PREPRINT TISON T., HEUSSAFF A., MASSA F., TURPIN I., NUNES R-F. (2014). Improvement in the predictivity of squeal simulations: uncertainty and robustness, Journal of Sound and Vibration, 333(15):3394–3412 http://dx.doi.org/10.1016/j.jsv.2014.03.011
Improvement in the predictivity of squeal simulations: uncertainty and robustness T. Tison(a,b,c) *, A. Heussaff(a,b,c), F. Massa(a,b,c), I. Turpin(a,d), R.F. Nunes(e) (a) PRES Univ Lille Nord de France, F-59000 Lille, France (b) Univ Valenciennes, LAMIH, F-59313 Valenciennes, France (c) CNRS, UMR 8201, F-59313 Valenciennes, France (d) Univ Valenciennes, LAMAV EA 4015, F-59313 Valenciennes, France (e) Daimler AG, Development Mercedes-Benz Passenger Cars, Sindelfingen, Germany Abstract The objective of this paper is to improve the predictivity of squeal simulations by introducing uncertainty and robustness concepts during simulations. Complex eigenvalue analysis is a traditional way to detect numerically the unstable modes that can be associated with extensive vibration and noise pollution. This simulation, for which associated computational times are compatible with the design phase, is known to be insufficiently predictive. We first propose a complete strategy that relies on the integration of random fields into the contact interface, complex eigenvalue calculations, probabilistic analysis and a robustness criterion. Next, this strategy is applied to study the instabilities of a complete industrial brake system. Experimental comparisons highlight the efficiency of the improved squeal detection methodology. Keywords: Squeal, Complex eigenvalue analysis, Uncertainty, Random fields, Robustness.
1.
Introduction
Brake noise issues have been investigated since the early 1930s. Yet, they are still barely understood today. According to J.D. Power and Associates surveys published in 2002, more than 60% percent of warranty claims concerning the brake system are due to brake noise. Higher consumer expectations make brake noise a priority topic. Friction induced vibration is the reason behind much of the noise pollution caused by brake noise. The most types of brake noise most frequently cited in the literature are groaning (frequencies under 100 Hz), moaning (100-500 Hz) and squealing (1-16 kHz). In order to avoid these instabilities, the industry has specific tools for performing efficient optimization. To numerically solve this type of problem, two analysis method categories based on the finite element method can be performed: a Complex Eigenvalue Analysis (CEA) [1], in which system equations are linearized around a stationary state whose stability is evaluated according to Lyapunov’s theory, or a nonlinear transient dynamic analysis [2, 3], which can be used to study the time evolution of mechanical solutions (e.g., velocities, * Corresponding author. Phone: +33 327511459, Fax : +33 327511317 E-mail addresses:
[email protected] (T. Tison)
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PREPRINT TISON T., HEUSSAFF A., MASSA F., TURPIN I., NUNES R-F. (2014). Improvement in the predictivity of squeal simulations: uncertainty and robustness, Journal of Sound and Vibration, 333(15):3394–3412 http://dx.doi.org/10.1016/j.jsv.2014.03.011
loads, contact node status). CEA requires less computational time. With appropriate computing facilities, a CEA can be of the order of one hour, even for a complete brake system,
* Corresponding author. Phone: +33 327511459, Fax : +33 327511317 E-mail addresses:
[email protected] (T. Tison)
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whereas transient analysis requires systematically many more than a day or can be unreachable for such systems. Therefore, CEA is more frequently used by industrials and is coupled with sensitivity and optimization tools. Nevertheless, the CEA method, which is characterized as a global approach, is known to be insufficiently predictive and highly dependent on a static equilibrium position, whereas nonlinear transient analysis can be used to study phenomenon more locally [1, 4]. For the latter method category, Vermot des Roches [2] proposes an interesting reduction technique associated with a modified nonlinear Newmark scheme. This method, which leads to a very compact model, was applied during sensitivity studies required for the development stage of an industrial brake system. However, brake squeal is a highly fugitive phenomenon [5, 6], and this variability is what makes brake squeal so difficult to predict and eliminate [6]. Furthermore, the environmental and operational conditions conducive to squeal are difficult to reproduce (temperature, pressure, excitation or braking history) and manufactured systems do not always have the same characteristics. To account for this variability and uncertainty, nondeterministic theories and methods that can be divided into two categories have been proposed: probabilistic theories [7, 8] and non-probabilistic theories (interval [9], convex [10] and fuzzy approaches [11]). From a probabilistic point of view, Nechak et al. [12] and Sarrouy et al. [13] recently proposed to introduce uncertainty into stability analysis using a polynomial chaos approach to analyze, for example, the effect of a friction coefficient on unstable modes. For non-probabilistic formalisms, Culla and Massi [5], and then Gauger et al. [14], used fuzzy formalism and a combinatory approach to evaluate respectively the influence of uncertainties on the modal behavior of a beam/disc model and an industrial brake model. The uncertainties were localized to the Young modulus and the friction coefficient of brake pads. The uncertain nature of the phenomenon is one of the main reasons why friction induced instabilities remain tricky to study. In this paper, a complete strategy based on integrating random fields at the contact interface, eigenvalue calculations, probabilistic analysis and a robustness criterion is proposed to improve the predictivity of squeal frequency simulations in a design phase. The aim is to build different families or classes of unstable modes and to study their robustness. A classification of each family’s participation in squeal is proposed. First, previous research on this topic and equations governing the stability problem are reiterated in Section 2. Then, a deterministic stability analysis of an industrial brake system is performed in Section 3. Numerically unstable modes are compared with those observed experimentally. Finally, a complete strategy, called Improved Squeal Detection Methodology (ISDM), is described and applied respectively in Sections 4 and 5. Conclusions and remarks about this work are given in Section 6. 