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Impact of contact stiffness heterogeneities on friction-induced vibration
International Journal of Solids and Structures 51 (2014) 1662–1669
Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Impact of contact stiffness heterogeneities on friction-induced vibration V. Magnier ⇑, J.F. Brunel, P. Dufrénoy Université Lille Nord de France, F-59000 Lille, France Lille1-LML, F-59655 Villeneuve d’Ascq, France CNRS, UMR 8107, F-59655 Villeneuve d’Ascq, France
a r t i c l e
i n f o
Article history: Received 28 June 2013 Received in revised form 27 November 2013 Available online 14 January 2014 Keywords: Friction-induced vibration Complex modal analysis Heterogeneity stiffness contact Effect of heterogeneity size
a b s t r a c t It is well-known that the occurrence of squeal depends on numerous phenomena still relatively unknown. Furthermore, squeal is affected by several factors on both the micro and macro scales. The present paper proposes some potential interactions between stiffness heterogeneities of the contact surface and the potential for triggering the squeal. Consequently, an analytical model has also been developed to highlight the impact of certain parameters on the squeal phenomenon. The contact surface has been modeled by connecting the disc with the pad via distributed springs (contact stiffness). From this model, a static equilibrium and a complex modal analysis were performed to determine respectively the pressure distribution and the natural frequencies of the system. In this configuration, the results demonstrate that the introduction of heterogeneities could change the dynamic behavior of the system. Moreover, the influence of the size of the heterogeneities was studied and results show that this parameter had an influence on the likelihood of squeal occurrence. The paper also points out a heterogeneities correlation length that is reasonable to consider for this special configuration. Such information is relevant when interpreting the occurrence of noise as a function to the sizes of the components of the brake pad and the morphology of the surfaces in contact. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In the transport industry, there sometimes occurs a loud noise or high-pitched squeal as the brakes are applied. The latter can be of great discomfort to the human ear because of its high frequency of greater than 1 kHz (Akay, 2002). The origin of this unwanted noise is issued from unstable dynamic behavior of the brake system due to frictional forces. During the past few decades, a significant number of theories have been developed regarding this instability phenomenon involving friction-induced vibration: stick-slip (Oden and Martins, 1985), negative friction coefficient sliding velocity slope (Mills, 1938), sprag-slip (Spurr, 1961) and mode lock-in (North, 1976). Stick-slip comes about when there are friction issues, sprag-slip is related to geometric effects from coupling between systems with different degrees of freedom, and mode lock-in is caused by selfexcited vibrations. In numerous cases of squeal in braking systems, the instability is linked to the mode lock-in mechanism (Jarvis and Mills, 1963). These different theories are associated to several ⇑ Corresponding author at: Université Lille Nord de France, F-59000 Lille, France. Tel.: +33 (0)3 28 76 73 57; fax: +33 (0)3 28 76 73 61. E-mail address: [email protected] (V. Magnier). http://dx.doi.org/10.1016/j.ijsolstr.2014.01.005 0020-7683/Ó 2014 Elsevier Ltd. All rights reserved.
numerical models which can enable to predict the squeal phenomenon. On the one hand, a model based on the real geometry using a finite element method has been developed to investigate the effects of system parameters, such as the coefficient of friction between the pad and the disc (Trichês et al., 2008), or the stiffness of the disc (Liu et al., 2007) etc. On the other hand, minimal models or low-order finite element models have been used to gain understanding of the role of each parameter, essentially with an analytical approach (Hoffmann et al., 2002), or for control (Massi et al., 2006). Even though these models are relatively well understood at the system scale, all the studies cited above have been carried out with homogeneous and mostly isotropic materials. However, it is well-know that braking depends on multi-scale and multiphysics phenomena which are still unknown to a large extent. Indeed, squeal is an interdisciplinary issue involving dynamics, tribology, acoustics, etc. largely conducted separately. Furthermore, squeal is affected by many different factors on both the micro and macro scales. Phenomena at small scales when it comes to both length (microscopic contact effects) and time (high-frequency vibrations), affect and are affected by phenomena at large scales (wear, behavior of the tribological triplet and dynamics of the entire brake system). Nevertheless, studies do not necessarily demonstrate the impact of the various scales (third body, bodies
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in contact, components, heterogeneities etc.) on the likelihood of squeal occurrence. Nevertheless, various brake studies take into account heterogeneities using mainly single-scale approaches to describe wear, damage etc. For example, at the third body scale, various methods enable to describe the surface dynamics during friction and wear and can show the formation of self-affine surfaces as found in modeling of real surfaces such as with the discrete element method (Nguyen et al., 2009; Renouf et al., 2011; Richard et al., 2008) or cellular automata (Popov and Psakhie, 2007; Dimitriev et al., 2008; Wahlstrom et al., 2011). An other contact scale which can find concerns heterogeneities introduced by machining defaults, mounting defects, etc. To model this scale, one of the first models was that proposed by Greenwood and Williamson (1966) in which the elastic contact between a rigid plane and spherical asperities was considered by modifying the contact stiffness. These statistical models have been enriched by many authors, e.g., (Bush et al., 1992; Whitehouse and Archard, 1970) and provide fast and accurately responses. The deterministic approaches have been developed to introduce a better geometrical description using mathematical functions to represent the asperities. Elastic, perfectly plastic (Chang et al., 1987) or elastoplastic behaviors (Zhao et al., 2000) as well as interactions between asperities (Zhao and Chang, 2001) have been integrated in models. Also from a geometrical viewpoint, but at a larger scale, Bonnay et al. (2011) took into account a thickness variation of the disc and showed the influence of this perturbation on the mode lock-in. Another scale of heterogeneities concerns the components presented in the friction pad. Indeed, modern brake pads are fabricated from composites comprising many different components (structural materials, matrix, filler and frictional additives). These heterogeneities play a key role for the local behavior through the local properties of each component (stiffness, bulk modulus etc.). Mbodj et al. (2010) and Peillex et al. (2008) studied the mechanical properties of the heterogeneities and considered the pad material using a homogenization technique while (Alart and Lebon, 1998) mixed static frictional contact and heterogeneous materials using an asymptotic homogenization technique. Recently, Magnier et al. (2011) proposed an analytical model with a contact interface including stiffness heterogeneities between pad and disc and demonstrated the influence of these kinds of heterogeneities on mode lock-in. Consequently, the objective of this paper has been to present an analytical model including stiffness heterogeneity in the contact. This numerical model was a simplified pin-on-disc setup inspired by a experimental arrangement developed at the Laboratoire de Mécanique de Lille (France). Focus was put on the influence of the heterogeneities and their sizes on the likelihood of squeal occurrence via mode lock-in. 2. Semi-analytical model 2.1. Description The semi-analytical model was inspired by an experimental setup developed in our lab. This specific experimental arrangement was simplified with a reduced number of parts to control the dynamic behavior, and the model comprised a thin plate, a pad housing, a friction pad and a disc as shown in Fig. 1(a). In the model, the disc was expressed by a vibration system with one degree of freedom (translational direction, namely yd ) and the pad was expressed by a vibration system with two degrees of freedom (translational and rotational directions, namely respectively yp and /p ) as shown in 1(b). This analytical model corresponds to that developed by Oura et al. (2009) with the exception that the rotation was not at the center of the pad.
