Uncertainty quantification applied to the mode coupling phenomenon

Journal of Sound and Vibration ∎ (∎∎∎∎) ∎∎∎–∎∎∎

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Uncertainty quantification applied to the mode coupling phenomenon Martin Treimer a,b,n, Baldur Allert a, Katrin Dylla a,b, Gerhard Müller b a b

BMW AG, Petuelring 130, 80788 München, Germany Technische Universität München, Lehrstuhl für Baumechanik, Arcisstraße 21, 80333 München, Germany

a r t i c l e i n f o

abstract

Article history: Received 5 January 2016 Received in revised form 8 October 2016 Accepted 12 October 2016 Handling Editor: A.V. Metrikine

In this study a method for the uncertainty quantification of friction induced vibrations based on the mode coupling phenomenon is shown. The main focus is the assessment of the phenomenon under consideration of uncertain input parameters for the robustness evaluation. Stability assessments of the system under parameter scatter are given. It is pointed out how this is implemented within the scope of the Finite Element method. On the basis of the Euler–Bernoulli beam as a proof-of-concept model a procedure for the assessment of the system's robustness is shown. An objective function is proposed and used to evaluate a design of experiment. By means of a regression analysis an indicator for the robustness of the system is given. Numerical results are presented on the basis of the Euler–Bernoulli beam and a Finite Element brake model. A universal procedure is shown, the approach of which can be used for robustness assessments in different fields of interest. The algorithm that has an optimal efficiency is validated by a comparison with an algorithm which has an optimal quality of prediction. The procedure is applied on the robustness' assessment of brake squeal. & 2016 Elsevier Ltd All rights reserved.

Keywords: Uncertainty quantification Mode coupling phenomenon Robustness Regression analysis Brake squeal Euler–Bernoulli beam

1. Introduction Brake squeal is an important topic in NVH (Noise, Vibration, Harshness) investigations during the development process of a car [1,2] since it can cause high warranty costs and customer complaints [3,4]. Next to the safety and performance issues of a brake system there are important requirements on acoustic and comfort aspects [2,5]. In New York City brake squeal is one of the ten biggest noise problems and causes more than one billion of warranty costs in the United States [6]. Although the first developments in the field of brake squeal reduction started in the 1930s [6,7] there is no generalised theory [4] for this complex phenomenon [8]. The excitation mechanisms for brake squeal are described in [9,10] and the corresponding minimal models in [11]. The noise is caused by a self-exciting oscillation [12] induced by friction [7,8,12]. This results in a mode coupling or lock-in [6,13] which causes high vibration related with noise radiation [4]. The mode coupling phenomenon is the considered aspect in this study. Brake squeal has to be distinguished from other noise emissions caused by brake systems [7]: Next to the brake squeal which occurs with frequencies ranging from 1 to 16 kHz [4] the most frequently cited noises are groaning for frequencies n

Corresponding author at: BMW AG, Petuelring 130, 80788 München, Germany. E-mail addresses: [email protected] (M. Treimer), [email protected] (B. Allert), [email protected] (K. Dylla), [email protected] (G. Müller). http://dx.doi.org/10.1016/j.jsv.2016.10.019 0022-460X/& 2016 Elsevier Ltd All rights reserved.

Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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below 100 kHz and moaning in a range between 100 and 500 kHz [7]. There are different theoretical and experimental possibilities for the description of brake squeal [12]. Brake squeal is evaluated typically in the frequency domain. Instead of a time-consuming transient analysis [14] the evaluation is carried out by reference to the system's eigenfrequencies [15]. A method for the simulation of brake squeal for different frequency domains is proposed in [9]. Here the complex eigenvalues are calculated by means of a Finite Element model which is validated through a comparison with experimental results of the whole system and the analysis of the singular components' eigenfrequencies. Based on this work the quality of prediction is improved by further development [11,16]. On the basis of these methods effective countermeasures for the abatement of brake squeal are defined. Fundamental investigations of variable input parameters regarding the sensitivities' assessment of friction induced vibrations are done in [17]. On the basis of a pin-on-disc model effects of small perturbations of the given conditions are analysed. For the sensitivity calculation the first-order perturbation method is used. A general robustness concept for a friction induced vibration is shown in [18]. Possibilities for the evaluation of a singular instability and the general system's robustness are given and a quantification of the robustness is defined. This is shown on a mechanical model consisting of masses, dampers and springs. Robustness evaluations for brake systems have become more and more important over the last years. In [7] the improved squeal detection method is shown for robustness assessments. This method is derived with focus on the surface of the brake pads' friction material. By means of a stochastic modelling the system's robustness is assessed referring to the roughness parameter. The assessment is done by the occurrence of unstable modes and shows a comparison between evaluations of the brake pad's topographical and structural aspects and material properties. A first analysis of the full brake system regarding robustness is described in [2,19]. This method is based on the determination of distributions for the particular parameters in the first step and a simulation of a small number of randomly chosen parameter combinations for the input parameters in the second step. By reference to these results a statement about the robustness of the system is generated and the parameters with the highest influences are identified. In [8] a combination of a design of experiment and a calculation of a regression model for the analysis of the robustness is shown. The evaluation takes different parameters of the brake pad into account. Prior to the calculation of the regression model for the design of experiment the parameters with the highest influence are identified on the basis of a fractional factorial design of experiment. After fitting the regression model to the simulation outputs the correlation is validated considering random input parameters. The procedure is applied on a simplified brake model. Based on this algorithm [4] describes an extended method for the robustness evaluation combined with an uncertainty optimisation. After defining interval parameters for the variation in the first step the regression model is calculated. On this basis a two layer optimisation is performed regarding the negative damping ratio as objective function. This leads to a reliability analysis and gives the opportunity for the optimisation of a single design parameter. The calculation of a regression model for uncertainty quantification is a common approach and also used in other fields [20]. Another assessment of the robustness of brake noise under input parameter variation is given in [21]. In this work a regression model is used to evaluate the system's robustness. Next to this analysis further applications of the surrogate model like optimisation and assessment of the reliability are discussed. The evaluation is based on a Latin-Hypercube Sampling and the assessment is done concerning the real parts of the system's eigenvalues. Apart from robustness evaluations on brake squeal there are also some analyses treating this topic in other fields. A possibility without the consideration of a design of experiment but with the aid of fuzzy parameters is shown in [22]. In [23] the descriptive equations for the calculation of a general vibroacoustic problem are extended by a stochastic part characterising the variation of the input parameters. For the implementation in the finite element code the stochastic part is reduced to a set of random algebraic equations. This procedure is based on the Polynomial Chaos [24], used for linear brake models [25] and is still under development [26]. In Section 2 of this paper a system of Euler–Bernoulli beams as a proof-of-concept model for the description of the relevant phenomena for brake squeal is shown. The third section gives a short overview over the calculation of brake squeal. After that the algorithm for the robustness evaluation is described. Section 5 combines the previous sections and gives a numerical example of the robustness algorithm by reference to the Euler–Bernoulli beam. A d-optimal design of experiment is compared with a full factorial one.

