Rapid solution for analysis of nonlinear fluid film force and dynamic behavior of a tilting-pad journal bearing-rotor system with turbulent and thermal effects

To analyze the nonlinear dynamics of a tilting-pad journal bearing (TPJB)-rotor system with high accuracy and speed, the database method (DM) is modified to rapidly determine the nonlinear fluid film force (NFFF) of a TPJB while considering turbulent and thermal effects. A high-accuracy, large-capacity NFFF database for a single pad is constructed by numerically solving the turbulent adiabatic hydrodynamic model for five equivalent state variables of the journal, which are discretized in the pad coordinates. The remaining variables are not discretized in the DM. A combined linear and parabolic interpolation polynomial based on the database is established to accurately calculate the NFFF of the tilting pads; thus, the NFFF of a four-pad TPJB is obtained in the bearing coordinates. The DM is applied to analyze and compare the nonlinear dynamic behavior of a water-lubricated TPJB-Jeffcott rotor system with and without turbulent and thermal effects. The present DM solution without these effects and the previous DM solution are shown to be consistent. The results demonstrate the importance of the flow regime and the negligibility of temperature increases in the nonlinear dynamics of a water-lubricated TPJB. This work contributes to the accurate and efficient analysis of the nonlinear dynamics of high-speed TPJBs and low-viscosity-fluid-lubricated TPJBs.


Introduction
Owing to good stability and adaptability, tilting-pad journal bearings (TPJBs) have been widely used in large rotating components, such as those in steam turbines, compressors, and nuclear reactor coolant pumps. Low-viscosity fluids, such as water, have been used as journal bearing lubricants in ship stern shafts, hydraulic turbines, and water pumps. In addition, water-lubricated TPJBs have been used in the nuclear reactor coolant pump of the third-generation AP1000 reactor. Because of the nonlinear effect of the fluid film force of a TPJB, the unstable behaviors of a TPJB-rotor system, such as subharmonic rotor resonance and pad fluttering, cannot be accurately predicted using a linear model based on stiffness and damping coefficients, even though the linear theory for TPJBs is basically complete. However, an increased workload and calculation time are to complete a dynamic analysis of a TPJB-rotor system using the nonlinear model represented by Reynolds equation solutions. Therefore, researchers are currently pursuing a rapid and accurate method for calculating the nonlinear fluid film force (NFFF) of a TPJB to enable a simple and rapid nonlinear dynamic analysis of TPJB-rotor systems.
Some researchers have used infinitely long and short journal bearing models to analyze the dynamic performance of a journal bearing-rotor system. Wang and Khonsari [1,2] presented an analytical NFFF expression for an axially grooved infinitely long journal  ,  ,   ,   Equivalent eccentricity ratio, attitude angle, dimensionless radial velocity, and tangential velocity of the journal max ( )   Maximum equivalent eccentricity ratio as a function of  ψ Clearance ratio c Radial clearance R Pad radius Re e , Re c , Re m Effective Reynolds number, critical Reynolds number, and mean Reynolds number n, ω Journal speed and angular speed μ 0 , μ m Dimensionless inlet dynamic viscosity and mean dynamic viscosity ϕ 1 , ϕ 2 Position angles at the inlet and outlet boundaries of the fluid film ρ p , h p , J Pad density, thickness, and moment of inertia bearing considering the inlet pressure and position and investigated the influence of inlet pressure and position on the nonlinear dynamic performance of the bearing-rotor system. Chang-Jian [3] analyzed the nonlinear dynamics of a gear pair system supported by journal bearings using a long bearing approximation. Avramov and Borysiuk [4] studied the nonlinear dynamics of an asymmetrical one-disk rotor-bearing system based on the short bearing theory, and Shi et al. [5] studied the nonlinear dynamic behavior of a vertical rotor-bearing system based on the short bearing theory. Dakel et al. [6] investigated the linear and nonlinear dynamic behavior of an on-board rotorbearing system using the short bearing model. Okabe and Cavalca [7] developed a short bearing model of a TPJB with turbulent effects and analyzed the dynamic behavior of a Jeffcott rotor-TPJB system with and without turbulent effects. Some researchers have reported an alytical models and calculation methods for the NFFF and pressure distribution and for nonlinear dynamic analysis of finite-length journal bearing-rotor systems. Zhang and Xu [8] proposed an NFFF model expressed by two symmetric positive definite matrices composed of three functions. Zhao et al. [9] investigated