In the model, K 1 and K 2 represent the Y-direction stiffness on each side of the thin plate. The stiffness of the friction pad was modeled by a parallel distribution of springs, where the stiffness of each spring ki was directly dependent on the material’s mechanical properties, i.e., ki ¼ E:S with hi is the height of each spring and S hi is equal to the discretization-dependent area of each spring. So, this parameter physically represent the stiffness of the bulk. Initially, the pad material was considered to be homogeneous with a Young modulus E fixed to 3000 MPa. The pad was assigned a height (hi ) of 10 mm before deformation. Table 1 presents the parameter values that were used in the computation. This analytical model was first computed in static equilibrium under conditions of pressure and sliding. This step rendered it possible to obtain the contact pressure distribution that was injected. This was done in the second step of the computation, in a complex eigenvalues analysis to determine the natural frequencies of the system. The dynamic equations were as follows:
8 n X > > € M ¼ K y ki yi y > d d d d > > > i¼1 > > > n < X € € 1 1 M p y2 2 y1 ¼ K 1 y1 þ y2 y x1 K 2 y2 þ y2 y x2 ki yi d d > > i¼1 > > > n n > X X > > y y y y y€ y€ > ki yi lh þ ki yi l : J 2 d 1 ¼ K 1 2 d 1 x21 K 2 2 d 1 x22 þ i¼1
i¼1
In this study, only the out-of-plane bending mode of the disc was considered. The assumed disc vibration was represented by a model with a single degree of freedom. The equivalent mass M d was set to be equal to the mass of the disc. The disc stiffness K d was calculated using the eigenfrequency determined with the finite element method and M d . M p represents the total mass, including the pad and the pad-housing, and J is the moment of inertia calculated at the center of mass of the plate spring. The spring coefficients, K 1 and K 2 were determined from M p ; J and the numerical pad frequencies. These two springs enabled to model the thin plate and achieve the two pad modes. y1 and y2 represent the elongation of the K 1 and K 2 springs during translation of the pad, and d is the distance between these two springs. yi represent the elongation of each spring i. All these parameters are presented in Table 2 and were inspired by the experimental set-up. Finally, x1 (resp. x2 ) corresponds to the distance between the center of rotation of the pad and the spring K1 K 1 (resp. K 2 ). After a quasi-static solution under sliding conditions, a complex eigenvalues analysis was performed and the results were written as k ¼ a þ jx. Here, k represents the eigenfrequency and x the real part. If the real part is zero, the mode is considered as stable. Conversely, a non-zero real part indicates an unstable mode. 2.2. Illustration of a semi-analytical model considering a homogeneous stiffness distribution This section illustrates how to determine the evolution of the system’s eigenfrequencies in relation to the coefficient of friction and to identify the unstable configurations. Indeed, it has been well established that the coefficient of friction is a relevant parameter in instability (Massi et al., 2006). The pad-housing and the thin plate were considered to be of steel with mechanical properties set to 210000 MPa for the Young modulus and 0.3 for the Poisson coefficient. For the pad, a value of 3000 MPa was taken for the isotropic Young modulus. For the semi-analytical model, the pad contact area was set to 30 20 mm2 with a discretization of 300 200 elements. The extremities of the springs K 1 and K 2 were submitted to a vertical displacement of 0.5 mm to obtain an equivalent normal force of 300 N. In the x-direction, a translation of the disc was applied to simulate the slip condition.
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Fig. 1. Semi-analytic contact model.
Table 1 Parameters used for the static equilibrium. K 1 (N/m) 256 10
3
K 2 (N/m)
d (mm)
256 103
20
Table 2 Parameters used for dynamical analysis. M d (kg) 3:88
K d (N/m) 4:85 10
Mp (kg) 8
0:07
K p (N/m) 6 10
5
J (kg.m2) 4:43 106
To illustrate the strategy of resolution, the analysis was carried out by ranging the values of the friction coefficient from 0.1 to 1. Fig. 2 shows the contact pressure distribution obtained by the static resolution with a coefficient of friction l fixed at 0.3. A non-uniform contact pressure distribution was obtained with a maximal value at the leading edge for all models. It can be noted that one spring along the thickness of the pad linearized the contact pressure distribution. Indeed, this result was due to the linear interpolation along the thickness pad direction. This semi-analytical model is considered reliable as compared to finite element calculations (Magnier et al., 2011). It is clear that the obtained contact pressure distribution is unlikely to be representative of a real case but it enables to simplify results in order to understand the role of each parameter. The advantage of this semi-analytical model is a reduced time computation which can render possible a fine discretization. The second step of the computation consists in carrying out a complex eigenvalue analysis for the determination of unstable modes. The four nodal diameter disc modes was the only one
Fig. 2. Pressure distribution for a 30 20 mm2 surface contact with
l = 0.3.
