To calculate the hysteresis friction many different approaches were made. In [4] the hysteresis friction between a rubber seal and piston surface is simulated by a two dimensional Finite-Element-Method. Input parameters are the nominal contact pressure of 1 MPa and a previous measured roughness of the piston surface, which is separated into multiple parts with a different wavelength λ. This separation is shown in figure 1b) and based on the method described in [3], in which the rubber friction on rough surfaces is calculated. But for this calculation the material behaviour and the roughness of the surface are needed. It is proposed to obtain the information through experiments, which take additional time. In [5] the calculation of rubber friction is based on a stochastic surface according to G REENWOOD and W ILLIAMSON which leads to the calculation of a root mean square of the height gradient of the surface. In combination with the material behaviour the calculation of a friction coefficient is possible. As mentioned, the material behaviour, especially damping and stress relaxation in cyclic deformations, is important. This viscoelastic behaviour is taken into account by considering the shear modulus as a complex modulus Aus Wissenschaft und Forschung 34 Tribologie + Schmierungstechnik · 70. Jahrgang · 6/ 2023 DOI 10.24053/ TuS-2023-0038 Introduction Rotary shaft seals (RSS) are used to seal rotating shafts in many different applications. Due to the wide variety of applications, many specific requirements for the development and use of RSS must be considered. These include knowledge of the different materials, the sealing task, possible process variations and experience. For the development of new shaft seals, experimental investigations are necessary to examine the operating behaviour under conditions close to the process. To reduce the time required, simulations can be used. However, knowledge of the boundary conditions or contact states is necessary for a correct result. Therefore, simulations are a useful tool for experienced designers to reduce the time required compared to experimental investigations. However, they also place great demands on the computing hardware. For these reasons, the Chair of Design and Drive Technology (KAt) at Paderborn University is developing a simplified calculation model that provides a sufficiently accurate statement regarding the operating behaviour of RSS with just a few parameters and at the same time offers a basis for the further development of shaft sealing systems. State of the art The elastomers used in shaft seals are special compounds which composition is known only to the manufacturer. Compared to steel, they are characterised by the nonlinear material behaviour shown in figure 1a): At low strains, the stress increases more than in the medium strain range. With greater strains, the stress then increases exponentially. The strains can be a multiple of 100 % [1, 2]. After reaching a strain maximum, the stress and strain decrease again. The difference in stress between stretching and retraction is caused by internal friction and is referred to as hysteresis friction [2]. Approximate simulation of the hysteresis friction of radial shaft seals Tobias Bührmann, Balázs Magyar* Eingereicht: 18.09.2023 Nach Begutachtung angenommen: 17.01.2024 Dieser Beitrag wurde im Rahmen der 64. Tribologie-Fachtagung 2023 der Gesellschaft für Tribologie (GfT) eingereicht. Applying an approach developed at the Chair of Design and Drive Technology the hysteresis friction of radial shaft seals (RSS) can be calculated in a good approximation. With a P RONY series the time and temperature dependent material behaviour of elastomers is calculated by determine the spring moduli and relaxation time of the dampers with suitable distribution functions. The time dependency is calculated from the surface area of the counter surface and shaft speed whilst the temperature dependency is calculated by a thermal distribution model. For validation the results are compared with experimental data. Keywords Radial shaft seal, Hysteresis friction, Calculation model, Numerical methods, Material model, Surface calculation Abstract * Tobias Bührmann, M.Sc. Orcid-ID: https: / / orcid.org/ 0009-0003-9116-4210 Prof. Dr.-Ing. Balázs Magyar Orcid-ID: https: / / orcid.org/ 0000-0003-1092-674X Paderborn University, Chair of Design and Drive Technology (KAt) Warburger Straße 100, 33098 Paderborn, Germany TuS_6_2023.qxp_TuS_6_2023 01.02.24 14: 18 Seite 34 G* with two parts: a storage modulus G' and a loss modulus G''(1) [6, 7, 8]. (1) G * Pa Complex shear G'' Pa Loss modulus modulus G' Pa Storage modulus Viscoelastic behaviour can be described by a P RONY series (Maxwell elements connected in parallel) according to figure 2a) [5, 7, 11]. Each element is characterised by a spring modulus G i and a relaxation time t r,i . The frequency-dependent excitation of the elements and the resulting material behaviour, figure 2b), can be calculated based on the geometry of the shaft and the seal as well as the surface roughness, since the same strain γ [5, 7, 8] is