the linear and nonlinear unbalance response of a symmetrical single-disk flexible rotor-bearing system. The NFFF was obtained by solving the Reynolds equation and energy equation using the finite element method (FEM) at each time step. Xia et al. [10] provided two tools, a Ritz model and a one-dimensional FEM approach, for calculating the NFFF based on the free boundary theory and variational method under the Reynolds boundary condition, which demonstrated high efficiency. Bastani and de Queiroz [11] applied correction functions to force models of infinitely long and short bearings and proposed a closed-form NFFF expression. Vignolo et al. [12] treated the Ocvirk number as an expandable parameter and proposed an approximate analytical solution of the Reynolds equation using the regular perturbation method. Sfyris and Chasalevris [13] added a set of particular solutions for the Reynolds equation to a general solution of the homogenous Reynolds equation using the method of separation of variables, thus obtaining a closed-form expression of lubricant pressure. Zhang et al. [14] expressed the pressure distribution of an infinitely long journal bearing as a circumferential separable function of pressure distribution and obtained an axial separable function of pressure distribution through the variational principle, thus providing an approximate analytical method for calculating the NFFF of a turbulent journal bearing. Hei et al. [15] obtained the NFFF by the method of separation of variables based on the variational principle and investigated the nonlinear dynamic behavior of a rod-fastening rotorbearing system. Abu-Mahfouz and Adams [16] expanded the NFFF of a pad in a Taylor series, obtained the instantaneous linearized stiffness and damping coefficients, which were updated at each time step, and analyzed the nonlinear vibration of an unbalanced rotor supported by a three-pad TPJB under an on-pad load and a non-static biasing load. Wang et al. [17] established an NFFF model of a single pad through the method of separation of variables under the dynamic π fluid film boundary condition, resulting in a NFFF model of a four-pad TPJB based on the pad assembly method.
In 2002, Chen et al. [18] proposed the database method (DM) to calculate the NFFF by introducing a speed parameter ranging from -1 to 1 to the manipulative Reynolds equation. The effectiveness of the DM was verified by analyzing the nonlinear dynamic characteristics of an elliptical bearing-rigid rotor system. The DM presents a new approach for rapidly and accurately analyzing the nonlinear dynamics of journal bearing-rotor systems. Qin et al. [19] constructed a NFFF database of hydrodynamic bearings based on neural networks and studied the nonlinear motion characteristics of an elliptical bearing-rigid rotor system. Jin et al. [20] analyzed the nonlinear dynamic behavior, critical speed, and threshold speed of a supercritical rotor-bearing system using the DM based on the Poincare transformation. Ying et al. [21] studied the nonlinear dynamic response of a Jeffcott rotor-TPJB system with and without pad inertial effects using the DM. Lü et al. [22] investigated the nonlinear dynamics of a Jeffcott rotor supported by fixed-tilting-pad journal bearings using the DM, and Hei et al. [23] analyzed the nonlinear dynamic behavior of a rod-fastening rotor supported by fixed-tilting-pad journal bearings based on the DM.
Based on the above research, the NFFF models and calculation methods can be categorized as follows: i) infinitely long and short bearing models; ii) analytical models of finite-length journal bearings; iii) numerical methods; iv) the DM. Because the DM is based on accurate numerical methods and interpolation methods, we expect that the DM can rapidly and accurately calculate the NFFF for finite-length journal bearings. However, the DM is generally applied with isothermal and laminar assumptions, which inevitably introduce errors in the calculation. These errors will increase for high-speed bearings and low-viscosity-fluid-lubricated bearings.
In this paper, turbulent and thermal effects are applied in the DM to rapidly and accurately calculate the NFFF of a TPJB. The nonlinear dynamic behaviors of a water-lubricated TPJB-Jeffcott rotor system with and without the turbulent and thermal effects are compared and analyzed using the DM. The primary novelty of this work is as follows: i) the previous DM is modified by considering turbulent and thermal effects, but these effects are not introduced as discretized variables in the present DM; ii) the database capacity is far larger, and the NFFF data in the database are more accurate than in the previous database; iii) a combined linear and parabolic interpolation polynomial based on the database is established to calculate the NFFF of a TPJB more accurately.

