On Solving One-Dimensional Partial Differential Equations With Spatially Dependent Variables Using the Wavelet-Galerkin Method

International audienceThe discrete orthogonal wavelet-Galerkin method is illustrated as an effective method for solving partial differential equations (PDE's) with spatially varying parameters on a bounded interval. Daubechies scaling functions provide a concise but adaptable set of basis functions and allow for implementation of varied loading and boundary conditions. These basis functions can also effectively describe C 0 continuous parameter spatial dependence on bounded domains. Doing so allows the PDE to be discretized as a set of linear equations composed of known inner products which can be stored for efficient parametric analyses. Solution schemes for both free and forced PDE's are developed; natural frequencies, mode shapes, and frequency response functions for an Euler-Bernoulli beam with piecewise varying thickness are calculated. The wavelet-Galerkin approach is shown to converge to the first four natural frequencies at a rate greater than that of the linear finite element approach; mode shapes and frequency response functions converge similarly

Circular Cylindrical Shell Made of Neo-Hookean-Fung Hyperelastic Material Under Static and Dynamic Pressure

The present study is devoted to the investigation of static and dynamic behavior of the three-layered composite shell made of hyperelastic material. Such a shell can be considered as a model of human aorta. Since soft biological materials are essentially nonlinear even in the elasticity zone, not only geometrical, but also physical nonlinearity should be taken into account. The physical nonlinearity of soft biological tissues is usually modeled by certain hyperelastic law. The law chosen for this study is the combination of the Neo- Hookean law, which describes the isotropic response at small strains, and Fung exponential law, that models the stiff anisotropic response of the collagen fibers at larger strains. Each of three shell layers has its own hyperelastic constants set. These constants are determined basing on experiential data [1]. The straindeflection relations are modeled with higher-order shear deformation theory [2]. Initially, the shell is preloaded with static pressure. Since the defection in our study is large we use the expression for pressure as a follower load [3]. The static problem is solved with the help of the local models method [4]. Afterwards, the free and forced dynamical response of the preloaded shell is studied both in vacuo and with still fluid inside. The modes of interest are the first axisymmetric mode and mode with two half-waves in circumferential direction (so-called collapse mode). It is found that static pressure decreases the dynamic nonlinearity and it is quite weak. At the same time, the presence of fluid makes the softening nonlinearity stronger as in case of shells of conventional material [5]

Etude de l'interaction modale rotor/stator dans un moteur d'avion

International audienceIn modern turbo machines such as aircraft jet engines, contact between the casing and bladed disk may occur through avariety of mechanisms : coincidence of vibration modes, thermal deformation of the casing, rotor imbalance, etc. Thesenonlinear interactions may result in severe damage to both structures and it is important to understand the physicalmechanisms that cause them and the circumstances under which they occur. In this study, we focus on the phenomenonof interaction caused by modal coincidence. A simple two-dimensional model of the casing and bladed disk structures isintroduced in order to predict the occurrence of the interaction phenomenon in terms of the rotation speed of the rotor.Each structure is represented in terms of its two $n_ d$ -nodal diameter vibration modes, which are characteristic of axi-symmetric structures and allow for travelling wave motions that may interact through direct contact. The equations ofmotion are solved first using an explicit time integration scheme in conjunction with the Lagrange multiplier method, andthen by the Harmonic Balance Method. Both methods generally agree well and exhibit two distinct zones of completelydifferent behaviors of the system. Finally, a second planar model is introduced in order to achieve a deeper understandingof the interaction phenomenon.Dans le domaine des turbo-machines comme les moteurs d’avion par exemple, les contacts structurels entre le carter etla roue aubagée peuvent avoir plusieurs origines : coïncidence vibratoire entre des modes, déformation thermique ducarter, apparition d’un balourd au niveau du rotor, etc. Ces interactions non linéaires peuvent fortement endommagerles structures et il est important de comprendre l’origine de ces mécanismes. Dans ce travail, nous nous concentronssur le phénomène d’interaction modale. Un premier modèle planaire du carter et de la roue aubagée est développé pourprédire les vitesses de rotation du moteur pour lesquelles le phénomène d’interaction peut exister. Chaque structure estdiscrétisée sur ses deux modes à $n_ d$ diamètres nodaux, caractéristiques des structures à symétrie cyclique et qui, combinés, permettent de construire des ondes de propagation qui interagissent par l’intermédiaire du contact. Les équations sont résolues grâce à un schéma d’intégration explicite couplé à la méthode des multiplicateurs de Lagrange, puis par la méthode de l’équilibrage harmonique. Les deux méthodes sont en accord et montrent qu’il y a deux zones distinctes de fonctionnement. Finalement, un deuxième modèle planaire est présenté pour comprendre plus en détail le phénomène d’interaction