considered in this study. The analysis was performed by ranging the values of the coefficient of friction from 0.1 to 1 while retaining the apparent contact length at 30 mm. The results of this step are plotted in Fig. 3. The eigenfrequency associated with the translation mode (blue curve) of the disc was slightly modified by the coefficient of friction. On the other hand, the eigenfrequency associated with the tilting (rotation around the z direction) mode of the pad (yellow curve) and the translation mode of the pad (red curve) was strongly affected by the contact surface. For a constant apparent contact length equal to 30 mm, the frequency of the tilting pad modes decreased as the value of coefficient of friction increased. This was mainly due to the diminution of the effective contact length induced by the coefficient of friction (Duboc et al., 2010). The tilting mode of the pad and the axial mode of the disc coalesced, which resulted in a coefficient of friction comprised between 0.3 and 0.4, and led to a mode lock-in mechanism and to an unstable mode being obtained with the real part different from zero. The eigenfrequencies associated with a friction coefficient equal to 0.3 were 7714 Hz for the translation pad mode and 4326 Hz for the 4-diameter disc mode and tilting pad mode. Thus, in this last case, where the pad was considered homogeneous, the system was unstable which could lead to squeal noise. 3. Introduction of a heterogeneous stiffness distribution 3.1. Procedure for the introduction of heterogeneous stiffness An extension of the previous case was investigated by introducing a distribution of heterogeneous stiffnesses with a Gaussian
Fig. 3. Mode coupling between the tilting pad mode and the 4-diameter disc mode (apparent contact length = 30 mm).
V. Magnier et al. / International Journal of Solids and Structures 51 (2014) 1662–1669
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by a simple Gaussian distribution. But, this simple function distribution enables us to understand the role of the heterogeneities of the stiffness. ðxgÞ2 1 f ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi e 2r2 : 2 2pr
Fig. 4. Gaussian function centered at 3000 MPa.
Fig. 4 presents a graph of the probability density function f ðxÞ. Several standard deviation values r ranging from 1 to 1000 MPa in the normal distribution have been used. For a fixed standard deviation r, the Young modulus varied from ð3000 3rÞ MPa to ð3000 þ 3rÞ MPa. In the following, two parameters were studied: the first was the standard deviation and the second was the size of the heterogeneities, ranging from 1 lm to 5 mm for the edge size. For each standard deviation and fixed patch size, one thousand random tests with a random dispersion of the Young modulus we are computed to establish a probabilistic study on the confusing modes. The Monte-Carlo method was used to obtain a stochastic approach. 3.2. Effect of the standard deviation
Fig. 5. Number of unstable cases vs. standard deviation for equal to 1,5 1,5 mm2.
l = 0.3 for a patch size
function f ðxÞ centered around the Young modulus of the homogeneous case (i.e., g = 3000 MPa). It is clear that in a real configuration, the distribution of stiffness probably cannot be represented
In this part of the study, the coefficient of friction was fixed at 0.3, the patch size was set to 1.5 1.5 mm2 and the standard deviation was ranged from 1 to 1000 MPa. Fig. 5 shows the number of unstable cases for each standard deviation. For a fixed patch size, when the standard deviation was close to 0 MPa, the results showed no influence on the dynamic aspect and the system was systematically in mode lock-in. This corresponds to the initial case where the material was considered as homogeneous. However, for a standard deviation greater than 60 MPa, some stable cases were systematically found. Further, the number of unstable cases (i.e., in stable case) increased with the standard deviation. For example, for the ultimate case where the standard deviation equaled 1000 MPa, 30% of cases led to mode lock-in which was the complete opposite of the tendency obtained in the homogeneous case. Thus, to explain this phenomenon, Fig. 6
Fig. 6. Pressure distribution with different standard deviation.
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show the contact pressure distribution associated respectively to a standard deviation of respectively 10 MPa, 500 MPa and 1000 MPa. Indeed, for small standard deviations close to 0 MPa, the pressure distribution was quasi-linear, just as in the homogeneous case (Fig. 2). But for a standard deviation greater than 60 MPa, the pressure distribution deviated from quasi-linearity and the pressure localization become influent. Indeed, the pressure localization was essentially concentrated to a patch where the stiffness was significant with a high level of pressure. The dispersion of the frequency of translation pad and the frequency in of the disc are shown Fig. 7 respectively for a variance fixed to 50 MPa, 500 MPa and 1000 MPa. It can be noted that the reference case with homogeneous material is placed on each figure. It can be noted that on 1000 random simulated cases treated with the variance equal to 1000 MPa, the frequency of translation pad mode is ranging from 7568 Hz to 7830 Hz; the frequency of the 4 diameter disc mode is ranging to 4286 Hz to 4387 Hz and finally the frequency associated to the tilting mode
is ranging from 4035 Hz to 4387 Hz (Fig. 7). The eigenfrequency associated to the translation mode of disc is slightly modified by the heterogeneities dispersion. Unlike, the eigenfrequencies associated to the tilting mode of the pad and the translation mode of the pad are more affected by the heterogeneities dispersion. Consequently, the results presented in this part were obtained for a patch size fixed at 1.5 1.5 mm2. A study dedicated to varying this parameter is discussed in the next section. 3.3. Effect of the size of the heterogeneities This section discusses the effect of the size of the heterogeneities, ranging from 1 1 l2 to 5 5 mm2 with a friction coefficient fixed at 0.3. Fig. 8(b) shows the percentage of unstable cases as a function of the patch size with different values of standard deviation (r = 3, 50, 100, 300, 500, 700, 1000 MPa). At first, for a fixed patch size, the increase in standard deviation enabled to avoid the homogeneous case and to increase the possibility of stable
Fig. 7. Dispersion of frequencies with different variance.
V. Magnier et al. / International Journal of Solids and Structures 51 (2014) 1662–1669
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a standard deviation set to 1000 MPa for different patch sizes (2 mm, 0.2 mm and 0.005 mm). The figure shows that the pressure localizations were more important when the patch size was large. In this configuration, the localization pressure was essentially concentrated on patches where the stiffness was significant, with a high level of pressure close to 1.8 MPa as opposed to 1.2 MPa for the homogeneous case. When the patch size decreased, the phenomenon of localization pressure was less important and the pressure distribution corresponded to that obtained in the homogeneous case. Indeed, the randomness of the stiffness dispersion enabled to avoid localization pressure and to evenly distribute the pressure on the global contact surface.
