Investigation on Pivot Rolling Motion Effect on the Behavior of Rocker Back Tilting Pad Journal Bearings

Abstract

The rolling motion of the pads with rocker back (RB) and ball and socket pivots is normally neglected in common software programs for the study of the rotor dynamic behavior of tilting pad journal bearings (TPJB). In other words, the theoretical contact point of the pivot is considered fixed. The aim of this work is to provide a novel way to implement in commercial software the effect of the variation of the circumferential coordinate of the theoretical contact point due to the pad rolling motion in RB TPJB. This is done by introducing an equivalent pivot rotational stiffness evaluated with an analytically derived formula, validated through finite element analysis. Such a stiffness is a function of the pad load and the radii of the contact pair, increasing with the load, the radii, and the degree of conformity of the contact. The static and dynamic characteristics of a five pad RB TPJB are then evaluated with a commercial software with and without the rotational stiffness contribution for two different pivot geometries. Non-negligible differences were found, particularly regarding the cross-coupled dynamic coefficients that show the higher sensitivity to the rotational stiffness. The inclusion of a pivot rotational stiffness among the data of commercial software for simulation of RB TPJB could contribute to fill the gap between numerical and experimental results.

1. Introduction

Tilting pad journal bearings (TPJB) are widely used in turbomachines. Their benefit is an improved rotor stability at higher speeds with respect to fixed-pad journal bearings, due to very low stiffness and damping cross-coupled coefficients. In Figure 1, some common pivot types are depicted: rocker back that is cylinder on cylindrical seat (hereafter referred as RB), ball and socket that is sphere on spherical seat (B&S), and flexural. The RB pivot has a theoretical line contact and typically undergoes a rolling motion with negligible sliding [1,2]. The B&S pivot has a theoretical point contact and may involve some sliding motion [3]. The main advantage of a B&S pivot is that it can allow some degree of axial shaft misalignment and comply with rotor sag. In a flexural pivot bearing, the pad and the bearing housing do not form a contact pair but are a single piece and their connection allows for elastic deformation and pad tilting [4]. The major advantages of flexural pivots are no contact wear at pad-housing interface and robustness to pad fluttering.

Geometrical parameters, materials, and surface treatments must be optimized in pivot design since they have a great influence on stress and wear due to the reduced and highly loaded contact areas. In addition, the characteristics of pivots can greatly influence the bearing rotor dynamic behavior with their stiffness and damping characteristics.

Some researchers have studied numerically the effect of pivot radial stiffness on the bearing characteristics [5,6] and on the journal nonlinear dynamics [7]. The radial stiffness of an RB or B&S pivot can be evaluated through classical Hertz theory if the contact is non-conformal [8]. However, Hertz’s formulas are not reliable for quasi-conformal contacts, as observed experimentally in [9], and whenever Hertz’s hypotheses are not satisfied and contact stiffness assessment is important, the latter should be evaluated more accurately [10]. Recently, Fang et al. proposed a theory for the estimation of radial stiffness of quasi-conformal B&S contact [11]. When the pivot radial stiffness is considered, the bearing direct stiffness coefficients are lower with respect to the rigid pivot models. Zhang et al. [12] numerically and experimentally studied the influence of pivot radial stiffness on the bearing characteristics and rotor stability in a vertical-rotor-bearing system.

Experimental works compared the behavior of RB and B&S TPJB showing the negligibility of cross-coupled stiffness terms in the former [1,2] along with lower power loss and pad temperature [13]. They attributed the results to friction in the B&S pivot. Therefore, while friction is usually neglected in numerical simulation of RB TPJB, quite a few papers can be found on modeling of B&S TPJB, including friction. Kim et al. [14] proposed a mathematical model for the B&S pad-pivot friction explaining its influence on the bearing characteristics. He et al. [15] showed that varying bearing loads, the pivot friction could result in non-synchronous pad tilting and journal vibrations. Kim et al. [16,17] focused on the pad-pivot friction-induced nonlinear rotor dynamic phenomena and bifurcations. The Stribeck curve model was applied for friction. Dyk et al. [18] proposed a detailed B&S model to predict TPJB dynamics, employing the Bengisu-Akay and Lu-Gre models to describe static and dynamic friction.

Hence, dynamic friction and sliding have been addressed by some papers but, to the authors’ knowledge, little attention has been given to the effect of static friction in RB pivots and to the consequent pad rolling motion. It is a common practice for bearing calculation software to assume a fixed theoretical pivot point [19]. However, the pad rolling motion changes the circumferential coordinate of the theoretical contact point. The purpose of this paper is to provide a way to implement the effect of RB pad rolling in a commercial bearing calculation software. This is done by introducing a fictitious equivalent rotational stiffness.

