A Beam Model for Tire Critical Speeds

1991 ◽  
Vol 19 (2) ◽  
pp. 113-120
Author(s):  
S. K. Clark
Keyword(s):  
Physical Model ◽  
Critical Speed ◽  
Wave Pattern ◽  
Beam Model ◽  
Material Damping ◽  
Critical Speeds ◽  
Naturally Occurring ◽  
Design Variables ◽  
Local Wave ◽  
Bending Wave

Abstract A brief review of tire critical speeds is given using the beam under tension as a physical model. In its most common form, this model visualizes the critical speed as that speed just sufficient to sustain a continuous sinusoidal bending wave in the tire tread band. A number of studies have been published modifying this concept by the introduction of material damping, centrifugal effects, and other characteristics, some of which aid in explaining the fact that the wave pattern observed experimentally is local and decays rapidly away from the contact patch. This paper presents a different view of the wave pattern needed for a critical speed to exist, namely that a naturally occurring local wave can arise independent of material damping and that as a practical fact, material damping may have little to do with the onset of the phenomenon. A discussion on the effect of tire design variables on critical speed is given based on the expressions derived here.

The Dynamic Analysis of SFD-Sliding Bearing Flexible Rotor System Based on Optimization Design

2012 ◽  
Vol 159 ◽  
pp. 355-360
Author(s):  
Ji Yan Wang ◽  
Rong Chun Guo ◽  
Xu Fei Si
Keyword(s):  
High Speed ◽  
Optimization Design ◽  
Critical Speed ◽  
Structural Parameters ◽  
Rotor System ◽  
Flexible Rotor ◽  
Optimization Analysis ◽  
Sliding Bearing ◽  
Critical Speeds ◽  
Design Variables

The paper establishes the mechanical model of SFD-sliding bearing flexible rotor system, adopting Runge-Kutta method to solve nonlinear differential equation, thus acquiring the unbalanced response curve and then gaining the first two critical speeds of the system. Meanwhile, the paper analyzes the sensitivity of the system on the first two critical speeds towards structural parameters, offering design variables to optimization analysis. Based on sensitivity analysis, genetic algorithm is employed to give an optimization analysis on critical speed, which aims to remove critical speed from working speed as much as possible. The critical speed ameliorates after the optimization which supplies theoretical basis as well as theoretical analysis towards the dynamic stability of high-speed rotor system and provides reference for the design of such rotor system.

Rotordynamic Analysis of a Rotor-Disk System Including a Synchronously Reduced Modal Truncation Method

2013 ◽  
Author(s):  
Timothy Dimond ◽  
Jawad Chaudhry ◽  
Matthew Wagner ◽  
Feng He ◽  
Jianming Cao ◽  
...  
Keyword(s):  
Critical Speed ◽  
Mode Shapes ◽  
Complete Analysis ◽  
Beam Model ◽  
Timoshenko Beam Model ◽  
Euler Beam ◽  
Critical Speeds ◽  
Unbalance Response ◽  
Rotor Bearing ◽  
Modal Truncation

There are many published works on rotordynamics which detail the types of analyses that are carried out: critical speeds, stability assessment, and forced response. The purpose of this paper is to present a more complete analysis of an existing, academic rotor/bearing model, taken from a textbook, more like it would be carried out in an industrial setting. The advantage is that all parameters of the rotor model are well known so that there are minimal uncertainties. However, some published papers on rotordynamics, as discussed in this work, present an incomplete analysis. For example, they may report the calculated critical speeds but leave out the critical speed plot and mode shapes in favor of the Campbell diagram. They may model a Bernoulli Euler beam model of the shaft and neglect the additional terms in the Timoshenko beam model. These papers may show some unbalance response plots for one disk in the model but not report on the amplification factor. This paper gives a much more complete rotordynamics analysis of this common rotor/bearing model than other works. The full undamped rotor analysis is presented, including critical speeds, critical speed map, and undamped mode shapes. The stability analysis presents the full set of eigenvalues including both forward and backward modes as well as the complex mode shapes. The differences between the Bernoulli Euler beam model and the full Timoshenko beam model are shown for this rotor. Full unbalance response plots, in the horizontal and vertical directions, are presented as well as the response along the semi-major axis. The unbalance response plots have calculated amplitudes, phase angles and amplification factors. In addition to the standard rotordynamic analyses, a synchronously reduced modal truncation method is presented. This method is better suited to automation, when compared to most truncation methods that require significant intervention by the analyst. The maximum error was on the order of 0.01%. It is hoped that future publications will present the more complete analysis shown for this rotor/bearing system.

