Influences on the vibration frequencies of tire tread blocks

Transcription

1 Influences on the vibration frequencies of tire tread blocks Matthias Kröger, Patrick Moldenhauer University of Technology Bergakademie Freiberg Institute of Machine Elementes, Design and Manufacturing Agricolastr. 1; Freiberg, Germany Tel: +49 (0) Abstract Tire noise is one of the main noise sources in urban areas and, therefore, a reduction is required. In order to find successful countermeasures, it is useful to understand the excitation mechanisms of the tire. One noise excitation mechanism results in stick-slip vibrations of the tread block during longitudinal acceleration or braking as well as severe cornering of the vehicle. The longitudinal or lateral slip of the tire causes friction induced self-excited vibrations of the tread blocks. The frequencies of the stick-slip vibrations depend on the geometry of the tread block, on the material and contact properties as well as on the process parameters like sliding velocity or normal load. The work presented here shows these dependencies using a numerically efficient tread block model which considers the block dynamics and the complex nonlinear contact effects in the tire/road system. 1 Introduction The tire/road contact excites the tire during rolling. This excitation causes noise which is one of the main noise sources in urban areas. Especially during acceleration, braking and severe cornering maneuvers the noise level of tires increases. Reasons for the noise can be oscillations of the tire tread blocks. During the passage through the contact patch the tread block first sticks on the road and in the second part of the contact the tread block slides. However, for some contact conditions, especially for certain slip and normal load conditions, the tread block shows self-excited stick-slip vibrations which are very noisy. To find countermeasures against the tire noise it is essential to get a substantiated understanding of the excitation mechanism and corresponding simulations. The latter can be used to design a tire with reduced self-excited vibrations. Typically the whole tire is modeled with FE, cp. Nackenhorst [1] or Brinkmeier [2], or by coupled elastic layers, cp. Kropp [3]. These models can describe the vibrations of the tire body very well in the low frequency range, e.g. up to 1000 Hz. In order to describe the high frequency range these models have to consider the geometry of the tread blocks and the complex contact properties. These strongly increase the complexity of the existing models resulting in a non-realistic computation time. Therefore, the existing models of whole tires consider the local contact properties with simplifications, e.g. with a constant friction coefficient. The model which is realized here describes one single tread block. Thereby, it is possible to use a very detailed description of the contact between tire and road, cp. Gäbel et al. [4], and to simulate high frequency vibrations. 4015

2 4016 PROCEEDINGS OF ISMA2010 INCLUDING USD2010 Fig. 1: Modular tread block model and a detail which shows the contact model 2 Tire tread block model The aim of the presented tread block model is to simulate high frequency oscillations which are e.g. selfexcited by the contact between tire and road. To fulfill this aim the model has to be very efficient with respect to computational calculation time. The tread block model is built up as a modular model which consists of four modules, see Fig. 1: A dynamic module (1), a friction module (2), a nonlinear contact stiffness module (3) and a wear module (4). The whole tire tread block model is validated by comparison with measured stick-slip vibrations, cp. Moldenhauer [5]. 2.1 Dynamic module The dynamic module is based on a 2D FE model. The degrees of freedom are reduced in order to limit the calculation time. A very efficient method for the reduction is the Craig/Bampton reduction, cp. Craig and Bampton [6]. With this method, which is a combined static and modal reduction method, all degrees of freedom of the contact nodes are still directly calculated as primary degrees of freedom. This avoids transformations at each time step. A further advantage of the Craig/Bampton reduction is the exact calculation of the static solution of the degrees of freedom. This is not the case for the original modal reduction method. 2.2 Friction module The main influencing parameters on the friction of rubber are the normal pressure p N and the sliding velocity v rel. This is considered in the friction model by an approximation of a measured friction characteristic. The unsteady friction characteristic μ at velocity v rel =0 is critical for the time efficiency of the simulation. Therefore, the unsteady behavior is smoothed by an arctan function. In case of typical process parameters the smoothing shows a negligible effect within the simulations. Nevertheless an exact sticking will not occur anymore, only a quasi-sticking with a very small relative velocity. The approximation of the local friction characteristic is described by (1)

3 TYRE/ROAD NOISE AND EXPERIMENTAL VALIDATION 4017 Fig. 2: Measured (grid) and approximated (semi-transparent) friction characteristic (left) in dependence of sliding velocity and pressure and comparison of the global contact stiffness (right) (Tread block rubber on Corundum 400) The parameters are identified by measurements. A comparison of the measured and approximated friction characteristic is shown in Fig. 2 left. 2.3 Non-linear contact stiffness module Due to the very rough road surface only a fraction of the nominal contact area contacts the tread block surface. In combination with the soft rubber material, this results in large local deformations of the tread block. This local effect influences strongly the global force-displacement characteristic on concrete and asphalt roads. A non-linear contact stiffness,, (2) is used to describe this local effect with spring compression u, cp. Fig. 1. A comparison of the measured contact stiffness and the simulation results is given in Fig. 2 right. The implementation of the described friction characteristic and the non-linear contact stiffness allows to simulate the contact adequately without a direct consideration of the very complex road roughness. 2.4 Wear module The wear of the tread block changes strongly the contact between the tread block and the road. Therefore, the wear has to be modeled as well to be able to predict the dynamic behavior under realistic tire operating conditions. Different wear laws exist in dependence of the influencing parameters like sliding velocity and normal load, e.g. from Fleischer [7], Archard [8], Hofstetter [9] or Viswanath and Bellow [10]. For realistic parameters all these models result in a similar shape of the tread block, cp. Fig. 3 right. Here, a generalized model of the wear rate η W as mass loss m W per sliding distance s in used, The parameters k i are identified by wear tests, see Fig 3 left. (3)