Fig. 8. Number of stable case vs patch size with different standard deviation.
cases. Indeed, for a low standard deviation, the pressure distribution converged to a quasi-linear pressure distribution and did not permit to completely change the dynamic behavior. However, when the standard deviation increased, the location of patches pressure are dominant in the system focuses on patches whose rigidity is more important. Secondly, for a fixed standard deviation, there was an increase of unstable cases as a function of the decrease in patch size, systematically leading to mode lock-in. For example, with a standard deviation set to 500 MPa, the number of unstable cases for a large patch size was close to 50% while it became 100% when the size of the edge patch was less than 0.4 0.4 mm2. Indeed, at this critical patch size or correlation length, the pressure distribution became quasi-linear and the pressure distribution determined in the homogeneous case was found once again. Fig. 9 illustrates the contact pressure distribution with
3.4. Friction coefficient equal to 0.2 and 0.6 In the previous example, where the coefficient of friction had been fixed at 0.3, the homogeneous case was in mode lock-in and the fact of introducing heterogeneities led to stable case. In this case, the coefficient of friction was fixed at 0.2 and 0.6, which meant that the homogeneous case was in stable case. As in the previous example, a heterogeneous stiffness dispersion was introduced in the model using a Gaussian distribution where the standard deviation was a parameter. Fig. 10(a) shows the percentage of unstable cases computed according to the patch size for varying standard deviations. For a friction coefficient equal to 0.6, no unstable cases were found for any of the standard deviations or patch sizes. Indeed, even when the pressure distribution was modified as in the previous cases, this was not enough to switch the system into mode lock-in. Admittedly, the eigenfrequencies of the system were too far to hope to obtain a mode lock-in.
Fig. 9. Pressure distribution for different patch sizes
r ¼ 1000 MPa and l ¼ 0:3.
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Fig. 10. Number of unstable cases vs patch size with varying standard deviations.
4. Conclusions and perspectives This paper describes the development of a complex eigenvalue analysis of a pin-on-disc system including friction contact. The study has included a heterogeneous distribution of material properties to investigate the influence of the contact distribution on mode lock-in. The frictional contact pressure distribution was linked to the springs of the model for which the stiffnesses were directly deduced from mechanical properties. The material pad was considered as heterogeneous by introducing a Gaussian distribution. Different standard deviations and varying patch sizes were examined, and for each, 1000 random heterogeneity dispersions were computed to reach a probabilistic study using a Monte-Carlos algorithm. The results demonstrated that the introduction of heterogeneities in the contact surface played a key role on the lock-in modes. Indeed, when starting from an unstable or stable configuration with a small standard deviation (configuration close to the homogeneous case), the tendency was the complete opposite of that for large standard deviations. Indeed, with a large standard deviation, a localization pressure appeared, causing a complete change of the contact condition which in turn could alter the dynamics of the system. However, for a small patch size, there was a tendency to converge to the homogeneous pressure distribution even for a large standard deviation. Admittedly, for small heterogeneities, their dispersion causes them not to have an influence on specific areas. However, these conclusions were only valid when the eigenfrequencies were close enough. All simulations presented in this paper showed that the contact pressure distribution could influence the occurrence of squeal. In this configuration with different assumptions, the results show an existence of a size limit for the heterogeneities below which mode lock-in does not have any effect on this specific system. In the future, the results have to be reconciled with experimental observations to demonstrate the influence of contact heterogeneities on the occurrence of squeal. In order to introduce realistic distributions, data can be extracted from experimental nano/micro/macro-indentation. Finally, the semi-analytical model should be extended to consider a variation and an evolution of gradient properties of the contact surface using for instance thermal aspects during a complete brake. Acknowledgments The present research was financed by the International Campus on Safety and Intermodality in Transportation, the Nord-Pas-deCalais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education
and Research, and the National Center for Scientific Research. The authors gratefully acknowledge the financial support of these institutions.
References Akay, A., 2002. Acoustics of friction. J. Acoust. Soc. Am., 1525–1548. Alart, P., Lebon, F., 1998. Numerical study of a stratified composite coupling homogeneization and frictional contact. Math. Comput. Model. 28 (4–8), 273– 286. Bonnay, K., Magnier, V., Brunel, J.F., Dufrenoy, P., De Saxcé, G., 2011. Modélisation d’un contact frottant avec prise en compte d’une imperfection surfacique et influence sur le crissement. In: 20ème congrès franais de Mécanique, Besanon, France. Bush, A.W., Gibson, R.D., Thomas, T.R., 1992. The elastic contact of a rough surface. Wear 153, 53–64. Chang, W.R., Etsion, I., Bogy, D.B., 1987. An elastic-plastic model for the contact of rough surfaces. J. Tribol. 109, 257–263. Dimitriev, A.I., Osterle, W., Klos, H., 2008. Numerical simulation of mechanically mixed layer formation at local contacts of an automotive brake system. Tribol. Trans. 51, 810–816. Duboc, M., Magnier, V., Brunel, J.F., Dufrénoy, P., Rejdych, G., Chancelier, T., 2010. Influence of contact conditions and pad geometry on disc brake squeal noise. JEF2010, pp. 247–254. Greenwood, J.A., Williamson, J.B.P., 1966. Contact of nominally flat surfaces. Proc. R. Soc. London A 295, 300–319. Hoffmann, N., Fischer, M., Allgaier, R., Gaul, L., 2002. A minimal model for studying properties of the mode-coupling type instability in friction induced oscillations. Mech. Res. Commun. 29, 197–205. Jarvis, J.P., Mills, B., 1963. Vibration induced by friction. In: Procceeding in the Institution of Mechanical Engineers, vol. 178, pp. 847–857. Liu, P., Zheng, H., Cai, C., Wang, Y.Y., Lu, C., Ang, K.H., Liu, G.R., 2007. Analysis of disc brake squeal using the complex eigenvalue method. Appl. Acoust. 68 (6), 603– 615. Magnier, V., Duboc, M., Brunel, J.F., Dufrénoy, P., 2011. Influence of heterogeneous contact between disc and pad on the brake squeal noise. SAE Brake Colloquium. Massi, F., Giannini, O., Baillet, L., 2006. Brake squeal as dynamic instability: an experimental investigation. J. Acoust. Soc. Am. 120, 1388–1399. Mbodj, C., Renouf, M., Baillet, L., Peillex, G., Berthier, Y., Jacquemard, P., 2010. Influence du contraste des propriétés élastiques des composites carbone/ carbone (C/C) sous conditions de contact frottant. Mécanique et Industries 11, 289–294. Mills, H.R., 1938. Brake squeal, Technical report N9000 B, Institution of Automobile Engineers. Nguyen, V.D., Fortin, J., Guessasma, M., Bellenger, E., Cogne, C., 2009. Heat transfer modelling by discrete element method. J. Mech. Mater. Struct. 4, 413–426. North, M.R., 1976. Disc brake squeal-a theoretical model. Inst. Mech. Eng. C38 (76), 169–176. Oden, J.T., Martins, J.A.C., 1985. Models and computational methods for dynamics friction phenomena. Comput. Methods Appl. Mech. Eng. 52, 527–634. Oura, Y., Kurita, Y., Matsumura, Y., 2009. Influence of dynamic stiffness in contact region on disk brake squeal. J. Environ. Eng. 4, 234–244. Peillex, G., Baillet, L., Berthier, Y., 2008. Homogenization in non-linear dynamics due to frictional contact. Int. J. Solids Struct. 45, 2451–2469. Popov, V.L., Psakhie, S.G., 2007. Numerical simulation methods in tribology. Tribol. Int. 40 (6), 916–923. Renouf, M., Massi, F., Fillot, N., Saulot, A., 2011. Numerical tribology of a dry contact. Tribol. Int. 44, 834–844, Issues 78.