present in each element due to the parallel connection. The first element of the P RONY series is a spring and represents the energy stored in the system. The other ele- ! " ! # $ ! %% ments connected in parallel are differently defined Maxwell elements. The amount of Maxwell elements varies. A balance must be struck between computing time and the necessary accuracy of the description of the material behaviour. But a minimum of six Maxwell elements should be used. For each element, the modulus G i and the relaxation time t r,i are to be determined. By summing up the respective storage and loss moduli, the material behaviour is described (2, 3) [5]. (2) (3) G' Pa Storage modulus G'' Pa Loss modulus G 0 Pa Storage modulus G k Pa Element specific of the first spring modulus ω 1/ s Excitation τ k S Element specific frequency relaxation time ! # ! & '! * , - * . , - * / *03 ! ## '! * , - * . , - * / *03 Aus Wissenschaft und Forschung 35 Tribologie + Schmierungstechnik · 70. Jahrgang · 6/ 2023 DOI 10.24053/ TuS-2023-0038 Figure 1: a) Stress-strain behaviour of an elastomer according [2], b) Deformation and friction behaviour of rubber for different surface wavelengths λ and sliding speeds v, according [3] Figure 2: a) Structure of the P RONY series, according [5], b) Excitation frequency-dependent material behaviour of elastomers, according [5] TuS_6_2023.qxp_TuS_6_2023 01.02.24 14: 18 Seite 35 tween the oil and air sides of the RSS. This is necessary because the material properties greatly influence the heat dissipation in particular. For the elastomer material, essential properties (geometry, density, hardness [9], heat capacity, thermal conductivity, ...) first need to be specified. The corrugations or the counter surface must be defined in a similar way. One difference lies in the specification of the counter surface condition, which is highly important for the later calculation. Furthermore, the operating conditions must be specified. These include ambient temperature and the speeds in the considered time range. Determination of initial state Based on this information, the calculations for the initial state are now carried out. This is valid for a fixed reference temperature and needs to be adjusted by the timetemperature displacement factor (4) for the current temperature at each time step in the later course of the calculation because the temperature in the sealing contact and in the lip itself will change and the material behaviour of elastomers is temperature dependent as shown in figure 3. The determination of the material behaviour of Aus Wissenschaft und Forschung 36 Tribologie + Schmierungstechnik · 70. Jahrgang · 6/ 2023 DOI 10.24053/ TuS-2023-0038 Since a linear model for the calculation of the material behaviour is used, the temperature dependence of the material behaviour of elastomers can be calculated using the time-temperature displacement factor a T of W ILLIAMS , L ANDEL and F ERRY [10] (4), as shown in figure 3. At temperatures below the glass transition temperature T g , a high shear modulus is present. At temperatures close to T g , internal binding forces in the material and thus the modulus decrease significantly [12]. Above T g up to the decomposition temperature T z , elastomers are in a rubber-elastic state in which deformations are largely reversible. The change in shear modulus is taken into account by using (4). (4) a T - Shift factor T K Temperature C 1,2 - Constant T S K Reference temperature Setting up the calculation model Based on the presented models a calculation model can be set up, which is divided into four main sections: determination of boundary conditions, determination of initial state, determination of operational behaviour and output. The results of each section are needed for further calculations in the following section. The complete structure of the calculation model is shown in figure 4 and explained in more detail below. Determination of boundary conditions In the first section, the boundary conditions for the calculation must be defined. First, the ambient media have to be defined, whereby a distinction must be made be- 4 5 6 7 8 3 9 9 : 8 9 9 : Figure 3: Temperature dependence of the shear modulus of an elastomer [12] Figure 4: Systematic presentation of the calculation model TuS_6_2023.qxp_TuS_6_2023 01.02.24 14: 18 Seite 36 the elastomer is of great importance and builds the core of this calculation model. This is mapped using the P RONY series described in the state of the art. Based on known material data from experimental investigations, it is possible to define the individual P RONY elements in a suitable manner. This is done at KAt with a self-developed approach using different statistical distribution functions for the spring elements and the dampers. For the spring elements, the proportional factors g i are calculated using a normalised Weibull distribution function (5), whereby the condition from (6) must always be fulfilled. (5) (6) G i Pa Element specific G *max Pa Maximum modulus modulus g i - Distribution factor i - Element b - Shape factor T - Position parameter The damper elements