# Optimization of a Minimal Model of Disc Brake

• Jan-Hendrik Wehner
• Dominic Jekel
• Rubens Sampaio
• Peter Hagedorn
Chapter
Part of the SpringerBriefs in Applied Sciences and Technology book series (BRIEFSAPPLSCIENCES)

## 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

Consider a rigid disc (moment of inertia $$\Theta$$, radius r, and thickness h) rotating with constant angular speed $$\Omega$$ around its center of mass with the angles $$q_1$$ and $$q_2$$ being minimal coordinates, cf. Fig. 3.1. Two pins representing the brake pads can move freely but under the actions of springs (stiffness k) and dampers (damping coefficient d) in the $$n_3$$-direction of a Newtonian reference frame and in frictional contact with the disc. Based on experimental tests conducted in [2] the friction is assumed to be of Coulomb type with a constant and isotropic coefficient $$\mu$$ in a first approximation. Detailed information and experimental results about a velocity dependent friction coefficient can be found in [3]. It is further assumed that there is only slip between the disc and the pins which is assured as long as the rotational speed of the disc is sufficiently large. The prestress in the pins is $$N_0$$, the supports of the rotating disc have the stiffness and damping properties $$k_t$$ and $$d_t$$, respectively. The parameters
\begin{aligned} \begin{aligned} h&=0.02\, \mathrm{m},\, r=0.13\,\mathrm{m},\, \Omega =5\pi \,\mathrm{s}^{-1},\, \mu =0.6,\, N_0=3000\,\mathrm{N},\, \Theta =0.16\,\mathrm{kg}\mathrm{m}^2,\\ k&=6.00\times 10^6\,\mathrm{N}/\mathrm{m},\, k_t=1.88\times 10^7\, \mathrm{N}\mathrm{m},\, d=5.0\,\mathrm{N}\mathrm{s}/\mathrm{m},\, d_t=0.1\,\mathrm{N}\mathrm{m}\mathrm{s} \end{aligned} \end{aligned}
(3.1)
are chosen to compare with [1].
In addition, to bring time-variance into the model, an asymmetric bearing and a mass point $$m_p$$ with the distance $$r_p$$ on the body fixed $$d_1$$-axis of the disc are assumed. In an actual brake, time-periodicity is due to ventilation channels. The equations of motion then have the perturbation parameters $${\kappa =\frac{k_{t2}-k_{t1}}{k_{t1}}>0}$$, $${\delta =\frac{ d_{t2}-d_{t1}}{d_{t1}}>0}$$, and $$\Theta _p=m_pr_p^2$$ leading to time-periodic matrices which may force the system to exhibit parametrically excited vibrations. If the bearing is symmetric, $$k_t$$ and $$d_t$$ are independent of direction, i.e. $$k_{t1}=k_{t2}=k_t$$, $$d_{t1}=d_{t2}=d_t$$, and hence $$\kappa =\delta =0$$. As outlined in [4], the matrices describing the equations of motion () become
