DOCTORAL THESIS 1985:41 D
BOUNDARY L U B R I C A T I O N IN SCREW-NUT TRANSMISSIONS
b y
LARS O. EKERFORS Division of Machine Elements
f O TEKNISKA
Lal HÖGSKOLAN I LULEÅ
LULEÅ UNIVERSITY O F TECHNOLOGY
1985:410
BOUNDARY LUBRICATION IN SCREW-NUT TRANSMISSIONS
av
LARS 0. EKERFORS
I n s t i t u t i o n e n f ö r Maskinteknik Avdelningen f ö r Maskinelement
AKADEMISK AVHANDLING
som med vederbörligt t i l l s t å n d av Tekniska Fakultetsnämnden vid Tekniska Högskolan i Luleå f ö r avläggande Sv teknisk doktors- examen kommer a t t o f f e n t l i g e n försvaras i Tekniska Högskolans hörsal E 246, E-huset, fredagen den 26 a p r i l 1985, kl 09.00.
BOUNDARY LUBRICATION IN SCREW-NUT TRANSMISSIONS
by
LARS 0. EKERFORS
D i v i s i o n of Machine Elements LULEÅ UNIVERSITY OF TECHNOLOGY
LULEÅ 1985
CONTENTS Page
ACKNOWLEDGEMENTS
ABSTRACT
1. INTRODUCTION 1 1.1 Background 1 1.2 Optimal function 1 1.3 Capability of performance 2
2. SYMBOLS 4
3. EXPERIMENTAL EQUIPMENT 7 3.1 Test r i g and t e s t object 7
3.2 Gauges 14 3.3 Recording equipment 16
4. THE COEFFICIENT OF FRICTION 17 4.1 Theoretical model 17 4.2 Experimental i n v e s t i g a t i o n s 25
4.3 Analysis of experimental r e s u l t s 29 4.4 Discussion and conclusions 43
5. THE HEAT CONDUCTION PROBLEM 48 5.1 Balance of developed power and heat 48
5.2 Theoretical model 49 5.3 The equation of heat conduction 50
5.4 Solution of the equation of heat conduction 52
5.5 Experimental investigations 63 5.6 Discussion and conclusions 71
6. DESIGN OF SCREW-NUT TRANSMISSIONS 74
6.1 C r i t e r i a 74 6.2 Numerical examples 80
7. REFERENCES 95
8. APPENDICES 97 Tables TI Constants of material 98
T2.T3 Trapezoidal screw threads, 99 extracts from Swedish Standards
T4-T23 Experimental r e s u l t s 107
Diagrams D1-D16 127 Computer programs C1-C4 143
ACKNOWLEDGEMENTS
The research work here presented has been carried out at the Depart- ment of Machine Elements, Luleå University of Technology.
The work has been f i n a n c i a l l y supported by the National Swedish Board f o r Technical Development (STU).
I would l i k e to take the opportunity to express my gratitude to Professor Bo Jacobson f o r his active i n t e r e s t , which has given r i s e to many stimulating and f r u i t f u l discussions.
Furthermore, I wish to thank Mr Sven-Erik Tiberg for his contributions i n connection with the construction of the electronic equipment and f o r his assistance in programming.
In chapter 5, which deals with the heat conduction problem, Dr Anders Grennberg has been of indispensable help in solving the heat conduc- t i o n equation and providing diagrams.
I should also l i k e to thank Professor Håkan Gustavsson for the i n t e r - esting discussion concerning the mathematical formulation of the heat conduction problem.
I thank Mr Allan Holmgren f o r his excellent help in the construction and c a l i b r a t i o n of the t e s t i n g equipment.
F i n a l l y , I wish to thank Miss Rose-Marie Lövenstig and Miss Gunnel Henriksson, who have typed the manuscripts so conscientiously.
ABSTRACT
This report deals with the function of screw-nut transmissions (power- screws). Two aspects of t h i s function have been investigated.
Owing to d i f f e r e n t running parameters, p r i m a r i l y s l i d i n g speed and average pressure between the s l i d i n g surfaces of the thread, the coef- f i c i e n t of f r i c t i o n and the e f f i c i e n c y w i l l vary w i t h i n wide l i m i t s . The running parameters can be summarized in a dimensionless number, the Sommerfeldt number S.
The problem, which has reference to boundary l u b r i c a t i o n , is solved by a theoretical model. The model i s based on two types of i n t e r a c t i o n between the s l i d i n g surfaces, namely s o l i d f r i c t i o n at asperity peaks and l i q u i d f r i c t i o n in the voids between the a s p e r i t i e s . An optimal i n t e r v a l of the Sommerfeldt number, where the c o e f f i c i e n t of f r i c t i o n i s at i t s minimum, has been established: 0.025 < S t < 0.042.
opt
As a r e s u l t of f r i c t i o n between the s l i d i n g surfaces, heat is deve- loped, which i s conducted through the material of the screw and nut.
The developed heat can cause high temperatures on the s l i d i n g surfaces of the thread.
The c a p a b i l i t y of performance is l i m i t e d by the development of high temperatures in the thread, where the running temperature of the actual l u b r i c a n t must not be exceeded.
P h y s i c a l l y , the phenomenon relates to heat conduction. A theoretical model is put forward. In the model the screw i s replaced by a c y l i n d r i c a l rod and a hollow cylinder corresponds to the nut. The equation of heat conduction i s stated and solved for the case of
steady state in the actual regions. I t is shown that an i n f i n i t e l y t h i n wall of the hollow cylinder i s the most severe case with a maxi- mum r i s e in temperature. The r e s u l t is presented in the form of a diagram with dimensionless temperature, rod speed and length of c y l i n d e r .
