Critical Speeds of Turbomachinery: Computer Predictions vs. Experimental Measurements—Part I: The Rotor Mass—Elastic Model

1987 ◽  
Vol 109 (1) ◽  
pp. 1-7 ◽  
Author(s):  
J. M. Vance ◽  
B. T. Murphy ◽  
H. A. Tripp
Keyword(s):  
Research Program ◽  
Critical Speed ◽  
Natural Frequencies ◽  
Computer Programs ◽  
Elastic Model ◽  
Experimental Measurements ◽  
Critical Speeds ◽  
Improve Accuracy ◽  
Speed Computer ◽  
Rotor Mass

This is the first part (Part I) of two papers describing results of a research program directed at verifying computer programs used to calculate critical speeds of turbomachinery. This research program was undertaken since questions existed about the accuracy of calculations for the second and higher critical speeds. Part I describes improvements in computer programs and data modeling that resulted from comparing measured and calculated “free-free” natural frequencies of several shafts and rotors. Program modifications to improve accuracy include consideration of the effect of disk/shaft attachment stiffness, revised treatment of the end masses, and an improved convergence. Modifications resulting from the study are applicable to many other damped and undamped critical speed computer programs.

Critical Speeds of Turbomachinery: Computer Predictions vs. Experimental Measurements—Part II: Effect of Tilt-Pad Bearings and Foundation Dynamics

1987 ◽  
Vol 109 (1) ◽  
pp. 8-14 ◽  
Author(s):  
J. M. Vance ◽  
B. T. Murphy ◽  
H. A. Tripp
Keyword(s):  
Critical Speed ◽  
Natural Frequencies ◽  
State Of The Art ◽  
Dynamic Properties ◽  
Experimental Measurements ◽  
Critical Speeds ◽  
Damping Coefficients ◽  
Speed Prediction ◽  
Bearing Stiffness ◽  
Stiffness And Damping

This is the second of two papers describing results of a research project directed at verifying computer programs used to calculate critical speeds of turbomachinery. This part describes measurements made to determine the characteristics of tilt-pad bearings and foundation dynamics. Critical speeds of a 166 kg laboratory rotor on tilt-pad bearings are then compared with predictions from a state-of-the-art damped eigenvalue computer program. Measured natural frequencies of a steam turbine are also compared with computer predictions. Accuracy of critical speed prediction is shown to depend on accuracy of 1) the “free-free” rotor models, 2) the bearing stiffness and damping coefficients, and 3) the dynamic properties of the foundation, which can be represented by an impedance that must be determined by experimental measurements.

Formation of Standing Waves in Radial Tires

1984 ◽  
Vol 12 (1) ◽  
pp. 44-63 ◽  
Author(s):  
Y. D. Kwon ◽  
D. C. Prevorsek
Keyword(s):  
High Speed ◽  
Critical Speed ◽  
Unit Length ◽  
Standing Waves ◽  
Rolling Resistance ◽  
Damping Coefficient ◽  
Circular Membrane ◽  
Critical Speeds ◽  
Steel Cord ◽  
The Individual

Abstract Radial tires for automobiles were subjected to high speed rolling under load on a testing wheel to determine the critical speeds at which standing waves started to form. Tires of different makes had significantly different critical speeds. The damping coefficient and mass per unit length of the tire wall were measured and a correlation between these properties and the observed critical speed of standing wave formation was sought through use of a circular membrane model. As expected from the model, desirably high critical speed calls for a high damping coefficient and a low mass per unit length of the tire wall. The damping coefficient is particularly important. Surprisingly, those tire walls that were reinforced with steel cord had higher damping coefficients than did those reinforced with polymeric cord. Although the individual steel filaments are elastic, the interfilament friction is higher in the steel cords than in the polymeric cords. A steel-reinforced tire wall also has a higher density per unit length. The damping coefficient is directly related to the mechanical loss in cyclic deformation and, hence, to the rolling resistance of a tire. The study shows that, in principle, it is more difficult to design a tire that is both fuel-efficient and free from standing waves when steel cord is used than when polymeric cords are used.

Vibration of Flexibly Mounted Gyroscopes

1973 ◽  
Vol 15 (3) ◽  
pp. 225-231
Author(s):  
L. Maunder
Keyword(s):  
Natural Frequencies ◽  
Critical Speeds ◽  
Supporting Structure ◽  
Vibrational Characteristics

Flexibility in the supporting structure of two-axis or single-axis gyroscopes is shown to have a radical effect on vibrational characteristics. The analysis determines the ensuing natural frequencies and critical speeds.

