EXPERIMENTAL RESEARCH AND MODELING OF NORMAL CONTACT STIFFNESS AND CONTACT DAMPING OF MACHINED JOINT SURFACES

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1 ADVANCES IN MANUFACTURING SCIENCE AND TECHNOLOGY Vol. 33, No. 3, 2009 EXPERIMENTAL RESEARCH AND MODELING OF NORMAL CONTACT STIFFNESS AND CONTACT DAMPING OF MACHINED JOINT SURFACES Konra Konowalski S u m m a r y The present paper presents a test stan use to measure normal contact stiffness of flat contact joints of steel surfaces subjecte to mechanical treatment. The test stan was use to etermine the epenence of normal contact isplacements on contact pressures. On the basis of the obtaine characteristics a non-linear moel of a contact joint was evelope. The moel escribes a epenence of normal contact stiffness an amping on mean pressures an the amplitue of ynamic loa with a harmonic course. The moel was base on the Kelvin-Voight moel wiely use in the literature on the subject taking into consieration the stiffness an amping variable. Keywors: contact stiffness, amping, experimental research, histeresis loops Baania oświaczalne oraz moelowanie sztywności stykowej normalnej i tłumienia w połączeniach elementów maszyn S t r e s z c z e n i e W pracy przestawiono stanowisko o baania stykowej sztywności normalnej płaskich połączeń stykowych i powierzchni elementów stalowych po obróbce mechanicznej. Wyznaczono zależność okształcenia stykowego normalnego o nacisku powierzchniowego. Na postawie uzyskanych charakterystyk opracowano moel nieliniowy połączenia stykowego, który opisuje zależność sztywności stykowej normalnej i tłumienia o śrenich wartości nacisku oraz amplituy obciążenia ynamicznego o przebiegu harmonicznym. Postawą opracowania jest moel ciała lepko-sprężystego Kelvina-Voighta, przy uwzglęnieniu zmiennej sztywności i tłumienia. Słowa kluczowe: sztywność stykowa, tłumienie, baania oświaczalne, pętle histerezy 1. Introuction Machines an mechanical evices are usually mae up of many elements an assemblies connecte with one another either immovable manner (bolte joints, shrink fit joints) or movable manner (slieways, bearings). These joints are essential to assure smooth running operation of machines, particularly when they are subjecte to ynamic loa. Owing to this fact the issue of joints has Aress: Konra KONOWALSKI, Ph.D. Eng., Faculty of Mechanical Engineering an Mechatronics Szczecin University of Technology, Szczecin, Polan, konra.konowalski@ps.pl

2 54 K. Konowalski been often analyze in various experimental research projects as well as wiely iscusse in theoretical papers [1-9]. Correct moeling of machine elements an their connections increases the precision of calculations an makes it possible to conuct a computer simulation of their behavior as early as at the esign stage. Methos of how to calculate an rationally shape iniviual machine elements taking into consieration their strength (especially volumetric strength) an stiffness have been well evelope an these methos have been for years use in esigning machines (e.g. FEM). The state-of-the-art knowlege about constructional connections an methos of how to calculate them is quite inaequate in the face of current an fast growing nees in this fiel. Many experiments an ay-to-ay practice have emonstrate [2-7] that they are not elements themselves, but their connections that are the weakest link in the structure of mechanical systems of many machines. Therefore, there is a nee to evelop reliable moels of such connections. In phenomenological approach moeling of contact joints is conucte on the basis of their static an ynamic characteristics. In every ay practice, very simplifie linear moels which escribe various contact phenomena are use for the necessary calculations. An this is true even though there are a number of experimental papers tackling the issue. This kin of simplification makes it impossible to properly unerstan physical phenomena taking place in contact joints. Neeless to say, this can also lea to gross errors. This is particularly true of contact joints which are weakly loae with normal forces, incluing joints of machines, prouction systems, measuring evices, etc. For practical calculations, FEM commercials coes such as ADINA [10], ANSYS [11], ABAQUS are usually use. The systems most often also contain proceures use to analyze contact phenomena. Two basic algorithms an their moifications are wiely use [12]. The first algorithm is the Lagrange Multiplayer Metho an the other the Penalty Function Metho. Both algorithms solve two basic problems, the first of which is how to etermine the volume an amount of the surface area, lines an points coming into contact. These parameters often change when a system unergoes ynamic loa changes. Owing to the fact that bounary conitions can unergo various changes, a contact phenomena shoul be treate as a non-linear phenomena. The other problem concerns etermining mutual influences of surfaces, lines or points coming into contact. This requires selecting reliable moels of contact joints an choosing their parameters. One of the basic parameters chosen by a system s user is contact stiffness. If this stiffness is too high, problems arise in reaching solution convergence or else we obtain matrix singularity of a system of equations. If the stiffness is too low, the solution may turn out to be imprecise or singular. These are typical problems any user of FEM has to face an overcome by means of a trial an error approach. In orer to make calculations easier, some FEM moels inclue proceures which automatically aopt stiffness of these elements making it possible to both achieve convergence an to maintain the necessary precision of their solutions. However, characteristics of contact