Etude des modes normaux non linéaires d'un arbre rotatif par la méthode des surfaces invariantes

For rotating shaft systems, the mechanism representing the supporting hydraulic bearings is inherently nonlinear even though the deformation of the rotating shaft can be modelled linearly. Complicated behaviors can occur in the whole system and an efficient reduced order model is important for the vibration analysis. In this paper, a rotating shaft supported by two short hydraulic bearings is studied in details. A simplify model of the bearings is applied so that the supporting forces can be expressed analytically. The shaft and the bearings are coupled by Craig-Bampton method to ensure a minimal degrees-of-freedom system model. A very new formulation for the manifolds is applied here to get the nonlinear normal modes of such a system with general damping, gyroscopic and stiffness matrices. Once those nonlinear modes are constructed the motion on the nonlinear manifolds is compared to a direct time simulation.Dans les systèmes intégrant des structures en rotation, les non linéarités proviennent souvent des supports, présents ici sous la forme de paliers lisses. Même si l'hypothèse de déformations linéaires au niveau de l'arbre est permise, le comportement de l'ensemble est non linéaire et un modèle simple est important pour la compréhension des phénomènes vibratoires. Une formule de Reynolds simplifiée pour le comportement hydrodynamique des paliers permet d'exprimer analytiquement les forces issues de la déformation du film d'huile. Les équations du mouvement sont ensuite discrétisées par la méthode de Craig-Bampton pour un nombre minimal de degrés de liberté. Une toute nouvelle formulation des surfaces invariantes permet de générer les modes normaux non linéaires d'un tel système ayant des matrices gyroscopique, d'amortissement et de raideur tout à fait générales et une simulation directe en temps est comparée aux résultats obtenus

Vibration of a Square Hyperelastic Plate Around Statically Pre-Loaded State

International audienceStatic deflection and free nonlinear vibrations of thin square plate made of biological material are investigated. The involved physical nonlinearity is described through Neo-Hookean, Mooney-Rivlin and Ogden hyperelastic laws; geometrical nonlinearity is modelled by Novozhilov nonlinear shell theory. The problem is solved by sequentially constructing the local models that describe the behavior of plate in the vicinity of a certain static configuration. These models are the systems of ordinary differential equations with quadratic and cubic nonlinear terms in displacement, which allows application of techniques used in analysis of thin-walled structures of physically linear materials. The comparison of static and dynamic results obtained with different material models is carried out

Nonlinear dynamics of slender inverted flags in uniform steady flows

International audienceA nonlinear fluid-elastic continuum model for the dynamics of a slender cantilevered plate subjected to axial flow directed from the free end to the clamped end, also known as the inverted flag problem, is proposed. The extension of elongated body theory to large-amplitude rotations of the plate mid-plane along with Bollay's nonlinear wing theory is employed in order to express the fluid-related forces acting on the plate, while retaining all time-dependent terms in both modelling and numerical simulations; the unsteady fluid forces due to vortex shedding are not included. Euler-Bernoulli beam theory with exact kinematics and inextensibility is employed to derive the nonlinear partial integro-differential equation governing the dynamics of the plate. Discretization in space is carried out via a conventional Galerkin scheme using the linear mode-shapes of a cantilevered beam in vacuum. The pseudo-arclength continuation technique is adapted to construct bifurcation diagrams in terms of the flow velocity, in order to gain insight into the stability and post-critical behaviour of the system. Integration in time is conducted using Gear's backward differentiation formula. The sensitivity of the nonlinear response of the system to different parameters such as the aspect ratio, mass ratio, initial inclination of the flag, and viscous drag coefficient is investigated through extensive numerical simulations. It is shown that for flags of small aspect ratio the undeflected static equilibrium is stable prior to a subcritical pitchfork bifurcation. For flags of sufficiently large aspect ratio, however, the first instability encountered is a supercritical Hopf bifurcation giving rise to flapping motion around the undeflected static equilibrium; increasing the flow velocity further, the flag then displays flapping motions around deflected static equilibria, which later lead to fully-deflected static states at even higher flow velocities. The results exposed in this study help understand the dynamics of the inverted-flag problem in the limit of inviscid flow theory

