Friction-induced vibrations due to coupling modes can cause severe damage and are recognized as one of the most serious problems in industry. In order to avoid these problems, engineers must find a design to reduce or to eliminate mode coupling instabilities in braking systems. Though many researchers have studied the problem of friction-induced vibrations with experimental, analytical and numerical approaches, the effects of system parameters, and more particularly damping, on changes in stableunstable regions and limit cycle amplitudes are not yet fully understood. The goal of this study is to propose a simple non-linear two-degree-of-freedom system with friction in order to examine the effects of damping on mode coupling instability. By determining eigenvalues of the linearized system and by obtaining the analytical expressions of the Routh-Hurwitz criterion, we will study the stability of the mechanical system's static solution and the evolution of the Hopf bifurcation point as functions of the structural damping and system parameters. It will be demonstrated that the effects of damping on mode coupling instability must be taken into account to avoid design errors. The results indicate that there exists, in some cases, an optimal structural damping ratio between the stable and unstable modes which decreases the unstable region. We also compare the evolution of the limit cycle amplitudes with structural damping and demonstrate that the stable or unstable dynamic behaviour of the coupled modes are completely dependent on structural damping.

In this paper, the influence of transverse cracks in a rotating shaft is analyzed. The paper addresses the two distinct issues of the changes in modal properties and the influence of crack breathing on dynamic response during operation. Moreover, the evolution of the orbit of a cracked rotor near half of the first resonance frequency is investigated. The results provide a possible basis for an on-line monitoring system. In order to conduct this study, the dynamic response of a rotor with a breathing crack is evaluated by using the Alternate Frequency/Time Domain approach. It is shown that this method evaluates the non-linear behaviour of the rotor system rapidly and efficiently by modeling the breathing crack with a truncated Fourier series. The dynamic response obtained by applying this method is compared with that evaluated through numerical integration. The resulting orbit during transient operation is presented and some distinguishing features of a cracked rotor are examined.

a b s t r a c tThis paper outlines the non-linear transient and stationary dynamics due to friction-induced vibrations in a disc brake model. Using a finite element model and the Continuous Wavelet Transform, the contributions of fundamental frequencies and harmonic components in non-linear transient and stationary dynamics are investigated for disc brake system subjected to single and multi-instabilities. Results from these non-linear analyses demonstrate the complexity of the contributions of different harmonic components in transient friction-induced vibrations with the coexistence of multi-unstable modes. One of the most important contributions of this study is to illustrate the limitation of stability analysis related to transient and stationary non-linear behaviors. Stability analysis around an equilibrium point can only be used as the first step in providing information on the onset and increase of self-excited disc brake vibrations. Consequently, a complete non-linear analysis is necessary to fully predict non-linear vibration and the contribution of unstable modes. This study shows that an under-estimation of the unstable modes observed in the non-linear time simulation can be calculated by the stability analysis. During transient vibrations, an additional unstable mode can appear. This instability is not predicted by the complex eigenvalues analysis due to the fact that linear conditions (i.e. the linearized stability around an initial equilibrium point) are not valid during transient and stationary oscillations. So new fundamental frequencies (linked to the appearance of the new unstable mode) can emerge in the signals due to the non-linear contact and loss of contact interactions at the frictional interface. Therefore, non-linear transient and stationary self-excited vibrations can become very complex and include more unstable modes than those predicted by a linearized stability analysis around a non-linear equilibrium point.

A damping strategy for blisks (integrally bladed disks) of turbomachinery involving a friction ring is investigated. These rings, located in grooves underside the wheel of the blisks, are held in contact by centrifugal loads and the energy is dissipated when relative motions between the ring and the disk occur. A representative lumped parameter model of the system is introduced and the steady-state nonlinear response is derived using a multiharmonic balance method combined with an AFT procedure where the friction force is calculated in the time domain. Numerical simulations are presented for several damper characteristics and several excitation configurations. From these results, the performance of this damping strategy is discussed and some design guidelines are given.

The influence of the presence of transverse cracks in a rotating shaft is analysed. The paper addresses the influence of crack opening and closing on dynamic response during operation. The evolution of the orbit of the cracked rotor near half and one-third of the first critical speed is investigated. The dynamic response of the rotor with a breathing crack is evaluated by expanding the changing stiffness of the crack as a truncated Fourier series and then using the Harmonic Balance Method. This method is applied to compute various parametric studies including the effects of the crack depth and location on the dynamic of a crack rotor. The evolution of the first critical speed, associated amplitudes at the critical speed and half of the critical speed, and the resulting orbits during transient operation are presented and some distinguishing features of a cracked rotor are examined.

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In this paper, the quantification of uncertainty effects on the variability of the non-linear response in rotor systems with multi-faults (such as unbalance, asymmetric shaft, bow, parallel and angular misalignments) is investigated. To take account of uncertainties in this kind of non linear problem, it is proposed to use the Harmonic Balance Method (HBM) with a Polynomial Chaos Expansion (PCE). The efficiency and robustness of the proposed methodology is demonstrated by comparison with Monte Carlo Simulations (MCS) for different kinds and levels of uncertainties.

International audienceBrake squeal is a friction induced instability phenomenon that has to be addressed during the development process. The mechanism is considered a mode coupling phenomenon also referred to as coalescence. The system eigenvalues have been computed using a technique based on the finite element method. The coalescence patterns were then determined in relation to the friction coefficient. The effects of damping on the coalescence patterns have been investigated. If the two modes involved in the coalescence are equally damped, a "lowering effect" that tends to stabilize the system is observed. If the two modes are not equally damped, both "lowering" and "smoothing" effects occur. If the "smoothing effect" prevails, added damping may act in an unintuitive way by destabilizing the system. To further study this point, stability areas have been plotted and a metric is proposed to find the most stable configuration in terms of damping distribution. In the squeal frequency range, coalescence patterns often involve more than two modes. In this case, the effect of damping is far more complicated since several modes are coupled both in terms of friction and damping