Self-Excited Vibrations of the Variable Mass Rotor//Fluid System (original) (raw)
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Lecture Notes in Mechanical Engineering, 2020
This paper presents a comparative study between the nonlinear dynamic behavior of an unbalanced rigid rotor in a short hydrodynamic bearing and of an unbalanced rigid rotor in a long hydrodynamic bearing. Two nonlinear mathematical models with two degrees of freedom are used in this investigation to predict the movement of the shaft and its bifurcations. Nonlinearity is introduced into models through analytical expressions of hydrodynamic forces. These analytical expressions are determined by the integration of the oil pressure distribution into the bearing using the short bearing approximation and the long bearing approximation. The numerical integration method is applied to determine the bifurcation diagrams using the rotor speed as a control parameter. In this study, Poincaré sections, frequency spectrum, motion orbits, and bifurcation diagrams are used to characterize the shaft motion. Several nonlinear phenomena such as jumping motion, multi-periodic oscillation, quasi-periodic motion and chaotic motion are predicted. It has been found that the effect of unbalance is very important on the stability threshold speed and on the amplitudes of the oscillations for the case of a short bearing. However, the unbalance effect is negligible in the case of a long bearing. Keywords: Hydrodynamic forces Á Long bearing Á Short bearing Á Numerical integration Á Stability analyses Á Nonlinear phenomena
Nonlinear dynamics of a support-excited flexible rotor with hydrodynamic journal bearings
Journal of Sound and Vibration, 2014
The major purpose of this study is to predict the dynamic behavior of an on-board rotor mounted on hydrodynamic journal bearings in the presence of rigid support movements, the target application being turbochargers of vehicles or rotating machines subject to seismic excitation. The proposed onboard rotor model is based on Timoshenko beam finite elements. The dynamic modeling takes into account the geometric asymmetry of shaft and/or rigid disk as well as the six deterministic translations and rotations of the rotor rigid support. Depending on the type of analysis used for the bearing, the fluid film forces computed with the Reynolds equation are linear/non-linear. Thus the application of the Lagrange's equations yields the linear/non-linear equations of motion of the rotating rotor in bending with respect to the moving rigid support which represents a non-inertial frame of reference. These equations are solved using the implicit Newmark time-step integration scheme. Due to the geometric asymmetry of the rotor and to the rotational motions of the support, the equations of motion include time-varying parametric terms which can lead to lateral dynamic instability. The influence of sinusoidal rotational or translational motions of the support, the accuracy of the linear 8-coefficient bearing model and the interest of the non-linear model for a hydrodynamic journal bearing are examined and discussed by means of stability charts, orbits of the rotor, time history responses, fast Fourier transforms, bifurcation diagrams as well as Poincaré maps.
2016
The paper covers the questions of modeling and research different hydrodynamic effects, that influence the occurrence of lateral vibrations in rotor systems with fluid-film bearings. The present research is aimed at developing rotor dynamics as one of the fields of science, as well as at developing the analysis and diagnostics methods of the dynamic condition of the rotor systems with fluid-film bearings. The results of the present research consist of a complex of the developed mathematical models and the numerical methods of solutions for centering and Magnus effects in fluid-film bearing and its influence on dynamics of rotor system. The significance of the results is determined by a wide range of applications in various designs of single-shaft rotor systems. The novelty of the obtained results is in the developed mathematical models that allow solving the analysis problems of lateral vibrations in the rotor systems with fluid-film bearings, given a nonlinear formulation of the pr...
Dynamics of Rotors on Hydrodynamic Bearings
Rotordynamic analysis is a crucial step in the development of rotor bearing systems in order to prevent rotor instabilities, excessive unbalance response, bearing overheating or other undesired phenomena. This study presents a rotordynamic analysis of a rotor supported by hydrodynamic bearings using Comsol Multiphysics. In this paper, the complexity of the model is gradually increased. Starting point of the analysis is the modal analysis of the rotor in free-free conditions, as it can be accurately validated by experiments. Once the rotor modal properties correspond with experimental values, the hydrodynamic bearing properties are investigated. A Reynolds model is set up to predict the lubricating film pressure distribution under shaft loading. Due to the cross coupling terms of the bearing stiffness coefficients, instability may occur. A point mass simulation of the shaft in a hydrodynamic bearing is able to predict the whirl frequency and amplitude of the bearing itself. Coupling the hydrodynamic bearings to a flexible shaft, the whirl can be seen to interact with the rotor natural frequencies.
MATEC Web of Conferences
This paper is focused on an analysis of rotating systems with fluid-film bearings, especially on their nonlinear behaviour in the course of developing fluid-induced instability. The studied system consists of Jeffcott rotor supported by a fluid-film bearing characterised by the Reynolds equation. The steady state response of the system is investigated by means of an approximate analytical solution of the Reynolds equation while the transient response of the system is investigated using a complex numerical solution. Results suggest that the rotor exercise a bounded chaotic motion if it becomes unstable. If the fluid-induced instability further develops, the motion actually becomes less chaotic and can be characterised as quasi-periodic.