The report ends with recommendations for how the results can be used f o r designing screw-nut transmissions. In t h i s context three numerical examples are given.
1
1 . INTRODUCTION
1.1 Background
The screw-nut transmission (power- or lead screws) is a machine ele- ment, which consists of a combination of screw and nut and is used f o r power transmission.
The screw-nut transmission transforms r o t a t i o n i n t o t r a n s l a t i o n or vice versa.
In the f i r s t case, great axial force i s produced and the motion is very accurate, even and easy to c o n t r o l . In the l a t t e r case, high r o t a t i o n a l speed w i l l be the r e s u l t .
The screw-nut transmission has many technical a p p l i c a t i o n s , such as t e s t machines for t e n s i l e s t r e s s , feed screws in l a t h e s , mechanical jacks and separators.
1.2 Optimal function
When transmitted power and power loss are moderate, the screw-nut transmission works with s a t i s f a c t o r y l u b r i c a t i o n w i t h i n a wide range of load and speed. In t h i s context, the macro-mechanical q u a l i t i e s of the surfaces of the thread flank are s i g n i f i c a n t .
Q u a l i t i e s such as the surface roughness and v i s c o s i t y of the l u b r i c a n t are relevant here.
Screw-nut transmissions work with r e l a t i v e l y low e f f i c i e n c y , so i t i s of great importance to f i n d t h e i r optimal running range.
2
The external load, i . e . the axial load, i s transferred to and d i s t r i - buted on the surfaces of the thread flanks as a pressure d i s t r i b u t i o n . When the surfaces of the thread flanks of the screw and nut s l i d e against each other f r i c t i o n appears, which manifests i t s e l f as a f r i c - t i o n a l f o r c e . By d e f i n i t i o n , the f r i c t i o n is represented by the so c a l l e d c o e f f i c i e n t of f r i c t i o n .
The screw-nut transmission works optimally when the combination of s l i d i n g speed and flank pressure results in as large a degree of e f f i - ciency as possible. This is equivalent to as small a c o e f f i c i e n t of f r i c t i o n as possible.
In t h i s thesis chapter 4 t r e a t s the c o e f f i c i e n t of f r i c t i o n as i t is affected by the s l i d i n g speed of the surfaces of the thread f l a n k s , contact pressure, surface roughness and v i s c o s i t y of the l u b r i c a n t .
1.3 Capability of performance
One of the most important f a c t o r s , l i m i t i n g the c a p a b i l i t y of perfor- mance of screw-nut transmissions, i s the increase of temperature which appears at the surfaces of the thread f l a n k s . This is caused by the f r i c t i o n , mentioned above, and arises when the surfaces of contact s l i d e on each other. Since s o l i d or grease l u b r i c a n t s are usually used, the generated f r i c t i o n heat cannot be abducted by c i r c u l a t i n g o i l , as is the case o f , for instance, radial journal bearings, gear pairs e t c .
The f r i c t i o n heat, i . e . the power l o s s , which i s dissipated by heat conduction through the threads in screw-nut transmissions, can cause
3
a r e l a t i v e l y high working temperature. This temperature determines the a b i l i t y of the l u b r i c a n t to form and maintain a s a t i s f a c t o r y bearing f i l m . In boundary l u b r i c a t i o n i t is found that when the temperature is raised there i s a c r i t i c a l temperature above which the f r i c t i o n and the surface damage increase markedly. [9]
In chapter 5 the heat conduction problem and the mechanism of how the temperature d i s t r i b u t i o n is influenced by supplied power are studied.
4
SYMBOLS
A t o t a l area of s l i d i n g surface - [m2]
Arø area of c i r c u l a r cylinder [m2]
A„ area of metallic contact [m21
Ar' geometry-dependent area [m2]
a radius of c i r c u l a r cylinder [m]
B = l / u ( 0 ) [0]
b outer radius of hollow c y l i n d e r , width of thread flank [m]
c s p e c i f i c heat [Ws/(kg-K)j dpdg constants of material [0]
F f r i c t i o n a l force [ N]
F_v axial force on nut [ Nl
ax 1 J
H height of hollow cylinder [m]
h average thickness of f i l m [m]
hQ f i l m thickness at beginning of asperity contact [m]
h* dimensionless f i l m thickness [0]
k slope of hydrodynamic l i n e [0]
1 average length of the a s p e r i t i e s [m]
L ordinate at o r i g i n of coordinates for hydrodynamic [0]
l i n e
M torque on nut [Nm]
N normal force [ N]
n r o t a t i o n a l speed [ r / m i n ] p average pressure [N/m ]
2 Phd hydrodynamic pressure in the l u b r i c a n t [N/m ]
p' average increase in pressure at the a s p e r i t i e s [N/m ]
T / d . ( l - d „2) ' [N/m2]
5
Q power [ W]
q f r i c t i o n a l power/area, transferred to screw and nut [W/m ]
r radial coordinate [m]
S = n v , Sommerfeld number [0]
h0 P
Sq Sommerfeld number at beginning of asperity contact, [0]
v = v0
s* = s/s0
s p i t c h of thread [m]
T temperature [K]
T* dimensionless temperature [0]
t time [s]
u axial speed [m/s]
u0 'ui •
Ug, dimensionless temperature in power expansion [0]
tangential speed [m/ s ]
v
x axial coordinate, tangential coordinate of s l i d i n g
surface [m]
x = £- 5 [0]
y coordinate normal to the s l i d i n g surface [m]
area r a t i o [0]
a
ß convection number [W/(m -K)]
ß* dimensionless convection number [0]
6 Dirac delta function [0]
[—] • — [0]
L cpJ ^ au L J
Y p r o f i l e angle [0]
o n e f f i c i e n c y , dynamic v i s c o s i t y [0],[Ns/m ]
e p i t c h angle [0]
X heat conduction number [W/(m-K)j
6
y c o e f f i c i e n t of f r i c t i o n [0]
S dimensionless axial coordinate [0]
p density, dimensionless radius [kg/m ] , [ 0 ]
normal stress [N/m2]
o
as y i e l d stress [N/m2]
T shear stress [N/m2]
Thd hydrodynamic shear stress [N/m2]
Ts y i e l d stress in shear [N/m2j
the d i s t r i b u t i o n function of the normal d i s t r i b u t i o n [0]
Subscripts:
p function of pressure v function of speed 1 c i r c u l a r rod 2 hollow cylinder
0 beginning of asperity contact
opp axial force and motion are in opposite directions eq axial force and motion are in equal directions
7
3. EXPERIMENTAL EQUIPMENT
3.1 Test r i g and t e s t object
A t e s t r i g has been designed and b u i l t . The central parts of the r i g are two p a r a l l e l , v e r t i c a l and r o t a t i n g screws where two nuts, the t e s t objects, can move along each screw.
