Optimization of a Minimal Model of Disc Brake
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Abstract
In this study, the latter idea of optimizing the damping matrix is explored further.
In this study, the latter idea of optimizing the damping matrix is explored further. In a first step, the minimal model of disc brake developed in [1] is analyzed. This model has two DOF so that the results can only give an insight into the stability behavior of an idealized brake.
3.1 Equations of Motion
3.2 Optimization Technique
Introducing modified damping parameters \(\tilde{d_t}=\alpha _1 d_t\) and \(\tilde{d}=\alpha _2 d\) yields the vector \(\hat{\mathbf {p}}=(\alpha _1,\alpha _2)^{\mathrm{T}}\) to be the set of parameters which is about to be optimized, cf. Sect. 2.5. Hence, damping due to the disc and damping due to the pins can be treated independently. Since the matrices (3.2) are timedependent, Floquet theory is applied and the monodromy matrix \(\mathcal {M}\) is calculated numerically. In order to make Floquet theory and CEA consistent, Eq. ( 2.22) is used to calculate the maximum real part of \(\lambda \) for a given \(\mu \). Setting \(\delta =0\), \(\kappa =0\), and \(\Theta _p=0\), both Floquet theory and CEA lead to the same results. Optimization problem ( 2.21) is sought using Matlab, where an insight into the pseudocode is given below. To keep it short, the FFT and the optimization of the damping ratio are not included.

for \(\alpha _1=\alpha _{1,\min }\) to \(\alpha _{1,\max }\)
 for \(\alpha _2=\alpha _{2,\min }\) to \(\alpha _{2,\max }\)

\(\hat{\mathbf {p}}=(\alpha _1,\alpha _2)^{\mathrm{T}}\)

calculate \(\varvec{\Gamma } \hat{\mathbf {p}}\)

find \(\tilde{\alpha }_1,\tilde{\alpha }_2\) satisfying \(\varvec{\Gamma } \tilde{\mathbf {p}} \le \mathbf {c}\), where \(\tilde{\mathbf {p}}=(\tilde{\alpha }_1,\tilde{\alpha }_2)^{\mathrm{T}}\)


end

end
 for each \(\tilde{\mathbf {p}}\)

\(\tilde{d_t}=\tilde{\alpha }_1d_t\)

\(\tilde{d} \ =\tilde{\alpha }_2d\)

calculate \(\mathcal M\) using ( 2.18)

calculate \(\mu _{\max }=\max \limits _i\mu _i\)


end

find \(\min \limits _{\tilde{\mathbf {p}}}\ \mu _{\max }\) and output \(\tilde{\alpha }_1=\alpha _{1,\mathrm{opt}}\), \(\tilde{\alpha }_2=\alpha _{2,\mathrm{opt}}\)

calculate \(\min \limits _{\hat{\mathbf {p}}}\max \limits _i \mathrm{Re}(\lambda _i)\) using ( 2.22)
3.3 Optimization Results
3.3.1 TimeInvariant Model
In this section, the system is assumed to be timeinvariant, i.e. the parameters \(\delta \), \(\kappa \), and \(\theta _P\) are set to zero and only selfexcitation due to circulatory terms may lead to unstable solutions, i.e. there is no parametric excitation. Applying the aforementioned optimization technique to determine the optimal weighting factors for the damping parameters \(d_t\) and d leads to the results in Fig. 3.2.
Since the damping ratio ( 2.20) is a favoured physical quantity to evaluate the rate of decay of damped oscillations, optimization ( 2.19) with \(\max \limits _{\hat{\mathbf p}} \min \limits _iD_i{\mathfrak {}}(\lambda _i)\) is applied with the constraints defined by (3.6). As shown in Fig. 3.5, the optimum at \(\alpha _{1,\mathrm{opt}}=1.16\) and \(\alpha _{2,\mathrm{opt}}=1.84\) matches the optimum point in Fig. 3.3. This is due to the fact that the frequency of a damped mode is nearly the same as in its undamped equivalent. However, the difference between the damped and the undamped frequency may have an influence when optimizing the stiffness instead of the damping parameters. The magnitude of the damping ratio can be explained by its definition. Since the magnitude of the frequency is \(\mathcal O(10^3)\) and the maximum real part is \(\mathcal O(10^{1})\), Eq. ( 2.20) results in a damping ratio being \(\mathcal O(10^{4})\).
3.3.2 TimePeriodic Model
3.3.3 Discussion
It is well known from practical experience that brake squeal depends on various parameters, especially on the prestress \(N_0\) and the friction coefficient \(\mu \). However, numerical investigations in this study show, that stability maps and contour plots are geometrically similar when varying these parameters. For example, setting \(N_0\) small, the real parts of the eigenvalues and hence the blue triangle, which represents the area of fluttertype instability, grow and vice versa. Setting \(N_0\) as small as possible within the velocity range where brake squeal occurs, would be a possibility to guarantee optimum damping parameters. The prediction of the model that large prestress forces act stabilizing, whereas small prestress forces act destabilizing matches with reality since the noise phenomenon mainly occurs at low vehicle speeds when the driver applies the brake gently.
Another key value is the angular velocity \(\Omega \). The maximum real part of \(\lambda \) grows with \(\Omega \) such that there exists a limit angular velocity above which the system is unstable [1]. This result seems to be a contradiction to practical experience. As shown in [3], the friction coefficient \(\mu \) depends on \(\Omega \) for high velocity values. Setting \(\mu \) as a linear decreasing function of the angular speed, i.e. \({\mu (\Omega )=\mu _0m\Omega }\), where \(m>0\), a decreasing maximum real part for growing \(\Omega \) can be observed. In this study, however, only small angular velocities are investigated and the restriction of a constant friction coefficient remains reasonable.
3.4 Traps and Shortcomings of CEA
References
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