Liapunov Stability Analysis of Elastic Shaft-Rigid Disk-Bearing Systems

1992 ◽  
Vol 59 (4) ◽  
pp. 946-954
Author(s):  
H.-Y. Huang ◽  
A. L. Schlack
Keyword(s):  
Direct Method ◽  
Rotating Shaft ◽  
Rigid Disk ◽  
Elastic Shaft ◽  
Coupling Stiffness ◽  
Modal Term ◽  
Direct Stiffness ◽  
Instability Regions ◽  
Arbitrary Location ◽  
Stiffness And Damping

A general method of analysis based on Liapunov’s direct method is presented for studying the dynamic stability of elastic shaft-rigid disk-bearing systems. A model comprised of a rigid disk rigidly attached at an arbitrary location along a flexible, rotating shaft which is mounted on two eight-component end bearings is used to develop stability criteria involving system stiffness and damping parameters. It is quantitatively shown by means of graphs for typical cases how the instability regions are reduced by (a) increasing the shaft dimensionless stiffness parameters, (b) increasing the bearing direct stiffness and damping parameters, (c) decreasing the bearing cross-coupling stiffness and damping parameters, (d) decreasing the mass ratio of the disk, and (e) increasing the disk’s radius ratio. These graphs present typical examples of the types of design information available to engineers through the equations provided in this paper. These graphs also verify that a two-modal term (N = 2) expansion is normally adequate to model the system deformations since the curves are not significantly altered by adding another term (N = 3) to the expansion. The critical value of the shaft dimensionless stiffness parameters is also studied.

Dynamic Stability of Elastic Rotor-Bearing Systems via Liapunov’s Direct Method

1991 ◽  
Vol 58 (4) ◽  
pp. 1056-1063 ◽  
Author(s):  
Abd Alla El-Marhomy ◽  
A. L. Schlack
Keyword(s):  
Dynamic Stability ◽  
Direct Method ◽  
Stability Criteria ◽  
Rotor Bearing ◽  
Coupling Stiffness ◽  
Direct Stiffness ◽  
Instability Regions ◽  
Stiffness And Damping ◽  
General Method ◽  
Liapunov's Direct Method

A general method of analysis based on Liapunov’s direct method is presented for studying the dynamic stability of elastic rotor-bearing systems. A model comprised of a continuous elastic shaft mounted on two 8-coefficient bearings is used to develop closed-form (series) stability criteria involving system stiffness and damping parameters. It is quantitatively shown by means of graphs how the instability regions are reduced by (a) increasing the shaft dimensionless stiffness parameters, (b) increasing the bearing direct stiffness and damping parameters, and (c) decreasing the bearing cross-coupling stiffness and damping parameters.

Effects of Disk Flexibility on Shaft Whirl Stability

1979 ◽  
Vol 101 (2) ◽  
pp. 298-303 ◽  
Author(s):  
F. J. Wilgen ◽  
A. L. Schlack
Keyword(s):  
Concentrated Mass ◽  
Stability Boundaries ◽  
Critical Speeds ◽  
Rigid Disk ◽  
Flexible Shaft ◽  
Elastic Shaft ◽  
Speed Stability ◽  
Whirl Speed ◽  
Arbitrary Location ◽  
Limiting Case

The effects of disk flexibility on the critical speeds of flexible shaft-disk systems is investigated by the method of Liapunov. The model consists of a flexible, continuous disk rigidly attached at an arbitrary location along a flexible, continuous shaft which is mounted on short, end bearings. Whirl speed stability boundaries are presented as functions of the disk flexibility parameter. These boundaries reduce to the limiting case of a shaft containing a concentrated mass at the point of disk attachment when the disk’s stiffness is very small, and approach the limiting case of an elastic shaft supporting a rigid disk as the stiffness increases.

A CFD Study on the Dynamic Coefficients of Labyrinth Seals

2012 ◽  
Author(s):  
Donghui Zhang ◽  
Chester Lee ◽  
Michael Cave
Keyword(s):  
High Speed ◽  
Fluid Dynamic ◽  
Labyrinth Seal ◽  
Leakage Flow ◽  
Labyrinth Seals ◽  
Dynamic Coefficients ◽  
Rotating Impeller ◽  
Coupling Stiffness ◽  
Direct Stiffness ◽  
Compressor Efficiency

Labyrinth seals are widely used in gas compressors to reduce internal leakage and increase the compressor efficiency. Due to the eccentricity between the rotating impeller and the stationary part as *well as the shaft whirling motion, forces are generated when the leakage flow passing through the cavities and the seals. For a lot of applications with high speed and pressure, these forces can drive the system unstable. Thus, predicting the forces accurately become a very important for compressor rotordynamic designs. A lot of research and studies has been done to the seals itself, including bulk flow method, computational fluid dynamic (CFD) and test measurement. The seal and leakage flow interaction forces can be predicted relatively accurate. But very few research treat the seal and cavities as one component interacting with the leakage flow and produce the forces. This paper presents results of CFD investigations on the dynamic coefficients of one typical impeller eye seal and front cavity. The CFD results show that large forces are generated in the front cavity due to circumferential uniform pressure distribution, which caused by the downstream labyrinth seal. The forces generated in the front cavity are more than in the front seal. It was found that the inertia, damping, and stiffness are proportional to average pressure. The cross-coupling stiffness increases with speed with power of 2 while the direct stiffness increases with speed with power of about 1.7.

