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Nguyen-Thai, Minh-Tuan; Wulff, Paul; Gräbner, Nils; Wagner, Utz von (2020). On the influence of

external stochastic excitation on linear oscillators with subcritical self-excitation and gyroscopic influence

with application to brake squeal. ZAMM – Journal of Applied Mathematics and Mechanics.

https://doi.org/10.1002/zamm.202000113

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Archived Versions.

Minh-Tuan Nguyen-Thai, Paul Wulff, Nils Gräbner, Utz von Wagner

On the influence of external stochastic

excitation on linear oscillators with

subcritical self-excitation and gyroscopic

influence with application to brake squeal

Accepted manuscript (Postprint)Journal article |

Received: 00 Month 0000 Revised: 00 Month 0000 Accepted: 00 Month 0000

DOI: xxx/xxxx

SPECIAL ISSUE PAPER

On the influence of external stochastic excitation on linear

oscillators with subcritical self-excitation and gyroscopic influence

with application to brake squeal

Minh-Tuan Nguyen-Thai*1| Paul Wulﬀ2| Nils Gräbner2| Utz von Wagner2

1Department of Applied Mechanics, Hanoi

University of Science and Technology,

Hanoi, Vietnam

2Chair of Mechatronics and Machine

Dynamics, Technische Universität Berlin,

Berlin, Germany

Correspondence

*Minh-Tuan Nguyen-Thai, Email:

tuan.nguyenthaiminh@hust.edu.vn

Summary

In a linear analysis for brake squeal, an unwanted type of sound in the kHz-range

produced during the braking process of vehicles, usually only the stability of the

system is examined. However, with the appearance of additional stochastic exci-

tation, the vibration of a linear system with subcritical self-excitation, i.e. having

self-excitation but due to damping still an asymptotically stable trivial solution, may

be large enough to produce a squeal sound. In this paper, this hypothesis of stochasti-

cally reinforced self-excitation is supported by a case study on a wobbling disk model

for brake squeal, which includes both circulatory and gyroscopic forces. For this

example, the Fokker-Planck equation is solved and numerical integrations are per-

formed. A short parameter study is carried out to examine the eﬀect of damping and

gyroscopic terms on these stochastically reinforced self-excitation. The results sug-

gest that this possibility should be considered additionally to classical explanations

of brake squeal.

KEYWORDS:

brake squeal, stochastic excitation, stochastically reinforced self-excitation, gyroscopic eﬀects

1INTRODUCTION

This special issue paper is the subsequent study of our report on a new hypothesis on the mechanism of brake squeal at the DSTA-

2019 Conference in Lodz[1] extending the therein examined linear model with gyroscopic terms and considering a minimal

model for brake squeal.

Brake squeal is an unwanted type of sound in the kHz-range produced during the braking process of vehicles. Despite the

existence of much literature discussing the cause and mechanism of brake squeal like the review papers[2,3], there has not been

any mathematical model for brake squeal that ﬁts every realistic observation and could be used for a predictive design. Since in

experimental observations, the frequency of brake squeal is almost independent on disk speed, this type of noise is commonly

assumed to be the result of self-excited vibration, not forced vibration. This assumption has led to many linear models with self-

excitation for brake squeal[4,5,6]. Stability analysis is applied to these models to predict, when the unwanted vibration occurs.

However, when the trivial solution of a linear system is unstable, the vibration amplitude will not be restricted and will increase to

inﬁnity as time goes on. This is not true in the reality of brake squeal, where the vibration displacement amplitudes of brake disk

and the brake pads lie in the micrometer-range. Thus, nonlinear models are derived to explain the limitation of the amplitudes,

and brake squeal is modeled as a limit cycle[7,8,9]. It is extremely hard to measure the nonlinear characteristics, especially those

2Minh-Tuan Nguyen-Thai ET AL.

of the frictional contacts[10,11,12], so that the validities of the nonlinear models are hard to conﬁrm. Normally, when the trivial

solution of a linear system is stable, brake squeal is considered not to happen, disregarding the transient process – the period

of time the system changes from an initial state to the steady state. The transient process is usually neglected because due to

the negative maximum real part of the eigenvalues, unwanted phenomena are not considered to be likely to happen during this

process and even if they happen, they are not expected to last long. However, if transient growth occurs and resets repeatedly,

it will aﬀect the behavior of the system signiﬁcantly. Transient growth is a phenomenon in which a growth in perturbation

happens at the beginning of the transient process before the decay ending in the steady state. Transient growth is well-known

in turbulence studies[13,14], but, to the knowledge of the authors, there has been only few publications in the ﬁeld of mechanical

engineering[15,16,17]. In[1], we presented a two degree of freedom (2-DOF) system that can undergo repeatedly transient growths.