4 4018 PROCEEDINGS OF ISMA2010 INCLUDING USD2010 * Fig. 3: Measured and approximated wear rates related to the friction coefficient η W = ηw / μ in dependence of sliding velocity and pressure (left) and side view of typical tread block after a sliding distance of s = 25 m (right) (Tread block rubber on Corundum 400) 3 Vibrations of sliding tread blocks A tread block shows vibrations during sliding on a road surface especially in the parameter range, where the friction coefficient has a negative gradient with respect to the sliding velocity. This is for the conditions investigated here e.g. the case for a normal pressure p N = 0.25 N/mm 2 and a sliding velocity v rel = 300 mm/s. The displacements of the run-in edge and the run-out edge are shown in Fig. 4 left. The limit cycles of different points at the run-in edge of the block are given in Fig. 4 right, cp. Fig. 1. The largest oscillations occur at the nodes which are closest to the contact area. The oscillation frequency of the stick-slip vibration is 2550 Hz. One important question for the design of a tire with low noise level is the parameter influence on the vibration amplitude and frequency. Therefore, parameter studies are performed with the simulation model. Thereby, the advantage of the numerical efficiency can be seen clearly, e.g. for the simulation of a sufficient time of 10 ms which gives about 25 vibration cycles using a maximum time step size of 10-6 s a standard PC (Intel Core 2 Duo) needs about 50 s. Fig. 4: Oscillating displacements (left) of different tread block points and limit cycles (right) of the tread block nodes at the run-in edge

5 TYRE/ROAD NOISE AND EXPERIMENTAL VALIDATION 4019 Fig. 5: Displacement versus time for different Young s modules (left) and different material damping factors (right) Fig. 5 left shows a variation of Young s modulus E between 15 N/mm 2 and 75 N/mm 2. The vibration amplitude decreases with increasing stiffness of the rubber while the stick-slip frequency increases. The influence of the material damping is depicted in Fig. 5 right using a damping matrix D which is proportional to the stiffness matrix C, D = β C. With an increasing material damping factor β from s to s the amplitude decreases while the frequency increases. For large damping factors the oscillations vanish. Furthermore, the mean sliding velocity has a large influence on the occurring vibrations due to the nonlinear dependency of the friction coefficient on the sliding velocity and due to the reduction of sticking time with increasing velocity. The displacement versus time and the limit cycles for different mean sliding velocities v between 50 mm/s and 500 mm/s are shown in Fig. 6. Only for mean sliding velocities between 100 mm/s and 400 mm/s oscillations can be observed. This is the range of velocities corresponding to a negative gradient of the friction coefficient with respect to sliding velocity, cp. Fig. 2 left. The limit cycles show that for v rel = 100 mm/s, 200 mm/s and 300 mm/s a stick-slip vibration occurs while for 400 mm/s there is no prominent sticking phase. Fig. 6: Displacement versus time (left) and limit cycle (right) for different mean sliding velocities

6 4020 PROCEEDINGS OF ISMA2010 INCLUDING USD2010 Fig. 7: Displacement versus time for different mean pressures p N (left) and different block geometries (right) with block height h of 6 mm and 10 mm and block length l of 15 mm and 45 mm The mean pressure in the contact influences the vibration behavior because with increasing normal load the friction force increases. Therefore, a parameter variation is conducted for normal pressures between p N = 0,08 N/mm² and p N = 0,63 N/mm². The largest amplitudes are observed at high normal pressures while the oscillations vanish for small pressures, see Fig. 7 left. Further the influence of the block geometry on the oscillations is important for tire development. The height h and the length l of the tread block are varied, see Fig. 7 right. The oscillation amplitude increases with increasing height, while it decreases with increasing block length. Further a decrease of the frequency can be observed if the amplitude increases. This is typical for stick-slip vibrations. One important issue concerning self-excited vibrations of tread blocks is the investigation of the stick-slip frequencies under variation of the slip velocity and tread block geometry. Simulations show a strong influence of these parameters, see Table 1. The stick-slip frequency f SS increases with increasing sliding velocity preliminary due to a shorter sticking phase. Above a certain critical sliding velocity the vibrations are vanished. The simulated frequencies do not exceed the first eigenfrequency f 0 of the free tread block system without contact. It can be noted that the frequency range for different sliding velocities and block geometries is very broad from 299 Hz up to 4717 Hz. Tread block h = 10 mm, l = 15 mm Tread block h = 6 mm, l = 15 mm Tread block h = 10 mm, l = 45 mm Tread block h = 6 mm, l = 45 mm f SS (v 0 = 80 mm/s) 299 Hz 1492 Hz 413 Hz 2128 Hz f SS (v 0 = 100 mm/s) 373 Hz 1961 Hz 565 Hz 2667 Hz f SS (v 0 = 200 mm/s) 787 Hz 3953 Hz 1355 Hz 4525 Hz f SS (v 0 = 300 mm/s) 1195 Hz 4546 Hz 2597 Hz 4717 Hz f SS (v 0 = 400 mm/s) 1759 Hz Hz - f SS (v 0 = 500 mm/s) 2123 Hz f SS (v 0 = 600 mm/s) 2304 Hz f Hz 4590 Hz 2876 Hz 4919 Hz Table 1: Influence of sliding velocity and geometry on the stick-slip frequency

