RATE- AND STATE-DEPENDENT FRICTION MODEL FOR ELASTOMERIC SEALS

RATE- AND STATE-DEPENDENT FRICTION MODEL FOR ELASTOMERIC SEALS Oliver Heipl*, Hubertus Murrenhoff* * Institute for Fluid Power Drives and Controls (IFAS) RWTH Aachen University Steinbachstraße 53, 5272 Aachen, Germany (E-mail: oliver.heipl@ifas.rwth-aachen.de) ABSTRACT Seals are crucial for the functionality of pneumatic linear actuators. They separate chambers with different pressure levels. Little air leakage which leads to reduced volumetric efficiency is often accepted. More important is the role of friction which decreases the efficiency caused by tribological contacts between seals and counter surfaces. Furthermore, friction leads to reduced precision. Static friction models are state of the art within the designing of pneumatic systems including cylinder devices. These friction models are based on a description of the Stribeck curve. But they are not able to predict dynamic friction instabilities like stick-slip, increasing break-away force due to dwell-time and hysteretic friction phenomena. The paper shows a simulation approach based on a powerful rate- and state-dependent friction model. Loads on the tribological contact between seal and rod are realised by a structural mechanics method. The numerical results are compared with experimental investigations. KEY WORDS Friction, Seal, Pneumatics, Elastomer, Thermo-Viscoelasticity NOMENCLATURE A : Area B : Left Cauchy Green strain tensor b : Contact width C, C 2 : WLF parameter C, C : Mooney-Rivlin parameter e, : Relaxation parameter F : Force I : Invariant m : Mass p : Pressure T : Temperature t : Time v : Velocity W : Strain energy density x : Stroke,, γ : Carlson-Batista parameters : Frictional state variable INTRODUCTION Due to the simple and robust design pneumatic linear devices are often used for automation purposes to generate reciprocating motion. To ensure the functionality seals are applied, see Figure. They are

located at the piston to separate the cylinder chambers, in the area of the end cushioning and at the rod to enable a pressure difference between chamber and ambience. At the rod additional wipers are used to avoid a contamination of the pneumatic system by dust. They can be carried out as single element or combined with the seal. F F, FC sgn( v ) F Fe F sgn( F ) S e S F C e v / v S S sgn( v ) F v ; v and F v ; v e F S ; otherwise (2) x i Piston Seals p m P Piston Guide A P x, x, x Damper Seals p 2 m R x o Rod Seal Rod Guide Wiper Figure : Schematic of a pneumatic linear device p A F L A R A little air leckage which results in a decrease of the volumetric efficiency is often accepted. In contrast, friction is one of the main problems of pneumatic cylinder devices. In consequence of the contact between the seals, wipers and guidance elements with the counter surfaces, friction is induced when relative motion occurs. The dissipative character of friction (kinetic energy is transferred to thermal energy) causes a decline of the pneumatic-mechanical efficiency as well as an interaction with the dynamics of the piston. An established tool in the design and development stage of pneumatic systems is the one-dimensional system simulation of the fluidic circuit. Hereby the dynamic properties of a pneumatic linear device can be investigated in an early stage of the development process. With ease, interaction between fluidic, mechanical and control system can be optimised. The mathematical description of the piston motion is based on a differential equation of second order, see Equation. The pressures in the cylinder chambers can be easily described by the pressure build up as a result of mass flow, volumetric changes as well as heat generation and transfer. However, the implementation of the friction into the simulation is very insufficient. mp mr x p AP p2ap AR pa AR FL FF () State of the art in present system simulation tools are static friction models. Normally, the so called Stribeck model is used which describes the fundamental relationship between friction and velocity. In Equation 2 a possible form to reproduce a Stribeck curve is shown [9]. In addition a linear scaling factor f to describe the influence of the pressure on the friction is generally considered, see Equation 3. F F, FF p f (3) By investigating measured friction of a pneumatic seal a more complex behaviour including hysteretic effects, frictional memory and non reversible friction characteristic is detectable, see Figure 2 and 3. Static linking between friction and velocity is insufficient for a realistic description [2,3,5,7,8]. Rather, a change in the friction characteristic is detectable due to the change off the operating conditions. Among others a significant influence of the temperature appears. Friction Force [N] Friction Force [N] 5 4 3 2 6 bar 4 bar 2 bar bar - -2-3 -4-5 -2 -.5 - -.5.5.5 2 Velocity [m/s] Figure 2: Pressure-dependent friction force 5 4 3 2 - -2-3 -4 85 C 55 C 25 C -5-2 -.5 - -.5.5.5 2 Velocity [m/s] Figure 3: Temperature-dependent friction force

