Unbalance Response of an Elastic Rotor in Damped Flexible Bearings at Supercritical Speeds

1971 ◽  
Vol 93 (2) ◽  
pp. 265-275 ◽  
Author(s):  
Neville F. Rieger
Keyword(s):  
Critical Speed ◽  
Mode Shapes ◽  
Rotor Speed ◽  
Flexible Rotor ◽  
Relative Stiffness ◽  
Unbalance Response ◽  
Fluid Film Bearings ◽  
Transmitted Force ◽  
Rotor Mass ◽  
Bearing System

The unbalance response of a uniform flexible rotor in fluid-film bearings has been analyzed for speeds up to 20 times the lowest rigid-bearing critical speed. Rotor mass and elasticity are distributed uniformly along the length of the rotor. A single radial speed-dependent force is used to represent the rotor unbalance. The rotor is assumed to operate in stable plain cylindrical bearings which are represented by direct and cross-coupled spring and damping forces. The influence of rotor speed, bearing operating eccentricity, relative stiffness of rotor and bearings, and unbalance location along rotor on the performance of the rotor-bearing system has been determined. Results are presented as charts of rotor maximum whirl amplitude and of bearing maximum whirl transmitted force for wide ranges of the foregoing parameters. Mode shapes at critical speeds are also included.

Rotor-Bearing Dynamics With Emphasis on Attenuation

1962 ◽  
Vol 84 (4) ◽  
pp. 491-498 ◽  
Author(s):  
J. W. Lund ◽  
B. Sternlicht
Keyword(s):  
Critical Speed ◽  
Reynolds Equation ◽  
Journal Bearings ◽  
Dimensionless Form ◽  
Damping Coefficients ◽  
Transmitted Force ◽  
Rotor Bearing ◽  
Bearing Dynamics ◽  
Rotor Mass ◽  
Bearing System

This paper presents a theoretical analysis of the dynamics of a rotor-bearing system. The analysis is quite general but because of space limitations only the symmetrical rotor supported in two plain cylindrical journal bearings is considered. Furthermore, the rotor mass is concentrated at midspan giving the rotor only one degree of freedom. Limiting the analysis to small amplitudes of rotor motion the components of the fluid film force are made linear with respect to journal amplitude and velocity. The resulting 8 coefficients, denoted spring and damping coefficients, are calculated from Reynolds equation and by coupling them with the rotor, the motion and the force transmitted to the bearing pedestal are obtained. Results are presented in dimensionless form for transmitted force and for critical speed.

Effect of Residual Shaft Bow on Unbalance Response and Balancing of a Single Mass Flexible Rotor—Part I: Unbalance Response

1976 ◽  
Vol 98 (2) ◽  
pp. 171-181 ◽  
Author(s):  
J. C. Nicholas ◽  
E. J. Gunter ◽  
P. E. Allaire
Keyword(s):  
Phase Angle ◽  
Response Speed ◽  
Rotor Speed ◽  
Flexible Rotor ◽  
Simple Method ◽  
Unbalance Response ◽  
Single Mass ◽  
Reverse Situation ◽  
Rigid Supports ◽  
Timing Mark

The effect of residual shaft bow on the unbalance response of a single mass rotor on rigid supports has been examined with a theoretical analysis. The analysis determined the amplitude, phase angle, and peak rotor response speed for various combinations of residual bow and unbalance. For most combinations the phase angle corresponding to the peak rotor response speed was significantly different from the 90 degrees observed in the conventional unbowed rotor. If the residual bow and unbalance were exactly out of phase, the rotor amplitude was zero for a rotor speed equal to the square root of the ratio of residual bow amplitude to unbalance eccentricity. The results of the study suggested a simple method for determining the relative amplitudes of residual bow and unbalance eccentricity based upon the motion of a timing mark on an oscilliscope screen. If the residual bow was less than the unbalance eccentricity, the timing mark moved first in the direction of rotor rotation as the speed is increased and then moved in the opposite direction at a speed less than the critical speed. In the reverse situation, the timing mark moved opposite to the direction of rotation as the speed is increased. At some speed above the critical, it reversed direction. Part II of this paper presents theoretical and experimental results for balancing of a single mass rotor with a residual bow.