\begin{aligned} \begin{aligned} \mathbf M&=\begin{bmatrix} \Theta&0 \\ 0&\Theta \\ \end{bmatrix}+\Theta _p\begin{bmatrix}\mathrm{sin}^2(\Omega t)&-\mathrm{sin}(\Omega t)\mathrm{cos}(\Omega t) \\ -\mathrm{sin}(\Omega t)\mathrm{cos}(\Omega t)&\mathrm{cos^{2}}(\Omega t) \\ \end{bmatrix},\\ \\ \mathbf D&=\begin{bmatrix}d_t(1+\frac{1}{2}\delta )+2dr^2+\frac{1}{2}\mu N_0\frac{h^2}{\Omega r}&-\frac{1}{2}\mu d h r \\ -\frac{1}{2}\mu d h r&d_t(1+\frac{1}{2}\delta ) \\ \end{bmatrix}-\Theta _p \Omega \begin{bmatrix}-\mathrm{sin}(2\Omega t)&\mathrm{cos}(2\Omega t) \\ \mathrm{cos}(2\Omega t)&\mathrm{sin}(2\Omega t) \\ \end{bmatrix}\\ {}&-\frac{1}{2}\delta d_t\begin{bmatrix}\mathrm{cos}(2\Omega t)&\mathrm{sin}(2\Omega t) \\ \mathrm{sin}(2\Omega t)&-\mathrm{cos}(2\Omega t) \\ \end{bmatrix},\\ \\ \mathbf G&=\begin{bmatrix}0&(2\Theta +\Theta _p)\Omega +\frac{1}{2}\mu dhr \\ -[(2\Theta +\Theta _p)\Omega +\frac{1}{2}\mu dhr]&0 \\ \end{bmatrix}, \\ \\ \mathbf K&=\begin{bmatrix}k_t(1+\frac{1}{2}\kappa )+2kr^2+N_0h&-\frac{1}{4}\mu r[2kh+N_0(4-\frac{h^2}{r^2})] \\ -\frac{1}{4}\mu r[2kh+N_0(4-\frac{h^2}{r^2})]&k_t(1+\frac{1}{2}\kappa )+(1+\mu ^2)N_0h \\ \end{bmatrix}-\frac{1}{2}\kappa k_t \begin{bmatrix}\mathrm{cos}(2\Omega t)&\mathrm{sin}(2\Omega t) \\ \mathrm{sin}(2\Omega t)&-\mathrm{cos}(2\Omega t) \\ \end{bmatrix}, \\ \\ \mathbf N&=\begin{bmatrix}0&\frac{1}{4}\mu r [2kh+N_0(4+\frac{h^2}{r^2})] \\ -\frac{1}{4}\mu r [2kh+N_0(4+\frac{h^2}{r^2})]&0 \\ \end{bmatrix}. \end{aligned} \end{aligned}
(3.2)
Setting $$m_p=0$$ the matrices simplify as shown in [4]. In order to get an insight into the different physical origins of damping and to formulate an optimization problem it is advantageous to decompose the damping matrix into $$\mathbf D=\mathbf D_{\mathrm{friction}}+\mathbf D_{\mathrm{pad}}+\mathbf D_{\mathrm{disc}}(t)+\mathbf D_{m_p}(t)$$, see [5], where
\begin{aligned} \begin{aligned} \mathbf D_{\mathrm{friction}}&=\begin{bmatrix}\frac{1}{2}\mu N_0\frac{h^2}{\Omega r}&-\frac{1}{2}\mu d h r \\ -\frac{1}{2}\mu d h r&0 \\ \end{bmatrix}, \quad \mathbf D_{\mathrm{pad}} =\begin{bmatrix}2dr^2&0 \\ 0&0 \\ \end{bmatrix}, \\ \mathbf D_{\mathrm{disc}}(t)&=\begin{bmatrix}d_t(1+\frac{1}{2}\delta )&0 \\ 0&d_t(1+\frac{1}{2}\delta ) \\ \end{bmatrix}-\frac{1}{2}\delta d_t\begin{bmatrix}\mathrm{cos}(2\Omega t)&\mathrm{sin}(2\Omega t) \\ \mathrm{sin}(2\Omega t)&-\mathrm{cos}(2\Omega t) \\ \end{bmatrix}, \\ \mathbf D_{m_{p}}(t)&=-\Theta _p \Omega \begin{bmatrix}-\mathrm{sin}(2\Omega t)&\mathrm{cos}(2\Omega t) \\ \mathrm{cos}(2\Omega t)&\mathrm{sin}(2\Omega t) \\ \end{bmatrix}. \end{aligned} \end{aligned}
(3.3)