3 Experimental research an moeling stiffness use in these calculations are often quite ifferent from those which are experimentally etermine. Bearing in min these facts, it is necessary to continue further experimental research into contact joints. It is also evient that new, more precise an nonlinear moels shoul be evelope for contact joints. These moels are necessary to properly unerstan current problems of compoun systems ynamics an to be able to solve these problems using computer simulation methos. This paper aims at eveloping an analyzing such a moel that coul be use in the fiel of machines ynamics. 2. Experimental research of flat contact joints loae with normal forces The experiments were performe on a test stan that was constructe specially for these purposes [7]. A iagram of the test stan is shown in Fig. 1. A typical feature of experimental research into contact joints is that even relatively high loa results in a very small contact isplacement, which amounts to anything between a few an several ozen micrometers, while their runs in the loa function are non-linear. Contact isplacement happening as a result of cyclically applie ynamic loa given a stable working loa amounts to 1 µm, or can even be smaller. A pair of samples which actually mae up the investigate contact joint (Fig. 1) was place on a plate 4, in a stiff frame. The samples were shape as cyliners with centrally locate holes. They were mae of steel C45, harness HRC. The investigate contact were the en faces of the samples, which Fig. 1. A iagram of the test stan use to investigate normal contact stiffness an amping

4 56 K. Konowalski were presse to each other with a force F(t). The contact surface of the samples ha a shape of a flat ring an the nominal area of the surface was A = 50 cm 2. Average surface pressure p was calculate as the ratio of force F(t) to the nominal area of the contact surface A. The contact surfaces of samples 1 an 2 ha been groun. Their parameters an their geometrical structure were measure. These measurements were performe on a computerize station esigne to measure geometrical state of surfaces an contours (SUPSGPiK) manufacture by Hommelwerke T8000. Ten measurements were mae for each groun surface. The obtaine mean values of the measure parameters are presente in Table 1. Table 1. The parameters of roughness of the specimens Parameter Displacements, µm Specimen 1 Specimen 2 R max R a R q The traces of surface treatment were place perpenicularly. The samples were subjecte to (static an ynamic) loa by means of a controlle, hyraulic system. The tests were carrie out in the range of constant pressure of p o = 0,5-2,5 MPa with harmonic runs with the amplitue of p a = 0,4 MPa. The total number of loa cycles in each trial was Then it was assume that the normal characteristics were stabilize. After having loae the contact interface up to the value of contact pressure p o (constant loa), it was assume that it was the zero level of ynamic pressure. Then, the reaings of the forces an isplacement in the test be were zeroe an following on from that the contact joint was loae, thus exciting harmonic runs with amplitue p a. The force applie on the investigate contact joint of the specimens was irectly measure with extensometers glue on the specially esigne an shape sample 2, an the relative contact isplacement δ(t) of samples 1 an 2 with an inuctive, non-touch islocation sensor 3, type Tr-20 manufacture by HBM. Signals were recore on a computer system equippe with S&H system an 16 channel transucer A/D of DAS 1800 type, manufacture by Keithley. TESTPOINT software, manufacture by CEC, was use to service the measuring stan. The isplacement time runs as well as the applie excitements were recore with a constant time step an the number of measuring steps, equal to all the experiments, amounte to The number of loa cycles in each trial was 128. The results of the experimental research were save as ASCII files. Files marke, for graphic presentation of runs an

5 Experimental research an moeling results of experiments an their appropriate processing, were exporte to EXCEL spreasheet application. 3. Results of experiments The experiments were conucte on samples whose contact area was grease with a thin layer of ŁT-43 grease. The recore fragments of the time runs of pressure an contact isplacement, given a mean pressure of p o = 0,5 MPa, amplitue of p a = 0,4 MPa an the frequency of the exciting force ω = 1 1/s are shown in Fig. 2a an 2b. Fig. 2. Time runs of pressures p (a), contact isplacement δ (b) an isplacement velocity v (c)

