## 4 Solution of Equations Using the Newmark \(\beta\) Method
Due to the high computational cost of regenerating the tangent stiffness \(\mathbf{K}^{(k-1)}\) at each iteration step for large structural systems, the modified Newton-Raphson method can also be employed for iteration. The modified Newton-Raphson method utilizes the tangent stiffness matrix \(\mathbf{K}(t)\) from time \(t\) throughout the entire iterative solution process at time \(t+\Delta t\). At this point, the iteration format (67) is revised to  
Figure 3:2  
Proceeding with the construction of constraint conditions using the Newmark Beta method, and based on the Lagrange mean value theorem, the velocity at time \(t+\Delta t\) is expressed as Figure 3:3  
The subscript \(i-1\) denotes the physical quantity at time \(t\), while the subscript \(i\) denotes the physical quantity at time \(t+\Delta t\). \(\dot{\mathbf{u}}^{*}\) represents the acceleration vector \(\ddot{\mathbf{u}}\) at some point within the interval \((t,t+\Delta t)\). The Newmark Beta method makes an approximate assumption given by  
Figure 3:4  
Thus, the velocity at time \(t+\Delta t\) can be expressed as  
Figure 3:5  
Derived from the Taylor series expansion, we have  
Figure 3:6  
In Equations (72) and (73), the parameters \(\delta\) and \(\beta\) are related to the accuracy and stability of the method. When \(\delta>1/2\), algorithmic damping is introduced, leading to perceived amplitude decay; when \(\delta<1/2\), negative damping occurs, causing the amplitude to grow during the integration process. Typically, the critical value \(\delta=1/2\) is chosen. With \(\delta=1/2\), setting \(\beta=0\) corresponds to the constant acceleration method, also known as the central difference method; \(\beta=1/4\) corresponds to the average acceleration method; and \(\beta=1/6\) corresponds to the linear acceleration method. Rewriting Equation (67) into expressions for velocity increment and acceleration increment, we obtain  
\[\left\{ \begin{array}{l}\dot{\mathbf{u}}_{i}=a_{1}(\mathbf{u}_{i}-\mathbf{u} _{i-1})+(1-a_{4})\dot{\mathbf{u}}_{i-1}-a_{5}\ddot{\mathbf{u}}_{i-1}\\ \ddot{\mathbf{u}}_{i}=a_{0}(\mathbf{u}_{i}-\mathbf{u}_{i-1})-a_{2}\dot{ \mathbf{u}}_{i-1}-(a_{3}-1)\ddot{\mathbf{u}}_{i-1}\end{array} \right. \tag{74}\]  
In the aforementioned equation,  
\[ \begin{array}{l}a_{0}=\frac{1}{\beta\Delta t^{2}},a_{1}=\frac{\delta}{\beta \Delta t},a_{2}=\frac{1}{\beta\Delta t}\\ a_{3}=\frac{1}{2\beta},a_{4}=\frac{\delta}{\beta},a_{5}=-\Delta t\Big{(}1- \frac{\delta}{2\beta}\Big{)}\end{array}  \tag{75}\]  
Rewritten in an iterative format, it becomes  
\[\left\{ \begin{array}{l}\dot{\mathbf{u}}_{i}^{(k)}=a_{1}\Big{(}\mathbf{u}_{i }^{(k-1)}-\mathbf{u}_{i-1}+\Delta\mathbf{u}^{(k)}\Big{)}+(1-a_{4})\dot{ \mathbf{u}}_{i-1}-a_{5}\ddot{\mathbf{u}}_{i-1}\\ \ddot{\mathbf{u}}_{i}^{(k)}=a_{0}\Big{(}\mathbf{u}_{i}^{(k-1)}-\mathbf{u}_{i-1 }+\Delta\mathbf{u}^{(k)}\Big{)}-a_{2}\dot{\mathbf{u}}_{i-1}-(a_{3}-1)\ddot{ \mathbf{u}}_{i-1}\end{array} \right. \tag{76}\]  
Substituting Equation (76) into (69) and rearranging, we arrive at  
\[\widetilde{\mathbf{K}}\Delta\mathbf{u}^{(k)}=\Delta\widetilde{\mathbf{F}}^{(k)} \tag{77}\]  
In this equation,  
\[\begin{array}{l}\widetilde{\mathbf{K}}=\mathbf{K}(t)+a_{0}\mathbf{M}+a_{1} \mathbf{C}\\ \Delta\widetilde{\mathbf{F}}^{(k)}=\mathbf{F}(t+\Delta t)-\mathbf{f}^{(k-1)}( t+\Delta t)\\ +\mathbf{M}\Big{[}a_{0}\Big{(}\mathbf{u}_{i-1}-\mathbf{u}_{i}^{(k-1)}\Big{  
Subsequently, the finite element model is solved by assembling the stiffness matrix \(\mathbf{K}\), mass matrix \(\mathbf{M}\), and gyroscopic matrix \(\mathbf{C}\) according to Equation (58). Here, \(\mathbf{K}\) includes only the linear part of the support stiffness, while the nonlinear components encompass the SFD oil film force and the nonlinear restoring force of the supports. These are applied to the supports in the form of loads as shown in Equations (59) and (60). It should be noted that \(\mathbf{K}\), \(\mathbf{M}\), and \(\mathbf{C}\) are all independent of time and are related only to the model's parameters such as length, radius, and density.