## 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. . Hence, damping due to the disc and damping due to the pins can be treated independently. Since the matrices (3.2) are time-dependent, Floquet theory is applied and the monodromy matrix $$\mathcal {M}$$ is calculated numerically. In order to make Floquet theory and CEA consistent, Eq. () 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 () 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.

Pseudocode of optimization problem ()
• 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 ()

• 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 ()

As can be seen in the pseudocode, the monodromy matrix is calculated numerically for every vector $$\tilde{\mathbf p}$$ within the admissible area using (), where the limit is set to $$m=100$$. This way of optimization has an advantage and a disadvantage. On the one hand, calculating the Floquet multipliers respectively the eigenvalues for every $$\tilde{\mathbf p}$$ guarantees the calculated optimum point to be a global optimum. A discussion about local and global maxima is not necessary. On the other hand, this may lead to large computing times which are mainly influenced by the range of $$\mathbf p$$ and the increments. In this example, the increments are chosen as $$\Delta \alpha _{1,2}=0.01$$ within a maximal range of $$\alpha _{1,2} \in [0,3]$$ which may be interpreted as technically relevant. The degree of precision of the weighting factors then is $$\pm 0.005$$.

## 3.3 Optimization Results

### 3.3.1 Time-Invariant Model

In this section, the system is assumed to be time-invariant, i.e. the parameters $$\delta$$, $$\kappa$$, and $$\theta _P$$ are set to zero and only self-excitation 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.