The axial load coming from a hydraulic jack i s transferred to the nuts by two beams, where the beams are pressed apart by the jack. The beams w i l l then in turn press against the four nuts.
Owing to the motion of the screws and the hydraulic j a c k , the t e s t ob- j e c t s w i l l be exposed to torque and axial load. The nuts are prevented from r o t a t i o n by an arrangement, here c a l l e d torque r i n g , which f a c i - l i t a t e s measurement of the torque figures 3.1.1 and 3 . 1 . 2 .
The two beams, the jack and the parts of the screws which are between the nuts thus form a closed system of forces, figure 3 . 1 . 3 . The screws, both ends of which are mounted in bearings, are coupled t o - gether p a r a l l e l l y by a chain transmission. The screws are operated by a continuously variable e l e c t r i c motor (ASEA LAC-315, 143 kW, 1800 rpm) via a gear p a i r with i n t e r s e c t i n g axes.
When the t e s t r i g is in operation the pair of beams perform r e c i p r o - cating motion with simultaneous loading of the hydraulic j a c k . The design of the t e s t r i g is shown in figures 3.1.4 and 3 . 1 . 5 .
When loading the nuts i t is important to see to that they are exposed to axial forces only. To eliminate the r i s k of an unbalanced load, the t r a n s f e r of axial forces to each nut is done via a spherical t h r u s t
8
r o l l e r bearing. To avoid transfer of torque to the support, the nut also rests on a t h r u s t ball bearing, f i g u r e 3 . 1 . 6 . The r e s u l t of t h i s combination of bearings is that the torque transferred to the torque r i n g i s very close to the t o t a l torque on the nut.
The material of the nuts i s t i n bronze, SIS 5465, hardness HB = 95, and the material of the screws is steel SIS 1672-01, hardness HV = 208.
According to measurements of the surface f i n i s h , the depth of p r o f i l e of the thread flanks is H = 1.6-2.3 ym t a n g e n t i a l l y and H = 2.6-2.7 ym r a d i a l l y .
The l u b r i c a n t applied i s an ordinary grease based on mineral o i l with EP a d d i t i v e s , such as l i t h i u m soap e t c . (commercial name ALEXOL HMP 2EP). The l u b r i c a n t has been analysed with respect to v i s c o s i t y in a r o t a t i o n viscosimeter of the type Rheotest, RV 2. The r e s u l t s , which include temperature and shear rate dependence of the dynamic viscos- i t y , are shown in the diagram, figure 3 . 1 . 7 .
9
screw
ring
Figure 3 . 1 . 1 . Arrangement f o r determining torque.
Figure 3.1.2. Assembled torque r i n g .
10
Figure 3.1.4. Diagram of the t e s t r i g .
11
Figure 3.1.5. Test r i g .
Figure 3.1.6. Diagram of arrangement of nut support.
12
Figure 3 . 2 . 1 . Beam with hydraulic j a c k .
14
3.2 Gauges
Experimentally determined quantities in the i n v e s t i g a t i o n were the axial force and torque on the nuts, r o t a t i o n a l speed of the screws and temperature of the nuts.
The axial forces were determined by measuring the load of the hydrau- l i c j a c k . Measurements were done by c a l i b r a t e d s t r a i n gauges attached to the ball attachments of the j a c k . See figures 3.1.3 and 3 . 2 . 1 .
The nut torque was measured by means of the torque ring mentioned e a r l i e r . This arrangement consists of a c i r c u l a r steel r i n g and a radial lever attached to the r i n g . The lever was equipped with one s t r a i n gauge on each side. Figures 3.1.1 and 3 . 1 . 2 .
The r o t a t i o n a l speed of the screws was determined using an optical counter.
The temperature of the nuts was measured with thermocouples. The thermo- couples were welded to the bottom of channels, which were r a d i a l l y d r i l l e d i n t o the wall of the nut to a depth of 1 mm from thread top.
The channels were placed one pitch of thread apart along a generatrix a t the outer surface of the nut. Figures 3.2.2 and 3 . 2 . 3 .
15
Figure 3.2.2. Cross-section of wall of nut.
Figure 3.2.3. Nut f i t t e d with connections f o r thermocouples.
16
3.3 Recording equipment
The experiments concerning the r e l a t i o n between the c o e f f i c i e n t of f r i c t i o n and pressure/speed resulted i n analogous measuring signals from the s t r a i n gauges, which were recorded by a d i g i t a l tape recorder PCM (pulse coded modulation). The accuracy of the tape recorder in the actual set up was better than 0.1% of maximum input s i g n a l .
