**3. Results**

## *3.1. Experimental Results*

Figure 4 shows the plots of the applied horizontal and vertical dynamic load, obtained in four different tests for the four-pad TPJB for two levels of static load, with the higher (L2) about double the lower one (L1).

**Figure 4.** Horizontal and vertical dynamic load plot for the four-pad tilting pad journal bearings (TPJB). Red and blue lines for the lower static load L1 and increasing dynamic/static load ratio, and green and yellow for the higher static load L2 and increasing dynamic/static load ratio.

The dynamic load plot is circular because it is a rotating vector with a constant amplitude controlled by means of the dynamic actuators with the feedback of the load sensor system. The dynamic load is indicated in the label as percentage of the static load. About five rotating load cycles are shown for each test. The corresponding stator orbits are shown in Figure 5. The zero values correspond to the central position of the bearing. The displacement fluctuation, related to the centrifugal force due to the shaft rotation, is noticeable particularly for the low static load conditions. It is evident that the orbit position is related to the static load level while its shape is greatly influenced by the dynamic/static load ratio, becoming more elliptical and closer to the linear orbits (Figure 5 case L2-3%) as the ratio

decreases. The shapes are similar to the ones theoretically found by direct integration of the Reynolds equation in References [9,11,12] for four-pad TPJBs.

**Figure 5.** Orbits generated by the rotating force for the four-pad TPJB. Red and blue lines for the lower static load L1 and increasing dynamic/static load ratio, and green and yellow for the higher static load L2 and increasing dynamic/static load ratio.

#### *3.2. Simple Analytical Models*

In order to give a possible explanation for the obtained orbits, the influence of different factors was evaluated by simulation with simple bearing models represented by linear and quadratic dynamic coefficients.

Analytical models with increasing complexity were devised assuming hydrodynamic forces having the following non-linear relations with displacements, obtained by adding quadratic terms to the commonly used linear relations, neglecting damping due to the low excitation frequency of the rotating load:

$$f\_{\mathbf{x}} = k\_{\mathbf{x}\mathbf{x}}d\_{\mathbf{x}} + k\_{\mathbf{x}y}d\_{\mathbf{y}} + k\_{\mathbf{x}\mathbf{x}^2}d\_{\mathbf{x}}^2 + k\_{\mathbf{x}y^2}d\_{\mathbf{y}}^2\tag{3}$$

$$f\_y = k\_{yx}d\_x + k\_{yy}d\_y + k\_{yx^2}d\_x^2 + k\_{yy^2}d\_y^2\tag{4}$$

The main objective of this work was not the identification of nonlinear coefficients but to preliminarily investigate the possibility of replicating the experimental nonlinear orbits with simple nonlinear models. Thus, the linear coefficients of Equations (3) and (4) were set equal to those obtained by the identification tests described in the previous section while the quadratic coefficients were determined by a trial and error procedure to ge<sup>t</sup> a good fit of the experimental orbits with analytical ones based on the proposed models.

The simplest analytical linear model takes into account only the direct stiffness coefficients in the linear relation with displacements:

$$d\_x = \frac{f\_x}{k\_{xx}},\ d\_y = \frac{f\_y}{k\_{yy}}.\tag{5}$$

The linear stiffness model takes into account both direct and cross-coupled stiffness coefficients in the linear relation with displacements:

$$d\_x = \frac{k\_{yy}f\_x - k\_{xy}f\_y}{k\_{xx}k\_{yy} - k\_{xy}k\_{yx}},\ d\_y = \frac{k\_{xx}f\_y - k\_{yx}f\_x}{k\_{xx}k\_{yy} - k\_{xy}k\_{yx}}.\tag{6}$$

*Machines* **2019**, *7*, 43

The simplest nonlinear model takes into account only direct linear and quadratic stiffness coefficients:

$$d\_x = \frac{-k\_{\rm xx} + \sqrt{k\_{\rm xx}^2 + 4k\_{\rm xx^2}f\_x}}{2k\_{\rm xx^2}}, \; d\_y = \frac{-k\_{yy} + \sqrt{k\_{yy}^2 + 4k\_{yy^2}f\_y}}{2k\_{yy^2}}\tag{7}$$

