1. Introduction
As a rotary system supported by hydrodynamic bearings operates, pressure distribution in the oil lubricant film of the bearing is generated, which bears the rotor weight as well as the unbalanced forces acting on the rotor and directly influences the dynamic behavior of the rotor (system stability, bending critical speed, vibration modes, etc.). The bearing behavior is influenced by the magnitude of the hydrodynamic forces, which can be expressed as a function of the rotor dynamic coefficients. According to Lund [
1], Vance [
2], and Dimarogonas [
3], there are four rotor dynamic coefficients of a bearing that represent the stiffness
and four others that represent the damping (
. The rotor dynamic coefficients of the fluid film in hydrodynamic bearings are calculated by analytical, numerical, and experimental approaches. Among the researchers specializing in this field, most experimentally estimate the rotor dynamic coefficients, specifically by using the response to the unbalance of the system due to the residual unbalance is always present in the rotary systems, which prevents the use of external equipment for exciting the system. Wengui Mao et al. [
4] presented an inverse method aimed at the identification of the rotor dynamic coefficients of a sliding rotor-bearing system with unbalance parameters, which consisted of the application of a dynamic-loading identification method along with the analysis of intervals. The identification of the dynamic parameters of the bearing could be formulated as the reconstruction of the force on the oil film with the experimental unbalance response. On the other hand, Changmin Chen et al. [
5] proposed a novel method to avoid ill-posed problems of the coefficient identification of circular journal bearings based on the unbalance response; focused on avoiding the ill-conditioned matrix problem, they proposed four complementary equations that are independent of the dynamic equations derived from the response to the unbalance. Two equations for the damping are derived from the Reynolds equation, and two others for the stiffness are derived from the Taylor expansion of the static forces acting on the bearing. With these four equations, the identification matrix has a low condition number, which allows for a more stable and reliable parameter identification.
Machinery, in general, plays a very important role in the industry; currently, different studies have been carried out on new methodologies for the diagnosis of failures in industrial machinery [
6,
7]. Recently Ke Zhao et al. [
8] designed a new transfer-learning framework called CWTWAE to solve the problem of rolling-bearing failure diagnosis with multi-source domains. There is vast information about rolling bearing, journal bearing and identification methods of unbalance and bearing dynamic parameters [
9,
10]. Colín Ocampo J. et al. [
11] proposed a methodology for the angular position identification of the unbalance force based on a two-degrees-of-freedom mathematical simplified model of a rotor with unequal principal moments of inertia of the shaft transverse section. They submitted this methodology in a numerical and experimental way, obtaining encouraging results. In addition, Jianfei Yao et al. [
12] proposed an integrated modal expansion/inverse problem methodology combined with an optimization procedure. This technique allowed for the identification of the axial location of the unbalance as well as its magnitude and phase. This technique was validated experimentally with acceptable error percentages. Lei Li et al. [
13] used a scale model and scale laws to identify unbalance for a full-size rotor system. They tested the proposed methodology numerically and experimentally, concluding that it is feasible to identify the unbalance values of the rotor with the proposed method. Recently, Aiming Wang [
14] presented the development of algorithms for the simultaneous identification of the unbalance and the bearing dynamic parameters; in both cases, the proposed algorithms were validated with experimental results. Seung Yoon On et al. [
15] developed a composite tilting-pad journal bearing using a hybrid pad structure composed of a carbon fiber/epoxy composite liner and backup metal to enhance the dynamic characteristics of the bearing system. The stiffness and damping parameters of the bearing fluid film were determined by a thermohydrodynamic analysis of lubrication using the finite difference method.
In the same way, investigations have also been carried out to identify the dynamic parameters of active magnetic bearings (AMBs) in a flexible rotor system. The AMBs support rotors using electromagnetic force rather than mechanical forces [
16,
17]. M. Asadi Varnusfaderani et al. [
18] developed an algorithm for identifying the parameters of flexible rotor systems equipped with smart magneto-rheological bearings. For the implementation of the identification algorithm, the finite element model of a flexible rotor system was equipped with magnetorheological squeeze-film dampers (MRSFDs). Eliott Guenat and Jürg Schiffmann [
19] built a test rig to experimentally identify the stiffness and damping coefficients of the Herringbone Grooved Journal Bearings (HGJBs) of a rotor perturbed by piezo-electric shakers; they concluded that the Narrow Groove Theory (NGT) tends to overestimate the stiffness and damping of the HGJB. At the rated speed, direct stiffness values and damping ratios were measured to be 38% and 27% lower, respectively, than the NGT prediction.
