 857 kg/m a specific heat capacity, c = i .88 t M/kg °C thermal conductivity, k = 1.4~ J /m s °C

The same oli was used for the bearings and the ioad jack.

The variabIe speed motor , equipped with an eplcycloidai speed reducer and invertibte pulleys, rotated the screw ir~, the two directions between 0.05 and 100 rad/s (higher speeds were obtained wit.hour the reducer) with a negligible periodic irregularity even at the lowest speeds. The hydraulic jack, equipped with a constanI flow valve. a constant pressure valve and a directional valve, ailows loads from 0 to 2000 kg on the screw in the two directions.

Measu foments

The quantities to be measured and the corresponding instrumentat ion were:

Volume flow of lubricant, measured at the inlet and outlet of the nut by variable area a~_d turbine flowmeters,

Axial loads and friction torques, measured on the nut by the dynamometer , equipped with strain gauge bridges;

Displacements, measured between screw and nut by micro-. meters and inductive transducers;

Pressures. measured in the supply central recess, m the lateral recesses, and atso in some points of the circuit by spring manometers and piezo-registive transducers;

Temperatures, measured along d~e circuit and it, a lateral recess by thermometers and thermocouples:

Rotational speed of the screw, checked by a precisio~ tachometer.

Tests, results and discussion Tests were made with the screw at rest, with increasing ioad using the hydraulic jack, and with the screw rotating with a constant ioad. All the tests were carried our. at a constant tempera~:ure of 30 °C measured at the nut in!or, giving an oii viscosity of 3.74 x 10-3kg s/m 2

Fig 5 shows dimensionless values o f pressures p ' from equation (9-A} in the central suppty recess, p,~ from equatlon

186 TRmBOLOGY international August 1979

A

15 12 5 V 4 ~ 5 2

I 1 A

y -8 • - - 7 6

b)g 3 Test rig." (1) carrying frame, (2) screw and nut support (5) dynamometer, (6) variable speed, motor, (11) slip joint, {12) hydraulic jack, (V) screw, (34} nut

t~)'g 4 Screw atzd dynamometer containing nut

(10-A) and pj from equation (1 l-A) in the side recesses, as functions of eccentricity ratio. These values were obtained from two tests with the screw at rest, blocking the pulley (9), which were made by supplying the nut with flow rates of 48.1 x 10-6ma/s and 96.2 x 10-6m3/s to which the theoretical supply pressures Po from equation (7-A) 10 x 104kg/m 2 and 20 x 104kg/m 2 correspond for e = O, ie no load.

The evident correlation between the experimental and theoretical values confirms the principle of self-regulating flow control. Rather high differences are evident, however, and increase with e. They are higher for Q = 48.1 k 10-6m3/s. Such differences are ascribed to three main causes: imperfect seals errors of machining and errors of coaxiality between screw and nut.

The side parts, the seals, of the nut do not entirely prevent leakage of the lubricant but act as hydraulic resistances. A qualitative evaluation, based on the comparison between

T~

P' Pi' Ps'

O © • Po = I X I0 5 Kg m -2

• • p = 2 x 10 5 Kg m -z [3

I I I I 0 O I O 2 0 3 0 4

q[ 0 5 0 6

Fig 5 Supply pressure p', pressures p~ and P's in inferior and superior recesses as functions o f eccentricity, e

TRIBOLOGY international August 1979 187

the heights and between the surfaces of these resistances and of the resistances (gaps) of the central part, has shown that the teakages througla the side resistance do not exceed the 2 3% of the total flow rate; this has been confirmed by measuring the flow at. the irJet and outlet of the nut.

The errors in machining of the screw and nut may be considered to be of two t}cpes, ie tolerances and errors of construction. The first, identifiable as errors on the plays, on the distalace between the threads and on the leads of the screw and nut, were measured and appeared to be small The errors of construction appeared to be large, especiaily in areas of contact between the central and side parts of the nut, where the threads end in a sharp edge (Fig 2). Such errors could be prevented by working the nut with greater care or making it in one piece.

The imperfect coaxiality between the screw and the nut Causes (even with e = 0) a decrease in gap height on one side and an increase on the other side, ultimately decreasing the discharge pressure of the lubricanto

The effect of the coaxial error was shown by the results in Fig 6~ which were obtained during testing of the screw and nut while held vertically by a simple structure (Fig 3-A) attached to the equipment, in this position, the screw tends to centre itself on the nut. Tests have been made for Q = I4~4 x 10-6m3/s, Pc = 3 x 104kg/m 2 .

The differences between experimental and theoretical values were Dss than from previous tests. A decrease in coaxial error

3

o_

o.