2. Euler–Bernoulli beam as a proof-of-concept model In order to depict the relevant phenomena related with brake squeal the brake disc and the two brake pads are modelled as Euler–Bernoulli beams. The contact is distributed and enforced by the penalty method. This proof-of-concept model cannot be used to analyse a specific brake system, but shall illustrate the general behaviour and offers a possibility to evaluate a system with low computational effort. It is used for a comparison between different designs of experiment. The excitation mechanisms are restricted to the so-called mode coupling or flutter phenomenon which can be identified in discretised models by an asymmetric stiffness matrix. This restriction should not restrain the validity of the model since mode coupling is often referred as the most crucial effect for brake squeal [5,27]. Also the analysis of brake systems using Finite Element models later in this work implies this effect to be the most significant one. Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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Fig. 1. Continuous beam model of a brake disc with two brake pads.

Most of the vibration pattern of the disc and the pads can be reproduced by this one-dimensional model without increasing the complexity. A more complex model based on Kirchhoff–Love plates would also be able to take radial vibrations into account. These vibration patterns are known to be less prone to brake squeal and will be neglected here. By varying the underlying parameters a large number of brake systems can be analysed in the following sections. A similar beam model to predict brake squeal is proposed in [28,29]. There the brake disc is modelled as a beam and the brake caliper including the brake pads is modelled as a system of springs and masses. In [30] another beam model is proposed representing the brake pads as simple springs suspended to a stiff support. The necessity of a distributed contact model in order to take the mode coupling phenomenon into account is shown in [31]. There are gyroscopic forces, radial and tangential friction damping, structural damping and follower-forces in the two dimensional plate model included. Finally the mode coupling phenomenon is identified being the most relevant phenomenon. This implies the strong focus on mode coupling in the present work without having disturbing effects by other excitation mechanisms. Also the tangential mode shapes show major influence compared to radial mode shapes. 2.1. System of differential equations The used set-up of Euler–Bernoulli beams is shown in Fig. 1. The following system of differential equations describes the substitute dynamic model of the brake system. The derivation of the equations is given in Appendix A. In the first region represented by the coordinates −l1 < ξ < − l2 and l2 < ξ < l1 where the disc is not in contact with the pads only the beam equation of the brake disc is relevant:

EI1x1′′′′ + kW x1 − A1ρ1 x¨1 = 0.

(1)

The second interval with the coordinates −l2 < ξ < l2 where the disc is in contact with the pads all three differential equations representing the brake disc and the two brake pads have to be fulfilled:

EI1x1′′′′ + kN μh1 (2x1′ − x2′ − x3′ ) + kN (2x1 − x2 − x3 ) + kW x1 − A1ρ1 x¨1 = 0, EI2 x2′′′′ + kN μh2 (x1′ − x2′ )

− kN (x1 − x2 )

− A2 ρ2 x¨2 = − p2 ,

EI2 x3′′′′ + kN μh2 (x1′ − x3′ )

− kN (x1 − x3 )

− A2 ρ2 x¨3 = p3 .

(2)

The close relation to a normal Euler–Bernoulli beam without contact remains obvious. 2.2. Application of the substitute model to a brake system In order to solve the system of differential equations the method of finite difference with linear order of convergence is applied [32] to Eqs. (1) and (2). This method discretises the continuous differential equations (1) and (2) into a system of equations of the form

(3)

Mx¨ + Cẋ + Kx = 0. The free parameters in the model are chosen with respect to the typical dimensions of a passenger car's brake system:

l1 = 900 mm,

A1ρ1 = 10 g mm−1,

kW = 500 N mm−1,

EI1 = 1010 N mm2 ,

h1 + h2 = 20 mm.

(4)

The length of the beam representing the brake disc is chosen according to the circumference of a 16–17 in disc and the mass equals the weight of the friction ring. The stiffness of the Winkler bedding kW and the bending stiffness EI1 are adjusted in order to fit to the Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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first eigenfrequencies of the brake disc. The distance between the neutral axis of brake disc and brake pads is half of the disc height plus the height of the friction material h1 + h2. All other parameters can be chosen using dimensionless parameters and can be varied later on. The reference values are equal to:

l2 = 0.05, l1

A2 ρ2 = 0.05, A1ρ1

kN = 100, kW

h2 = 0.5, h1

EI2 = 0.04, EI1

μ = 0.5.

