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 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.

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.

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.

Vibration and Dynamic Stability of Pre-Twisted Thick Cantilever Beam Made of Functionally Graded Material

2015 ◽  
Vol 15 (04) ◽  
pp. 1450058 ◽  
Author(s):  
Sukesh Chandra Mohanty ◽  
Rati Ranjan Dash ◽  
Trilochan Rout
Keyword(s):  
Dynamic Stability ◽  
Power Law ◽  
Twist Angle ◽  
Stability Boundary ◽  
Parametric Instability ◽  
Functionally Graded ◽  
Load Factor ◽  
Graded Material ◽  
The Stability ◽  
Power Law Index

In the present work, the vibration and dynamic stability of functionally graded ordinary (FGO) pre-twisted cantilever Timoshenko beam has been investigated. Finite element shape functions are established from differential equations of static equilibrium. Expressions for element stiffness and mass matrices are obtained from energy considerations. Floquet's theory is used to establish the stability boundary. The material properties along the thickness of the beam are assumed to vary according to the power law. The effects of power law index and pre-twist angle on the natural frequencies and dynamic stability of the beam have been investigated. Increase in pre-twist angle enhances the stability of the beam for first mode whereas it makes the beam more prone to parametric instability for the second mode. The increase in power law index is found to have a detrimental effect on the stability of the beam. The chance of parametric instability is enhanced with the increase in static load factor.

Dynamic Stability Characteristics of Axially Accelerated Beams

2006 ◽  
Vol 321-323 ◽  
pp. 1654-1658 ◽  
Author(s):  
Hong Hee Yoo ◽  
Sung Jin Eun
Keyword(s):  
Dynamic Stability ◽  
Natural Frequencies ◽  
Equations Of Motion ◽  
Stability Diagram ◽  
Accelerated Motion ◽  
Stability Diagrams ◽  
Divergence Type ◽  
The Stability ◽  
Direct Numerical Integration ◽  
Free Beam

Dynamic stability of axially accelerated beams is investigated in this paper. The equations of motion of a fixed-free beam undergoing axially accelerated motion are derived. Unstable regions due to the acceleration are obtained by using the Floquet’s theory. Stability diagrams are presented to illustrate the influence of the acceleration characteristics. Large unstable regions of flutter type instability exist around the first, twice the first, and twice the second bending natural frequencies. Divergence type instability also occurs when the acceleration exceeds a certain value. The validity of the stability diagram is confirmed by direct numerical integration of the equations of motion.

Stability Boundary of Non-Linear Orbits Within Clearance Circle of Journal Bearings

1993 ◽  
Vol 115 (3) ◽  
pp. 303-307 ◽  
Author(s):  
M. M. Khonsari ◽  
Y. J. Chang
Keyword(s):  
Stability Analysis ◽  
Journal Bearing ◽  
Stability Boundary ◽  
Journal Bearings ◽  
Initial Condition ◽  
Bearing Clearance ◽  
Linearized Stability ◽  
Non Linear

The transient, nonlinear study of journal bearing stability requires tracking the locus of the shaft center as a function of time. It is shown that there exists a boundary within the bearing clearance circle outside of which any initial condition would yield an unstable orbit. This is shown to be the case for operating speeds that are well below the threshold of instability according to the linearized stability analysis.

Dynamic Stability of a Spinning Timoshenko Beam Subjected to a Moving Mass-Spring-Damper Unit

2010 ◽  
Author(s):  
T. H. Young ◽  
M. S. Chen
Keyword(s):  
Dynamic Stability ◽  
Timoshenko Beam ◽  
Multiple Scales ◽  
Equations Of Motion ◽  
Axial Direction ◽  
Longitudinal Axis ◽  
Moving Mass ◽  
The Stability ◽  
The Galerkin Method ◽  
Mass Spring

This paper investigates the dynamic stability of a finite Timoshenko beam spinning along its longitudinal axis and subjected to a moving mass-spring-damper (MSD) unit traveling in the axial direction. The mass of the moving MSD unit makes contact with the beam all the time during traveling. Due to the moving MSD unit, the beam is acted upon by a periodic, parametric excitation. In this work, the equations of motion of the beam are first discretized by the Galerkin method. The discretized equations of motion are then partially uncoupled by the modal analysis procedure suitable for gyroscopic systems. Finally the method of multiple scales is used to obtain the stability boundaries of the beam. Numerical results show that if the displacement of the MSD unit is equal to only one of the two transverse displacements of the beam, very large unstable regions may appear at main resonances.