OBJECT OF RESEARCH Static friction models which are implemented in present one-dimensional system simulation tools do not allow considering the dynamic friction phenomena as well as the influence of the design of the sealing system. Furthermore the influence of the temperature on friction is neglected. Enhancing the description of friction within the system simulation by a novel approach based on a coupled simulation between system simulation, dynamic friction model and structural mechanics, a more accurate modelling including friction instabilities and time dependent effects is possible. Integration of structural mechanics allows considering the influence of the seal shape, the dimension of the groove as well as the seal material on friction. MODELLING APPROACH The modelling approach is based on a system simulation model, a friction model and a structural mechanics model. The interaction between the three partial models is figured out in Figure 4. Pre-Simulation System Simulation Temperature Material Time Lubricant Structural Mechanics Model Load FN Friction Force Friction Model FN x FR System Simulation Model Shape Dynamics Figure 4: Modelling approach Surface Pressure The system simulation model describes the fluidic, mechanical and control properties of the investigated system. For this purpose the commercial system simulation tool DSHplus is used. An exemplary system simulation model is shown in Figure 5. A pneumatic cylinder without any internal friction models is controlled by a valve. The programmable logic controller (PLC) triggers the valve with a typical driving cycle. A signal input is used as well as a signal output to couple the system simulation model with the friction model. Separating system simulation and friction model allows using different system simulation tools. To determine the friction force, values (e.g. temperature, pressure and velocity) from the system simulation are transferred to the friction model via the signal output. Loading of the pneumatic cylinder by a friction force occurs via the signal input. Figure 5: System simulation model The friction model uses the values from the system simulation tool and calculates the friction forces based on empirically and physically motivated equations. This partial model characterises the friction in a microscopic way which means that the description is limited to the contact area. The implemented rate- and state-dependent friction model is based upon the model by Carlson and Batista [,4,] which describes the fundamental physics in the easiest possible way. The basic principle of this model is a frictional state variable to model the transition from dry friction to a full hydrodynamic friction. The change of the frictional state variable in a range from (dry friction) to (hydrodynamic friction) satisfies Equation 4. The state term on the left side implies melting of the film defined by a time constant. The rate term on the right side describes the change of the film due to shear stresses. The factor α correlates to a characteristic length in which the changes take place. In addition a load on the contact is taken into account by the factor F N. The load is dependent on the installation situation, the shape and the material of the seal as well as the operating conditions like pressure and temperature. FN x (4) FN A progression of the frictional state variable for a simple periodic motion is shown in Figure 6. With increasing velocity a transition between dry friction and hydrodynamic friction occurs (lower loop) while with decreasing velocity the value of the frictional state variable increases again and the portion of boundary friction grows. The transition between the different states effects a hysteretic behaviour. By increasing the load F N a rather weakly developed transition between dry and hydrodynamic friction occurs (upper loop).

