DETC2005-84616 MULTIPLE SOLUTIONS IN AN AMPLITUDE LIMITED JEFFCOTT ROTOR INCLUDING RUBBING AND STICK-SLIP EFFECT

ABSTRACT The non-linear behaviour of rub-impact rotors have been studied in several papers. In such systems rich dynamics have been found together with the coexistence of solutions within some specific parameter ranges. In this paper an attempt is made to find all stable solutions for an amplitude limited Jeffcott rotor including rubbing and stick-slip effect. The recently suggested "multi bifurcation diagram method" is used to find and extract stable sets of bifurcation diagrams. A system is chosen where the linear stationary amplitude only exceeds the clearance in a narrow region near the natural frequency. Therefore large regions in frequency are expected to have only the linear stationary response. The results show that it is only for very low frequencies that one single solution exists. Even though periodic motions are dominant, there exist large ranges in frequency with quasi-periodic or chaotic motions. For the studied cases, three coexisting stable solutions are most common. In one case as many as four stable solutions was found to coexist. For rotors with large clearances (no impacts necessary) it is still possible to find several coexisting motions. For all cases the stick motion is the most severe one with large amplitudes and high backward whirl frequencies. In real situations the consequence of this stick motion is machine failure. These high amplitude motions were found to be stable over large frequency ranges. From the stability analysis it was found that this rolling motion can be avoided by low spin speed, low contact stiffness, low coefficient of friction, small ratio of disc radius/clearance or high damping ratio. In a design situation the parameters are seldom known with high accuracy. Therefore, it is of interest to know all solutions for parameter intervals. The multi-bifurcation diagram can be used in such situations to design a robust machine or at least be prepared for unwanted dynamics

Les choix électoraux des petits commerçants et artisans en 1967:L'importance des variables contextuelles

Sur la base d'enquêtes réalisées en 1967, il apparaît que le comportement électoral des petits commerçants et artisans (PCA) dépend essentiellement de leur position intermédiaire entre la bourgeoisie et le prolétariat. Leur statut d'indépendant l'emporte sur leurs caractéristiques populaires et motive leur hostilité au communisme et leur vote pour la droite (toutefois, les artisans, plus proches des ouvriers par la nature de leur travail, votent plus fréquemment pour la gauche). Cette tendance générale peut être accentuée ou affaiblie selon les caractéristiques politiques et sociales du contexte local. Comme tous les groupes socio-professionnels, les PCA sont influencés par ce contexte local, et leur vote de droite diminue dans les départements où le PC est électoralement puissant. Il augmente au contraire quand l'importance du groupe ouvrier, représentant une menace sociale et politique, les rejette du côté de la bourgeoisie et de la droite. Inversement, leur vote de gauche s'élève quand le groupe des agriculteurs est important, et qu'ils s'identifient par contraste aux « petits ». C'est enfin dans les départements du Sud de la France où se maintiennent de concert petit commerce, artisanat et agriculture traditionnels, que les PCA votent le plus pour la gauche

Experimental and Numerical Simulation of Unbalance Response in Vertical Test Rig with Tilting-Pad Bearings

In vertically oriented machines with journal bearing, there are no predefined static radial loads, such as dead weight for horizontal rotor. Most of the commercial software is designed to calculate rotordynamic and bearing properties based on machines with a horizontally oriented rotor; that is, the bearing properties are calculated at a static eccentricity. For tilting-pad bearings, there are no existing analytical expressions for bearing parameters and the bearing parameters are dependent on eccentricity and load angle. The objective of this paper is to present a simplified method to perform numerical simulations on vertical rotors including bearing parameters. Instead of recalculating the bearing parameters in each time step polynomials are used to represent the bearing parameters for present eccentricities and load angles. Numerical results are compared with results from tests performed in a test rig. The test rig consists of two guide bearings and a midspan rotor. The guide bearings are 4-pad tilting-pad bearings. Shaft displacement and strains in the bearing bracket are measured to determine the test rig’s properties. The comparison between measurements and simulated results shows small deviations in absolute displacement and load levels, which can be expected due to difficulties in calculating exact bearing parameters

Analysis of multiple solutions in bifurcation diagrams to avoid unexpected dynamics

In mechanical applications it is essential that unexpected dynamics are avoided. The industry wants to build reliable machines that are not sensitive to initial conditions. Therefore, a simple method has been developed to extract all sets of stable bifurcation diagrams. The method gives a designer a good overview of possible dynamics and thereby the possibility to select a safe operating region. The method is described and demonstrated with a rub-impact rotor. The practical usage of this method is to help the designer to determine if parameter ranges exist where coexistent solutions will appear. Thereby one can design the system to work in parameter ranges where only one acceptable solution exists.Godkänd; 2009; 20091202 (joa