The dotted pattern ➁ represents the admissible area, where $$\varvec{\Gamma }\hat{\mathbf {p}}\le \mathbf c$$ is satisfied. Since the weighting factors can be varied independently, a rectangle is the simplest way of setting the constraints which are chosen as
\begin{aligned} \begin{aligned} \varvec{\Gamma }=\begin{bmatrix}1&0 \\ 0&1 \\ -1&0 \\ 0&-1 \\ \end{bmatrix}, \quad \mathbf c=\begin{bmatrix}3 \\ 3 \\ 0 \\ 0 \\ \end{bmatrix}. \end{aligned} \end{aligned}
(3.4)
The white area, [➂ $$\cup$$ ➁] $$\setminus$$ ➀, including a subset of the admissible area, predicts the stable solution, whereas in the blue area ➀, the solution is unstable and of flutter-type. If only Coulomb damping is active, i.e. $$\alpha _1=\alpha _2=0$$, the solution is unstable. The green bars compare the optimized and non-optimized maximum real part and the weighting factors, respectively. The red dot denotes the optimum $$\tilde{\mathbf {p}}$$. Since there is only one coherently blue area of flutter-type instability, which has the form of a triangle, neither adding damping in the disc nor in the pads, in this case, can destabilize the system. Larger values of $$\alpha _1$$ and $$\alpha _2$$ within the admissible area always stabilize or make more stable the equilibrium position. Therefore, the optimum is characterized by the largest possible values $$\alpha _{1,\mathrm{opt}}=\alpha _{2,\mathrm{opt}}=3$$.
Thinking economically, an additional cost function may further reduce the scope of the admissible area. Similar to the operations research optimization, a cost function could have the form
\begin{aligned} c_1\frac{\tilde{d}_t}{d_t}+c_2\frac{\tilde{d}}{d} \le c_{\mathrm{max}}, \end{aligned}
(3.5)
where $$c_1$$ and $$c_2$$ represent the costs of adding damping, while the maximum costs, which may be due to additional economic aspects, are limited by $$c_{\mathrm{max}}$$. Setting for example $$c_1=1000$$, $$c_2=1000$$, and $$c_{\mathrm{max}}=3000$$ yields $$\alpha _1+\alpha _2\le 3$$. Due to physical aspects, the damping parameters have to be larger or equal than zero, i.e. $$\alpha _1\ge 0$$, $$\alpha _2\ge 0$$, which is equivalent to $$-\alpha _1\le 0$$ and $$-\alpha _2\le 0$$. Finally, the constraints can be written as
\begin{aligned} \varvec{\Gamma }=\begin{bmatrix}1&1 \\ -1&0 \\ 0&-1\end{bmatrix}, \quad \mathbf c=\begin{bmatrix}3 \\ 0 \\ 0\end{bmatrix}, \end{aligned}
(3.6)
where the admissible area is a triangle that excludes the optimal point calculated above. In this case, the optimum point is $$\alpha _{1,\mathrm{opt}}=1.16$$ and $$\alpha _{2,\mathrm{opt}}=1.84$$, cf. Fig. 3.3. With regard to decomposition (3.3) and to the constraints (3.6), it therefore can be concluded that it is reasonable to shift some damping from the disc to the pins for system () with the matrices (3.2) and the parameters (3.1) to be optimized for stability.
In Fig. 3.4, a contour plot combined with a gradient field provides information on how to interpret these results. The contours represent the values of the maximum real part depending on the weighting factors $$\alpha _1$$ and $$\alpha _2$$, while the arrows represent the gradient of the maximum real part as a function of $${\hat{\mathbf p}=(\alpha _1,\alpha _2)^{\mathrm{T}}}$$, i.e. $$\frac{\partial }{\partial \hat{\mathbf p}}[{\max \limits _i} {\text {Re}}\,(\lambda _i)]$$ and thus indicate the most beneficial direction to make more stable the equilibrium solution.
It becomes apparent that adding damping in the pads is only worthwhile up to $$\alpha _{2,\mathrm{lim}} \approx 2$$, whereas damping in the disc is always advantageous. Evaluating the gradient field with respect to the calculated optimum point in Fig. 3.2, i.e. $$\alpha _{1,\mathrm{opt}}=\alpha _{2,\mathrm{opt}}=3$$, more damping in the pads than $$\tilde{d}_t=\alpha _{2,\mathrm{lim}}d_t$$ may not be beneficial. Choosing any $$\alpha _2>\alpha _{2,\mathrm{lim}}$$ makes nearly no difference with regard to the maximum real part, whereas a larger value of $$\alpha _1$$ is always purposeful. For this reason, damping in the disc can be given a higher priority. Evaluating Fig. 3.3 at $$\alpha _{1,\mathrm{opt}}=1.16$$ and $$\alpha _{2,\mathrm{opt}}=1.84$$ shows that the optimum value $$\alpha _{2,\mathrm{opt}}$$ is a more precise numerical calculation of $$\alpha _{2,\mathrm{lim}}$$, i.e. $$\alpha _{2,\mathrm{opt}}=\alpha _{2,\mathrm{lim}}=1.84$$. For the constraints
\begin{aligned} \varvec{\Gamma }=\begin{bmatrix}1&1 \\ -1&0 \\ 0&-1\end{bmatrix}, \quad \mathbf c=\begin{bmatrix}c \\ 0 \\ 0\end{bmatrix}, \end{aligned}
(3.7)
with $$c>1.84$$, the optimum value for damping in the pins is always $${\alpha _{2,\mathrm{opt}}=1.84}$$. Additionally, the condition $$\alpha _1+\alpha _2=c$$ has to be satisfied in order to minimize the maximum real part. Hence, the optimum value for damping in the disc can be determined by $$\alpha _{1,\mathrm{opt}}=c-1.84$$. Setting
\begin{aligned} \varvec{\Gamma }=\begin{bmatrix}1&1 \\ -1&0 \\ 0&-1\end{bmatrix}, \quad \mathbf c=\begin{bmatrix}2 \\ 0 \\ 0\end{bmatrix} \end{aligned}
(3.8)
yields $$\alpha _{1,\mathrm{opt}}=0.16$$, which is the same result obtained in [6]. The points of intersection between the two skewed constraint lines and the horizontal line $${\alpha _{2}=1.84}$$ in Fig. 3.4 illustrate the position of the two optimum points discussed above. It is important to mention that, in most cases, the optimum point is located at a corner of the admissible area, whereas in the example above the optimum is located at an edge. Certainly, these results are not based on analytical calculations and are to be understood as numerical approximations in a finite area. However, no counterexamples for these results in $$(\alpha _1, \alpha _2) \in [0,100]\times [0,100]$$ could be found in this study.

Since the damping ratio () is a favoured physical quantity to evaluate the rate of decay of damped oscillations, optimization () 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. () results in a damping ratio being $$\mathcal O(10^{-4})$$.

