Analytical approach to the stepped multi-span rotor-bearing system with isotropic elastic boundary conditions

Highlights

• A new method for modeling stepped multi-span rotor-bearing systems is proposed.

• The internal damping effect is also considered.

• This method is applicable to rotor systems with complicated geometries and isotropic elastic boundary conditions.

• The analytical solutions match well with FEM solutions.

Abstract

This paper strives to derive the analytical solutions for the dynamic analysis of stepped multi-span rotor system, where the rotating shaft's internal damping effect is also considered. Specifically, the steady-state whirl analysis and free vibration analysis is given herein. For the steady-state whirl analysis, firstly, the stepped shaft is spilt to several uniform segments and their governing equations are established by Hamilton principle. Subsequently, by applying variable separation method and Laplace transform method to each segment's governing equation, their steady-state responses in terms of determinate interpolation functions multiplied by unknown boundary constants are derived. Then based on the transfer matrix method, each segment's boundary constants are determined with consideration of the compatibility conditions of each two adjacent segments and the end boundary conditions. For the free vibration analysis, one only need to remove the forced term in the steady-state form of transfer matrix and the rotor's characteristic equation will be obtained. Through the theoretical derivation, it is found that this analytical approach can be generalized to any isotropic elastic boundary conditions. To validate the proposed method, two case studies are proposed, where the finite element method is used as benchmark method. For the first one, the influence of internal damping on a uniform rotor's damped whirl characteristics and its stability is analyzed. The damped whirl characteristics analysis reveals that viscous internal damping result in the destabilization of forward modes as long as the spinning speed becomes higher than the critical speed. This phenomenon is also demonstrated by the time responses analysis. For the second one, the dynamic responses of a two-span three steps rotor-bearing system with isotropic viscoelastic boundary conditions are determined. All of simulated results show a great agreement between analytical results and FEM results.

Introduction

Rotor-bearing systems such as various types of turbines, gear boxes, generators, and rotating shafts, are of the most applicable systems in the modern world. Imposed by operational and economic needs in recent decades, the gradually demand for more powerful rotors makes them operate above the second and sometimes higher critical speed. In this case, many problems will be encountered. For example, much greater centrifugal imbalance force will be generated as the operating speed becomes higher, even if the rotor eccentricity is small. Such imbalance force will lead to rotor vibrations and usually bring about fatigue. Therefore, the dynamic analysis of rotor system always attracts the researchers’ attention. Especially as the rotors and its boundary conditions become more and more complicated, the importance of developing analytical or approximate methods for dynamic analysis becomes increasingly prominent. Up to now, many researchers have contributed to this research topic.

Actually the dynamic analysis of rotor system can be boiled down to the problem of solving partial differential equations (PDE) with simple or complex boundary conditions, i.e., the boundary value problem. In the field of structural dynamics or rotordynamics, one kind of the most encountered PDE is the fourth-order PDE, which usually appears as the governing equations of continuous system. To solve it, the widely used approximate method are assumed modes method [1], [2], [3], [4], [5], [6], [7] and finite element method (FEM) [8], [9], [10], [11], [12], [13], [14]. The other methods like Differential transform technique [15] and Homotopy perturbation method [16] are also the available methods to solve PDE, whereas compared with these two methods, it is more convenient to use assumed modes method or FEM for the rotor system. In addition, it should be mentioned that the existence and uniqueness of solutions of boundary value problem is also the critical problem which must be considered before solving the PDE [17]; however, for simplicity, we assume that the solution of the rotor's PDE is existing and unique in this paper.