are to be determined by calculating the respective relaxation time t r,i . Due to the analogy to the discharge behaviour of a capacitor from electrical engineering, it is assumed that a damper relaxes according to a decreasing exponential function. However, it must be considered that for a complete representation of the relaxation behaviour of an elastomer, several dampers with different relaxation times must be present and that these are influenced by the respective spring element connected in series. This relationship can be considered by the ratio of the respective deformation in a springdamper model shown in figure 5. Based on the analogy to the series connection of a resistor and a capacitor, there is an angular offset of 90° be- ; $ < 9 > $ 9 ? @A3 B AC D E F G '! D H 3 '; D ! IJK " H 3 ! IJK " tween the ideal spring and the ideal damper. If the deformation diagram describes the state at time t = t 0 , i.e. in the case of abrupt deformation of the system, given an infinitely soft damper and an infinitely hard spring, the deformation γ would be present completely in the damper. The system would be immediately relaxed, as the introduced deformation and the associated deformation energy in the damper were irreversibly converted into heat. The angular displacement would be δ = 0° and the relaxation time t r = 0 s. With an infinitely hard damper and an infinitely soft spring, the deformation would be completely in the spring. The damper would be undeformed. As a result, the angular displacement would be δ = 90° and the relaxation time t r would be infinite. At the same time, if the deformations in the damper and the spring are equal, the angular displacement is δ = 45°. Accordingly, the spring-damper models can be defined on the basis of the angular displacement. For this purpose, a distribution function independent of the modulus distribution must be selected. Here, the beta distribution (7, 8) is used, as it offers sufficient possibilities to represent experimentally determined material behaviour due to its two parameters and the normalised form. (7) (8) ab - Parameter u - Normalized element numbers Based on the analogy to the discharge process of a capacitor, it is assumed that with equal deformation fractions γ s = γ d , the relaxation behaviour of the damper corresponds to the decreasing functio⁄n f (x) = e -t . According to figure 5, the following applies: tan (δ = 45°) = γ spring ⁄ γ damper =1. If this function is now supplemented by the factor tan (δ) as an additional expo- L M . N 6O < P JA3 . P @A3 QP K & N 6O < P JA3 . P @A3 QP 3 & Aus Wissenschaft und Forschung 37 Tribologie + Schmierungstechnik · 70. Jahrgang · 6/ 2023 DOI 10.24053/ TuS-2023-0038 Figure 5: Analogy between mechanics and electrical engineering TuS_6_2023.qxp_TuS_6_2023 01.02.24 14: 18 Seite 37 help of a deformation matrix and the simplified material model Neo-Hook, and the previously determined material behaviour, a stress state can now be calculated in a good approximation. By mounting the RSS, the sealing lip is expanded to the diameter of the shaft and quasi statically deformed. In addition, there is a relative movement between the sealing lip and the surface of the counter surface during operation. Because of the necessary roughness of the surface, a frequency-dependent deformation of the sealing lip occurs in addition to the static deformation. In order to be able to take this excitation into account in a suitable manner, it is necessary to map the rough surface using a suitable calculation model and to derive relevant parameters from it. In this case, the calculation is simplified by assuming a rough surface of the counter surface and an ideal smooth surface of the elastomer. In addition, it assumed that the rough surface can be represented statistically accurately by a normal distribution of roughness. For the calculation, the surface is divided into N elements of equal size in order to save calculation time. Aus Wissenschaft und Forschung 38 Tribologie + Schmierungstechnik · 70. Jahrgang · 6/ 2023 DOI 10.24053/ TuS-2023-0038 nent, a parameterisable function for describing the relaxation behaviour of the damper as a function of the angular displacement δ is obtained. Using suitable numerical methods, a straight line is now defined according to figure 6, the zero crossing of which can be calculated. This numerically determined time corresponds to the element-specific relaxation time t r,i . The procedure is shown for three elements in figure 6. With the help of this procedure, the spring-damper elements of the P RONY series can be defined independently of each other. Furthermore, it is possible to build up a material-specific database through the different parameters. In addition to the material behaviour, the deformation during the assembly of the RSS must be calculated. As large deformations are possible with elastomers and the deformation behaviour is not linear, a simplified assumption is made at this point: Since only relatively small strains (<10 %) are involved in the sealing lip during assembly of the RSS, this is assumed to be linear. In addition, the