*2.1. Mathematical Model of the Rotor-Bearing System*

The FE method is used to obtain the mathematical model of the multiple DOF rotorbearing system. The shaft is modelled with a finite element type beam with four DOF per node, two lateral displacements and two rotations (beam deflections), as illustrated in Figure 1.

**Figure 1.** Beam-like finite element for the modelling of the rotor-bearing shaft.

The nodal displacement vector is defined as

$$\{\delta\} = \{u\_1, w\_1, \psi\_1, \theta\_1, u\_2, w\_2, \psi\_2, \theta\_2\}^T \tag{1}$$

where superscript *T* denotes the transposed vector.

Displacement and rotations corresponding to the movement along X and Z directions are

$$\begin{aligned} \{\delta\_{\boldsymbol{w}}\} &= \{\boldsymbol{u}\_{1}, \boldsymbol{\psi}\_{1}, \boldsymbol{u}\_{2}, \boldsymbol{\psi}\_{2}\}^{\mathrm{T}}\\ \{\delta\_{\boldsymbol{w}}\} &= \{\boldsymbol{w}\_{1}, \theta\_{1}, \boldsymbol{w}\_{2}, \theta\_{2}\}^{\mathrm{T}} \end{aligned} \tag{2}$$

The mathematical model of the multiple DOF rotor-bearing system with excitation by unbalanced mass is given by [2]

$$[M]\left\{\ddot{\delta}\right\} + \left[\mathcal{C}\left(\dot{\phi}\right)\right]\left\{\dot{\delta}\right\} + \left[K(\ddot{\phi})\right]\{\delta\} = \dot{\phi}^2 \left\{F\_{\mathfrak{u}(1)}(\phi)\right\} + \ddot{\phi}\left\{F\_{\mathfrak{u}(2)}(\phi)\right\} \tag{3}$$

with

$$\begin{aligned} F\_{\mathfrak{u}(1)} &= m\_{\mathfrak{u}} d(\sin(\phi + \mathfrak{a}) + \cos(\phi + \mathfrak{a})) \\ F\_{\mathfrak{u}(2)} &= m\_{\mathfrak{u}} d(\sin(\phi + \mathfrak{a}) - \cos(\phi + \mathfrak{a})) \end{aligned}$$

where *mu*, *d* and *α*, are mass, eccentricity and angular position of system unbalance, respectively, .. *<sup>φ</sup>* and . *φ* are angular acceleration and velocity of the rotor-bearing system, respectively, and *<sup>φ</sup>* <sup>=</sup> . *φt*. Moreover, {*δ*} is a vector with all the nodal displacements, [*M*] is the global mass matrix of the system, - *C* . *φ* is the global damping matrix that includes gyroscopic effects as a function of the rotational velocity . *φ*[*C*2] and [*C*1] that represents the damping in the supports, - *K* .. *φ* is the global stiffness matrix constituted by [*K*1], [*K*2], which include the supports and rotor stiffness, respectively, and .. *φ*[*K*3], which is a stiffness term as a function of the rotational acceleration of the system. Finally, , *Fu*(1)(*φ*) and , *Fu*(2)(*φ*) are the components of the centrifugal force vector caused by the unbalanced mass. Shape functions for the beam type finite element and a detailed definition for matrices in Equation (3) are provided in Appendix A.

Stiffness and damping matrices provided by the bearings are obtained by determining the generalized forces that these elements exert on the rotor shaft. After applying the virtual work principle to the bearing model shown in Figure 2, forces acting on the rotor can be expressed in a matrix form as [40]

$$
\begin{Bmatrix}
\begin{Bmatrix} F\_{\dot{u}\_{i}} \\ F\_{\dot{w}\_{i}} \end{Bmatrix} = - \begin{bmatrix} k\_{xx} & k\_{xz} \\ k\_{zx} & k\_{zz} \end{bmatrix} \begin{Bmatrix} u\_{i} \\ w\_{i} \end{Bmatrix} - \begin{bmatrix} c\_{xx} & c\_{xz} \\ c\_{zx} & c\_{zz} \end{bmatrix} \begin{Bmatrix} \dot{u}\_{i} \\ \dot{w}\_{i} \end{Bmatrix} \tag{4}
$$

where *i* denotes the nodal location of the bearing inside the rotordynamic systems. Matrices from the right side of Equation (4) are stiffness and damping matrices corresponding to system supports [*K*1] and [*C*1], respectively.

**Figure 2.** Stiffness and damping parameters in bearings [40].