(5)

3. Stability analysis The assessment of the system's stability based on the Lyapunov theory [33,34] is carried out on the basis of a quasi-static analysis [3,7,11] applied to the proof-of-concept model. The system's stability is assessed on the basis of an eigenvalue analysis. The describing equation (3) for the simplified system of the Euler–Bernoulli beam is given in the following notation:

(6)

Mx¨ + Cẋ + Kx = 0.

The asymmetric parts of the stiffness matrix K are linked to the friction mechanism and can cause instabilities. They are responsible for the mode coupling phenomenon. Eq. (6) is given in the state space by:

Aẏ = By, with y =

( ), A = ( x ẋ

(7) C M

M 0

) and B = (

−K 0

0 M

).

Based on the approach

y = Ψ e st ,

(8)

in combination with Eq. (7) the complex eigenvalue problem is given,

(B − λ k A) Ψk = 0 with 1 ≤ k ≤ 2n.

(9)

where λ represents the eigenvalue, Ψ the corresponding eigenvector and n the number of degrees of freedom of the Finite Element system. The eigenvalues are normalised by the following condition for the guarantee of the uniqueness [7]:

Ψk⊤ AΨk = 1.

(10)

Prior to the calculation of the complex eigenvalue problem a subspace is created. For this the undamped and symmetric system is considered and all asymmetric terms are neglected. The eigenfrequencies from (9) set up the basis for the computation of the complex frequencies. For that reason the matrices are projected onto the subspace,

M ⋆ = Ψ ⊤MΨ , C ⋆ = Ψ ⊤CΨ , K ⋆ = Ψ ⊤KΨ .

(11)

With this projection the following equation is solved:

M ⋆x¨ + C ⋆ẋ + K ⋆x = 0.

(12)

Unstable frequencies are characterised by a positive real part of the eigenvalue and stable modes by a negative one. This analysis is used for the prediction of brake squeal and is considered in the proof-of-concept model of the Euler–Bernoulli beam.

4. Algorithm for the robustness evaluation In the following an algorithm for the robustness assessment of the prediction of brake squeal on the basis of the real parts of the eigenvalues is presented. For the consideration of the input parameter scatter the Finite Element model of a brake system is described by a surrogate model. This is created by the usage of design of experiments. The surrogate model is used for the system's assessment. Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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4.1. General procedure During the first step of the algorithm the relevant parameters are identified. The variables with a high influence to the propensity of brake squeal are chosen and additional parameters with a large scatter behaviour are taken into account. The parameters' boundaries of variation and distributions are described. These are based on experimental evaluations of different brake systems. Different distributions for each parameter are taken into account, a uniform or a normal distribution. Based on the minimal and maximal values of the scatter and the distribution of the parameters an experimental design is created. Several designs are possible, for example full factorial or d-optimal, see Section 4.2. The choice depends on the parameter quantity, the sampling points and the prediction quality desired, with regard to the calculation effort. In the next step the design of experiment is performed numerically on the basis of Finite Elements. For the later evaluation each experiment is assigned a value which identifies the occurrence of brake squeal. This is called objective function and is described in Section 4.3. Based on the assigned scalar value the design of experiment is evaluated. For an analytic discussion a regression analysis is performed. The resulting steady function is analysed regarding the behaviour on parameter changes, the dependencies and the influence factors. For the verification of the system's reliability the correlation coefficient between numerical results and the results obtained on the basis of the analytical model is assessed. The interpretation of the results is carried out with the overall scatter of the objective function, the parameter sensitivities regarding the objective function variation and an instability probability index. 4.2. Different methods for design of experiment In a full factorial design [35,36] all chosen parameter settings are combined among each other. Thus the number of experiments n is given by:

n = sk,

(13)

with k being the number of parameters and s the parameter levels. With a full factorial design information about all sensitivities for every parameter is given and the calculation of a detailed regression model is possible. For a high number of parameters this leads to a prohibitive calculation effort. The d-optimal design [36–38] is a reduction of the full factorial design to the minimum number of experiments for calculating the regression model of a certain order. A consideration of all effects and interactions is possible. Unlike a LatinHypercube Sampling the values for the sampling points are not randomised and remain fixed. They are chosen before creating the design of experiment. For a second order model the minimum number of experiments is given by:

2p +

p (p − 1) + 1, 2

(14)

with p being the number of the main effects multiplied by 2 because of the quadratic terms. The number of the interactions between the parameters is given by p (p −1) . The last part is for the description of the constant term. The model is based on the 2

quasi-linear polynomial ansatz

y = Xβ + ε ,

(15) Rn × m

with y being vector of the objective, X ∈ the design matrix constructed by the candidates for the experimental design, β the vector of unknown model parameters and ε the estimation error. The dimension n stands for the number of parameters and m for the number of experiments. To get the minimal and optimal design exchange algorithms are performed with the aim to minimise the expression det (X⊤X)−1 based on a fixed number of experiments. The experiments in X are replaced iteratively until the best configuration is found. This is done by costly calculation algorithms [39,40] with the aim of the improvement of the design's quality of prediction. A full-factorial design is reduced to a minimum number of experiments but keeping the assessment's possibility of all effects included in the model. The realised number of experiments is larger than the minimal one. This is needed to illustrate the variance in the experiments' results and is a compromise between the calculation effort and the quality of prediction. Alternatively Monte Carlo algorithms [41–44] can be carried out. 4.3. Objective function The analysis of the brake system is carried out by assessing the real Re and the imaginary parts Im of the eigenvalues by the following objective function qsim :

qi =

| Re (λ i ) − Re (λ i + 1)| − | Im (λ i ) − Im (λ i + 1)| . 1 (Im (λ i ) + Im (λ i + 1)) 2

λi

(16)

For all neighbouring modes, i = 1, … , m − 1, with m being the maximal mode number, the quotient qi is calculated. The Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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objective function for one experiment is described by:

qsim =

max qi .