The information was transferred to an ABC-80 microcomputer v i a a n A / D - transducer, accuracy 0.1%, and f i n a l l y stored through an input program on disks, see Appendix C l .
In the experiments in connection with the heat conduction problem, the signals from the thermocouples and the s t r a i n gauges were transferred
via an A/D-transducer d i r e c t l y to the ABC-80 and stored on d i s k s . The input program i s shown in Appendix C3.
Stored information was processed using the programs given in Appendix C2 and C4. The r e s u l t was w r i t t e n on a l i n e p r i n t e r .
The t r a n s f e r i s shown in f i g u r e 3 . 3 . 1 .
I s t r a i n gauges!
thermo- couple
1 ine p r i n t e r
Figure 3 . 3 . 1 . Diagram to show the t r a n s f e r and processing of experimental r e s u l t s .
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4. THE COEFFICIENT OF FRICTION
4.1 Theoretical model
A model has been assumed in which the dependence of the c o e f f i c i e n t of f r i c t i o n on load and speed has been taken into account. The load is here represented as the average pressure on the surfaces of thread f l a n k s .
4.1.1 f_ujidlamentaj_s
By d e f i n i t i o n F
U = N
according to [2] and [ 8 ] .
According to [ 1 ] , [3] and [16] the f r i c t i o n force i s
F = Ar. xs + ( A - Arhn d
and the normal force
N = Ar- as + ( A - Ar) ph d = A.p
where
p is the average pressure on the surfaces of the thread flanks Pn d is the hydrodynamic pressure in the l u b r i c a n t
os is the y i e l d pressure of the s o f t e r metal
Ts is the y i e l d stress in shear of the s o f t e r metal is the hydrodynamic shear stress
A is the t o t a l s l i d i n g surface
A is the surface with m e t a l l i c contact.
18
Let
o (4.1)
This gives
[16] (4.2)
The area of the surface of contact is among other things dependent on v and p.
This dependence is divided into
v A v
The i r r e g u l a r i t y , i . e . the a s p e r i t i e s , of the softer surface is taken i n t o consideration.
The area of the surface of metallic contact i s a function of the o i l f i l m thickness, h.
This function i s influenced p a r t l y by the geometry of every single asperity and p a r t l y by the d i s t r i b u t i o n of the heights of the asperi- t i e s .
(4.3)
4.1.2 J_he influence of_sj)eed
According to (4.1)
(4.4)
Assume that the a s p e r i t i e s are conical or pyramidal and that t h e i r d i s t r i b u t i o n over the surface of the flank is normal. [ 3 ] , [ 2 0 ] .
19
This can now be w r i t t e n as
Ar = Ar' [ l - k . * ( z ) ] = Ar' . * where
A ' is the area of contact depending on geometry o>fz) is the d i s t r i b u t i o n f u n c t i o n of the normally
d i s t r i b u t e d asperity peaks k is a f a c t o r of correction
$ = l-k«<t>(z)
Figure 4 . 1 . 1 . Pyramidal a s p e r i t y .
From figure 4.1.1 i s obtained
A v hQ
Putting i n equation (4.4) gives
a = ( J l ) = (1 - ^ - )2. * (4.5)
v A v hQ
The r e l a t i o n between f i l m thickness, h, and s l i d i n g speed, v, i s ob- tained from dimensional analysis of Reynolds' equation [18]
d_ ( h3 dp_) . 6 n v dh dx dx dx I ( h3 £ l ) - n v H o
1 1 1
where 1 i s the average length o f a s p e r i t i e s
p' is the average increase i n pressure at the asperities
20
hO
With constant 1 , p ' , n a n d — w e have
h3~ v or
h - ^
h* = f - = £ - ) (4.6)
hO vO
where the subscript Q indicates the s t a r t i n g contact of a s p e r i t i e s .
4.1.3 The jn_f1 uence of_ay_erage_pressure
According to v. Mises and Bowden, Tabor [1] and [2]
Z.A 2 _ 2 _ H 2
+dlT - as - dlTs when v = 0 , a = ag.
While si iding a decreases due to T * 0.
Only the average pressure is taken into consideration, i . e . the hydro- dynamic pressure, which is dependent on the s l i d i n g speed, is disregarded.
Then one obtains
_ N N _ p
o - — - - J-—
(A„)_ a 'A a r p p p
Furthermore, a i s assumed to be inversely proportional to the d i s t r i b u - t i o n function of the normally d i s t r i b u t e d a s p e r i t i e s
o = - P —
While s l i d i n g T = T * , which is assumed to be independent of p and
T * = d2xs [8]
I n s e r t i n g these expressions for o and x i n t o v. Mises equation gives
(JL-)2 • dl (d2xs)2 = a /
21
From t h i s one obtains
a = - P — (4.7) P $.p*
where
p* = os / l - d? 2' = xs / d ^ l - d g2) ' (4.8)
According to ( 4 . 3 ) , and a f t e r i n s e r t i n g ( 4 . 5 ) , (4.6) and ( 4 . 7 ) , one obtains
a = a . « = £ - [ 1 - ( ^ - )1 / 3]2 (4.9)
P P v0
4.1.4 J_he hydrody_nami_c_shear_s^res^
The r e l a t i o n s h i p for internal f r i c t i o n in a viscous f l u i d as proposed by Newton [14] is
*M (4.10) where y is the coordinate normal to the s l i d i n g surface. Inserting
(4.6)
h = h0( ^ )1 / 3
° v0 gives
^hd = ^ V (^)_1/3 (4-U )
nO v0
4.1.5 The equation of the c o e f f i c i e n t of f r i c t i o n
Regroup the r e l a t i o n (4.2)
= a( I i . I M ) + 1™ - a ll (1 Z M ) + I M
P P P P Ts p
22
Thd
The r a t i o « 1 i s disregarded and i n s e r t i o n of equations (4.9) and
(4.10) gives
„ « I i [ i - ( V - )1 / 3]2 + m (4.12)
vo
P* 1 vn hp
Introduce the Sommerfeld number, S = 2 ^ - , and equation (4.8) nnp
1 n iv ^ l / 3n2 A -v ,-1/3.
u . _ = L _ [ i - ( f r - ] + % ) - - S
/ d ^ i - d g2) vo vo
Also introduce S* = | - and / d ^ l - d ^ ) ' = B
S0 Then one obtains
y(S*) = 1 (1 - S *1 / 3)2 + S *2 / 3. SQ (4.13)
B u
The equation is v a l i d in the regime of boundary l u b r i c a t i o n ,
0 < S* < 1.