V. Magnier et al. / International Journal of Solids and Structures 51 (2014) 1662–1669 Richard, D., Iordanoff, I., Renouf, M., Berthier, Y., 2008. Thermal study of dry sliding contact with third body presence. J. Tribol. 130 (3). Spurr, R.T., 1961. A theory of brake squeal. Proc. Inst. Mech. Eng., 33–52. Trichês, M.J., Gerges, S., Jordan, R., 2008. Analysis of brake squeal noise using the finite element method: a parametric study. Appl. Acoust. 69 (2), 147–162. Wahlstrom, J., Soderberg, A., Olofsson, U., 2011. A cellular automaton approach to numerically simulate the contact situation in disc brakes. Tribol. Lett. 42 (3), 253–262.
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Contents lists available at ScienceDirect
International Journal of Solids and Structures journal homepage: www.elsevier.com/locate/ijsolstr
Impact of contact stiffness heterogeneities on friction-induced vibration V. Magnier ⇑, J.F. Brunel, P. Dufrénoy Université Lille Nord de France, F-59000 Lille, France Lille1-LML, F-59655 Villeneuve d’Ascq, France CNRS, UMR 8107, F-59655 Villeneuve d’Ascq, France
a r t i c l e
i n f o
Article history: Received 28 June 2013 Received in revised form 27 November 2013 Available online 14 January 2014 Keywords: Friction-induced vibration Complex modal analysis Heterogeneity stiffness contact Effect of heterogeneity size
a b s t r a c t It is well-known that the occurrence of squeal depends on numerous phenomena still relatively unknown. Furthermore, squeal is affected by several factors on both the micro and macro scales. The present paper proposes some potential interactions between stiffness heterogeneities of the contact surface and the potential for triggering the squeal. Consequently, an analytical model has also been developed to highlight the impact of certain parameters on the squeal phenomenon. The contact surface has been modeled by connecting the disc with the pad via distributed springs (contact stiffness). From this model, a static equilibrium and a complex modal analysis were performed to determine respectively the pressure distribution and the natural frequencies of the system. In this configuration, the results demonstrate that the introduction of heterogeneities could change the dynamic behavior of the system. Moreover, the influence of the size of the heterogeneities was studied and results show that this parameter had an influence on the likelihood of squeal occurrence. The paper also points out a heterogeneities correlation length that is reasonable to consider for this special configuration. Such information is relevant when interpreting the occurrence of noise as a function to the sizes of the components of the brake pad and the morphology of the surfaces in contact. Ó 2014 Elsevier Ltd. All rights reserved.
1. Introduction In the transport industry, there sometimes occurs a loud noise or high-pitched squeal as the brakes are applied. The latter can be of great discomfort to the human ear because of its high frequency of greater than 1 kHz (Akay, 2002). The origin of this unwanted noise is issued from unstable dynamic behavior of the brake system due to frictional forces. During the past few decades, a significant number of theories have been developed regarding this instability phenomenon involving friction-induced vibration: stick-slip (Oden and Martins, 1985), negative friction coefficient sliding velocity slope (Mills, 1938), sprag-slip (Spurr, 1961) and mode lock-in (North, 1976). Stick-slip comes about when there are friction issues, sprag-slip is related to geometric effects from coupling between systems with different degrees of freedom, and mode lock-in is caused by selfexcited vibrations. In numerous cases of squeal in braking systems, the instability is linked to the mode lock-in mechanism (Jarvis and Mills, 1963). These different theories are associated to several ⇑ Corresponding author at: Université Lille Nord de France, F-59000 Lille, France. Tel.: +33 (0)3 28 76 73 57; fax: +33 (0)3 28 76 73 61. E-mail address: [email protected] (V. Magnier). http://dx.doi.org/10.1016/j.ijsolstr.2014.01.005 0020-7683/Ó 2014 Elsevier Ltd. All rights reserved.