Dynamic characterization of a mechanical coupling for a rotating shaft

2010 ◽  
Vol 225 (3) ◽  
pp. 604-616 ◽  
Author(s):  
A T Tadeo ◽  
K L Cavalca ◽  
M J Brennan
Keyword(s):  
Frequency Response ◽  
Response Functions ◽  
Dynamic Characterization ◽  
Rotating Shaft ◽  
Frequency Response Functions ◽  
Mechanical Coupling ◽  
Lumped Parameter ◽  
Parameter Coupling ◽  
Stiffness And Damping

This article concerns the dynamic characterization of a flexible coupling that connects two co-axial shafts. Four different lumped parameter coupling models from the literature are investigated to see which model could best predict the dynamic behaviour of the coupling. The finite-element method was used to model the rotor dynamic system incorporating the coupling. Frequency response functions from this model were compared with measured frequency response functions from the rotor test rig with the shaft and coupling rotating at a specific speed. Parameters from the model were adjusted to minimize an objective function involving the measured and predicted frequency response functions. It was found that the simplest model of the coupling that could reasonably represent the coupling involves rotational (bending) stiffness and damping.

Experimental Analysis of Angled Injection Aerostatic Hybrid Bearings

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Julian Le Rouzic ◽  
Mihai Arghir
Keyword(s):  
Equations Of Motion ◽  
Cross Coupling ◽  
Impulse Turbine ◽  
Rotordynamic Coefficients ◽  
Counter Rotation ◽  
Theoretical Investigations ◽  
Coupling Stiffness ◽  
Direct Stiffness ◽  
The Cross ◽  
Dynamic Excitations

Counter-rotation angled injection employed for aerostatic hybrid bearings reduces the cross coupling stiffness that may lead to whirl–whip instabilities at high rotation speeds. The benefits of counter-rotation injection have been known for years. Theoretical investigations were performed for water or air fed hybrid bearings but experiments were conducted only for water fed bearings. The present work is the first effort dedicated to angled injection in air fed hybrid bearings. The tests were performed for a simple rotor supported by two identical hybrid bearings. The hybrid bearings are provided with small size, shallow pockets and are fed with air via counter-rotation-oriented orifice type restrictors. An impulse turbine fed with air entrains the rotor. An impact gun applies dynamic excitations and the rotordynamic coefficients are identified from the equations of motion of the rotor. Different air feeding pressures are tested as well as high rotational speeds. Compared to the dynamic characteristics of radial injection hybrid bearings, the direct stiffness of counter-rotation injection bearings has slightly lower values and the direct damping is higher but the main impact is the drastic reduction of the cross-coupling stiffness that may have even negative values.

scholarly journals Rotordynamic Force Coefficients of Pocket Damper Seals

2006 ◽  
Author(s):  
B. Ertas ◽  
A. Gamal ◽  
J. Vance
Keyword(s):  
Dynamic Pressure ◽  
Mechanical Impedance ◽  
Impedance Method ◽  
Force Density ◽  
Frequency Dependent ◽  
Force Coefficients ◽  
Direct Stiffness ◽  
Cross Axis ◽  
Positive Direct ◽  
Stiffness And Damping

This paper presents measured frequency dependent stiffness and damping coefficients for 12 and 8 bladed pocket damper seals (PDS) subdivided into 4 different seal configurations. Rotating experimental test are presented for inlet pressures at 69 bar (1,000 psi), a frequency excitation range of 20–300 Hz, and rotor speeds up to 20,200 rpm. The testing method used to determine direct and cross-coupled force coefficients was the mechanical impedance method, which required the measurement of external shaker forces, system accelerations, and motion in two orthogonal directions. In addition to the impedance measurements, dynamic pressure responses were measured for individual seal cavities of the 8 bladed PDS. Results of the frequency dependent force coefficients for the 4 PDS designs are compared. The conclusions of the test show that the 8 bladed PDS possessed significantly more positive direct damping and negative direct stiffness than the 12 bladed seal. The results from the dynamic pressure response tests show that the diverging clearance design strongly influences the dynamic pressure phase and force density of the seal cavities. The tests also revealed the measurement of same-sign cross-coupled (cross-axis) stiffness coefficients for all seals, which indicate that the seals do not produce a de-stabilizing influence on rotor-bearing systems.

scholarly journals Nonlinear Resonance Responses of Electromechanical Integrated Magnetic Gear System