The idea and results are summarized as follows.

Firstly, a set of homogenous linear ordinary diﬀerential equations (ODE) is considered



𝐱+𝐃

𝐱+ (𝐊+𝐍)𝐱=𝟎,(1)

where Dis the symmetric positive deﬁnite damping matrix and Kthe symmetric and also positive deﬁnite stiﬀness matrix. The

Nmatrix is skew-symmetric and implies the appearance of non-conservative circulatory forces, which means there is possibility

for the system to exhibit self-excited vibration. Here, we consider parameter cases where the trivial ﬁxed point is asymptotically

stable, thus the term “subcritical self-excitation” is used.

Then, to model uncertainties that may aﬀect the real system, a stochastic excitation is added and Eq. (1) becomes



𝐗𝑡+𝐃

𝐗𝑡+ (𝐊+𝐍)𝐗𝑡=𝛔𝜉𝑡.(2)

The expectation is that the stochastic excitation may continuously restart the transient process, leading to repeatedly transient

growths that maintain over time a large enough, but bounded, vibration considered as squeal sound. In this paper, this cause of

vibration is called stochastically reinforced self-excitation, since the transient growth is probably the consequence of the (sub-

critical) self-excitation. To conﬁrm this eﬀect, we consider two systems: system (2) and a respective system without circulatory

forces, i.e. no subcritical self-excitation, but having the same negative maximum real part of the eigenvalues. It turns out that

under an appropriate stochastic excitation, the vibration level of the system with subcritical self-excitation is much larger than

that in the no self-excitation case as described in[1].

This paper extends these results by adding a 𝐆matrix, representing the appearance of gyroscopic forces, to the considered

equations. Such equations can be derived from a minimal model for brake squeal with a wobbling disk.

Note that a linear model is usually the result of linearization from an original nonlinear model. In brake squeal, there is the

possibility that a stable limit cycle co-exists with a stable equilibrium but also stable limit cycles and unstable equilibria as

well as solely stable equilibria occur depending on parameters[7],[12],[18]. In simpliﬁed considerations, the equations linearized

with respect to the equilibrium are investigated and asymptotic stability of the equilibrium is interpreted as non-existence of

squeal, which is in the nonlinear case only true if the basin of attraction of this equilibrium is not left. One may expect that

small initial conditions (i.e. small deviations from equilibirum) are in the basin of attraction of the stable equlibrium; however,

transient growth according to self-excitation and without the help of stochastic excitation is proved to make a solution even

with small initial conditions able to ﬁnally reach a stable limit cycle[16].The actual paper describes that even in the case of non-

leaving the basin of attraction of the stable equilibrium, vibrations comparable to squeal may occur in the presence of external

noise. Nevertheless, it is possible that stochastically reinforced self-excitation may also bring the solution to the attraction of a

coexisting stable limit cycle in the nonlinear case. However, this problem is out of the scope of this paper and may be investigated

in the future.

2A 2-DOF EDGKN SYSTEM WITH STOCHASTIC EXCITATION FOR BRAKE SQUEAL

Consider a low-DOF brake system model with wobbling disk and frictional point contacts (Fig. 1). This model includes a rigid

wobbling disk and a pair of massless pads elastically suspended along both normal (associated with out-plane vibration) and

circumferential (associated with in-plane vibration) directions with respect to the disk[12]. This model is an extension of the

minimal model introduced in[6] while such an extension with massless pads performing circumferential movement can also

already be found e.g. in[19,20]. A more complicated pad suspension system based on[6] is found in[8].

Minh-Tuan Nguyen-Thai ET AL.3

FIGURE 1 A low-DOF brake system model with wobbling disk and frictional point contacts (Fig.5.1-[12]).

