International Journal of Rotating Machinery

Volume 2016 (2016), Article ID 7817134, 22 pages

http://dx.doi.org/10.1155/2016/7817134

## An Investigation on the Dynamics of High-Speed Systems Using Nonlinear Analytical Floating Ring Bearing Models

General Electric, Rugby CV212NH, UK

Received 9 May 2016; Revised 1 July 2016; Accepted 5 July 2016

Academic Editor: Ryoichi Samuel Amano

Copyright © 2016 Athanasios Chasalevris. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The scope of this paper is to investigate the dynamics of a rotor-bearing system of high-speed under recently developed analytical bearing models. The development of a theory that can yield the dynamic response of a high-speed system without short/long bearing approximation and without time-consuming numerical methods for the finite-length bearing model is the outcome of this work. The rotor system is introduced as a rigid body so that the dynamics of the system are influenced only from the nonlinear bearing forces which are introduced with closed form expressions. The outcome is a system of nonlinear equations and its solution produces the dynamic response of the high-speed system using exact analytical solution for the bearing forces. The transient dynamic response of the system is evaluated through the wide range of rotating speed and under different bearing solutions including short bearing approximation, presenting the subsynchronous components that are developed when instabilities occur. Time-frequency analysis of the resulting response time-series is presented and the outcome is compared with that obtained from numerical solution of the bearing lubrication and with the short bearing approximation model.

#### 1. Introduction

Numerous simulations considering different geometric and physical parameters of the floating ring bearing elements can be required in the design of high-speed systems. Unfortunately, the lack of the possibility of expressing the dynamic response of a rotating system in direct relevance to the bearing design parameters leads to numerous case studies considering transient response analysis under various combinations of the bearing geometry, interpreted to variation of bearing radial clearance, and of bearing width through a specified range of values. The nonlinear response of such systems usually contains additional sub/superharmonics that are responsible for leading the system beyond the demands for maximum response amplitude, bearing eccentricities, power losses, acoustic emission, and so forth. More specifically, the developed eccentricities of the bearings are of crucial importance, as they are assumed to be the first indication for system durability considering its millions of run-ups and run-downs in applications of automotive turbochargers.

The variation of bearing design parameters consists of a set of run-up simulations that can consider some hundreds of individual geometric configurations for an initially developed turbosystem with dynamic response and other operational characteristics to be evaluated in each of them. Then, the outcome of that case study, that is, mainly the response time-series at selected points of the rotor-bearing system, is postprocessed with decompositions in time and frequency domain, orbits are plotted at selected speeds, and other operational parameters are extracted for further analysis, to note their tendency regarding each bearing configuration. The high evaluation capacity of modern computers and the implementation of developed numerical tools and theories for rotor-bearing dynamics in commercial software allow simulating such systems and offer confident results in relatively reasonable time.

In a run-up of an automotive turbocharger, three domains of rotational speed usually provoke the three corresponding instabilities at the inner and outer fluid films of the floating ring bearings [1, 2]. There is also total instability at even higher speeds that soon leads to system failure [3]. The three subsynchronous components of the response, referred to as sub-1, sub-2, and sub-3, are developed due to oil whirl/whip instability and are influenced in their amplitude and duration from the bearing’s physical and geometric parameters. The channel through which the bearing’s geometric and physical configuration influences the bearing’s instability is the resulting impedance and friction forces/moments; these phenomena have been extensively explained in [1–4]. Therefore, the model for evaluating oil film pressure distribution and then impedance forces and moments from the fluid film to reaction parts (shaft, ring, and housing) crucially defines the rotational speed in which instability will occur (or not) and the amplitude with which the subsynchronous/subharmonic response will be developed. It is important to remember that the presence and amplitude of the subcomponents are not only influenced by the bearing’s properties; various parameters contribute to this, such as unbalance magnitude, elastic component deformations, internal damping mechanisms (material damping, frictional damping between compressor wheel and shaft, etc.), and preloading stresses [5, 6].

The possibilities of incorporating floating ring bearing simulation into rotor dynamic algorithms are many and are extensively discussed in the literature. The numerical solution of the Reynolds equation for both inner and outer oil films has been performed with numerical methods such as FDM, FEM, and CFD [7–13]. Such methods are quite accurate but, even after the development of very efficient solvers, they are considered slower in comparison to direct formulas for the bearing’s impedance forces yielded from short/long bearing approximation. However, the incorporation of holes in the floating ring connecting the inner and the outer film leaves no room for nonnumerical treatment of the Reynolds full equation. The short/long bearing approximation has been widely used and the literature features many rotor dynamic investigations of turbosystems under this assumption [14–20]. However, the length-to-diameter ratio presented in the inner and outer film of the floating ring bearings is not always below or near ; in automotive turbochargers, very frequently, the outer fluid film is defined with a ratio near or even higher than .