A contour plot with the weighting factors being in the extended intervals $${0<\alpha _1<30}$$ and $${0<\alpha _2<1000}$$ shows that adding damping may also lead to larger maximum real parts of the eigenvalues, cf. Fig. 3.6. Every contour line has a turning point at $$\alpha _2\approx 120$$, so damping in the pads in general may not make the system more stable. Numerical investigations using arbitrarily large weighting factors show, however, that adding damping in this brake model cannot destabilize the system. For $$\alpha _2>120$$ the maximum real part converges asymptotically to zero. Choosing $$\alpha _1=\alpha _1^*$$ arbitrarily and optimizing the damping in the pads yields $$\alpha _{2,\mathrm{opt}}\approx 120$$ which can be assumed to be a global optimum in $$\alpha _2 \in \mathbb {R}^+$$. At this point, damping in the pads would have to be around 120 times higher than the initial value d given in (3.1) for which reason a technical relevance of this global optimum is questionable.

### 3.3.2 Time-Periodic Model

In this section, damping of the brake model is optimized for the more general case of time-periodic matrices. As can be seen in (3.2), the parameters $$\delta$$, $$\kappa$$, and $$\Theta _p$$ activate time-variance and may force the system to parametrically excited vibrations. In comparison to the case of symmetric damping in the bearing, numerical investigations suggest that asymmetric damping ($$\delta \ne 0$$) nearly has no influence on the stability. Thus, only the perturbation parameters $${\kappa \ne 0}$$ and $${m_p \ne 0}$$ are discussed which are chosen to compare with [7], i.e.
\begin{aligned} \delta =0,\, \kappa =0.02,\, \Theta _p=-0.001. \end{aligned}
(3.9)
A negative moment of inertia respectively a particle with negative mass can be interpreted as a hole in the disc which, in realistic brakes, has the purpose of draining water off to improve the braking properties. Consider the hole having the distance $$r_p=0.1\mathrm{m}$$ to the center of the disc and the radius $${r_h=0.0025\mathrm{m}}$$. The disc is made out of steel (density $${\rho _{\text {steel}}\approx 7.9 \ \mathrm{g/cm^3}}$$) and is $$h=0.02 \mathrm{m}$$ thick. Since $$r_h\ll r_p$$, the approximation of the hole to be a mass point can be used. Calculating the moment of inertia $${\Theta _p=\frac{1}{2}\rho _{\text {steel}}\pi r_h^2 h r_p^2}$$ yields $${\Theta _p=\mathcal {O}(10^{-3})}$$.
With the linear constraints $$\varvec{\Gamma }$$ and $$\mathbf c$$ as in (3.4) the optimum still lies in the upper right corner of the admissible range, where $$\alpha _{1,\mathrm{opt}}=\alpha _{2,\mathrm{opt}}=3$$. Compared to the time-invariant case, the region of instability (blue triangle) grows. However, setting $$\kappa$$ sufficiently large, e.g. $$\kappa =0.07$$, this domain disappears completely. Using constraints (3.6) results in a different optimum point which is at $${\alpha _{1,\mathrm{opt}}=0}$$ and $${\alpha _{2,\mathrm{opt}}=3}$$, cf. Fig. 3.7. The course of the contour lines and the direction of the arrows in Fig. 3.8 are similar compared to Fig. 3.4. The line $$\alpha _2=\alpha _{2,\mathrm{lim}}$$, where increasing $$\alpha _2$$, i.e. adding damping in the pads, starts being no longer worthwhile, moves upwards from $$\alpha _2\approx 1.84$$ to $$\alpha _{2}\approx 6.3$$. Hence, adding damping in the pads is more efficient than in the time-invariant case. The optimization of the damping ratio according to () yields the same results.

### 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 flutter-type 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. $${\mu (\Omega )=\mu _0-m\Omega }$$, 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

