Nonlinear dynamic analysis of a rotor-bearing-seal system under two loading conditions

Highlights

• Coupling effects of oil-film force coupled seal force are analyzed under two loading conditions.

• The first and second mode instability rules are analyzed under two loading conditions.

• The second loading condition can delay the first mode whips to some extent.

• Nonlinear seal force can restrain oil film instability and improve the instability speed.

• The transformation of the self-excited vibration energy can be observed.

Abstract

The operating speed of the rotating machinery often exceeds the second or even higher order critical speeds to pursue higher efficiency. Thus, how to restrain the higher order mode instability caused by the nonlinear oil-film force and seal force at high speed as far as possible has become more and more important. In this study, a lumped mass model of a rotor-bearing-seal system considering the gyroscopic effect is established. The graphite self-lubricating bearing and the sliding bearing are simulated by a spring-damping model and a nonlinear oil-film force model based on the assumption of short bearings, respectively. The seal is simulated by Muszynska nonlinear seal force model. Effects of the seal force and oil-film force on the first and second mode instabilities are investigated under two loading conditions which are determined by API Standard 617 (Axial and Centrifugal Compressors and Expander-compressors for Petroleum, Chemical and Gas Industry Services, Seventh Edition). The research focuses on the effects of exciting force forms and their magnitudes on the first and second mode whips in a rotor-bearing-seal system by using the spectrum cascades, vibration waveforms, orbits and Poincaré maps. The first and second mode instability laws are compared by including and excluding the seal effect in a rotor system with single-diameter shaft and two same discs. Meanwhile, the instability laws are also verified in a rotor system with multi-diameter shaft and two different discs. The results show that the second loading condition (out-of-phase unbalances of two discs) and the nonlinear seal force can mainly restrain the first mode instability and have slight effects on the second mode instability. This study may contribute to a further understanding about the higher order mode instability of such a rotor system with fluid-induced forces from the oil-film bearings and seals.

Introduction

Modern rotating machines, such as turbines, compressors, generators, are designed for high speed, flexibility and efficiency. In order to avoid unstable vibrations at higher operating speeds, more and more attention has been paid on the self-excited vibration, which is induced by the interaction between the rotor and surrounding fluid. Fluid-induced forces mainly include the force from the oil-film bearings and seals. It can lead to significant alternating stresses in the rotor, the high-level vibration, the rubbing between the rotor and the stator and the potential damage of the rotating machinery eventually. So the research on the mechanism of fluid–solid interaction in the rotor-bearing-seal system is of great importance for modern rotating machines.

In earlier studies, the linear stiffness and damping coefficients are widely adopted to simulate the dynamic characteristics of bearings and seals [1], [2]. However, the observed phenomena show that the bearing and seal fluid forces present strong nonlinearity. And the linear model will fail to analyze the nonlinear dynamic behaviors of the rotor-bearing-seal system under some conditions, such as the large perturbed motion of the journal.

In order to simulate the nonlinearity of the sliding bearing better, some nonlinear oil-film force models have been proposed, such as in papers [3], [4], [5]. Based on Capone model, Adiletta et al. [6] analyzed the possible chaotic motions resulted from the nonlinear response of bearings; Jing et al. [7], [8] studied the nonlinear dynamic behaviors of a rotor-bearing system considering the oil whip phenomenon; de Castro et al. [9] researched the system instability threshold influenced by the unbalance, rotor arrangement form and bearing parameters; Ding et al. [10] analyzed the non-stationary dynamic responses of the system during speed-up with a constant angular acceleration for a multi-bearing rotor. Based on the non-steady nonlinear oil-film force model presented by Zhang [5], Ding et al. [11] analyzed non-stationary processes of a rotor-bearing system by taking the rotating angular speed as control parameter. In his another paper [12], Ding et al. studied dimension reductions of a continuous rotor system by the standard Galerkin method and the nonlinear Galerkin method, and his results revealed that transitions or bifurcations of the rotor whirl from being synchronous to nonsynchronous as the unstable speed was exceeded. Zhang et al. [13] presented a mathematical model and a computational methodology to simulate the complicated flow behaviors of the journal microbearing in the slip regime, and their investigation showed that the rotor motion was stable with half-frequency whirling when the system located in the lower stability region, and the rotor had high-frequency whirling when the system located in the upper stability region.