(17)

i = 1, … , m − 1

In practical applications with several instabilities in different frequency bands qsim is calculated with dependency of the respective instabilities of interest. Therefore after the first the second maximum and so on is chosen. For instabilities the objective function qi is equal to the damping ratio of unstable modes di [27] which is a conventional squeal index:

di = −

2 Re (λ i ) . Im (λ i )

(18)

This belongs to the mode coupling phenomenon and its associated relations when two modes are coupled, Re (λ i ) = − Re (λ i + 1) and Im (λ i ) = Im (λ i + 1). That means:

qi =

| Re (λ i ) + Re (λ i )| − | Im (λ i ) − Im (λ i )| 2 Re (λ i ) = = |di | . 1 Im (λ i ) (Im (λ i ) + Im (λ i )) 2

(19)

An advantage of the objective function towards the damping ratio is the indication of stable modes with a negative value unequal to zero. The objective function allows a continuous evaluation of the mode coupling phenomenon even if the modes are not paired.

4.4. Regression model The aim of a regression model is the generation of a surrogate model of first or second order out of experimental data. The experimental results are based on a design of experiment X and are given in the vector Y . The surrogate model is based on the experimental design X and expressed with the linear approach:

^ Y = a + bX,

(20)

^ ^ with Y being the vector containing the analytical results, a and b the coefficients of a linear function. The analytical results Y are fitted to the experimental results Y by adjusting the coefficients a and b. The requirement for the estimation is the minimisation of the ordinary least squares N

ols =

∑ (Yi − Y^i)2.

(21)

i=1

^ For comparison of the fitted function Y and the experimental data Y the coefficient of determination R2 is calculated,

R2 =

^ N ∑i = 1 (Yi − Yi )2 N ∑i = 1 (Yi − Y¯ )2

, (22) 2

with R2 ∈ [0, 1] and Y¯ representing the average of Yi . R is a correlation coefficient between the experimental and analytical data. A R2 near to 1 indicates a good fit.

5. Numerical results The algorithm for the robustness evaluation is applied on different systems. At first the Euler–Bernoulli beam is chosen because it has a low computation effort. Secondly a brake system modelled by Finite Elements is shown.

5.1. Euler–Bernoulli beam The algorithm for the mode coupling phenomenon's robustness evaluation is performed on the Euler–Bernoulli beam described in Section 2. For the given parameters the system is marginally stable. So a robustness evaluation is necessary. In the first step the parameters for the variation are identified. In this case the given dimensionless parameters from Section 2 are taken and the notation of Table 1 is assumed. For the definition of the parameter variations the nominal values are considered. These are obtained by the fitting of the Euler–Bernoulli beam of Section 2 and Appendix A to a real brake system: Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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Table 1 The notation used for the evaluation of the Euler–Bernoulli beam, see Eq. (5). Physical formula

Description

Notation

l2 l1 A2 ρ2 A1ρ1

Length ratio between pad and disc

l

Mass ratio between pad and disc

ϱA

EI2 EI1 kN kW h2 h1 μ

Bending stiffness ratio between pad and disc Contact stiffness ratio between pad and disc

EI

Thickness ratio between pad and disc

h

Friction value between pad and disc

mu

l2 = 0.05, l1

A2 ρ2 = 0.05, A1ρ1

EI2 = 0.04, EI1

kN = 100, kW

h2 = 0.5, h1

μ = 0.5.

pen

The robustness assessment is performed with a variation of ±10% around the nominal value. This assumption is made because for this proof-of-concept model no data or measurements are available. This leads to a simple and automated generation of the design of experiment. All parameters are taken into account with the same variation and the sampling points are distributed in equidistant intervals. In this case the design of experiment is constituted on a d-optimal plan based on five sampling points for all variables. Out of a total of 15,625 combinations 50 experiments are chosen for the d-optimal design. To verify the approach of the experiments' amount reduction the results of the 50 experiments (Appendix B) are compared with a full factorial design consisting of all 15,625 possible configurations. The output of the experiments, which are calculated with Matlab, are the 13 first eigenfrequencies of the system. Higher mode numbers are related to eigenfrequencies above the threshold of audibility. In case of an instability the eigenvalue has a positive real part. Because of the mode coupling phenomenon there exists a second eigenvalue with the same frequency and a negative real part. An example for the output of the system is given in Table 2. In the next step every experiment is matched with the objective function calculated with Eq. (17). The evaluation is done for i = 1, … , m − 1, m = 13, and the result is given by qsim . If an experiment contains instabilities, the objective function is positive. For example in Table 2 qsim is represented by the difference between λ12 and λ13,

qsim =

| − 187 − 187| − |79, 372 − 79, 372| = 0.00471. 1 (79, 372 + 79, 372) 2

(23)

