Laboratoire de Tribologie et Dynamique des Systèmes

This paper presents a method of mistuning identification for turbomachinery blisks (integrally bladed disks) from measurements of the system modes and natural frequencies. The procedure is based on the “Best Achievable Eigenvectors” of all measured modes simultaneously combined with a regulation technique. Four illustrative numerical simulations, based on a reduced-order model of the blisk, are given which demonstrate that this technique produces acceptable mistuning identification. To do so, a finite element model of the bladed disk and a computational reduced-order modelling technique, based on component-mode substitution method and combined with a cyclic characteristic of the blade assembly, are developed. Moreover, sensibility coefficients of the mistuning parameter with respect to measured data are derived.

The influence of the presence of transverse cracks in a rotating shaft is analyzed. 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.

This paper presents the research devoted to the study of instability phenomena in non-linear model with a constant brake friction coefficient. Indeed, the impact of unstable oscillations can be catastrophic. It can cause vehicle control problems and component degradation. Accordingly, complex stability analysis is required. This paper outlines stability analysis and centre manifold approach for studying instability problems. To put it more precisely, one considers brake vibrations and more specifically heavy trucks judder where the dynamic characteristics of the whole front axle assembly is concerned, even if the source of judder is located in the brake system. The modelling introduces the sprag-slip mechanism based on dynamic coupling due to buttressing. The non-linearity is expressed as a polynomial with quadratic and cubic terms. This model does not require the use of brake negative coefficient, in order to predict the instability phenomena. Finally, the centre manifold approach is used to obtain equations for the limit cycle amplitudes. The centre manifold theory allows the reduction of the number of equations of the original system in order to obtain a simplified system, without loosing the dynamics of the original system as well as the contributions of non-linear terms. The goal is the study of the stability analysis and the validation of the centre manifold approach for a complex non-linear model by comparing results obtained by solving the full system and by using the centre manifold approach. The brake friction coefficient is used as an unfolding parameter of the fundamental Hopf bifurcation point.

Friction-induced vibrations are a major concern in a wide variety of mechanical systems. This is especially the case in aircraft braking systems where the problem of unstable vibrations in disk brakes has been studied by a number of researchers. Solving potential vibration problems requires experimental and theoretical approaches. A non-linear model for the analysis of mode aircraft brake whirl is presented and developed based on experimental observations. The non-linear contact between the rotors and the stators, and mechanisms between components of the brake system are considered.Stability is analyzed by determining the eigenvalues of the Jacobian matrix of the linearized system at the equilibrium point. Linear stability theory is applied in order to determine the effect of system parameters on stability.

This paper presents a model of fully flexible bladed rotor developed in the rotating frame. An energetic method is used to obtain the matrix equations of the dynamic behaviour of the system. The gyroscopic effects as well as the spin softening effects and the centrifugal stiffening effects, taken into account through a pre-stressed potential, are included in the model. In the rotating frame, the eigenvalues' imaginary parts of the latter matrix equation give the Campbell diagram of the system and its stability can be analysed through its associated eigenvalues' real parts. The turbo machine casing is also modelled by an elastic ring in the rotating frame through an energetic method. Thus, in some rotational speed ranges the contact problem between the rotor and the stator can be treated as a static problem since both structures are stationary to each other. Prior to the study of the complete problem of contact between the flexible blades of the rotor and the flexible casing, a simple model of an elastic ring having only one mode shape, excited by rotating loads is developed in the rotating frame too, in order to underline divergence instabilities and mode couplings. Then, the complete problem of frictionless sliding contact between the blades and the casing, without rubbing, is studied. The stable balanced static contact configurations of the structure are found as function of the rotational speed of the rotor. Finally, the results are compared to these of the simple model of rotating spring-masses on an elastic ring, showing good adequacy. The present model of rotor appears thus particularly adapted to the study of blades-casing contacts and highlighted an unstable phenomenon near the stator critical speed even in case of frictionless sliding.

