L. Jezequel

Analysis of non-linear dynamical systems by the normal form theory

Abstract A method is proposed for calculating the periodic solutions of non-linear mechanical systems with analytical non-linearities. The Jordan normalization procedure for the case of non-linear autonomous systems is described and generalized to dampened harmonically excited oscillators. Non-linear modes for Hamiltonian systems are introduced; normal forms simplify the analysis of bifurcation. It is shown how to extend, the modal synthesis procedure: the proposed non-linear modes obtained from free vibrations are used to construct a superposition technique to describe the forced response of harmonically excited systems. These results are tested for one- and two-degrees-of-freedom systems with cubic non-linearities. The results are compared with expressions obtained by classical analytical methods (averaging or multiple scales methods) or Runge-Kutta numerical methods.

A dynamic Lagrangian frequency-time method for the vibration of dry-friction-damped systems

Abstract A new frequency–time domain procedure, the dynamic Lagrangian mixed frequency–time method (DLFT), is proposed to calculate the non-linear steady state response to periodic excitation of structural systems subject to dry friction damping. In this formulation, the dynamic Lagrangians are defined as the non-linear contact forces obtained from the equations of motion in the frequency domain, with the adjunction of a penalization on the difference between the interface displacements calculate by the non-linear solver in the frequency domain and those calculated in the time domain from the non-linear contact forces, thus accounting for Coulomb friction and non-penetration conditions. The dynamic Lagrangians allow one to solve for the non-linear forces between two points in contact without using artifacts such as springs. The new DLFT method is thus particularly well suited to handling finite element models of structures in frictional contact, as it does not require a special model for the contact interface. Dynamic Lagrangians are also better suited to frequency-domain friction problems than the traditional time-domain method of augmented Lagrangians. Furthermore, a reduction of the non-linear system to relative interface displacements is introduced to decrease the computation time. The DLFT method is validated for a beam in contact with a flexible dry friction element connected to ground, for frictional constraints that feature two-dimensional relative motion. Results are also obtained for a large-scale structural system with a large number of one-dimensional dry-friction dampers. The DLFT method is shown to be accurate and fast, and it does not suffer from convergence problems, at least in the examples studied.

Analysis of squeal noise and mode coupling instabilities including damping and gyroscopic effects

Abstract This paper deals with an audible disturbance known as automotive clutch squeal noise from the viewpoint of friction-induced mode coupling instability. Firstly, an auto-coupling model is presented showing a non-conservative circulatory effect originating from friction forces. Secondly, the stability of an equilibrium is investigated by determining the eigenvalues of the system linearized equations. The effects of the circulatory and gyroscopic actions are examined analytically and numerically to determine their influence on the stability region. Separate and combined effects are analyzed with and without structural damping and important information is obtained on the role of each parameter and their interactions regarding overall stability. Not only is structural damping shown to be of primary importance, as reported in many previous works, this article also highlights a particular relationship with gyroscopic effects. A method of optimizing both the stability range and its robustness with respect to uncertainty on system parameters is discussed after which practical design recommendations are given.

Analysis of friction and instability by the centre manifold theory for a non-linear sprag-slip model

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.

Piano soundboard: structural behavior, numerical and experimental study in the modal range

Abstract The low frequency broadband vibrational behavior of a piano soundboard is considered. Attention is focused on the ability of finite element models and analytical models to predict precisely the behavior of such a complicated structure—especially with its orthotropy and rib effects. In order to validate these abilities, an experimental modal analysis, considered as the reference, is compared first with a numerical calculation and then with an analytical modeling of the modal basis of the same soundboard. The high structural complexity of the soundboard exceeds the analytical capabilities, but agreement is very good for the numerical model, in the frequency domain, and equally in the spatial one. The final aim is to generate a numerical tool for designing and optimizing piano soundboards.

Methods to reduce non-linear mechanical systems for instability computation

SummaryNon-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.

Energetics of axisymmetric fluid-filled pipes up to high frequencies

Abstract The energetics of motions of axisymmetric fluid-filled pipes are presented in this paper, in view of high-frequency modelling. This study deals in particular with derivations of local energy equations well suited for the prediction of averaged response of coupled fluid–structure systems. The derivation of the latter requires special manipulation of the kinematic dynamics based here on the notion of propagation modes. Thus, the focus is on the Donnell–Mushtari cylindrical shell with an internal acoustic fluid, a typical example of waveguides with multiple transmission mechanisms. “Exact” and statistical approaches are developed for this system. A state-space representation is first proposed; it allows the characterization of propagating modes in a general manner. This propagating content then leads to the formulation of the local energy approach for this canonical problem.

The Role of Damping and Definition of the Robust Damping Factor for a Self-Exciting Mechanism With Constant Friction

This paper presents a linear two-degree-of-freedom model in order to analyze friction-induced instabilities that are governed by modal interaction. The role of structural damping on flutter instability is undertaken, and the effects of the structural damping ratio between the stable and unstable modes are investigated in order to clarify and to explain the mechanical process of flutter instability. In certain conditions, it is demonstrated that the merging scenario and the unstable mode may change due to this structural damping ratio. Discussions not only demontrate the role of strutural damping and the associated mechanical process but also define the robust damping factor in order to avoid design errors and to reduce flutter instability.

A simple shear wall model taking into account stiffness degradation

The damage undergone by reinforced concrete structures affects their modal characteristics. The study consists of constructing an original simplified model for a heavily reinforced shear wall relying on modal characteristic changes related to a damage increase. The elaboration of this simplified model is realised in several stages. The first stage is devoted to the inelastic finite element analysis of a shear wall which was tested under pseudo dynamic loading. Then, by applying a set of ideal excitations to the two-dimensional finite element modelling, the identification of the decrease of the fundamental frequency as a function of a damage variable is carried out. The third stage is concerned with constructing a new uniaxial dynamic model by introducing directly the expression of the decrease of the fundamental frequency into the equation of motion. Finally, the validity of the simplified modelling is assessed by comparing the numerical results obtained from the finite element analysis with those derived from the simplified analysis, for different types of seismic excitation.

Active rubber mounts by means of piezoelectric actuators, experimental work

This paper proposes an experimental mock-up which aims to validate a new concept of a piezo-rubber mount. This new concept is based upon the combination of an electrically-monitored active piezoelectric block with a passive rubber mount. An adaptive control law is then used for an off-line identification and control of the force transmissibility. The new hybrid mount is shown to be very efficient and significantly improves the transfer between disturbances and receiving components.