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Essais & Simulations n°116

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Mesures et Methodes de

Mesures et Methodes de Mesure vg is the Eulerian sliding speed, n and t subscripts stand for the normal and tangential projections of a field on the contact interface respectively. Moreover, to deal with the unilateral contact, a non- regularized Signorini law is chosen: Fig. 4. Finite element model of TGV brake system where g is the initial gap at the contact interface. The main advantage of the Signorini law results in the fact that it does not require the introduction of a coefficient such as contact stiffness that would require measurement and should be difficult to estimate. By using classical finite element discretization of the problem with linear elements on the potential contact zone leads, the nonlinear dynamics problem may be written in a discrete form as follows (see [10] for details): where M , K and C are the classical mass, stiffness and damping matrices of the system. f and rc define the generalized force and contact reaction respectively. First of all, validation of the finite element model versus experiments is performed by applying a classical modal analysis. The contact reaction r c , the displacement u and the velocity u¯ verify the contact and friction laws defined in (eq. 1) and (eq. 2) at each mesh node. Classically, a reformulation of these contact and friction laws can be rewritten in terms of projections on the negative real set ( projx ) and on the Coulomb cone ( proj K ) is used to facilitate the numerical implementation in the treatment of the contact state [11] where ρ nu and ρ t are two arbitrary positive scalars called normal displacement augmentation parameter and tangential augmentation parameter respectively [9]. 3.2 Stability analysis In order to predict the occurrence of self-excited vibrations, a classical stability analysis can be performed. This approach can be divided into two parts. The first step is the static problem: the steady-state operating point for the full set of non-linear equations is obtained by solving them for the equilibrium point. This equilibrium point is obtained by solving the nonlinear static equations for a given net brake pressure. Then, one obtains the linearized equations of motion by introducing small perturbations about the equilibrium point into the non-linear equations [2, 8]. Stability consists on computing the complex modes and the complex eigenvalues associated to the linearized problem in the frequency range of interest. Solving this problem is achieved by using the Residual Iteration Method [12]. The complex eigenvalues λ = a + iω of provide information about the local stability of the equilibrium point. The TGV brake system is stable if all the real parts a of the eigenvalues are negative, and unstable if we have one or more eigenvalues having a positive real part. The imaginary part of these eigenvalues represents frequencies of Fig. 5. Stability of TGV brake system (red: unstable modes, blue: stable modes) Fig. 6. Mode shapes of the unstable modes (with the frequency and growth rate) unstable complex modes that correspond to squeal frequencies. The stability of the system is given on (Fig. 5). 9 unstable modes (with positive divergence) are detected. (Fig. 6) shows the mode shapes of these 9 unstable modes. We can see that the modes with the most important growth rate appear from pads modes (near 2050 Hz and 2760 Hz with a growth rate of 9.47% and 6.44% respectively). Essais & Simulations • MARS 2014 • PAGE 18

Centre technique en corrosion corrosion marine biocorrosion Etudes sur mesure Qualification de matériaux, de revêtements (anticorrosion, antifouling), biocides Expertises Identification de l’origine des dommages causés par la corrosion. Préconisation d’actions correctives Conseils Accompagnement dans le choix de matériaux, leur assemblage et méthode de protection Fig. 7. Comparison between numerical simulation (black line) and experiments (red lines) 3.3 Nonlinear self-excited vibration and comparison with experiments As previously explained in [9] and [13], the stability analysis may lead to an under- estimation or an over-estimation of the unstable modes observed in the non-linear time simulation due to the fact that linear conditions (i.e. the linearized stability around an initial equilibrium point) are not valid during transient oscillations. So the non-linear transient self-excited vibrations can become very complex and include more or less unstable modes due to the non-linear contact and loss of contact interactions at the frictional interface. Therefore, a numerical resolution of the complete nonlinear system has to be performed in addition to the stability analysis to estimate the nonlinear behaviour of the solution far from the sliding equilibrium. Since the instability of the sliding equilibrium may lead to strongly nonlinear events with contact and no-contact states at the different frictional interfaces between each pad and the disc, a first-order φ -method time integration scheme [10] is developed for the computation of the transient solution. A typical brake squeal spectrum obtained via numerical simulation is presented and compared with measurements in (Fig. 7). There is a good agreement, although slight differences may be noticed. Moreover, non-linear squeal vibrations can become complex with appearances of new frequency peaks in the signals (in comparison with a stability analysis). For example, a new resonance peak is predicted near 4000Hz. It may correspond to the second harmonic component of the unstable mode at 2050Hz. This demonstrated that squeal is composed of not only fundamental frequencies of unstable modes (i.e. eigenvalues via stability analysis) but also harmonic components and new contributions due to the coexistence of several fundamental frequencies. Recherche &Développement Programme de R&D collaboratifs sur l’étude de l’interface matériau / biofilm, fouling 02 33 01 83 40 - Essais & Simulations • MARS 2014 • PAGE 19

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