The perturbation equations are derived in[12] and read



𝐌

𝐱+

𝐁

𝐱+

𝐂𝐱 =𝟎,(3)

where



𝐌=[Θ 0

0 Θ ],(4)



𝐁=⎡⎢⎢⎢⎣

𝑑𝑡+ (𝑑1+𝑑2)𝑟2+𝜇2

𝑘𝑁0ℎ

2(𝑑1

𝑘𝑖𝑝1

+𝑑2

𝑘𝑖𝑝2)ΦΩ𝑑

𝜇𝑘𝑁0𝑟(𝑑1

𝑘𝑖𝑝1

+𝑑2

𝑘𝑖𝑝2)− (𝑑1+𝑑2)𝜇𝑘ℎ𝑟

2− ΦΩ𝑑𝑑𝑡⎤⎥⎥⎥⎦

,(5)



𝐂=⎡⎢⎢⎢⎣

𝑘𝑡+𝑁0ℎ+ (𝑘1+𝑘2)𝑟2+𝜇2

𝑘𝑁0ℎ

2(𝑘1

𝑘𝑖𝑝1

+𝑘2

𝑘𝑖𝑝2)𝜇2

𝑘𝑁0ℎ

2(ℎ− (1 + 𝜇2

𝑘)𝑁0

𝑘𝑖𝑝1

+𝑁0

𝑘𝑖𝑝2)

−𝜇𝑘𝑟(2𝑁0−𝑁0𝑘1

𝑘𝑖𝑝1

−𝑁0𝑘2

𝑘𝑖𝑝2

+𝑘1ℎ

2+𝑘2ℎ

2)𝑘𝑡+ (1 + 𝜇2

𝑘)𝑁0(ℎ−𝑁0

𝑘𝑖𝑝1

−𝑁0

𝑘𝑖𝑝2)⎤⎥⎥⎥⎦

.(6)

In the equations, ℎis the thickness of the brake disk and 𝑟is the eﬀective braking radius. Θand Φare the moments of inertia

of the brake disk with respect to its in-plane and out-of-plane symmetry axes, respectively. The disk is elastically supported by

two rotational springs of stiﬀness 𝑘𝑡and rotational dampers of damping coeﬃcient 𝑑𝑡. The pads are supported in the normal

direction with respective to the disk by linear springs of stiﬀnesses 𝑘1and 𝑘2, and linear dampers of damping coeﬃcients 𝑑1and

𝑑2. The in-plane supports of the pads are linear springs of stiﬀnesses 𝑘𝑖𝑝1and 𝑘𝑖𝑝2. The coeﬃcient of kinetic friction between the

pads and the disk is 𝜇𝑘and Ω𝑑is the present macroscopic disk angular velocity, which is hereafter called in short as disk speed.

Eq. (3) can be rewritten as



𝐱+𝐁

𝐱+𝐂𝐱 =𝟎,(7)

where

𝐁=1



𝐁,𝐂=1



𝐂.(8)

4Minh-Tuan Nguyen-Thai ET AL.

Matrix 𝐁can be split into a symmetric matrix 𝐃(damping matrix) and a skew-symmetric matrix 𝐆(gyroscopic matrix),

while matrix 𝐂can be split into a symmetric matrix 𝐊(stiﬀness matrix) and a skew-symmetric matrix 𝐍(circulatory matrix).

Eq. (7) then reads



𝐱+ (𝐃+𝐆)

𝐱+ (𝐊+𝐍)𝐱=𝟎,(9)

where

𝐃=1

2(𝐁+𝐁⊤),𝐆=1

2(𝐁−𝐁⊤),(10)

𝐊=1

2(𝐂+𝐂⊤),𝐍=1

2(𝐂−𝐂⊤),(11)

System (9) is correspondingly called an EDGKN system, a special case of an MDGKN system[21]. The replacement of M by

the identity matrix E eases the below-mentioned parameter study. It also marks the fact that the original mass matrix in equation

(3) is just a scaled identity matrix. Not all 2-DOF linear models for brake squeal have this property, e.g. the model described in[4].

The gyroscopic term also does not appear for some other models[4,5] and the equations of motion just form an MDKN system.

