## 2 Mathematical Method
### The Dynamic Coefficients of Tilting Pad Bearings  
For tilting pad bearings, without considering the radial displacement of pivot, the perturbation variables \(\widetilde{\epsilon}_{d}\), \(\widetilde{\varphi}_{d}\) and \(\widetilde{\delta}_{d}\) are involved in perturbed equation Equation (12). Taking partial derivative of Equation (12) to \(\widetilde{\epsilon}_{d}\), \(\epsilon_{0}\widetilde{\varphi}_{d}\) and \(\widetilde{\delta}_{d}\) yields three partial differential equations for \(P_{\epsilon}\), \(P_{\varphi}\) and \(P_{\delta}\) instead of \(\widetilde{p}_{d}\). Meanwhile, according to Equation (6) the corresponding \(\widetilde{h}_{d}\) is replaced by \(H_{\epsilon}=\cos(\theta-\varphi_{0})\), \(H_{\varphi}=\sin(\theta-\varphi_{0})\) and \(H_{\delta}=\sin(\beta_{l}-\theta)\), respectively. Then the numerical integration below is suitable for the coordinate system in Figure 1.  
\[ \begin{array}{ll}-\frac{1}{2}\int_{-1}^{1}\int_{\theta_{L}}^{\theta_{T}}P_{ \bullet,l}\cos\theta d\theta d\lambda=k_{x\bullet,l}+i\Omega c_{x\bullet,l}\\ -\frac{1}{2}\int_{-1}^{1}\int_{\theta_{L}}^{\theta_{T}}P_{\bullet,l}\sin\theta d \theta d\lambda=k_{y\bullet,l}+i\Omega c_{y\bullet,l}\end{array}  \tag{16}\]  
where the subscript \(\cdot\) represents \(\epsilon\), \(\varphi\), and \(\delta\). The leading edge of the \(l\)th pad \(\theta_{L}\) has a relation with pivot offset \(\gamma\), pad's angular extent \(\theta_{\text{prad}}\), and \(\beta_{l}\) as \(\theta_{L}=\beta_{l}-\gamma\theta_{\text{prad}}\). The relation between the \(l\)th pad's trailing edge \(\theta_{T}\) and its leading edge \(\theta_{L}\) is \(\theta_{T}=\theta_{L}+\theta_{\text{prad}}\). Equation (17) is employed to get dimensionless dynamic coefficients caused by shaft whirling in the coordinate system in Figure 1.  
\[\left[\begin{array}{c}k_{xx\_l}\\ k_{xy\_l}\\ \left[ \begin{array}{c}k_{yx\_l}\\ k_{yy\_l}\\ \end{array} \right]=\mathbf{A_{T}}\left[ \begin{array}{c}k_{yx\_l}\\ k_{xy\_l}\\ \end{array} \right];\left[ \begin{array}{c}c_{xx\_l}\\ \sin\varphi_{0}\\ \end{array} \right]=\left[ \begin{array}{cc}\cos\varphi_{0}&-\sin\varphi_{0} \\ \sin\varphi_{0}&\cos\varphi_{0}\\ \end{array} \right]\left[ \begin{array}{c}k_{xx\_l}\\ k_{xy\_l}\\ \end{array} \right]\\ \left[ \begin{array}{c}k_{yx\_l}\\ k_{yy\_l}\\ \end{array} \right]=\mathbf{A_{T}}\left[ \begin{array}{c}k_{yz\_l}\\ k_{yy\_l}\\ \end{array} \right];\left[ \begin{array}{c}c_{xx\_l}\\ c_{xy\_l}\\ \end{array} \right]=\mathbf{A_{T}}\left[ \begin{array}{c}c_{yz\_l}\\ c_{yy\_l}\\ \end{array} \right]=\mathbf{A_{T}}\left[ \begin{array}{c}c_{yz\_l}\\ c_{yy\_l}\\ \end{array} \right] \tag{17}\]  
The dimensionless dynamic equation of the pad corresponding to moment to the pivot is  
\[\bar{\bar{J}}\delta_{d,l}=\overline{F}_{y,l}\cos\beta_{l}-\overline{F}_{x,l} \sin\beta_{l} \tag{18}\]  
The subsequent process for deriving the dimensionless equivalent dynamic coefficients of the tilting pad bearing \(K_{ij}\) and \(C_{ij}\left(l,j=x,y\right)\) is similar to [16]. The dimensionless definition form is taken as [25]. \[ \begin{array}{l}K_{ij}=\frac{\mu_{\omega}\omega RL}{W\overline{\varphi}^{2}} \sum\limits_{l=1}^{5}k_{ij\_l}-p\left(k_{i\delta\_j}k_{\delta j,l}-\Omega^{2}c _{i\delta\_j}c_{i\delta j,l}\right)-\Omega^{2}q\left(k_{i\delta\_l}c_{i\delta j,l}+k_{\delta j,l}c_{i\delta\_l}\right)\\ C_{ij}=\frac{\mu_{\omega}\omega RL}{W\overline{\varphi}^{2}}\sum\limits_{l=1} ^{5}c_{ij\_l}-p\left(k_{i\delta\_l}c_{j\delta j,l}+k_{\delta j,l}c_{i\delta \_l}\right)-\Omega^{2}q\left(c_{i\delta\_l}c_{\delta j,l}-\frac{k_{i\delta\_ l}c_{j\delta j,l}}{\Omega^{2}}\right)\end{array}