scholarly journals The Dynamic Analysis of Two-Rotor Three-Bearing System

2015 ◽  
Vol 2015 ◽  
pp. 1-15
Author(s):  
Jianfei Yao ◽  
Jinji Gao ◽  
Ya Zhang ◽  
Weimin Wang
Keyword(s):  
Dynamic Behavior ◽  
Critical Speed ◽  
Amplitude Spectrum ◽  
Element Model ◽  
Nonlinear Phenomena ◽  
Motion Patterns ◽  
Unbalance Response ◽  
Mass Eccentricity ◽  
Coupling Stiffness ◽  
Bearing System

A finite element model considering the shear effect and gyroscopic effect is developed to study the linear and nonlinear dynamic behavior of two-rotor three-bearing system named N+1 configuration with rub-impact in this paper. The influence of rotational speed, eccentric condition, and the stiffness of coupling on the dynamic behavior of N+1 configuration and the propagation of motion are discussed in detail. The linear rotordynamic analysis included an evaluation of rotor critical speed and unbalance response. The results show that the critical speed and unbalance response of rotors are sensitive to coupling stiffness in N+1 configuration. In the nonlinear analysis, bifurcation diagram, shaft-center trajectory, amplitude spectrum, and Poincaré map are used to analyze the dynamic behavior of the system. The results of the research transpire that these parameters have the great effects on the dynamic behavior of the system. The response of the system with rub-impact shows abundant nonlinear phenomena. The system will exhibit synchronous periodic motion, multiperiodic motion, quasiperiodic motion, and chaotic motion patterns under rotor-stator rub interaction conditions. The dynamic response is more complicated for flexible coupling and two mass eccentricities than that of system with rigid coupling and one mass eccentricity.

scholarly journals Blade Loss Simulation of Multi Mass Rotor Model With Nonlinear Tilt Pad Journal Bearings

1995 ◽  
Author(s):  
R. K. Gadangi ◽  
A. B. Palazzolo
Keyword(s):  
Natural Frequency ◽  
Transient Response ◽  
Critical Speed ◽  
Journal Bearings ◽  
Rigid Rotor ◽  
Flexible Rotor ◽  
Equilibrium Location ◽  
Rotor Model ◽  
Limiting Value ◽  
Bearing System

Prediction of rotor vibrations due to large imbalance requires nonlinear solution of the supporting bearings. This paper presents a methodology and results for the effects of large, sudden imbalance on the response of a multi mass rotor model supported on tilt pad journal bearings. For a given imbalance, response is obtained for rotor speeds below, above and at the rotor natural frequency. The maximum peak to peak amplitude is larger at the critical speed than at a speed above or below the critical. The imbalance response is compared with two other methods used for predicting the transient response of a rotor bearing system. The rigid rotor and nonlinear bearing model shows a response similar in shape to that obtained with a flexible rotor and nonlinear bearing model, but the magnitude is different, which reached a limiting value as the imbalance was increased. The flexible rotor and linearized bearing model predicts a similar trend as the flexible rotor and nonlinear bearing model, with increasing speed for a given imbalance, but the shape and magnitude of the orbit is completely different. The motion of rotor to static equilibrium location for the flexible rotor and nonlinear bearing model showed oscillations which diminished with time, while the rigid rotor and nonlinear bearing model does not show any oscillations.

Multilevel Optimization of the Geared Rotor-Bearing System for Multi-Objectives With Critical Speed Constraints

2004 ◽  
Author(s):  
T. N. Shiau ◽  
J. R. Chang ◽  
W. B. Lu
Keyword(s):  
Critical Speed ◽  
Transmission Error ◽  
Mesh Stiffness ◽  
Error Response ◽  
Critical Speeds ◽  
Gear Mesh ◽  
Unbalance Response ◽  
Multilevel Algorithm ◽  
Gear Mesh Stiffness ◽  
Bearing System

This paper presents the multi-objective optimization of a geared rotor-bearing system with the critical speeds constraints by using an efficient multilevel algorithm. The weight of each rotor shaft, the unbalance response, and the response due to the transmission error are minimized simultaneously under the critical speed constraints. The design variables are the inner radii of the shaft, the stiffness of bearings, and the gear mesh stiffness. The finite element method (FEM) is employed to analyze the dynamic characteristics and the method of feasible direction (MFD) is applied in the optimization of the single objective stage. Based on the sensitivity analysis that gear mesh stiffness has almost no influences on the critical speeds of the uncoupled modes of two shafts, an efficient multilevel algorithm composed of system and subsystem levels is developed. In the cycle of iteration, the minimization of the shaft weight is performed in the subsystem level with the critical speed constraints of only uncoupled modes of two shafts and the unbalance response and the transmission error response are reduced in the system level with the critical speed constraints of only coupled modes. It is indicated from the numerical results that the shaft weight, the unbalance response, and the transmission error response via the multilevel technique (ML) are all reduced much more than those via the weighting method (WM) and the goal programming method (GPM).

