Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 41, 1, pp. 107-118, Warsaw 2003 INFLUENCE OF THE LUBRICATING AGENT ON THE PROPERTIES OF CONTACT JOINTS Janusz Skrodzewicz Institute of Manufacturing Engineering, Technical University of Szczecin e-mail: Janusz.Skrodzewicz@ps.pl The paper presents nonlinear mathematical models describing the pro- perties ofdryand lubricatedcontact joints loaded in thenormaldirection within the range 0.5-2.5MPa.The surfaces of jointswere coatedwith an assembly paste andwith a hydraulic oil. The dependences of the energy dissipation coefficient Ψ as a function of the load for applied lubricating agents were presented. The structure of the models was determined on the basis of the experimentally determined spectral-response characteri- stics of the relative displacements of the nominal contact of the surface and of the contact load in the normal direction. The forms of the func- tional factors of models were estimated using the methods of the linear regression analysis. Key words: contact joint, nonlinear models, energy dissipation 1. Introduction Contact joints weakly loaded in the normal direction frequently occur in various technical applications e.g., they are commonly found in connections of movable elements of technological equipments. The physical phenomena describing the motion in the zone of such connections are characterised by significant nonlinearity. The application of different types of lubrication for the contact joints with various surface roughness may considerably change the character of their deformability and damping. Thereby, there is a need to develop mathematical models which describemore precisely the properties of lubricated contact joints than those used until now. 108 J.Skrodzewicz 2. Estimation of nonlinear mathematical models of lubricated contact joints 2.1. The test stand The experimental tests were carried out on a specially designed and con- structed stand, which enables the realization of static and dynamic loadings in the normal directions. The block diagram of the stand is given in Fig.1 (Skrodzewicz, 1999; Skrodzewicz and Gutowski, 2000). The samples for inve- stigationsweremade of steel 45. The surfaces forming the investigated contact joints weremachined by themethod of cylindrical grinding, and their nominal area of contact amounted to A=50cm2. Hardness of the samples was within the range 15-50 HRC. The surfaces of the contact joint were machined in the manner, which allows one to achieve a markedly different roughness. Rough- ness of the more precisely machined surface was characterised by a value of the indicator Rq = 0.86µm. The roughness indicator of the second surface was Rq = 2.37µm. The contact joints were placed in such a way that the mutual location of the traces of machining of the examined surfaces was per- pendicular to each other. Before the actual experiment, the dry contact joints had been stabilised by loading them with 2000 cycles with the average value of the loading σ0 =2.5MPa and the amplitude σm =2.25MPa. Fig. 1. Block diagram of the system for recording and processing signals The experiment was carried out for the dry joint, and subsequently for the joint coated with NUTO 46 hydraulic oil. After careful washing of the joint with birol it was coated with the assembly paste OKS200 containing MoS2. All measurements were carried out in temperature 22± 1 ◦C. Before each measurement, the contact joint had been stabilised by loading it with Influence of the lubricating agent... 109 1000 cycles with the average value of the loading σ0 =2.5 and the amplitude σ0 =2.25MPa,which resulted in squeezing out of the excess of the lubricating agent. 