Taylor-vortex instability and annulus-length effects

1976 ◽  
Vol 75 (1) ◽  
pp. 1-15 ◽  
Author(s):  
J. A. Cole
Keyword(s):  
Critical Speed ◽  
Taylor Vortex ◽  
Vortex Instability ◽  
Critical Speeds ◽  
Taylor Vortices ◽  
Torque Measurements ◽  
Concentric Cylinders ◽  
Theoretical Predictions ◽  
Range Of Values ◽  
Visual Observations

Critical speeds for the onset of Taylor vortices and for the later development of wavy vortices have been determined from torque measurements and visual observations on concentric cylinders of radius ratios R1/R2 = 0·894–0·954 for a range of values of the clearance c and length L: c/R1 = 0·0478–0·119 and L/c = 1–107. Effectively zero variation of the Taylor critical speed with annulus length was observed. The speed at the onset of wavy vortices was found to increase considerably as the annulus length was reduced and theoretical predictions are realistic only for L/c values exceeding say 40. The results were similar for all four clearance ratios examined. Preliminary measurements on eccentrically positioned cylinders with c/R1 = 0·119 showed corresponding effects.

Determination of the Critical Operating Speeds of Planar Mechanisms by the Finite Element Method Using Planar Actual Line Elements and Lumped Mass Systems

1979 ◽  
Vol 101 (2) ◽  
pp. 210-223 ◽  
Author(s):  
S. Kalaycioglu ◽  
C. Bagci
Keyword(s):  
Dynamic Systems ◽  
Natural Frequencies ◽  
Dynamic Equilibrium ◽  
Large Error ◽  
Free Vibrations ◽  
Lumped Mass ◽  
Critical Speeds ◽  
Rotating Shafts ◽  
Line Elements ◽  
Kinematic Motion

It has been a well-established fact that dynamic systems in motion experience critical speeds, such as rotating shafts and geared systems whose undeformed reference geometry remain the same at all times. Their critical speeds are determined by their natural frequencies of considered type of free vibrations. Linkage mechanisms as dynamic systems in motion change their undeformed geometries as function of time during the cycle of kinematic motion. They do also experience critical operating speeds as rotating shafts and geared systems do, and their critical speeds are determined by the minima of their natural frequencies during a cycle of kinematic motion. Such a minimum occurs at the critical geometry of a mechanism, which is the position at which the maximum of the input power is required to maintain the instantaneous dynamic equilibrium of the mechanism. Actual finite line elements are used to form the global generalized coordinate flexibility matrix. The natural frequencies of the mechanism and the corresponding mode vectors (mode deflections) are determined as the eigen values and eigen vectors of the equations of instantaneous-position-free-motion of the mechanism. Method is formulated to include or exclude the link axial deformations, and apply to any number of loops having any type of planar pair. Critical speeds of planar four-bar, slider-crank, and Stephenson’s six-bar mechanisms are determined. Experimental results for the four-bar mechanism are given. Effect of axial deformations and link rotary inertias are investigated. Inclusion of link axial deformations in mechanisms having pairs with sliding freedoms is seen to predict critical speeds with large error.

Estimating Critical Speeds of Stepped Shafts with Flexible Bearings

1971 ◽  
Vol 8 (03) ◽  
pp. 327-333
Author(s):  
R. H. Salzman
Keyword(s):  
High Speed ◽  
Critical Speed ◽  
Design Stage ◽  
Normal Range ◽  
Critical Speeds ◽  
Stepped Shaft ◽  
Bearing Stiffness ◽  
Variable Support ◽  
Method Show ◽  
Good Agreement

This paper presents a semi-graphical approach for finding the first critical speed of a stepped shaft with finite bearing stiffness. The method is particularly applicable to high-speed turbine rotors with journal bearings. Using Rayleigh's Method and the exact solution for whirling of a uniform shaft with variable support stiffness, estimates of the lowest critical speed are easily obtained which are useful in the design stage. First critical speeds determined by this method show good agreement with values computed by the Prohl Method for the normal range of bearing stiffness. A criterion is also established for determining if the criticals are "bearing critical speeds" or "bending critical speeds," which is of importance in design. Discusser E. G. Baker

Study of In-Plane Motions in a Thin Rotating Disk

2000 ◽  
Author(s):  
Moreshwar Deshpande ◽  
C. D. Mote
Keyword(s):  
Critical Speed ◽  
Natural Frequencies ◽  
Rotation Speed ◽  
Rotating Disk ◽  
Travelling Wave ◽  
Linear Strain ◽  
Strain Measure ◽  
Nodal Diameter ◽  
Plane Oscillations ◽  
Disk Energy

Abstract A model for the in-plane oscillations of a thin rotating disk has been derived using a nonlinear strain measure to calculate the disk energy. This accounts for the stiffening of the disk due the radial expansion resulting from its rotation. The corresponding non-dimensionalized natural frequencies are seen to depend only on rotation speed and have been calculated. The radially expanded disk configuration is linearly stable over the range of rotation speeds studied here. The sine and cosine modes for all nodal diameters couple to each other at all nonzero rotation speeds and the strength of this coupling increases with rotation speed. This coupling causes the reported frequencies of the stationary disk to split. The zero, one and two nodal diameter in-plane modes do not have a critical speed corresponding to the vanishing of the backward travelling wave frequency. The use of a linear strain measure in earlier work incorrectly predicts instability of the rotating equilibrium and the existence of critical speeds in these modes.

