**1. Introduction**

Tilting pad journal bearings (TPJB) are widely used in turbomachinery because of their stability characteristics at high speeds. For design purposes it is essential to know the coe fficients that characterize their dynamic behavior. These coe fficients are usually obtained with the assumption of small amplitude motions with respect to the static equilibrium position, which allows linearization according to the well-known Lund model [1] and many di fferent experimental procedures have been proposed to identify them [2,3].

For plain journal bearings, Qiu [4] has experimentally verified that the error in the estimation of dynamic coe fficients not considering nonlinearity is negligible when the perturbation displacement amplitude is in the order of magnitude of 0.02 of radial clearance (c) and suggests displacement amplitudes not exceeding 0.05c to contain the error within 2.5%.

However, in industrial practice, the amplitude of the shaft vibrating motion can reach about 0.1c in operating conditions, violating the hypothesis of small perturbations. Moreover, in experimental tests for the identification of linearized dynamic coe fficients, the choice of imposing small perturbations conflicts with the need for measuring small displacements with reduced errors.

From an analytical/numerical point of view, several authors have tackled the problem of nonlinearity, especially in plain journal bearings. The approach consists of the direct integration of the Reynolds equation [5] and the description of the oil-film forces with a larger number of dynamic

coefficients with respect to the classical linearized eight ones, by retaining more terms of the Taylor series expansion [6,7]. The proposed nonlinear models proved to represent well the nonlinear effects within the considered journal bearings. A parametric analysis to study the sensitivity of non-linear forces in journal bearings of different types is reported in Reference [8]. Results are presented for a two-lobe elliptical bearing showing the effects of several parameters on size, shape, and orientation of the elliptical orbits produced by synchronous vibrations.

As far as the TPJB is concerned, the literature offers less contributions compared to plain journal bearings. From the analytical/numerical point of view, nonlinear effects were studied by direct integration of the Reynolds equation in References [9–14], showing how the rotor orbit increasingly deviates from the theoretical elliptical orbit of the linear case as the ratio of dynamic and static load amplitudes increases. The effect of different unbalanced loads on a four-pad TPJB in the load-between-pad (LBP) configuration was investigated considering thermal and deformation effects in Reference [9]. Slightly triangular journal orbits were found with a dynamic load about 70% of the static one while nearly quadrilateral orbits were found with a dynamic load greater than the static one. The position and the shape of the orbits appeared to be influenced by thermal effects on both viscosity and pad deformations, while the influence of the elastic deformations appeared to be smaller. Thermo-elasto-hydrodynamic theory should be used to predict more accurate results as shown also in Reference [10]. The effect of the liner compliance on the nonlinear dynamic behavior of the same TPJB used in References [9,10] supporting vertical and horizontal rotors was investigated in Reference [11]. Almost square orbits were found when the unbalanced load was applied to the vertical rotor (with zero static load) whilst three-lobed orbits were found with a dynamic load 50% of the static one applied to the horizontal rotor. Particularly for the horizontal configuration, shape and size of the orbits are influenced by several factors such as liner thickness and material, preload, pivot offset, and radial clearance. Three-lobed orbits were also found for a TPJB with two loaded pads in LBP configuration [12,13] and five-lobed orbits for a five-pad bearing [14] also with an approximated analytical solution using Fourier series developments. Other researchers have proposed models with an increased number of coefficients identifying them from the dynamic response obtained by direct integration [15].

From the experimental point of view different TPJBs were tested in Reference [16] with two different test rigs detecting typical non-linear behaviors. Symptoms of a non-linear behavior of the pad journal bearings of two industrial machines were found in Reference [17]. The dynamic coefficients of three nonlinear models were identified with a different number of coefficients (28, 24, 36) on a five-pad TPJB bearing, with a 100 mm diameter [18]. The results of the three nonlinear models and the linear one are in good agreemen<sup>t</sup> regarding the identified linear terms. Moreover, the identified stiffness and damping coefficients appear to decrease with increasing dynamic force amplitude with a reduction of up to 60% for direct stiffness in the case of a dynamic force increase from 5% to 30% of the static load.

Following a preliminary investigation on possible nonlinear effects in a large size TPJB [19], the present paper focuses on the nonlinear response of tilting pad journal bearings to harmonic excitation observed during the experimental procedure for the identification of the linear dynamic coefficients. In particular it uses the tests to ascertain the linear and nonlinear range of displacements prior to the dynamic characterization campaign. Nonlinear effects related to the dynamic/static load ratio should in fact be considered in the experimental identification procedure and accounted for or avoided. There are very few experimental results published on this topic and in particular on large size TPJB. De Falco et al. [16] and Chatterton et al. [18] have dealt with the problem with smaller bearings and different test rig configurations. Moreover, as far as the authors are aware, the application of an asynchronous rotating force instead of harmonic forces with constant directions, with different dynamic/static load ratios, is new. The tests are performed on a unique experimental apparatus realized for large size journal bearings operating at high peripheral speeds and static loads, with single tone or multi-tone dynamic loads. The design criteria are described in Reference [20]. The main characteristics of the realized bench and the first experimental results obtained during the commissioning with a

four-pad TPJB are reported in [21,22]. Results obtained under stationary and slow variable conditions are particularly reported in Reference [21]. The di fferent systems related to the test rig are shown in more detail in Reference [22] together with the procedures adopted for both static and dynamic tests.