2. Simulation of squeal frequency instabilities 2.1. Mode coupling phenomenon For many years, a decreasing coefficient of friction with increasing sliding velocity was the only explanation for what triggers brake squeal. Thereafter, engineers and researchers suggested that structural mechanisms may also be root causes of friction induced vibration. Today, many theories and models are available in the literature. All these models, which come from various domains, aim to explain the root causes of friction related dynamic instabilities and to predict the conditions under which oscillation may occur. For example, one of the most famous theories is the stick-slip. Observations of the oscillation of the system reveal the existence of two distinct phases. The first phase presents no relative motion of the sliding bodies and is known as sticking. The second phase shows the
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relative motion of the bodies in contact and is known as sliding. Moving back and forth between these two states leads to stick-slip oscillation. Otherwise, in the sprag-slip theory, vibrations are caused by multiple degrees-of-freedom interactions. The sprag situation leads to component buckling. After the components have been sufficiently deformed, they come loose and return to the original contact situation. Spragging in an elastic system can lead to sprag-slip limit cycles. This structural mechanism highlights the fact that friction induced vibration can be predicted without a decreasing coefficient of friction. Finally, most articles consider brake squeal to be vibration caused by mode coupling phenomena, which is also discussed in this paper. North developed a two degrees-of-freedom model [15], which considered not only the geometrical characteristics of brake components but also the characteristics of the friction interface. Millner extended [16] North’s model and developed a six degrees-of-freedom model for a fixed caliper disc brake system. He observed that brake squeal triggering is dependent on friction coefficients, the mass and stiffness parameters of the brake assembly and the contact properties of the pad-piston interface. Writing the equations of motion of these systems leads to an eigenvalue problem and adding friction to the system leads to the coupling of some degrees of freedom. If an eigenvalue has a positive real part, i.e., , the system becomes unstable and its response to a perturbation is exponentially increasing oscillation. This mechanism is referred to as mode coupling and leads to flutter instability. Roots of couplings are assumed to be stiffness matrix asymmetries resulting from the existence of follower forces. Today, numerous publications on the frequency analysis of instability can be found in the literature. One cited the works of Fritz et al. [17], which studied the flutter instabilities of a finite element model for a whole automotive brake corner. A complex eigenvalue analysis of the system is undertaken to assess brake stability as a function of the friction coefficient. Different kinds of damping spreading over the modes have been studied and Fritz pointed out that if the ratio of damping between two modes is sufficient, an increase in damping tends to make the brake unstable for a lower friction coefficient value. Lorang et al. [18] studied the squealing behavior of a TGV brake system considering the asymmetry of both the damping and stiffness matrices by means of a complex eigenvalue analysis of the problem. They found good agreement between the numerical analysis and the experiments. However, they stated that the phenomenon is not yet completely explained. Duffour and Woodhouse [19] investigated the linear stability of a pin-on-disc, which contains sliding frictional contact at a single point. Contrary to traditional complex eigenvalue approaches, Duffour and Woodhouse studied the stability of the system and the mode coupling phenomenon in terms of transfer functions. Like Fritz et al., they observed that damping can cause new unstable regions of the parameter space to appear. These frequency analyses can be completed through the calculation the limit cycle to determine the amplitude of instability by performing analyses at the Hopf bifurcation point. Due to its ease of implementation and its good results, the Harmonic Balance Method (HBM) is an interesting method to consider [20]. The next subsection recalls the main equations governing stability by considering modecoupling phenomena. 2.2. Stability problem In the context of finite element models, the equation governing the dynamics of the system is as follows: ̈
̇
(1)
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where , , are respectively the mass, damping and stiffness matrices of different components, ̈ , ̇ and the acceleration, velocity and displacement vectors of nodal quantities, and and the nonlinear contact loads dependent upon displacement at the contact interface and the external load respectively. The nonlinear contact loads can be broken down into two contributions, noted and , associated with normal and tangential contact effects: (2) The stability analysis is broken down into two calculations. The first one considers a nonlinear static analysis defined by Eq. (3): (3) where is the contact load dependent upon static equilibrium displacement. The aim is to determine the static equilibrium position corresponding to the application of external loads. In the case of a brake system, this step corresponds to the application of pressure on pads by the pistons of the brake system. Only the normal contact loads are calculated. The tangential contact loads are determined using the normal contact loads by considering a permanent sliding state and are expressed by Eq. (4): (4) where and are respectively the constant friction coefficient and the relative sliding velocity vector between the bodies in contact, which is determined by considering the rotation of the brake system disk. Then, Eq. (1) is linearized around the static equilibrium position, calculated previously using a perturbation technique, such as: ̅
(5)
where ̅ is a small perturbation of static displacement vector By inserting Eq. (5) into Eq. (1), we obtain: ̅̈
̅̇
.
