Stability and Damped Critical Speeds of Rotor-Bearing Systems

1975 ◽  
Vol 97 (4) ◽  
pp. 1325-1332 ◽  
Author(s):  
P. N. Bansal ◽  
R. G. Kirk
Keyword(s):  
Rotor System ◽  
System Stability ◽  
Mode Shapes ◽  
Computational Time ◽  
System Matrix ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Analytical Technique ◽  
Rotor Bearing ◽  
Response Amplification

This paper describes an analytical technique to calculate the damped critical speeds and the instability threshold speed of multimass rotor-bearing systems. Necessary equations are developed to study the effect of bearing as well as bearing support flexibility and damping on the system stability, thereby enhancing the current state of the art. Included in the analysis are the effects of linearized disk gyroscopic moments, shear deformation, and speed dependent bearing characteristics. The method of solution is based on the Transfer Matrix approach and uses complex variable notation to develop the overall system matrix. Mutter’s quadratic interpolation technique is employed to extract the complex eigenvalues of the rotor system and the corresponding mode shapes are found by back substitution. The analysis has been programmed for digital computer solution. Computational time is saved by eliminating from the polynomial the complex conjugates of the roots already found. Numerical overflow/underflow is controlled via scale factors. In addition to calculating the damped critical speeds, the computer program also provides information about the undamped frequencies, peak response frequencies, response amplification factors, and logarithmic decrements of the system. The accuracy of the predictions of the program has been verified and is shown to be acceptable for typical rotor systems. The results of an extensive investigation of an intershaft journal bearing instability in a dual rotor system are summarized. The stability map for this system is presented and clearly indicates the effect of rotor radial misalignment on system stability.

scholarly journals An analytical investigation for optimizing the support stiffness and positions of the bearings of a flexible rotor system

2020 ◽  
Author(s):  
Jing Liu ◽  
Changke Tang
Keyword(s):  
Optimization Design ◽  
Rotor System ◽  
Mode Shapes ◽  
Rotational Inertia ◽  
Fe Model ◽  
Flexible Rotor ◽  
Fe Method ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Flexible Rotor Systems

Abstract The support stiffness and positions of the bearings can greatly affect the vibrations of flexible rotor systems. However, most previous works only focused on the effect of the support stiffness of the bearings on the critical speeds of the rigid rotor systems or modal characteristics including natural frequencies and mode shapes, which missed the combine effects of the support stiffness and positions of the bearings. To overcome this issue, an analytical dynamic model of a flexible rotor system based on the finite element (FE) method is proposed. The model considers the support stiffness of the bearings and rotational inertia of the rotor system. The frequency equation of the rotor system is established for solving the critical speeds. The critical speeds and modal deformations of the system from the presented model and the numerical model based on a commercial software are compared to verify the effectiveness of the proposed FE model. The effects of the support stiffness and positions of the bearings on the critical speeds of the flexible rotor system are analyzed. The results show that the critical speeds are positively correlated with the support stiffness. The critical speeds of the flexible rotor system are also greatly affected by the support positions of the bearing. This study can provide some guidance for the optimization design method of the support stiffness and positions of the bearings in the flexible rotor systems.

Dynamic Analysis of Geared Rotor-Bearing Systems by the Transfer Matrix Method

1995 ◽  
Author(s):  
Siu-Tong Choi ◽  
Sheng-Yang Mau
Keyword(s):  
Transfer Matrix ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vibration Analysis ◽  
Transmission Error ◽  
Mode Shapes ◽  
Rotor Systems ◽  
Gear Mesh ◽  
Mass Unbalance ◽  
Rotor Bearing

Abstract In this paper, an analytical study of the dynamic characteristics of geared rotor-bearing systems by the transfer matrix method is presented. Rotating shafts are modeled as Timoshenko beam with shear deformation and gyroscopic effects taken into account. The gear mesh is modeled as a pair of rigid disks connected by a spring-damper set and a transmission-error exciter. The transfer matrix of a gear mesh is developed. The coupling motions of the lateral and torsional vibration are studied. In free vibration analysis of geared rotor systems, natural frequencies and corresponding mode shapes, and the whirl frequencies under different spin speeds are determined. Effects of bearing stiffness, isotropic and orthotropic bearings, pressure angle of the gear mesh are studied. In steady-state vibration analysis, responses due to the excitation of mass unbalance and the transmission error are studied. Parametric characteristics of geared rotor systems are discussed.

scholarly journals Dynamics of a rotor system coupled with water-lubricated rubber bearings

2018 ◽  
Vol 232 (23) ◽  
pp. 4263-4277 ◽  
Author(s):  
YF Shi ◽  
M Li ◽  
GH Zhu ◽  
Y Yu
Keyword(s):  
Elastic Deformation ◽  
Dynamic Characteristics ◽  
Hydrodynamic Lubrication ◽  
Dynamic Performance ◽  
Elastohydrodynamic Lubrication ◽  
Rotor System ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Mass Unbalance ◽  
Rubber Bearings