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.

Calculations and Experiments on the Unbalance Response of a Flexible Rotor

1967 ◽  
Vol 89 (4) ◽  
pp. 785-796 ◽  
Author(s):  
J. W. Lund ◽  
F. K. Orcutt
Keyword(s):  
Critical Speed ◽  
Distributed Parameter ◽  
Distributed Parameter System ◽  
Flexible Rotor ◽  
Parameter System ◽  
Damping Coefficients ◽  
The Third ◽  
Tilting Pad ◽  
Speed Range ◽  
Bearing System

The results of a combined analytical and experimental investigation of the unbalance vibrations of a rotor are presented. The analysis applies to a general rotor-bearing system in which the dynamic bearing forces are represented by four spring coefficients and four damping coefficients. The rotor can be represented as either a lumped or a distributed parameter system, and gyroscopic moments are included. In general, the unbalance whirl motion of the rotor will be elliptical. The analysis has been programmed for a digital computer to obtain results for comparison with the experimental data. The test rotor is a uniform, flexible shaft with heavy wheels mounted at the ends and in the middle. The rotor is supported in two silicone fluid-lubricated, tilting-pad bearings. The rotor amplitude caused by an induced unbalance has been measured over a speed range of 3000 to 24,000 rpm for three different rotor configurations, obtained by removing one or both end wheels. This speed range extends to or through the third critical speed for each of the rotor configurations. The results are compared with the theoretical values and, in general, the agreement is found to be good.

scholarly journals Chaos of flexible rotor system with critical speed in magnetic bearing based on the improved precise Runge–Kutta hybrid integration

2018 ◽  
Vol 10 (9) ◽  
pp. 168781401880085 ◽  
Author(s):  
Xi Fang ◽  
Dongbo Zhang ◽  
Xiaoyu Zhang ◽  
Huachun Wu ◽  
Fei Gao ◽  
...  
Keyword(s):  
Critical Speed ◽  
Integration Method ◽  
Magnetic Bearing ◽  
Rotor System ◽  
Kutta Method ◽  
Hybrid Integration ◽  
Flexible Rotor ◽  
Runge Kutta Method ◽  
Runge Kutta ◽  
Bearing System

Magnetic rotor-bearing system has drawn great attention because of its several advantages compared to existent rotor-bearing system, and explicit Runge–Kutta method has achieved good results in solving dynamic equation. However, research on flexible rotor of magnetic bearing is relatively less. Moreover, explicit Runge–Kutta needs a smaller integral step to ensure the stability of the calculation. In this article, we propose a nonlinear dynamic analysis of flexible rotor of active magnetic bearing established by using the finite element method. The precise Runge–Kutta hybrid integration method and the largest Lyapunov exponent are used to analyze the chaos of the rotor system at the first- and second-order critical speed of the rotor. Experiment on chaos analysis has shown that compared with the explicit Runge–Kutta method, the precise Runge–Kutta hybrid integration method can improve the convergence step of calculation significantly while avoiding iterative solution and maintain high accuracy which is four times the increase of the integral step.

Modal Response of a Flexible Rotor in Fluid-Film Bearings

1974 ◽  
Vol 96 (2) ◽  
pp. 525-533 ◽  
Author(s):  
J. W. Lund
Keyword(s):  
Fluid Film ◽  
Mode Shapes ◽  
Flexible Rotor ◽  
First Order ◽  
Fluid Film Bearings ◽  
Modal Response ◽  
Governing System ◽  
System Equations ◽  
Multistage Compressor ◽  
Modal Method

An analysis is presented for calculating the response of a general flexible rotor in fluid-film bearings to forced and transient excitation. It is a modal method where the governing system equations are transformed by means of normal coordinates into a set of decoupled, first-order equations which can be solved in closed form. The transformation is based on the orthogonal complex modal functions (“mode shapes”) associated with the eigenvalues of the system. The method has been applied to an industrial multistage compressor. Numerical results are given for the response to selected unbalance distributions and, also, the transient response to a shock pulse.

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