Stability Analysis of a Whirling Rigid Rotor Supported by FDBs Considering Five Degrees of Freedom of a General Rotor-Bearing System

2014 ◽  
Author(s):  
M. H. Lee ◽  
J. H. Lee ◽  
G. H. Jang
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Time Varying ◽  
Algebraic Equations ◽  
Dynamic Coefficients ◽  
Mass Unbalance ◽  
Whirling Motion ◽  
Rotor Bearing ◽  
The Stability ◽  
Bearing System

A rotor supported by fluid dynamic bearings (FDBs) has a whirling motion by centrifugal force due to the mass unbalance or by the flexibility of shaft. This whirling motion also generates periodic time-varying oil-film reaction and dynamic coefficients even in case of the stationary grooved FDBs. This paper proposes a method to determine the stability of a whirling rotor supported by stationary grooved FDBs considering five degrees of freedom of a general rotor-bearing system. Dynamic coefficients are calculated by using the finite element method and the perturbation method, and they are represented as periodic harmonic functions by considering whirling motion. Because of the periodic time-varying dynamic coefficients, the equations of motion of the rotor supported by FDBs can be represented as a parametrically excited system. The solution of the equations of motion can be assumed as the Fourier series so that the equations of motion can be rewritten as simultaneous algebraic equations with respect to the Fourier coefficients. Hill’s infinite determinant is calculated by using these algebraic equations in order to determine the stability. The stability of the FDBs decreases with the increase of rotational speed. The stability of the FDBs increases with the increase of whirl radius, because the average and variation of Cxx increase faster than those of Kxx. The proposed method is verified by solving the equations of motion by using the forth Runge-Kutta method to determine the convergence and divergence of whirl radius.

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.

Pointwise Optimal Control of Dynamical Systems Described by Constrained Coordinates

1999 ◽  
Vol 121 (4) ◽  
pp. 594-598 ◽  
Author(s):  
V. Radisavljevic ◽  
H. Baruh
Keyword(s):  
Optimal Control ◽  
Dynamical Systems ◽  
Feedback Control ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Control Law ◽  
Algebraic Equations ◽  
The Stability ◽  
The Mathematical Model

A feedback control law is developed for dynamical systems described by constrained generalized coordinates. For certain complex dynamical systems, it is more desirable to develop the mathematical model using more general coordinates then degrees of freedom which leads to differential-algebraic equations of motion. Research in the last few decades has led to several advances in the treatment and in obtaining the solution of differential-algebraic equations. We take advantage of these advances and introduce the differential-algebraic equations and dependent generalized coordinate formulation to control. A tracking feedback control law is designed based on a pointwise-optimal formulation. The stability of pointwise optimal control law is examined.

Calculation of Dynamic Coefficients in a Hydrodynamic Bearing Considering Five Degrees of Freedom for a General Rotor-Bearing System

1999 ◽  
Vol 121 (3) ◽  
pp. 499-505 ◽  
Author(s):  
G. H. Jang ◽  
Y. J. Kim
Keyword(s):  
Degrees Of Freedom ◽  
Disk Drive ◽  
Spindle Motor ◽  
Hydrodynamic Bearing ◽  
Thrust Bearings ◽  
Dynamic Coefficients ◽  
Rotor Bearing ◽  
The Moment ◽  
Wedge Effect ◽  
Bearing System

A complete method is presented to calculate the stiffness and the damping coefficients in a hydrodynamic bearing considering five degrees of freedom for a general rotor-bearing system. Perturbation equations are obtained from Reynolds equation by assuming the small amplitude motion of a bearing center, and are solved by the finite element method. Their characteristics due to eccentricity and misalignment are investigated for herringbone groove journal and thrust bearings in the spindle motor of a hard disk drive. This research shows that the dynamic coefficients increase with increasing the misalignment as well as the eccentricity due to the wedge effect. It also shows that the moment coefficients, which have been neglected in most of the previous analyses, are of significant magnitude in a journal bearing and have even bigger values for the thrust bearing when they are compared with the ball bearing in the same type of a spindle motor.