Mathematical model of the TPJB-rotor system
A four-pad TPJB-Jeffcott rotor system is taken as the research object, as shown in Fig. 1. A brake springdamper is set on the outlet edges of the third and fourth pad backs of the TPJB to avoid fluttering of the two unloaded pads, including the rub impact between the pads and the journal. The dimensionless dynamic equations of the TPJB-rotor system can be expressed as follows: where M j and M d are the masses of the journal and disk, respectively; ( are the displacements and accelerations, respectively, of the journal and disk in the X and Y directions; FX and FY are the NFFF of the TPJB in the X and Y directions, respectively; K is the rotor stiffness; G is the gravitational acceleration; ε d is the unbalanced mass eccentricity ratio of the disk; τ is a time variable; i M and J are the torque and moment of inertia of the i-th pad, respectively; K δ and C δ are the stiffness and damping of the spring-damper, respectively; δ i , δ i ′, and δ i ″ are the tilting angle, tilting velocity, and tilting acceleration of the i-th pad, respectively; α is the pad arc angle; and Pc is the pivot coefficient. All parameters are dimensionless, and the dimensional ratios utilized in this paper are shown in the Appendix.
For the single pad shown in Fig  be written as follows: where H is the fluid film thickness; P is the fluid film pressure; D is the pad diameter; L is the pad length;  is the dynamic viscosity; K ϕ and K λ are turbulent flow coefficients; and ϕ and λ are circumferential and axial coordinates, respectively. The dimensionless fluid film thickness equation can be expressed as follows: where A is a preload factor; β is the pivot position angle; ε is the journal eccentricity ratio; θ is the journal attitude angle; and ψ = c/R is the clearance ratio, where c is the radial clearance and R is the pad radius. Equation (3) can be transformed as an equivalent expression without A and δ i : After substituting Eq. (4) into Eq. (2), Eq. (2) can be derived as follows: where  ,  ,   , and   are the equivalent eccentricity ratio, attitude angle, radial velocity, and tangential where T is the fluid film temperature and τ c is the turbulent flow coefficient. The Vogel viscosity-temperature relationship is adopted as follows: where μ and T are the dynamic viscosity and temperature of the fluid film, respectively, with dimensionless forms of  and T . For a water film, t 1 , t 2 , and t 3 have been calculated as 2.5016 × 10 −5 , 569.21773, and 134.17565, respectively, through a fitting method.
The appropriate values of K ϕ , K λ , and τ c are given by the following [24,25]: where Re e is the effective Reynolds number, which can be expressed for three types of flow regime conditions determined by the critical Reynolds number Re c , including the laminar flow regime, transitional flow regime, and turbulent flow regime. Re e can be expressed as follows: where ρ is the fluid density; ω is the journal angular speed;  Re c 41.2 R c ; and Re m = ρωRc/μ m is the mean Reynolds number. μ m is the mean dynamic viscosity, expressed as follows: where μ 0 is the inlet dynamic viscosity and ϕ 1 and ϕ 2 are the position angles at the inlet and outlet boundaries of the fluid film, respectively.
The dimensionless NFFF and torque of the pad can be written as follows: Then, the dimensionless NFFF of the four-pad TPJB is

Boundary conditions and computation
For the pressure field, the classical Reynolds boundary conditions are adopted for Eq. (5): For the temperature field, the fluid film temperature at the inlet boundary is usually set as the inlet temperature 0 T , and heat is assumed to be completely transmitted by the pad under reverse flow conditions. Thus, the boundary conditions for Eq. (6) are given as follows: n i ), are not included in the database, the NFFF will be evaluated by an interpolation method using the existing NFFF at the nodes closest to ( where where cfx1-72 and cfy1-72 are the polynomial coefficients of FX i and FY i , respectively, which can be calculated from the database using the NFFF at the nodes closest to ( The NFFF of the other pads can also be conveniently obtained from the database using a coordinate transformation of the journal attitude angle. Equation (16) is a combined linear and parabolic interpolation polynomial with five equivalent state variables, and the NFFF can be rapidly calculated using an analytical expression based on the database instead of using numerical iterative methods. The process for accessing the NFFF database is shown in Fig. 4

Numerical results and discussion
A precise time-integration method is adopted to solve Eq. (1). In this study, the time step for direct numerical integration is specified as  2500 , and the computational data under the steady state condition are extracted and analyzed. The calculation is performed by MATLAB using an Intel Core i7-5500U at 2.40 GHz. The structural and operational parameters of the TPJB-rotor system are listed in Table 1, and the expression of the pad moment of inertia is given in the Appendix. Because water has been employed in the nuclear reactor coolant pump of the third-generation AP1000 reactor, water is chosen as the TPJB lubricant to study the effects of turbulence and temperature.

