Lyapunov’s Stability of Nonlinear Misaligned Journal Bearings

1995 ◽  
Vol 117 (3) ◽  
pp. 576-581 ◽  
Author(s):  
P. G. Nikolakopoulos ◽  
C. A. Papadopoulos
Keyword(s):  
Journal Bearing ◽  
Direct Method ◽  
Equations Of Motion ◽  
Orbital Stability ◽  
Element Formulation ◽  
Nonlinear Stiffness ◽  
Lyapunov Direct Method ◽  
Rotor Bearing ◽  
The Stability ◽  
General Method

In this paper the stability of nonlinear misaligned rotor-bearing systems is investigated, using the Lyapunov direct method. A finite element formulation is used to determine the journal bearing pressure distribution. Then the linear and nonlinear stiffness, damping, and hybrid (depending on both displacement and velocity) coefficients are calculated. A general method of analysis based on Lyapunov’s stability criteria is used to investigate the stability of nonlinear misaligned rotor bearing systems. The equations of motion of the rigid rotor on the nonlinear bearings are used to find a Lyapunov function using some of these coefficients, which depend on L/D ratio and the misalignment angles ψx, ψy. The analytical conditions for the stability or instability of some examined cases are given and some examples for the orbital stability are also demonstrated.

Lyapunov’s Stability of Non-Linear Misaligned Journal Bearings

1994 ◽  
Author(s):  
Padelis G. Nikoiakopouios ◽  
Chris A. Papadopouios
Keyword(s):  
Journal Bearing ◽  
Direct Method ◽  
Equations Of Motion ◽  
Orbital Stability ◽  
Element Formulation ◽  
Lyapunov Direct Method ◽  
Non Linear ◽  
Rotor Bearing ◽  
The Stability ◽  
General Method

In this paper the stability of non-linear misaligned rotor-bearing systems is investigated, using the Lyapunov direct method. A finite element formulation is used to determine the journal bearing pressure distribution. Then the linear and nonlinear stiffness, damping and hybrid (depended on both displacements and velocity) coefficients are calculated A general method of analysis based on Lyapunov’s stability criteria is used to investigate the stability of non-linear misaligned rotor bearing systems. The equations of motion of the rigid rotor on the non-linear bearings are used to find a Lyapunov function using some of the above coefficients, which are depending on L/D ratio and the misalignment angles ψa, ψr. The analytical conditions for the stability or instability of some examined cases are given and some examples for the orbital stability are also demonstrated.

Theoretical Prediction of Journal Center Motion Trajectory

1976 ◽  
Vol 98 (4) ◽  
pp. 620-628 ◽  
Author(s):  
D. V. Singh ◽  
R. Sinhasan ◽  
S. P. Tayal
Keyword(s):  
Theoretical Prediction ◽  
Dynamic Characteristics ◽  
Journal Bearing ◽  
Equations Of Motion ◽  
Motion Trajectory ◽  
Rotor Bearing ◽  
The Stability ◽  
Insight Into ◽  
Bearing System

By discretizing time and numerically integrating the equations of motion either for the linearized or the nonlinear journal bearing system, the locus of journal center can be predicted in the wake of a disturbance which upsets the equilibrium. From this locus, not only the stability of the system can be readily checked but also a greater insight into the dynamic characteristics of the rotor bearing systems can be obtained. Systems of solid bearings and porous bearings with journal bearing axes parallel as well as skewed have been studied.

Finite Journal Bearing With Nonlinear Stiffness and Damping. Part 2: Stability Analysis

1982 ◽  
Vol 104 (2) ◽  
pp. 406-411 ◽  
Author(s):  
E. Hashish ◽  
T. S. Sankar ◽  
M. O. M. Osman
Keyword(s):  
Stability Analysis ◽  
Journal Bearing ◽  
Linear Equations ◽  
Orbital Stability ◽  
Nonlinear Stiffness ◽  
Stability Behavior ◽  
Supply Pressure ◽  
The Stability ◽  
Stiffness And Damping ◽  
High Possibility

Stability analysis is performed on the linearized as well as the actual nonlinear finite bearing equations using the improved mathematical models for the hydrodynamic forces that are presented in Part I of this investigation. The results of the analysis using the linear equations show a significant trend, different from previous investigation, with respect to different L/d ratios and therefore can be considered as modified stability curves for the finite bearing. The nonlinear analysis, based on numerical integration of the equation of motion, is carried out for the commonly used L/d = 1. Details on the stability behavior of the finite bearing are established, including the orbital stability regions. It is also found that under certain light loading conditions, the supply pressure can introduce a high possibility of orbital stability to the system.