Online Combustor Stability Margin Assessment Using Dynamic Pressure Data

2004 ◽  
Author(s):  
Tim Lieuwen
Keyword(s):  
Dynamic Stability ◽  
Environmental Changes ◽  
Dynamic Pressure ◽  
Stability Boundary ◽  
Stability Margin ◽  
Operating Conditions ◽  
Margin Assessment ◽  
Pressure Oscillations ◽  
Acoustic Damping ◽  
The Stability

This paper describes a strategy for determining a combustor’s dynamic stability margin. Currently, when turbines are being commissioned or simply going through day to day operation, the operator has no idea how the dynamic stability of the system is affected by changes to fuel splits/operating conditions unless, of course, pressure oscillations are actually present. We have developed a methodology for ascertaining stability margin from passive monitoring of the acoustic pressure. This method consists of signal processing and analysis that determines a real-time measure of combustor damping. When the calculated damping is positive, the combustor is stable. When the damping goes to zero, the combustor approaches its stability boundary. Changes in the stability margin of each of the combustor’s stable modes due to tuning, aging or environmental changes can then be monitored through online analysis of the pressure signal. This paper outlines the basic approach used to quantify acoustic damping and demonstrates the technique on combustor test data.

scholarly journals Stability analysis of long hydrodynamic journal bearings based on the journal center trajectory

Friction ◽  
2020 ◽  
Author(s):  
Yu Huang ◽  
Haiyin Cao ◽  
Zhuxin Tian
Keyword(s):  
Stability Analysis ◽  
Equilibrium Point ◽  
Journal Bearing ◽  
Stability Boundary ◽  
Initial Point ◽  
Journal Bearings ◽  
Bearing Surface ◽  
Rotating Speed ◽  
Stability Threshold ◽  
The Stability

AbstractIn this study, we observe that there are two threshold speeds (stability threshold speed and second threshold speed) for the long journal bearing, which is different for the short bearing. When the rotating speed is below the stability threshold speed, the stability boundary nearly coincides with the clearance circle, and the journal center gradually returns to the equilibrium point after being released at an initial point. If the rotating speed is between the stability threshold speed and the second threshold speed, after being released at an initial point, the journal center converges to a contour containing the equilibrium point. In this situation, for a higher rotating speed, the corresponding contour is also larger. When the rotating speed exceeds the second threshold speed, the journal gradually moves towards the bearing surface after being released at an initial point.

THERMOMECHANICAL STABILITY ANALYSIS OF FUNCTIONALLY GRADED THIN-WALLED CANTILEVER PIPE WITH FLOWING FLUID SUBJECTED TO AXIAL LOAD

2011 ◽  
Vol 11 (03) ◽  
pp. 513-534 ◽  
Author(s):  
M. HOSSEINI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Axial Force ◽  
Volume Fraction ◽  
Equations Of Motion ◽  
Stability Boundary ◽  
Functionally Graded ◽  
Flowing Fluid ◽  
Thin Walled ◽  
The Stability ◽  
Compressive Axial Force

In this study, the thermomechanical stability of functionally graded thin-walled cantilever pipes conveying flow and loading by compressive axial force is investigated. The governing equations of motion and boundary conditions are derived via the Hamilton's variational principle. The thin-walled structure is formulated based on Rayleigh's theory. Moreover, quasi-steady flow pressure loadings and steady surface temperature are considered and the temperature gradient through the wall thickness of the pipe is included. The partial differential equations of the pipe are transformed into a set of ordinary differential equations using the extended Galerkin method. Finally, having solved the resulting thermal-structural-fluid eigenvalue system of equations, the effects of the compressive axial force, fluid speed, fluid mass ratio, volume fraction index of functionally graded materials (FGMs), and temperature change through the thickness of the pipe on the stability boundary are investigated. Numerical comparisons are also performed with the available data in the literature and good agreement is observed.

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