circular disc model with constant thickness shown in figure 7 is used for the calculation. With the Figure 6: Function of element specific relaxation and for calculating the element-specific relaxation time using the initial tangents Figure 7: a) Mounted RSS; b) Circular disc model for calculating the deformation state TuS_6_2023.qxp_TuS_6_2023 01.02.24 14: 19 Seite 38 Employing a suitable iterative procedure, surfaces as shown in figure 8 can be generated and surface characteristics are determinable. With the help of the surface and the calculated shear modulus of the elastomer, a coefficient of friction according to (9) can now be calculated in a good approximation and a statement about the existing hysteresis friction is made [7]. (9) ∇ zq - Root mean square | G * | Pa Complex of the gradient of modulus the surface G'' Pa Loss modulus Since the material behaviour of the elastomer is strongly dependent on temperature, a suitable thermal model of the sealing lip must be developed. Due to the low thermal R STU ! ## V ! " V conductivity of the elastomer, a sufficiently accurate discretisation of the sealing lip is also necessary. Based on this, the heating per element is calculated by balancing the heat flows. It must be taken into account that for the first element, the frictional power is assumed to be the incoming heat flow. For the other elements, heat conduction through the sealing lip applies. The complete thermal model is shown in figure 9. Determination of operating behaviour The subsystems described so far converge in the following calculation step and establish the connections shown in figure 10. First, the current state of the sealing lip is recorded by recording the temperature present for the current time step. For the first time step, this corresponds to the ambient temperature defined at the beginning. The WLF displacement factor is determined for this temperature. Furthermore, the operating state is recorded, or in this case the existing speed. With the two results and the data Aus Wissenschaft und Forschung 39 Tribologie + Schmierungstechnik · 70. Jahrgang · 6/ 2023 DOI 10.24053/ TuS-2023-0038 Figure 9: Thermal model of the sealing lip Figure 8: Surface model in sealing contact TuS_6_2023.qxp_TuS_6_2023 01.02.24 14: 19 Seite 39 between the measurements is the frictional torque of the RSS. The validation was carried out using a RSS with the data given in table 1. Aus Wissenschaft und Forschung 40 Tribologie + Schmierungstechnik · 70. Jahrgang · 6/ 2023 DOI 10.24053/ TuS-2023-0038 already available, the material behaviour for this time step is then determined. With the counter surface characteristics and the complex material behaviour, the friction value can now be determined, which, in conjunction with the material behaviour, is required to calculate the stress state and resulting frictional power. Based on the heat flow balance in figure 9, the temperature distribution in the sealing lip is calculated. The calculations are carried out for each time step, whereby the above-mentioned variables are also stored for each time step. Output Upon reaching the last time step, individually definable variables are put out (i.e.: friction torque, friction value, friction power, temperature field, ...). Experimental validation To validate the described calculation model, frictional torque measurements for a specified operating situation were performed on a test rig. Since a low frictional torque is expected, the experiment for each RSS consists of two measurements. The first is without a mounted shaft sleeve therefore the RSS has no contact to the shaft and the frictional torque of the test rig is measured. For the second measurement the shaft sleeve is mounted, so the RSS has contact with the shaft sleeve. The difference Figure 10: Systematic representation for the calculation of the operating behaviour Figure 11: Test rig for measuring the frictional torque of radial shaft seals Table 1: Data radial shaft seal Material: NBR Nominal diameter: 20 mm Size: 40 x 20 x 7 Shore A Hardness: 70 The test rig shown schematically in figure 11 is available for the experimental investigation. During measurements, the RSS is inserted into a seat connected to the bearing housing. When the shaft sleeve is mounted on the shaft, the RSS expands to the diameter of the shaft sleeve. Thus, it is possible to measure the frictional torque for deviations from the nominal diameter. Since the aim of this experiment is the measurement of the hysteresis friction of the RSS, the contact between the lip and the surface is dry and the spring of RSS has been removed. For the measurement, the drive accelerates to 1,000 rpm within three seconds and decelerates to standstill again after 60 seconds. The short time allows a measurement bevor the lip of the seal is destroyed due to overheating because of the dry contact. The RSS is replaced after the experiment. Measurements were taken TuS_6_2023.qxp_TuS_6_2023 01.02.24 14: 19 Seite 40 for three different shaft diameters: 19.75 mm, 20 mm (nominal dimension) and 20.25 mm. Figure 12 shows the results of the calculation model