Survey on stiffness and damping of machine tool elements
Citation for published version (APA):
Hijink, J. A. W., & van der Wolf, A. C. H. (1973). Survey on stiffness and damping of machine tool elements. (TH Eindhoven. Afd. Werktuigbouwkunde, Laboratorium voor mechanische technologie en werkplaatstechniek : WT rapporten; Vol. WT0312). Technische Hogeschool Eindhoven.
Document status and date: Published: 01/01/1973
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~IPA
SURVEY ON STIFFNESS AND DAMPING OF MACHINE TOOL ELEMENTS
by
J.A.W. Hijink and A.C.H. van der Wolf
Department of ~1echanical Engineering
Eindhoven University of Technology, the Netherlands
•
Papep to be ppesented to the 23pd GENERAL ASSEMBLY of CIRP,
Bled, Yugoslavia. August 1973.
SUMMARY
The stiffness and damping of elements as bearings, slideways, joints etc. has to be known 1n order to predict completely the behaviour of a machine tool by means of Computer-Aided-Design. This paper aims to draw up an inventory of the work which has been done on this stiffness and damping in the several institutes.
1. INTRODUCTION
In Computer-Aided-Design of machine tool structures it is almost common practice to consider the machine tool as con-sisting of a number of beam elements. In this way of thinking it is assumed that the connection between two elements of the model of the machine tool is rigid for every degree of freedom,
or in case of a hinge is rigid for some degrees of freedom. As. a result the division of a machine tool in elements is often more directed to "easy modelling" than to "reality".
The input for the computer program consists of the geometrical properties of the elements and the material properties such as modulus of elasticity, shear modulus and specific mass. The damping in the material can be neglected.
Also the co-operative work in Computer-Aided-Design of machine tools within CIRP Technical Committee Ma follows the procedure mentioned above·).
This way of modelling passes the reality of the machine tool which is built up out of a number of elements connected by means of elements as joints, bearings and slideways. In a correct model these connecting elements should be represented by springs
and dampers.
Although the mod·elling mentioned above is basicly wrong, the results of the analysis are not always useless. Up to now, there are roughly three reasons for leaving out of the model springs and dampers representing the connecting elements
1st there are rather few - reliable - numerical values of these springs and dampers available,
2nd the way in which these data are available, is largely unsuitable for direct use in CAD,
*)
COWLEY, A Co-operative work in Computer-Aided-Design in the CIRP., J - 2
'-3rd the introduction of damping in the existing programs, makes these programs more compl icated - and last but not
least - more expensive.
The aim of this publication is to draw up an inventory of what has been done on stiffness and damping of the connecting
parts of machine tools. The inventory concerns mainly the work of the following members and their co~workers :
Professor J.G. Bollinger,
University of Wisconsin, Madison, U.S.A.,
Professor F. Koenigsberger and Dr. R. Bell,
University of Manchester, Great Britain,
Professor H. Opitz,
T.H. Aachen, West Germany,
Professor J. Peters and Mr. P. Vanherck,'
university of Louvain, Belgium,
Professor A.C.H. van der Wolf and Mr. J.A.W. Hijink,
2. CATEGORIES OF CONNECTING ELEMENTS
2.1 Machine tool joints
On the static and dynamic behaviour of constructions joints have an important influence. A very good overall view on the work done untill this moment can be found in the survey paper on joints by BACK, BURDEKIN and COWLEY (]). They describe how almost all the work on joints was concentrated upon the surface stiffness, which depends upon: the materials in contact, hardness, machining process, surface roughness, relative orientation of the surface layers, size of the contact area, flattness deviation and the contact pressure. DOLBEY, BELL (2) et. al. give for a number of materials and surface conditions values with which the surface stiffness can be calculated. Experiments in order to obtain these data were often done on small testpieces which were rather rigid and up to now there is no model in which all the factors influencing the surface stiffness are considered.
BACK (3) however did incorporate the surface stiffness characteris-tics into non rigid finite elements models of a number of different joints. He describes three methods to enclose the surface stiffness between the two surfaces of the joint, particulary the hydrostatic-, plate- and spring method. Using the plate- or spring method the surface is considered as a separate non linear element and with an iterative way the deflections and pressure distribution are calculated. A very good correlation was found between the theoretical and experimental deflec-tions. A very important conclusion of Back is that for most of the joints the surface deflections contribute only for approximately 10 percent to the overall deflection (see Fig.I). This means that it is almost impossible to predict the stiffness of a joint only by taking into account the surface stiffness. However calculating every joint with the finite elements method will be expensive.