numerical models which can enable to predict the squeal phenomenon. On the one hand, a model based on the real geometry using a finite element method has been developed to investigate the effects of system parameters, such as the coefficient of friction between the pad and the disc (Trichês et al., 2008), or the stiffness of the disc (Liu et al., 2007) etc. On the other hand, minimal models or low-order finite element models have been used to gain understanding of the role of each parameter, essentially with an analytical approach (Hoffmann et al., 2002), or for control (Massi et al., 2006). Even though these models are relatively well understood at the system scale, all the studies cited above have been carried out with homogeneous and mostly isotropic materials. However, it is well-know that braking depends on multi-scale and multiphysics phenomena which are still unknown to a large extent. Indeed, squeal is an interdisciplinary issue involving dynamics, tribology, acoustics, etc. largely conducted separately. Furthermore, squeal is affected by many different factors on both the micro and macro scales. Phenomena at small scales when it comes to both length (microscopic contact effects) and time (high-frequency vibrations), affect and are affected by phenomena at large scales (wear, behavior of the tribological triplet and dynamics of the entire brake system). Nevertheless, studies do not necessarily demonstrate the impact of the various scales (third body, bodies
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in contact, components, heterogeneities etc.) on the likelihood of squeal occurrence. Nevertheless, various brake studies take into account heterogeneities using mainly single-scale approaches to describe wear, damage etc. For example, at the third body scale, various methods enable to describe the surface dynamics during friction and wear and can show the formation of self-affine surfaces as found in modeling of real surfaces such as with the discrete element method (Nguyen et al., 2009; Renouf et al., 2011; Richard et al., 2008) or cellular automata (Popov and Psakhie, 2007; Dimitriev et al., 2008; Wahlstrom et al., 2011). An other contact scale which can find concerns heterogeneities introduced by machining defaults, mounting defects, etc. To model this scale, one of the first models was that proposed by Greenwood and Williamson (1966) in which the elastic contact between a rigid plane and spherical asperities was considered by modifying the contact stiffness. These statistical models have been enriched by many authors, e.g., (Bush et al., 1992; Whitehouse and Archard, 1970) and provide fast and accurately responses. The deterministic approaches have been developed to introduce a better geometrical description using mathematical functions to represent the asperities. Elastic, perfectly plastic (Chang et al., 1987) or elastoplastic behaviors (Zhao et al., 2000) as well as interactions between asperities (Zhao and Chang, 2001) have been integrated in models. Also from a geometrical viewpoint, but at a larger scale, Bonnay et al. (2011) took into account a thickness variation of the disc and showed the influence of this perturbation on the mode lock-in. Another scale of heterogeneities concerns the components presented in the friction pad. Indeed, modern brake pads are fabricated from composites comprising many different components (structural materials, matrix, filler and frictional additives). These heterogeneities play a key role for the local behavior through the local properties of each component (stiffness, bulk modulus etc.). Mbodj et al. (2010) and Peillex et al. (2008) studied the mechanical properties of the heterogeneities and considered the pad material using a homogenization technique while (Alart and Lebon, 1998) mixed static frictional contact and heterogeneous materials using an asymptotic homogenization technique. Recently, Magnier et al. (2011) proposed an analytical model with a contact interface including stiffness heterogeneities between pad and disc and demonstrated the influence of these kinds of heterogeneities on mode lock-in. Consequently, the objective of this paper has been to present an analytical model including stiffness heterogeneity in the contact. This numerical model was a simplified pin-on-disc setup inspired by a experimental arrangement developed at the Laboratoire de Mécanique de Lille (France). Focus was put on the influence of the heterogeneities and their sizes on the likelihood of squeal occurrence via mode lock-in. 2. Semi-analytical model 2.1. Description The semi-analytical model was inspired by an experimental setup developed in our lab. This specific experimental arrangement was simplified with a reduced number of parts to control the dynamic behavior, and the model comprised a thin plate, a pad housing, a friction pad and a disc as shown in Fig. 1(a). In the model, the disc was expressed by a vibration system with one degree of freedom (translational direction, namely yd ) and the pad was expressed by a vibration system with two degrees of freedom (translational and rotational directions, namely respectively yp and /p ) as shown in 1(b). This analytical model corresponds to that developed by Oura et al. (2009) with the exception that the rotation was not at the center of the pad.
In the model, K 1 and K 2 represent the Y-direction stiffness on each side of the thin plate. The stiffness of the friction pad was modeled by a parallel distribution of springs, where the stiffness of each spring ki was directly dependent on the material’s mechanical properties, i.e., ki ¼ E:S with hi is the height of each spring and S hi is equal to the discretization-dependent area of each spring. So, this parameter physically represent the stiffness of the bulk. Initially, the pad material was considered to be homogeneous with a Young modulus E fixed to 3000 MPa. The pad was assigned a height (hi ) of 10 mm before deformation. Table 1 presents the parameter values that were used in the computation. This analytical model was first computed in static equilibrium under conditions of pressure and sliding. This step rendered it possible to obtain the contact pressure distribution that was injected. This was done in the second step of the computation, in a complex eigenvalues analysis to determine the natural frequencies of the system. The dynamic equations were as follows:
8 n X > > € M ¼ K y ki yi y > d d d d > > > i¼1 > > > n < X € € 1 1 M p y2 2 y1 ¼ K 1 y1 þ y2 y x1 K 2 y2 þ y2 y x2 ki yi d d > > i¼1 > > > n n > X X > > y y y y y€ y€ > ki yi lh þ ki yi l : J 2 d 1 ¼ K 1 2 d 1 x21 K 2 2 d 1 x22 þ i¼1
i¼1
In this study, only the out-of-plane bending mode of the disc was considered. The assumed disc vibration was represented by a model with a single degree of freedom. The equivalent mass M d was set to be equal to the mass of the disc. The disc stiffness K d was calculated using the eigenfrequency determined with the finite element method and M d . M p represents the total mass, including the pad and the pad-housing, and J is the moment of inertia calculated at the center of mass of the plate spring. The spring coefficients, K 1 and K 2 were determined from M p ; J and the numerical pad frequencies. These two springs enabled to model the thin plate and achieve the two pad modes. y1 and y2 represent the elongation of the K 1 and K 2 springs during translation of the pad, and d is the distance between these two springs. yi represent the elongation of each spring i. All these parameters are presented in Table 2 and were inspired by the experimental set-up. Finally, x1 (resp. x2 ) corresponds to the distance between the center of rotation of the pad and the spring K1 K 1 (resp. K 2 ). After a quasi-static solution under sliding conditions, a complex eigenvalues analysis was performed and the results were written as k ¼ a þ jx. Here, k represents the eigenfrequency and x the real part. If the real part is zero, the mode is considered as stable. Conversely, a non-zero real part indicates an unstable mode. 2.2. Illustration of a semi-analytical model considering a homogeneous stiffness distribution This section illustrates how to determine the evolution of the system’s eigenfrequencies in relation to the coefficient of friction and to identify the unstable configurations. Indeed, it has been well established that the coefficient of friction is a relevant parameter in instability (Massi et al., 2006). The pad-housing and the thin plate were considered to be of steel with mechanical properties set to 210000 MPa for the Young modulus and 0.3 for the Poisson coefficient. For the pad, a value of 3000 MPa was taken for the isotropic Young modulus. For the semi-analytical model, the pad contact area was set to 30 20 mm2 with a discretization of 300 200 elements. The extremities of the springs K 1 and K 2 were submitted to a vertical displacement of 0.5 mm to obtain an equivalent normal force of 300 N. In the x-direction, a translation of the disc was applied to simulate the slip condition.