A second non-linear model takes into account linear direct and cross-coupled stiffness coefficients and quadratic direct stiffness coefficients. First of all, the force and displacement components of Equations (1) and (2) were expressed in polar coordinates:

$$f\_x = F\cos(\theta + \varphi), \ f\_y = F\sin(\theta + \varphi), \tag{8}$$

$$d\_x = R\cos(\theta), \ d\_y = R\sin(\theta). \tag{9}$$

After squaring the terms of both Equations (3) and (4), they were summed obtaining an equation of the fourth degree of *R* that can be solved analytically as a function of θ:

$$\begin{aligned} R^4 \Big[ k\_{\rm xx}^2 \cos^4(\theta) + k\_{yy}^2 \sin^4(\theta) \Big] \\ + R^3 \Big[ 2k\_{\rm xx} \cos^2(\theta) \Big( k\_{\rm xx} \cos(\theta) + k\_{\rm xy} \sin(\theta) \Big) + 2k\_{yy} \sin^2(\theta) \Big( k\_{\rm yx} \cos(\theta) + k\_{\rm yy} \sin(\theta) \Big) \Big] \\ + R^2 \Big[ \left( k\_{\rm xx} \cos(\theta) + k\_{\rm xy} \sin(\theta) \right)^2 + \left( k\_{\rm yx} \cos(\theta) + k\_{\rm yy} \sin(\theta) \right)^2 \Big] - F^2 = 0. \end{aligned} \tag{10}$$

A rotating load was imposed as a sum of two sinusoidal functions with the same amplitude and a 90◦ phase shift, and the corresponding displacements were calculated. The case L1-36% (low static load, high load ratio) was chosen as reference case due to its extreme dynamic load conditions. For the sake of comparison, the same experimentally identified stiffness coefficients for the four-pad TPJB were used in all the analytical models while, for the quadratic coefficients, values tuned to fit the experimental orbits were adopted for the *x* and *y* directions for all nonlinear models. The value zero in the diagrams corresponds to the static equilibrium position. Further non-linear models were obtained by considering the dependence of stiffness coefficients from the actual vertical load, sum of the static load, and the vertical component of the rotating load, according to a fit of experimental results obtained for different static vertical loads and small dynamic ones. In such a case, a phase angle of a few degrees between the displacement vector and the force vector was included, affecting the model related to Equation (10). The orbits obtained with the different analytical models are presented in Figure 6, with dashed blue lines representing those of the constant stiffness models and red lines representing those of the load dependent stiffness models.

Comparing the orbits of Figure 6, one can observe that if only direct stiffness coefficients are included the orbits are necessarily circular due to the same bearing stiffness along the orthogonal directions. The inclusion of the linear cross-coupled stiffness coefficients in the model modifies the orbit shape in a tilted slightly elliptical one. Including second order direct stiffness coefficients in the model yields an orbit with a shape more similar to the experimental one with three lobes. It seems that including cross-coupled stiffness coefficients in this latter model makes it more adaptable but does not bring significant improvements unless a specific optimization procedure is performed. The inclusion of coefficient load dependence produces a vertical shift of all the orbits, an increase of the ellipticity for the simpler models, and more pronounced lobes for the nonlinear ones.

#### *3.3. Comparison of Analytical and Experimental Results*

In order to evaluate the capability of analytical models to simulate the actual bearing behavior, experimental loads and corresponding linear stiffness coefficients were implemented in the models, tuning the quadratic coefficients, to obtain orbits to be compared with the experimental ones. Figure 7 shows a comparison of orbits calculated with different load dependent stiffness models and experimental ones for three different load ratios.

**Figure 6.** Calculated orbits for the four-pad TPJB for four different models: (**a**) linear constant and load dependent direct stiffness coefficients only; (**b**) linear constant and load dependent direct and cross-coupled stiffness coefficients; (**c**) nonlinear constant and load dependent direct stiffness coefficients only; (**d**) nonlinear constant and load dependent direct stiffness coefficients with linear constant and load dependent cross-coupled stiffness coefficients. Case L1-36%.

The more complex nonlinear model is omitted at this point because its optimization deserves a further in-depth analysis and thus it is left to future development. It is evident especially at high load ratios that the nonlinear model overcomes the limitations of the linear ones and succeeds in replicating the experimental orbit shape. At lower load ratios the differences are less marked as the orbits tend to a more elliptical shape.

For very low load ratios, that is for reduced values of the dynamic force the orbit tends to become more circular and all models, linear and nonlinear, give similar results, as shown in Figure 7c for the case L2-3%, thus indicating a linear behavior. Note that in this case, since both displacement and force experimental values are very small, the measurement relative error increases and it is also quite difficult to control the rotating force vector at such load ratios. Those are the main reasons for the discrepancy of experimental and analytical orbits shown in Figure 7c.

**Figure 7.** Calculated and experimental orbits of the four-pad TPJB for three different models with load dependent direct stiffness coefficients and two loads (L2>L1): (**a**) L1-36%, (**b**) L2-17%, (**c**) L2-3%.