Stability Analysis of a Hydrodynamic Journal Bearing With Rotating Herringbone Grooves

2003 ◽  
Vol 125 (2) ◽  
pp. 291-300 ◽  
Author(s):  
G. H. Jang ◽  
J. W. Yoon
Keyword(s):  
Journal Bearing ◽  
Equations Of Motion ◽  
Fourier Series Expansion ◽  
Algebraic Equations ◽  
High Rotational Speed ◽  
Dynamic Coefficients ◽  
Stiffness Coefficients ◽  
Hydrodynamic Journal Bearing ◽  
The Stability ◽  
Herringbone Grooves

This paper presents an analytical method to investigate the stability of a hydrodynamic journal bearing with rotating herringbone grooves. The dynamic coefficients of the hydrodynamic journal bearing are calculated using the FEM and the perturbation method. The linear equations of motion can be represented as a parametrically excited system because the dynamic coefficients have time-varying components due to the rotating grooves, even in the steady state. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill’s infinite determinant of these algebraic equations. The validity of this research is proved by the comparison of the stability chart with the time response of the whirl radius obtained from the equations of motion. This research shows that the instability of the hydrodynamic journal bearing with rotating herringbone grooves increases with increasing eccentricity and with decreasing groove number, which play the major roles in increasing the average and variation of stiffness coefficients, respectively. It also shows that a high rotational speed is another source of instability by increasing the stiffness coefficients without changing the damping coefficients.

scholarly journals Stability of Fractional Differential Equations with New Generalized Hattaf Fractional Derivative

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Khalid Hattaf
Keyword(s):  
Stability Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Lyapunov Direct Method ◽  
Science And Engineering ◽  
Fractional Differential ◽  
The Stability

This paper aims to study the stability of fractional differential equations involving the new generalized Hattaf fractional derivative which includes the most types of fractional derivatives with nonsingular kernels. The stability analysis is obtained by means of the Lyapunov direct method. First, some fundamental results and lemmas are established in order to achieve the goal of this study. Furthermore, the results related to exponential and Mittag–Leffler stability existing in recent studies are extended and generalized. Finally, illustrative examples are presented to show the applicability of our main results in some areas of science and engineering.

Basin of Attraction of the Simplest Walking Model

2001 ◽  
Author(s):  
A. L. Schwab ◽  
M. Wisse
Keyword(s):  
Equations Of Motion ◽  
Basin Of Attraction ◽  
Mapping Method ◽  
Walking Robots ◽  
Natural Energy ◽  
Walking Motion ◽  
The Stability ◽  
The Basin Of Attraction ◽  
General Method ◽  
Small Disturbances

Abstract Passive dynamic walking is an important development for walking robots, supplying natural, energy-efficient motions. In practice, the cyclic gait of passive dynamic prototypes appears to be stable, only for small disturbances. Therefore, in this paper we research the basin of attraction of the cyclic walking motion for the simplest walking model. Furthermore, we present a general method for deriving the equations of motion and impact equations for the analysis of multibody systems, as in walking models. Application of the cell mapping method shows the basin of attraction to be a small, thin area. It is shown that the basin of attraction is not directly related to the stability of the cyclic motion.

On the Dynamic Stability of Self-Aligning Journal Gas Bearings

1977 ◽  
Vol 99 (4) ◽  
pp. 434-440 ◽  
Author(s):  
M. J. Cohen
Keyword(s):  
Dynamic Stability ◽  
Journal Bearing ◽  
Equations Of Motion ◽  
Stability Boundary ◽  
Initial Condition ◽  
Small Motion ◽  
Gas Bearings ◽  
Loading Case ◽  
The Stability ◽  
Small Disturbances