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Fig. 1. Semi-analytic contact model.
Table 1 Parameters used for the static equilibrium. K 1 (N/m) 256 10
3
K 2 (N/m)
d (mm)
256 103
20
Table 2 Parameters used for dynamical analysis. M d (kg) 3:88
K d (N/m) 4:85 10
Mp (kg) 8
0:07
K p (N/m) 6 10
5
J (kg.m2) 4:43 106
To illustrate the strategy of resolution, the analysis was carried out by ranging the values of the friction coefficient from 0.1 to 1. Fig. 2 shows the contact pressure distribution obtained by the static resolution with a coefficient of friction l fixed at 0.3. A non-uniform contact pressure distribution was obtained with a maximal value at the leading edge for all models. It can be noted that one spring along the thickness of the pad linearized the contact pressure distribution. Indeed, this result was due to the linear interpolation along the thickness pad direction. This semi-analytical model is considered reliable as compared to finite element calculations (Magnier et al., 2011). It is clear that the obtained contact pressure distribution is unlikely to be representative of a real case but it enables to simplify results in order to understand the role of each parameter. The advantage of this semi-analytical model is a reduced time computation which can render possible a fine discretization. The second step of the computation consists in carrying out a complex eigenvalue analysis for the determination of unstable modes. The four nodal diameter disc modes was the only one
Fig. 2. Pressure distribution for a 30 20 mm2 surface contact with
l = 0.3.
considered in this study. The analysis was performed by ranging the values of the coefficient of friction from 0.1 to 1 while retaining the apparent contact length at 30 mm. The results of this step are plotted in Fig. 3. The eigenfrequency associated with the translation mode (blue curve) of the disc was slightly modified by the coefficient of friction. On the other hand, the eigenfrequency associated with the tilting (rotation around the z direction) mode of the pad (yellow curve) and the translation mode of the pad (red curve) was strongly affected by the contact surface. For a constant apparent contact length equal to 30 mm, the frequency of the tilting pad modes decreased as the value of coefficient of friction increased. This was mainly due to the diminution of the effective contact length induced by the coefficient of friction (Duboc et al., 2010). The tilting mode of the pad and the axial mode of the disc coalesced, which resulted in a coefficient of friction comprised between 0.3 and 0.4, and led to a mode lock-in mechanism and to an unstable mode being obtained with the real part different from zero. The eigenfrequencies associated with a friction coefficient equal to 0.3 were 7714 Hz for the translation pad mode and 4326 Hz for the 4-diameter disc mode and tilting pad mode. Thus, in this last case, where the pad was considered homogeneous, the system was unstable which could lead to squeal noise. 3. Introduction of a heterogeneous stiffness distribution 3.1. Procedure for the introduction of heterogeneous stiffness An extension of the previous case was investigated by introducing a distribution of heterogeneous stiffnesses with a Gaussian
Fig. 3. Mode coupling between the tilting pad mode and the 4-diameter disc mode (apparent contact length = 30 mm).
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by a simple Gaussian distribution. But, this simple function distribution enables us to understand the role of the heterogeneities of the stiffness. ðxgÞ2 1 f ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi e 2r2 : 2 2pr
Fig. 4. Gaussian function centered at 3000 MPa.
Fig. 4 presents a graph of the probability density function f ðxÞ. Several standard deviation values r ranging from 1 to 1000 MPa in the normal distribution have been used. For a fixed standard deviation r, the Young modulus varied from ð3000 3rÞ MPa to ð3000 þ 3rÞ MPa. In the following, two parameters were studied: the first was the standard deviation and the second was the size of the heterogeneities, ranging from 1 lm to 5 mm for the edge size. For each standard deviation and fixed patch size, one thousand random tests with a random dispersion of the Young modulus we are computed to establish a probabilistic study on the confusing modes. The Monte-Carlo method was used to obtain a stochastic approach. 3.2. Effect of the standard deviation
Fig. 5. Number of unstable cases vs. standard deviation for equal to 1,5 1,5 mm2.
l = 0.3 for a patch size
function f ðxÞ centered around the Young modulus of the homogeneous case (i.e., g = 3000 MPa). It is clear that in a real configuration, the distribution of stiffness probably cannot be represented
In this part of the study, the coefficient of friction was fixed at 0.3, the patch size was set to 1.5 1.5 mm2 and the standard deviation was ranged from 1 to 1000 MPa. Fig. 5 shows the number of unstable cases for each standard deviation. For a fixed patch size, when the standard deviation was close to 0 MPa, the results showed no influence on the dynamic aspect and the system was systematically in mode lock-in. This corresponds to the initial case where the material was considered as homogeneous. However, for a standard deviation greater than 60 MPa, some stable cases were systematically found. Further, the number of unstable cases (i.e., in stable case) increased with the standard deviation. For example, for the ultimate case where the standard deviation equaled 1000 MPa, 30% of cases led to mode lock-in which was the complete opposite of the tendency obtained in the homogeneous case. Thus, to explain this phenomenon, Fig. 6
Fig. 6. Pressure distribution with different standard deviation.
