Non-linear dynamic of rotor-stator system with non-linear bearing clearance

The study deals with a rotor-stator contact inducing vibration in rotating machinery. A numerical rotor-stator system, including a non-linear bearing with Hertz contact and clearance is considered. To determine the non-linear responses of this system, non-linear dynamic equations can be integrated numerically. However, this procedure is both time consuming and costly to perform. The aim of this Note is to apply the Alternate Frequency/Time Method and the 'path following continuation' in order to obtain the non-linear responses to this problem. Then the orbits of rotor and stator responses at various speeds are investigated

On a new harmonic selection technique for harmonic balance method

International audienceThis paper is intended to present a new harmonic selection technique when solving nonlinear dynamic systems with the harmonic balance method. This technique belongs to the class of method called the adaptive harmonic balance method (AHBM). The harmonic selection is based on the use of a tangent predictor and relies on a stepwise regression procedure that allows for a dynamic management of the number of selected harmonics via an addition or removal procedure. The efficiency of this method relative to the classical harmonic balance method (HBM) is then evaluated through examples; this later step will indicate that AHBM can significantly reduce the number of variables, thus leading to computational time savings without deteriorating solution quality

On the Use of the Proper Generalised Decomposition for Solving Nonlinear Vibration Problems

International audienceThis paper presents the use of the so called Proper Generalized Decomposition method (PGD) for solving nonlinear vibration problems. PGD is often presented as an a priori reduction technique meaning that the reduction basis for expressing the solution is computed during the computation of the solution itself. In this paper, the PGD is applied in addition with the Harmonic Balance Method in order to find periodic solutions of nonlinear dynamic systems. Several algorithms are presented in order to compute nonlinear normal modes and forced solutions. Application is carried out on systems containing geometrical nonlinearity and/or friction damping. We show that the PGD is able to compute a good approximation of the solutions event with a projection basis of small size. Results are compared with a Proper Orthogonal Decomposition method showing that the PGD can sometimes provide an optimal reduction basis relative to the number of basis components

Harmonic Balance-Based Approach for Quasi-Periodic Motions and Stability Analysis

Quasi-periodic motions and their stability are addressed from the point of view of different harmonic balance-based approaches. Two numerical methods are used: a generalized multidimensional version of harmonic balance and a modification of a classical solution by harmonic balance. The application to the case of a nonlinear response of a Duffing oscillator under a bi-periodic excitation has allowed a comparison of computational costs and stability evaluation results. The solutions issued from both methods are close to one another and time marching tests showing a good agreement with the harmonic balance results confirm these nonlinear responses. Besides the overall adequacy verification, the observation comparisons would underline the fact that while the 2D approach features better performance in resolution cost, the stability computation turns out to be of more interest to be conducted by the modified 1D approach

Computing multiple periodic solutions of nonlinear vibration problems using the harmonic balance method and Groebner bases

International audienceThis paper is devoted to the study of vibration of mechanical systems with geometric nonlinearities. The harmonic balance method is used to derive systems of polynomial equations whose solutions give the frequency component of the possible steady states. Groebner basis methods are used for computing all solutions of polynomial systems. This approach allows to reduce the complete system to an unique polynomial equation in one variable driving all solutions of the problem. In addition, in order to decrease the number of variables, we propose to first work on the undamped system, and recover solution of the damped system using a continuation on the damping parameter. The search for multiple solutions is illustrated on a simple system, where the influence of the retained number of harmonic is studied. Finally, the procedure is applied on a simple cyclic system and we give a representation of the multiple states versus frequency

Harmonic Balance-Based Approach for Quasi-Periodic Motions and Stability Analysis

Quasi-periodic motions and their stability are addressed from the point of view of different harmonic balance-based approaches. Two numerical methods are used: a generalized multidimensional version of harmonic balance and a modification of a classical solution by harmonic balance. The application to the case of a nonlinear response of a Duffing oscillator under a bi-periodic excitation has allowed a comparison of computational costs and stability evaluation results. The solutions issued from both methods are close to one another and time marching tests showing a good agreement with the harmonic balance results confirm these nonlinear responses. Besides the overall adequacy verification, the observation comparisons would underline the fact that while the 2D approach features better performance in resolution cost, the stability computation turns out to be of more interest to be conducted by the modified 1D approach

Nonlinear Harmonic Analysis of a Blade Model Subjected to Large Geometrical Deflection and Internal Resonance

International audienceThis paper is devoted to the study of the nonlinear harmonic response of an industrial blade model subjected to large geometrical deflection. A reduction procedure is performed on the blade model using the linear normal modes of the structure. Geometrical nonlinear effects are taken into account by considering cubic and quadratic stiffnesses in the dynamical reduced model. Reduced nonlinear stiffness coefficients are computed with the STiffness Evaluation Procedure (STEP) and periodic solutions are sought using the Harmonic Balance Method (HBM) coupled to a pseudo-arclength continuation. Along with the harmonic response, a bifurcation analysis is performed to compute both turning and branching points. Specific attention is paid to the internal resonance phenomenon. 2 to 1 internal resonance occurred during the frequency response analysis close to the first and second modes of the reduced model. Mode coupling phenomena occurred during the harmonic analysis and secondary branches of solutions were obtained from branching point bifurcations