An actual pivot rotational stiffness is implemented in most software for the simulation of flexural pivots by defining it on the basis of the beam theory or by means of structural analysis [20,21]. To the best of the authors’ knowledge, this is the first attempt to consider the rolling motion of an RB pivot through the introduction of an equivalent rotational stiffness and exploit the capabilities of the software to deal with flexure pivot features. The applicability of this method is restricted to tilting motion without sliding, as sliding is not modeled here. The no-sliding hypothesis is reasonable for a typical RB pivot, while it may be improper for a B&S pivot.

The formula for RB pivot equivalent rotational stiffness evaluation is derived in Section 2.1 and validated through finite element (FE) analysis in Section 2.2. The results of static and dynamic analyses carried out on a journal bearing with different RB pivot geometries are reported in Section 3. The peculiar aspects of the proposed approach and the obtained results are discussed in Section 4. In Section 5, the main findings are summarized and some conclusive remarks are provided.

2. Materials and Methods

2.1. Analytical Approach

Let us consider the 2D model of an RB pivot in Figure 2.

Here, $O_h$ is the housing center, $O_p$ is the center of the pad bore, $O_b$ is the center of the pad back surface or pivot surface. $O_p$ coincides with $O_h$ when preload is zero and tilt angle is zero. In Figure 2a, the theoretical contact point is $P_1$ which coincides with $B_1$. In Figure 2b, the pad tilts by an angle:

$${\alpha }_t={\alpha }_b-{\alpha }_h$$

(1)

where ${\alpha }_b=\widehat{B_1{O}_b{B}_2}$ and ${\alpha }_h=\widehat{P_1{O}_h{P}_2}$. In Figure 2b, the theoretical contact point is $P_2$ which coincides with $B_2$. It is assumed that the pad tilts without sliding. Let $\xi$ and $\eta$ be a reference system fixed with respect to the pad. Neglecting viscous forces, the resultant of a generic pressure distribution is a force $F$ passing through the point $O_p$. In addition, to satisfy moment equilibrium of the pad, the resultant $F$ must pass through the theoretical pivot point $P_2$. Therefore, the angle $\gamma$ between the resultant $F$ and the axis $\xi$ is different from zero. The computed pressure distribution should produce a null resultant moment with respect to point $P_2$, with this point depending on the tilting angle.

However, the used bearing calculation software does not consider the motion of the theoretical contact point; in particular, the resultant moment is computed with respect to point $B_1$. In Figure 2b, the resultant moment of hydrodynamic forces with respect to the point $B_1$ is:

$$M=\left(R_p+t\right)F_{\eta }$$

(2)

where $R_p=O_{p}A$ is the pad bore radius, $t=A{B}_1$ is the pad thickness, $F_{\eta }$ is the component of the force $F$ along $\eta$. The displacement of the theoretical contact point can thus be included into the program by introducing an equivalent rotational stiffness $K_{rot}$, such that:

$$\left(R_p+t\right)F_{\eta }=K_{rot}{\alpha }_t$$

(3)

The angle $\gamma$ satisfies the following geometrical relation:

$$\tan \gamma =\frac{R_b\sin {\alpha }_b}{R_b\cos {\alpha }_b+R_p+t-R_b}$$

(4)

where $R_b=O_b{B}_2=O_b{B}_1$ is the pad back radius. Moreover:

$$\tan \gamma =F_{\eta }/F_{\xi }$$

(5)

where $F_{\xi }$ is the force component along $\xi$. In addition, the rolling without sliding motion imposes:

$$R_b{\alpha }_b=R_h{\alpha }_h$$

(6)

where $R_h=O_h{P}_2=O_h{P}_1$ is the housing radius. Therefore, after some manipulation of the above Equations, $K_{rot}$ can be written as:

$$K_{rot}=F_{\xi }\frac{\left(R_p+t\right)R_b{R}_h\sin {\alpha }_b}{\left(R_h-R_b\right)\left(R_b\cos {\alpha }_b+R_p+t-R_b\right){\alpha }_b}$$

(7)

which, for small tilting angles, reads:

$$K_{rot}\cong F_{\xi }\frac{R_h{R}_b}{R_h-R_b}$$

(8)

The pad back is here assumed circular for the sake of simplicity; however, if it were not circular, the procedure is generalizable by employing in Equation (8) a pad back radius $R_b\left({\alpha }_t\right)$ as a function of the tilting angle.