̅
̅
(6)
The nonlinear contact loads can be developed as a first order Taylor series, as defined in Eq. (7): (
̅)
(
)
(
)̅
(7)
where ( ) is the linearized nonlinear load Jacobian matrix. Eq. (6) can be rewritten by introducing Eq. (7) as follows: ̅̈
̅̇
̅
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(
)̅
(8)
Finally, Eq. (1) is approximated using the linearized system defined in Eq. (8). The solutions of this equation are highly dependent on the static equilibrium position, and thus on the contact conditions. Matrix ( ) integrates the effect of friction and is asymmetric. To define the state space Eq. (10), one can use the following trivial identity Eq. (9): ̅̇
̅̇
(9)
̇ with
̅ [ ̇ ], ̅
[
] and
(10) (
[
)
]
The second step involves performing the so-called Complex Eigenvalue Analysis (CEA). The complementary solution to the homogenous, second order, matrix differential equation Eq. (8) is in the form of (11) Substituting Eq. (11) into Eq. (10), an associated complex eigenvalue problem can be defined: (
)
with
(12)
where and are respectively the kth complex eigenvalue and eigenvector, n is the number of degrees of freedom of the FE model. To ensure the uniqueness of eigenvectors, a normalization condition is added: (13) To solve this complex eigenvalue problem, the first step is to extract the eigenvectors of the symmetric undamped system in order to create a subspace onto which the original asymmetric damped system can be projected. Next, the reduced problem is solved using the QZ method or residue method [2]. The real parts of the eigenvalues allow the damping rates to be quantified while the imaginary parts indicate the mode frequencies. The stability of solutions is studied in the Lyapunov sense. Unstable modes present positive real parts while stable modes present negative real parts. It is important to remember that this analysis does not provide the vibration amplitude of the system, but rather only the potential unstable frequencies. In the next section, a stability analysis is performed on an industrial brake system using a CEA. 3. Stability analysis of an industrial brake system 3.1. Description and validation of a numerical model A finite element model of a complete automotive brake corner was created on Abaqus. The system, shown in Figure 1, consists of a vented nonperforated rotor, a brake system (fixed calipers, pistons and linings) and a front axle system (hub, knuckle, steering, lower and upper arms). The model is made of approximately 460,000 nodes and 320,000 elements, i.e., has
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approximately 4 million degrees of freedom. The finite element models of the axle components and of the calipers are made of modified second order (quadratic) interpolation continuum solid elements while the rotor, the pistons, the pads and the wheel hub are made of first order (linear) interpolation continuum solid elements. Moreover, all the mechanical contacts are modeled using the Lagrange and Augmented Lagrange methods. Surface-to-surface formulation and small sliding conditions are generally adopted (eg. pads/caliper and pads/pistons interfaces). Only the pad-to-disc interface is modeled using the node-to-surface formulation and the small sliding condition. The pressures on the pistons are represented by a uniform load. The boundary conditions are modeled using spring elements and kinematic coupling constraints.
Fig. 1. Finite element model of the automotive brake corner
The main components of the corner assembly were measured separately by means of standard experimental modal analysis using three dimensional scanning laser vibrometry. Preliminary measurements of the frequency range of interest led to a measurement bandwidth of 4 kHz and a frequency resolution of 1.25 Hz. Above this frequency it became difficult to obtain accurate estimates of the frequency response functions. Regarding the linings, the modal testing was performed up to 10 kHz. The numerical and experimental modes of each brake corner component were paired using a minimum Modal Assurance Criterion (MAC) of 80% in order to calibrate and validate each associated finite element model independently. The Elastic moduli of each component were corrected in order to minimize the root mean square of the relative frequency deviations. Except for the linings, the material properties were modeled using elastic isotropic models, whose final parameter values were found by means of gradient based optimizations. The densities of the components were experimentally determined. The linings were modeled using a fully anisotropic formulation. Hence, ultrasonic measurements [21] were first performed to define the lining shear and tensile moduli.
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Table 1 Frequency deviation and percentage of paired modes for each component.
Main components (Name)
Percentage of paired modes (with MAC>80%)
Frequency deviations (Root mean square)
Knuckle
86%
2.81%
Lower arm
100%
5.00%
Caliper
100%
0.97%
Disc
50%
0.81%
Pads
100%
3.40%
Tension rod
100%
1.01%
Table 1 gives, after updating and for each component of the brake corner, the percentage of paired modes with MAC values superior to 80% as well as the root mean square of the relative frequency deviation. Validation was achieved up to 4 kHz for the main components and 10 kHz for the brake linings. The single components validation shows fairly good results with a mean frequency deviation of 2.34%, a maximal deviation of 11.5% for the lower arm at 1400 Hz and a mean value of 89% for paired modes for the six measured components of the automotive brake corner. For the disc, measurement points were concentrated on the friction surface and only out of plane modes were considered. These are the major reasons why only half of the number of modes shape of the disc has been correlated with MAC values superior to 80%. The validation of the upper arm was not achieved due to noisy measurements which did not enable satisfactory identification to be achieved. In order to identify the modal properties of the complete automotive brake corner assembly, partial experimental modal analyses of the axle system alone and of the mounted brake disc were performed. The system was mounted on a frame. The first analysis did not yield usable results. Extensive degradation of the coherence functions and of the repeatability of the measurements was observed. The second analysis of the mounted brake disc was performed using 296 measurement points mainly concentrated on the disc and the caliper and was achieved up to 3 kHz. The system was excited by means of an electro-dynamical shaker (the excitation point is situated on the inner cheek of the disk), periodic chirp and pseudo random signals. Regarding the coherence function, the reproducibility of the measurements and the results of the cross correlation analysis, the results of this second analysis were satisfactory and can be used to perform a partial validation of the model. A minimum modal assurance criterion of 70% was assumed for the modal matching. Of the studied frequency band, 33% of the experimental modes were paired with a root mean square of the relative frequency deviations of about 12%. Figure 2 represents the experimental and numerical mode shapes at 1820.6Hz and 1803.4Hz respectively and Table 2 represents a comparison of the frequencies of the seven paired mode shapes of the mounted brake disc only.
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Fig. 2. Comparison of mode shapes of the mounted disc at 1.8 kHz: (a) experimental and (b) numerical. Table 2 MAC and Frequency errors of the seven paired modes.