The report presents an investigation of the dynamic stability behaviour of self-aligning journal gas bearings when subjected to arbitrary small disturbances from an initial condition of operational equilibrium. The method is based on an approach similar to the nonlinear-ph solution of the author for the quasi-static loading case but the equations of motion of the journal are the linearized forms for small motion in the two degrees (translational) of freedom of the journal center. The stability domains for the infinite journal bearing are presented for the whole of the eccentricity (ε) and rotational speed (Λ) ranges for any given bearing geometry, in the shape of stability boundaries in that domain. It is shown that a given bearing will be stable within a corridor in the (ε, Λ) parametral domain having as its lower bound the so called “half-speed” whirl stability boundary and as its upper bound another whirling instability at a higher characteristic (relative) frequency, the instability occurs generally at the higher eccentricities and lower rotational speeds.

Stability of the High-Speed Journal Bearing Under Steady Load: 1—The Incompressible Film

1962 ◽  
Vol 84 (3) ◽  
pp. 351-357 ◽  
Author(s):  
M. M. Reddi ◽  
P. R. Trumpler
Keyword(s):  
High Speed ◽  
Journal Bearing ◽  
Orbital Motion ◽  
Hydrodynamic Force ◽  
Nonlinear Differential Equations ◽  
Equations Of Motion ◽  
Static Equilibrium ◽  
The Stability ◽  
External Loads ◽  
Steady Load

The phenomenon of oil-film whirl in bearings subjected to steady external loads is analyzed. The journal, assumed to be a particle mass, is subjected to the action of two forces; namely, the external load acting on the bearing and the hydrodynamic force developed in the fluid film. The resulting equations of motion for a full-film bearing and a 180-deg partial-film bearing are developed as pairs of second-order nonlinear differential equations. In evaluating the hydrodynamic force, the contribution of the shear stress on the journal surface is found to be negligible for the full-film bearing, whereas for the partial-film bearing it is found to be significant at small attitude values. The equations of motion are linearized and the coefficients of the resulting characteristic equations are studied for the stability of the static-equilibrium positions. The full-film bearing is found to have no stable static-equilibrium position, whereas the 180-deg partial-film bearing is found to have stable static-equilibrium positions under certain parametric conditions. The equations of motion for the full-film bearing are integrated numerically on a digital computer. The results show that the journal center, depending on the parametric conditions, acquired either an orbital motion or a dynamical path of increasing attitude terminating in bearing failure.

scholarly journals Asymptotic Stability of Distributed-Order Nonlinear Time-Varying Systems with the Prabhakar Fractional Derivatives

2020 ◽  
Vol 2020 ◽  
pp. 1-8 ◽  
Author(s):  
MohammadHossein Derakhshan ◽  
Azim Aminataei
Keyword(s):  
Asymptotic Stability ◽  
Fractional Derivatives ◽  
Direct Method ◽  
Time Varying ◽  
Lyapunov Direct Method ◽  
Fractional Differential Systems ◽  
Time Varying Systems ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

In this article, we survey the Lyapunov direct method for distributed-order nonlinear time-varying systems with the Prabhakar fractional derivatives. We provide various ways to determine the stability or asymptotic stability for these types of fractional differential systems. Some examples are applied to determine the stability of certain distributed-order systems.

Nonconservative Stability of Gyroscopic Rotor Systems

1993 ◽  
Author(s):  
Hwang-Kuen Chen ◽  
Der-Ming Ku ◽  
Lien-Wen Chen
Keyword(s):  
Direct Method ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Follower Force ◽  
The Finite Element Method ◽  
Nonconservative System ◽  
Rotor Systems ◽  
Eigenvalue Sensitivity ◽  
The Stability ◽  
Flutter Load

Abstract The stability behavior of a cantilevered shaft, rotating at a constant speed and subjected to a follower force at the free end, is studied by the finite element method. The equations of motion for such a gyroscopic system are formulated by using deformation shape functions developed from Timoshenko beam theory. The effects of translational and rotatory inertia, gyroscopic moments, bending and shear deformations are included. In order to determine the critical load of the present nonconservative system more quickly and efficiently, a simple and direct method that utilizes the eigenvalue sensitivity with respect to the follower force is introduced. The numerical results show that for the present nonconservative system, the onset of flutter instability occurs when the first and second backward whirl speeds are coincident. And also, due to the effect of the gyroscopic moments, the critical flutter load decreases as the rotational speed increases.

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