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

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.

scholarly journals Hybrid Gas Bearings With Controlled Supply Pressure to Eliminate Rotor Vibrations While Crossing System Critical Speeds

2008 ◽  
Author(s):  
Luis San Andre´s ◽  
Keun Ryu
Keyword(s):  
Critical Speed ◽  
Pressure Control ◽  
Sudden Increase ◽  
Critical Speeds ◽  
Gas Bearings ◽  
Supply Pressure ◽  
Simple Strategy ◽  
Tilting Pad ◽  
Bearing Stiffness ◽  
Bearing Force

Micro-turbomachinery (MTM) implements gas bearings in compact units of enhanced mechanical reliability. Gas bearings, however, have little damping and wear quickly during transient rub events. Flexure pivot tilting pad bearings offer little or no cross-coupled stiffnesses with enhanced rotordynamic stability; and when modified for hydrostatic pressurization, demonstrate superior rotordynamic performance over other bearing types. External pressurization stiffens gas bearings thus increasing system critical speeds, albeit reducing system damping. Most importantly, measurements demonstrate that external pressurization is not needed for rotor super critical speed operation. In practice, the supply pressure could be shut off at high rotor speeds with substantial gains in efficiency. The paper introduces a simple strategy, employing an inexpensive air pressure regulator to control the supply pressure into the hybrid bearings, to reduce or even eliminate high amplitudes of rotor motion while crossing the system critical speeds. Rotor speed coast-down tests with the pressure controller demonstrate the effectiveness of the proposed approach. A simple on-off supply pressure control, i.e. a sudden increase in pressure while approaching a critical speed, is the best since it changes abruptly the bearing stiffness coefficients and moves the system critical speed to a higher speed. A rotordynamic analysis integrating predicted bearing force coefficients forwards critical speeds in agreement with the test results. Predicted rotor responses for the controlled supply conditions show an excellent correlation with measured data. The experiments validate the predictive tools and demonstrate the controllable rotordynamic characteristics of flexure pivot hybrid gas bearings.

Critical Speeds, Divergence, and Flutter Instability in High-Speed Planetary Gears

2012 ◽  
Author(s):  
Christopher G. Cooley ◽  
Robert G. Parker
Keyword(s):  
High Speed ◽  
Critical Speed ◽  
Numerical Results ◽  
Planetary Gears ◽  
Critical Speeds ◽  
Flutter Instability ◽  
Divergence Instability ◽  
Translational Mode

The structured properties of the critical speeds and associated critical speed eigenvectors of high-speed planetary gears are given. Planetary gears have only planet, rotational, and translational mode critical speeds. Divergence instability is possible at speeds adjacent to critical speeds. Numerical results verify the critical speed locations. Divergence and flutter instabilities are investigated numerically for each mode type.

scholarly journals Hybrid Gas Bearings With Controlled Supply Pressure to Eliminate Rotor Vibrations While Crossing System Critical Speeds

2008 ◽  
Vol 130 (6) ◽  
Author(s):  
Luis San Andrés ◽  
Keun Ryu
Keyword(s):  
Critical Speed ◽  
Pressure Control ◽  
Sudden Increase ◽  
Critical Speeds ◽  
Gas Bearings ◽  
Supply Pressure ◽  
Simple Strategy ◽  
Tilting Pad ◽  
Bearing Stiffness ◽  
Bearing Force

Microturbomachinery implements gas bearings in compact units of enhanced mechanical reliability. Gas bearings, however, have little damping and wear quickly during transient rub events. Flexure pivot tilting pad bearings offer little or no cross-coupled stiffnesses with enhanced rotordynamic stability; and when modified for hydrostatic pressurization, demonstrate superior rotordynamic performance over other bearing types. External pressurization stiffens gas bearings thus increasing system critical speeds, albeit reducing system damping. Most importantly, measurements demonstrate that external pressurization is not needed for rotor supercritical speed operation. In practice, the supply pressure could be shut off at high rotor speeds with substantial gains in efficiency. This paper introduces a simple strategy, employing an inexpensive air pressure regulator to control the supply pressure into the hybrid bearings, to reduce or even eliminate high amplitudes of rotor motion while crossing the system critical speeds. Rotor speed coast-down tests with the pressure controller demonstrate the effectiveness of the proposed approach. A simple on-off supply pressure control, i.e., a sudden increase in pressure while approaching a critical speed, is the best since it changes abruptly the bearing stiffness coefficients and moves the system critical speed to a higher speed. A rotordynamic analysis integrating predicted bearing force coefficients forwards critical speeds in agreement with the test results. Predicted rotor responses for the controlled supply conditions show an excellent correlation with measured data. The experiments validate the predictive tools and demonstrate the controllable rotordynamic characteristics of flexure pivot hybrid gas bearings.

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.