Equation (8) is worth some additional discussion. Firstly, the hypothesis of small tilting angle is typically reasonable in real tilting pad journal bearings, as the tilting angles are of the order of ${10}^{-3}$ radians. Then, the equivalent rotational stiffness is proportional to the equivalent radius of the contact and to the pad normal load, which can be computed as:

$$F_{\xi }=\underset{z_1}{\overset{z_2}{{\displaystyle \int }}}\underset{{\theta }_1}{\overset{{\theta }_2}{{\displaystyle \int }}}p\left(\theta ,z\right)\cos \theta R_{p}d\theta dz$$

(9)

where $p\left(\theta ,z\right)$ is the pressure field, $z$ is the axial coordinate, and $\theta$ is the circumferential coordinate, with $\theta =0$ corresponding to the $\xi$ axis.

Furthermore, in the limiting case where $R_b=0$, the contact point does not move, reducing the equivalent rotational stiffness accordingly to zero. In the limiting case where $R_b=R_h$, a fixed pad is obtained, that is, a pad with $K_{rot}\to \infty$. All the other cases are in between these two.

Lastly, the hypothesis of no sliding may limit the applicability of Equation (8). To evaluate its applicability, let the static friction coefficient between pad and housing be $f$. Then, to avoid sliding:

$$\tan \left(\widehat{O_b{B}_2{O}_p}\right)\le f$$

(10)

must hold.

From geometrical considerations, the following relationship holds:

$$\widehat{O_b{B}_2{O}_p}={\alpha }_b-\gamma$$

(11)

Assuming small tilting angles, Equation (10) can be written as:

$${\alpha }_b-\gamma \le f$$

(12)

while Equation (4) can be written as:

$$\gamma \cong \frac{R_b{\alpha }_b}{R_p+t}$$

(13)

Therefore, with the use of Equations (1), (6), (13), Equation (12) can be rewritten as:

$${\alpha }_t\le f\frac{\left(R_p+t\right)\left(R_h-R_b\right)}{\left(R_p+t-R_b\right)R_h}$$

(14)

Equation (14) evaluates the maximum tilting angle before the pad starts sliding.

From the preload definition $m{C}_r=\overline{O_p{O}_h}$, where $m$ is the preload and $C_r$ is the pad-journal radial clearance, the following relation:

$$R_p+t=R_h+m{C}_r$$

(15)

can be written and used in Equation (14) to obtain:

$${\alpha }_t\le f\frac{\left(R_h+m{C}_r\right)\left(R_h-R_b\right)}{\left(R_h-R_b+m{C}_r\right)R_h}$$

(16)

Equation (16) is not satisfied when $R_h=R_b$, or for small pivot clearances: $\left(R_h-R_b\right)\ll m{C}_r$. However, considering typical geometrical values for an RB pivot, and a friction coefficient of $f=0.15$, the no-sliding condition is generally largely satisfied. This justifies the use of Equation (8) for a typical RB pivot.

Equation (14) can also be applied to a B&S pivot. However, the seat radius must be used for $R_h$ instead of the housing radius. Moreover, Equation (16) is not valid if the seat radius is used in place of $R_h$, as Equation (15) is no longer valid. It can be shown by Equation (14) that a B&S pivot is much more likely to slide, due to its lower pivot clearance and lower seat radius.

2.2. Finite Element Analysis

The hypothesis of rigid bodies needed to derive Equation (8) can be relaxed using finite element analysis. Static structural simulations were performed with the commercial software ANSYS 2021 R2. Simulations were set as bidimensional in plane strain, to reduce computational cost. This corresponds to a bearing with $L/D\to \infty$. An example of the FE models of the contact pair is reported in Figure 3.

The material chosen for the simulations was linear elastic steel, with Young’s modulus equal to $2\times {10}^{11}$ Pa and Poisson’s modulus equal to $0.3$. The contact was modeled as frictional, with normal Lagrange penalty formulation. The interface treatment was set to “adjust to touch”. The mesh was refined in proximity of the contact. The sensitivity of the results to the mesh was checked. Large deflections were allowed. Points on the outer radius of the lower body were fixed. A uniform pressure and tilting moment were incrementally applied to the upper surface of the upper body. Then, the tilting moment was gradually decreased to check the reversibility of the motion. An example of loading history is shown in

Table 1. Example of loading history.

Time Step	Pressure [MPa]	Moment [N·mm]
1	0.1	0
2	0.2	0
…	…	…
10	1	0
11	1	50
12	1	100
…	…	…
30	1	1000
31	1	950
…	…	…
50	1	0

. The pressure is increased in steps of 0.1 MPa up to 1 MPa; then, the tilting moment is increased in steps of 50 N·mm up to 1000 N·mm.