Consider about the real structures like rods, beams and rotating shafts’ physical characteristics, the transfer matrix method [18], [19], [20], [21] is also frequently used. For the assumed modes method, many research papers have illustrated its practicability. Ouyang et al. [2] presented a dynamic model for the vibration analysis of a rotating Timoshenko beam subjected to a three-directional load moving in the axial direction. Al-Solihat et al. [4] investigated the three-dimensional nonlinear dynamics and force transmissibility characteristics of a flexible shaft-disk rotor system which is supported by two viscoelastic elements at the end. All of their governing equations were namely discretized by assumed modes method. Hence, one could see that no matter for the linear or nonlinear case, the assumed modes method is very convenient to be used. However, it has the disadvantage that the mode shapes are difficult to be assumed if the boundary conditions become complicated. To deal with this problem, one can use the FEM. As the most widely used approximate method, since Nelson et al. [8] systematically proposed the FEM of rotor-bearing system, FEM has shown its power. Han and Chu [10] used FEM to build FE model for the flexible rotor-bearing system under time-varying base angular motions, and then, analyzed such system's parametric instability. Based on the FEM and Timoshenko beam theory, Ribeiro et al. [11] presented a methodology to determine the optimal constructive form for any viscoelastic support. Their research aimed at minimizing the unbalance frequency response of the rotor system using a hybrid optimization technique (genetic algorithms and Nelder–Mead method). No matter what, these above mentioned approximate methods have their own characteristics and are applicable to different situations.

On the other hand, the analytical methods [22], [23], [24], [25], [26], [27], [28], [29], [30] also always attract the researchers’ attention, because they are very convenient to be used to analyze the effect of parameters on the system's behavior. The analytical solutions can directly reveal the system's physical essence. Yang [26] developed a new analytical method to derive the exact transient response of a stepped system with any number of components and subject to arbitrary external, boundary and initial excitations. Özsßahin et al. [27] derived the sub-segment frequency response functions analytically using non-self-adjoint system characteristics, for the purpose of investigating dynamic behavior of multi-segment rotor-bearing system. For the non-rotating case, Li et al. [28] developed the steady-state Green's functions for forced vibrations of Timoshenko beams with damping effects.

All of above mentioned analytical methods are perfect, however, it seems that there is little research paper focused on the analytical solution of the stepped multi-span rotor-bearing system, let alone with consideration of the rotating shaft's internal damping effect. Despite that the internal damping [31], [32], [33] has significant influence on the rotor's whirl frequencies, unbalance response, instability thresholds, etc. Until recently, Wang [29] derived the analytical solution of an eccentric rotor by applying Laplace transform and proposed a rotor unbalance identification method based on that solution. Mereles et al. [30] used the Laplace transform to derive the mode shapes of the complex rotor system with multiple-stepped cross-sections, several disks and bearings. These references enlighten us that it is possible to apply the Laplace transform to analyze the rotor's dynamic characteristics.

Thus, in this paper, we make effort to develop the analytical solutions for the dynamic analysis of stepped multi-span rotor-bearing system. To fulfil this task, in Section 2, the shaft is firstly spilt to several uniform segment. Then by applying Hamilton principle, each segment's governing equation is derived. In Section 3, by applying the variable separation method and Laplace transform, each segment's steady-state response in terms of determinate interpolation functions multiplied by unknown boundary constants is derived. Then according to the compatibility condition of each two adjacent segments and end boundary conditions, the generalized transfer matrix of steady-state whirl analysis for the entire rotor is obtained. If the forced term in such matrix is removed, the characteristic equation which is used for whirl characteristic analysis can also be derived. Furthermore, through the expression of transfer matrix, one can find that this analytical approach is applicable to many boundary condition, as long as the boundary is isotropic or symmetric. To validate the proposed method, in Section 4, the finite element method is used as the benchmark method for comparison. Here, two case studies are provided. For the first one, the influence of internal damping on a single-span uniform rotor's stability and damped whirl characteristics are analyzed. For the second one, the damped whirl characteristics, steady-state whirl and global responses of a two-span three steps rotor system with viscoelastic boundary condition are determined. The simulated results show that the analytical results are coinciding exactly with the FE results, which demonstrates that the proposed method is appropriate to analyze the dynamic characteristics of stepped multi-span rotor-bearing system.