Imbalance Response of a Rotor Supported by Hybrid Air Foil Bearings

2011 ◽  
Author(s):  
Daejong Kim ◽  
Prajwal Shetty ◽  
Donghyun Lee
Keyword(s):  
High Speed ◽  
Critical Speed ◽  
Hydrodynamic Instability ◽  
High Reliability ◽  
Load Capacity ◽  
Damping Ratio ◽  
Foil Bearings ◽  
Supply Pressure ◽  
Bearing Stiffness ◽  
Frequency Ratios

Air foil bearings (AFB’s) are widely used in small to midsized turbomachinery. They are simple in construction, offer very low drag friction, and have very high reliability at high speed operations. This paper presents experimental imbalance response of a 4.84 kg rigid rotor (operating below bending critical speed) supported by two hybrid air foil bearings with 50 mm in diameter. The concept of “hybrid” in this paper utilizes the hydrostatic augmentation of the load capacity during the start up and shut down. The hybrid air foil bearings were designed with three top foils for enhanced stability. Imbalance responses in cylindrical mode are presented up to 44,000rpm with different supply pressures. As the supply pressure is increased from 2.67 to 4 bar, the bearing stiffness increases slightly, resulting in slightly larger vibration (and reduced damping ratio) during the trans-critical speed operation. Hydrodynamic instability was observed with whirl frequency ratios of about 0.17∼0.2 depending on the supply pressures. Tests were also conducted to investigate the effect of supply pressure on the rotordynamic stability. The test results show that the hybrid operation is very effective to suppress the subsynchronous vibrations at high speeds.

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.

Modal Analysis of the Cylinder Mechanism Based on ANSYS Workbench

2013 ◽  
Vol 561 ◽  
pp. 496-500
Author(s):  
De Gong Chang ◽  
Hong Yun Dou ◽  
Fu Qin Yang ◽  
Rong Zong
Keyword(s):  
Modal Analysis ◽  
High Speed ◽  
Optimization Design ◽  
Critical Speed ◽  
Structural Characteristics ◽  
Safety Evaluation ◽  
Mode Shapes ◽  
Bearing Stiffness ◽  
Working Principle ◽  
Ansys Workbench

According to the working principle and structural characteristics of cylinder mechanism, the simplified model is built by the Pro/E software. Then we carry on the modal analysis for cylinder mechanism combined with ANSYS Workbench [1], and obtain the natural frequency and mode shapes of all steps. It provides the basis for optimization design and safety evaluation of cylinder mechanism. Finally, we study the influence of bearing stiffness on the critical speed, which provides the basis for the research of high-speed cylinder.

Turbomachinery Dual Rotor-Bearing System Analysis

2002 ◽  
Author(s):  
Hsiao-Wei D. Chiang ◽  
Chih-Neng Hsu ◽  
Wes Jeng ◽  
Shun-Hsu Tu ◽  
Wei-Chen Li
Keyword(s):  
Finite Element ◽  
Transfer Matrix ◽  
Critical Speed ◽  
System Analysis ◽  
Design Parameters ◽  
Critical Speeds ◽  
Rotor Design ◽  
Rotor Bearing ◽  
Bearing Stiffness ◽  
Bearing System

It is very common for aircraft engines to have dual rotor or even triple rotor designs. Due to the complexity of having multiple rotor design, the transfer matrix methods have used in the past to deal with multiple rotor-bearing systems. However, due to transfer matrix method’s assumptions, sometimes resulted in numerical stability problems or root-missing problems. The purpose of this paper is to develop a systematic theoretical analysis of the dynamic characteristics of turbomachinery dual rotor-bearing systems. This dual rotor-bearing system analysis will start with a finite element (FEM) rotor-bearing system dynamic model, then using different methods to verify the analysis results including critical speed map and bearing stiffness. In an inertia coordinate system, a general model of continuous dual rotor-bearing systems is established based on a lagrangian formulation. Gyroscopic moment, rotary inertia, bending and shear deformations have been included in the model. From a point of view of the systematic approach, a solution of the finite element method is used to calculate the critical speeds by several different methods, which in turn can help to verify this dual rotor-bearing system approach. The effects of the speed ratio of dual rotors on the critical speed will be studied, which in turn can be used as one of the dual rotor design parameters. Also, both critical speeds are in effect functions of dual rotor speeds. Finally, the bearing stiffness between high speed and low speed shafts not only affect the critical speeds of the dual rotor system, but also affect the mode shapes of the system. Therefore, the bearing stiffness in between is of even greater importance in turbomachinery dual rotor or multiple rotor design.

Whirling with Spring Bearings and Rough Snubbers

1963 ◽  
Vol 67 (625) ◽  
pp. 66-67
Author(s):  
B. Irons
Keyword(s):  
Gas Turbine ◽  
High Speed ◽  
Critical Speed ◽  
Peak Amplitude ◽  
Amplitude Response ◽  
Critical Speeds ◽  
Bearing Load

To an increasing extent, aero gas turbine manufacturers are supporting high speed rotors on spring bearings, in order to escape the consequences of lightly damped and inconveniently placed critical speeds. While experience has been generally good, engineering doubts periodically arise.(a) By introducing the flexible bearings, the critical speed is reduced, say, from 9,000 r.p.m. to 3,000 r.p.m. At 3,000 r.p.m. a peak amplitude response is experienced, although the bearing load is comparatively low. (Some spring bearing designs incorporate damping as an accidental feature, many do not, and very few have damping designed into them.) The peak amplitude at 3,000 r.p.m. can rub the seals or overstress the spring, and to prevent this the bearing amplitude is restricted by a circular stop known here as the “snubber.”

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