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show the contact pressure distribution associated respectively to a standard deviation of respectively 10 MPa, 500 MPa and 1000 MPa. Indeed, for small standard deviations close to 0 MPa, the pressure distribution was quasi-linear, just as in the homogeneous case (Fig. 2). But for a standard deviation greater than 60 MPa, the pressure distribution deviated from quasi-linearity and the pressure localization become influent. Indeed, the pressure localization was essentially concentrated to a patch where the stiffness was significant with a high level of pressure. The dispersion of the frequency of translation pad and the frequency in of the disc are shown Fig. 7 respectively for a variance fixed to 50 MPa, 500 MPa and 1000 MPa. It can be noted that the reference case with homogeneous material is placed on each figure. It can be noted that on 1000 random simulated cases treated with the variance equal to 1000 MPa, the frequency of translation pad mode is ranging from 7568 Hz to 7830 Hz; the frequency of the 4 diameter disc mode is ranging to 4286 Hz to 4387 Hz and finally the frequency associated to the tilting mode
is ranging from 4035 Hz to 4387 Hz (Fig. 7). The eigenfrequency associated to the translation mode of disc is slightly modified by the heterogeneities dispersion. Unlike, the eigenfrequencies associated to the tilting mode of the pad and the translation mode of the pad are more affected by the heterogeneities dispersion. Consequently, the results presented in this part were obtained for a patch size fixed at 1.5 1.5 mm2. A study dedicated to varying this parameter is discussed in the next section. 3.3. Effect of the size of the heterogeneities This section discusses the effect of the size of the heterogeneities, ranging from 1 1 l2 to 5 5 mm2 with a friction coefficient fixed at 0.3. Fig. 8(b) shows the percentage of unstable cases as a function of the patch size with different values of standard deviation (r = 3, 50, 100, 300, 500, 700, 1000 MPa). At first, for a fixed patch size, the increase in standard deviation enabled to avoid the homogeneous case and to increase the possibility of stable
Fig. 7. Dispersion of frequencies with different variance.
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a standard deviation set to 1000 MPa for different patch sizes (2 mm, 0.2 mm and 0.005 mm). The figure shows that the pressure localizations were more important when the patch size was large. In this configuration, the localization pressure was essentially concentrated on patches where the stiffness was significant, with a high level of pressure close to 1.8 MPa as opposed to 1.2 MPa for the homogeneous case. When the patch size decreased, the phenomenon of localization pressure was less important and the pressure distribution corresponded to that obtained in the homogeneous case. Indeed, the randomness of the stiffness dispersion enabled to avoid localization pressure and to evenly distribute the pressure on the global contact surface.
Fig. 8. Number of stable case vs patch size with different standard deviation.
cases. Indeed, for a low standard deviation, the pressure distribution converged to a quasi-linear pressure distribution and did not permit to completely change the dynamic behavior. However, when the standard deviation increased, the location of patches pressure are dominant in the system focuses on patches whose rigidity is more important. Secondly, for a fixed standard deviation, there was an increase of unstable cases as a function of the decrease in patch size, systematically leading to mode lock-in. For example, with a standard deviation set to 500 MPa, the number of unstable cases for a large patch size was close to 50% while it became 100% when the size of the edge patch was less than 0.4 0.4 mm2. Indeed, at this critical patch size or correlation length, the pressure distribution became quasi-linear and the pressure distribution determined in the homogeneous case was found once again. Fig. 9 illustrates the contact pressure distribution with
3.4. Friction coefficient equal to 0.2 and 0.6 In the previous example, where the coefficient of friction had been fixed at 0.3, the homogeneous case was in mode lock-in and the fact of introducing heterogeneities led to stable case. In this case, the coefficient of friction was fixed at 0.2 and 0.6, which meant that the homogeneous case was in stable case. As in the previous example, a heterogeneous stiffness dispersion was introduced in the model using a Gaussian distribution where the standard deviation was a parameter. Fig. 10(a) shows the percentage of unstable cases computed according to the patch size for varying standard deviations. For a friction coefficient equal to 0.6, no unstable cases were found for any of the standard deviations or patch sizes. Indeed, even when the pressure distribution was modified as in the previous cases, this was not enough to switch the system into mode lock-in. Admittedly, the eigenfrequencies of the system were too far to hope to obtain a mode lock-in.
Fig. 9. Pressure distribution for different patch sizes
r ¼ 1000 MPa and l ¼ 0:3.
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Fig. 10. Number of unstable cases vs patch size with varying standard deviations.
4. Conclusions and perspectives This paper describes the development of a complex eigenvalue analysis of a pin-on-disc system including friction contact. The study has included a heterogeneous distribution of material properties to investigate the influence of the contact distribution on mode lock-in. The frictional contact pressure distribution was linked to the springs of the model for which the stiffnesses were directly deduced from mechanical properties. The material pad was considered as heterogeneous by introducing a Gaussian distribution. Different standard deviations and varying patch sizes were examined, and for each, 1000 random heterogeneity dispersions were computed to reach a probabilistic study using a Monte-Carlos algorithm. The results demonstrated that the introduction of heterogeneities in the contact surface played a key role on the lock-in modes. Indeed, when starting from an unstable or stable configuration with a small standard deviation (configuration close to the homogeneous case), the tendency was the complete opposite of that for large standard deviations. Indeed, with a large standard deviation, a localization pressure appeared, causing a complete change of the contact condition which in turn could alter the dynamics of the system. However, for a small patch size, there was a tendency to converge to the homogeneous pressure distribution even for a large standard deviation. Admittedly, for small heterogeneities, their dispersion causes them not to have an influence on specific areas. However, these conclusions were only valid when the eigenfrequencies were close enough. All simulations presented in this paper showed that the contact pressure distribution could influence the occurrence of squeal. In this configuration with different assumptions, the results show an existence of a size limit for the heterogeneities below which mode lock-in does not have any effect on this specific system. In the future, the results have to be reconciled with experimental observations to demonstrate the influence of contact heterogeneities on the occurrence of squeal. In order to introduce realistic distributions, data can be extracted from experimental nano/micro/macro-indentation. Finally, the semi-analytical model should be extended to consider a variation and an evolution of gradient properties of the contact surface using for instance thermal aspects during a complete brake. Acknowledgments The present research was financed by the International Campus on Safety and Intermodality in Transportation, the Nord-Pas-deCalais Region, the European Community, the Regional Delegation for Research and Technology, the Ministry of Higher Education
and Research, and the National Center for Scientific Research. The authors gratefully acknowledge the financial support of these institutions.
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