The seal force models have also been developed by many researchers. Alford [14] derived the formula of the gas exciting force first to research on the stability of aeroengine. The Alford model could explain some basic phenomena, but it was a simple linear one. Based on Hirs' turbulent lubrication equations, Childs [15] derived dynamic coefficient expressions for high-pressure annular seals typical of neck-ring and interstage seals employed in multistage centrifugal pumps. Muszynska [16], [17] proposed a simple model of nonlinear fluid dynamic force generated in the seal based on the results of a series of experiments.

Based on Muszynska nonlinear seal force model, Li et al. [18] determined the empirical parameters of gas exciting force of the Muszynska model by using the results of computational fluid dynamics (CFD); Li et al. [19] analyzed dynamic behaviors of an unbalanced rotor-seal system with sliding bearings based on Floquet theory and the bifurcation theorem; Hua et al. [20] established a nonlinear model of rotor-seal system and investigated the nonlinear behavior of the unbalanced rotor-seal system by using an efficient and high-precision direct integration method; Ding et al. [21] investigated a symmetric rotor/seal system and analyzed the Hopf bifurcation of the system; Wang et al. [22] established a nonlinear mathematical model for orbital motion of the rotor system under the influence of leakage flow through an interlocking seal.

Considering the coupled effect of the nonlinear oil-film force and seal force, many researchers studied the dynamic behaviors of the rotor-bearing-seal system. Cheng et al. [23], [24] and Shen et al. [25], [26] investigated nonlinear dynamic behaviors of a rotor-bearing-seal coupled system by using the nonlinear oil-film forces obtained under the short bearing theory and Muszynska nonlinear seal force model. Based on an unsteady oil-film force proposed by Zhang and Muszynska seal force model, Li et al. [27], [28] established a new dynamic model of a rotor system by the Hamilton principle and the finite element method, and they analyzed the coupled effects of the nonlinear oil-film force, the nonlinear seal force, and the mass eccentricity of the disc; Wang et al. [29] studied the nonlinear coupling vibrations excited by a labyrinth seal and two air-film journal bearings through numerical simulations for high-speed centrifugal compressors.

It should be noted that in all the above researches, only the first mode instability (oil whirl/whip) was concerned. In fact, the operating speed of the rotating machinery often exceeds the second or even higher order critical speed to pursue higher efficiency. Thus, the second mode instability can appear when the operating speed approaches or exceeds twice the second-order critical speed according to the literatures [[16], [30], [31], [32]]. The researches about the coupled effects of the nonlinear oil-film force and seal force on the second mode instability have not been found. In this paper, influences of the nonlinear oil-film force coupled with seal force on the first and the second mode instabilities of a rotor-bearing-seal system, attached with two discs, are investigated. A nonlinear oil-film force model under short bearing assumption [3], [4] and Muszynska seal force model [16], [17] are adopted. Numerical integrations are used to get the solutions because of the nonlinearity of oil-film and seal forces. Spectrum cascade, vibration waveform, orbit and Poincaré map are applied to analyze various nonlinear phenomena and system unstable processes.

This paper consists of five sections. After this introduction, in Section 2, mathematical model of a rotor-bearing-seal system is established considering the nonlinearity of sliding bearing and seal. In Section 3, the first and second mode instability laws are compared by including and excluding the seal effect in a rotor system with single-diameter shaft and two same discs (simulation 1), and the effects of rotating speeds and eccentricities of disc on system instability are discussed under two loading conditions. Then, for examining the universality of simulation 1, Section 4 analyzes the instability laws in a rotor system with multi-diameter shaft and two different discs (simulation 2) under two loading conditions by adopting the same analysis method as that in Section 3. Finally, some conclusions are drawn in Section 5.