## 1 Introduction  
A tilting pad journal bearing (TPJB) is known as a structure in which the lower surface of the pad executes a tilt motion on the upper surface of the pivot, with respect to the rotation of the shaft. This tilt motion suppresses the cross-coupled stiffness of the bearing and improves stability [1]. As a result, TPJBs have been widely selected in many modern turbomachinery. The typical structure of a pivot can be categorized into the rocker back (cylindrical) type and the ball-socket (spherical) type. Various researchers have analyzed the differences in the performance of the two structures. [PERSON] et al. [2, 3] conducted a series of experiments to measure the eccentricity ratios, dynamic stiffness, and damping force coefficients of the two types of pivots in the TPJB system. They found variations in the attitude angles and cross-coupled stiffness between the two types of TPJB and concluded that the higher pad-pivot friction in the ball and socket pivot was the primary contributor to the differences. [PERSON] and [PERSON] [4, 5] measured the differences in the pad temperature profiles and the power loss in a five-shoe TPJB of the two types of pivots. [PERSON] et al. [6] developed a numerical model to investigate the static and dynamic characteristics of TPJBs by considering the pivot stiffness of the two types of pivots. The primary reason for this difference is the difference between the line contact of the cylindrical type and point contact of the spherical type. This implies that the point contact may induce a larger friction force between the pad and pivot, thereby affecting the dynamic states. [PERSON] and [PERSON] [7] introduced an equivalent spring model composed of a fluid lubricant film and a pivot connected in series to reflect the stiffness of TPJB's ball socket type pivot. The TPJB's stiffness and damping coefficients, which were modified using the equivalent spring model, were in good agreement with the test results, but showed a very large difference in the journal eccentricity. [PERSON] and [PERSON] [8] developed a three-dimensional high fidelity finite element model for the spherical pivot type TPJB, which can simulate the effect of pivot stiffness. They found that the rigid pivot yields the highest temperature in the lubricant, pad, and shaft, providing the increased thermal expansion and the decreased minimum film thickness ratio. [PERSON] [9] tried to experimentally verify the influence of pivot friction on the shaft tracking motion of pad. By analyzing the frequency components of the pad and shaft, the lack of sufficient pad tracking motions was observed in the heavier loading conditions. [PERSON] and [PERSON] [10] introduced an analytical friction model for the spherical pivot type of TPJB and calculated the friction force and moment between the pad and pivot. They presented numerical results which indicated that the pad-pivot friction can induce differences in the pad pressure and attitude angles of the journal in the TPJB system. He [11] conducted a numerical study by considering a conformal contact condition between the pad and spherical pivot surfaces. The results showed that the sinusoidal motions of the pads changed to triangular wave motions with the friction condition. In particular, [PERSON] and [PERSON] [12] performed a broad range of nonlinear studies to identify the instability onsets in autonomous and nonautonomous rotor systems, including the effects of pad-pivot friction. They analyzed nonlinear behaviors such as sub-synchronous whirls, quasi-periodic responses, Hopf bifurcation, and Neimark-Sacker bifurcations, induced by pad-pivot friction. Nevertheless, the advantage of TPJB is that it has various design parameters that are available for variation and performance, the result was limited to a single bearing geometry and did not include the effects of the bearing design parameters to pad-pivot friction to nonlinear rotordynamic bifurcation.