Experimental frequencies (Hz)
Numerical Frequencies (Hz)
MAC (%)
Frequency errors (%)
542.40
615.90
78
13.6
551.30
652.00
70
18.3
658.50
774.70
72
17.6
668.40
800.70
80
19.8
1151.00
1022.20
90
11.2
1820.60
1803.40
88
0.9
2496.70
2651.70
85
6.2
Despite the fact that the validation of the models of the main components did not show any problems, the assembly composed of numerous joints presenting backlashes and nonlinear behavior make the validation of the automotive brake corner more difficult. In addition, a detailed analysis of the boundary conditions has shown the strong influence of the frame, which is at the root of the low correlation. However, the results of the modal analysis of the mounted disc led to an acceptable partial validation of the brake assembly. 3.2. Dynamometer results The squealing behavior of the brake assembly was studied on a dynamometer using a squeal noise test matrix. The squeal noise test matrix simulates real braking conditions and covers four different types of lining wear that represent typical states of contact evolution (new linings, brake linings of normal consumers, linings used in sport braking conditions and
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temperature induced lining fade). The dynamometer runs consist of approximately 2,000 real braking conditions (500 for each type of lining wear) with a temperature range of 50°C to 550◦C (for the fade section), a caliper pressure range of 0 to 160 bar and a vehicle velocity range of 0 to 100 km/h. Brake squeal acquisition was performed using microphones and accelerometers. The microphone used for the measurements is located 10 cm outboard from the wheel hub face along the centerline of the axle and 50 cm above and perpendicular to the centerline of the axle. Sounds that exceed a minimum sound pressure level threshold of 70 dB (A) are defined as squeal. Figures 3 and 4 present the results of the squeal noise matrix test. The scatter plots represent the maximum sound level and squeal frequency recorded during the braking events while the line plots represent the temperature of the disc. From these figures, it can be seen that the squeal behavior of the brake is weakly influenced by the type of lining wear. In particular, brakes show a highly robust squeal at 1850 Hz.
Fig. 3. Squeal noise matrix results: sound pressure level (dot), temperature (line).
Fig. 4. Squeal noise matrix results: squeal frequencies at the maximal sound amplitude (dot), temperature (line).
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Fig. 5. Acquisitions during squealing: (a) sound pressure and (b) acceleration.
Figure 5 shows sound pressure and acceleration acquisitions during squealing with easily discernible main frequencies. The main squeal frequency is found at 1850 Hz with 1,355 occurrences over 70 dB. Of these 1,355 occurrences, 1,018 are over 100 dB. Two secondary frequencies are found at 3250 Hz and 3650 Hz. 3.3. Discussion on system instabilities Following the experimental tests, a stability analysis of the automotive brake corner was conducted according to the methodology described in section 2. The analysis was performed using the nominal (or deterministic) updated model. In this case, the modes are computed using perfect surfaces and nominal values for all model parameters. The nominal model exhibits only two instabilities at 3.2 kHz and 4.4 kHz and does not present any instability at approximately 2 kHz, which is dominant on the dynamometer. This result highlights a well known complex eigenvalue analysis situation: the method and the associated model correspond to a simplification of the real situation (i.e., braking) that can lose some potentially unstable modes and fail to discriminate computed ones. Current transient analyses suffer from the same real braking simulation limitations but are generally more predictive [3]. Nevertheless the transient simulation has the disadvantage of being very time consuming, especially given the size of the model presented above. As a result, the CEA is mostly used by the industry because its low numerical cost allows for an efficient combination of sensitivity and optimization analyses to be performed. In conclusion for this third section, table 3 summarizes the most prominent instabilities shown by both experimental and numerical deterministic studies. Table 3 Deterministic studies: most prominent unstable frequencies
Dynamometer results
Deterministic numerical results
1850, 3250 and 3650 Hz
3200 and 4400 Hz
In the following section, we propose a new strategy to enhance the predictivity of CEA.
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4. Improved Squeal Detection Methodology 4.1. Influence of surface properties The impact of surface properties on brake squeal phenomena is clearly known and already published in the literature. Sherif [22] studied the impact of the topography of pad-to-disc interfaces on the appearance and disappearance of brake squeal. Sherif was one of the first to introduce the notion of contact stiffness into the domain of sliding systems. Contact stiffness is defined as the result of the deformation of asperities at the contact interface. Sherif studied the impact of this parameter on squeal triggering and concluded that friction induced instabilities are clearly linked to contact stiffness. In the 1990’s, Eriksson [23] focused on the relationships between the evolution and state of the contact interface and the appearance of squeal. The author was a pioneer in introducing and investigating the nature of contact plateaus. He explained that plateaus are made of two main parts: a primary plateau and a secondary plateau. The primary plateau is made of structural pad components and the secondary plateau is an accumulation of other elements (e.g., wear particles) under the first plateau (e.g., the pad fiber). The author showed that the characteristics of the real contact surface (limited to these plateaus) considerably affect squeal. The formation and degradation of contact plateaus is shown to be a very rapid local process that enables a shape adaptation of the contact interface at a microscopic level and leads to extensive local deformations, stresses and variations of the local tribological properties of the sliding surfaces. Chen [24] carried out experiments using flat-flat and flat-ball contacts and identified four surface topography states. He explored the influence of surface topography on squeal recurrence and concluded that surfaces with squeal are distinctly different from those without squeal. Areas with squeal are characterized by adhesively joined asperities, while areas without squeal are characterized by abrasive wear topography. Moreover, areas where squeal is present are rougher than those in which squeal has not occurred. On a more macroscopic scale, Bergman [25] studied the squeal of an industrial brake system whose disc was shot-blasted with small particles (causing evenly distributed, superficial, 100-μm wide pits). Bergmann concluded that no squeals are generated when the coefficient of friction is below a critical value and that the coefficient of friction must depend on other parameters, such as pad and disc geometry, state of wear, stiffness or frictional properties at both macroscopic and microscopic levels. Likewise, Ahmed [26] investigated the influence of macroscopic contact interface using slotted and non-slotted pads as well as coated and non-coated discs. In order to reduce brake squeal, Hammerstroem [27] proposed to induce spiral shaped modifications to the brake disc surface topography. The results showed a significant reduction in noise generation. By using lumped models, Rusli [28] investigated the effects of the topography of the sliding surfaces on the normal and tangential contact stiffness and on self-excited vibration. He made the same observations as Sherif. Hetzler [29] recently presented statistical modeling of the contact interface and its impact on brake squeal. The interface was modeled after Greenwood and Williamson using half-space theory and the principle of minimum complementary potential energy. Brake squeal was investigated on a minimal model (brake pad and disc). Hetzler showed that contact stiffness has a significant influence on the vibrational behavior of the system. Oura [30] studied the distribution of the contact stiffness of a simplified brake system. He concluded that the pressure dependency of the contact stiffness is a cause of squeal. Abu Bakar [31] modeled a partial brake assembly and Söderberg [32] a full brake automotive corner assembly in order to study the evolution of the pad-to-disc interface with different loads and operating conditions (wear state evolutions). Their methods simulate only steady state pressure distribution conditions and wear was defined according to generalizations