Static and Dynamic Behavior of Circular Cylindrical Shell Made of Hyperelastic Arterial Material

International audienceStatic and dynamic responses of a circular cylindrical shell made of hyperelastic arterial material are investigated. The material is modeled as a combination of Neo-Hookean and Fung hyperelastic materials. Two pressure loads are implemented: distributed radial force and deformation-dependent pressure. The static responses of the shell under these two different loads differ essentially at moderate strains, while the behavior is similar for small loads. The main difference is in the axial displacements that are much larger under distributed radial forces. Free and forced vibrations around pre-loaded configurations are analyzed. In both cases the nonlinearity of the single-mode (driven mode) response of the pre-loaded shell is quite weak but a resonant regime with co-existing driven and companion modes is found with more complicated nonlinear dynamics

Nonlinear Normal Modes of a Rotating Shaft Based on the Invariant Manifold Method

International audienceThe nonlinear normal mode methodology is generalized to the study of a rotating shaft supported by two short journal bearings. For rotating shafts, nonlinearities are generated by forces arising from the supporting hydraulic bearings. In this study, the rotating shaft is represented by a linear beam, while a simplified bearing model is employed so that the nonlinear supporting forces can be expressed analytically. The equations of motion of the coupled shaft-bearings system are constructed using the Craig-Bampton method of component mode synthesis, producing a model with as few as six degrees of freedom (d.o.f.). Using an invariant manifold approach, the individual nonlinear normal modes of the shaft-bearings system are then constructed, yielding a single d.o.f. reduced-order model for each nonlinear mode. This requires a generalized formulation for the manifolds, since the system features damping as well as gyroscopic and non-conservative circulatory terms. The nonlinear modes are calculated numerically using a nonlinear Galerkin method that is able to capture large amplitude motions. The shaft response from the nonlinear mode model is shown to match extremely well the simulations from the reference Craig-Bampton model

A reduction technique for mistuned bladed disks with superposition of large geometric mistuning and small model uncertainties

International audienceA new method for vibration analysis of mistuned bladed disks is presented. The method combines two previously reported modeling techniques in order to study the effects of small random parameter variation on geometrically mis-tuned bladed disks. It is based on the observation that the nominal projections usually possess a certain degree of ro-bustness tolerating small perturbations. Hence the subspace spanned by a sufficient number of compensated tuned system normal modes can be used to repeatedly project models with small random parameter variation. The method is validated numerically on an industrial bladed disk model, both free and forced responses are compared with full model finite element analysis. The benchmark cases show acceptable accuracy while retaining low computational cost to build and evaluate obtained ROM

Axisymmetric deformations of circular rings made of linear and Neo-Hookean materials under internal and external pressure: A benchmark for finite element codes

International audienceThe axisymmetric deformations of thick circular rings are investigated. Four materials are explored: linear material, incompressible Neo-Hookean material and Ogden's and Bower's forms of compressible Neo-Hookean material. Radial distributed forces and a displacement-dependent pressure are the external loads. This problem is relatively simple and allows analytical, or semi-analytical, solution; therefore it has been chosen as a benchmark to test commercial finite element software for various material laws at large strains. The solutions obtained with commercial finite element software are almost identical to the present semi-analytical ones, except for the linear material, for which commercial finite element programs give incorrect results.Highlights: Linear, incompressible and compressible Neo-Hookean materials are used. Analytical benchmark solution to test commercial programs. Two commercial FE programs give incorrect results for large strains and linear elastic material.</ol