2.3.1. Algebraic Identifier with Constant Operation Velocity

As pointed out above, it is necessary to have a mathematical model for the dynamic behavior of the rotor-bearing system to develop algebraic identifiers. From this model and through an algebraic manipulation of the equations, estimators for the unknown parameters are obtained.

If a constant rotational velocity of the system is considered, Equation (3) can be written as

$$\{M\}\left\{\ddot{\delta}\right\} + \left[\mathbb{C}\_1 + \Omega \mathbb{C}\_2\right] \{\dot{\delta}\} + \left[\mathbb{K}\_1 + \mathbb{K}\_2\right] \{\delta\} = m\_\mu d\Omega^2 \{\sin(\Omega \mathbf{t} + \alpha) + \cos(\Omega \mathbf{t} + \alpha)\} \tag{7}$$

Now, Equation (7) is multiplied by *t* <sup>2</sup> and, after that, the result is twice integrated with respect to time, giving

$$\int \int^{(2)} [\mathcal{K}\_1] t^2 \{\delta\} + [\mathcal{C}\_1] \left[ \int t^2 \{\delta\} - 2 \int^{(2)} t \{\delta\} \right] = \int^{(2)} m\_\hbar d\Omega^2 \{\sin(\Omega t + \alpha) + \cos(\Omega t + \alpha) \} t^2 \tag{8}$$

where 0 (2) *f*(*t*) denotes iterated integrals. Furthermore, bearing stiffness and damping terms to be identified are included in [*K*1] and [*C*1], respectively. Therefore, after the integration of the left side of Equation (8) and an algebraic treatment, the following expression can be obtained

$$\begin{aligned} &\int^{(2)} \left[ [M] \left\{ \ddot{\delta} \right\} + [\text{C}\_1 + \Omega \text{C}\_2] \left\{ \dot{\delta} \right\} + [\text{K}\_1 + \text{K}\_2] \{\delta \} \right] t^2 \\ &= \int^{(2)} \left[ 2\Omega \text{C}\_2 t - 2M - \text{K}\_2 t^2 \right] \{\delta \} + \int^{\cdot} [4M - \Omega \text{C}\_2 t] t \{\delta \} \\ &- \{M\} t^2 \{\delta\} + \int^{(2)} m\_\mu d\Omega^2 \{\sin(\Omega t + \alpha) + \cos(\Omega t + \alpha) \} t^2 \end{aligned} \tag{9}$$

Equation (9) can be separated into individual equation systems for each node where the bearings are located. These equations can be presented in the form

$$
\begin{bmatrix} k\_{xx} & k\_{xx} \\ k\_{zx} & k\_{zz} \end{bmatrix} \int \prescript{(2)}{}{t^2} \left\{ \begin{array}{c} u\_i \\ w\_i \end{array} \right\} + \begin{bmatrix} c\_{xx} & c\_{xz} \\ c\_{zx} & c\_{zz} \end{bmatrix} \left( \int \begin{array}{c} t^2 \left\{ \begin{array}{c} u\_i \\ w\_i \end{array} \right\} - 2 \int \begin{array}{c} \begin{array}{c} u\_i \\ w\_i \end{array} \right\} \right) = \left\{ \begin{array}{c} b\_{xi} \\ b\_{wi} \end{array} \right\} \tag{10}
$$

To solve Equation (10) an equal number of equations and unknowns is needed. For this, Equation (10) is successively integrated three times in order to obtain the missing equations, which are written as

$$
\int \begin{bmatrix} k\_{xx} & k\_{xz} \\ k\_{zx} & k\_{zz} \end{bmatrix} \int^{(3)}\_{l} t^{2} \begin{Bmatrix} u\_{i} \\ w\_{i} \end{Bmatrix} + \begin{bmatrix} c\_{xx} & c\_{xz} \\ c\_{zx} & c\_{zz} \end{bmatrix} \left( \int^{(2)}\_{l} t^{2} \begin{Bmatrix} u\_{i} \\ w\_{i} \end{Bmatrix} - 2 \int^{(3)}\_{l} t \begin{Bmatrix} u\_{i} \\ w\_{i} \end{Bmatrix} \right) = \int \begin{Bmatrix} b\_{ii} \\ b\_{ii} \end{Bmatrix} \tag{11}
$$