After the calculation of the objective function of all experiments the results are linked with the design of experiment. The regression model is generated on the basis of the calculated values. There are six independent input variables and the objective function represents the dependent variable. The regression is based on a quadratic model with the consideration of interactions. All terms of second order are contained in the model including bilinear terms. This leads to an analytical surrogate model with 28 terms, six for the linear and six for the quadratic influence factors, 15 interactions and one constant. For the d-optimal design the illustration in Fig. 2 shows a cut through the six dimensional plane of the regression model for the parameter settings with their boundaries. In this case they are represented by the nominal values. For comparison the parameter processes for the full factorial design are shown in Fig. 3. Table 2 Example output for one experiment out of the design of experiment. Some stable modes and an unstable mode λ13 are shown. Eigenvalues

Real part

Imaginary part

λ1 λ2 ⋮ λ11 λ12 λ13

0 0 ⋮ 0
187 187

7056 7419 ⋮ 55,659 79,372 79,372

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Fig. 2. Visualisation of the parameter progression for the d-optimal design. The cut through the regression function is shown for nominal parameter settings. The dots at the bottom mark the sampling abscissa. They are not linked with qsim .

Fig. 3. Visualisation of the parameter progression for the full factorial design. The cut through the regression function is shown for nominal parameter settings. The dots at the bottom mark the sampling abscissa. They are not linked with qsim .

The validation of the quality of prediction is done with the aid of R2 according to Eq. (22). In this case the calculated value is R2 = 0.981 for the d-optimal and R2 = 0.962 for the full factorial design. So the results are trustworthy and a good prediction of the parameter progression is possible. The interpretation of the calculated regression models is carried out on the basis of three different assessments: 1. total scatter and distribution of the objective function, 2. parameter sensitivities on variations of the objective function, 3. calculation of an instability probability index. In Fig. 4 the distribution of the values calculated for the models' objective function are shown. The values qsimmin = − 0.017 and qsimmax = 0.008 for the d-optimal respectively qsimmin = − 0.014 and qsimmax = 0.009 for the full factorial design represent the optimal values for the surrogate models in the given parameter intervals. The distribution of the objective function is generated by a Monte Carlo sampling using 10,000 sampling points. A uniform distribution is taken into account for all the input parameters. Because the experiments have positive and negative objective functions the system is not robust concerning the prevention of instabilities. In Fig. 5 the parameter sensitivities for variations in the objective function are shown. The analysis of the influencing factors is carried out by consideration of the regression model in dependency of a single variable. If a term of the regression model does not contain the considered variable, it is neglected for the calculation. After the calculation of all influencing factors they are normalised. For a robust system the influences of the parameter scatter results should be minimised. For this purpose the variance of the most influencing parameter is reduced or components are redesigned in order to minimise the Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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Fig. 4. Description of the distribution for the objective function designed with the Monte Carlo sampling. On the left is the d-optimal and on the right the full factorial design.

Fig. 5. Visualisation of the parameter sensitivities causing changes in the objective function. A higher bar displays a greater influence to variations in the objective function. The values are normalised according to the parameter with the highest sensitivity. On the left is the d-optimal and on the right the full factorial design.

impact of the considered parameter. In the given example the parameter l has the highest influence to the objective function for both designs of experiment. In reality this parameter has little scatter. However, in applications on brake systems it is used for modifications to prevent squeal by chamfering as shorter pads reduce the risk of brake squeal. Out of the histogram for the characterisation of the system regarding the robustness of instabilities the instability probability index is calculated. Based on a Monte Carlo approach with 10,000 sampling points parameter combinations are tested to find stable and unstable configurations for the system. Thereof the index is calculated. For the given system with the d-optimal design 31.4% of the combinations produce instabilities and 68.6% are stable and for the full factorial design 32.0% are unstable while 68.0% are stable. For the given instabilities there is the risk of noise emission. The d-optimal design's calculation effort is

1 300

of the full factorial one. The assessments of the designs show little dif-

ferences in the sensitivity analysis while the results of the parameter processes and the instability index are close. For the full factorial design the processes are more linear than in the d-optimal one but with the same impact on the objective function. Because of these results the following brake model is calculated on the basis of a d-optimal design for the saving of computation time. 5.2. Brake model After the application of the algorithm to the proof-of-concept model of an Euler–Bernoulli beam a brake system is considered. The robustness analysis is based on the Finite Element model shown in Fig. 6 whose simulation is done with Abaqus. The model has about 600,000 degrees of freedom and consists of tetrahedron, pentahedron and hexahedron elements. The considered parameters are the stiffness and the density of the brake disc's friction ring, the friction value and the Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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Fig. 6. The Finite Element model used for the brake system.

Fig. 7. Visualisation of the parameter progression. The cut through the regression function is shown for nominal parameter settings. The dots at the bottom mark the sampling abscissa. They are not linked with qsim .

brake pressure. The nominal values for cast iron are based on the literature [45] and assumptions for variations are made. The brake pressure and the friction value represent customer relevant ranges. The variations are given by the following: Stiffness brake disc, E (GPa) 90–110 Density brake disc, ϱ (tm−3) 7.1–7.2 0.2–0.7 Friction value, μ Brake pressure, p (bar) 1–30 The nominal value is represented by the mean value of the intervals and for all parameters a uniform distribution is considered. For the calculation of the regression model a d-optimal design of experiment with 30 simulations is performed. A full factorial design is not practical due to computation effort. The objective function represents an instability at 1.7 kHz and is calculated in this area using Eq. (17). The resulting regression model is visualised in Fig. 7. Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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Fig. 8. Description of the distribution for the brake model's objective function designed with the Monte Carlo sampling.

Fig. 9. Visualisation of the parameter sensitivities causing changes in the objective function for the brake model. A higher bar displays a greater influence to variations in the objective function. The values are normalised according to the parameter with the highest sensitivity.