A novel semi-analytical method for the dynamics of nonlinear rotor-bearing systems
Mechanism and Machine Theory, 2014
A semi-analytical simulation of a rotor bearing system that consisted of a multi-segment continuous rotor and plain fluid film bearings with finite length is developed in this paper in order to investigate the system's response under the current proposal of simulation. The rotor is simulated using the continuous medium theory and the bearings with finite length follow a very recent analytical simulation that incorporates the analytical solution of the Reynolds equation for the plain, finite journal bearing. The boundary conditions combine the rotor's shearing force and the fluid film forces at the points where the bearings are located, which are expressed analytically with direct integration of the pressure distribution function. A case study of simulating a multi-segment shaft is used in order to compare the current simulation with corresponding simulations using continuous rotor mounted on linear bearings, or finite bearings numerically simulated using the finite difference method. The time response, the amplitude of the response and the phase during passage through resonance are evaluated for these three cases of bearing simulation using numerical procedures and the differences are notified and remarked.
Fluid Induced Instability of Rotor Systems with Journal Bearings
The paper deals with stability of the rotor vibration in a journal bearing. The vibration signal, describing the rotor motion, is a complex signal. The real part of this kind of signals is a rotor displacement in the X-direction while the imaginary part is a displacement in the perpendicular direction, as we say in the Y-direction. A tool for analysis is a full spectrum, which results from the Fourier transform of the complex signal. The full multispectra of the rotor run up and coast down are employed to evaluate a magnitude as a function of the rotor rotation speed. The multispectrum slices serve to verification of the simplified mathematical model of a rotor systém and to analyze the rotor vibration using a procedure based on the Nyquist stability criterion. As it is well known the self-excited vibration, called fluid induced vibration, occurs when the rotor rotation speed crosses a certain threshold.
Nonlinear responses of externally excited rotor bearing system
MATEC Web of Conferences, 2014
A mathematical model incorporating higher order deformation in bending is developed to investigate the nonlinear behavior of rotor. Transverse harmonic base excitation is imparted to rotor system and Euler-Bernoulli beam theorem is applied with effects such as rotary inertia, gyroscopic effect, higher order large deformations, rotor mass unbalance and dynamic axial force. Discretization of the kinetic and strain (deformation) energies of the rotor system is done using the Rayleigh-Ritz method. Second order coupled nonlinear differential equations of motion are obtained using Hamilton's principle. Nonlinear dynamic response of the rotor system is obtained by solving above equations using the method of multiple scales. This response is examined for resonant condition. It is concluded that nonlinearity due to higher order deformations and variations in the values of different parameters like mass unbalance and shaft diameter significantly affects the dynamic behavior of the rotor system. It is also observed that the external harmonic excitation greatly affects the dynamic response.
As immersed rotors vibrate in a viscous media such as fluid, a considerable amount of damping may be generated due to the interaction phenomena between the rotor components and the fluid media. Such damping is depending on many factors such as; fluid drag, fluid friction, turbulence, vortex and so on. Immersed rotors find their application in many engineering fields such as Marines machines, gear box, turbine and pumps. In the present work, a mathematical model is attempted to investigate the dynamical behavior immersed rotor. The model takes into account the effects of the most rotordynamic parameters, namely; fluid drag, damping and stiffness of bearing, unbalance and gyroscopic effects of the attached disc, and elastic bending and internal damping of rotor shaft. Four types of fluid are employed as a fluid immersing media which are; Air, Water, SAE 20 and SAE 40 oils. The experimental apparatus includes a sample rotor with single disc and plastic fluid container. Two proximate sensors are employed for measuring the unbalance response and orbits shapes under different rotor speeds, and discs size and locations. Modal analysis is employed for solving the governing equation of vibration motion. To check the validity of the mathematical model the theoretical results are compared with the experimental results. It is found that; the theoretical results are in a good agreement with the experimental ones, where the maximum error is not exceeded (6.8 %), and that; the fluid damping can highly reduce the peak amplitude of the unbalance response (up to 60 %) however, it has slight effect on the critical speeds which are highly affected by the size and location of the attached disc.
A nonlinear, dynamic, continuous, damped model for rotor-bearing systems
Strong nonlinearities in mechanical systems usually lead to the formulation of dynamical systems by ways expressing the physical phenomena in an adequate ap-proximation. In this paper the mechanical system is a rotating shaft mounted on finite fluid film bearings. The fact that the rotor model follows the Rayleigh equations of motion and that the fluid film forces are calculated using the Reynolds equation keep the model presented in this paper far away from any approximation usually made to-wards simplification. The current dynamical system is obtained due to the variable boundary conditions that are functions of time, or more generally are functions of the response that the rotor obtains during operation. So the system of equations of motion is reformed at any time step. The strong non-linearity in this model is due to the fluid film forces that are confronted as a feedback in the journal (rotor) response. The fluid film forces are obtained for each journal displacement and veloci...