#### **2. Materials and Methods**

Figure 1 shows a picture and a schematic drawing of the test bench. The rotor, supported by rolling bearings, is driven by an electric motor connected to a gearbox with a gear ratio of six. A torque meter measures the driving torque. The test bearing housing is floating, and the static load and the dynamic ones are applied to it by three hydraulic actuators. The static load acts upwards in the vertical direction while the dynamic loads are applied in mutual orthogonal directions, at 45◦ with respect to the vertical one (Figure 2). The dynamic actuators can work one at a time or simultaneously. In the second case, if they operate with equal amplitude in phase, they can produce a vertical force (y direction), in antiphase a horizontal force (x direction), and in quadrature a rotating force. Three pitch stabilizers, placed at 120◦ around the bearing, provide the bearing housing with axial constraints that can be adjusted to align the bearing with respect to the rotor. Table 1 summarizes the main characteristics of the test rig.

**Figure 1.** Photograph and schematic drawing of the test bench.

**Figure 2.** Drawing of the test cell.


**Table 1.** Main characteristics of the experimental apparatus.

Load cells and instrumented stingers, capable of measuring dynamic loads, measure all significant forces acting on the bearing housing while high-resolution proximity sensors measure the relative displacements of the bearing housing and the rotor in the directions of the dynamic actuators shown in Figure 2 (U and V directions shown in Figure 3). Eight sensors are employed, placed on two parallel planes perpendicular to the bearing axis. Four accelerometers measure the acceleration of the stator at the mid-section in the direction of the dynamic actuators.

Three different oil supply systems are used for the TPJB, the actuators, and the multiplication gearbox.

Tests are managed by a very complex control and data acquisition system. Up to 30 high-frequency signals (forces and torque, displacements, rotational speed, accelerations) can be acquired and sampled at up to 100 kHz, while up to 60 low-frequency (quasi-static) signals (temperatures, pressures, and flow-rates in the main and auxiliary lubrication systems and motor electric current) can be acquired and sampled at 1 Hz. A high sampling rate means an accurate description of signals but also a large amount of stored data; however, since the identification tests are rather short, data storage is not a

problem. High-frequency signals are also averaged every second and their mean values are stored together with the low-frequency ones.

**Figure 3.** Test cell displacement and acceleration sensor positions.

The test articles consisted of a four-pad (without offset) and a five-pad (with offset) TPJBs provided by an industrial partner. The bearings had a 280 mm inner diameter and were tested in the load between pads configuration. No additional data will be given about the bearings and their characteristics due to a non-disclosure agreemen<sup>t</sup> with the company.

Two main types of test are usually carried out with the test apparatus:


In the first test a rotating force vector applied to the bearing housing is generated by sinusoidal signals with 90◦ phase difference. The oil is not supplied, and the shaft is not rotating. The force must be increased until the polygon shaped orbit does not change.

In the second test the forces applied during excitation can contain one ("single tone" test) or more ("multitone" test) frequency components. As dynamic coefficients can vary with frequency, excitation tones are chosen below and above the rotational frequency avoiding harmonics and test bench resonances. The required synchronous values are then obtained by interpolation to avoid imbalance disturbance. Multitone tests with *n* frequency components have been proven to provide the same results of *n* single tone tests but in the time of a single test. For the identification of the dynamic linear coefficients, two tests with linearly independent excitations are required for each excitation frequency. The two tests consist in two distinct excitations, vertical (subscript *y*) and horizontal (subscript *x*), respectively, obtained using the dynamic actuators in in-phase (subscript *f)* and anti-phase (subscript *a)* operation modes. The bearing impedance matrix **H**, expressed in terms of stiffness (*k*) and damping (*c*) coefficients:

$$
\begin{bmatrix} H\_{xx} & H\_{xy} \\ H\_{yx} & H\_{yy} \end{bmatrix} = \begin{bmatrix} k\_{xx} & k\_{xy} \\ k\_{yx} & k\_{yy} \end{bmatrix} + i\alpha \begin{bmatrix} c\_{xx} & c\_{xy} \\ c\_{yx} & c\_{yy} \end{bmatrix} \tag{1}
$$

where ω is the excitation frequency, is determined in the frequency domain, using the Fast Fourier Transform (FFT), by multiplying the [2 × 2] bearing force complex matrix by the corresponding inverse displacement complex matrix:

$$
\begin{bmatrix} H\_{xx} & H\_{xy} \\ H\_{yx} & H\_{yy} \end{bmatrix} = \begin{bmatrix} F\_{bxf} & F\_{bxa} \\ F\_{byf} & F\_{bya} \end{bmatrix} \begin{bmatrix} D\_{xf} & D\_{xa} \\ D\_{yf} & D\_{ya} \end{bmatrix}^{-1} \tag{2}
$$

where *Fb* indicates the amplitude of the force transform and *D* indicates the amplitude of the displacement transform. Synchronous coefficients are obtained at the shaft rotational frequency. The stiffness dynamic coefficients have shown to be practically independent of frequency for a wide range starting from quasi-static conditions. They are the basis of the simplified models that will be presented in Section 3.2.

In addition to such tests, in order to ascertain the linear and nonlinear range of displacements prior to the dynamic characterization campaign, and also to have a reference stiffness estimate, another procedure has been developed. The stator is subjected to a slowly rotating force vector generated by equal amplitude sinusoidal forces with a 90◦ phase angle difference, applied by the two dynamic actuators. The force low rotational speed allows to consider the damping coefficients negligible. This work focuses on this latter test procedure that makes it possible to evidence nonlinear effects by detecting the loss of the typical elliptical orbit related to linear bearing film stiffness. In the tests reported in this work the shaft rotational speed was set at 1000 rpm while the frequency of the rotating load was set at 1 round every 100 s (i.e., 0.01Hz). Different static load levels were applied in load-between-pad configuration combined with different dynamic load levels ranging 3% to 36% of the static load. Forces and displacements were recorded at a rate of one sample/s.