2.2. Formulation of the mathematical model The formulation and identification of the mathematical model was per- formed for two independent variables σ0 and σm establishing the values of the preliminary load for the variable σ0 at 5 levels with the values: 0.5MPa, 1MPa, 1.5MPa, 2MPa and 2.5MPa, respectively. Next, each level of the va- riable σ0 was divided into 6 parts, which resulted in determining 5 values of the input signal amplitudes simultaneously determining the values of the ran- ge for the variable σm in such away that σm1 =∆σm ≈ 0.165σ0 achieved the value σm1 ≈ 0.83σ0 at the fifth level. As a result of those divisions a 25-point even experimental plan was obtained. A variable component of the input si- gnal of the frequency of 1Hz in the form similar to the harmonic signal was applied. The structure of the mathematical model coupling the relative displace- ment of the contact joint with its load in the form of the system of nonlinear equations was selected σ(t)=σ0+σm sin(ωt) (2.1) x(σ0,σm, t)= a1(σ0,σm)sin[ωt+ϕ(σ0)]+a2(σ0,σm){cos2[ωt+ϕ(σ0)]−1}− −a3(σ0,σm)sin3[ωt+ϕ(σ0)]−a4(σ0,σm){cos4[ωt+ϕ(σ0)]−1} where σ0 >σm. As a result of the performed analyses, the best fitting model from among simple mathematical ones was the model in the form a1(σm,σ0)=α11+α12 σm σl0 +α13 (σm σ0 )2 (2.2) For the remaining components, the best fitted model proved to be the multi- plicative one in the form ak(σm,σ0)=αk (σm σ0 )k π 2 (2.3) for 2¬ k¬ 4. The model of the component representing damping for one independent variable σ0 was established in the form ϕ(σ0)=ϕ0+γ logσ0 (2.4) 110 J.Skrodzewicz 2.3. Experimental results As a result of the estimation for the grinding, dry and lubricated surfaces, the following values of the components of the model given by equation (2.1) were obtained: — dry surface a1(σm,σ0)= 0.021+0.205 σm σ0.250 +0.25 (σm σ0 )2 a2(σm,σ0)= 0.135 (σm σ0 )π a3(σm,σ0)= 0.06 (σm σ0 ) 3π 2 a4(σm,σ0)= 0.018 (σm σ0 )2π ϕ(σ0)=−0.004+0.011logσ0 rad — surface coated with assembly paste OKS 200 a1(σm,σ0)= 0.0134+0.35 σm σ0.3750 +0.255 (σm σ0 )2 a2(σm,σ0)= 0.14 (σm σ0 )π a3(σm,σ0)= 0.05 (σm σ0 ) 3π 2 a4(σm,σ0)= 0.012 (σm σ0 )2π ϕ(σ0)=−0.022+0.047logσ0 rad We did not manage, however, to obtain an accurate mathematical model with the structure of functions (2.2) and (2.3) for the surface of the contact joint, which was coated with the hydraulic oil. The reason for this failure was a non-monotonic character of the functions coupling σm with σ0, which was distinctly demonstrated by the numerical values of the coefficients a1(σm) of themodels of the contact joints coatedwith theoil.A trial of application of the formulated two-variable model would lead to further considerable compilation of the forms of equations (2.2) and (2.3). Thus, the mathematical models for one independent variable σm were estimated with a sufficient accuracy assuming σ0 as a parameter. For: Influence of the lubricating agent... 111 —the surface coated with hydraulic oil NUTO 46 for σ0 =0.5MPa a1(σm)=−0.05+1.42σm−1.17σ 2 m a2(σm)= 1.0σ π m a3(σm)= 0.68σ 3π 2 m a4(σm)= 0.9σ 2π m —the surface coated with hydraulic oil NUTO 46 for σ0 =1MPa a1(σm)= 0.08−0.04σm+0.81σ 2 m a2(σm)= 0.145σ π m a3(σm)= 0.022σ 3π 2 m a4(σm)= 0.0005σ 2π m —the surface coated with hydraulic oil NUTO 46 for σ0 =1.5MPa a1(σm)= 0.025+0.36σm+0.16σ 2 m a2(σm)= 0.043σ π m a3(σm)= 0.0045σ 3π 2 m a4(σm)= 0.0004σ 2π m —the surface coated with hydraulic oil NUTO 46 for σ0 =2MPa a1(σm)= 0.02+0.3σm+0.125σ 2 m a2(σm)= 0.018σ π m a3(σm)= 0.0012σ 3π 2 m a4(σm)= 0.00005σ 2π m —the surface coated with hydraulic oil NUTO 46 for σ0 =2.5MPa a1(σm)= 0.04+0.265σm+0.09σ 2 m a2(σm)= 0.01σ π m a3(σm)= 0.0006σ 3π 2 m a4(σm)= 0.00001σ 2π m and ϕ(σ0)=−0.012+0.034logσ0 rad for 0.5¬σ0 ¬ 2.5MPa Graphical images of the nonlinear mathematical models illustrating the deformability of the contact joints and the phase planes of themotion in these joints are presented in Fig.2. A distinct difference can be seen between the characteristic of the contact joint coatedwith theoil and the remainingcontact joints. For this reason, the experiment with the contact joint coated with the oil was repeated, and the same result was obtained. The contact joint coated with the oil caused a pronounced weakening of the dynamics of motion as a function of σ0, and demonstrated definitely the greatest nonlinearity. 