The distribution of the objective function is calculated in an equal way to the Euler–Bernoulli beam with 10,000 sampling points and includes positive and negative terms, see Fig. 8. Therefore the same stability criteria as for the Euler–Bernoulli beam are considered. A positive objective function indicates an instability and a negative one a stable system. So the system can have stable and unstable configurations. The instability probability is 41.3% and 58.7% of the parameter configurations are stable. The system is not robust concerning the prevention of instabilities. In Fig. 9 the parameter sensitivities are shown. The stiffness of the brake disc has the highest influence to the objective function. For an improvement of the system to get a lower instability probability it is useful to make restrictions for this parameter. In the given example a restriction of the brake disc's Young's modulus E to the interval 90–100 GPa reduces the instability probability to 7.25%. The technical realisation is a compromise between commercial costs and expected warranty claims. At the end the question arises: whether it is possible to link the instability probability to the noise emission of the brake systems in cars. Since the algorithm is mostly performed during the brake system's development there are no studies from a relevant number of cars available. In future feedback from the customers has to be evaluated to get a correlation between the instability probability in simulation and the probability of noise emission in cars.

6. Conclusion An algorithm for a robustness evaluation of uncertain input parameters is carried out. It is based on a d-optimal design of experiment and calculates a regression model out of the available data. The requirement for this is an identification of the varying parameters and their limits beforehand. Afterwards the regression model is used to evaluate the system with statistical methods to generate trustworthy assessments about the system's stability under parameter variations according to their distributions. The algorithm is based on a method for the detection of the mode coupling phenomenon with an analysis of the brake system's eigenfrequencies by conducting a Finite Element calculation. For a numerical example an Euler–Bernoulli beam is considered. The modelling of the beam with finite differences is shown and linked with the stability analysis of a brake system concerning squeal. The Euler–Bernoulli beam is used to validate the usage of the d-optimal design of experiment. For the robustness evaluation some varying parameters are identified and their limits are determined. The calculation of the regression model is performed on the basis of a design of experiment with 50 simulations for the Euler–Bernoulli beam and 30 simulations for the brake model representing the variations of the parameters. These results for the Euler–Bernoulli beam are compared with a full factorial design to check the quality of prediction of the d-optimal design. The interpretation of the results is performed on the total scatter of the objective function, the parameter sensitivities and their effects on the objective function plus the calculation of an instability probability index. Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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Acknowledgement This research was done in a cooperation of the TU Munich and the BMW Group.

Appendix A. Description of the Euler–Bernoulli beam The Euler–Bernoulli beam is used as a proof-of-concept model to compare a full factorial and a d-optimal design of experiment. The describing system of the differential equations is derived in a detailed way.

A.1. Modelling The brake disc is modelled as an Euler–Bernoulli beam with the length 2l1, the height 2h1, the bending stiffness EI1 and the density ρ1 as shown in Section 2. Periodic boundary conditions are assumed. This modelling enables to represent all vibration patterns without nodal circles, see [9] for an overview of brake disc vibration patterns. The suspension of the brake disc's friction ring by the bearing and the knuckle is modelled as an elastic Winkler-bedding with the stiffness parameter kW. Due to the important influence of the stiffness of the pad's material [46] the brake pads are also modelled as Euler– Bernoulli beams with the lengths 2l2 = 2l3, the heights 2h2 = 2h3, the bending stiffnesses EI2 = EI3 and the densities ρ2 = ρ3. The stiffness in normal direction contributes to the contact stiffness kN. The beams representing brake pads have variable lengths but they are always shorter than the beam representing the brake disc. The analysis of the operational deflection shapes has been carried out on the basis of laser interferometric measurements. Thus movements at the end of the brake pads imply Neumann boundary conditions. That means that there are no external forces and moments at the ends of the beams. Analogously the displacement x2 respectively x3 and the rotation x2′ respectively x3′ will have non-zero values. The interface between disc and pads is modelled by a contact stiffness in the normal direction kN taking the axial stiffness of the brake pads into account. The friction is modelled as Coulomb-friction with the friction coefficient μ. The friction force acting in the tangential direction is applied at the surface of the beams.

A.2. Equilibrium of forces The forces and moments applying at the three beams are shown in Fig. A1 at a differential element of length dξ . This results in:

Fig. A1. Equilibrium at the beam elements of infinitesimal length dξ .

Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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Mi = −EIi xi″ ξ + 1 dξ ,

Mi− = −EIi xi″ ξ − 1 dξ ,

Q i+ = −EIi xI‴ ξ + 1 dξ ,

Q i = −EIi xI‴ ξ − 1 dξ ,

2

2

−

2

2

FNu = kN (x1 − x2 ),

FTu = μFNo,

FNl = kN (x3 − x1),

FTl = μFNu,

FIi = Ai ρi x¨ i ,

13

FW = kW x1 (i = 1, 2, 3).