To calculate bolted joints PLOCK (4) gives a method starting from the stiffness of a flange-bolt combination and the surface stiffness of the flanges. Knowing the outline of the joint and the distribution of the bolts, it is possible to calculate the total contact area the tension in
4
-of the flanges can not be calculated and the results as shown "in Fig.2, are on~y satisfactory for rather stiff flanges and bolts placed almost in line with the walls.
BACK, BURDEKIN, COWLEY (1) and GROTH (5) drew the following conclu-sions about the damping in joints
- the dynamic and static stiffness for dry surfaces in contact is the same, the damping is negligible,
- the introduction of a lubricant adds damping to the structure, the damping coefficient is independant of the normal load,
- the damping coefficient for joints with metallic contact depends on the pressure distribution of the surface, surface condition and viscosity of the lubricant,
the mechanism of the normal damping is analogous to squeeze film damping of a lubricant film.
About damping in joints almost no data are available, and up to now it is impossible to predict the damping coefficient when geometry, pressure and viscosity are known.
2.2 Plain Slideways
The static and dynamic properties normal to the sliding motion of plain slideways are almost the same as those of joints. The same difficulties and possibilities in astablishing values for the normal stiffness occur.
In the direction of sliding the dynamic characteristics are mainly determined by the damping caused by friction, the mass of the carriage and the stiffness and damping of the drive.
The damping caused by slideway friction is nonlinear and may be either positive or negative, which means stable sliding or self-excited oscilla-tions caused by stick-slip. BELL and BURDEKIN (6), (7), (8), (9) have published the dynamic behaviour of a testrig representative of a machine tool. They concluded that dynamic measurements are essential to obtain values for the damping. Besides the mass of the carriage, the stiffness of the drive and the coefficient of friction, the carriage speed has an important effect on the stability and damping of the carriage (see Fig.(3)). Because the stability and damping is a complex function of a number of
the development of a complex analytical model. The influence of time dependent variables are difficult to accomodate in definite terms.
BRITTON and BELL (10) examined the influence of design parameters on the stability of feed drives. Their principal conclusions were :. - the drive stiffness is a dominant factor,
- the influence of lubricant viscosity is important,
- the drive natural frequency has no primary influency on the stability of sliding motion. The drive stiffness and table mass must be considered as separate parameters.
- Effective design solutions may be achieved when the critical velocity is below the minimum feed speed or when the amplitude of vibration is not detrimental to machine performance.
To determine the resonance curves of machine tool feed drives for a
number of coefficients of friction and sliding speeds KALS (II) and
HOOGENBOOM (12) give analogue models. The only difficulty here is to obtain proper values for the coefficient of friction.
2.3 Roller bearings and roller guideways
Based on the Hertz theory Palmgren developed a set of formulae to calculate the deflections of all common types of bearings subjected to radial or axial loading. In these formulae the only unknowns are the number of rolling
elements, the dimensions of the rolling elements and their contact angle. The formulae give only a fair result for bearings without clearance and no preload.
GUNTHER (13) did research on cylindrical-roller bearings. He examined the influence of radial clearance, accuracy and preload on the radial stiffness of the bearings and made nomograms for two types of bearings.
Thrustbearings, which are an important part of the feeddrives of NC machines, have been examined by BELL and KIMBER (14), (15). They concluded
that the use of high preloads ensures higher stiffness of the bearing. Bearings with a relatively large number of rolling elements give a higher stiffness but a reduction of the load carrying capacity. In the case of double thrustbearings the influence of temperature is a secondary factor.
On the clamping stiffness of taperroller bearings some work has been done by ELSERMAN (16) DEBRABANDERE and VAES (17). Tests were carried out
6
-with a rotating and nonrotating bearing loaded by a static moment. Experiments showed that, due to frictional resistance, the clamping stiffness of a nonrotating bearing is 20-30 times higher than the clamping stiffness of a rotating bearing (see Fig.4). Between 300 and 1100 cycl/min the clamping stiffness does not change. A theoretical model is developed to calculate the clamping stiffness as a function
of the preload.