It is already known that, depending on the parameters, the considered system can be either unstable, so that self-excited

vibration predominates the damping, or stable, i.e. the self-excitation is subcritical[1]. This fact makes the system a candidate

to test the abovementioned hypothesis. For stochastically reinforced self-excitation to occur, an external stochastic excitation is

added to the model, whose equations then read as follows



𝐗𝑡+ (𝐃+𝐆)

𝐗𝑡+ (𝐊+𝐍)𝐗𝑡=𝛔𝜉𝑡.(12)

There are several candidates for this random excitation, e.g. the roughness of the contact surfaces, the frictional behavior or

other sources from the surrounding. For simplicity for these fundamental investigations, the noise 𝜉𝑡is assumed to be a scalar

Gaussian white noise with zero mean and the vector 𝛔contains their intensity coeﬃcients.

3COMPARISON OF SELF-EXCITED AND NON-SELF-EXCITED SYSTEMS IN THE

PRESENCE OF STOCHASTIC EXCITATION

To carve out the diﬀerence in the behavior of the EDGKN system and a suitable EDK system, we consider the following case.

The considered EDGKN system has subcritical self-excitation, while the EDK system has no self-excitation, provided that the

stiﬀness and damping matrices are both positive deﬁnite. If under the same stochastic excitation, the vibration largeness of the

EDGKN system is much higher than that of the EDK system, it might be because the stochastic excitation eﬀectively reinforces

the self-excitation. However, this statement is not reasonable if the EDGKN system has a higher negative maximum real part

of the eigenvalues than the EDK system, because in that case one may claim that this diﬀerence might be the reason for the

diﬀerence in the vibration largeness. This is one reason why the two systems within one comparison should have the same

negative maximum real part of the eigenvalues. The systems that share the same maximum real part of the eigenvalues are

hereafter called maximum-real-part-of-the-eigenvalues-companions (MRECs). Another reason to focus on the maximum real

part of the eigenvalues is to prove that the sign of the maximum real part of the eigenvalues should not be the only concern when

considering linear mathematical brake squeal models.

Therefore we consider an EDGKN system, and a corresponding EDK system that has the same stiﬀness matrix and the

damping matrix is scaled from that of the EDGKN system by



𝐱+𝛼𝐃

𝐱+𝐊𝐱 =𝟎.(13)

Adjusting the damping scale factor 𝛼, one might get a non-circulatory non-gyroscopic MREC of the given EDGKN system,

i.e. they have the same (negative) maximum real part of the eigenvalues. Adding stochastic excitation also in this case yields



𝐗𝑡+𝛼𝐃

𝐗𝑡+𝐊𝐗𝑡=𝛔𝜉𝑡.(14)

Minh-Tuan Nguyen-Thai ET AL.5

The systems from (12) and (14) are in the following called stochastically excited maximum-real-part-of-the-eigenvalues-

companions (SEMRECs).

As a measure for the intensity of vibrations we consider the marginal probability density function (PDF) of state variables.

The larger the marginal PDF deviates from zero, the larger the intensity of the vibration of the corresponding system is.

The probability density function can either be calculated using numerical integration (Monte-Carlo simulation) or by solving

the corresponding Fokker-Planck equation which is in general a challenging task, but here comparably simple, as we have linear

systems with Gaussian excitation. In both cases, the second-order SDEs (12) and (14) are rewritten as ﬁrst-order systems of the

form

d𝐐𝑡=𝐀𝐐𝑡+𝐠d𝑊𝑡,(15)

where 𝐐𝑡is the vector of the random state processes

𝐐𝑡=[𝐗𝑡



𝐗𝑡].(16)

𝑊𝑡is the Wiener process corresponding to 𝜉𝑡, and the other matricies are determined as follows

𝐀=[𝟎 𝐄

−𝐂−𝐁],(17)

𝐠=⎡⎢⎢⎣

𝛔⎤⎥⎥⎦

.(18)

The 𝐁and 𝐂matrices in both EDGKN and EDK cases can be found by rewriting the system in the form of equation (7).

The diﬀusion matrix is deﬁned as

𝐇=𝐠𝐠⊤.(19)

The PDF 𝑝(𝐪)of the stationary process 𝐐𝑡can be found by solving the stationary Fokker-Planck equation associated with Eq.

