**1. Introduction**

Over the past few decades, several numerical approximations on the dynamic behavior analysis for rotordynamic systems have been developed. Among these approximations, the most popular approach is the finite element (FE) method because it is highly efficient and convenient for modelling diverse physical systems. According to Koutromanos [1], with this method a complex region that defines a continuous system is discretized with simple geometrical forms called finite elements. The material properties as well as the governing relationships are taken into consideration for these elements and expressed in terms of unknown values on the element boundaries. After an assembly process and consideration of the loads and boundary conditions, an equation system is obtained. The solution for these equations provides the approximated behavior of the continuous system. At the start of the 1960s, engineers used the FE method to obtain approximated solutions for problems related to stress analysis, fluid flows, heat transfers and other areas. However, the FE method was not applied to rotordynamics until a decade later. Through the 1970s, diverse efforts were made to incorporate effects of rotational inertia, gyroscopic moments, axial load, shear deformation and internal damping, as pointed out in [2]. Recently, Shen et al. [3] remarked on the importance of including the effects of rotational inertia in

99

**Citation:** Mendoza-Larios, J.G.; Barredo, E.; Arias-Montiel, M.; Baltazar-Tadeo, L.A.; Landa-Damas, S.J.; Tapia-Herrera, R.; Colín-Ocampo, J. An Algebraic Approach for Identification of Rotordynamic Parameters in Bearings with Linearized Force Coefficients. *Mathematics* **2021**, *9*, 2747. https:// doi.org/10.3390/math9212747

Academic Editor: Carlo Bianca

Received: 15 September 2021 Accepted: 25 October 2021 Published: 29 October 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

the finite elements used to model and analyze rotordynamic systems, in order to have a more general and appropriate kinematic and dynamic description of rotating structures supported by bearings with stiffness and viscous damping characteristics.

Through the bearing characterization in rotor-bearing systems, the rotordynamic stiffness and damping coefficients can be determined. Physical insight of these parameters is essential for the correct modelling of every rotordynamic system as they are important factors in determining the system's dynamical behavior. In general, when a rotor-bearing system is studied, stiffness and damping coefficients of the bearings are unknown, meaning it is therefore necessary to implement a methodology to determinate them. According to Tiwari [4], Matsushita et al. [5] and Bre ´nkacz [6], there are eight rotordynamic coefficients in bearings, four for stiffness (two directs and two crossed) and four for damping (two directs and two crossed). Nowadays, rotor-bearing systems can be modelled in a very precise way by using modern modelling techniques. However, accurately estimating the dynamic parameters through theoretical models is still a challenge because it is difficult to accurately model every phenomenon affecting the dynamic behavior of the bearings. This problem has led to the development of novel numerical and experimental techniques for dynamic parameter identification [4,6,7]. Tiwari and Chougale [8] developed an algorithm to estimate the dynamic parameters of active magnetic bearings as well as the residual rotor unbalance. The proposed algorithm is based on the least squares technique in frequency domain. Moreover, Xu et al. [9] presented a novel identification approach for estimating bearing dynamic parameters based on the transfer matrix method. Stiffness and damping parameters of an active magnetic bearing were determined by minimizing the error between the unbalance response calculated by the transfer matrix approach and the experimental approach. Mao et al. [10] also proposed a method for identifying bearing dynamic parameters in flexible rotor-bearing systems by minimizing the quadratic error between the numerical and experimental results of the vibration response caused by system unbalance. There are several investigations on methods for identifying unbalance and bearing dynamic parameters [11–15]. Recently, Wang et al. [16] presented the development of algorithms for the simultaneous identification of unbalance and bearing dynamic parameters. In both cases, the proposed algorithms were validated by comparison with experimental data. Additionally, in [17], the authors estimated the rotordynamic coefficients of a controllable floating ring bearing with a magnetorheological fluid (MRF) showing that the magnetic field-induced, field-dependent viscosity of the MRF changes the stiffness and damping bearing coefficients, which can be used to modify the dynamic behavior of the rotor-bearing system. In 2020, Kang et al. [18] used the Kalman filter to estimate the bearing dynamic coefficients of a flexible rotor-bearing system. The rotor system is modeled with Timoshenko beam elements, but the imbalance force considered in the dynamic model is calculated for a constant rotational velocity condition. More recently, in 2021, Chen et al. [19] proposed a method to simultaneously identify the parameters of the oil-film bearings and active magnetic bearings/bearingless motors AMBs/BELMs along with the residual unbalanced forces during the unbalanced vibration of the rotor. The proposed method requires independent rotor responses and control currents to form a regression equation to estimate all of the unknown parameters. Independent rotor responses are realized by changing the PID control parameters of the AMBs/BELMs. The finite element method is used to model the system by using Timoshenko beam elements, and both numerical and experimental results are presented at a unique operation velocity of 2400 rpm. Taherkhani and Ahmadian [20] used the Bayesian approach to an appropriate parameter selection procedure and suitable sampling strategy for stochastic model updating to investigate variability in the dynamic behavior of a complex turbo compressor supported by hydrodynamic bearings, leading to successful parameter identification results. Brito Jr. et al. [21] presented an experimental method to estimate the direct and cross-coupled dynamic coefficients of tilting-pad journal bearings of vertical hydro-generators. The method employs only the shaft radial relative vibrations, and the bearing radial absolute vibrations originated by the hydro-generator residual unbalance. The authors affirmed that the vibration measurements required by the

