Modal analysis of continuous asymmetrical rotor-bearing systems

doi:10.1016/0022-460X(92)90359-6

Journal of Sound and Vibration

Volume 152, Issue 2, 22 January 1992, Pages 245-262

https://doi.org/10.1016/0022-460X(92)90359-6 Get rights and content

Abstract

Modal analysis of an asymmetrical rotor-bearing system which consists of asymmetrical Rayleigh shafts, asymmetrical rigid disks and isotropic bearings is performed. The rotor includes the effects of rotary inertia and gyroscopic moment. A solution method for the vibration analysis of a rotating uniform asymmetrical shaft is developed. The whirl speeds and mode shapes of the uniform asymmetrical shaft are investigated, and the effects of the boundary conditions and rotor asymmetry on the modal properties and stabilities are examined. The resonances of asymmetrical rotor-bearing systems in forward and backward precessions are also discussed.

References (12)

C.W. Lee et al.
Modal analysis of a distributed parameter rotating shaft
Journal of Sound and Vibration
(1988)
C.W. Lee et al.
Modal analysis of continuous rotor-bearing systems
Journal of Sound and Vibration
(1988)
H.D. Taylor
Critical-speed behavior of unsymmetrical shafts
Journal of Applied Mechanics
(1940)
W.R. Foote et al.
Critical speeds of a rotor with unequal flexibilities, mounted in bearings of unequal flexibility
Journal of Applied Mechanics
(1943)
S.H. Crandall et al.
On the stability of a rotor with rotationally unsymmetric inertia and stiffness properties
Journal of Applied Mechanics
(1961)
E.H. Hull
Shaft whirling as influenced by asymmetric stiffness
Journal of Engineering for Industry
(1961)

There are more references available in the full text version of this article.

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Briend et al. [13] proposed a new finite element (FE) model to analyze the effect of simple and combined sinusoidal rotational and translational motions of the support as well as that of a mass unbalance excitation on the behavior stability of an on-board rotor-bearing system. On the other hand, for analytical approaches [16–23], it seems to have no unified method. There are some researchers dedicated to this work.
In this paper, a method based on Green's function is proposed for the dynamic analysis of a rotor-bearing system with electromechanically coupled boundary conditions. The rotor system is supported by two ring-shaped piezoelectric dampers, which has been manufactured in our previous research. Based on the piezoelectric shunt resonant circuits, e.g., current flowing shunt circuit, the rotor system's boundary condition will become complicated and electromechanically coupled. The Laplace transform method is applied to solve the partial governing equations with such boundary conditions and the steady-state whirl Green's function is derived subsequently. Since the Green's functions are fundamental solutions of the system, the steady-state whirl solutions can be obtained by applying the superposition principle. Owing to the solutions’ concise form, it is convenient and suitable for analysis and computation. Validation of the proposed method is demonstrated by comparison with the finite element method (FEM). The simulated results show that the analytical solutions possess high accuracy and the piezoelectric damper has significant damping performance.
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Nonetheless, a more general approach can be used by considering the boundary values of the eigenfunctions in the vectors {Φi(x)} and {Ψi(x)}. For more information, the reader is referred to [12,29,30]. In this section, two numerical examples are shown for the illustration of the method proposed.
The mathematical modeling of rotor systems is a challenging task, mainly due to the complex nature of such systems, that might posses multiple rotors, bearings, disks and more. Most approaches in modeling of these systems are based on the well established transfer matrix and finite elements methods. In this work, an alternative method based on the distributed parameter or continuous model is presented. The method presented here can be applied to complex rotor system with multiple-stepped cross-sections and several disks and bearings. The procedure is to obtain the eigenfunctions of the system directly from the equations of motion, thus considering disks and isotropic bearings, as well as damping, on the eigenfunctions. With these functions, which also give the mode shapes, the well known partial differential equations are discretized leading to uncoupled first order differential equations for the modal coordinates, enabling close-form solutions to be obtained. The main difference of the present method from other distributed parameters approaches is that the equations are obtained in the time domain, instead of relying on inverse frequency transforms. Two numerical examples, with a simple and a complex geometry, of rotor systems are presented to evaluate the model.
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Grybos [16] considered the effect of the shear deformation and rotary inertia of a rotor on its critical speeds. Jei and Leh [17] investigated the whirl speeds and mode shapes of a uniform asymmetrical Rayleigh shaft with asymmetrical rigid disks and isotropic bearings. Kim et al. [18] studied the free vibration of a rotating tapered composite Timoshenko shaft.
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Analytical modeling of asymmetric multi-segment rotor - Bearing systems with Timoshenko beam model including gyroscopic moments
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However; in the literature there exist limited numbers of studies that concentrate on the analytical modeling of rotor systems. For the continuous beam model of rotor systems, Lee et al. [11–13] proposed modal analysis solution based on non-self-adjoint system characteristics. Similarly, Wang and Kirkhope [14,15] proposed eigensolution and modal analysis for the undamped rotor systems and applied perturbation analysis for the damped cases.
In this study, analytical modeling and an analysis approach for asymmetric multi-segment rotor – bearing systems are presented. Timoshenko beam model which includes the effect of gyroscopic moments is employed for modeling rotor segments. Instead of applying FEM, sub-segment Frequency Response Functions (FRFs) are obtained analytically, and sub-segment FRFs obtained are coupled by using receptance coupling method. Bearing properties are included into system dynamics by employing structural modification techniques. The proposed analytical model is verified by using FEM approach. It is shown that using analytical model and receptance coupling, compared with FEM, reduces computational time drastically without losing accuracy.
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2009, Mechanism and Machine Theory
In this paper the free vibrations of an in-extensional simply supported rotating shaft with nonlinear curvature and inertia are considered. Rotary inertia and gyroscopic effects are included, but shear deformation is neglected. To analyze the free vibrations of the shaft, the method of multiple scales is used. This method is applied to the discretized equations, and directly to the partial differential equations of motion, which demonstrates the same results. An expression is derived which describes the nonlinear free vibrations of the rotating shaft in two transverse planes. It is found that in this case, both forward and backward nonlinear natural frequencies are being excited. The results of perturbation method are validated with numerical simulations.

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Modal analysis of continuous asymmetrical rotor-bearing systems

Abstract

Journal of Sound and Vibration

Journal of Sound and Vibration

Critical-speed behavior of unsymmetrical shafts

Journal of Applied Mechanics

Critical speeds of a rotor with unequal flexibilities, mounted in bearings of unequal flexibility

Journal of Applied Mechanics

On the stability of a rotor with rotationally unsymmetric inertia and stiffness properties

Journal of Applied Mechanics

Shaft whirling as influenced by asymmetric stiffness

Journal of Engineering for Industry