112 J.Skrodzewicz Fig. 2. Deformability characteristics and phase plane of motion in the tested contact joint for σ m =0.425MPa and 0.5<σ0 < 2.5MPa Influence of the lubricating agent... 113 Fig. 3. Deformability characteristics and phase plane of motion in the tested contact joint coated with oil 114 J.Skrodzewicz 2.4. Determination of the energy dissipation coefficient The estimation of the energy losses in the nonlinear contact joint requires the elaboration of the method of its calculation. The main assumption in this process is its similarity to the linear models (Skrodzewicz, 1999). In the case of nonlinear contact joints, the most appropriate measure of the energy dissipation seems to be a dimensionless energy dissipation coefficient Ψ, which expresses the ratio of the dissipated and potential energy. The definition is proposed in the integral form. If the experiment applies a periodic excitation signal F(t)= n ∑ k=0 Fk sin(kωt+φk) (2.5) then the output signal, in which the nonlinear object is examined, can be obtained also in a periodic form x(t)= n ∑ k=0 xk sin(kωt+ δk) (2.6) The energies Eh and Ep are represented by the appropriate areas of the plot of the relative displacements of the surfaces which constitute the contact in the function of the load, see Fig.4. Applying definite integrals, one can calculate the area of the curved tra- pezium in the parametric form as follows S= t2 ∫ t1 y(t) dt (2.7) where y(t)=F(t) dx(t) dt Applying this formula, the definition of the Ψ coefficient (for nonlinear case) canbe transformed into four identical integral equations describedby equation (2.8). Separating the integration limits, the following is obtained s1 = T 4 ∫ 0 y(t) dt s2 = T 4 ∫ T 2 y(t) dt s3 = 3T 4 ∫ T 2 y(t) dt s4 = 3T 4 ∫ T y(t) dt (2.8) for T =2π/ω. Influence of the lubricating agent... 115 Fig. 4. Graphical interpretation of the integral definition of the energy dissipation coefficient Assuming notation Epu = 1 2 (s1+s2) Epl = 1 2 (s3+s4) Ehu = s1−s2 Ehl = s3−s4 (2.9) the potential energy for the nonlinear system can be presented in the form Ep = 1 2 (Epu+Epl) (2.10) when the energy of losses in this system is Eh =Ehu+Ehl (2.11) Hence, the energydissipation coefficient Ψ canbedescribedby the relationship Ψ = Eh Ep (2.12) It should be noticed that the presented definition of the energy dissipation coefficient in the linear case is reduced to the relationship consistent with the Kelvin-Voigt linear model. Making use of relationship (2.12), the dependence Ψ = f(σ0) was estimated as: 116 J.Skrodzewicz — for the contact joint coated with the assembly paste Ψ(σ0)= 0.137−0.293logσ0 — for the contact joint coated with the hydraulic oil Ψ(σ0)= 0.075−0.214logσ0 — for the dry contact joint Ψ(σ0)= 0.025−0.062logσ0 Based on the experimental results, it can be practically assumed that the values of the energy dissipation coefficients Ψ for the presented nonlinear models are very close to the values which could be calculated on the basis of relationship (2.13) obtained from the Kelvin-Voigt linear model (for small values of the angle ϕ) Ψ(σ0)= 2π tan[ϕ(σ0)]≈ 2πϕ(σ0) (2.13) The characteristics illustrating the dependence of the energy dissipation coef- ficient Ψ on σ0 are shown in Fig.5. Fig. 5. Characteristics of the energy dissipation coefficient Ψ of the tested contact joints 3. Summary The phenomena occurring in the contact joint take place at the boundary of micro and nanometers, and they are characterised by strong nonlinearity. Influence of the lubricating agent... 