In the regime S* > 1 , i . e . hydrodynamic l u b r i c a t i o n , the r e l a t i o n between u and S* is assumed to be l i n e a r ,
u = kS*+L
The derivative of u ( S * ) , equation (4.13), f o r S* = 1 gives the slope of the "hydrodynamic l i n e " .
4 H - = 2 ( l - S *1 / 3) ( - I ) S * "2 / 3 + i Sn- S * ~1 / 3 (4.14)
dS* B 3 3 0
I n s e r t i n g S* = 1 in the l i n e a r r e l a t i o n above and (4.14), we get
— =dS* 3 k = 4 s 0 n and
^ = k = i s „ (4.15)
23
Thus
and
S0 = f S0 + L
L = j S0 (4.16)
Combining the assumed l i n e a r r e l a t i o n , (4.15) and (4.16) gives
„(S*) = JL (2S*+1) (4.17)
S* > 1
Static f r i c t i o n is obtained by p u t t i n g in S* = 0 in (4.13)
y(0) = I (4.18) B
In the regime S* < 1 n has a minimum, and = 0 in equation (4.14) dS*
gives
S - S = S° 3 (4.19)
0pt (1+BS0)3 and
S0
u . = — (4.20)
mi n / 1+BS0
The constants B and SQ are e m p i r i c a l l y determined.
Example Karlebo Handbok [10] gives f o r the combination steel/bronze s t a t i c f r i c t i o n y(0) = 0,18
s l i d i n g f r i c t i o n y ( l ) = 0,10
24
This gives
u . = — 9 A = 0,064
Mmi n
0,18 f o r
S = 0,1 (1 + ° ^ - )3
0,18
0,027
Figure 4.1.2 shows the curve of the c o e f f i c i e n t of f r i c t i o n according to the theoretical model. The parameters B and SQ are taken from the numerical example.
s* = S/SQ
boundary l u b r i c a t i o n 1 hydrodynamic l u b r i c a t i o n
Figure 4 . 1 . 2 . The c o e f f i c i e n t of f r i c t i o n according to the t h e o r e t i c a l model.
25
4.2 Experimental investigations
4.2.1 Test series
Two s e r i e s , 'l and 2, were performed i n compliance with t h i s p r i n c i p l e . Speed, pressure, and supplied power per m2, q! , appears from table 4 . 1 . The tests were carried out by simultaneous recording of axial load, torque and r o t a t i o n a l speed of every screw/nut combination.
Four combinations with thread TR 80x10 were tested according to the f o l l o w i n g
Nut H
1 0.04 m 0.060 m 2 "- 0.050 m 3 " - . 0.055 m 4 " - 0.045 m
0.12 m
Figure 4 . 2 . 1 . Dimensions of nut.
In order to cover as large a range as possible of the parameter (v/p) with a l i m i t e d number of t e s t s , combinations of (v/p) were chosen such as to be d i s t r i b u t e d on c i r c l e s in a v-p diagram.
0 , 1 0 , 2 0 , 3 v m / a
Figure 4 . 2 . 2 . P r i n c i p l e f o r choosing v / p - combinations.
26
Table 4.1
Series 1
v : 0-0.2 m/s p : 0-1 MPa q' : 0-15 kW/m2
Series 2
v : 0-0.3 m/s p : 1-1.5 MPa q' : 0-30 kW/m2
4.2.2 Prrjcessing_o f_ tejs t_res_ul Jt
In each s p e c i f i c t e s t the torque of the nut varies to a r e l a t i v e l y high degree when the nut moves along the screw. Example from a t e s t diagram i s shown in figure 4 . 2 . 3 . The variations can amount to about 50% of the mean of the torque and are mainly caused by i r r e g u l a r i t i e s of the threadflanks. One can observe that the v a r i a t i o n s form a r e - peated p a t t e r n , the parts of which perpetually w i l l be encountered when the nuts are moving along the screws and that every part of the
screws thus shows i t s own typical p a t t e r n . This means that every part of the threadflanks of the screw has a s p e c i f i c character of i t s own.
Considering the variations mentioned above, the torque i s treated as a stochastic v a r i a b l e , which motivates c a l c u l a t i o n of the mean. Conse- quently estimation of error i s not relevant in t h i s context. This i s done by the equipment described in part 3.3. The analogous output signal of the torque is hereby d i g i t i z e d , and the arithmetic average i s c a l c u l a t e d . The data for such a c a l c u l a t i o n are taken from those parts of every t e s t which are considered representative. This i m p l i e s , among other t h i n g s , that only measurements of the t o t a l tested length of each screw, reduced by 10% from each end p o s i t i o n , are included in the calculated average.
rx3 •^1
Figure 4.2.3. Variations in the torque of the nut. Example from a test diagram.
28
E f f i c i e n c i e s , n, and c o e f f i c i e n t s of f r i c t i o n , u, are then calculated from averages of torque, M, and mutually related axial f o r c e s , F , by
ax the following formulas.