Dynamic behaviour is significantly important in the design of large rotor systems supported on water-lubricated rubber bearings. In this study, the mathematical model of elastohydrodynamic lubrication of the bearing is established based on the theory of hydrodynamic lubrication after considering the elastic deformation of rubber, and the dynamic characteristics of water-lubricated rubber bearings are analysed under small perturbation conditions according to the load increment method and the finite difference method. Next, the differential equation of rotor systems coupled with the water-lubricated rubber bearing is deduced using Lagrange’s approach, and its critical speeds, stability, and unbalanced responses are analysed in detail. The numerical results show that several parameters, such as the eccentricity, length–diameter ratio, and clearance of bearing and the rotating speed of the rotor, have a great impact on the dynamic performance of water-lubricated rubber bearings, and this influence cannot be ignored, especially in the case of large eccentricity ratios. The dynamic characteristics of rotor systems guided by water-lubricated rubber bearings reveal that the critical speeds are much lower than the ones under the rigid supports because of the elastic deformation, and they also indicate that the rotor system supported on water-lubricated rubber bearings has a weaker stability. In addition, the steady-state responses of the rotor system are analysed when the mass unbalance of the propeller exists, and the effect of the thickness of the rubber liner is also considered.

A Hybrid Method for the Vibration Analysis of Rotor-Bearing Systems

1991 ◽  
Vol 205 (2) ◽  
pp. 131-137 ◽  
Author(s):  
R Firoozian ◽  
H Zhu
Keyword(s):  
Finite Element ◽  
Transfer Matrix ◽  
Hybrid Method ◽  
Transfer Matrix Method ◽  
Matrix Method ◽  
Vibration Analysis ◽  
Mode Shapes ◽  
Rotating Shaft ◽  
Critical Speeds ◽  
Rotor Bearing

The transfer matrix method together with a digital computer form the foundation of the dynamic analysis of rotor-bearing systems. The properties of each segment of the rotating shaft are expressed in simple matrix form and the overall dynamic behaviour is then obtained by successive multiplication of the element matrices. The main drawback associated with this method is the numerical instability in calculating natural frequencies for complex systems. The finite element method, on the other hand, uses the element stiffness and mass matrices to form the global equation of motion for the complete system. This avoids the numerical problems of the transfer matrix method at the expense of the computer memory requirements. The new method described in this paper combines the transfer matrix and finite element techniques to form a powerful algorithm for vibration analysis of rotor-bearing systems. It is shown that the accuracy improves significantly when the transfer matrix for each shaft segment is obtained from finite element techniques. The accuracy and efficiency of the hybrid method are compared with the transfer matrix method for a simply supported uniform rotating shaft where an analytical solution for the critical speeds and mode shapes is available. The method is then applied to a flexibly supported uniform shaft and a non-uniform shaft with a large disc to show the capability of the method for finding the critical speeds of complex rotor-bearing systems.

Reduced-Scale Model Design for a High-Speed Rotor System Based on Similitude Theory

2019 ◽  
Author(s):  
Biao Zhou ◽  
Haotian Liang ◽  
Hui Miao ◽  
Chaoping Zang
Keyword(s):  
High Speed ◽  
Structural Engineering ◽  
Rotor System ◽  
Scale Model ◽  
Mode Shapes ◽  
Design Parameters ◽  
Critical Speeds ◽  
Operation Speed ◽  
Reduced Scale ◽  
Similitude Theory

Abstract Reduced-scale models are often established based on similitude theory as an alternative to the direct experimental observation on the prototype, which is usually oversized or requires unacceptable expenses. Much insight into the similitude theory applied to various fields in structural engineering, vibration and impact problems has been gained to date. However, the efficient dynamic similarity design of complex rotors remains elusive. This paper is devoted to developing a reduced-scale model based on similitude theory from a high-speed rotor system prototype. Three critical speeds within the range of operating speeds characterize this flexible rotor. A reduced scaling design strategy for the complex rotor system is proposed as a two-step scheme. Similarity conditions relating the critical design parameters (such as rotor geometry, support stiffness, etc.) between the reduced-scale model and the prototype are derived. The scaling factors are accordingly determined by a dimensional analysis in combination with the governing equation of rotordynamics. This leads to a downsized rotor model with distorted geometric configuration whose operation speed is efficiently narrowed down. Dynamic similitude is assured by proportionally scaling down the three critical speeds while the rotor mode shapes still maintain high correlation between the prototype and downscaled model. The resultant reduced-scale model of the rotor system will practically guide the construction of the essential part of a whole engine dynamics test rig for laboratory use.