The moment behavior was set to “rigid” so that the upper surface of the upper body could only move rigidly. This was set to remove any ambiguity in the definition of the tilting angle. The tilting angle was computed through the upper surface displacement. Approximately 1.5 M nodes were employed.

2.3. Rotor Dynamic Analysis

Simulations were made using XLHydroDyn™, a commercial software developed by Rotating Machinery Analysis, Inc. The software computes bearing static and dynamic characteristics and can deal with flexural pivots. The properties of the simulated bearing, schematically shown in Figure 4, are reported in

Table 2. Properties of the simulated bearing.

Property	Value	Unit
Bearing Diameter	90	mm
Pad Machined Diametral Clearance	0.4	mm
Bearing Outer Diameter	130	mm
Pad Material	Steel
Journal Material	Steel
Shell Material	Steel
Lubricant	ISO VG 32
Oil Supply Temperature	50	°C
Number of pads	5
Load type	Load Between Pads
Pad Arc	52°
Pad Offset	0.6
Preload	0.55
Pad axial length	45	mm
Housing radius, Rh	65
Pivot back radius, Rb	58 or 64	mm

.

Two configurations with different pivot curvature radii were compared. The pivot curvature radius and the housing radius were used to calculate $K_{rot}$ as in Equation (8). Since $K_{rot}$ depends on the pad normal load $F_{\xi }$, the simulation was repeated iteratively until convergence on the value of $K_{rot}$ was reached for all pads. The acceptable relative error was set to ${10}^{-5}$ and quite few iterations were necessary. The convergence criterion on $K_{rot}$ was not present in the software and was implemented by the authors through an outer loop. The software has its own convergence criterion consisting in a relative error criterion on journal eccentricity that was set to ${10}^{-4}$. The generalized Reynolds equation is solved by discretizing the pad; 50 and 11 elements per pad were employed, respectively, in the circumferential and axial direction for pressure and temperature calculation. Pad and pivot mechanical and thermal deformations were not considered to isolate the contribution of the rolling motion. Turbulence coefficients were not considered as well.

Once the static equilibrium was determined, the dynamic coefficients were calculated by the software by perturbing the static equilibrium condition. When the operating point is perturbed, $K_{rot}$ is not updated, but its value, dependent on the static pad load, affects the perturbed equilibrium quantities because it is included in the software calculations as for the flexural pivot.

3. Results

3.1. Finite Element Analysis

The tilting behavior as a function of the applied moment $M$ is shown in Figure 5. The corresponding analytical relationship is: $M=K_{rot}{\alpha }_t$, where the equivalent rotational stiffness $K_{rot}$ is given by Equation (8).

For the explored cases, the analytical formula coincides with the results of the finite element analysis. Therefore, the bodies deformability has a negligible effect. The most critical condition is that with the highest load $F_{\xi }$, which is the one shown in Figure 5. The small difference between the loading and the unloading phase is attributed to the numerical error. In a different set of simulations, the friction coefficient was changed, but it had negligible influence on the results. Figure 5b follows the example loading history shown in Table 1.

In Figure 6, the equivalent von Mises stress is shown at two different time steps, one before the tilting moment is applied and the other when its maximum is reached. A shift of the contact area is visible between Figure 6a,b but, apart from the shift, the stress distribution seems only slightly affected by the moment.

The software also gives as output the contact status: sticking, sliding or no contact. Sticking prevails where contact occurs while the sliding area is negligible in agreement with the reversibility of the loading and unloading phases observed in Figure 5.

3.2. Rotor Dynamic Analysis

The results of the calculations of the bearing characteristics using XLHydroDyn™ are reported for three pad configurations with $R_h=65$ mm. K0 is the reference configuration with no equivalent rotational stiffness; KA is the configuration with $R_b=58$ mm; KB is the configuration with $R_b=64$ mm. In Figure 7, the non-dimensional eccentricity, defined as the journal eccentricity over the bearing radial clearance, is reported.

Figure 7 shows the journal static eccentricity for increasing applied load, with similar incremental eccentricity variations for all the examined configurations. However, while little difference between configurations K0 and KA is observed, a significant difference in the attitude angle can be seen between configurations KA and KB.

The results of the calculations of the bearing dynamic coefficients are shown in Figure 8, Figure 9, Figure 10 and Figure 11.

Figure 8 shows the direct dynamic coefficients as a function of rotating speed for a given load. Their trend with speed is similar for all configurations. They increase with $K_{rot}$ with a maximum increase at lower speeds of KB with respect to K0 of about 30% for direct stiffness and 10–20% for direct damping.