6 58 K. Konowalski The time runs of contact isplacement are asymmetrical to the static state of equilibrium. After ifferentiating these runs, time velocity runs v (t) were obtaine. They can be seen in Fig. 2c. On the basis of the results obtaine in the experiments a characteristic showing epenence of contact isplacement δ on contact pressure p was etermine. The characteristic is shown in Fig. 3. Given the establishe working conitions, uner loa an relief of the contact joint, these characteristics run along various lines an create a close hysteresis loop. The area insie the loop efines the energy issipate in one, close loa an relief cycle. In orer to etermine this area, the loa curve p loa, the relief curve p unloa an the centre line p centre locate between them were escribe with a 4 th egree polynomial: a1δ + a2δ + a3δ a4δ p = a + (1) where: a 0 -a 4 are constant parameters (Table 2). Table 2. The values of parameters for the hysteresis loop shown in Fig. 3 Contact Constant parameters pressure a 0 a 1 a 2 a 3 a 4 p loa p centre 0 1, , , ,87416 p unloa The changes in isplacement pace of this contact joint epening on the values of contact isplacement are presente in the phase portrait (Fig. 4). Fig. 3. Dynamic hysteresis loop obtaine for the contact joint lubricate with a thin layer of Łt43 grease

7 Experimental research an moeling Fig. 4. Phase portrait of the investigate contact. 4. Contact stiffness an amping characteristics Until now joints between various elements have been substitute in calculations with massless iscrete linear spring-amping noes which can be escribe with two coefficients which efine ynamic stiffness an amping. Real joints in multi-mass systems are usually substitute with massless springamping noes, which meet the criteria set by Kelvin-Voight (Fig. 5a). In this kin of a moel, a stable hysteresis loop (at harmonic loa) has the shape of elipse ABCD (Fig. 5b). Fig. 5. Scheme (a) an mechanical characteristic for Kelvin-Voight moel (b) In a contact joint total instantaneous contact pressures p c are the sum of static pressure p o an ynamic pressure p, which lea to static δ 0 an ynamic contact isplacement δ. In orer to prevent a separation of the surface of the

8 60 K. Konowalski contact, the amplitue of contact pressure p a shoul not excee static pressure values p o. Dynamic contact isplacement is symmetrical to the static state of equilibrium (Fig. 5b). The values of energy of spring strain uring loa an relief proceure are equal to each other an they amount to E p = E pu. In Kelvin-Voight moel, the ynamic static stiffness is a constant value which can be given by formula: k 2 pa = δ K V = const. (2) The issipation of vibration energy in Kelvin-Voight moel for one vibration cycle (Fig. 5b) is efine by a coefficient of vibration energy ψ KV which can be escribe as: = E ψ KV (3) E pu where: E is the energy issipate in one cycle of ynamic loa an which is etermine by the area of hysteresis loop shape as ellipse, E pu is the energy of spring strain which correspons to maximum amplitue of eflection from the state of equilibrium efine by pressures p o. The ynamic hysteresis loops etermine uring experimental research for a real contact are very ifferent from the shape of ellipse (Fig. 5). Normal characteristics of contacts show asymmetry of isplacement from the static state n of equilibrium δ 0 efine by pressures p o. The values of spring energy E an n E p (Fig. 6b) corresponing to maximum eflection from the state of equilibrium are in this case ifferent. In practice, in calculations of such contact joints, Kelvin-Voight moel is also use. On the basis of non-linear characteristics obtaine in experiments, equivalent values of contact stiffness an amping are etermine. In the range of small amplitues of ynamic loa, the equivalent contact stiffness is etermine as the tangent of the inclination angle of the tangent line to AOC line, at point O (Fig. 7): pu k tan p = tg( a2) = δ p= p0 (4)

9 Experimental research an moeling Fig. 6. Scheme showing a contact between flat, machine surfaces (a) an its mechanical characteristic (b) k sec At higher amplitues of ynamic loa, the equivalent contact stiffness can be etermine as the tangent of the inclination angle of the secant AC to the axis δ (Fig. 7): k sec 2pa = tg( a1) = (5) δ Fig. 7. A iagram of ynamic hysteresis loop etermine on the basis of experimental research of the real contact joint