Dynamics of some vibro-impacting systems with amplitude constraints

This thesis concerns the dynamics of some vibro-impacting systems with fixed or moving amplitude constraints. It is based on and includes five papers, marked A to E. Simple models of three different vibro-impacting systems with applications in the fields of impact hammers, granular flow and disk brakes in vehicles are analysed. A 2-DOF (two-degree-of-freedom) model of a threshold-limited impact hammer is studied (Paper A). The stability of a class of periodic motions is analysed. For some parameter values these periodic motions are found to be qualitatively similar to the ones observed for a corresponding 1-DOF system. At other parameter combinations, however, new kinds of periodic or chaotic motions can be observed. For low damping, phenomena resembling antiresonance for linear systems can also be observed. Granular shear flows show a transitional behaviour in the rapid flow regime as the shear speed or the concentration of the grains is varied. The motion can, for example, change from smooth and orderly to erratic and turbulent. Some aspects of this transitional behaviour in granular shear flow are studied numerically, analytically and experimentally (Papers B, C and D). Simple vibro-impacting models are suggested to get some analytical insight into the dynamics of shear layers. Results from a 1-DOF model show that for high forcing frequencies, which correspond to high shear speeds, periodic as well as chaotic motions can exist, whereas, for low forcing frequencies the vibrations are completely damped out to a stationary state (Paper B). Stability of this stationary state is studied analytically (Paper C), and experimentally (Paper D), where the motions of granular particles in a transparent shear cell are followed by using video techniques. For low shear speeds a single shear layer adjacent to the bottom boundary of the shear cell is observed. As the shear speed is increased, a transition to a random like state involving many layers is found to occur. In order to understand the phenomenon of squeal in disk brakes, a 3-DOF model is suggested to simulate the dynamics of a brake pad. The region of contact between the brake pad and the disk is described by using a coefficient of friction and distributed stiffness. The brake pad is allowed to have adjustable support locations and possibilities of impacts with its surroundings. The equilibrium state of the pad is determined by using a static analysis. The assumption is that the instability of this stationary state is a possible explanation of squeal, therefore, the stability is analysed in detail. Examples of different kinds of pad motions are presented. A rich variety of motions are found to exist including periodic, seemingly chaotic, stationary behaviour in slip, vibrations with full contact with the disk, stick-slip and impacts.Godkänd; 1995; 20070426 (ysko

Dynamics of some vibro-impacting systems with amplitude constraints

This thesis concerns the dynamics of some vibro-impacting systems with fixed or moving amplitude constraints. It is based on and includes five papers, marked A to E. Simple models of three different vibro-impacting systems with applications in the fields of impact hammers, granular flow and disk brakes in vehicles are analysed. A 2-DOF (two-degree-of-freedom) model of a threshold-limited impact hammer is studied (Paper A). The stability of a class of periodic motions is analysed. For some parameter values these periodic motions are found to be qualitatively similar to the ones observed for a corresponding 1-DOF system. At other parameter combinations, however, new kinds of periodic or chaotic motions can be observed. For low damping, phenomena resembling antiresonance for linear systems can also be observed. Granular shear flows show a transitional behaviour in the rapid flow regime as the shear speed or the concentration of the grains is varied. The motion can, for example, change from smooth and orderly to erratic and turbulent. Some aspects of this transitional behaviour in granular shear flow are studied numerically, analytically and experimentally (Papers B, C and D). Simple vibro-impacting models are suggested to get some analytical insight into the dynamics of shear layers. Results from a 1-DOF model show that for high forcing frequencies, which correspond to high shear speeds, periodic as well as chaotic motions can exist, whereas, for low forcing frequencies the vibrations are completely damped out to a stationary state (Paper B). Stability of this stationary state is studied analytically (Paper C), and experimentally (Paper D), where the motions of granular particles in a transparent shear cell are followed by using video techniques. For low shear speeds a single shear layer adjacent to the bottom boundary of the shear cell is observed. As the shear speed is increased, a transition to a random like state involving many layers is found to occur. In order to understand the phenomenon of squeal in disk brakes, a 3-DOF model is suggested to simulate the dynamics of a brake pad. The region of contact between the brake pad and the disk is described by using a coefficient of friction and distributed stiffness. The brake pad is allowed to have adjustable support locations and possibilities of impacts with its surroundings. The equilibrium state of the pad is determined by using a static analysis. The assumption is that the instability of this stationary state is a possible explanation of squeal, therefore, the stability is analysed in detail. Examples of different kinds of pad motions are presented. A rich variety of motions are found to exist including periodic, seemingly chaotic, stationary behaviour in slip, vibrations with full contact with the disk, stick-slip and impacts.Godkänd; 1995; 20070426 (ysko

Influence of the thrust bearing on the natural frequencies of a72-MW hydropower rotor

The thrust bearing is an essential element of a hydropower machine. Not only does itcarry the total axial load but it also introduces stiffness and damping properties in the system.The focus of this study is on the influence of the thrust bearing on the lateral vibrations of theshaft of a 72-MW propeller turbine. The thrust bearing has a non-conventional design with alarge radius and two rows of thrust pads. A numerical model is developed to estimate naturalfrequencies. Numerical results are analyzed and related to experimental measurements of arunaway test.The results show the need to include the thrust bearing in the model. In fact, the vibrationmodes are substantially increased towards higher frequencies with the added properties fromthe thrust bearing. The second mode of vibration has been identified in the experimentalmeasurements. Its frequency and mode shape compare well with numerical results

Comparison of contact dynamics in bladed Jeffcott rotors

Godkänd; 2013; 20130726 (flothi

Numerical Estimation of Torsional Dynamic Coefficients of a Hydraulic Turbine

The rotordynamic behavior of a hydraulic turbine is influenced by fluid-rotor interactions at the turbine runner. In this paper computational fluid dynamics (CFDs) are used to numerically predict the torsional dynamic coefficients due to added polar inertia, damping, and stiffness of a Kaplan turbine runner. The simulations are carried out for three operating conditions, one at about 35% load, one at about 60% load (near best efficiency), and one at about 70% load. The runner rotational speed is perturbed with a sinusoidal function with different frequencies in order to estimate the coefficients of added polar inertia and damping. It is shown that the added coefficients are dependent of the load and the oscillation frequency of the runner. This affect the system's eigenfrequencies and damping. The eigenfrequency is reduced with up to 65% compared to the eigenfrequency of the mechanical system without the fluid interaction. The contribution to the damping ratio varies between 30–80% depending on the load. Hence, it is important to consider these added coefficients while carrying out dynamic analysis of the mechanical system