Comparison of present DM and previous DM [18]
In this section, the research object is a single pad of the water-lubricated TPJB, whose pivot position angle is 180°. A basic comparison of the present DM and the previous DM is presented in Table 2. It is observed that the present DM, which considers turbulent and thermal effects, has a very large database capacity and a large construction cost. Figure 5 compares the dimensionless NFFF calculated by the present DM and those obtained by the FDM and the previous DM. The solution obtained from the present DM with turbulent and thermal effects is in good agreement with the FDM solution with the same effects. However, the values are larger than those for the previous DM solution in some regions because the previous DM is based on isothermal and laminar assumptions.
Moreover, the present DM solution without turbulent or thermal effects is in good agreement with the previous DM solution, demonstrating that the constructed database is suitable. Figure 6 presents ten cycles of  transient journal center orbits calculated by the present DM and those determined by the previous DM at 3,000 rpm. The present DM solution without turbulent or thermal effects is in good agreement with the previous DM solution, while the present DM solution with these effects is closer to the bearing center and has a smaller orbit amplitude than the previous DM solution. The calculation costs for ten cycles of transient journal center orbits for the present DM and previous DM are shown in Table 3. It is observed that the calculation cost for the present DM with turbulent and thermal effects is approximately nine times larger than that with the previous DM; however, this cost is acceptable for performing nonlinear dynamic analysis. Figure 7 shows the journal center orbits with and without turbulent and thermal effects. From Fig. 7, it can be seen that the orbit centers are approximately located at the Y axis of the TPJB and move closer to the TPJB center as the speed increases. Moreover, the journal orbit amplitude increases as the speed increases.

Synchronous rotor motion with and without turbulent and thermal effects
The journal orbit amplitudes with turbulent and thermal effects are smaller than those without these effects, except at 1,000 rpm. At 6,000 rpm, the journal orbit amplitude with turbulent and thermal effects is nearly three times smaller than the orbits without these effects. The orbits at 2,000 rpm-6,000 rpm with turbulent and thermal effects are much closer to the TPJB center than those without these effects. Figure 8 shows the disk center orbits with and without turbulent and thermal effects. From Fig. 8, it can be seen that the trends for the disk center orbits are similar to those for the journal center orbits. However, the disk orbit amplitudes are larger than    the journal due to the rotor flexibility. The disk orbit amplitudes with turbulent and thermal effects are slightly smaller than those without these effects. The temperature field of the fluid film on the first pad at 1,000 rpm and 6,000 rpm without an unbalanced mass is shown in Fig. 9. It can be seen that the fluid film temperature is almost constant in the λ direction and increases almost linearly with increasing ϕ. The temperature increases by 2.5 °C at 6,000 rpm and by only 0.2 °C at 1,000 rpm. The temperature rise of the fluid film for the water-lubricated TPJB-rotor system at 1,000 rpm-6,000 rpm is too small (0.2 °C -2.5 °C ) to have a substantial effect; thus, only the flow regime plays an important role in the dynamic analysis. For 1,000 rpm, Re e = 0 and K ϕ = K λ = τ c = 1, which correspond to laminar flow conditions; thus, the same axis orbit is obtained regardless of whether turbulent and thermal effects are considered. For 2,000 rpm-6,000 rpm, K ϕ > 1, K λ > 1, and τ c > 1, indicating that the TPJB-rotor system is in a transitional flow or even the turbulent flow Fig. 9 Temperature field of the fluid film on the first pad at 1,000 rpm and 6,000 rpm without an unbalanced mass. regime, which embodies in a higher dynamic viscosity. The increase in fluid viscosity will lead to an increased NFFF; thus, the calculated orbits are closer to the bearing center. Figures 10 and 11 show the journal center orbits and Poincaré maps for six representative speeds with and without turbulent and thermal effects, respectively. At