With respect to time-periodic systems, it should be noted that using CEA, as it is normally done in the automotive industry, leads into a trap. In [7], it is shown that the eigenvalues of a time-dependent matrix $$\mathbf A(t)$$, calculated at some frozen times over one period, provide no information about the stability. The system may be asymptotically stable although there exist eigenvalues with positive real part. Similarly, the system may be unstable although all eigenvalues calculated at discrete time steps have negative real part. Let $$\lambda (t)$$ be the solution of the equation $$\mathrm{det}(\mathbf A(t)-\lambda (t) \mathbf I)=0$$. Then, it is not possible to draw conclusions on the Floquet multipliers $$\mu$$ from $$\lambda (t)$$, i.e.
\begin{aligned} \exists \ \tilde{t} \in [0,T] \ \exists \ i\in [0,2n]: \mathrm{Re}(\lambda _{i}(\tilde{t}))>0 \not \Rightarrow \ \exists \ \mu _j:|\mu _j|>1, \end{aligned}
(3.10)
respectively
\begin{aligned} \forall \ \tilde{t} \in [0,T] \ \forall \ i \in [0,2n]:\mathrm{Re}(\lambda _i(\tilde{t}))<0 \not \Rightarrow \ \forall \ \mu _j:|\mu _j|<1. \end{aligned}
(3.11)
Conclusion (3.10) is derived using a time-periodic system with one DOF. The two complex conjugated eigenvalues can be represented by $$\lambda _1(t)=a(t)+\mathrm{i}b(t)$$ and $$\lambda _2(t)=a(t)-\mathrm{i}b(t)$$. The Liouville formula is
\begin{aligned} \mathrm{det}[\varvec{\Phi }(t)]=\mathrm{det}[\varvec{\Phi }(0)]\mathrm{e}^{\int _0^t \mathrm{tr}[\mathbf A(\tau )]\mathrm{d\tau }}. \end{aligned}
(3.12)
Without loss of generality, the initial conditions may be set equal to the identity matrix and the time interval of the integral in (3.12) is chosen as one period T. Furthermore, the trace and the determinant of a matrix are known to be the sum and the product of its eigenvalues, respectively. As a result, Eq. (3.12) simplifies to
\begin{aligned} \mathrm{e}^{2\int _0^Ta(t)\mathrm{dt}}=\mu _1 \mu _2. \end{aligned}
(3.13)
Using the mean value theorem of integration $$\int _0^Ta(t)\mathrm{d}t=a(\xi )T$$ for $$\xi \in [0,T]$$ Eq. (3.13) can be rewritten as
\begin{aligned} \mathrm{e}^{2a(\xi )T}=\mu _1 \mu _2. \end{aligned}
(3.14)
It is possible that the mean value $$a(\xi )$$ is negative, while there exists any $$a(\tilde{t})>0$$. Adopting this situation yields
\begin{aligned} \mu _1 \mu _2<1. \end{aligned}
(3.15)
If the Floquet multipliers are assumed to be complex, i.e. $$\mu _1\mu _2=|\mu _1|^2=|\mu _2|^2$$, it follows
\begin{aligned} |\mu _1|=|\mu _2|<1. \end{aligned}
(3.16)
Therefore, the system may be asymptotically stable although there are positive real parts at all times during one period T.
Expanding (3.12) to an n-dimensional problem leads to similar results. In this more general case, the Liouville formula becomes
\begin{aligned} \mathrm{e}^{\int _0^T \sum _{i=1}^n \lambda _i(t)\mathrm{d}t}=\prod _{i=1}^{n}\mu _i. \end{aligned}
(3.17)
Suppose the eigenvalue $$\lambda _n$$ to be larger than zero during one period, i.e. $${\lambda _n(t)>0 \ \forall \ t \in [0,T]}$$. Splitting $$\lambda _n$$ from the integral Eq. (3.17) becomes
\begin{aligned} \mathrm{e}^{\int _0^T \sum _{i=1}^{n-1} \lambda _i(t)\mathrm{d}t+\int _0^T\lambda _n(t)\mathrm{d}t}=\prod _{i=1}^{n}\mu _i. \end{aligned}
(3.18)
If the condition
\begin{aligned} \int _0^T \sum _{i=1}^{n-1} \lambda _i(t)\mathrm{d}t+\int _0^T\lambda _n(t)\mathrm{d}t<0 \end{aligned}
(3.19)
is satisfied, Eq. (3.18) becomes
\begin{aligned} \prod _{i=1}^{n}\mu _i<1. \end{aligned}
(3.20)
On the one hand, the product of n Floquet multipliers being smaller than one may contain factor values larger than one or only Floquet multipliers smaller than one and no conclusion about the largest Floquet multiplier can be drawn. On the other hand, consider the case