(12) and (14)

∑

𝑖=1

𝜕

𝜕𝑞𝑖[𝑝(𝐪)

∑

𝑗=1

𝑎𝑖𝑗 𝑞𝑗]−1

∑

𝑖=1

∑

𝑗=1

𝜕2

𝜕𝑞𝑖𝜕𝑞𝑗[𝑝(𝐪)ℎ𝑖𝑗 ]= 0 ,(20)

where 𝑎𝑖𝑗 and ℎ𝑖𝑗 are the elements in row 𝑖and column 𝑗of matrix 𝐀and matrix 𝐇, respectively.

Since we have a linear system with Gaussian excitation, the corresponding solution is also Gaussian. Following[22], the

solution has the form

𝑝(𝐪) = 1

(2𝜋)2|𝚲|1

exp (−1

2𝐪⊤𝚲−1𝐪),(21)

with mean value zero vector and covariance matrix 𝚲. The mean value vector is a zero vector while the covariance matrix is to

be found. Eq. (21) yields to the following algebraic equation[23]

[(𝐄⊗𝐀) + (𝐀⊗𝐄)]vec(𝚲) + vec(𝐇) = 0 ,(22)

where ⊗denotes the Kronecker product and vec(⋅)denotes vectorization operator.

Solving (22) for 𝚲and computing (21), one obtains 𝑝(𝐪). Marginal PDFs 𝑝𝑋1and 𝑝𝑋2are also Gaussian and can be calculated

6Minh-Tuan Nguyen-Thai ET AL.

𝑝𝑋1(𝑥1) = 𝑝𝑄1(𝑞1) =

∞

∫

−∞

∞

∫

−∞

∞

∫

−∞

𝑝(𝐪) d𝑞2d𝑞3d𝑞4,(23)

𝑝𝑋2(𝑥2) = 𝑝𝑄2(𝑞2) =

∞

∫

−∞

∞

∫

−∞

∞

∫

−∞

𝑝(𝐪) d𝑞1d𝑞3d𝑞4.(24)

The corresponding standard deviations of the marginal PDFs can be determined as

𝜎1=√Λ11 , 𝜎2=√Λ22 ,(25)

where Λ11 and Λ22 are the ﬁrst two element on the main diagonal of the covariance matrix 𝚲.

The stochastic-excitation ratios between the SEMRECs are deﬁned as

𝜌1=𝜎1−𝐸𝐷𝐺𝐾𝑁

𝜎1−𝐸𝐷𝐾

, 𝜌2=𝜎2−𝐸𝐷𝐺𝐾𝑁

𝜎2−𝐸𝐷𝐾

.(26)

If either 𝜌1or 𝜌2or both are signiﬁcantly higher than 1, a stochastically reinforced self-excitation is considered to be eﬀective.

4COMPARISON OF VIBRATIONAL BEHAVIOR OF EDGKN AND EDK SYSTEMS

SUBJECTED TO EXTERNAL STOCHASTIC EXCITATION

4.1 A case study

Consider the SDE (12) that is derived from the perturbation equation (3). The parameters are chosen as follows

ℎ= 0.05 m;𝑟= 0.13 m; Θ = 0.16 kg m2; Φ = 2Θ ; 𝑘𝑡= 1.88 × 107Nm ;𝑑𝑡= 0.1Nm s ;

𝑘1= 6 × 106Nm ;𝑘2= 6 × 106Nm ;𝑑1= 5 Ns/m ;𝑑2= 5 Ns/m ;𝑘𝑖𝑝1= 6 × 106Nm ;

𝑘𝑖𝑝2= 6 × 106Nm ;𝜇𝑘= 0.6 ; 𝑁0= 3000 N; Ω𝑑= 5𝜋rad/s

Most of the physical parameters are taken from the literature[6,12]. The obtained system matrices are

𝐃=[8.5682 −0.059719

−0.059719 0.625 ]s−1 ,(27)

𝐆=[0 31.476

−31.476 0 ]s−1 ,(28)

𝐊=[1.1876 × 108−7.3072 × 104

−7.3072 × 1081.175 × 108]s−2 ,(29)

𝐍=[0 7.3178 × 104

−7.3178 × 1040]s−2 .(30)

The corresponding trivial solution is stable with the maximum real part of the eigenvalues 𝜆= −2.1451 s−1. The EDK MREC

of the form (13) is obtained with the damping scale factor 𝛼= 6.6933. Though the two MRECs are both stable, their behaviors

are diﬀerent given the same initial condition 𝑥= [1,0]⊤rad: the EDK system shows decaying vibration with a short transient

process (Fig. 2b) while the EDGKN shows a beating phenomenon and maintains much higher amplitudes during the same period

(Fig. 2a). Nevertheless, conventional linear analysis for brake squeal treats them equally according to the same maximum real

part of the eigenvalues. The beating phenomenon has been found also in EDKN system[15].