10 62 K. Konowalski The equivalent values of contact stiffness etermine from formulae (4) an (5) are treate as constant values an they o not take into account their changes in the whole range of ynamic loa (±p a ) (Fig. 7). From the course of the line AOC it can be seen that the contact stiffness k is a changeable variable, which epens on the current ynamic isplacement values δ. At points A an C (Fig. 7) it can be much ifferent from the values etermine at point O. In orer to increase the precision of calculations, ynamic contact stiffness can be given by the following polynomial: p = = (6) δ s śro 2 3 k a1 a2 δ a3 δ a4 δ The values of issipation energy of vibrations at the real contact an in the investigate non-linear moel shoul also be equal. In orer to achieve this, it is suggeste that the relative coefficient of energy issipation for the non-linear moel of a contact joint shoul be etermine from the following formula: s n 2E ψ = (7) E + E n pu n p where: E n the energy issipate in one ynamic loa cycle an it is etermine by the area of hysteresis loop (Fig. 6b), E, E n p n pu the energy of spring strain (Fig. 6b) for a contact joint with a non-linear characteristics. 5. A moel of a contact joint subjecte to normal harmonic loa by: For Kelvin-Voight moel, ynamic pressure p(t) at the contact can be given K V p( t) = k δ ( t) + h v( t) (8) where: δ (t) is the normal contact isplacement, µm, v (t) the spee of contact s isplacement [µm/s], K V k h the amping coefficient, MPa s/µm. the constant normal stiffness, MPa/µm,

11 Experimental research an moeling After applying ynamic loa with a harmonic course p(t), the amplitue p a an the frequency ω, to the contact joint we get, accoring to formula: ( ) sin( ω ) p t = p t (9) stable time runs of contact isplacement, which are escribe by formula: where: a ( t) K sin( t ) δ = ω φ (10) K = K V ( k ) p 2 a ( h ω) 2 + (11) ω h φ = arctg K V k (12) By eliminating from equations (9) an (10) time t we obtain a epenence of ynamic pressures p on the normal contact isplacement δ. This epenence is non-linear an in one, close cycle of loa an relief of a contact it takes the shape of an ellipse. The area of this ellipse is the measure of energy issipation at the contact. Losses of vibration energy is often characterise by the relative coefficient of vibration amping ψ (formula 3). In Kelvin-Voight moel, the following epenence takes place between coefficients h i ψ KV : KV h = K V k KV ψ 2 π ω (13) After substituting (13) into (11) an (12), we obtain: K = k K V p a ψkv 1+ 2π 2 (14) ψkv φ = arctg 2 π (15)

12 64 K. Konowalski In orer to etermine ynamic characteristics of a non-linear moel it is necessary to know the epenence of normal contact stiffness coefficient an the relative coefficient of energy issipation on contact pressure p. Experimental research has shown that these epenencies are non-linear an that they epen on mean contact pressures p 0 an loa amplitue p a. A iagram showing a moel of such a boy with variable characteristics of contact stiffness an amping is presente in Fig. 6. On the basis of conucte experimental research on a contact loae with mean pressure p 0 = 0,5-2,5 MPa an ynamic loa with amplitue p a = 0,4 MPa, the epenence of normal contact stiffness coefficient on contact isplacement δ given by formula (6), can be expresse as the function of ynamic pressure p: s k = b p + b p + b p + b (16) s where: k normal contact stiffness expresse, MPa/µm, p contact pressure expresse, MPa, b 1 -b 4 coefficients which epen on mean surface pressure p 0 an pressure amplitue p a. s The course of changes of normal contact stiffness k, within the range of amplitue p a = ±0,4 MPa for various values of mean contact pressure can be seen in Fig. 8. The relative coefficient of amping Ψ is etermine on the basis of the area of hysteresis loop, accoring to formula (7). Fig. 8. Depenence of contact stiffness s k on contact pressures

13 Experimental research an moeling After substituting relation (16) into (13) an (8) ( k K V s = k ) an relation (7) into (13) ( ψ KV = ψ ) we obtaine a non-linear moel of a contact, which can be given by equation: s k ψ p( t) = k δ( t) + v( t) 2τ ω s (17) In orer to compare ynamic characteristics for a moel of linear contact an non-linear contact caculations were performe, in which the following ata were assume: p 0 = 0,5 MPa, p a = 0,4 MPa, ω = 1 1/s. For the linear moel a constant value of contact stiffness was assume an this value was efine by formula (4) k = MPa/µm. It escribes normal contact stiffness with tan ynamic pressures p = 0. The relative coefficient of amping was assume on the basis of experimental research Ψ = The contact characteristics obtaine in the calculations were compare with the characteristics obtaine in the experiments. They are presente in Fig Fig. 9.The course of contact isplacements for the linear an non-linear moels obtaine in the experiments The course of contact isplacements obtaine in the calculations for the non-linear moel of the contact (line 2, Fig. 9) coincie with the results obtaine in the experiments (line 3, Fig. 9). The assumption of a constant contact stiffness in the calculation moel (line 1, Fig. 9.) leas to symmetrical runs of contact

