Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 45, 1, pp. 33-40, Warsaw 2007 OCCURRENCE OF STICK-SLIP PHENOMENON Jan Awrejcewicz Paweł Olejnik Department of Automatics and Biomechanics, Technical University of Lodz e-mail: awrejcew@p.lodz.pl; olejnikp@p.lodz.pl A self-excited systemwith unilateral friction contactmodelled by a two degrees-of-freedommechanical system,where the normal force varies du- ring displacementof a blockhas been studied.The constructed real labo- ratory rig approximates the investigated system, and it includes feedback reinforcementof the friction forceacting onthevibratedblock.Someme- thods of data acquisition and data handling procedures are proposed for experimentally observed results and data collection. A novel static fric- tion force model for both positive and negative velocities of the base is proposed. Key words: stick-slip dynamics, friction forcemodel, numerical analysis, experiment 1. Introduction Relative sliding of two solid bodies is a non-equilibriumprocess, where kinetic energy of motion is transferred into energy of irregular microscopic motion. This dissipative process creates the dry friction phenomenon. The phenome- nological laws of dry friction like Coulomb’s laws are well-known, and there is well established theory in applied physics (Bowden and Tabor, 1954) related to this subject. The simplest models describe friction as a function of the velocity diffe- rence of sliding bodies. Such models like Coulomb’s friction ones are called static models. In fact, Coulomb’s dry friction laws simplify very complex be- havior which involves mechanical, plastic, and chemical processes (Singer and Pollock, 1992). However, experimentally observed differences in application of Coulomb’s law are often found (Awrejcewicz and Delfs, 1990a,b; Galvanet- to et al., 1995). Computer simulations of motion of mechanical systems with friction are difficult because of the strongly nonlinear behavior of the friction 34 J. Awrejcewicz, P. Olejnik force near zero velocity and the lack of a universally considered frictionmodel. For rigid bodies with dry friction, the classical Coulomb law of friction is usu- ally applied to engineering contact problems exactly because of its simplicity. It can explain several phenomena associated with friction and it is common- ly used for friction compensation (Friedland and Park, 1991). A well known velocity-limited frictionmodel given byOden andMartins (1985) uses a smo- oth quadratic function when the sliding velocity is nearly zero. The value of the limiting velocity is very important, but there is no standard method for its estimation. The problem of modeling of friction forces is not solved, because physi- cal and dynamical effects are not sufficiently understood. There are twomain theoretical approaches to model dry friction interfaces: the macro-slip and micro-slip approaches (Feeny et al., 1998). In the micro-slip approach, a rela- tively detailed analysis of the friction interfaces should bemade. In this case, investigations can provide accurate results onlywhen the preload between the interfaces is very high. In the macro-slip approach, the entire surface is assu- med to be either sliding or sticking. The force necessary to keep sliding at a constant velocity depends on the sliding velocity of the contact surfaces.With this respect, friction laws of smooth and non-smooth velocity functions have been cited in Awrejcewicz and Olejnik (2003a,b), Popp et al. (1996). There is a lack of works which take into account problems of experimen- tally observed velocity depending friction force models. The paper by Brandl and Pfeiffer (1999) deals with the measurement of dry friction. A tribometer was developed to identify both sticking and sliding friction coefficients. The so called Stribeck-curve has been determined for any material in the contact zone. Similarly, a multi degrees-of-freedom model of friction was investigated byBogacz andRyczek (1997), where an experimentally observed friction cha- racteristic expresses the kinetic friction force as a function of relative velocity of motion. 2. Laboratory rig A laboratory rig (Olejnik, 2002) designed for observations and experimental research of friction effects including the friction force measurement (Awrej- cewicz and Olejnik, 2003a) has been constructed and investigated as well. A general view togetherwith indication of component parts and some connectors like coil springs is schematically shown in Fig.1. The self-excited system pre- sented in Fig.1 is equivalent to the real experimental rig (see Fig.2) in which the block mass m is moving on the belt in the direction x1, and where the angle body represented by the mass moment of inertia J is rotating around Occurrence of stick-slip phenomenon 35 Fig. 1. The analysed 2-DOF system Fig. 2. The laboratory rig point swith respect to theangle direction.Theanalysed systemconsists of the following parts: two bodies are coupled by linear springs k2 and k3; the block on the belt is additionally coupled to the fixed base using a linear spring k1; the angle body is excited only by spring forces; there are no extra mechanical actuators; rotational motion of the angle body