Avoidance of brake squeal by a separation of the brake ... - tuprints

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2 Avoidance of brake squeal by breaking symmetries with j > i. The stiffness matrix is diagonalized and therefore the disc’s modal asymmetry is represented by ∆k, which can be expressed by ∆k = 2π∆ω 2 , (2.13) with the difference between between two neighboring eigenfrequencies ∆ω = ω l − ω k . This difference is induced by the rotating asymmetry of the brake disc, where asymmetry in the sense used here refers to geometric modifications of the disc or modifications in the mass and/or stiffness distribution. If this asymmetry is not present, ∆ω = 0 and the eigenfrequencies under consideration ω k and ω l are the double eigenfrequencies ω i = ω j . The damping matrix, however, is not guaranteed to be diagonal, therefore the terms ∆d and ˜d account for the most general structure of the damping matrix. The following first order approximation of the stability boundary given by ∂|ρ j | ∂ǫ ∣ under the consideration of Eqs. (2.9) and (2.8) can be deduced from ǫ=0 Eq. (2.11): ( d 0 = − 2 + ∆d ) +Re 4 (√ ∆d 2 16 + ˜d 2 4 + n2 4ω 2 j − g2 4 − ∆k2 16ω 2 j ( ∆d∆k + i − gn ) ) . 8ω j 2ω j (2.14) This is the general case of the stability boundary achieved by perturbation theory and could directly be evaluated numerically. However, for the next steps the gyroscopic terms g will be neglected and the damping matrix will be assumed to be diagonal with ∆d = 0 and ˜d = 0. This enables an analytical evaluation of Eq. (2.14) and therefore an analytical approximation of the stability boundary. The influence of the gyroscopic terms and of the structure of the damping matrix will be discussed in section 2.2.4 based on numerical evaluations. Still, these numerical studies show that the most important parameters influencing the stability are covered by the analytical approximation presented in the following. For g = 0, ∆d = 0 and ˜d = 0, Eq. (2.14) simplifies to 24 0 = −4d 2 ω 2 j −∆k2 +4n 2 = s(d,∆k,n,ω j ). (2.15)
2.2 Splitting of the brake rotor’s eigenfrequencies by asymmetry The steady-sliding state of the system is stable for s < 0 and unstable for s > 0, since the derivative of the Floquet multiplier then directs into the stable region or into the unstable one, respectively (see Eq. (2.8)). Thus, the damping d and the split between neighboring eigenfrequencies ∆k act as stabilizing, while n destabilizes. This mathematical derivation justifies the experimental evidence that splitting eigenfrequencies of the brake rotor helps to avoid squeal, which is an important basis for the following steps and the structural optimization of the brake rotor. For given circulatory terms n and damping d, a certain split of eigenfrequencies (represented by ∆k) is necessary to stabilize the solution of the equations of motion leading to a minimal required grade of modal asymmetry of the brake rotor. Since the brake disc can have many pairs of double eigenfrequencies or closely spaced ones, it is necessary to introduce a split of eigenfrequencies for all pairs in a certain frequency range to assure stability. In the next section an estimate of the minimal necessary distance between eigenfrequencies is presented and the frequency range in which this has to be achieved. These are the criteria for the design of squeal free brakes. 2.2.3 Design criteria for squeal free brakes The analysis in this section has been published originally in [125]. The minimal necessary distance ∆f between eigenfrequencies of the rotor can directly be achieved by transformation of Eq. (2.15) under consideration of Eq. (2.13). This leads to ∆f = 1 4 2π √ 1 ( ) n2 π 2 −4π 2 d 2 fj 2 (2.16) with the double eigenfrequency f j . If the brake system under consideration is known and modeled in detail and therefore the circulatory terms n and the damping d are known to a certain precision, ∆f can be calculated directly. However, it is highly desirable to be able to estimate ∆f before a final brake design is known, in order to conduct a structural optimization to achieve this split to avoid squeal in the final brake disc. Therefore, in the following, an estimate for the minimal necessary distance will be derived. 25
Page 1: MECHANIK FORSCHUNGSBERICHT Avoidanc
Page 4 and 5: Wagner, Andreas Avoidance of brake
Page 6 and 7: Meinem ehemaligen Kollegen Gottfrie
Page 9 and 10: Kurzfassung Bremsenquietschen stell
Page 11 and 12: Contents 1 Introduction 1 1.1 Excit
Page 13 and 14: 1 Introduction The technical progre
Page 15 and 16: 1.1 Excitation mechanism of brake s
Page 17 and 18: 1.2 Modeling and analysis of squeal
Page 19 and 20: 1.3 Brake squeal countermeasures be
Page 21: 1.4 Outline of the thesis concept a
Page 24 and 25: 2 Avoidance of brake squeal by brea
Page 51 and 52: 3 Efficient modeling of brake discs
Page 53 and 54: 3.1 A suitable modeling approach fo
Page 59 and 60: 3.2 Longitudinal vibrations of an i
Page 61 and 62: 3.2 Longitudinal vibrations of an i
Page 63 and 64: 3.3 Free vibrations of a rectangula
Page 71 and 72: 3.4 Application to brake discs beha
Page 73: 3.4 Application to brake discs a) b
Page 76 and 77: 4 Structural optimization of automo
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4 Structural optimization of automo
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5 Conclusions After a short introdu
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Bibliography [10] Barthold, F.J.; S
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Bibliography [33] Grandhi, R.: Stru
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Bibliography [74] Massi, F.; Gianni
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Bibliography [96] Schumacher, A.: O
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Bibliography [117] Takahashi, S.: V
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Bisher sind in dieser Reihe erschie
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Band 20 Finite-Element-Modelle zur
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Brake squeal is a high-pitched nois
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