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.

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.

Characteristics of Herringbone-Grooved, Gas-Lubricated Journal Bearings

1965 ◽  
Vol 87 (3) ◽  
pp. 568-576 ◽  
Author(s):  
J. H. Vohr ◽  
C. Y. Chow
Keyword(s):  
Differential Equation ◽  
Journal Bearing ◽  
Groove Depth ◽  
Radial Force ◽  
Journal Bearings ◽  
Speed Ratio ◽  
Plain Bearing ◽  
Relative Speed ◽  
Whirl Speed ◽  
Limiting Case

A differential equation is obtained for the smoothed “overall” pressure distribution around a herringbone-grooved, gas-lubricated journal bearing operating with a variable film thickness. The equation is based on the limiting case of an idealized bearing for which the number of grooves approaches an infinite number. A numerical solution to the differential equation is obtained valid for small eccentricities. This solution includes the case where the journal is undergoing steady circular whirl. In addition to the usual plain bearing parameters L/D, Λ, and whirl speed ratio ω3/(ω1 + ω2), the behavior of a grooved bearing also depends on four additional parameters: The groove angle β, the relative groove width α, the relative groove depth H0, and a compressibility number, Λs, which is based on the relative speed between the grooved and smooth members of the bearing. Results are presented showing bearing radial force and attitude angle as functions of β, α, H0, Λs, Λ, and whirl speed ratio.

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.

Critical Speeds of Shaft-Disk Systems Subjected to Longitudinal Forces

1997 ◽  
Author(s):  
Lien-Wen Chen ◽  
Hong-Cheng Sheu
Keyword(s):  
Axial Force ◽  
Longitudinal Force ◽  
Numerical Results ◽  
Follower Force ◽  
Numerical Techniques ◽  
Disk System ◽  
Critical Speeds ◽  
Whirl Speed ◽  
The Right ◽  
Left Portion

Abstract The critical speeds of a spinning Timoshenko shaft with an intermediate attached disk subjected to a longitudinal force are analytically solved. The expressions of whirl speed equations for hinged-hinged, hinged-clamped, clamped-hinged, and clamped-clamped rotors are given respectively. The critical speeds of each shaft-disk system are sought from its corresponding whirl speed equation by using simple numerical techniques. The effects of the disk location and the longitudinal force on the critical speeds of the shaft-disk systems are investigated. Numerical results reveal that if the disk locates in the left portion of the shaft, both the primary forward and backward critical speeds for the rotor subjected to a follower force are larger than those subjected to an axial force with the same magnitude. The results are contrary while the disk locates in the right portion of the shaft.

Stability Analysis of Rotor-Bearing Systems Using Component Mode Synthesis

1980 ◽  
Vol 102 (2) ◽  
pp. 352-359 ◽  
Author(s):  
D. A. Glasgow ◽  
H. D. Nelson
Keyword(s):  
Stability Analysis ◽  
General Problem ◽  
Component Mode Synthesis ◽  
Internal Coordinates ◽  
System Equations ◽  
Speed Stability ◽  
Whirl Speed ◽  
Comparative Results ◽  
Rotor Bearings ◽  
Mode Truncation

A method of component mode synthesis is presented for the analysis of multishaft rotor-bearings systems. The motion of each component of the system is described as the superposition of constraint modes associated with boundary coordinates and constrained precessional modes associated with internal coordinates. The constrained precessional modes for each component are truncated and the reduced component equations are assembled to yield a set of system equations. The nonsymmetric nature of the general problem requires the utilization of biorthogonality relations between right and left vector sets in order to decouple the component precessional modes. The method is developed for damped whirl speed/stability analysis and comparative results are presented for various levels of mode truncation for two example systems.

A Study of the Parametrically Excited Behavior of Passenger and Freight Railway Vehicles Using Linear Models

1991 ◽  
Vol 113 (2) ◽  
pp. 336-338 ◽  
Author(s):  
J. Lieh ◽  
I. Haque
Keyword(s):  
Parametric Resonance ◽  
Parametric Excitation ◽  
Floquet Theory ◽  
Linear Models ◽  
Stability Boundaries ◽  
Parametrically Excited ◽  
Critical Speeds ◽  
Railway Vehicles ◽  
Principal Parametric Resonance ◽  
The Stability

This paper presents a study of the parametrically excited behavior of passenger and freight vehicles on tangent track due to harmonic variations in conicity using linear models. The effect of primary and secondary stiffnesses on parametric excitation is also studied. Floquet theory is used to find the stability boundaries. The results show that wavelengths associated with conicity variation that are in the vicinity of half the kinematic wavelengths of the vehicles can lead to significant reductions in critical speeds. Results also show that the primary and warp stiffnesses can affect the severity of principal parametric resonance depending on the vehicle models and magnitude of stiffnesses chosen.

Stability of a Four-Bar Linkage with Flexible Coupler

1971 ◽  
Vol 13 (4) ◽  
pp. 237-242 ◽  
Author(s):  
M. R. Smith ◽  
L. Maunder
Keyword(s):  
Transverse Vibration ◽  
Homogeneous Equation ◽  
Hill’S Equation ◽  
Stability Boundaries ◽  
Inertia Effects ◽  
Digital Computation ◽  
Equation Analysis ◽  
Hill's Equation ◽  
Speed Stability ◽  
Four Bar Linkage

The undamped transverse vibration of a flexible coupler in a crank and rocker four-bar linkage is shown to be governed by an inhomogeneous Hill's equation. Analysis of the corresponding homogeneous equation shows that the vibration may be unstable and that at a prescribed input speed stability depends on the inertia effects of the follower as well as on the characteristics of the coupler. Stability boundaries may be determined by digital computation, and an illustrative chart is worked out for a particular case.

Asymptotic Modal Analysis of Structure Borne Noise in Cross Beams

1989 ◽  
Vol 111 (4) ◽  
pp. 472-479
Author(s):  
R. M. Chi
Keyword(s):  
Modal Analysis ◽  
Equations Of Motion ◽  
Intersection Point ◽  
Point Load ◽  
Upper And Lower Bounds ◽  
Analysis Technique ◽  
Coupling Constraint ◽  
Coupled Beams ◽  
Arbitrary Location ◽  
Limiting Case

The dynamic response (structure borne noise) of two cross beams intersecting perpendicularly to each other is studied for a point load applied vertically at an arbitrary location on one beam. The modal analysis technique is used to analyze the bending and torsional responses of the system of coupled beams. The modal equations of motion are derived from Lagrange’s equations with the coupling constraint at the beam intersection point included via Lagrange multipliers. For the limiting case of a large number of participating modes, an asymptotic modal analysis is performed resulting in simple estimates of the upper and lower bounds of the beam responses. These bound estimates are compared with the corresponding SEA results.

Sign in / Sign up

Export Citation Format

Share Document