How to Play a Disc Brake: A Dissipation-Induced Squeal

Kirillov, Oleg N.

doi:10.4271/2008-01-1160

Cited by 4 publications

(8 citation statements)

References 47 publications

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“…Nevertheless, an axially symmetric rotor with an anisotropic stator as well as an asymmetric rotor with an isotropic stator can be described as an autonomous non-conservative gyroscopic system (Genta, 2007). Unless the vented disk or the disk with specially manufactured symmetry-breaking pattern (Fieldhouse et al, 2004) is considered, the model of an axially symmetric rotor with an anisotropic stator is reasonable for the description of the disk brake as well as for the description of other sound-emitting rotating elastic bodies of revolution in frictional contact such as the drum brake or the glass harmonica (Kirillov, 2008a).…”

Section: Disk Brake As An Axially Symmetric Rotor With An Anisotropicmentioning

confidence: 99%

Perspectives and obstacles for optimisation of brake pads with respect to stability criteria

Kirillov¹

2009

IJVD

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Structural modifications of the brake components is an evident passive countermeasure directed on elimination of propensity of the brake to a squeal. The assignment of the unstable eigenvalues into the stable part of the complex plane by the structural optimisation of the non-conservative rotor system requires, however, careful investigation of the doublets in the Campbell diagram. Such an investigation substantially illuminates the perspectives and obstacles for the structural optimisation of brake pads with respect to stabilising eigenvalue assignment and provides explicit formulas, which allow one to assign desirable stable and unstable modes of the coupled system selectively.

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Section: Disk Brake As An Axially Symmetric Rotor With An Anisotropicmentioning

confidence: 99%

Perspectives and obstacles for optimisation of brake pads with respect to stability criteria

Kirillov¹

2009

IJVD

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“…According to the ideas going back to von Neumann and Wigner [1], in the multiparameter operator families, eigenvalues with various algebraic and geometric multiplicities can be generic [12]. In some applications additional symmetries yield the existence of spectral meshes [33] in the plane 'eigenvalue versus parameter' containing infinite number of nodes with the multiple eigenvalues [10,15,30,38]. As it has been pointed out already by Rellich [2] sensitivity analysis of multiple eigenvalues is complicated by their nondifferentiability as functions of several parameters.…”

Section: Introductionmentioning

confidence: 99%

“…As it has been pointed out already by Rellich [2] sensitivity analysis of multiple eigenvalues is complicated by their nondifferentiability as functions of several parameters. Singularities corresponding to the multiple eigenvalues [12] are related to such important effects as destabilization paradox in near-Hamiltonian and near-reversible systems [5,18,25,26,27,35], geometric phase [31], reversals of the orientation of the magnetic field in MHD dynamo models [32], emission of sound by rotating continua interacting with the friction pads [38] and other phenomena [23].…”

Section: Introductionmentioning

confidence: 99%

Perturbation of multiparameter non-self-adjoint boundary eigenvalue problems for operator matrices

Kirillov

2008

Preprint

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We consider two-point non-self-adjoint boundary eigenvalue problems for linear matrix differential operators. The coefficient matrices in the differential expressions and the matrix boundary conditions are assumed to depend analytically on the complex spectral parameter λ and on the vector of real physical parameters p. We study perturbations of semi-simple multiple eigenvalues as well as perturbations of non-derogatory eigenvalues under small variations of p. Explicit formulae describing the bifurcation of the eigenvalues are derived. Application to the problem of excitation of unstable modes in rotating continua such as spherically symmetric MHD α 2 -dynamo and circular string demonstrates the efficiency and applicability of the theory.

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“…1(b) for the case of n = 2 degrees of freedom. When, however, all types of forces are involved in the perturbation, then according to [5,6] the real parts of the eigenvalues originated after the splitting of the double eigenvalue iω 1 for n = 2 are Reλ = (−trD/4 ± |c| + Rec)δ/4 with…”

Section: Conical Zones Of the Subcritical Flutter Instability Inducedmentioning

confidence: 99%

“…The distribution of the doublets as a function of s is usually different for various bodies of revolution. For example, ω s = s corresponds to the spectrum of free vibrations of a circular string [5].Separating time by the substitution x = u exp(λt) into (1), we arrive at the eigenvalue problem, whose eigenvalues for δ = κ = ν = 0 together with their complex conjugates λ ± s = iω s ± isΩ and λ ± s = −iω s ∓ isΩ form a spectral mesh in the plane (Ω, Imλ), see Fig. 1(a).…”

mentioning

confidence: 99%

Dissipation‐induced subcritical flutter in the acoustics of friction

Kirillov

2008

Proc Appl Math and Mech

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We consider a gyroscopic system under the action of small dissipative and non-conservative positional forces, which has its origin in the models of rotating elastic bodies of revolution in frictional contact such as the singing wine glass or the squealing disc/drum brakes. The spectrum of the unperturbed gyroscopic system forms a spectral mesh in the plane 'frequency versus gyroscopic parameter' with double semi-simple purely imaginary eigenvalues at the nodes. In the subcritical range of the gyroscopic parameter the eigenvalues involved into the crossings have the same Krein signature and thus their splitting due to changes in the stiffness matrix, which break the rotational symmetry of the body, cannot produce complex eigenvalues and, therefore, flutter. We establish that perturbation of the gyroscopic system by the dissipative forces with the indefinite matrix can lead to the subcritical flutter instability even if the rotational symmetry is destroyed. With the use of the perturbation theory of multiple eigenvalues we explicitly find the linear approximation to the domain of the subcritical flutter, which turns out to have a conical shape. The orientation of the cone in the three dimensional space of the parameters, corresponding to gyroscopic, damping, and potential forces, is determined by the sign of an explicit expression involving the entries of both the damping and potential matrices. With the use of a time-dependent coordinate transformation we demonstrate that the conical zones of flutter for the original autonomous system coincide with the zones of the subcritical parametric resonance of the rotationally symmetric flexible body with the load moving in the circumferential direction. Conical zones of the subcritical flutter instability induced by the indefinite dampingWe consider a linear autonomous non-conservative gyroscopic system with dissipationwhere Ω is the gyroscopic parameter, G = diag(J, 2J, . . . , nJ) = −G T and P = diag(ω 2 1 I, ω 2 2 I, . . . , ω 2 n I) = P T are the matrices of gyroscopic and potential forces withand the dissipative, conservative, and non-conservative perturbations with the real matrices D = D T , K = K T , and N = −N T are controlled by the parameters δ, κ, and ν, respectively. The transformation x = Az := exp(−ΩGt)z yields an equivalent to (1) potential system with the periodic perturbation, seeFor n = 2 the matrix N(t) := A −1 NA = N and the periodic stiffness and damping matrices areEquation (1) frequently originates after linearization and discretization of continuous models of rotating flexible bodies of revolution in frictional contact and describes their small oscillations in the stationary frame, whereas (3) describes them in the rotating frame [2][3][4]. Due to the rotational symmetry the eigenvalues ω 2 s , s = 1, 2, . . . , n, of the matrix P are double semi-simple. The distribution of the doublets as a function of s is usually different for various bodies of revolution. For example, ω s = s corresponds to the spectrum of free vibrations of a circular string...

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How to Play a Disc Brake: A Dissipation-Induced Squeal

Cited by 4 publications

References 47 publications

Perspectives and obstacles for optimisation of brake pads with respect to stability criteria

Perspectives and obstacles for optimisation of brake pads with respect to stability criteria

Perturbation of multiparameter non-self-adjoint boundary eigenvalue problems for operator matrices

Dissipation‐induced subcritical flutter in the acoustics of friction

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