117 The contact joint coated with the oil was found to be the most difficult for a mathematical description among the examined joints. This joint causes a pronounced drop in the dynamics of motion as a function of σ0, and decisive- ly demonstrates the greatest nonlinearity. The complexity of the phenomena occurring is such a joint illustrates Fig.3. The changes of deformability for σ0 =0.5MPa reassemble a slightly unfolded fanwith the deformability decre- asing in the functionof σm.On the contrary, for σ0 =1MPa thedeformability as a function of σm increasedmore strongly.This fact explains the opposite si- gns of the components a1(σm) for σ0 =0.5MPa and a1(σm) for σ0 =1MPa. The changes of deformability as a function of σm were significantly smaller for the value of the parameter σ0 =1.5MPa. These phenomenamay by utilized for the control, in a limited range, of the deformability of the contact joint as a function of the static load σ0 as well as the amplitude of the dynamic load σm. 4. Conclusion Finding an answer to the question: what is the influence of relation of the type of surface roughnesswith the physical properties of the lubricating liquid (viscosity?) on the properties of the contact joints (especially the deformabili- ty) seems to be important.When do the pastes possess the similar properties as lubricating liquids on other types of roughness in the contact area? Does any of the roughness indicators determines the best the type of such a sur- face? Or, maybe, one should look for another, better indicator? The answer to these questions would allow the designers of the technological equipment to control (in a limited range) the deformability of the contact joints occur- ring inmany constructions in the natural way, andwould enable optimization of these constructions with regard to e.g. their vibrostability. The thus per- formed optimization would not require significant changes to the equipment structure, but only an appropriate selection of the constant surface, the type of its roughness and the lubricating agent. For this reason, the recognition and description of the phenomena-taking place in the lubricated joints seems to be extremely interesting. References 1. AndrewC., Cockburn J.A.,WaringA.E., 1967,Metal surfaces in contact undernormal forces: somedynamic stiffness damping characteristics,Proc. Inst. Mech. Engrs., London, 182, Part 3K, 92-100 118 J.Skrodzewicz 2. ChmielewskiK., Skrodzewicz J., 1999,Planowanieeksperymentówdlanie- liniowego modelu opisującego połączenia stykowe słabo obciążone, VIII Kon- ferencja Naukowo-Techniczna, Metrologia w Technikach Wytwarzania Maszyn, Szczecin, 23-29, ISBN 83-87423-28-9 3. Petuelli G., 1983, Theoretische und experimentelle Bestimmung der Ste- ifigkeits und Dampfungseigenschaften normalbellasteter Fugestellen. Diss. THAachen 4. Skrodzewicz J., 1999, Estimation of nonlinear mathematical model of the contact joint based on experimental data, Fourth International Conference on Computational Methods in Contact Mechanics, Stuttgart, 525-534, ISBN: 1-85312-694-2 5. Skrodzewicz J., Gutowski P., 2000, Nonlinear mathematical models of weakly loaded contact joints, Journal of Theoretical and Applied Mechanics, 38, 4, 781-785 Wpływ czynnika smarującego na właściwości połączeń stykowych Streszczenie W pracy przedstawiono nieliniowe modele matematyczne opisujące właściwości połączeń stykowychsuchych i smarowanychobciążonychwkierunkunormalnymwza- kresie 0.5-2.5MPa.Powierzchnie połączeń pokryto pastąmontażową i olejemhydrau- licznym.Podanozależnościwspółczynnika rozproszeniaenergii Ψ wfunkcji obciążenia dla zastosowanych czynników smarujących. Strukturęmodeli określono na podstawie eksperymentalnie wyznaczonych charakterystykwidmowych sygnałów przemieszczeń względnych nominalnych powierzchni styku i obciążenia styku w kierunku normal- nym. Postaci czynników funkcyjnychmodelu wyestymowano stosującmetody analizy regresji liniowej. Manuscript received May 7, 2002; accepted for print October 30, 2002