To provide an effective antifriction guideway, rolling element guide-ways can be used. DE FRAINE (18), (19) studied the static stiffness of ballslideways and its relation with the load eccentricity. He showed that
the load-capacity decreases very quickly with the eccentricity; the decrease in the stiffness is not as critical.
Because of the fact that below a certain frequency the dynamic stiffness of roller guideways without dampers is lower than the static stiffness, HALLOWES and BELL (20) added fluid film dampers to the guideways and became good dynamic results. The results depend on the thickness of the squeeze film and the viscosity of the oil but a general increase in the dynamic . stiffness is always attained.
2.4 Hydrostatic bearings and -guideways •.
In the past decades many constructions for hydrostatic thrust- and journal bearings have been developed. However, the design was almost only concerned with the static behaviour i.e. stiffness, maximum load and flowrates.
BOTTCHER, EFFENBERGER and OPITZ (21}, (22) and also COWLEY and KHER (23) investigated the dynamic behaviour of hydrostatically supported spindle-bearing systems. Bottcher used a more complicated spring-damper model for
the bearing (Fig.5), while Cowley used a one spring-damper model. They showed that it is possible to optimize the stiffness and damping for the spindle-bearing system (Figs. 6 and 7). By only stiffening the bearings the system can get a very bad dynamic response.
The damping and stiffness of a hydrostatic bearing can be changed in a wide range by varying the pressure and/or dynamic viscosity of the oit.
In this respect the hydrostatic bearing has much advantage on the roller bearing.
the work done on the dynamic behaviour of thrust and journal bearings including squeeze film damping, compressibility and inertia effects.
On hydrostatic guideways PORSCR (25) did some research on the static stiffness of guideways with different frames.
2.5 Aerostatic bearings and -guideways.
WARNECKE (26) has given a very good survey on the static- and dynamic characteristics of all kinds of aerostatic Bearings. Aerostatic bearings have advantage when used for very low friction and high accuracy.
Attain-ing a sufficient load capacity and stiffness are the most difficult
problems, caused in particular by the low dynamic stiffness and pneumatic instability.
8
-3. DAMPERS
In machine tool structures dampers have a special function. Only when the dynamic behaviour of the machine tool leaves a certain range, one will try to change this dynamic behaviour with the help of a damper.
So in CAD a damper never can be implemented in the first stage of a design. Only when the dynamic behaviour is known one can give the proper dimensions to the damper.
Two kinds of dampers can be distinguished the passive damper and active damper.
3.1 Passive dampers.
The passive dampers can be subdivided in the following systems a) a viscous damper between two machine parts,
b) damped added mass, c) Lanchester damper, d) Impact damper.
In the cases a and b the fluid film damper is often used as damping element. VANHERCK (27) gives a description of the dimensioning of flat and circular fluid film dampers. Also plastics can be used as element with damping and spring characteristics. With the help of special dynamic
CAD programs these dampers can be implemented in the machine tool struc-tures in order to see how the results are.
The impact damper can only be calculated with help of an analogue computer.
3.2 Active dampers.
BECKENBAUER (28) developed an active damper for machine tools. When using an active damper, energy is added to the total system. Therefore
4. CONCLUSIONS
Much work has been done on the static and some on the dynamic behaviour of "connecting" elements in machine tool structures.
In spite of this work, only for the static stiffness of roller bearings, and hydrostatic- and aerostatic bearings data are available for direct use in CAD. For the normal stiffness of joints and plain slideways every joint has to be calculated with a special purpose program. On damping there is some knowledge on hydrostatic and aerostatic bearings and guide-ways which could be used in CAD.
The knowledge of the damping of joints and plain slideways is very specific and there is no general model available for all kinds of joints and guide-ways.
In table 1 a survey of the usefullness of the different categories is given.
As fields of research are recommended investigations into :
1st the stiffness of joints and guideways, in a general way,
2nd the dynamic stiffness and damping of joints and plain guideways,
3rd the dynamic stiffness and damping of roller bearings.
Remark : The references are only from the institutes mentioned in the
introduction. Of course there has been done much more research on this
subject within other institutes. Particulary in ref.
(1), (24)and (26)
to
-REFERENCES
I. BACK, N., BURDEKIN, M. and COWLEY, A.: Review of the research
on fixed and sliding joints.
Proc. 13th MTDR Conf., Birmingham (1972).