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of the well-known Archard’s wear law. The Abu Bakar model takes into account a realistic representation of the initial surfaces of pads at a macroscopic level. Söderberg used a simpler model with a flat surface definition for the pads and an isotropic material definition for the lining. Abu Bakar experimentally and numerically studied the influence of wear on squeal triggering. He interpreted the fugitive nature of brake squeal as a permanent evolution of the topography of the contact interface. He concluded that the evolution of pressure distribution is an explanation for the evolution of squeal level and frequency. In order to complete these investigations, Yue [33] studied the influence of secondary contact interfaces, i.e., outer pad-tocaliper, inner pad-to-piston and pad-to-abutment contact areas, and showed that they can also significantly affect brake squeal and are found to be critical in both low and high frequency squeals at low or high brake pressures. 4.2. Modification of equilibrium position Although frequency stability computations supply important information about the behavior of a nonlinear system, as described in the literature mentioned in section 2.1, the validity range is unknown and clearly depends on the stationary state. As shown in section 2.2, the static equilibrium position is directly related to the description of the contact surface. Thus, a modification of the contact surface can introduce a new distribution of normal pressure, and subsequently a new state of stiffness matrix asymmetry. Hetzler [29] presented the evolution of stability behavior as a function of the shape of pressure contact, for example, equal or half cosine-shaped pressure distribution. Research on the contact pressure distribution between discs and pads of brake systems has been carried out by a great number of scientists and engineers. The study of this parameter is becoming obvious and essential to brake research in order to predict the wear behavior of pads and discs, and thus, their topographic evolutions according to certain parameters (such as pressure, sliding velocity or time, which are the most cited in the literature). In the literature, most of the authors who investigated wear mechanisms [31, 32] either developed their own frictional law or integrated existing or modified models adapted from empirical equations into their computations. To define the wear behavior and surface evolution of sliding systems, Müller and Ostermeyer [34, 35] employed a mesoscopic particle approach. To take the different evolutions and the correlation between points of brake pad contact surfaces into account, we propose to define stochastic processes to model experimentally observed, realistic surfaces. Random fields are particularly well suited for this type of modeling. The complete methodology for modeling the variability of brake lining contact interfaces is described by Heussaff et al in [36]. For the convenience of the reader, only the main equations about random fields are mentioned in the following subsection. 4.3. Definition of random fields To generate random fields, several ways have already been proposed and discussed in the literature [37, 38]. In this paper, the Covariance Matrix Decomposition (CMD) method is considered. This method, which relies on the use of a deterministic spatial function based on the decomposition of the covariance matrix, show good accuracy regarding our goals and easy implementation. Given the experimental observations [36], the methodology for building random fields shown in figure 6 depends on pad profiles, which are considered topographically as Roughnesses (R) superimposed on more general curvatures called Wavinesses (W) and longrange deviations called Forms (F). To build output finite element mesh (FE-mesh) for pad surfaces, three steps are necessary:
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The surfaces need to be discretized into a set of random variables, Gaussian random fields need to be generated using Karhunen–Loève decomposition, Researched non-Gaussian fields (given by measurements) need to be calculated using a nonlinear transformation of normal distributed profiles.
Fig.6. Methodology for modeling the variability of brake pad topography.
The different quantities used to evaluate the different spatial properties of the profiles are the maximum amplitude of the forms (fourth degree polynomials) of the raw profiles, noted , the root mean square of the heights of the zero-mean wavinesses and roughnesses, noted and , the 6.7%, 30.9%, 69.1% and 93.3%-quantiles of the extracted wavinesses, noted ,, and the ratio of the maximum heights to the 93.3%-quantile of the profile wavinesses, noted . These data provide the information needed to compute the autocovariance and Johnson transformation functions. Let a probability space and a parameter set be given. A random field is then a finite or real valued function which, for every fixed is a measurable function of . The n-dimensional space , with will be considered in the following. The dependency on the underlying probability space will usually be suppressed throughout the text . For a fixed , the function is a non-random function of . This deterministic function is usually called a sample path or a realization. The parameter u is called the coordinate (or position) by standard terminology. In this context the formal definition simply means: A random field on (i.e. ) is a function whose values are random variables for any . A one-dimension random field is usually called a stochastic process and the parameter is then denoted in reference to the time . The term random field is usually used to stress that the dimension of the coordinate is higher than one. }, where is the field By considering a 2-dimensional Gaussian field { domain of , a random field can be written in the form: (14)
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A Gaussian random field (that is random field involving Gaussian probability density functions of the variables) is completely defined by its mean and variance-covariance function noted . The theorem of the orthogonal decomposition of random fields states that any Gaussian field described by a continuous covariance function can be written as: ∑
(15)
with represents mutually uncorrelated and standard normal random variables and functions defined from the covariance function . The representation in series of a random field consists on creating a separation of the space and the random variables of the field . Several representations in series are available in the literature in particular, the Karhunen–Loève decomposition method (KL). The KL expansion of a random field is based on the spectral decomposition of its covariance function. The set of deterministic functions, over which any realization of the field is expanded, is defined by the eigenvalue problem: (
∫
)
(
)
(16)
Eq. (16) is a Fredholm integral equation expressed here for two-dimensional random fields. Any realization of the random field can be expanded over the orthogonal basis formed by the set of functions as follows: ∑
(17)
where and represent respectively the eigenvalues and the eigenvectors of the Fredholm problem. In practice, the eigenvalues are ordered in a descending series and the sum is truncated so as to keep the most influential modes. As previously mentioned, the generated surfaces are Gaussian random fields. So, to obtain non-Gaussian surfaces, a non-linear Johnson transformation is applied to the profile: (