$$
\begin{bmatrix} k\_{\text{xx}} & k\_{\text{xz}} \\ k\_{\text{zx}} & k\_{\text{zz}} \end{bmatrix} \int^{(4)} \boldsymbol{t}^2 \left\{ \begin{array}{c} \boldsymbol{w}\_{\text{i}} \\ \boldsymbol{w}\_{\text{i}} \end{array} \right\} + \begin{bmatrix} \boldsymbol{c}\_{\text{xx}} & \boldsymbol{c}\_{\text{xz}} \\ \boldsymbol{c}\_{\text{zx}} & \boldsymbol{c}\_{\text{zz}} \end{bmatrix} \left( \int^{(3)} \boldsymbol{t}^2 \left\{ \begin{array}{c} \boldsymbol{w}\_{\text{i}} \\ \boldsymbol{w}\_{\text{i}} \end{array} \right\} - 2 \int^{(4)} \boldsymbol{t} \left\{ \begin{array}{c} \boldsymbol{w}\_{\text{i}} \\ \boldsymbol{w}\_{\text{i}} \end{array} \right\} \right) = \int^{(2)} \left\{ \begin{array}{c} \boldsymbol{b}\_{\text{ui}} \\ \boldsymbol{b}\_{\text{ui}} \end{array} \right\} \tag{12}
$$

$$
\begin{bmatrix} k\_{\text{xx}} & k\_{\text{xz}} \\ k\_{\text{zx}} & k\_{\text{zz}} \end{bmatrix} \int^{(5)}\_{I} t^{2} \left\{ \begin{array}{c} u\_{i} \\ w\_{i} \end{array} \right\} + \begin{bmatrix} c\_{\text{xx}} & c\_{\text{xz}} \\ c\_{\text{zx}} & c\_{\text{zz}} \end{bmatrix} \left( \int^{(4)}\_{I} t^{2} \left\{ \begin{array}{c} u\_{i} \\ w\_{i} \end{array} \right\} - 2 \int^{(5)}\_{I} t \left\{ \begin{array}{c} u\_{i} \\ w\_{i} \end{array} \right\} \right) = \int^{(3)} \left\{ \begin{array}{c} b\_{\text{zz}} \\ b\_{\text{zz}} \end{array} \right\} \tag{13}
$$

From Equations (10)–(13), a linear system equation is obtained for each node where the bearings are located. These equations can be expressed as

$$\begin{bmatrix} A\_s(t) \end{bmatrix} \begin{Bmatrix} \Theta\_s \end{Bmatrix} = \begin{Bmatrix} b\_s(t) \end{Bmatrix} \tag{14}$$

where {Θ*s*} <sup>=</sup> {*kxx kxz kzx kzz cxx cxz czx czz* }*<sup>T</sup>* denotes the transposed vector of parameters to be identified and [*As*(*t*)], {*bs*(*t*)} are 8 × 8 and 8 × 1, respectively.

As can be observed in Figure 2, eight parameters are required to define stiffness and damping characteristics provided by the system supports. This is because in order to obtain the terms of [*As*(*t*)] and {*bs*(*t*)} in Equation (14), eight simultaneous equations involving the unknown support parameters are required to obtain their magnitudes.

From Equation (14) it can be concluded that vector {Θ*s*} is identifiable if, and only if, the dynamic system trajectory is persistent. That is to say, the trajectories or dynamic system behaviors satisfy the condition *det*[*As*(*t*)] = 0. In general, this condition is maintained at least in a small interval (*t*0, *t*<sup>0</sup> + ] where is a positive and sufficiently small value [29]. Then, the linear system Equation (14) is solved to obtain the algebraic identifier for determining the stiffness and damping parameters of rotor-bearing support with constant operation velocity.

$$\{\Theta\_s\} = [\mathcal{A}\_s]^{-1} \{b\_s\} \;\forall t \in (t\_{0\prime} t\_0 + \epsilon]. \tag{15}$$

It is important to mention that to identify the support parameters, lateral vibration measurements at the node and the nodal slopes are required. Moreover, similar information from the adjacent node is also needed. The nodal slopes can be calculated by numerical approximation using the lateral displacements from two adjacent nodes.