is damped using virtual actu- ators characterizing air resistance (marked by constants c1 and c2); damping of the block is neglected; it is assumed that the angle of rotation of the angle body is small and within interval (+5,−5) degrees (in this case, the rotation is equivalent to the linear displacement y1 of the angle body legs); the belt is moving with a constant velocity vb and there is no deformation of the belt in the stick-slip contact zone. 36 J. Awrejcewicz, P. Olejnik Non-dimensional equations governing thedynamics of our investigated sys- tem have the following form ẋ1 =x2 ẋ2 =−x1−α −1 1 [η1(x2+y2)−y1−T(vw)] ẏ1 = y2 ẏ2 =α −1 2 (−β3y1−η12y2−x1−η1x2) (2.1) where: x=(x1,x2), y=(y1,y2) are state-space variables of the block and the angle body, respectively; ω is the periodicity of themass m; the friction force T(vw) is described by Eq. (4.3); vw =x2−vb is the relative velocity between the bodies [α1,α2,β1,β2,β3,η1,η2,η12] = [ω 2m,ω2Jr−2,k1 + k2,µ0k3,k2 + k3,c1ω,c2ωµ0,ω(c1+ c2)]k −1 2 are remaining parameters. A non-stretchable 25mm thick belt (made of hard rubber) is placed on solid uniform shafts and then supported by means of a flat bar at the place, where the examined block vibrates. The propeller shaft is installed in a flo- atingmanner to avoid belt’s tension, and the rig is essentially equippedwith a direct current commutator motor (PZTK 60-46 J suitable to use in cross-feed drives of numerically controlled machines) supplied by voltage 30V and ulti- mate current load equal to 4.1A. Stabilisation and control ofmotor rotational velocity is additionally accomplished by means of the RN12 regulator (this device works in the system also as an amplifier). The system variables like displacement of the block and rotation of the equal-leg angle are quantified by using of non-sticking measurement method, where the laser proximity switch is assigned to the displacementmeasurement of the block; Hall-effect device (principle of operation is based on changes of magnetic field) is used to determine rotation of the equal-leg angle, respecti- vely. It should be emphasized that both the quoted devices have linear cha- racteristics of the measured quantity versus voltage output. Additionally, all construction parts of the described laboratory rig are fixed to a stable frame. 3. Measurement Analog signals incoming from themeasuringdevices are processed bydynamic acquisition using test instrumentsmadebyNational Instruments and coopera- ted through a PCI card (PCI-6035E with chasis SCXI-1000, -1321 and -1302) with the LabView professional software. The commutator motor is equipped with a rotary-impulse converter whose output is transformed to linear veloci- ty of the belt. Acquisition and data handling is in the LabView environment Occurrence of stick-slip phenomenon 37 made thanks to composition of a special block-wire-block diagram.The stored data are indicated on panel-situated scaleable charts. Disturbances of the whole structure, noise in electrical circuits, and other additional effects have significant influenceonaccuracyof anymeasured signal. Therefore, some signals are filtered digitally (e.g. elliptic topology), and then real differentiation preventing formation of high peaks is applied as follows yn = ayn−1+k0xn+k1xn−1 (3.1) where yn is a value of thederivative at thepoint n;yn+1 denotes thepreceding value (the algorithm starts from n = 1 when y0 = 0); xn, xn+1 – values of the differentiate function of displacement or velocity, when the computational process is associated with acceleration; a, k0, k1 are integration constants. 4. Investigations Results of measurements are obtained following the methodology described in Section 3. Appropriately transformed equations of motion can be used for calculation of the friction force after a real time measurement of state varia- bles of the investigated system. Characteristics of the friction force T in the nondimensional system versus relative velocity vw between moving the belt andblock for positive andnegative velocities of the belt vb are shown inFig.3. In the case of T+ branch, the equation of friction force dependence describing Fig. 3. Experimental characteristics of the friction force the friction force model for positive relative velocity (see Fig.3a) have the following form T+ =Ts−|vw| Ts−Tmin vw,max (4.1) 38 J. Awrejcewicz, P. Olejnik where Ts is a static friction force, vw,max is the maximum positive relative velocity. The T − branch can be described by a second order exponentially decay function describing the friction forcemodel for negative relative velocity (see Fig.3b) of the following form T − =Ts+A1exp ( − |vw|−vw,min t1 ) +A2exp ( − |vw|−vw,min t2 ) (4.2) where vw,min is the maximum negative relative velocity; A1, A2, t1, t2 are constant values. Therefore, the main multivalued function describing friction force changes occurring in our investigated 2-DOF system with the variable normal force is T(vw)=        T+ sgnvw if vw > 0 T − sgnvw if vw < 0 |Ts| if vw =0 (4.3) With respect to the validation of the estimated static frictionmodel, Eq. (4.3), a special numerical integration algorithm has been used after comparisons of experimental results with their numerical counterparts. The based onHénon’s method numerical procedure describing the stick-slip phenomenon in the con- tact zone, which is extremely useful to locate the stick to slip and slip to stick transitions in non-smooth systems, has been applied (Awrejcewicz and Olej- nik, 2002a,b, 2003a,b). Nondimensional parameters of T(vw) characteristics obtained by both measurement and identification are presented in Table 1. Tabela 1. Parameters of the nondimensional friction model Ts Tmin vw,max A1 A2 t1 t2 T+ 3.63 0.86 0.27 – – – – T − −5.94 −1.42 −0.28 3.2345 2.8736 0.0342 0.3053 Numerical analysis with the implementation of the derived friction force dependency has yielded the results presented in Fig.4. 