as well as the measurement data of a friction torque measurement for the RSS described in table 1. In general, the results are consistent, especially when expanded to the nominal diameter. The higher friction torque measured at the beginning can be attributed to adhesion effects. These are favoured by the dry sealing contact and the increasing static pressure with expansions beyond the nominal diameter. The difference between the experimental data for the expansion of the RSS to 20.25 mm and the calculated torque are due to the simplified modelling of the nonlinear strain-stress deformation. The higher frictional torque leads to a higher frictional power and thus to greater heating. After around 28 seconds, the measured frictional torque approaches the calculated value, but it can be assumed that the sealing lip is already damaged. The results also clearly show the influence of temperature on frictional torque and material behaviour. The temperature curve in the sealing lip for expansion to the nominal diameter is shown in figure 13a). Due to the low thermal conductivity of the material, the sealing lip heats up particularly strongly in sealing contact, reaching temperatures of over 100 °C relatively quickly due to the dry friction. At the outer part of the sealing lip, however, the temperature rises only slightly over time. The average temperature, which is relevant for the calculation of the material behaviour, continues to increase slightly after an initial sharp rise. Figure 13b) shows the material behaviour at reference temperature. Using the calculation model, this can be calculated for each time step and each temperature. Aus Wissenschaft und Forschung 41 Tribologie + Schmierungstechnik · 70. Jahrgang · 6/ 2023 DOI 10.24053/ TuS-2023-0038 Figure 12: Frictional torque measurement for various expansions of the RSS Figure 13: a) Calculated temperature in the RSS, b) Time-dependent material behaviour at reference temperature of 20 °C TuS_6_2023.qxp_TuS_6_2023 01.02.24 14: 19 Seite 41 Number 8. American Institute of Physics, Maryland. 2001. [4] Wangenheim, M.: Untersuchung zu Reibmechanismen an Pneumatikdichtungen. Dissertation. Gottfried Wilhelm Leibniz Universität Hannover. 2012. [5] Popov, V. L.: Contact Mechanics and Friction. Physical Principles and Applications. Second Edition. Springer- Verlag, Berlin Heidelberg. 2017. DOI: 10.1007/ 978-3- 662-53081-8 [6] Lazan, B. J.: Damping of materials and members in structural mechanics. Pergamon Press Inc., New York. 1968. [7] Wrana, C.: Polymerphysik. Eine physikalische Beschreibung von Elastomeren und ihre anwendungsrelevanten Eigenschaften. Springer-Verlag, Berlin Heidelberg. 2014. DOI: 10.1007/ 978-3-642-45076-1 [8] Findley, W.N.; Lai, J. S.; Onaran, K.: Creep and Relaxation of nonlinear viscoelastic materials. North-Holland Publishing Company, Amsterdam. 1976. [9] Kunz, J.; Studer, M.: Determining the Modulus of Elasticity in Compression via Shore A Hardness. In: Kunststoffe international 06/ 2006. Carl Hanser Verlag, München. 2006. [10] Williams, M. L.; Landel, R. F.; 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/ 1955. American Chemical Society, Washington D.C. 1955. [11] Jennewein, B.: Integrierter Berechnungsansatz zur Prognose es dynamischen Betriebsverhaltens von Radialwellendichtringen. Dissertation. Technische Universität Kaiserslautern. 2016. [12] Eyerer, P.; Hirth, T.; Elsner, P.: Polymer Engineering. Technologien und Praxis. Springer-Verlag, Berlin Heidelberg. 2008. DOI: 10.1007/ 978-3-540-72419-3 Aus Wissenschaft und Forschung 42 Tribologie + Schmierungstechnik · 70. Jahrgang · 6/ 2023 DOI 10.24053/ TuS-2023-0038 Summary By means of the calculation model described here, the hysteresis friction of RSS can be calculated for a known operating situation. In addition, statements regarding material behaviour, temperature or the coefficient of friction can be made. The advantage of the calculation model is the simple handling. Only a few parameters are needed to make a statement with a good approximation. For new developments, parameter studies are also possible regarding the design of the counter surface, choice of material, minimum expansion required to achieve a sufficiently large pressure in the sealing contact or the expected temperature development. The calculation model is a good alternative to more complex simulations because a statement regarding the required variables is possible within a few seconds. Especially for the further development of sealing technology for high-speed applications, this calculation model offers a good basis for efficiently designing sealing systems in a resource-conserving way. References [1] Baumann, J. T.: Fatigue, Stress, and Strain of Rubber Components. A Guide for Design Engineers. Carl Hanser Verlag, Munich. 2008. [2] Schwarzl, F. R.: Polymermechanik: Struktur und mechanisches Verhalten von Polymeren. Springer Verlag, Berlin Heidelberg. 1990. DOI: 10.1007/ 978-3-642-61506-1 [3] Persson, B. N. J.: Theory of rubber friction and contact mechanics. In: Journal of chemical Physics. Volume 115, TuS_6_2023.qxp_TuS_6_2023 01.02.24 14: 19 Seite 42