Dry friction Stroke [m] State Variable [-]. -..8.6.4 [,] which bases on a strain energy function, see Equation 6. B is the left Cauchy Green strain tensor and tr(b) represents the first principal invariant I of that tensor, while tr(b - ) stands for the second principal strain invariant I 2. The parameters C and C have to be determined by curve fitting for each material. C trb C trb 3 kdet( B ) 2 3 W (6) Fluid friction.2-2 -.5 - -.5.5.5 2 Velocity [m/s] Figure 6: Progression of the frictional state variable The friction force is calculated based on the frictional state variable, see Equation 5. A distinction between static F a and dynamic frictional components F b takes place. In addition, the third term of the equation provides the viscous friction which is assumed as linearly dependent to the piston velocity. The parameter describes the fluid properties, primarily the (equivalent-)viscosity of grease. The factor F N and the contact width b are interfaces to the structural mechanics again. F F Θ F F b γ x FF a b b N (5) The structural mechanics model is the third partial model and provides the loading on the tribological contact in a macroscopic way using a Finite Element Method (FEM). First geometrically simple O-ring seals are applied to simplify the modelling. Due to this a fully parameterisable axial symmetric seal model was build up. The FE-model is able to consider the fundamental physics of elastomeric seals. In contrast to steel elastomeric materials does not possess a linear stress-strain relation even at low loading. Measured quasi-static stress-strain relations are shown in Figure 7 for two typical sealing materials for pneumatic applications. Stress [MPa] 6 5 4 3 2 uniaxial tensile test Mooney-Rivlin NBR 8 Shore A NBR 72 Shore A The third term of the strain energy function represents a thermal expansion which is equivalent to a change in the third invariant I 3, see Equation 7. T I3 det( F ) 3 T T (7) If temperature is varied and especially when temperature is low relative to the glass transition temperature the thermo-viscoelasticity of the sealing material has to be taken into account [6,2]. In this case the history-dependent behaviour requires a strain energy function which considers not only the actual but also the past values of loading, see Equation 8 and 9. Ŵ W log at,tg N tˆ t Î; Î2 ei { exp( )} (8) i iat, TG C T TG (9) C T T 2 G Using this constitutive framework for elastomeric seals FE analysis can be performed. As an exemplary result a cooling process is shown in Figure 8. The temperature of an O-ring seal which is assembled in a groove is going down from room temperature to -9 C. The cooling rate is small enough that the sealing forces are in a thermodynamic equilibrium until the glass transition temperature T G. Below the glass transition temperature the thermal shrinkage is less adapted which leads to a much stronger sealing force reduction. At about -8 C the contact between seal and groove disappear and a gap results. Afterwards the forces at the rod increases due to thermal shrinkage. A more detailed discussion of the thermo-viscoelastic properties of sealing materials can be found in [6]...2.3.4.5.6.7 Strain [-] Figure 7: Stress-strain relation for elastomers The hyperelastic material behaviour can be modelled by using a widely spread approach by Mooney and Rivlin

max. Contact Pressure [kpa] 3 2 - -2-3 2 C Groove K T 6 s Rod T G p max, groove p max, rod -8 C 2 3 4 5 6 7 Time [s] Figure 8: Cooling process The three introduced partial models (system simulation model, friction model and structural mechanics model) are integrated in a coupled simulation using MATLAB/Simulink, see Figure 9. Due to the reason that a finite element simulation would dominate the calculation time compared to the system simulation the structural mechanics is solved by using characteristic diagrams. These characteristic diagrams are generated in a pre-simulation. Chamber Pressure Stroke.75.5.25 Low Load High Load.25.5.75 Time Figure : Simulation at different loads VALIDATION.75.5.25 Experimental investigations are necessary to parameterise and validate the friction model. For this purpose a test rig for pneumatic piston seals was used. The measurement principle is shown in Figure. The piston including the test seal is at rest and the tube is driven by a crank mechanism. To pressurise the test seal a second piston is applied. Measuring at different temperature levels is possible by