Finite Element Modelling of a Generic Rotor-Bearing System and Experimental Validation

2015 ◽  
Author(s):  
A. Rehman ◽  
K. S. Ahmed ◽  
F. A. Umrani ◽  
B. Munir ◽  
A. Mehboob ◽  
...  
Keyword(s):  
Finite Element ◽  
Dynamic Response ◽  
Finite Element Model ◽  
Element Model ◽  
Mode Shapes ◽  
Efficient Operation ◽  
Critical Speeds ◽  
Rotor Bearing ◽  
The Impact ◽  
Bearing System

The design and development of the rotating machinery require a precise identification of its dynamic response for efficient operation and failure prevention. Determination of critical speeds and mode shapes is crucial in this regard. In this paper, a finite element model (FEM) based on the Euler beam theory is developed for investigating the dynamic behavior of flexible rotors. In-house code in Scilab environment, an open source platform, is developed to solve the matrix equation of motion of the rotor-bearing system. The finite element model is validated by the impact hammer test and the dynamic testing performed on the rotors supported on a purpose-built experimental setup. Bearing stiffness is approximated by using the Hertzian contact theory. Obtaining the critical speeds and mode shapes further improves the understanding of dynamic response of rotors. This study paves way towards advanced research in rotordynamics in Faculty of Mechanical Engineering, GIK Institute.

Effect of Damping on the Lateral Critical Speeds of Rotor-Bearing Systems

1976 ◽  
Vol 98 (2) ◽  
pp. 505-513 ◽  
Author(s):  
Pranabesh De Choudhury ◽  
Stephen J. Zsolcsak ◽  
Eugene W. Barth
Keyword(s):  
High Speed ◽  
Rotating Machinery ◽  
Mode Shapes ◽  
Response Analysis ◽  
Rigid Support ◽  
Critical Speeds ◽  
Rotor Bearing ◽  
System Design Approach ◽  
Analytical Tools ◽  
Bearing System

Rigid support lateral critical speeds along with undamped system critical speeds and mode shapes are presented for typical rotor-bearing systems. The steady-state unbalance response analysis presented shows the effect of fluid-film bearing damping on the rotor response. Experimental results show reasonably good correlation with analytical results. The investigation shows that a rational rotor-bearing system design approach can be made for high-speed rotating machinery using the analytical tools.

Constrained Optimization of a Rotor-Bearing System Based on an Evolutionary Algorithm

2012 ◽  
Author(s):  
Donghua Wang ◽  
Wanyou Li ◽  
Zhigang Liu ◽  
Zhansheng Liu
Keyword(s):  
Evolutionary Algorithm ◽  
Optimal Solution ◽  
Population Based ◽  
Rotor System ◽  
System Configuration ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Elite Strategy ◽  
Practical Design ◽  
Bearing System

How to modify a rotor system configuration to adjust some critical speeds distribution for safety and achieve the least change of system configuration at the same time, is a focus in the rotordynamics field. An existing method is introduced and its disadvantages are analyzed. To overcome these difficulties smoothly, a more robust mathematical model for optimal design of critical speeds distribution is presented, with more constraints considered. And a Population-based Evolutionary Algorithm with Elite Strategy (PEAES) is developed to solve the proposed optimization model. The results of study on a rotor system in different cases show that the proposed method can find the optimal solution and may be applicable in the practical design process of rotor systems.

scholarly journals A Method for Predicting the Influences of Bearing Support Stiffness and Position on the Vibrations of a Flexible Rotor System

2021 ◽  
Vol 26 (4) ◽  
pp. 287-295
Author(s):  
Jing Liu ◽  
Changke Tang
Keyword(s):  
Optimization Design ◽  
Design Method ◽  
Frequency Equation ◽  
Mode Shapes ◽  
Rotational Inertia ◽  
Flexible Rotor ◽  
Fe Method ◽  
Critical Speeds ◽  
Rotor Systems ◽  
Flexible Rotor Systems

The bearing support stiffness and position can greatly affect the vibrations of flexible rotor systems (FRSs). However, most previous works only focused on the effect of the bearing support stiffness on the critical speeds or modal characteristics including the natural frequencies and mode shapes of rigid rotor systems (RRSs). The previous studies missed the combined effects of the bearing support stiffness and position. To overcome this issue, an analytical method of a FRS based on the finite element (FE) method is proposed. Our model considers the bearing support stiffness and rotational inertia of FRS. The frequency equation of FRS is established for solving the critical speeds. The critical speeds and modal deformations of FRS from our model and the numerical model based on a commercial software are compared to verify the effectiveness of the presented method. The effects of the bearing support stiffness and position on the critical speeds of FRS are analyzed. The results show that the critical speeds are positively correlated with the bearing support stiffness. The critical speeds of FRS are also greatly affected by the bearing position. This study can provide some guidance for the optimization design method of bearing support stiffness and position in FRSs.

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.

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