(A.1)

A.3. Principle of virtual displacements For the static case the inertia forces are not considered. δWstat =

l1

∫−l1

bedding disc

bending disc

+

l3

∫−l3

EI3 x3″ δx3″ 

l3

∫−l3

EI2 x2″ δx2″ 

′ +  kN (x1 −  x (x1 − x kN μ (x1 − x2 ) δ ( − h1x1′ − d2 x 2 ) δ  2) +  2 ) dξ

bending upper pad

upper contact, normal direction

upper contact, tangential direction

′ +  kN (x3 − x (x3 −  x kN μ (x3 − x1 ) δ (h 1) δ  1) +  1x1′ + d3 x 3 ) dξ + 

bending lower pad

−

l2

∫−l2

EI1x1″ δx1″ +  kW x1δx1 dξ + 

p x3 3 δ 

lower contact, normal direction

dξ =

l2

∫−l2

l1

∫−l1 (EI1x1⁗ + kW x1) δx1dξ +

lower external forces

p x2 2 δ 

dξ

upper external forces

lower contact, tangential direction

⎡⎣ EI1x1″ δx1′ − EI1x1‴ δx1⎦⎤ l1 −l1 periodic boundary conditions

l2

l3

+

l −kN μh1 (x1 − x2 ) δx1⎤⎦ 2 + ∫ (kN (x1 − x3 ) + kN μh1 (x1′ − x3′ )) δx1dξ ∫−l2 (kN (x1 − x2 ) + kN μh1(x1′ − x2′ )) δx1dξ + ⎡⎣ −l 3 −l2

+

⎡⎣ −kN μh1 (x1 − x3 ) δx1⎦⎤ + −l 3

Neumann boundary conditions l3

l2

∫−l2 (EI2 x2⁗ − kN (x1 − x2 ) + kN μh2 (x1′ − x2′ ) + p2 ) δx2 dξ

Neumann boundary conditions

+

⎡⎣ EI2 x2″ δx2′ − EI2 x2‴ δx2 − kN μh2 (x1 − x2 ) δx2 ⎤⎦ l2 + −l2 

+

⎡⎣ EI3 x3″ δx3′ − EI3 x3‴ δx3 − kN μh3 (x1 − x3 ) δx3 ⎤⎦ l 3 . −l 3 

l3

∫−l3 (EI3 x3⁗ − kN (x1 − x3 ) + kN μh3 (x1′ − x3′ ) − p3 ) δx3 dξ

Neumann boundary conditions

(A.2)

Neumann boundary conditions

For the dynamic modelling in addition the inertia forces are considered.

δWdyn = δWstat −

l1

∫−l

1

A ρ1 x¨1δx1 dξ − 1 

l2

∫−l

disc

2

A ρ2 x¨2 δx2 dξ − 2 

l3

∫−l

upper pad

3

A ρ3 x¨3 δx3 dξ . 3  lower pad

(A.3)

The brake systems installed in vehicles mostly contain two brake pads of similar shape. The analysis of brakes containing similar brake pads is not part of this work. So all parameters of the upper pad can be chosen identically to those of the lower pad, i.e.

l2 = l3,

EI2 = EI3,

h2 = h3,

A2 = A3 ,

ρ2 = ρ3 .

(A.4)

A.4. System of differential equations The equilibrium equations result in the following system of differential equations: Interval 1: −l1 < ξ < − l2 and l2 < ξ < l1. Here only the beam equation of the brake disc is relevant:

EI1x1⁗ + kW x1 − A1ρ1 x¨1 = 0.

(A.5)

Interval 2: −l2 < ξ < l2. All three differential equations representing the brake disc and the two brake pads have to be fulfilled:

EI1x1⁗ + kN μh1 (2x1′ − x2′ − x3′ ) + kN (2x1 − x2 − x3 ) + kW x1 − A1ρ1 x¨1 = 0, EI2 x2⁗ + kN μh2 (x1′ − x2′ )

− kN (x1 − x2 )

− A2 ρ2 x¨2 = − p2 ,

EI2 x3⁗ + kN μh2 (x1′ − x3′ )

− kN (x1 − x3 )

− A2 ρ2 x¨3 = p3 .

(A.6)

In order to obtain the homogenous solution of the system of differential equations the external forces vanish, p2 = p3 = 0. Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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Fig. A2. Periodic boundary conditions, forces and moments at the brake disc at ξ = ± l1.

A.5. Boundary conditions and transition conditions The periodic boundary conditions of the brake disc at ±l1 are shown in Fig. A2:

−EI1x1″ ξ =−l = −EI1x1″ ξ = l ,

x1 ξ =−l1 = x1 ξ = l1 ,

1

x1′ ξ =−l = x1′ ξ = l , 1

1

− EI1x1‴ ξ =−l

1

1

= −EI1x1‴ ξ = l .

(A.7)

1

The transitional conditions of the brake disc at ±l2 are given by:

x1 ξ =−l2 = x1 ξ =−l2 ,

x1 ξ = l2 = x1 ξ = l2 ,

x1′ ξ =−l = x1′ ξ =−l ,

x1′ ξ = l = x1′ ξ = l ,

2

2

2

2

−EI1x1″ ξ =−l = −EI1x1″ ξ =−l , 2

2

−EI1x1″ ξ = l = −EI1x1″ ξ = l . 2

(A.8)

2

Due to the missing continuity of the third derivative x1‴ at ±l2 in Eq. (A.9) a distinction has to be made between the left-hand side limit values (e.g. x1‴ ξ=−l −), the functional values (e.g. x1‴ ξ=−l2 ) and the right-hand side limit values (e.g. x1‴ ξ=−l + ). This 2

2

discontinuity in the third derivative of the disc is caused by a concentrated friction force at the end of the brake pads:

−EI1x1‴ ξ =−l − = −EI1x1‴ − kN μh1 ( 2x1 − x2 − x3 ) 2

−EI1x1‴ ξ = l + = −EI1x1‴ − kN μh1 ( 2x1 − x2 − x3 ) 2

ξ =−l2+

ξ = l2−

,

.