heating the complete test assembly. Test Seal Tube Velocity System Simulation p Force Sensor Drive Figure : Measurement principle Friction Force Structural Mechanics Frictional State Variable Figure 9: Coupling of the partial models A result of a coupled simulation is shown in Figure. A differential cylinder where the friction is only located at the rod seal is controlled with a rectangular signal. The motion characteristic changes as well as the sealed pressure dependent on the loading with different temperature levels. At low temperature the piston reacts a little bit faster and more dynamic due to lower contact forces and in succession lower friction as shown before compared to a higher temperature. Comparisons between simulated and measured friction forces are shown in Figure 2 for room temperature and Figure 3 for a temperature of 55 C. The simulated friction forces are much smoother then the measured one. Reasons are small form deviations of the tube and vibrations of the test assembly caused by the crank mechanism. Friction Force [N] 6 4 2-2 Simulated Measured -4-6 -5-25 25 5 Stroke[mm] Figure 2: Comparison at room temperature

Friction Force [N] 6 4 2-2 -4 Simulated Measured -6-5 -25 25 5 Stroke[mm] Figure 3: Comparison at 55 C CONCLUSION The motion characteristics of pneumatic cylinder devices are significantly influenced by seal friction. In present one-dimensional system simulation tools static friction models are state of the art. To consider rate- and state-dependent friction a dynamic friction model is presented. Coupling of system simulation with a powerful friction model and a structural mechanics model shows accurate results. Especially influences on the friction process (e.g. temperature and seal shape) can be considered with this novel method. Using measurements with different parameters of the tribosystem like different roughnesses will be necessary to expand the database which is used for parameterising the friction model. ACKNOWLEDGEMENT The IGF-project 6229 N/ of the Forschungsvereinigung Forschungskuratorium Maschinenbau e.v. FKM, Lyoner Straße 8, 6528 Frankfurt am Main is supported via the AiF in the context of the program to support the Joint Research and Development (IGF) by the Federal Ministry of Economics and Technology because of an enactment of the German Federal Parliament. In: Physical Review E 53 (996), No. 4, pp. 453-465 5. Dahl, P.: A solid friction model. In: Technical Report TOR-58, The Aerospace Corporation, El Segundo (968), pp. 37-38 6. Heipl, O. ; Murrenhoff, H. ; Achenbach, M.: Friction Modelling for Pneumatic Actuator Seals Regarding Structural Mechanics. In: Proceedings of the 6th International Sealing Conference (ISC), Stuttgart, 2, pp. 4-52 7. Hess, D. and Soom, A.: Friction at a Lubricated Line Contact Operating at Oscillating Operation Conditions. In: Journal of Tribology 2 (99), No., pp. 47-52 8. Olsson, H. ; Åström, K. ; Canudas de Wit, C. ; Gäfvert, M. and Lischinsky, P.: Friction Models and Friction Compensation. In: European Journal of Control 4 (998), No. 3, pp. 76-95 9. Persson, B. N.: Theory of friction: on the origin of the stick-slip motion of lubricated surfaces. In: Chemical Physics Letters 254 (996), No. -2, pp. 4-2. Rivlin, R. S., Saunders, D. N.: Large Elasticity Deformations of Isotropic Materials VII: Experience on the Deformation of rubber. In: Phil. Trans. Roy. Soc. Lond. Math. Phys. Sci. 243 (95), No. 865, pp. 25-288. Treloar, L. R.: The Physics of Rubber Elasticity. Third Edition. Oxford : Clarandon Press, 975 2. Williams, M. L. ; Landel, R. F. and Ferry, J. D.: The Temperature Dependence of Relaxation Mechanisms in Amorphous Polymers and Other Glass-forming Liquids. In: Journal of the American Chemical Society 77 (955), No. 4, pp. 37-377 REFERENCES. Achenbach, M. and Papatheodorou, T.: Modeling of Friction Phenomena in Pneumatic Cylinders. In: Proceedings of the 6 th International Fluid Power Conference (IFK), Volume 2, 28, pp. 279-288 2. Al-Bender, F. ; Lampaert, V. and Swevers, J.: Modeling of dry sliding friction dynamics - From heuristic models to physically motivated models and back. In: Chaos 4 (24), No. 2, pp. 446-46 3. Canudas de Wit, C. ; Olsson, H. ; Åstrom, K. J. and Lischinsky, P.: A New Model for Control of Systems with Friction. In: IEEE Transations on Automatic Control 4 (995), No. 3, pp. 49-425 4. Carlson, J. M. and Batista, A. A.: Constitutive relation for the friction between lubricated surfaces.