The short bearing approximation will give a different pressure distribution, especially for high eccentricities and high eccentricity rate of change compared to the pressure distribution evaluated numerically. However, the difference in the final outcome, that is, impedance force, is not really relatively big, so the approximation can still be acceptable for the demands of rotor-bearing design. Considering further the bearing’s operating parameters, such as power loss in the lubricant, shearing stresses developed in it, resulting torque from the oil to the ring, and increment of ring speed, the method of the evaluation of pressure distribution can lead to different results regarding the occurrence of instabilities and their characteristics. This paper aims to highlight the evaluated ring speed under different solutions for pressure distribution.

As mentioned above, the evaluation time of bearing forces is an important issue for an efficient design procedure. For a rotor dynamic simulation of, for example, 100 cases of bearing geometric configurations, operating temperatures, and other parameters, the evaluation of transient response can be a matter of days (without postprocessing). Given that a simulation of a transient run-up of an automotive application turbocharger should be performed up to the rotating speed of 250 kRPM or even 300 kRPM and that the resulting response time history should be evaluated for 10 s–15 s, the length of time histories exceed usually the 1E6 samples (extrapolation is performed before postprocessing) under constant time interval, for example, s. With a Matlab® Solver ode15s method [21, 22], the bearing routine would be called around 2E7 times (depending on the stiffness of the mathematical system). A subroutine for the bearing’s dynamics does not last more than 0.02 s with an FDM solution (using ADI elimination with relatively weak convergence criteria) and 0.005 s for a short bearing approximation solution; further details will be given below. The evaluation time dedicated to the bearing routine during a DoE (e.g., 100 cases) can be drastically reduced if the bearing impedance terms are returned faster. Therefore, a suggestion for a floating ring bearing model consisting of analytical closed form expressions would contribute to increasing evaluation speed and considering the finite-length bearing geometry would contribute to the rotor dynamic analysis of high-speed systems in terms of accurate bearing simulation.

This paper does not aim to delve into the mechanisms of instabilities in turbosystems; this issue has been thoroughly discussed in recent [1–6, 11–14] and past literature [7–10]. It aims to use two theoretical models for floating ring bearing simulation that are based on a recent achievement of analytical solution of the Reynolds equation in finite-length journal bearings [23–25] and in the application on the floating ring bearings of finite length [26–28]. This analytical solution was implemented on the floating ring bearing with two different theoretical models; the first leads to an exact analytical solution and the second to an approximate analytical solution [26, 27]. Both theoretical models consist of four SL problems, with one of the four to be treated differently in each of the models. The first theoretical model, the exact analytical solution, uses PSM to solve the one SL problem and to create the resulting pressure distribution in each of the four fluid films (two floating ring bearings are always incorporated). The second, the approximate analytical solution, uses Bessel functions for the solution of the one SL problem. This paper presents shortly the analytical treatment of the Reynolds up to the point that the solution is split in the four SL problems, referring always to the recent work [26]. The solutions for the bearing are incorporated in a rotor dynamic algorithm that simulates a rotor-bearing system consisting of a rigid rotor carrying its components (compressor/turbine wheels) and two floating ring bearings [28]. The motions of the rotor and of the floating rings are expressed using 10 DOF and the set of equations consists of 10 2nd-order ODEs that are converted to 20 1st-order ODEs.

Both theoretical bearing models (exact and approximate) express the impedance forces and moments of the bearing in closed form expressions. The comparison of the system’s dynamic behaviour among different bearing models also incorporates results with a short bearing model and results with a finite bearing model solved with FDM.

The results incorporate evaluated time histories for a set of geometric configurations of the rotor-bearing system; two systems are simulated covering the range of dimensions and consequently of operating speeds for the automotive turbochargers of passenger cars. The bearing eccentricities, the compressor wheel response, and the trajectories of the rotor and the floating rings during run-up are of major interest and are compared among different methods of bearing simulation. Time-frequency decomposition of the compressor wheel response is performed in order to extract characteristics of the subsynchronous response. The results among the different bearing models were found either in good agreement to each other or in divergence regarding the instability thresholds and they are discussed extensively. The pressure distribution in the bearing fluid films under various operating conditions that has been compared among the different bearing models [26] is the main source of the differences. The resulting ring speed is also compared as further interpretation of the different instability thresholds of the bearing models.

#### 2. Motion Equations of the Rotor-Bearing System

The elastic deformations of a rotor in a high-speed system such as automotive turbochargers do not reach severe magnitudes. This does not mean that the simulation of the rotor as an elastic body is out of any consideration [4, 12, 14]. In this work, in order to highlight the influence of journal bearings on the dynamics of the high-speed system, the rotor is considered as a rigid body [16–18].

In this analysis, a rigid rotor and two rigid wheels, with their masses concentrated at each rotor end, are fixed to each other and rotate at a rotational speed around the rotor line, as shown in Figure 1. The three components (CW, TW, and rotor) are assumed to have a total mass , a total moment of inertia (polar) with respect to the rotor centre line, and a total diametric moment of inertia with respect to the vertical and horizontal axes that pass through the centre of mass of the system; see Figure 1. A global rectangular coordinate system has its centre at the system reference line at the plane of the centre of mass of the system; see Figure 1. Since the system is not assumed to obtain any axial displacement (hypothesis of plane orbits), the coordinate system is fixed and with respect to this system all the lateral displacements will be defined.