\begin{aligned} \int _0^T\sum _{i=1}^n \lambda _i(t)\mathrm{d}t=0. \end{aligned}
(3.21)
\begin{aligned} \prod _{i=1}^{n}\mu _i=1 \end{aligned}
(3.22)
and it can be concluded that either there exists at least one Floquet multiplier with an absolute value larger than one or each Floquet multiplier has an absolute value equal to one. Hence, the system either is weakly stable or unstable. Consequently, if
\begin{aligned} \int _0^T\sum _{i=1}^n \lambda _i(t)\mathrm{d}t>0 \end{aligned}
(3.23)
is satisfied, the system must be unstable. As a result, condition (3.19) is necessary for a time-periodic system to be asymptotically stable. In most cases, this condition is fulfilled for damped systems since it is equivalent to
\begin{aligned} -\int _0^T\mathrm{tr}[\mathbf A(t)]\mathrm{d}t=\int _0^T \mathrm{tr}[\mathbf M^{-1}(t)\mathbf D(t)]\mathrm{d}t>0 \end{aligned}
(3.24)
which is satisfied if $$\mathbf D(t)>0 \ \forall \ t \in [0,T].$$ Furthermore, since $${\int _0^T\mathrm{tr}[\mathbf A(t)]\mathrm{d}t=0}$$, if $$\mathbf D(t)=0 \ \forall t \in [0,T]$$, it can be seen with regard to condition (3.21), that an undamped time-periodic system, similar to the case of time-invariant systems, cannot be asymptotically stable; it may only be weakly stable.
The error that occurs using CEA for some frozen times in technically time-periodic models is shown in Fig. 3.9. The parameter $$\beta _1$$ on the abscissa is the weighting factor associated with the stiffness $$k_t$$ of the disc and the parameter $$\beta _2$$ on the ordinate is the weighting factor associated with the stiffness k of the pins, both of which are are chosen as in (3.1). The comparison between the stability maps computed using Floquet theory and those applying the concept of frozen times demonstrate the trap. The former predicts circle-like areas of instability, whereas the latter yields misleading results. At certain times, e.g. at $$t=0$$, a fan-shaped instability area appears. Within the time interval $${\frac{2}{10}T<t<\frac{8}{10}T}$$ this area totally disappears and reappears for $${t\in (\frac{8}{10}T,T]}$$.
Since (3.17) is equivalent to
\begin{aligned} \mathrm{e}^{\int _0^T \mathrm{tr}[-\mathbf M^{-1}(t)\mathbf D(t)]\mathrm{dt}}=\prod _{i=1}^{n}\mu _i, \end{aligned}
(3.25)
it can be concluded that the structure of a time-dependent damping matrix plays also an important role with regard to stability. Consider the equations of motion of the wobbling disc with the matrices (3.2) and $$m_p=0$$. To ensure the system to be either unstable or weakly stable the damping matrix $$\mathbf D$$ must satisfy
\begin{aligned} \mathrm{tr}[\mathbf M^{-1}\mathbf D(t)]=0. \end{aligned}
(3.26)
Introducing the mass and damping matrix from (3.2) yields
\begin{aligned} \delta =-2\frac{d r^2}{d_t}-\frac{\mu N_0 h^2}{2d_t \Omega r}-2. \end{aligned}
(3.27)
As discussed in Sect. 3.3.2, numerical investigations suggest that asymmetric damping properties only have little influence on the stability behavior of the investigated brake model. Especially, a destabilization of the system, when $${\delta \ne 0}$$, is not possible because $$\delta >0$$ and (3.27) cannot be satisfied simultaneously. However, since the system may be unstable (or stable), if $$\int _0^T\mathrm{tr}[\mathbf M^{-1}\mathbf D(t)]\mathrm{d}t<0$$, there may be unknown values $$\delta$$ which have a more significant influence on the system’s stability.

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## Authors and Affiliations

• Jan-Hendrik Wehner
• 1
Email author
• Dominic Jekel
• 2
• Rubens Sampaio
• 3
• Peter Hagedorn
• 2
1. 1.WeinheimGermany