Figure 9 shows the cross-coupled dynamic coefficients as a function of rotating speed for a given load. A much higher sensitivity to the choice of pivot back radius compared to the direct coefficients can be observed, as well as order of magnitude increases and changes of sign, with respect to the reference case.

Figure 10 shows the direct coefficients as a function of specific load for a given rotating speed. Their trend with load is similar for all configurations. They increase with $K_{rot}$ with a maximum increase of KB with respect to K0 of less than 30% for direct stiffness at higher loads, and of about 10–15% for direct damping at intermediate loads.

Figure 11 shows the cross-coupled dynamic coefficients as a function of specific load for a given rotating speed. Their trend with load is similar for all configurations. A much higher sensitivity to the choice of pivot back radius compared to the direct coefficients can be observed as well as order of magnitude increases and changes of sign with respect to the reference case.

The destabilizing energy [20,22] acting on the rotor is proportional to $K_{xy}-K_{yx}$. In the explored range, there was a notable increase in the destabilizing energy with larger rotational stiffness. In configuration K0, the destabilizing energy was negative; in configuration KA, the destabilizing energy was positive; from configuration KA to KB, the increase was approximately fivefold.

4. Discussion

The pad tilting and corresponding pivot rolling motion causes a shift of the pivot-housing contact area. If the pivot-housing degree of conformity is small, such an effect can be neglected as it is done in most bearing computational tools and in the literature. However, where stress and durability concerns lead to higher degrees of conformity, it should be considered because it can have consequences on the bearing performance. The contact shift means a new pivot angular position within the housing and a new offset of the pressure resultant on the pad. Moreover, each pad has a different configuration that depends on the load acting on it, and the bearing loses its symmetry with respect to the load line direction with the effects on bearing characteristics depicted in Dang et al. [23].

As described in Section 2 the effect of pivot shift can be obtained through a fictitious equivalent rotational stiffness obtaining the actual offset of the pad pressure resultant due to rolling. The difference with flexural pivot bearing is that, in the proposed RB bearing model, the rotational stiffness is load dependent and, consequently, each pad has its own. This last feature can be implemented in XLHydroDyn™.

The value of $K_{rot}$, for a given housing radius, depends on the choice of pad rocker back radius. Further, a lower clearance between the pivot and the housing increases the absolute value of the cross-coupled coefficients, which is detrimental to the bearing stability. This can be especially critical for the stability of flexible, high-speed rotors. On the other hand, a greater conformity helps to reduce contact stresses, fatigue, and wear. Therefore, an optimal trade-off exists between rotor dynamic stability and pivot durability.

As shown in the previous section, including equivalent rotational stiffnesses produces non-negligible differences in the static and dynamic bearing performance for higher degrees of pivot-housing conformity. Compared to cross-coupled coefficients, direct coefficients are less influenced by the pivot rotational stiffness and their values are relatively close to those of the reference configuration in the investigated range. Cross-coupled coefficients show instead a much higher sensitivity to $K_{rot}$. Their increase with it can be easily explained with the fact that the pad with finite rotational stiffness represents an intermediate situation between the pad perfectly free to rotate ($K_{rot}=0$) and the fixed pad ($K_{rot}=\infty$).

5. Conclusions

In this paper a novel way to include the pivot rolling motion in RB TPJB modeling was proposed. An equivalent pivot rotational stiffness was formulated as a function of the pad load and radii of the contact pair. Such stiffness increases with the pad load, the radii, and the degree of conformity. The results of the analytical rigid body approach were confirmed by FE simulations due to the negligible local deformation compared to the rigid body motion.

The TPJB static and dynamic characteristics were evaluated with a commercial software including the equivalent rotational stiffness $K_{rot}$ and non-negligible differences were found with the results obtained with no such stiffness, particularly for a higher degree of pivot-housing conformity. In particular,

• journal eccentricity and attitude angle were affected by $K_{rot}$;
• cross-coupled coefficients, as well as the destabilizing energy, showed high sensitivity to $K_{rot}$;
• direct coefficients were less influenced by $K_{rot}$, and their values were relatively close to those of the reference configuration in the investigated range.

Therefore, the results indicate that the designer should select a trade-off value of pad rocker back radius, considering both rotor dynamic stability and pivot durability.

The proposed approach should contribute to fill the gaps that are still found between theoretical and experimental results in RB TPJB characterization. Future work will be devoted to such a comparison. Furthermore, a similar approach will be extended to B&S TPJB; although in such a case, pivot and seat surfaces are much more conformal and sliding friction might prevail.