7 TYRE/ROAD NOISE AND EXPERIMENTAL VALIDATION 4021 Fig. 8: Different contact phases of a rolling tire (top) and simulated deformations of tread block (bottom) 3.1 Vibrations of a tread block during rolling The contact of a tread block of a rolling tire with the road can be separated in four phases: run-in, sticking phase, sliding phase and snap-out, see Fig. 8 top. If the trajectory of the flattened tire belt during rolling is added to the block model the behavior of a tread block on a rolling tire can be examined as well. The simulated deformation of the tread block for different stages in the contact patch is depicted in Fig. 8 bottom. In the sliding phase stick-slip vibrations can occur depending on the slip rate and the vehicle velocity. An example is given in Fig. 9 showing displacement and velocity versus contact coordinate x Tr. Fig. 9: Lateral displacements (left) and velocities (right) of the run-in edge during the passage through the contact zone

8 4022 PROCEEDINGS OF ISMA2010 INCLUDING USD Conclusions The tire noise is one of the main noise sources in urban areas. This paper has shown a model which describes the deformations and oscillations of tread blocks during sliding as well as in the case of rolling of the tire. Due to the negative gradient of the friction coefficient with respect to the sliding velocity friction induced vibrations can occur. The resulting stick-slip vibrations of the tread blocks are in the acoustically relevant frequency range. A parameter study is performed to show the influences of the contact conditions as well as of the design parameters on vibration amplitude and frequency. One important observation is the dependency of the stick-slip frequency on sliding velocity. The stick-slip frequency is not the free eigenfrequency of the tread block. The eigenfrequency describes only the maximum frequency of the stick-slip vibrations for high sliding velocities. In this case the sticking phase is very small. For higher sliding velocities the vibrations vanish or oscillations without a sticking phase occur. The simulation model and the parameter study give the possibility to design a tire in such a way that the tire noise is minimized whereby the other tire performances have to be considered. Acknowledgements The investigations published in this report received financial support from the German Research Foundation (DFG) as a part of the collaborate research unit DFG FOR 492, Dynamic Contact Problems with Friction of Elastomers, to whom we extend our thanks. References [1] NACKENHORST, U.: Rollkontaktdynamik - Numerische Analyse der Dynamik rollender Körper mit der Finite Elemente Methode. Universität der Bundeswehr, Hamburg : Habilitation, [2] BRINKMEIER, M.: Modellierung und Simulation der hochfrequenten Dynamik rollender Reifen. Leibniz Universität Hannover, Dissertation, [3] KROPP, W.: Structure-born Sound on a Smooth Tyre. In: Applied Acoustics 26 (1989), No. 3, pp [4] GÄBEL, G.; MOLDENHAUER, P.; KRÖGER, M.: Lokale Effekte zwischen Reifen und Fahrbahn. In: ATZ Automobiltechnische Zeitschrift 110 (2008), Nr. 6, pp [5] MOLDENHAUER, P.: Modellierung und Simulation der Dynamik und des Kontakts von Reifenprofilblöcken. Technische Universität Bergakademie Freiberg, Dissertation, [6] CRAIG, R.; BAMPTON, M.: Coupling of Structures for Dynamic Analyses. In: AIAA Journal 6 (1968), Nr. 7, pp [7] FLEISCHER, G.: Energetische Methode der Bestimmung des Verschleißes. In: Schmierungstechnik 4 (1973), Nr. 9, pp [8] ARCHARD, J. F.: Contact and Rubbing of Flat Surfaces. In: Journal of Applied Physics [1] 24 (1953), Nr. 8, pp [9] HOFSTETTER, K.: Thermo-mechanical Simulation of Rubber Tread Blocks During Frictional Sliding. Technische Universität Wien, Dissertation, [10] VISWANATH, N.; BELLOW, D.G.: Development of an equation for the wear of polymers. In: Wear (1995), Nr. 1, pp