Additional Investigations Into the Natural Frequencies and Critical Speeds of a Rotating, Flexible Shaft-Disk System

2007 ◽  
Author(s):  
Lyn M. Greenhill ◽  
Valerie J. Lease
Keyword(s):  
Natural Frequencies ◽  
Slenderness Ratio ◽  
Rotor Dynamics ◽  
Axial Position ◽  
Disk System ◽  
Apparent Contradiction ◽  
Critical Speeds ◽  
Shaft Flexibility ◽  
Gyroscopic Effects ◽  
Lumped Masses

Traditional rotor dynamics analysis programs make the assumption that disk components are rigid and can be treated as lumped masses. Several researchers have studied this assumption with specific analytical treatments designed to simulate disk flexibility. The general conclusions reached by these studies indicated disk flexibility has little effect on critical speeds but significantly influences natural frequencies. This apparent contradiction has been reexamined by using axisymmetric harmonic finite elements to directly represent both disk and shaft flexibility along with gyroscopic effects. Results from this improved analysis show that depending on the thickness-to-diameter (slenderness) ratio of the disk and the axial position of the disk on the shaft, there are significant differences in all natural frequencies, for both forward and backward modes, including synchronous crossings at critical speeds.

Frictionally Excited Thermoelastic Instability in Automotive Drum Brakes

1999 ◽  
Vol 122 (4) ◽  
pp. 849-855 ◽  
Author(s):  
Kwangjin Lee
Keyword(s):  
Coefficient Of Friction ◽  
Material Properties ◽  
Critical Speed ◽  
Hot Spot ◽  
Frictional Heating ◽  
Friction Material ◽  
Thermoelastic Instability ◽  
Critical Speeds ◽  
Drum Brake ◽  
The Coefficient Of Friction

Thermoelastic instability in automotive drum brake systems is investigated using a finite layer model with one-sided frictional heating. With realistic material properties of automotive brakes, the stability behavior of the one-sided heating mode is similar to that of the antisymmetric mode of two-sided heating but the critical speed of the former is higher than that of the latter. The effects of the friction coefficient and brake material properties on the critical speeds are examined and the most influential properties are found to be the coefficient of friction and the thermal expansion coefficient of drum materials. Vehicle tests were performed to observe the critical speeds of the drum brake systems with aluminum drum materials. Direct comparisons are made between the calculation and measurement for the critical speed and hot spot spacing. Good agreement is achieved when the critical speeds are calculated using the temperature-dependent friction material properties and the reduced coefficient of friction to account for the effect of intermittent contact. [S0742-4787(00)01503-4]

scholarly journals Dynamical analysis of hollow-shaft dual-rotor system with circular cracks

2020 ◽  
pp. 146134842094828
Author(s):  
Yang Yongfeng ◽  
Wang Jianjun ◽  
Wang Yanlin ◽  
Fu Chao ◽  
Zheng Qingyang ◽  
...  
Keyword(s):  
Critical Speed ◽  
Equations Of Motion ◽  
Harmonic Balance Method ◽  
Circular Crack ◽  
Rotor System ◽  
Resonance Phenomenon ◽  
Dynamical Analysis ◽  
Dynamical Response ◽  
Critical Speeds ◽  
Hollow Shaft

In this paper, we considered a dual-rotor system with crack in shaft. The influence of circular crack in hollow shaft on dynamical response was studied. The equations of motion of 12 elements dual-rotor system model were derived. Harmonic balance method was employed to solve the equations. The critical speed and sub-critical speed responses were investigated. It was found that the circular crack in hollow shaft had greater influence on the first-backward critical speed than the first-forward critical speed. Owing to the influence of crack, the vibration peaks occurred at the 1/2, 1/3 and 1/4 critical speeds of the rotor system, along with a reduction in sub-critical speeds and critical speeds. The deeper crack away from the bearing affected the rotor more significantly. The whirling orbits, the time-domain responses and the spectra were obtained to show the super-harmonic resonance phenomenon in hollow-shaft cracked rotor system.

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