IO_

2

P' P i ' Ps ~

& & Pc : 3 x IO 4 Kg m -2

i

Pi

A P s'

0 0.1 0.2 0 . 5 0 4 0 5 0 . 6

Fig 6 Pressures ~ , p[ and P's as functions o f e with the screw in a vertical position (Fig 3-A)

may be obtained by careful assembly; yet a ~'etative displace- ment of the axes of the helicoidal couple may be necessary in some machines when the slide ~o which the nut is connected approaches the ways >:o which the screw ~s connected for example in machine .:oois with hydrostatic slideways driven by screws and nu~s. Relative dispiacement of this kind may be satisfied by normal hydrostatic screws and nuts with rectangu!ar threads (for examp!e 4's).

Fig 7 shows dimensionless values o f p ' from eq,aa~ien (I i-A) as functions of e for the three cases shown. The differences between experimental and theoretical vNues increase w~t5 e For e = 0.5 the maximum dimensional experimenta! ".,ames

SCFew Pc = 3 x 1 0 4 Kg ,~-2 verhcotDosll}on

o Pc = 0 ^ 0 4 Kg . ~ - 2 " Screw ~ - o Pc = 2 0 x 04 Kg m -2 > nonzon'toI DOS~IIOR

3 -

/ / /

/ /

/

/ /

/

0 Oi 02 05

s

Fig 7 Load capacity as funStion uj" ~ ~

i

/

/ /

/ /

0 4 ~ 5 0 6

0 D C ~ U

o,. I

d . - r - 5 D Ps Pc = 2 0 X 04 Kg - 2

Tc~ i ' O ~3 ~ ~ = 0 48 rod s-

J Ps

I

¢

Fig 8 Pressures p~, p~ ~nd P's as fu~.et~ons o f screw rotation angle

188 T R I B O L O G Y in ternat iona l August 1979

of P are respectively: P = 98.5 kg for Q = 14.4 x lO-6m3/s, P = 289 kg for Q =48.1 x 10 -6m 3 / s a n d P = 6 2 0 k g f o r Q = 9 6 . 2 x 10-6m3/s.

Fig 8 shows values of pressures p ' , p{, p~ as functions of rotation angle 4) of the screw rotating at 0.148 rad/s. The test was perfornred for Q = 96.2 x 10-6mr/s, po = 20 x 104 kg/m 2 , for a medium vahae of eccentricity, e = 0.3, and for a load P = 620 kg.

The theoretical value from equation (5-A) of friction torque wasM r = 0.0292 x 10 -s kg m; the real value, influenced by the side parts of the nut which act as high pressure seals, is nearly double. Such a va, lue would be bound to increase with low pressure seals, which collect the lubricant drawn from the high pressure seals. With seals of low friction material (eg ptfe) and of axial length not greater than 0.25 Pc friction torques of the order of those for the ball screw may be obtained.

During testing the increase in temperature between the nut inlet and outlet was always less than 1 °C.

Results of subsequent tests will be published shortly.

Summary A self-regulated hydrostatic screw and nut with trapezoidal threads was designed and manufactured on the basis of previous results in accordance with the optimum theoretical operating conditions. A test rig for hydrostatic screws and nuts in which the screw moves axially while the nut is at rest has been described. This facilitated both construction of the rig and the nut feed, which is effected through holes in the dynamometer where the nut is engaged. Analysis of the results of a series of tests on the self-regulating ilow control screw and nut have verified the principle of self-regulation.

Acknowledgement This work was supported by a contribution from the CNR.

r 2 ~ rs41 l + 1 1J { [ l - I r4 /4q4r3 /4 E 1 -(r31 X /~o ~1 - e V s l J \ r s l - - ]

1 r2 r4 121+l~Q_/2 [1 (r2 12 }02 , r +~T r5 { [ 1 -(r?-,_l ,rs! -V3-1 - + ~n (5 -A)

47T 4 ~bc'r rs ! rs r

In the above expressions Fc, r is given by:

£nr4 l p2c( l 1 ) rs - 8rr ~ r4 ~ - rs ~

=~n,-2 l ( 1 1 ) (6-A, r3 _87r 2 p2 r2: - rs2"

and 0c,r is given by:

Po (r2 rs ) >

In dimensionless torm, Po is defined as:

<, O l l -rr /~ n cos#0 h 3 t:c,r (7 -A)

which is the feed pressure for e = 0, and

1 1" 7 = -~ u O c o ¢ 0

x rs 2 r3 ~/"3 ] ] l 3 { [ I - ( ::: )2 I -( r5 ~ [1 _(r~ t 2 ]

,) } - 0c , . (S -A) _rr rs r5 /" 3 "

L6

Appendix 1

Fig I-A 2 , shows tile self-regulating screw and nut.