2. DaLBEY, M.P. and BELL, R.: The contact stiffness of joints at
low apparent interface pressures. Annals of the CIRP. Vol. XVIV (1970).
3. BACK, N.: Deformations of machine tool joints.
Ph. D. Thesis, University of Manchester (1972).
4 . . PLOCK, R.: Untersuchung und Berechnung des elastostatischen
Verhaltens von ebenen Mehrschraubenverbindungen~
Dissertation T.H. Aachen (1972).
5. GROTH, W.H.: Die Dampfung in verspannten Fugen und Arbeits-'
fuhrungen von Werkzeugmaschinen. Dissertation T.H. Aachen (1972).
6. BELL, R. and BURDEKIN, M.: The frictional damping of plain
slideways for small fluctuations of the velocity of sliding. Proc. 8th MTDR Conf. Manchester (1967).
7. BELL, R. and BURDEKIN, M.: A study of the stick-slip motion of
machine tool feed drives.
Proc. Instn. Mech. Engrs. 184, Pt I (1969 - 1970).
8. BELL, R. and BURDEKIN, M.: Plain Slideways.
Proc. Instn. Mech. Engrs. 184, Pt I (1969 - 1970).
9. BELL, R. and BURDEKIN, M.: The influence of slideway materials
and lubricants on the dynamic characteristics of plain slideways. Instn. Mech. Engrs. Tribology Convention (t971).
10. BRITTON, D.R. ,and BELL, R.: The influence of designparameters on the stability of sliding motion of machine tool feed drives. Proc. 11th MTDR Conf., Birmingham (1970).
II. KALS, H.J.J.: Dynamic stability in cutting.
12. HOOGENBOOM, A.J.: Some dynamic aspects of the coulomb friction combined with relative velocity.
Report WT 248, Eindhoven, University of Technology (1970). 13. GUNTHER, D.: Untersuchung der Federung von
Hauptspindel-lagerungen in Werkzeugmaschinen. Dissertation, T.H. Aachen (1964).
14. BELL, R. and KIMBER, E.: The axial stiffness of
thrust-bearing assemblies for N.C~ machine tool feed drives.
Proc. 10th MTDR Conf., Manchester (1969).
15. KIMBER, E.: The static axial stiffness of rolling element bearing assemblies.
M.Sc. Dissertation, University of Manchester (1968). 16. ELSERMANS, M.: Survey on the study concerning the static
clamping stiffness of taper roller bearings.
Report 72 R I, University of Louvain, Dept. Mech. Eng. (1972).
17. DEBRABANDERE, E. and VAES, E.: Experimentele en theoretische bepaling van de inklemmingsstijfheid van kegellagers.
Report 72 E 2, University Lo~vain, Dept. Mech. Eng. (1972).
18. DE FRAINE, J.: Optimalisatie van het concept en het gebruik van kogelgeleidingen uitgaande van de vervormingsanalyse. Thesis Louvain (1968).
19. DE FRAINE, J.: The choice of ballslide ways for machine tools. Proc. 12th MTDR Conf., Manchester (1971).
2b~ HALLOWES, J.G.M. and BELL, R.: The dynamic stiffness of
anti-friction rollerguideways.
Proc. 13th MTDR Conf., Birmingham (1972).
21. BOTTCHER, R., Untersuchungen uber das dynamische Verhalten hydrostatischen Spindellagerungen
Dissertation, T.H. Aachen (1968).
22. OPITZ, H., BOTTCHER, R. and EFFENBERGER, W.: Investigation on the dynamic behaviour of hydrostatic spindle bearing systems. Proc. 10th MTDR Conf., Manchester (1969).
12
-23. COWLEY, A. and KHER, A.K.: The dynamic characteristics of a hydrostatically supported spindle bearing system.
Proc. 10th MTDR Conf. Manchester (1969).
24. KOENIGSBERGER, F. and COWLEY, A.: Dynamics of hydrostatic bearing systems.
Instn.Mech.Engrs. and Instn.Prod.Engrs.
Joint Conf. on Externally Presurized Bearings, Londen (1971). 25. PORSCH, G.: Uber die Steifigkeit hydrostatischer Fuhrungen
unter besonderer Berucksichtigung eines Umgriffes. Dissertation, T.H. Aachen (1969).
26. WARNECKE, H.J.: Konstruktion und Eigenschaften aerostatischer Lager und Fuhrungen.
Annals of the CIRP Vol. XXI (1972).
