**4. Discussion**

The results obtained for the four-pad TPJB that has equal direct stiffness coefficients and equal cross-coupled ones indicate that only models including second order direct stiffness coefficients can replicate the characteristic shape of the experimental orbit.

The non-elliptical shape of the orbits is reflected in the appearance of multiple frequency peaks in the Fast Fourier Transform (FFT) of the displacement signal as shown in Figure 8a (case L1-36%). As the excitation is a single tone force, the FFT content of the displacement with multiple frequencies indicates non-linear or coupled phenomena. Figure 8b shows the results of the same analysis performed for the L2-3% case whose orbit is surely more elliptical (Figure 7c). Note that the shaft rotational frequency (16.67 Hz) is not present in these diagrams focused on the low frequency zone. Figure 9 shows the experimental orbit of the case L1-36% compared with the orbits obtained filtering the results at the force rotational frequency (1X) and twice (2X) and three times (3X) the fundamental frequency. While the 1X harmonic component of displacement corresponds indeed to the linear orbit for low load ratios (case L2-3%, Figure 7c), the 1X filtered ellipse observed for a higher load ratio (case L1-36%) in Figure 9 appears with a tilt that is not justified with a linear model considering the negligible linear damping and cross-coupled stiffness coefficients, thus indicating the apparent effects of nonlinearity. Note that all data recorded during the rotation of the force vector are plotted in Figure 9b (four cycles in this case) instead of the averaged values as in Figure 7. This provides better evidence of the signal fluctuations due the difficulties in controlling and measuring low values of forces and displacements as mentioned above.

**Figure 8.** Fast Fourier Transform (FFT) of the X displacement L1-36% (**a**) and L2-3% (**b**) cases.

**Figure 9.** Experimental and filtered orbits for the cases L1-36% (**a**) and L2-3% (**b**).

At the end of the tests some geometrical differences among the pads were also found. It is worth mentioning that geometrical errors can also influence the results, as reported for example in Reference [18].

In order to confirm the previous findings, the case of another TPJB was analyzed. It was quite different from the previous one as it had five offset pads of the same size. It underwent the same tests of the four-pad TPJB in the LBP configuration. Unlike the four-pad TPJB this bearing has quite different direct stiffness coefficients in the *x* and *y* directions (*kyy* greater than *kxx*), and that has obviously a remarkable impact on the orbit shapes. Figure 10 presents some experimental orbits for the five-pad TPJB for different load ratios and two different static load levels. The load L3 is about 60% of L2, so a little greater than the load L1 used for the four-pad TPJB. Figure 11 presents calculated and experimental orbits for three different models with load dependent direct stiffness coefficients. The difference in direct stiffness coefficients produces the ellipticity even of the orbits of the simpler models but the coefficient load dependence causes a distortion of the ellipse, though there is still a difference in its orientation compared to the experimental one. Again, when the load ratio is small, as in cases L3-5% and L2-3% of Figure 10, the orbit is elliptical and, as shown in Figure 11b, quite close to the classical linear model predicted one. Moreover, the model with quadratic coefficients produces an orbit more similar to the experimental one, particularly noticeable for larger load ratios. Nonetheless there is still

margin for an optimized tuning of the estimated quadratic coefficients. Better results can be expected from a best fit optimization involving all dynamic coefficients, including the linear ones. The set of linear coefficients obtained by linear identification could be the starting solution of the nonlinear identification procedure that will be the object of future work.

**Figure 10.** Orbits of the five-pad TPJB generated by the rotating force. Cyan, red, and blue lines for lower static load L3 and increasing dynamic/static load ratio, and green and yellow for higher static load L2 and increasing dynamic/static load ratio.

**Figure 11.** Calculated and experimental orbits of the five-pad TPJB for three different models with load dependent stiffness coefficients and two loads (L3<L2): (**a**) L3-26%, (**b**) L2-3%.

The peculiar three lobe orbit shape, predicted by simulations [9,11,12] for horizontal rotors and found experimentally in the present work, has been ascribed to the number of pads involved in the bearing reaction to the load. With four pads, with a high static load between pads and a rotating load not overcoming the static one, the bearing behavior is similar to that with only two pads, like the one described in Reference [12]. The same behavior occurs with five pads with load between the pads, as can be deduced observing the results reported in this section. 3X components in the journal orbit, in addition to 1X and 2X, have been also reported in Reference [18] for a five-pad TPJB on a floating shaft configuration test rig. Unfortunately, comparison can be only qualitative because experimental conditions are quite different.