14 66 K. Konowalski isplacements. When the contact is loae, contact isplacement is bigger for the linear moel than for the non-linear moel. When the contact is relieve, isplacements are smaller from those obtaine for the non-linear moel. Extreme values of isplacement pace, in loa an relief of the contact have the same values (Fig. 10). However, an increase of the pace in a given time, i.e. acceleration, is ifferent in loa an relief moes. The non-linear moel of the contact more precisely represents time runs of contact isplacement than the linear moel. This is also evient in the phase graph (Fig. 11). Fig. 10. Velocity courses of the linear an non-linear moels an those obtaine in the experiments Fig. 11. Phase graphs for the linear an non-linear moels an those obtaine in the experiments

15 Experimental research an moeling Hysteresis loops for the non-linear moel an those etermine in the experiments practically coincie with each other (Fig. 12). This moel can well escribe, both quantitatively an qualitatively, the real stiffness an amping in the conitions of preset ynamic loa. Fig. 12. Hysteresis loops for the linear an non-linear moels an those obtaine in the experiments 6. Conclusion On the basis of the conucte experiments it was foun that in the range of small contact pressures the epenence of contact eformations on the contact pressures is non-linear. At stable harmonic loa conitions, in one close cycle of loa an relief of the contact, these characteristics create close hysteresis loops. The area within hysteresis loop can be the basis on which it is possible to etermine amping in the non-linear moel of the contact. When contact stiffness variable (etermine on the basis of the experiments) is introuce into the moel, the time runs of contact isplacement both calculate an obtaine in the experiments practically coincie. However, the non-linear moel of the contact is not fully equivalent as far as velocities of contact eformations go. The assumption of a constant value of ynamic contact stiffness in the calculations gives, as a result, significant eflections of contact eformation runs an velocity in comparison with the results obtaine in the experiments. References [1] Y. Ren, C.F. Bears: Ientification of effective linear joints using coupling an joint ientification techniques. ASME Journal of Vibration an Acoustics, 120 (1998),

16 68 K. Konowalski [2] W.P. Fu, Y.M. Huang, X.L. Zhang, Q. Guo: Experimental investigation of ynamic normal characteristics of machine joint surfaces. ASME, Journal of Vibration an Acoustics, 122 (2000), [3] K. Marchelek: Dynamika obrabiarek. WNT, Warszawa [4] K. Konowalski: Experimental investigations of contact joint uner ynamic loa. Proc. Fifth Inter. Conf. on Computational Methos in Contact Mechanics, Wessex Institute of Technology, Escuela Superior e Ingenieros, University of Seville, WitPress, Southampton Boston 2001, [5] X. Shi, A.A. Policarpou: Measurement an moeling of normal contact stiffness an contact amping at the meso scale. ASME, Journal of Vibration an Acoustics, 127(2005). [6] K. Gruziński, K. Konowalski: Baania charakterystyk mechanicznych połączeń stykowych przy obciąŝeniach ynamicznych. Cz. I. Postawy baań oświaczalnych i stanowisko baawcze. Archiwum Technologii Maszyn i Automatyzacji, 22 (2002)2, [7] K. Konowalski, K. Gruziński: Baania charakterystyk mechanicznych połączeń stykowych przy obciąŝeniach ynamicznych. Cz. II. Baania oświaczalne. Archiwum Technologii Maszyn i Automatyzacji, 22 (2002)2, [8] G. Pugliese, S.M.O. Tavares, E. Ciulli, L.A. Ferreira: Rough contacts between actual engineering surfaces, Part II Contact Mechanics. Wear, 264(2008), [9] U. Sellgren, S. Björklun, S. Anersson: A finite element-base moel of normal contact between rough surfaces. Wear, 254(2003), [10] ADINA R&C Inc.: Automatic ynamic incremental nonlinear analysis. Theory an moeling guie, Report ARD 02-7, Boston [11] Documentation for ANSYS, Guie to the ANSYS Documentation Contact Technology Guie, Release [12] P. Wriggers: Computational Contact Mechanics. John Wiley&Sons. Lt., New York Receive in October 2009