Motion and axial force directions opposite:
_ Fa x 's _ (l-n)cosy
2 t t « M n/tane+tane
Motion and axial force directions equal:
2tmM (l-n)cosy n u =
F «s n» tane+l/tane ax
Obtained c o e f f i c i e n t s of f r i c t i o n are presented in Appendix, T4-T11 and Dl-016. Diagrams of u-values obtained from experiments with oppo- s i t e or equal directions of motion and axial force respectively are thus given separately. The reason f o r t h i s is that one can observe c e r t a i n differences in the r e l a t i o n s of y-v/p in the two types of motion, and t h i s in turn can be an i n d i c a t i o n of s i g n i f i c a n t physical d i s s i m i l a r i t i e s in the way of f u n c t i o n i n g . The f o l l o w i n g subscripts are used,
opp opposite d i r e c t i o n s of motion and axial force eq equal d i r e c t i o n s of motion and axial f o r c e .
No s i g n i f i c a n t difference between the results of series 1 and 2 can be observed, so the results are shown together in the same diagrams.
29
4.3 Analysis of experimental r e s u l t s
The parameters S Q and 1/B are e m p i r i c a l l y determined from the n - ( v / p ) - diagram by graphical c o n s t r u c t i o n .
I t appears from equation (4.16) that the ordinate at o r i g i n of c o o r d i - nates for the "hydrodynamic" l i n e is
Sq is graphically determined by f i t t i n g a s t r a i g h t l i n e to the points of measurement, which are judged to be w i t h i n the hydrodynamic regime.
The distance L is then measured, i . e . the i n t e r s e c t i o n of the s t r a i g h t l i n e and the y - a x i s . Figure 4.3.1 and 4 . 3 . 2 . This then gives
S0 = 3 L
The s t a t i c f r i c t i o n y(0) = 1/B is obtained from u -p. The equations (4.18) and (4.20) give
p ( 0 ) = l / B = — _ i (4.21)
um l-n i s determined from the curve, which is adjusted to the point of measuring in the boundary l u b r i c a t i o n regime. Figure 4 . 3 . 2 .
This method of determination of the c o e f f i c i e n t of s t a t i c f r i c t i o n is preferable to d i r e c t measurement in the diagram, since i t i s "imposs- i b l e to f i n d a d i s t i n c t point corresponding to p ( 0 ) . Nor w i l l d i r e c t determination from experiments give acceptable values since t h i s is associated with great p r a c t i c a l d i f f i c u l t i e s .
30
my Fax, 1.0
Nut no. 4
0.8
0.6
0.4
0.2
L= 173~S.
r
s7"
—
—
2.0 4.0 6.0
3 ( v / p ) 140
2a—.1* -,(?=
8.0 IQjO
v / p E-6 m /NS
Figure 4 . 3 . 1 . Graphical determination of SQ and the slope of the "hydrodynamic" l i n e .
my F a xo pP
0.20
Nut no. 4
0.16
0.12
0.08
0.04
0.4
/ y
y y y
< ^
v / p ^ O , > y
)2 = 0.31
0.8 1.2
Figure 4 . 3 . 2 . Graphical determination of SQ and minimum c o e f f i c i e n t of f r i c t i o n .
1.6 2.0 v/p E-6-m3/Ns
31
Transformation of the variable (v/p) i s done according to the f o l 1 owing.
D e f i n i t i o n
S = f . (1)
ho P Derivation
3S = i - . 3(1)
h0 P
Solve f o r n _ aS
Develop the derivative
3S _ as as* ap 3(1) aS* 3y 3(1)
By d e f i n i t i o n
S* = — which leads to - — = S
SQ 3S*
and from (4.15)
Insertion gives
S = 1.5 • — — • (—) . 3(1) P or
s* =hl . 3 a— (1)
so 3(1) P
o
3 U - = ! sn one obtains l £ L = i i *
aS* 3 u 3y S0
32
Here — i s the slope of the "hydrodynamic" l i n e and i s measured i n
the y-(-jj-) diagram. See figure 4 . 3 . 1 .
Graphically determined and calculated values of the parameters S Q and — are given in table 4 . 3 . The table also gives measured values
of um l-n and corresponding values of (v/ p )o p T/
In table 4.4 calculated values of y(0) according to (4.21) and values of S ^ according to (4.19) and (4.20) are presented.
<: 3
s 0 - V m i n
0p t " (1 + B S0)3 " S02
In the figures 4.3.3-4.3.10 t e s t r e s u l t s in dimensionless form and curves of the theoretical model are combined. The t e s t results are also presented in Appendix, T12-T15.
Table 4.3 Graphically determined parameters.
Nut no 1 Nut no 2 Nut no 3 Nut no 4
F a XoP P Fax eq
Fa x opp
Fax eq
Fax opp
Fax eq
Fax opp
Fax eq L = S0/3 0.031 0.031 0.041 0.040 0.053 0.030 0.031 0.031
S0 0.093 0.093 0.123 0.120 0.158 0.090 0.093 0.093
wmin 0.062 0.064 0.086 0.079 0.086 0.069 0.061 0.067
( v /P > o p t [m3/Ns]-10~6
0.16 0.23 0.25 0.22 0.16 0.23 0.23 0.23
8u/a(J) [Ns/m3]103
101 80 107 89 104 98 96 94
34
my Fax 0.20
0.16 ^
0,12
0.08 r-
0.04
opp Nut no. 1
0.08 0.12 0.16
Figure 4 . 3 . 3 . Theoretical curve of c o e f f i c i e n t of f r i c t i o n and experimental r e s u l t s .