The stiffness and damping parameters of the fluid film bearings change with the speed of the rotor, the viscosity provided by the type of oil used, pressure changes, and temperature changes, among other factors. Muhammad Imran Sadiq et al. [
20] evaluated bio-oils and mineral-based oils in terms of their stiffness and damping coefficients for bearing applications. The approach they used was analytical and experimental. Rotodynamic coefficients are determined with analytical expressions that are a function of the eccentricity ratio, which is determined using the dynamic viscosity of the oil. This viscosity is obtained experimentally for different temperatures. Hussein Sayed and T.A. El-Sayed [
21] carried out an investigation on the dynamics and stability of rotors supported on journal bearings; this analysis was based on the second order stiffness and damping coefficients present in the bearings. They used a flexible rotor model supported on two symmetrical journal bearings. Bearing parameters were identified using the direct solution of the Reynolds equation and using the time-dependent second-order perturbation method.
In addition, Michel Fliess and Hebertt Sira Ramírez [
22] presented an approach known as algebraic identification, which allows for the development of identifiers addressing the online determination of unknown parameters supported by differential algebra along with operational calculus based on the mathematical model of the mechanical system.
Later, Mendoza Larios G. et al. [
23], in a numerical approach, developed an estimator for the rotor dynamic coefficients by applying algebraic identification, in which the Finite Element Method was used to numerically obtain the response of the rotary system by using a multiple degrees-of-freedom (DOF) model. The proposed identifier requires the lateral displacements and the slope of the node located at the support to identify, as well as the slope of the adjacent node. The complexity of the experimental measurements of the nodal slopes complicates the implementation of the method. Recently, Baltazar Tadeo L. et al. [
24] used the algebraic identification method to determine the magnitude and angular position of the unbalance in an asymmetric rotor-bearing system, taking as a basis for the algebraic identifier a mathematical model of an asymmetric rotor-bearing system of multiple degrees of freedom using active balancing disks. They tested the proposed identifier numerically and experimentally, demonstrating that it is possible to reduce the vibration amplitudes under resonance conditions of an asymmetric rotor by more than 95%.Therefore, Nango [
25], Beltrán Carvajal et al. [
26], Arias et al. [
27], and Mendoza Larios G. et al. [
28] proposed identifiers by using algebraic identification for the estimation of the parameters in rotor dynamic systems; both numerical and experimental results showed that the identification is quickly achieved and exhibits high robustness with regard to the parametric uncertainty. One of the main advantages of algebraic identification is that it provides identification relationships regardless of the initial conditions of the system and only requires the system response as the input data. The parameter identification is conducted online either in continuous or discrete time.
As can be seen, there are different investigations in the field of roto-dynamics where algebraic identification has been used to estimate some sought parameter; however, it has not been used to identify the dynamic coefficients of a pressurized bearing experimentally. Therefore, in the present work, a mathematical model was developed to identify the direct rotodynamic coefficients of a pressurized bearing at constant speed in a rotor-bearing system. The mathematical model was developed by applying the algebraic identification technique to a simplified two-degree-of-freedom model of a rotor-bearing system. Numerical and experimental simulations were carried out to test the functioning of the proposed algebraic identifier. It should be noted that this identifier of the direct rotodynamic coefficients only needs the lateral displacements of the node where the support to be identified is located. This is a great advantage since in the experimental part, the instrumentation is very easy to implement, and a minimum number of sensors is required. Another advantage of the proposed identifier is that it does not require the implementation of any external device to introduce some excitation force to the rotodynamic system since, as previously mentioned, the identification is made with the response of the system due to the unbalance, which is always present in all rotodynamic systems. The proposed algebraic identifier can be used in other types of bearings; in this case, it was used for a pressurized bearing, considering different operating speed and constant pressure. This means that it can be used to identify the dynamic parameters of conventional bearings, where their dynamic coefficients are considered constant, or magnetic bearings, where their properties change depending on the current supplied to the support.