The supply' pressure p is

t) =/)s + l)i ( l-A)

t)s and l)i being the pressures in the 'superior' (upper)and in tile "inferior" (lower) recesses

3 C) I i 1 ])S = - g /'/ tl COS40 / lo 3 (1 + e) 3 Fc,r ' (2-At

3 O I l 1 P i = - ,7 ~ n cos40 tto:~ ( l_c i3- / :or (B-A)

The load capacity, P, is

p = 3 1 2 /a O cos40

rs : 1 1 _ i t 4 12 )2 xho3" [ ( i - ) ) 3 - ( i + e ) 3 1 { I l , r s , ]_( ,r ;

[ l -( r2 12 I - ' r 2 (~n F4 - ~ l l r ~ 2 } r3 ! 2rr a rs rs r 0c,r (4-A)

The friclion torque Mr is

rr 1 -: COS 0

Fig I-A Self regulating Jlow-control hydrostatic screw and Illl[

TRIBOLOGY international August 1979 189

Equations ( l -A) , (2-A), (3-A) and (4-A) respectively become

Po 2 ( l - e ) 3 + ( l + e ) ~ ' (9-A)

Ps 1 1 P; Po 2 ( i + e) 3 ' (10-A)

_ P i 1 1 P'i ( 11 - t )

po - 2~ (1 - &

p, P _ 1 i 1 i ] P7 2 (1 - e) 3 (1 +e ) 3 ~ ( I2-A)

Appendix 2 Dynamometer

The dynamometer (5) (Fig 2-A) consists of a central body, with tubular side arms of small diameter and thickness. The nut is supplied through the arms. These are encased in thin-walled sleeves (5 ') with a larger diameter than the arms and so have a larger moment of inertia. The sleeves serve as stiffening when screwed on the collars at the ends of the arms and locked with bushes (5"). The dynamometer is connected ~¢ith the stanchion (4) to the support (2) of the nut (see also Fig 3).

Two strain-gauge bridges, one to measure loads and the " other to measure friction torques, are at tached to each arm.

Calibration of the dynamometer was made with fluid at pressure in the arms, and with disengaged sleeves for low values of ioads and torques and with engaged sleeves for high values; a large range o f values may be measured accurately.

Owing to the characteristics of the equipment and the dynamometer , the lat ter measures all and only the axial load P with which the screw loads the nut and all and only the viscous friction torque M r between the screw and the nut.

The orders of magnitude of P and M r were:

=~f 0.03 - 10 kg with stiffening sleeves disengaged P L 5 1000 kg engaged

r

~J 0.003 - 1 kg m disengaged Jvd 7-

=L0.5 i00 kg m engaged

5' 5" QI2 j2 4 2 _ _ ~

Fig 2-A D y n a m o m e t e r f o r loads and to~yues," the nu t is f e d through the d y n a m o m e t e r

Fig 3-A Structure for holding the screw zn a vertical posi t ion

Appendix 3

Structure for self-centring tests

A simpie vertical structure, show~, in Fig 3-A. was made for self-centring tests on the screw in the nut. The screw, which is prevented from rotating by a bar is loaded by weights added at i~s bot tom. The sides of the threads are inclined° so the screw tends to centre itself in (he nut, especialiy as the load increases.

References

!. Bassani R. A new opposed-pad hydrostatic bearing: the ~ow self-regulating bearing. Proc. 2rid Nat. AIMETA Congresx, fI/ /]974 255. Also in MECC.. X/]975) 107

Bassani R. The flow self-regulating hydrostatic screw and nut_. Atti I~t. Mace. Un. Pisa No. 2 "1977]

R u m b a u g e r J .M. and Wertwijn G. Hydrostatic lead screw~. Mach Des. (11 April) (1968) 218

Lombard J~ and Moisan A. Caract~ristiques statiquea et dynamiques d'un syst~me vis-ecrou hydrostatique. Ann CtRP XVIL r '1970) 521

El-Sayed H.R. ~qd Khataan H.A. Study of performance of power screw-nuz systems. Wear. 39. ]5 H976J

Bassani R and Culta C. Static and dynamic behaviour of an experimentai self-regulated hydrostatic screw and nut~ Unpublished work

2 .

3. e

4.

5,

6.

This paper was presented at the 1st International Con Cere~ee: Frontiers in Lubricant Technology, Naples. 4 - 7 April. !9 78

190 TRJBOLOGY international August 1979