Non-linear dynamic of rotor–stator system with non-linear bearing clearance

International audienceThe study deals with a rotor-stator contact inducing vibration in rotating machinery. A numerical rotor-stator system, including a non-linear bearing with Hertz contact and clearance is considered. To determine the non-linear responses of this system, non-linear dynamic equations can be integrated numerically. However, this procedure is both time consuming and costly to perform. The aim of this Note is to apply the Alternate Frequency/Time Method and the 'path following continuation' in order to obtain the non-linear responses to this problem. Then the orbits of rotor and stator responses at various speeds are investigated

Dynamique non-linéaire d'une roue de turbine Basse Pression soumise à des excitations structurales d'un turboréacteur

La prise en compte des couplages dynamiques entre les différents organes constituant une turbomachine s inscrit dans le processus d optimisation des designs moteur. L amélioration des performances des turboréacteurs passe souvent par l utilisation d architectures multi-rotors. Dans le contexte des moteurs avec une architecture bi-rotor, des résultats d essais expérimentaux montrent qu il est nécessaire de considérer, dès la conception, l influence de la dynamique de l arbre Haute Pression (HP) sur les aubages de l arbre Basse Pression (BP). Dans ce cadre d étude, un premier modèle simplifié de bi-rotor aubagé est développé dans le repère tournant lié au rotor BP. Ce modèle est composé de deux rotors modélisés par des équivalents poutres - masses - ressorts et d une roue aubagée constituée d aubes souples modélisées par des poutres encastrées sur un disque rigide. Desnon-linéarités de type jeu radial avec contact au niveau des paliers sont également considérées et la réponse des aubes soumises à des excitations multi-fréquentielles de type balourd BP et HP est analysée. La présence de non-linéarités dans le système conduit à mettre en oeuvre des algorithmes adaptés, basés sur des techniques de résolution dans le domaine fréquentiel avec l évaluation des efforts non-linéaires dans le domaine temporel. Afin d avoir une meilleure description de la dynamique de la roue aubagée, une méthode spécifique de couplage est proposée, permettant de coupler un modèle réduit de roue aubagée 3D à un modèle simplifié de bi-rotor. Une démarche adaptée à la modélisation de la roue aubagée en symétrie cyclique est implémentée afin de considérer des non-linéarités de type contact en tête d aube. La méthode de couplage proposée est ensuite illustrée sur un exemple simple puis validée dans un cadre linéaire et non-linéaire. Enfin, cette méthode de couplage est appliquée au cas d une structure industrielle, constituée d un modèle d ensemble simplifié représentatif d un moteur et d un modèle éléments finis d une roue de turbine BP. Les résultats obtenus mettent en évidence le couplage entre la dynamique d ensemble et la dynamique de la roue aubagée et permettent de prédire la réponse non-linéaire des aubes de turbine BP en présence d une excitation multi-fréquentielle, dans des configurations de co-rotation et de contra-rotation.The design and optimization process of high efficiency turbomachinery has become a major challenge and a topical issue at both industrial and research levels. Performance improvement has motivated the use of multi-shaft architecture in engines. In the context of dual-shaft aircraft engines, the interaction between dynamics occurring within shafts and bladed disks seems to play an important role at the design stage. The present research work deals with the coupling of these components involving several unbalances in the dynamic response of blades. Within this framework, a simplified analytical model of a bladed dual-shaft developed in the rotating frame is presented. The dual-shaft is modelled by spring - mass- beam systems and connected to a bladed disk composed of a set of flexible blades modelled by Euler-Bernoulli beams clamped in a rigid disk. Nonlinearities coming from bearings are also considered and modelled as a radial clearance and contact stiffness. Considering nonlinearities requires the implementation of dedicated algorithms and specific resolution techniques in the frequency domain as well as the computation of nonlinear forces in the time domain. The nonlinear response of blades subjected to unbalances excitations is investigated and analysed. To have a finer description of the bladed disk dynamics, a specific coupling method is proposed allowing to connect a bladed disk finite element model with the simplified dual-shaft model. A cyclic symmetry approach well-suited to the nonlinear dynamics of bladed disks is developed in order to consider blade tip contact nonlinearities. Performances of the proposed method are illustrated through an academic example and validated in both linear and nonlinear settings. Eventually, the coupling technique is applied to a complex industrial case involving a classical simplified dual-shaft model and a finite element model of the low pressure turbine bladed disk. Numerical results clearly demonstrate the coupling between dynamics and enable to predict the nonlinear response of low pressure turbine blades to several unbalances, for both co-rotating and counter-rotating engines.LYON-Ecole Centrale (690812301) / SudocSudocFranceF