Brake squeal noise is still an issue since it generates high warranty costs for the automotive industry and irritation for customers. Key parameters must be known in order to reduce it. Stability analysis is a common method of studying nonlinear phenomena and has been widely used by the scientific and the engineering communities for solving disc brake squeal problems. This type of analysis provides areas of stability versus instability for driven parameters, thereby making it possible to define design criteria. Nevertheless, this technique does not permit obtaining the vibrating state of the brake system and nonlinear methods have to be employed. Temporal integration is a well-known method for computing the dynamic solution but as it is time consuming, nonlinear methods such as the Harmonic Balance Method (HBM) are preferred. This paper presents a novel nonlinear method called the Constrained Harmonic Balance Method (CHBM) that works for nonlinear systems subject to flutter instability. An additional constraint-based condition is proposed that omits the static equilibrium point (i.e. the trivial static solution of the nonlinear problem that would be obtained by applying the classical HBM) and therefore focuses on predicting both the Fourier coefficients and the fundamental frequency of the stationary nonlinear system.The effectiveness of the proposed nonlinear approach is illustrated by an analysis of disc brake squeal. The brake system under consideration is a reduced finite element model of a pad and a disc. Both stability and nonlinear analyses are performed and the results are compared with a classical variable order solver integration algorithm.Therefore, the objectives of the following paper are to present not only an extension of the HBM (CHBM) but also to demonstrate an application to the specific problem of disc brake squeal with extensively parametric studies that investigate the effects of the friction coefficient, piston pressure, nonlinear stiffness and structural damping.

This paper presents the stability analysis of a system composed of rotating beams on a flexible, circular fixed ring, using the Routh–Hurwitz criterion. The model displayed has been fully developed within the rotating frame by use of an energy approach. The beams considered possess two degrees of freedom (dofs), a flexural motion as well as a traction/compression motion. In-plane deformations of the ring will be considered. Divergences and mode couplings have thus been underscored within the rotating frame and in order to simplify understanding of all these phenomena, the dofs of the beams will first be treated separately and then together. The dynamics of radial rotating loads on an elastic ring can create divergence instabilities as well as post-critical mode couplings. Moreover, the flexural motion of beam rubbing on the ring can also lead to mode couplings and to the locus-veering phenomenon. The presence of rubbing seems to make the system unstable as soon as the rotational speed of the beams is greater than zero. Lastly, the influence of an angle between the beams and the normal to the ring's inner surface will be studied with respect to system stability, thus highlighting a shift frequency phenomenon.

Non-linear dynamical structures depending on control parameters are encountered in many areas of science and engineering. In the study of non-linear dynamical systems depending on a given control parameter, the stability analysis and the associated non-linear behaviour in a near-critical steady-state equilibrium point are two of the most important points; they make it possible to validate and characterize the non-linear structures. Stability is investigated by determining eigenvalues of the linearized perturbation equations about each steady-state operating point, or by calculating the Jacobian of the system at the equilibrium points. While the conditions and the values of the parameters which cause instability can be investigated by using linearized equations of motion studies of the non-linear behaviour of vibration problems, on the other hand, require the complete non-linear expressions of systems. Due to the complexity of non-linear systems and to save time, simplifications and reductions in the mathematical complexity of the non-linear equations are usually required. The principal idea for these non-linear methods is to reduce the order of the system and eliminate as many non-linearities as possible in the system of equations. In this paper, a study devoted to evaluating the instability phenomena in non-linear models is presented. It outlines stability analysis and gives a non-linear strategy by constructing a reduced order model and simplifying the non-linearities, based on three non-linear methods: the centre manifold concept, the rational approximants and the Alternating Frequency/Time domain method. The computational procedures to determine the reduced and simplified system via the centre manifold approach and the fractional approximants, as well as the approximation of the responses as a Fourier series via the harmonic balance method, are presented and discussed. These non-linear methods for calculating the dynamical behaviour of non-linear systems with several degrees-of-freedom and non-linearities are tested in the case of mechanical systems with many degrees-of-freedom possessing polynomial non-linearities. Results obtained are compared with those estimated by a classical Runge-Kutta integration procedure. Moreover, an extension of the centre manifold approach using rational approximants is proposed and used to explore the dynamics of non-linear systems, by extending the domain of convergence of the non-linear reduced system and evaluating its performance and suitability.