Dynamic Analysis of a Motorized Spindle With Externally Pressurized Air Bearings

2015 ◽  
Vol 137 (4) ◽  
Author(s):  
Chundong Xu ◽  
Shuyun Jiang
Keyword(s):  
Dynamic Characteristics ◽  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Forced Response ◽  
Air Bearings ◽  
Motorized Spindle ◽  
Static And Dynamic Characteristics ◽  
Rotating Speed ◽  
Rotor Bearing ◽  
Bearing System

The purpose of this paper is to investigate the dynamic characteristics of a motorized spindle with externally pressurized air bearings. The externally pressurized air bearings consist of a journal bearing and a double pad thrust bearing with orifice restrictors. The equations of motion for the rotor-bearing system are established considering five degrees-of-freedom (DOF). The perturbation method and the finite difference method are introduced to calculate the static and dynamic characteristics of the air bearings; and the effects of the rotating speed and tilt angle of the rotor on the dynamic characteristics of the air bearings are analyzed. With the dynamic coefficients of the air bearings and the 5DOF rotor-dynamic model obtained, the stability, the unbalance response, and the forced response of the rotor-bearing system are investigated. Finally, the static and dynamic characteristics of the spindle are verified by an experimental study.

Parametric Study of Stability Criteria for Rotor Bearing Model With Viscoelastic Support

2017 ◽  
Author(s):  
S. Chandraker ◽  
J. K. Dutt ◽  
H. Roy
Keyword(s):  
Classical Method ◽  
Equations Of Motion ◽  
Weight Ratio ◽  
High Strength ◽  
Point Of View ◽  
Stability Criteria ◽  
Engineering Structures ◽  
Rotor Bearing ◽  
The Stability ◽  
Bearing System

In the last few decades, intensive research has been carried out on viscoelastic materials. Among them, most importantly polymers and composites thereof find extensive applications in engineering structures and rotors primarily due to quite high strength to weight ratio in comparison with metals. In dynamic modeling of rotor bearing system, incorporation of damping is very important as stationary (external) damping always helps in stability, however rotary damping (internal) promotes instability of rotors above a certain speed. Therefore for modeling point of view, it is very important to consider both internal or external damping effect. For this reason, the dissipation mechanism has been handled in such a way that it provides proper forces irrespective of its presence in a stationary or a rotary frame. Also in present work, both classical method and operator multiplier method are suggested to derive the equations of motion. The analysis also shows the stability zones of the rotor bearing system for various parametric values of different viscoelastic supports. It is found that choosing a right viscoelastic support can increase the stability criteria of the system to some extent.

Effect of Flexible Damped Bearing-Supports on the Dynamic Stability of High-Speed Turbochargers

2013 ◽  
Author(s):  
A. Alsaeed ◽  
G. Kirk ◽  
S. Bashmal
Keyword(s):  
Dynamic Stability ◽  
Dynamic Characteristics ◽  
High Speed ◽  
Element Model ◽  
Dynamic Coefficients ◽  
Speed Range ◽  
Rotor Bearing ◽  
The Stability ◽  
The Finite Element Model ◽  
Bearing System

The aim of this study is to analytically design flexible damped bearing-supports in order to improve the dynamic characteristics of the rotor-bearing system. The finite-element model of the turbocharger rotor with linearized bearing dynamic coefficients is used to solve for the logarithmic decrements and hence the stability map. The design process attempts to find the optimum dynamic characteristics of the flexible damped bearing-support that would give best dynamic stability of the rotor-bearing system. The method is successful in greatly improving the dynamic stability of the turbocharger and may also lead to a total linear stability throughout the entire speed range when used besides the enhanced-performance hydrodynamic bearings.

Stability Analysis of a Rotating System Due to the Effect of Ball Bearing Waviness

2002 ◽  
Vol 125 (1) ◽  
pp. 91-101 ◽  
Author(s):  
G. H. Jang ◽  
S. W. Jeong
Keyword(s):  
Parametric Excitation ◽  
Ball Bearing ◽  
Equations Of Motion ◽  
Contact Forces ◽  
Fourier Series Expansion ◽  
Time Varying ◽  
Algebraic Equations ◽  
Rotating System ◽  
Mathieu’S Equation ◽  
The Stability

This research presents an analytical model to investigate the stability due to the ball bearing waviness in a rotating system supported by two ball bearings. The stiffness of a ball bearing changes periodically due to the waviness in the rolling elements as the rotor rotates, and it can be calculated by differentiating the nonlinear contact forces. The linearized equations of motion can be represented as a parametrically excited system in the form of Mathieu’s equation, because the stiffness coefficients have time-varying components due to the waviness. Their solution can be assumed as a Fourier series expansion so that the equations of motion can be rewritten as the simultaneous algebraic equations with respect to the Fourier coefficients. Then, stability can be determined by solving Hill’s infinite determinant for these algebraic equations. The validity of this research is proven by comparing the stability chart with the time responses of the vibration model suggested by prior research. This research shows that the waviness in the ball bearing generates the time-varying component of the stiffness coefficient, whose frequency is called the frequency of the parametric excitation. It also shows that the instability takes place from the positions in which the ratio of the natural frequency to the frequency of the parametric excitation corresponds to i/2 i=1,2,3,….

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.

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