5. Conclusions The experimental trajectory shown in Fig.4 is rather rotated and has an ir- regular sticking phase. Such irregularity describesmicro-stick and -slip condi- tions usually prevailing in the real contact zone of cooperated surfaces. The numerical trajectory is satisfactorily close to its experimental counterpart re- corded for the investigated dynamical system. The sticking velocity is almost the same, but only some distinguishable differences are observed in the sliding Occurrence of stick-slip phenomenon 39 Fig. 4. Similarity of the experimental and numerical phase trajectories of the block for T − friction characteristic phase. Additionally, the carried out comparison has proved that our analysed system is non-symmetric. Investigations on the real laboratory rig have been supportedby numerical analysis allowing one tomodel and then todescribe the feedback reinforcement of the friction force (model of T − branch). To sum up, the T − friction force model is suggested to be applied in systems, where the normal force acting between cooperated surfaces is fluctuated. Acknowledgement This work has been supported by the PolishMinistry of Science andHigher Edu- cation for years 2005-2008 (grant No. 4T07A03128) References 1. Awrejcewicz J., Delfs J., 1990a,Dynamics of a self-excited stick-slip oscil- lator with two degrees of freedom, Part I: Investigation of equilibria,European J. Mech. A/Solids, 9, 4, 269-282 2. Awrejcewicz J., Delfs J., 1990b,Dynamics of a self-excited stick-slip oscil- lator with two degrees of freedom, Part II: Slip-stick, slip-slip, stick-slip trans- itions, periodic and chaotic orbits,European J. Mech. A/Solids, 9, 5, 397-418 3. Awrejcewicz J., Olejnik P., 2002a, Calculating Lyapunov exponents from an interpolated time series,Proc. XX Symposium –Vibrations in Physical Sys- tems, Błażejewko, 94-95 4. Awrejcewicz J., Olejnik P., 2002b, Numerical analysis of self-excited by friction chaotic oscillations in two-degrees-of-freedom system using Hénonme- thod,Machine Dynamics Problems, 26, 4, 9-20 40 J. Awrejcewicz, P. Olejnik 5. Awrejcewicz J., Olejnik P., 2003a, Numerical and experimental investiga- tions of simple non-linear system modelling a girling duo-servo brake mecha- nism,Proc. ASME Design Eng. Tech. Conf., DETC2003/VIB-48479 6. Awrejcewicz J., Olejnik P., 2003b, Stick-slip dynamics of a two-degree-of freedom system, Int. J. Bif. and Chaos, 13, 4, 843-861 7. BogaczR., RyczekB., 1997,Dry friction self-excitedvibrations analysis and experiment,Eng. Trans., 45, 3/4, 487-504 8. Bowden F.P., Tabor D., 1954,Friction and Lubrication, Oxford University Press 9. Brandl M., Pfeiffer F., 1999, Tribometer for dry friction measurement, Proc. ASME Design Eng. Tech. Conf., DETC99/VIB-8353 10. Feeny B., Guran A., Hinrichs N., Popp K., 1998, A historical review on dry friction and stick-slip phenomena,App. Mech. Rev., 51, 321-344 11. Friedland B., Park Y.-J., 1991, On adaptive friction compensation, Proc. 30st IEEE Conf. Decision and Control, 2899-2902 12. GalvanettoU., Bishop S.R., Briseghella L., 1995,Mechanical stick-slip vibrations, Int. J. Bif. and Chaos, 5, 3, 637-657 13. 13 Oden J.T., Martins J.A.C., 1985,Models and computationmethods for dynamic friction phenomena,Comp. Methods App. Mech. Eng., 52, 527-634 14. Olejnik P., 2002,Numerical and Experimental Analysis of Regular and Cha- otic Self-Excited Vibrations in a TwoDegrees-of-FreedomSystemwith Friction, Ph.D. Thesis, Technical University of Lodz 15. PoppK., HinrichsN.,OestreichM., 1996,Dynamics with Friction,World Scientific, London 16. Singer I.L., Pollock H.M., 1992, Fundamentals of Friction: Macroscopic and Microscopic Processes, Kluwer Academic Publishers, Dordrecht O zjawisku typu stick-slip Streszczenie Analizie poddany jest układ samowzbudny z jednostronnym kontaktem ciernym, który zamodelowano jako układ mechaniczny o dwóch stopniach z tarciem oraz ze zmienną siłą nacisku wywieraną na podstawę podczas przemieszczenia się badane- go ciała sztywnego. Skonstruowane stanowisko doświadczalne przybliża w pewnym stopniu rozważany układ i zawiera sprzężenie ruchu drgającego ciała z siłą normalną pochodzącą od nacisku. Zgodnie z obserwacjami eksperymentalnymi i dla zmierzo- nych sygnałów podano metody akwizycji danych oraz procedury ich obsługi. Zapro- ponowano nowy, statycznymodel siły tarcia dla odpowiednich dodatnich i ujemnych prędkości przemieszczania się podstawy. Manuscript received August 3, 2006; accepted for print August 18, 2006