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.

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.

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

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.

Lund’s Contribution to Rotor Stability: The Indispensable and Fundamental Basis of Modern Compressor Design

2001 ◽  
Author(s):  
Edgar J. Gunter
Keyword(s):  
Critical Speed ◽  
Space Shuttle ◽  
Numerical Procedure ◽  
Complex Eigenvalue ◽  
Critical Speeds ◽  
Complex Roots ◽  
Full Speed ◽  
Order Of Magnitude ◽  
Excitation Mechanisms ◽  
Turbine Design

Abstract In the early design of compressors and turbines, it was of importance to locate the rotor critical speeds so as to insure that a turbo rotor would not be operating at a critical speed. As compressor and turbine design became more sophisticated, a more detailed analysis of the rotor damped critical speeds and rotor log decrements was necessary. With compressors and turbines operating at higher speeds, under high power levels, and at multiples of the first critical speed, the problem of stability or self-excited whirling is of paramount importance. The onset of self-excited motion may lead to large amplitudes of motion with possible destructions of the rotor or the inability of the compressor to operate at peak power levels. Encountering this phenomenon with a compressor on an off-shore oil platform can mean millions of dollars of lost production. Self-excited whirling can be caused by fluid-film bearings, seals, Alford-type cross-coupling forces, or internal shaft friction, to name a few of the excitation mechanisms. The problem of computing the approximate values of the rotor log decrement under full speed and loading conditions requires the solution of a complex eigenvalue problem. The computation of the rotor complex roots is an order of magnitude more difficult than the problem of undamped critical speed calculations. J. W. Lund presented the first practical numerical procedure for computing turbo-rotor log decrements. The mathematical transfer matrix method pioneered by Lund has allowed industry to develop and stabilize a vast array of rotating machinery leading to the savings to industry of millions of dollars. Without the procedures of Lund, for example, it would not have been possible to resolve stability problems encountered with both the hydrogen and oxygen space shuttle pumps. This paper briefly presents some of the attributes of the Lund stability procedure and its unique characteristics.

A slender ship moving at a near-critical speed in a shallow channel

1995 ◽  
Vol 291 ◽  
pp. 263-285 ◽  
Author(s):  
Xue-Nong Chen ◽  
Som Deo Sharma
Keyword(s):  
Numerical Procedure ◽  
Wave Theory ◽  
Wave Pattern ◽  
Slender Body Theory ◽  
Petviashvili Equation ◽  
Theoretical Predictions ◽  
Local Wave ◽  
Sinkage And Trim ◽  
Step Algorithm ◽  
Shallow Channel

The problem solved concerns a slender ship moving at a near-critical steady speed in a shallow channel, not necessarily in symmetric configuration, involving the special phenomenon of generation of solitary waves. By using the technique of matched asymptotic expansions along with nonlinear shallow-water wave theory, the problem is reduced to a Kadomtsev–Petviashvili equation in the far field, matched with a nearfield solution obtained by an improved slender-body theory, taking the local wave elevation and longitudinal disturbance velocity into account. The ship can be either fixed or free to squat. Besides wave pattern and wave resistance, the hydrodynamic lift force and trim moment are calculated by pressure integration in the fixed-hull case; running sinkage and trim, by condition of hydrodynamic equilibrium in the free-hull case. The numerical procedure for solving the KP equation consists of a finite-difference method, namely, fractional step algorithm with Crank–Nicolson-like schemes in each half step. Calculated results are compared with several published shipmodel experiments and other theoretical predictions; satisfactory agreement is demonstrated.

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.

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