Non-linear dynamics due to friction induced vibrations in a complex aircraft brake model are investigated. This paper outlines a non-linear strategy, based on the center manifold concept and the rational in order to evaluate the non-linear dynamical behaviour of a system in the neighbourhood of a critical steady-state equilibrium point. In order to obtain time–history responses, the complete set of non-linear dynamic equations may be integrated numerically. But this procedure is both time consuming and costly to perform when parametric design studies are needed. So it is necessary to use non-linear analysis: the center manifold approach and the rational approximants are used to obtain the limit cycle of the non-linear system and to study the behaviour of the system in the unstable region. Results from these non-linear methods are compared with results obtained by integrating the full original system. These non-linear methods appear very interesting in regard to computational time and also necessitate very few computer resources.

This paper presents a robust damage assessment technique for the nondestructive detection and size estimation of open cracks in beams. The damage detection, based on the constitutive relation error updating method, is used for the identification of the crack's location and size in a simply-supported beam. The transverse open crack is modeled through the introduction of the flexibility due to the presence of the crack, i.e. by reducing the second moment of area of the element at the crack's location.This identification algorithm is illustrated through numerical examples involving different positions and sizes of a transverse open crack. We show that the detection of damage and the identification of the crack's size and position can be achieved with satisfactory precision, even if 20% noise has been added to the simulations and less than 5% of all displacements have been measured.

In this paper, a non-linear strategy, based on the centre manifold, the rational approximants and the alternating frequency/time domain method has been developed, in order to study the non-linear dynamical behaviour of a system in the neighbourhood of a critical steady state equilibrium point. The stability analysis and the non-linear dynamics of a complex braking system with a non-linear rotor/stator contact are presented. Moreover, one of the most important steps of this paper is the determination of the non-linear behaviour and the limit cycle amplitudes of this complex system. In order to conduct this study, the dynamic response is evaluated by using applying the centre manifold, the rational approximants and the alternating frequency/time domain method, that permit to obtain rapidly and efficiently the non-linear behaviour of the system. The dynamic response obtained by applying this method is compared with that evaluated through numerical integration.

Friction-induced vibration is still a cause for concern in a wide variety of mechanical systems, because it can lead to structural damage if high vibration levels are reached. Another effect is the noise produced that can be very unpleasant for end-users, thereby making it a major problem in the field of terrestrial transport. In this work the case of an aircraft braking system is examined. An analytical model with polynomial nonlinearity in the contact between rotors and stators is considered.Stability analysis is commonly used to evaluate the capacity of a nonlinear system to generate friction-induced vibrations. With this approach, the effects of variations in the system parameters on stability can be easily estimated. However, this technique does not give the amplitude of the vibrations produced. The integration of the full set of nonlinear dynamic equations allows computing the time-history response of the system when vibration occurs. This technique, which can be time-consuming for a model with a large number of degrees of freedom (dof), is nevertheless necessary in order to calculate the transient-state behavior of the system. The use of a continuous wavelet transform (CWT) is very suitable for the detailed analysis of the transient response. In this paper, the possibilities of coexistence of several instabilities at the same time will be examined. It will be shown that the behavior of the brake can be very complex and cannot be predicted by stability analysis alone.

This study aims at clarifying the phenomenological roots of an acoustical disturbance known as “clutch squeal noise”. A nonlinear two-degrees-of-freedom model is introduced in order to illustrate some basic phenomena leading to self-generated vibrations. The damping of the system as well as both circulatory and gyroscopic actions are included in order to highlight their respective influence and the destabilization paradox. Results are obtained on the stability range of the equilibrium, the nature of the Hopf bifurcation, the limit cycle branches and their stability. A dynamic extension of the destabilization paradox is proposed and some non-periodic behaviours are identified too.

The aim of this paper is to present a damage assessment technique for the non-destructive detection and sizing of multiple open cracks in beams. The constitutive relation error updating method is used for the identification of the location and the size of multi-cracks in a simply supported beam.The present identification method is illustrated through numerical examples including double and triple cracks. Moreover, the efficiency and robustness of the proposed method is demonstrated through various numerical simulations in regard to the non-dimensional crack depth and the crack location.It is demonstrated that the constitutive relation error updating method can detect the number of cracks on the beam and can estimate both the crack positions and sizes with satisfactory precision, even if 10% or 20% noise levels has been added to the simulations, and only few degrees of freedom are used for the identification procedure.

In this paper, the influence of transverse cracks in a rotating shaft is analysed. 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 nonlinear behaviour of the rotor system rapidly and efficiently by modelling 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.

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 stable-unstable 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.