estimation method could be a major problem in low-speed machines (less than 400 rpm). Although any type of bearing provides stiffness and damping forces that may depend on the operation speed and many other factors, linearized force coefficients are widely used to model the reaction forces from fluid film bearings. These linear force coefficients are derived from the assumption of small amplitude motions about an equilibrium position [22] and have been used to study the dynamic responses and analyze the stability of rotor systems supported by oil-lubricated tilting-pad bearings, cylindrical bearings and foil bearings, as pointed out in [23] and references within. Recently, Dyk et al. [24] presented diverse linearization methods in the stability analysis of rotating systems supported on floating ring bearing (FRBs), demonstrating the usefulness of the linear force coefficients to predict the dynamic behavior of non-linear systems such as turbochargers supported by FRBs.

There is also substantial literature on parameter identification and estimation methods. Most of these schemes are essentially asymptotic, recursive or complex [25–27], and, according to Arias-Montiel et al. [28], these methods lead to unrealistic implementations. Over the past few years, another method of parametric identification called algebraic identification has been successfully implemented in a wide range of engineering applications [29]. The algebraic identification method is based on differential algebra and operational calculus for developing estimators in determining unknown system parameters from a mathematical model. These estimations are carried out on-line in continuous or discrete time. An advantage of algebraic identification over other methods is that it provides identification expressions that are completely independent of the initial system conditions. Algebraic identification has been used for parameter and signal estimation in linear and non-linear vibrational mechanical systems [30–39]. Numerical and experimental results show that algebraic identification is extremely robust against parameter uncertainty, frequency variations, measurement errors and signal noise. Additional information on the algebraic identification robustness and other advantages and disadvantages of this method are highlighted by Sira-Ramírez et al. in [29].

In this work, a novel methodology for developing two mathematical models for identifying the unknown stiffness and damping parameters of bearings in multiple degreeof-freedom (DOF) rotor-bearing systems is proposed. This methodology is based on the algebraic identification technique. Developed identifiers are obtained based on an FE model for a multiple DOF rotor-bearing system that considers the effects of rotational inertia, gyroscopic effects, shear deformations, internal damping and linear forces attributable to stiffness and damping parameters of the supports. Estimators are developed for two different operation conditions of the rotor-bearing system: constant and variable rotational speed. Analysis and evaluation of the proposed identifiers is carried out by numerical results showing the viability for applying algebraic identification techniques for the rotordynamic coefficients in rotor-bearing systems.