27. VANHERCK, P.: Dimensioning of fluid film dampers. Annals of the CIRP Vol. XVII (1969).
28. BECKENBAUER, K.: Entwickelung und Einsatz eines aktiven Dampfers zur Verbesserung des dynamischen Verhaltens von Werkzeugmaschinen. Dissertation, T.H. Aachen (1970).
~
Stiffness Usefullness Damping Usefullnessin CAD in CAD
Machine known, but by means of something not usefuH
tool only specific special fini te known
knowledge of element
pro-joints static s tiff- grams
ness
Plain slide some speci- by means of something not yet,
fic know- special fini te known by to specific
ways
ledge element pro- means of
grams analogue
models
Roller bea- much know- usefull unknown
-rings and ledge about
guideways static, some
abou t dynamic
Hydrostatic much know- useful! some usefull
bearings and ledge about knowledge
-guideways static, some
abou t dynamic
Aerostatic much know- usefull some useful!
bearing and ledge about knowledge
-guideways static,some
about dynamic
Dampers
-
-
known only passivedampers
- 14 _ . _ . _ r -I
1 ---- .- -
---I ScQI~ I • , • ,o
10 20 30 40mm---
... x Measured deflectionsFig. 1. Deformations ( 1000x) of a joint using the spring method ( after BACK
(3) )
15000 30000
E
z
.0 ~z
10000....
u. 20000 e Ql measured Ql E calculated u---I - 0 0 E
-
01 C e E "C_.-.---l - e 0 ~5000
z
10000o
~_~O~ ---L.. _ _ _ _ : ' - - - ' : - - - - ' - - - 'a
1 2 4 Deformat ion {5 (~m)Fig. 2. Comparison between calculated and measured deflections, for different preloads F on
v
l( fero~e5'cs/Ca>[ iron 5 x 4
l
Tufnol/Cast iron 514l~
3 • 312k~
+
~;t---:~A)(~_'::_~~!2._..!'r.J, ~L!~-.!'
""6
v..-,1.-.~
-::2 5 10 15 I 5 ' 10 x A"I5 v -I _I L E' SLIDING Vf.LOCITY--mm!s --"'_.-L< IS -I .-SLlO:NG c; ~ ~co,t iron!Nitridedr
2f- l ( .,I :...
~.
o
~ J . _ .~~.1~_ _~ _Il.
5 10 15 VfLOCITY-mm/s;
~
Nit' id,dI
NIl< Id,da[---- .
-"t----."•
/0
---L1;
-I SLIDING VELOCIT v-mm/s
A .8·4Hz
• 14.6Hz
l( 19·7 Hz
Fig.
3.
Slideway damping characteristics for differentslideway materials and Tonna 27 as lubricant
( after BELL
(9) )
1200 bearing30212 TODD 800 600 ~oo.
non rotating spindle
200
2000 ~ooo
rotating spindle
:
:
6000 8000 10000 12000 Fa(N)~
.
Fig.
4.
The clamping stiffness of a taperroller- 16 -0 2 3 6 7 8 9 10 11 12 13 Y
~
I,
i ,,
Oy12 ~cs" b ...J.Fig.
5.
Hydrostatically supported spindle with itsequivalent calculable model (after BOTTCHER (21) )
• x
~
-~
-
----~
Wd50N,2l2W--~
oL--5;:-;ObO;--~--:Cl0~6C-;o;----;'500 20'00 2500 3000Compressibility stiffness of the front bearing
KC'
Jim
..
] c. 8 ~E..
~ v ~ 4 c: o III...
It: I~ 12 HO,60/Jm IJ:3110'~s/m2o Taken from frequen{:y rtsponse )( Taken from curve of decay - Theoret ical
I I I I I ---l
500 1000 1500 2000 2500 3000
Compressibility sl itfnes5 of front bearing Kc'
Jim
00
Fig.
6.
Damping ratio and resonance amplitude of asystem as a function of the compressibility stiffness of the front journal bearing
(after BOTTCHER (21) )
r
.12~
.1Ol
, i ·08:-,'2"'I
!" 06,~ '" I .!: i c. E ·0. t-oi
c 02 L 1- .05,um/N 2-·04~m/N 3 - ,02~mrN 4··01,um/N DampIng constant. C (N 1m)Fig.