36
37
38
39
40
s
Figure 4 . 3 . 9 . Theoretical curve of c o e f f i c i e n t of f r i c t i o n and experimental r e s u l t s .
42
43
4.4 Discussion and conclusions
The c o e f f i c i e n t of f r i c t i o n (4.13) has been deduced with the assump- t i o n that the average increase in pressure at the a s p e r i t i e s , p ' , and the v i s c o s i t y , n, are constant. However, i t should be possible to study the v a r i a t i o n in the c o e f f i c i e n t of f r i c t i o n , u , according to the parameter S i r r e s p e c t i v e of the value of p' or p, as u is a func- t i o n of the r a t i o v/p and not of v and p separately [ 2 ] , [ 1 5 ] . Experi- ments that have been carried out also indicate t h i s .
The roughness of the s l i d i n g surfaces has influence on the f r i c t i o n [ 4 ] . However, the experimental r e s u l t s confirm the assumption that the t h e o r e t i c a l model i s independent of the d i s t r i b u t i o n of the asperi- t i e s . The s i g n i f i c a n t roughness parameter is probably the depth of p r o f i l e which is included in the equation of the c o e f f i c i e n t of f r i c - t i o n (4.13) ,(4.17) in the constant SQ.
Concerning the assumed constancy of the v i s c o s i t y i t should be stated t h a t t h i s assumption i s not c o r r e c t .
I t - i s true that the temperature dependence of v i s c o s i t y of l u b r i c a t i n g greases is less pronounced than that of the corresponding base o i l s
[ 1 1 ] . Nevertheless, the v i s c o s i t y varies by more than 100% with present v a r i a t i o n s of temperature, see f i g u r e 3 . 1 . 7 .
However, the r e s u l t s of the present experiments with various powers and coherent increases in temperature indicate that the c o e f f i c i e n t of f r i c t i o n i s not affected by increasing temperature. This is an obser- vation also made by H i r s t and Hollander; the f r i c t i o n remains constant with r i s i n g temperature u n t i l a c r i t i c a l temperature i s attained above which i t rises rapidly [ 9 ] .
44
This phenomenon may be explained according to the f o l l o w i n g .
Increasing transfer of power gradually causes such an increase in tem- perature that the v i s c o s i y , n, w i l l be a f f e c t e d . When normal l u b r i - cants are used, the v i s c o s i t y decreases when the temperature is r a i s e d . Simultaneously, however, as the carrying capacity of the l u b r i c a n t de- creases, the f i l m thickness, h, w i l l also decrease. This, in turn causes a tendency of the r a t i o n/h to a t t a i n a constant value, thus r e s u l t i n g in a r e l a t i v e l y small influence on the hydrodynamical shear s t r e s s .
According to equation (4.12) the c o e f f i c i e n t of f r i c t i o n can be w r i t t e n
p*L vQ j hp
in which the r a t i o n/h only appears in the l a s t term.
On the basis of the argument above, i t i s clear that decreasing v i s c o s i t y w i l l not a f f e c t the c o e f f i c i e n t of f r i c t i o n to a larger extent.
A closer analysis of the r e l a t i o n between supplied power and c o r r e - sponding increase in temperature i s given in chapter 5.
Analytical r e l a t i o n s and experiments show that screw-nut transmissions have a way of functioning t h a t , in many respects, are reminiscent of o i l lubricated journal bearings. In p a r t i c u l a r , the r e l a t i o n of u-S shows t h i s . This r e l a t i o n has the same c h a r a c t e r i s t i c appearance as journal bearings and is mentioned by a number of authors [ 5 ] , [ 1 6 ] .
An important difference in the way of functioning between the journal bearing and the screw nut transmission is the increase in f i l m thickness in connection with increasing S-values, which cannot be achieved in screw-nut transmissions, i . e . a purely hydrodynamical behaviour can never be achieved. This i s clear from the f o l l o w i n g .
45
With purely hydrodynamical l u b r i c a t i o n the following holds true when
S > S Q and a = 0.
From the equations (4.2) and (4.10) and the d e f i n i t i o n of S , one obtains
u - — . S
h0 P and the d e r i v a t i v e
3p = n_
3(1) K
This r e l a t i o n is sometimes called the "Petrov asymptote" [ 5 ] , [ 1 6 ]
In screw-nut transmissions the following holds true when S > S Q .
Transform the d e r i v a t i v e
3y _ j ) y _ > _3_S*_ # 3S
3(1) 3S* 3S 3(1)
According to equation (4.15)
3y . 2 f 3S* 3 0 -
D e f i n i t i o n s and d i f f e r e n t i a t i o n give
S * = ^ , ^L-L. a n d
S0 3S S0
s = — . (-) , 1 L _ = iL
h0 P 3(1) h0
I n s e r t i n g the derivatives in the transformed expression gives 3ja = 1 JJ_
3(1) 3 hQ
which is a l i n e a r r e l a t i o n , but the l i n e does not pass through the o r i g i n .
46
When the results given in diagrams 4 . 3 . 3 - 4 . 3 . 1 0 are s t u d i e d , one obser- ves that the c o e f f i c i e n t of f r i c t i o n has a minimum w i t h i n a l i m i t e d i n t e r - v a l . The middle of the i n t e r v a l corresponds to So p t according to chapter 4 . 3 . Within t h i s i n t e r v a l , the screw-nut transmission operates at maximum e f f i c i e n c y . One can also note that the position and the size of the i n t e r - val are almost independent of the level of c o e f f i c i e n t of f r i c t i o n .
SQpt cannot be determined exactly owing to the semi-empirical charac- t e r of the theoretical model. However, i t is possible to estimate the l i m i t s of SQ p t with the obtained values given in table 4 . 4 as a s t a r t i n g p o i n t . This i s shown in table 4 . 5 .