The rest of the document is organized as follows: in
Section 2, two models are analyzed, the model of a rotor-bearing system with multiple degrees of freedom and the simplified model with two degrees of freedom, and a comparison is made between the responses of both models. In this same section, the algebraic identifier of the direct dynamic coefficients of a bearing is developed.
Section 3 describes the simulation of the proposed identifier, numerically and experimentally. In this section, the identified coefficients are validated with the help of the models developed in the previous section.
Section 4 shows a discussion section. Finally,
Section 5 shows the conclusion of the research work presented.
4. Discussion
Different simulations were carried out to test the functioning of the proposed algebraic identifier of rotodynamic coefficients. In the numerical simulation, the vibration response used by the identifier is obtained from the multiple-degrees-of-freedom model of Equation (3). It is important to highlight that the proposed identifier only uses the lateral displacements of a single node to identify the coefficients; in this case, it was node 11. This is a great advantage since other identifiers based on a more complex model need the entire displacement vector, i.e., the lateral vibration and the slope nodes of the node which you want to identify and of two adjacent nodes [
23,
24]. This is complex to achieve in the experimental part: First, the nodal slopes cannot be measured directly; it is best to approximate them with the lateral displacements. Second, instrumenting is complex since ten sensors would be used just to obtain the necessary displacements to identify the coefficients, which is a disadvantage. Meanwhile, the proposed identifier requires only two sensors (see
Figure 10) to acquire the necessary displacements to identify the rotodynamic coefficients in an acceptable manner. This pair of sensors is placed in the position of the support to be identified (node 11).
It is important to highlight that when identifying the rotodynamic coefficients based solely on the unbalance response, there is an ill-conditioned matrix problem, which increases the algorithm complexity and decreases the precision of the identification results. It must be remembered that this algebraic identification methodology results in a system of equations (Equation (23)) where the matrix
A(
t) and
P(
t) change as a function of time; for this reason, the condition number of said matrices also changes. The condition number of a matrix is used to quantify its level of ill conditioning. If the condition number is close to 1, the matrix is said to be well conditioned. If the condition number is significantly greater than 1, the matrix is said to be ill conditioned. In this case, small variations in the data can produce large variations in the results. It is common to find ill-conditioned matrices in inverse problems, as is the case of the proposed identifier. However, with the identified parameters, it is possible to reproduce the dynamics of the system with very few differences (see
Figure 13 and
Figure 14). Taking this into account, an identifier of the direct rotodynamic coefficients was proposed with a reduced model, which has systems of 2 × 2 equations (see Equation (18)), resulting in a more stable identifier compared with more complex models with 8 × 8 matrices and with a higher condition number.
Figure 18 shows the condition number of the matrices
(blue line) and
(red line), calculated with the experimental measurements of the displacements at a speed of 2400 rpm. It can be seen how the number of conditions tends to decrease after a short time and ends up oscillating between a large range of values. Even with this behavior, the proposed algebraic identifier fulfills its function of estimating the rotodynamic parameters, and with these identified parameters, it is possible to reproduce the dynamics of the rotodynamic system acceptably (see
Figure 14). It can be seen in
Figure 14 that the greatest difference between the experimental responses and the numerical one obtained with the identified dynamic coefficients is found at speeds close to resonance (4500–5000 rpm), as observed in the numerical simulation represented in
Figure 3, where both models are compared. These differences are mainly because Equation (12) is taken as the basis for the development of the proposed identifier Equation (17) in a simplified model of two degrees of freedom, while in the experimental part, there is a rotor-bearing system with multiple degrees of freedom. This is a limitation of the identifier. Another limitation is that the developed identifier only identifies the direct coefficients and misses the crossed coefficients, which are essential to more accurately reproduce the dynamics of the system. It is very important to monitor the crossed coefficients since it is known that certain values can cause instabilities in the rotor-bearing system.