## 5 Mathematical Model of the Unsteady State of the Bearing
### Method for Calculating the Criteria for the Dynamic Quality of a Bearing  
When calculating the dynamic characteristics of the bearing, the transfer function (TF) (101) of the dynamic compliance of the bearing was represented in the form  
\[K(s)=\frac{\overline{\Delta H}}{\overline{\Delta F}}=\frac{b_{0}+b_{1}s+b_{2} s^{2}+\ldots+b_{m}s^{m}}{1+a_{1}s+a_{2}s^{2}+\ldots+a_{n}s^{n}}, \tag{102}\]  
where \(n>0\), \(m>0\), \(n>m\) and \(a_{i}\) and \(b_{i}\) are coefficients. The denominator of (102) is the characteristic polynomial (CP) of a linear dynamic system with distributed parameters, since its dynamics are described by differential equations. The speed and stability margin of the system were estimated by the roots of the characteristic equation [35]. To calculate the coefficients (102), a method was developed that provides the representation of the TF in the form (102) and allows the calculation of dynamic criteria with a predetermined accuracy. Since (101) was obtained by a numerical method, in order to represent the TF in the form (102), rational interpolation (101) was performed [25]. The existing methods of rational interpolation are based on the solution of a linear system of equations for the TF coefficients, which contains \(n+m\) equations for the unknown CP coefficients. The system is usually solved by the Gauss method, which has a cubic order of complexity \((n+m)^{3}\), where the order of complexity of a computational method means the time complexity of the algorithm that implements it [26]. In order to speed up the calculations, a method for finding the coefficients has been developed, which has a quadratic order of complexity m (n + m). If necessary, the method allows us to find only the CP coefficients. Then, the order of complexity of the system is \(n^{2}\). In all cases, this method works an order of magnitude faster than the Gaussian method. First, the difference in the degrees of the polynomials of the TF \(p=n-m\) is determined by the formula  
\[\underset{p\rightarrow\infty}{lim}s^{p}K(s)=\frac{b_{m}}{a_{n}}\
eq 0, \tag{103}\]  
where \(p=n-m\). Usually, \(p=1-2\), so we can find this limit and \(p\) rather quickly. The number of unknown coefficients (102) \(k=n+m\). Specifying \(m\), we define \(n=m+p\) and find the coefficient  
\[b_{0}=K(0). \tag{104}\]  
Let us rewrite (102) in the form  
\[a_{1}+a_{2}s+\ldots+a_{n}s^{n-1}+\Gamma(s)\Big{(}b_{1}+b_{2}s+\ldots+b_{m}s^{m -1}\Big{)}=\Lambda(s)/s, \tag{105}\]  
where \(\Gamma(s)=\frac{-1}{K(s)}\), and \(\Lambda(s)=-b_{0}-\Gamma(s)\). We calculate \(e=exp\left(-\frac{2\pi i}{k}\right)\), where \(i\) is the imaginary unit, and put \(s_{1}=1\), \(s_{j}=es_{j-1}\), \(j=2,3,\ldots,k\), \(\Psi_{j}=K(s_{j})\). Substituting \(s=s_{j}\), (\(j=1,2,\ldots,k\)) in (105), we obtain a system of linear equations for unknown coefficients (102), which can be written in matrix form:  
\[Mx=y. \tag{106}\]  
We write the matrix \(M\) in the form  
\[M=\Phi U, \tag{107}\]  
where \(U\) is the matrix, and \(\Phi\) is the matrix of the discrete Fourier transform [36], which has the previously known inverse matrix \(\Phi^{-1}\). Multiplying \(\Phi^{-1}\) by (107), we transform System (35) into the form  
\[Ux=z, \tag{108}\]  
where, for \(m>0\), the _matrix_\(U\) has a cellular structure of the form  
\[U=\left[ \begin{array}{cc}E&C\\ 0&D\end{array} \right],\]  
\(E\) and \(0\) are the unit and zero size matrices and \(C\) and \(D\) are Toeplitz matrices [37]. Equation (108) splits into two independent systems, one of which is a system of equations for the coefficients \(b_{i}\) of a TF with an asymmetric Toeplitz matrix of a special form [38]. Such a system can be solved by fast methods of complexity \(m^{2}\). The latter include the methods of Trench, Berlekamp-Massey and Euclid [38; 39].