Table 4 . 5
^ o p t ^ m i n 'Sopt'mv ^opt^max calculated 0 . 0 2 6 0 . 0 3 4 0 . 0 4 2
graph.det. 0 . 0 2 5 0 . 0 3 1 0 . 0 4 1
Calculation of the l i m i t s of S Q by combining the greatest and smallest values of um i-n and S Q i s not correct owing to the f a c t that the com- bination umjN/ S Q i s s p e c i f i c to each nut.
When c a l c u l a t i n g c o e f f i c i e n t s of f r i c t i o n numerically i t i s necessary to use relevant values of the constants B and S Q . As a basis for the estimation of B and S Q , the greatest, smallest and mean values are represented in table 4 . 6 . The values are taken from tables 4 . 3 and 4 . 4 .
Table 4 . 6
min mv max
1 / B 0 . 1 8 0 . 2 3 0 . 3 0
Sn 0 . 0 9 0 0 . 1 0 1 0 . 1 2 3
47
From table 4.3 one can observe that the t e s t r e s u l t , for nut no 3 running opposite Ff l X, shows the value SQ = 0.158. This value d i f f e r s from the other to such a great extent that i t cannot be considered representative. The value has been omitted and thus does not influence mean and maximum values.
48
5. THE HEAT CONDUCTION PROBLEM
5.1 Balance of developed power and heat
The f r i c t i o n and coherent development of power i s p r i m a r i l y located at the i r r e g u l a r i t i e s at the surfaces, a s p e r i t i e s , which appear on the thread f l a n k s . The asperities cause local peaks of temperature, which are q u i c k l y quenched to the ambient temperature. The local peaks of temperature can reach about 1000°C but they have a very short duration of 0.1 ms or l e s s . [6] , [ 1 2 ] .
In t h i s context, the ambient regions are the threads themselves and a zone, the boundary layer, consisting of the contact surface of the thread flanks and the i n t e r j a c e n t l u b r i c a n t .
The temperature of the contact surface region gradually increases as a r e s u l t of development of power along the contact surface.
The balance of power can be w r i t t e n
QF + Q + Qe ax
i s the t o t a l supplied power i s the power to move a x i a l force / qdA
A
is the f r i c t i o n a l power/unit area absorbed by nut and screw i s the f r i c t i o n a l power transported away from the nut with the l u b r i c a n t .
Q t o t
Where 0t o t
ax Q
49
5.2 Theoretical model
H
hollow cylinder
rod (1) (2)
3 b
Figure 5 . 2 . 1 . Rod and hollow cylinder in the mathematical model.
The model consists of a c i r c u l a r c y l i n d r i c a l rod, corresponding to the screw, which slides through a hollow c i r c u l a r c y l i n d e r , corresponding to the nut in the screw-nut transmission, at constant speed with simulta- neous development of power. The zone where the development of power takes place i s , in the model, represented by the c i r c u l a r c y l i n d r i c a l contact surface between the rod and the hollow c y l i n d e r . In the screw- nut transmission t h i s contact surface corresponds to the -zone that includes the threads of the nut and the screw.
The motion of the screw-nut transmission, r o t a t i o n and t r a n s l a t i o n , is replaced by pure t r a n s l a t i o n . This is a s i m p l i f i c a t i o n which should not influence the fundamental process.
The f o l l o w i n g r e l a t i o n is taken i n t o account
u = v • tan8
where v i s the mean peripheral speed of the thread flank e is the mean pitch angle of the thread
50
I f the part of the power that is transported out of the hollow c y l i n - der by the l u b r i c a n t is not taken i n t o consideration, the f o l l o w i n g i s val i d ,
q = q j + q2 [W/m2] (5.1)
where q^ the part of the power conducted i n t o the c y l i n d r i c a l rod q2 the part of the power conducted through the hollow c y l i n d e r .
5.3 The equation of heat conduction
The d i f f e r e n t i a l equation of heat conduction in an i s o t r o p i c medium can be w r i t t e n [13]
pC( H + IT • vT) - XV2T = q • S ( r - a ) [W/m3l
1 +• L J
t is the time [s]
T i s the absolute temperature [K]
Q is the heat f l u x [W/m2] u is the v e l o c i t y vector [m/s]
P i s the density [kg/m3]
X i s the c o e f f i c i e n t of thermal conductivity [W/(m-K)]
c is the heat capacity [Ws/(kg-K)]
and
S(r-a) = 0 f o r r * a
/ 6 ( r - a ) d r = 1 a-
[1/m]
[0]
51
With c y l i n d r i c a l coordinates and considering the c i r c u l a r symmetry:
V2T + I I - ( r H ) 3x r 3r 3r we get
,3T 3T. , , 3 1 , 1 3 , I T , , . , , o p c ( — + u — ) - x[—j + (r — ) ] = q 6 ( r - a )
3t 3x 3x r 3r 3r
The rod, region 1 , 0 l r i a
3T, ST. 32T, , 3T,
P lc ( — i + u - 1 ) - X j l — « i + -— (r — L ) ] = 0 ( 5 . 2 ) 31 3x 3x r 3r 3r
Boundary layer, a" < r < a+
4 = °
3X
a+ a T A+ 1 a a T A +
f pc — dr - / x i - ( r - ) d r = / q6(r-a)dr a- at a" r 3r 3r a"
- x 3 T /+
T j = T2
3T1 3T9
x ( — i ) - x2( — ) = q(x) [W/mz] (the equation ( 5 . 3 )
3 r r=a 3r r=a of power)
which means that
A i £ dx - A2 ^ dx = / q ( x ) d x = _ L ( Qt o t- QF a x) ( 5 . 4 )