(
)
)
(18)
where , , and are calculated using four quantiles, previously mentioned, and for all real , we define the inverse translation function as:
{
(19)
The final random field is obtained by adding the variable , including the form , to field . Form consists of polynomial functions, whose parameters are random variables. (20) In this study, the discretization that governed the random field and the FE mesh which represents a lining surface are the same (note that due to the FE mesh refinement, squares of
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about 1mm side, the roughness is not considered). The values given by S(u) which represent the height of the surface nodes are then directly transferred to the FE mesh. The following subsection presents the proposed Improved Squeal Detection Methodology which relies on complex eigenvalue analyses and random description of the contact interface. 4.4. Description of the ISDM To implement the methodology, numerical Designs Of Experiments (DOE) are performed with consideration for the variability of brake lining surfaces. The spatial properties of the waviness and form of a brake pad are stochastically considered by means of random fields as well as the mechanical properties of the contact interface. Numerical DOE can be performed using different schemes. However, to limit the number of computations and to ensure good representation of the real variability, the design space is explored thanks to Latin Hypercube Sampling (LHS). For each sample point of the LHS, random parameters allow to build the description of the pads/disc contact interface. The topographical aspect is managed by a random field (see section 4.3). The structural aspect is managed directly by random parameters of the contact law (see section 5.1). A traditional complex eigenvalue analysis is then run and a post-process of the unstable modes is performed. The result of the entire process (all samples) is a collection of sets of unstable modes. Direct analysis of a stability diagram with the results of all simulations does not truly give complementary information. On the contrary, such a graph makes analysis difficult due to the amount of information needed to be analyzed. To overcome this difficulty, we propose to build families of unstable modes by aggregating results of the same type. The calculated solutions correspond to complex modal quantities. Therefore, it seems natural to aggregate similar unstable mode shapes by means of the MAC criterion to form families of unstable modes. The MAC criterion is mentioned below. | |
| ||
|
(21)
Where and are two unstable modes of the coupled system and ()H the complex conjugate transpose (Hermitian) operator. In practice, the process of creating families of unstable modes corresponds formally to a classification problem. Families of unstable modes will be referred to as classes of instability or simply classes. The classification process takes place in three steps. Step 1: Search uncorrelated mode shapes among all unstable modes. The number of uncorrelated mode shapes corresponds to the initial number of classes. Step 2: For each uncorrelated mode shape, search for satisfactory correlations between unstable modes and for aggregation of the correlated modes to form initial classes. The first and second steps introduce a MAC Threshold (MacT). Step 3: Aggregate initial classes if they share some identical unstable modes and delete initial classes that contain few modes. At this point, two thresholds are introduced. The aggregation threshold (AggT) corresponds to a minimum acceptable percentage of correlation between two classes and the suppression threshold (SuppT) corresponds to an acceptable minimum number of unstable modes per class. It is defined as a percentage of the number of DOE. This last step provides the final classes of instability for further analysis.
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When classes of instability are composed, it is possible to evaluate the dispersion of the real and imaginary parts of the eigenvalues (growth rate and circular frequency) for each family. The underlying idea is to identify modes that could lead to real instability by selecting classes corresponding to low dispersions. In other words, the objective is to find some stability (or robustness) among classes of unstable modes. The assumption is that the robustness of an unstable mode increases the confidence that it will appear if the occurrence is important due to the number of DOE samples. The procedure is described here with a non-deterministic representation of the contact interface that is perfectly compatible with a non-deterministic description of the material (or geometrical) parameters of the components of the braking system. This solution has been introduced, for example, [39] to represent variations of the material properties of the disc and pads according to temperature variations. 5. Application of the ISDM 5.1. Characterization of numerical contact surfaces and contact law As already mentioned in section 3.2, the squeal behavior of a braking system does not show high dependence on pad-to-disc interface evolution (i.e., different types of lining wear). Therefore, there was no distinction between the different wear states for the simulations. Quantities used to characterize the uncertain parameters correspond to experiments on normal consumer pad types (10 pads were used). The coefficient of friction is considered to be constant since its impact on mode coupling is well known, and we focus only on the topographical and structural aspects of the contact interface. In order to limit the number of uncertain parameters, five typical uncertain parameters are considered: the amplitude of the form, the 93.3%-quantile of the waviness, the two parameters of the penetration-pressure curves of the regularized contact law and the third body thickness. The contact law curves are made of the nonlinear part of the experimental indentation curves [36] and are approximated by (22) and by means of the Levenberg-Marquardt algorithm for nonlinear least squares. (
)
(22)
In the above equation, is the ith contact penetration, the associated contact load, and , the parameters of the approximation, are used to define the FE regularized contact laws. Since the number of indentation measurements per pad does not allow to identify spectral properties, a and b are considered to be global random variables. Hence, at each probability outcome, a regularized contact law is defined and applied to the complete surface of the lining. The statistical distributions of the parameters are not sufficiently known and are assumed to be uniform. Table 4 gives information about the minimum and maximum values of the five parameters. The height of the third body is generally considered to be less than 30 m [40, 41, 42]. Finally, to complete the description of the parameterization, brake line pressure and vehicle velocity are considered constant and are fixed at 20 bar and 5 km/h respectively. These values correspond to a theoretical nominal pressure of 0.68 MPa and a velocity at the contact interface of about 0.73 m/s. The values were defined according to the bench test results. Table 4 Range of uncertain DOE parameters.
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Form (mm)
Waviness (mm)
Third body (mm)
Contact law: parameter
Contact law: parameter (mm)
Minimum
0.0005
0.0052
0.0001
3.06
0.0052
Maximum
0.0995
0.0498
0.0299
14.94
0.0448
Figure 7 presents two brake pad profiles and pressure distributions obtained with the first steps (static analyses) of the simulations. Good agreement with experimental pressure distribution can also be seen. Experimental pressure distributions were obtained under static conditions with a pressure sensitive film (Fuji film) introduced between the pad and the cleaned disc from the third body. In order to avoid large deviations between inboard and outboard linings, the same parameters are used to generate both lining surfaces, but with two distinct realizations.