(A.9)

The Neumann boundary conditions of the brake pads have to be modified so that non-zero external forces and moments can be eliminated:

−EI2 x2″ ξ =−l = 0, 2

−EI2 x2″ ξ = l = 0, 2

−EI2 x3″ ξ =−l = 0, 2

−EI2 x3″ ξ = l = 0, 2

−EI2 x2‴ − kN μh2 (x1 − x2 ) ξ =−l = 0, 2

−EI2 x2‴ − kN μh2 (x1 − x2 ) ξ = l = 0, 2

−EI2 x3‴ − kN μh2 (x1 − x3 ) ξ =−l = 0, 2

−EI2 x3‴ − kN μh2 (x1 − x3 ) ξ = l = 0. 2

(A.10)

A.6. Application of the beam model to a brake system As described in Section 2 the continuous differential equations are transformed in the following system of equations:

(A.11)

Mx¨ + Cẋ + Kx = 0. See Table A1 for the description of the matrices. Like shown in Section 2 the reference values are equal to:

l2 = 0.05, l1

A2 ρ2 = 0.05, A1ρ1

kN = 100, kW

h2 = 0.5, h1

EI2 = 0.04, EI1

μ = 0.5.

(A.12)

Two vibrational mode shapes of the system are exemplarily plotted in Fig. A3 with respect to the circumferential coordinate ξ. The fifth out of plane mode related to an eigenfrequency of 12,611 Hz is unstable due to the excitation by friction. This leads to a positive real part of the eigenvalue and an exponentially increasing vibration amplitude. The real and imaginary parts of the mode shape vector are both nonzero, they describe a travelling wave on the brake disc. The system does not Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i

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Table A1 Matrices and vectors for the brake squeal calculation and their properties. Mass matrix for the description of acceleration proportional forces, real-valued, symmetric and positive definite Damping matrix for the description of velocity proportional forces, real-valued, symmetric and positive definite Stiffness matrix for the description of displacement proportional forces, real-valued and non-symmetric Acceleration vector Velocity vector Displacement vector

M C K x¨ ẋ x

Fig. A3. Shapes of the unstable mode of sixth order at 12,611 Hz (left) and marginally stable mode of first order at 1180 Hz (right): The solid line represents the real part while the dotted line represents the imaginary part. The brake pads are printed above and below the brake disc.

vibrate in phase neither exists any point without movement. The first out of plane mode related to an eigenfrequency of 1180 Hz is marginally stable. The system vibrates in phase which can be seen by the vanishing imaginary part of the mode shape vector. In a three dimensional model this vibration pattern equals to a tilting friction ring which does not imply any strains there. In contrast the one dimensional beam model also implies bending strains for this mode shape.

Appendix B. Design of experiment

No.

l

ϱA

EI

pen

h

μ

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

0.05 0.0525 0.0525 0.0525 0.05 0.05 0.055 0.0525 0.0475 0.0525 0.045 0.045 0.05 0.055 0.055 0.05

0.055 0.0525 0.05 0.045 0.05 0.0525 0.05 0.045 0.0525 0.0525 0.0475 0.045 0.0525 0.05 0.0525 0.045

0.044 0.044 0.04 0.036 0.044 0.042 0.042 0.042 0.04 0.04 0.042 0.036 0.044 0.036 0.038 0.04

90 105 100 90 105 110 100 110 95 105 105 100 105 90 95 95

0.45 0.55 0.45 0.55 0.45 0.45 0.475 0.5 0.525 0.45 0.475 0.45 0.5 0.525 0.45 0.5

0.45 0.525 0.475 0.55 0.475 0.525 0.525 0.45 0.45 0.55 0.5 0.55 0.45 0.525 0.45 0.5

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16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50

0.045 0.045 0.045 0.05 0.0525 0.045 0.05 0.045 0.055 0.055 0.055 0.055 0.0475 0.055 0.05 0.055 0.055 0.045 0.0475 0.055 0.0525 0.055 0.045 0.0475 0.045 0.045 0.0525 0.045 0.0475 0.0475 0.0475 0.0475 0.045 0.055

0.0475 0.05 0.0475 0.0525 0.055 0.055 0.045 0.055 0.045 0.055 0.055 0.0475 0.045 0.055 0.0475 0.055 0.045 0.045 0.05 0.045 0.055 0.0475 0.045 0.055 0.055 0.055 0.05 0.0475 0.0525 0.045 0.05 0.055 0.0475 0.045

0.036 0.038 0.044 0.04 0.036 0.04 0.038 0.044 0.042 0.04 0.042 0.038 0.036 0.036 0.044 0.044 0.044 0.044 0.036 0.044 0.036 0.038 0.038 0.036 0.038 0.036 0.042 0.036 0.04 0.038 0.042 0.044 0.044 0.036

110 90 90 90 95 110 100 95 90 100 90 110 90 110 105 110 90 95 110 110 90 100 105 100 105 95 110 90 90 110 95 110 110 100

0.475 0.475 0.55 0.5 0.45 0.5 0.475 0.45 0.525 0.475 0.55 0.45 0.45 0.55 0.525 0.525 0.475 0.5 0.55 0.55 0.55 0.525 0.525 0.45 0.55 0.55 0.525 0.55 0.475 0.45 0.55 0.5 0.55 0.5

0.55 0.475 0.55 0.5 0.525 0.45 0.45 0.55 0.475 0.525 0.55 0.55 0.45 0.55 0.55 0.475 0.525 0.45 0.45 0.525 0.475 0.45 0.5 0.475 0.5 0.55 0.5 0.45 0.55 0.5 0.5 0.55 0.45 0.475

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Please cite this article as: M. Treimer, et al., Uncertainty quantification applied to the mode coupling phenomenon, Journal of Sound and Vibration (2016), http://dx.doi.org/10.1016/j.jsv.2016.10.019i