2018 ◽  
Vol 2018 ◽  
pp. 1-16 ◽  
Author(s):  
Xiu-hong Hao ◽  
Hong-qian Zhu ◽  
Deng Pan
Keyword(s):  
Natural Frequency ◽  
Magnetic Force ◽  
Nonlinear Differential Equations ◽  
Nonlinear Resonance ◽  
Magnetic Coupling ◽  
Dominant Frequency ◽  
Electromechanical Integrated ◽  
Coupling Stiffness ◽  
Magnetic Gear ◽  
Stiffness And Damping

Nonlinear differential equations for an electromechanical integrated magnetic gear (EIMG) system are developed by considering the nonlinearity in the magnetic force of the system components. Expressions for the main resonances and superharmonic resonances are obtained for output wave frequencies close to the natural frequency and half the natural frequency of the derived EIMG system. The response laws are discussed in detail. The magnetic coupling stiffness among the components is found to exhibit distinct nonlinearity, leading to strong main resonances and superharmonic resonances. Smaller values of the magnetic coupling stiffness and damping result in larger response amplitudes and transient responses that slowly decay to zero. When the main resonances and superharmonic resonances occur, the dominant frequency of the response is the natural frequency of the derived EIMG system, and the amplitudes of different components of the resonance display large differences.

scholarly journals Influence of stiffness characteristics of a railway track on output parameters in a multibody model

2020 ◽  
Vol 318 ◽  
pp. 01003
Author(s):  
Ján Dižo ◽  
Miroslav Blatnický
Keyword(s):  
Indirect Method ◽  
Ride Comfort ◽  
Direct Method ◽  
Point Of View ◽  
Railway Track ◽  
Railway Vehicle ◽  
Running Safety ◽  
Multibody Model ◽  
Stiffness Characteristics ◽  
Stiffness And Damping

The article is aimed at the research of the influence of stiffness characteristics included in a model of a railway track, which is the part of a multibody system. The other part of the multibody model is a railway vehicle. Authors are focused on the investigation of response of some selected output parameters under various values of input of stiffness and damping coefficients. The interaction of a railway vehicle and a railway track is studied. A passenger railway vehicle has been chosen for presented research. Outputs parameters are chosen in the passenger ride comfort point of view and the running safety point of view. The passenger ride comfort can be evaluated either by the direct method, when a real vehicle runs on a track and passengers evaluate a vehicle by means of their feelings during the ride or by means of the indirect method, when accelerometers are used for measuring accelerations in various positions of a tested wagon and subsequently values of accelerations are processed in required way. Then, the ride comfort is calculated and indexed by means of ride comfort indices. In the presented work, the indirect method has been used. In the computer multibody model of the wagon accelerations on a floor have been detected and the mean ride comfort for a person is assessed. The ride safety is most often determined by waveforms of vertical wheel forces, lateral wheel forces and the derailment quotient.

On the Dynamical Behavior of Rotating Shafts Driven by Universal (Hooke) Couplings

1958 ◽  
Vol 25 (1) ◽  
pp. 47-51
Author(s):  
R. M. Rosenberg
Keyword(s):  
Critical Speed ◽  
Dynamical Behavior ◽  
Rotating Shaft ◽  
Critical Speeds ◽  
Elastic Shaft ◽  
Rotating Shafts

Abstract The system considered here is a massless, uniform elastic shaft carrying at its mid-point a disk (having mass) and supported at the ends by universal (Hooke) joints. The purpose of this investigation is to examine the effect of Hooke-joint angularity (as obtained by design, or from faulty alignment) on the bending stability of the rotating shaft. It is found that separate investigations are required for shafts not transmitting axial torques and for those required to transmit torques. Each gives rise to instabilities which are absent when the Hooke joint is straight. In the absence of axial torques, the shaft develops unsuspected mild critical speeds at odd integer submultiples of the “familiar” critical speed found with a straight Hooke joint. When the shaft is required to transmit moderate axial torques, the joint angularity produces true instabilities near all integer submultiples of the familiar critical speed. Surprisingly, these instabilities vanish for sufficiently large axial torques.

On the Dynamic Behavior of Hybrid Journal Bearings

1976 ◽  
Vol 98 (1) ◽  
pp. 90-94 ◽  
Author(s):  
S. M. Rohde ◽  
H. A. Ezzat
Keyword(s):  
Dynamic Behavior ◽  
High Frequency ◽  
Journal Bearings ◽  
Damping Coefficients ◽  
Stiffness Coefficients ◽  
Bearing Stiffness ◽  
Coupling Stiffness ◽  
High Frequency Excitation ◽  
Frequency Excitation ◽  
Stiffness And Damping

An analysis of the dynamic behavior of hybrid journal bearings is presented. The analysis accounts for the compressibility of the lubricant in the bearing recesses and supply line. Results show that when the journal is subjected to high frequency excitation the bearing stiffness and damping can change drastically. The behavior is characterized by a “break frequency” beyond which the bearing stiffness increases sharply. This is accompanied by a rapid decrease in bearing damping. It is also shown that the cross-coupling stiffness coefficients are reduced at high excitation frequencies. The asymptotic behavior of the stiffness and damping coefficients is examined at both ends of the frequency spectrum.

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