Fig.7. (a,b): two realizations of brake lining profiles (mm), (c,d) : corresponding pressure distributions (Mpa) and (e) pressure-sensitive film measurement.
Further analyses of the calculated pressure distribution shows that the mean real contact area is about 30% of the nominal one and that 85% of the samples present a real contact area less than 50% of the nominal one. This may explain why the mean pressure at the real contact interface is 2 MPa, despite the fact that the theoretical nominal pressure is 0.68 MPa. Moreover in fewer than 10% of the cases, the maximal pressure at the contact interface can reach a mean
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value of 15 MPa and a maximal value of about 60 MPa. The high local stresses can be responsible for the rapid evolution of the contact interface geometry. 5.2. Uncertain and robustness analyses The process described in section 4.4 is applied on the industrial brake system presented in section 3.1. An LHS composed of 100 samples was built using the five random parameters defined in the preceding section. Of the 100 simulations, 98 presented unstable modes. For all simulations, there were a total number of 439 unstable modes. The minimum number of unstable modes for a simulation was one and the maximum number was nine. Figure 8 shows the AutoMAC calculated before and after the classification process. The first treatment is only limited to a sorting of unstable frequencies in ascending order. Groups of unstable modes are discernible but difficult to define precisely, particularly when the mode number (i.e., frequency) increases. The classification process was performed using a MAC threshold of MacT=80%, an aggregation threshold of AggT=40% and a suppression threshold of SuppT=20%. These criteria allow seven classes of instability to be identified. They are clearly visible on the right AutoMAC of Figure 8. Of the total number of unstable modes, 298 remain after classification since classes with fewer than 20 unstable modes are suppressed because they are considered to occur with low frequency. The maximum percentage of correlation between modes of the same class is given by the first class (81%), the minimum by the fifth class (43%) and the mean correlation percentage is 60%, which means that classes aggregate modes with a good level of identical shape.
Fig.8. AutoMAC on unstable modes: (a) before classification and (b) after classification (random surfaces case).
Figure 9 shows the instability diagram (since only unstable modes are represented), where all unstable modes (small grey cross), the seven classes (circle, star and square markers) and the results of the deterministic study (the black stars labeled ‘nom’) and experimental study (dotted lines) are superimposed.
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Fig.9. Instability diagram of the deterministic, experimental and non-deterministic studies - random surfaces case.
An initial observation reveals good correspondence between the results of the experimental study and those of the study that take the variability of the brake lining into account. In particular, the unstable frequency of approximately 1850 Hz, which is dominant on the dynamometer, is easily retrieved with ISDM. This is not the case with the nominal model. The second observation concerns the large number of unstable modes found in the 3000 – 4000 Hz frequency band. This zone also corresponds to the occurrence of two unstable modes found experimentally but with a lower frequency than the unstable mode at 1850 Hz. If the analysis is stopped at this point, one can conclude that the results of the ISDM are in better agreement with the experimental ones than the results of the deterministic approach. However, the frequency approach’s lack of discrimination persists. To overcome this limitation, the robustness of classes of instability can be evaluated. The idea is to link the robustness (in terms of stability) of a class to the confidence of its appearance. The result of this evaluation is represented in Figure 10, where both the dispersion in terms of frequency and growth rate and the occurrence of each class are represented.
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Fig.10. Dimensionless variability of the 7 classes of instability and related occurrences - random surfaces case.
By looking at Figure 10, it is clear that the most robust class is the first one with the lowest sensitivity to brake lining variation and the most frequent occurrence (94%). The first class is characterized by a mean frequency of 1900 Hz with a standard deviation of approximately 3Hz. We notice that the robustness analyses do a good job of exhibiting the most crucial unstable mode found in practice, and engineers must focus their attention on this unstable mode. Except for the second and third classes, for which the occurrence and the stability are relatively important, the other classes, which have a low occurrence and a higher sensitivity to brake lining variation, should not present any interest for the designer. Moreover, the important variation in the frequencies and growth rates of the last four classes could explain the transient aspect of brake squeals during bench measurements. Consequently, the results of the robustness analyses are in line with the experimental measurements and provide new information about the influence of the macroscopic aspects of surfaces on mode coupling phenomena. 5.3. Discussion The previous section highlighted the impact of brake lining surfaces, numerically modeled by random fields, on the stability properties of the system. Introducing these variations into the design step is of utmost importance, and the results obtained are more predictive than other uncertainty models (such as material property models) often published in the literature. Contact surface modifications directly impact static equilibrium position and mode coupling. To highlight this affirmation, a second study was performed, which assumed coefficients of variation of 5% for the Young moduli of the main components (disk, pad, caliper). Moreover, perfect surfaces and hard contact formulations were used for the second analysis so that only the
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impact of the elastic moduli on the stability of the system was considered. The remaining parameters are the same as for the first study. Again, the LHS is composed of 100 samples, but in this case gives a total of 96 unstable modes (versus 439 in the first study). The minimum number of unstable modes for a simulation was again one, but the maximum was only three. The ISDM is conducted exactly in the same way, except for the suppression threshold (suppT) set at 5% of the number of DOE samples due to the low number of unstable modes. A greater percentage would excessively suppress classes of instability. Figure 11 shows the MAC matrix before and after ISDM application.
Fig.11. AutoMAC on unstable modes: (a) before classification and (b) after classification - Young moduli case.
Here again, a first class of instability appears clearly, even before ISDM application, and in this class of instability, unstable modes have only been sorted according to increasing frequencies. However, the five other classes do not appear clearly before ISDM application. An examination of the results of the robustness analysis in Figure 12 shows the limitation of a simulation with material parameter randomness.