these speeds, the orbits have distinct nonlinear characteristics, and a substantial difference is observed between the nonlinear characteristics with and without turbulent and thermal effects. At 6,500 rpm and 6,800 rpm, the journal motion is synchronous when turbulent and thermal effects are considered (see Figs. 10.1 and 10.2), while the journal exhibits period-4 motion and period-3 motion, respectively, when turbulent and thermal effects are neglected (see Figs. 11.1 and 11.2). At 7,600 rpm, the journal exhibits period-2 motion regardless of whether turbulent and thermal effects are considered (see Figs. 10.3 and 11.3), but there exist significant differences, such as in the amplitude and form, between the two orbits. At 8,200 rpm, a period-3 motion arises for the journal in the system when turbulent and thermal effects are considered (see Fig. 10.4); however, the motion of the journal is approximately 4-periodic when turbulent and thermal effects are neglected, which can be obtained from the four points of the Poincaré map (see Fig. 11.4). At 8,300 rpm, the journal displays period-7 motion when turbulent and thermal effects are included (see Fig. 10.5), while a period-2 motion arises for the journal when turbulent and thermal effects are ignored (see   Fig. 10.6). However, in Fig. 11.6, the motion trajectory is highly disordered and the Poincaré mapping points are clustered in groups; thus, the journal exhibits chaotic motion when turbulent and thermal effects are neglected. Figures 12 and 13 show the time domain waveforms and the spectrograms for the four pads at 6,500 rpm with and without turbulent and thermal effects, respectively. The frequency ratio in the spectrogram refers to the ratio of pad vibration frequency to rotor rotational frequency. As shown in Fig. 12, when turbulent and thermal effects are considered, the motions of the four pads are all synchronous, and the vibration frequencies of the four pads primarily match the rotor rotational frequency. Figure 13 shows that when turbulent and thermal effects are neglected, a subsynchronous vibration of the four pads arises in the system, which can be observed from three fluttering frequency components (frequency ratios of 1/4, 1/2, and 3/4) lower than the rotor rotational frequency in 12 Friction | https://mc03.manuscriptcentral.com/friction the spectrograms. Figures 14 and 15 show time domain waveforms and spectrograms for the four pads at 8,900 rpm with and without turbulent and thermal effects, respectively. Figure 14 shows that when turbulent and thermal effects are considered, seven obvious fluttering frequency components arise (frequency ratios of 1/8, 1/4, 3/8, 1/2, 5/8, 3/4, and 7/8) in the spectrograms, indicating that the subsynchronous vibration of the four pads is stronger and more complex than that at 6,500 rpm. As shown in Fig. 15, when turbulent and thermal effects are ignored, there exist numerous irregular, continuous, and broadband frequency components in the spectrograms in addition to the main rotor's rotational frequency; thus, chaotic vibrations may occur for the four pads in the system.

Conclusions
The current work builds upon the DM to calculate the NFFF of a TPJB while considering turbulent and thermal effects. An NFFF database with high accuracy and large capacity is compiled, and a combined linear and parabolic interpolation polynomial of the NFFF for a single pad is constructed. The DM is applied to perform nonlinear dynamic calculations of a waterlubricated TPJB-rotor system. The results show that the present DM solution is in good agreement with the FDM solution when turbulent and thermal effects are considered and agrees with the previous DM solution when considering laminar and isothermal effects. Significant differences arise in the results with respect to nonlinear dynamic behavior, such as synchronous motion, period-doubling or sub-synchronous motion, and chaotic motion, for a water-lubricated TPJB-rotor system with and without turbulent and thermal effects. For synchronous motion, the journal center orbit is closer to the TPJB center, and the orbit amplitude is generally smaller when considering turbulent and thermal effects compared with the results obtained without turbulent or thermal effects. These differences are primarily caused by the flow regime, rather than the temperature increase of the water film.