Minh-Tuan Nguyen-Thai ET AL.7

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-1.0

-0.5

0.0

0.5

1.0

t(s)

x×105(rad)

0.00 0.05 0.10 0.15 0.20 0.25 0.30

-1.0

-0.5

0.0

0.5

1.0

t(s)

x×105(rad)

a) EDGKN system: Beating phenomenon b) EDK system: Decaying vibration without beating

FIGURE 2 Time history for the deterministic MRECs.

-1 0 1

x1×105(rad)

pX1×10-5

EDK

EDGKN

-0.5 0 0.5

0.5

1.5

2.5

x2×105(rad)

pX2×10-

EDK

EDGKN

a) b)

FIGURE 3 Marginal PDFs 𝑝𝑋1and 𝑝𝑋2of the SEMRECs.

Performing the deviation comparison for the corresponding SEMRECs (12) and (14) with the intensity vector 𝛿=

[0,0,0.1,0]⊤s−2, one gets the stochastic-excitation ratios

𝜌1= 3.2271 >1, 𝜌2= 6.6971 >1.(31)

Therefore, the stochastically reinforced self-excitation is eﬀective for both state variables. The marginal PDF plots also conﬁrm

that in the EDGKN case the state variables distribute in much wider ranges (Fig. 3a, 3b). To further illustrate the mechanism of

brake squeal, the time responses of the two SEMRECs to a speciﬁc excitation are calculated using the Euler-Maruyama method.

It can be seen clearly in Fig. 4 that the vibration in the EDGKN case are much larger so it is more likely to produce brake

squeal than in the EDK case. The frequency characteristic of the responses 𝑋1𝑡and 𝑋2𝑡for the EDGKN case is visualized by

the plot of the absolute values of corresponding transfer functions 𝐻1(𝑓)and 𝐻2(𝑓)in Fig. 5, the peaks of the plots lie in the

kHz-range, ﬁtting the characteristic of brake squeal. The contour plot in Fig. 6 shows that the dominating frequencies are nearly

independent on the disk speed, which is a typical property of disk brake squeal[24] as illustrated by measurement results in Fig.

7 (taken from[18], Fig. 42).

This case study shows that, though still having a negative maximum real part of the eigenvalues, an EDGKN system with

stochastic excitation may exhibit a vibration that has some similar properties to a measured vibration during brake squeal. It

means stochastically reinforced self-excitation should be considered additionally to the well-known maximum real part of the

eigenvalues when analyzing brake squeal through mathematically models.

8Minh-Tuan Nguyen-Thai ET AL.

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

t(s)

X×105(rad)

X2t

X1t

0.0 0.5 1.0 1.5 2.0

-1.0

-0.5

0.0

0.5

1.0

t(s)

X×105(rad)

X2t

X1t

a) EDGKN system b) EDK system

FIGURE 4 Realizations of the processes 𝑋1𝑡and 𝑋2𝑡for the SEMRECs with the same excitation intensity

1 1 1 1 1



0





10



f(s-1)

|H| ×105

|H2|

|H1|

FIGURE 5 Transfer functions |𝐻1(𝑓)|and |𝐻2(𝑓)|of the EDGKN system

4.2 Parameter study

It is complicated to do a full parameter study for the EDGKN system, since the number of parameters is as high as 8 and the

number of physical parameters is even higher. To simplify the problem, the number of parameters should be reduced.

Starting from the matrices (27)-(30), doing a scale in time and simplifying the numeric values, one obtains the following

dimensionless matrices

𝐃=[0.0008 0

0 0.00006 ],(32)

𝐆=[0 0.003

−0.003 0 ],(33)

𝐊=[1 −0.0006

−0.0006 0.99 ],(34)

𝐍=[0 0.0006

−0.0006 0 ].(35)

The considered intensity vector is 𝛿= [0,0,1,0]⊤.