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Fig.12. Dimensionless variability of the 6 classes of instability and related occurrence - Young moduli case.
As expected, the variability of unstable frequencies is more affected due to the modification of component stiffness. The relatively equal variability of the growth rate no longer helps in terms of class robustness and confidence in the appearance of the associated squeal frequency. Moreover, the occurrence associated with each class of instability once again highlights the unstable 1900 Hz frequency, and one can see the high density of unstable modes in the 3000-3400 frequency band. However, the occurrence values are very low, with just 21% for the first class and less than 8% for the five other classes. Due to the latter, the designer cannot be confident regarding the appearance of a particular unstable frequency. The numerical analyses of the brake system clearly show that the macroscopic aspects are significant on every floor. The appearance of unstable modes manifests as high sensitivity to the variability of pad-to-disc properties. The changes in sliding surfaces strongly influence the mode coupling phenomenon with different eigenfrequency and growth rate variations. The robustness of these values associated with a high number of occurrences allows the designer to decide on the importance of an unstable frequency. As a consequence, the proposed methodology improves the selectivity of the complex eigenvalue analysis. In contrast, simulations that use Young moduli as uncertain parameters do not give such clear results. Nevertheless, these parameters can be added to the first study in order to take into account production discrepancies to represent a family of brake systems rather than a nominal brake system. To fully confirm the validity of the brake lining study, the following paragraph presents an investigation using acoustic holography. 5.4. Verification and validation In order to validate both the dynamometer and ISDM observations with random contact surfaces as well as to more accurately analyze the squealing behavior of the brake system,
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acoustic holography measurements were performed. Figure 13 presents a view of the dynamometer device, in which a microphone array composed of 96 microphones is set up at the rear of the brake assembly. In order to limit sound reflection, the measurements were made in a semi anechoic chamber.
Fig.13. Side view of the acoustic holography test bench.
Two holographic prediction planes were defined at the back of the caliper and parallel to the friction surface of the disc. Predictions were performed according to the KirchhoffHelmholtz integral equation. To compare acoustic holography to the Complex Eigenvalue Analysis and identify the unstable modes at the root of the squealing, the direct Boundary Elements Method (BEM) is performed by using the LMS Virtual Lab software. This method allows us to compute the normalized sound pressure in a plane equivalent to the holographic plane. Thus, the measured and computed pressure distributions can be compared and squealing modes can be identified and validated. The BEM input consists of the finite element eigenvectors and eigenfrequencies, the air density and the speed of sound. The output of the BEM is the normalized sound pressure distribution on the FE input mesh and at the output plane.
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Fig.14. (a) Holography prediction on the side of the brake system at 3250 Hz and (b) normalized sound pressure on the equivalent plane, computed using the direct BEM method and the 3200 Hz complex eigenvector
Figure 14 presents a comparison of the predicted and computed sound pressures. The computation is based on the 3.2 kHz unstable FE-mode and the results are given for the rear of the brake system. The figure shows that the pads greatly contribute to the acoustic radiation. The acoustic holography and the spectrogram of the squeal allow four squeal frequencies to be identified: 1850 Hz (up to 119 dB), 2900 Hz (up to 63 dB), 3250 Hz (up to 74 dB) and 3650 Hz (up to 89 dB). Table 5 compares the experimental and computational paired frequencies. Table 5 Comparison of experimental and numerical squealing frequencies. Experimental squealing Numerical unstable modes (Hz) frequencies (Hz) 1850 1901 3100 3095/3123 3200 3241 3600 4002
Frequency deviations (%) 2,75 0,16/0,74 1,30 16,70
Modes 1 and 3 present the high acoustic dominance of the disc and the pads respectively. However, there is no dominance of any single component for modes 2 and 4. The frequency deviation results are satisfactory compared with the static modal validation results for the complete assembly. One reason for this is the dominance of some main components during squeal, which leads to a frequency deviation level equivalent to that of the single component validation. For mode 2, two numerical frequencies are proposed. Due to the complexity of the hologram and the presence of similar FE-eigenvectors in this frequency domain, the authors did not clearly identify the unstable mode responsible for the 3.0 kHz squeal. However, we cannot rule out the fact that this squeal could be caused by several unstable modes. Nevertheless, the acoustic holography helps identify the eigenmodes responsible for squeal in a fairly efficient way. The process gives us more information about the acoustical behavior of brake systems and increases our degree of confidence in the results provided by the complex eigenvalue analyses coupled with a random description of the contact interface.
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6. Conclusion The purpose of this research was to improve the prediction of finite element models by studying the dynamic instabilities that lead to squeal. Brake systems are highly complex, highly variable structures. Moreover, the literature review showed that the nature and evolution of the contact interface is a key factor involved in the appearance and disappearance of brake noise. Hence, we proposed to focus on the impact of the evolution of the contact interface on brake squeal and to define a strategy called Improved Squeal Detection Methodology to predict squeal based on the nondeterministic modeling of the contact interface and the use of complex eigenvalue analyses. The proposed methodology is applied here using an industrial brake finite element model. The surfaces of the linings were studied and statistically modeled using stochastic processes and then, the impact of the variability of the pad-to-disc interfaces was introduced in the finite element model of the brake system. Sensitivity and robustness analyses were then used to identify the most relevant unstable modes, which have the greatest chance of appearing. The numerical analyses clearly showed significant brake lining variability compared with the variability introduced with material parameters. The sliding surface changes strongly influence the appearance of unstable modes. The use of nondeterministic simulations allowed us to better identify the most relevant instabilities and the consideration of the evolution of the sliding interfaces led to a more predictive model consistent with the bench observations. Acknowledgments The present research was supported by the International Campus on Safety and Intermodality in Transportation of the Nord-Pas-de-Calais Region, the European Community FEDER, the Regional Delegation for Research and Technology, the Ministry of Higher Education and Research, the National Center for Scientific Research and DAIMLER AG. The authors gratefully acknowledge the support of these institutions. References [1] [2] [3]
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