Since the gyroscopic terms are of interest, matrix G should be parameterized

Minh-Tuan Nguyen-Thai ET AL.9

    



f(s-1)

Ωd(rad/

)

d

-73.6

-70.4

-67.2

-64.0

-60.8

-57.6

-54.4

-51.2

-48.0

FIGURE 6 Transfer function |𝐻1(𝑓)|of the EDGKN system at diﬀerent disk speeds.

FIGURE 7 Measured results from a brake test rig. Vibration amplitude of the disk (squeal intensity) over the frequency and

the rotational speed of the disk (taken from[18], Fig. 42)

𝐍=[0 0.0008𝛾

−0.0008𝛾0],(36)

where 𝛾is deﬁned as the ratio between the second element of the ﬁrst row of 𝐆and the ﬁrst element of 𝐃

𝛾=𝑔12

𝑑11

.(37)

The matrix 𝐃is also parameterized

𝐍=[0.0008 0

0 0.0008𝛽],(38)

where 𝛽is deﬁned as the ratio between the damping coeﬃcient in the second equation and that in the ﬁrst one

10 Minh-Tuan Nguyen-Thai ET AL.

0.0 0.2 0.4 0.6 0.8 1.0

ρ2

ρ1

FIGURE 8 Stochastic-excitation ratios dependence on 𝛽.

0.0 0.2 0.4 0.6 0.8 1.0

ρ2

ρ1

FIGURE 9 Stochastic-excitation ratios dependence on 𝛾.

𝛽=𝑑22

𝑑11

.(39)

When not otherwise indicated, the parameters are set as 𝛽= 3.7 and 𝛾= 0.075.

It can be seen in Fig. 8 that as 𝛽becomes lower, both the stochastic-excitation ratios increase signiﬁcantly. Taking a look

back at (5), this means lowering 𝑑𝑡– the rotational damping of the wobbling disk – may increase the intensity of the resulting

vibrations.

The dependence of the stochastic-excitation ratios on 𝛾is shown in Fig. 9. When 𝛾reaches 4.2, 𝜌1peaks at 37 and 𝜌2at 11.

Considering (5), (8), (10) and (37), 𝛾and the disk speed Ω𝑑have a direct monotonic relationship so there is no upper bound

for 𝛾; however, if 𝛾is too high, the original EDGKN system is unstable so the deviation comparison is no longer valid. The

stochastic-excitation ratios decrease when 𝛾increases beyond 4.2 but do not drop below 6.

Fig. 10 shows the dependence of the stochastic-excitation ratios on both 𝛾and 𝛽, where 𝛽takes all positive values below 1

and 𝛾varies from 0 to 15. For all values of 𝛾, the stochastic-excitation ratios increase rapidly when 𝛽approaches 0, just similar

to the case shown in Fig. 8.

5CONCLUSIONS AND OUTLOOK

In this paper a stochastically excited EDGKN system derived from a 2-DOF model for brake squeal has been proved to possibly

have much larger vibration than an EDK system with the same negative maximum real part of the eigenvalues and under the

Minh-Tuan Nguyen-Thai ET AL.11

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

ρ1

0.4

0.8

1.2

1.6

2.0

2.4

2.8

3.2

3.6

0 2 4 6 8 10 12 14

0.0

0.2

0.4

0.6

0.8

1.0

ρ2

a) b)

FIGURE 10 Stochastic-excitation ratios with varied 𝛾and 𝛽for a) the ﬁrst state variable and b) the second state variable .

same stochastic excitation. It means that when considering brake vibrations based on linear analysis, one should consider not

only the stability of the system but also hypothesized stochastically reinforced self-excitation. A partial parameter study has

shown that the ratio between the main damping coeﬃcients and the gyroscopic terms do aﬀect the stochastic-excitation ratios.

Thus, future work might focus on the optimization to avoid brake squeal. Further parameter study might also be performed to

contribute more understanding to the stochastically excited EDGKN system.

Beyond stable linear systems, the combination and interaction of stochastically reinforced self-excitation, instability, and

nonlinearities might be examined in future work.

ACKNOWLEDGMENTS

At the early stage of this study, the ﬁrst author acknowledged the ﬁnancial support by Project 911 (from Vietnamese Govern-

ment) and a supplemental scholarship from DAAD.

CONFLICT OF INTEREST

The authors declare no potential conﬂict of interests.

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