## 3 The Hydrodynamic Forces and Introduction into the Model and Solution
At \(r=R\), the boundary parameters at the driveshaft surface tank walls and bottom are evaluated as  
\[\Rightarrow P_{rr}(r,\overline{\theta},t)=-P+2\mu_{f\bar{l}}\frac{\partial u }{\partial r}\bigg{|}_{r=R}=-iR\rho\Omega^{2}(r_{1}+\zeta(t))\cos\overline{ \theta}e^{i\Omega t} \tag{24}\]  
\[p_{r\overline{\theta}}=\mu_{f\bar{l}}\bigg{(}\frac{\partial v}{\partial r} \bigg{)}=-\rho\frac{\partial\Psi_{2}}{\partial t}\Rightarrow p_{r\overline{ \theta}}(r,\overline{\theta},t)=-\frac{2i\Omega^{2}\rho(r_{1}+\zeta(t))K_{1}( kR)\sin(\overline{\theta})e^{i\Omega t}}{kK_{0}(kR)} \tag{25}\]  
where \(\mu_{f\bar{l}}\) represents the fluid viscosity; therefore, the hydrodynamic pressure distribution will lead eventually to the hydrodynamic forces acting on the driveshaft-stator, and it is obtained by integrating the pressure distribution over the projected driveshaft length \(L\) as  
\[F(r,\overline{\theta},t)=LR\int\limits_{0}^{2\pi}\big{(}-P_{rr}\cos\overline{ \theta}+P_{r\theta_{1}}\sin\overline{\theta}\big{)}d\overline{\theta} \tag{26}\]  
It is assumed that the driveshaft angular speed \(\Omega\) coincides with whirling fluid speed \(\omega\) and, integrating the first term by parts and using Equations (24) and (25), yields the following result  
\[F(r,\overline{\theta},t)=-i\pi\rho LR^{2}\Omega^{2}(r_{1}+\zeta(t))e^{i\Omega t }\bigg{(}1+\frac{2K_{1}(kR)}{R^{2}kK_{0}(kR)}\bigg{)} \tag{27}\]  
The obtained hydrodynamic force acting on the shaft is  
\[ \begin{array}{l}F_{fluid}(r,\overline{\theta},t)=(r_{1}+\zeta(t))(\pi\rho LR^ {2})\Omega^{2}\Upsilon_{o}\times\\ \big{[}(\pi-4\gamma+4\ln(2)+3-4\ln(R\sqrt{\Omega/v})\cos\Omega t+(4\gamma-4 \ln(2)-3+4\ln(R\sqrt{\Omega/v}))\sin\Omega t\big{]}\end{array}  \tag{28}\]  
Due to the position of the eccentric mass and the geometric centre, the immersed driveshaft rotor has a static unbalance (linear eccentricity-\(e\)). The hydrodynamic force component can be reduced over the projected area to  
\[ \begin{array}{l}F_{real}(r,\overline{\theta},t)=m_{0}\Upsilon_{o}(\pi- \Upsilon_{1})\Omega^{2}r_{1}\cos\Omega t+m_{0}\Upsilon_{o}\Upsilon_{1}\Omega^{ 2}r_{1}\sin\Omega t+m_{0}\Omega^{2}e\Upsilon_{o}\Upsilon_{1}\sin\Omega t\\ +m_{0}\Omega^{2}e\Upsilon_{o}(\pi-\Upsilon_{1})\cos\Omega t\end{array}  \tag{29}\]where \(m_{0}=\pi\rho LR^{2}\) represents the constant mass of the fluid displaced by length \(L\) of the shaft \(\Upsilon_{1}=4\gamma-4\ln(2)-3+4\ln(R\sqrt{\Omega/v})\) and \(\gamma\) is the Euler constant. \(\Upsilon_{o}\) represents the dimensionless partial amplitude of vibration and can be found in [11]. In many cases of flow-induced forced vibration, the response estimation can be broken down into independent equations of motion such as the added mass and the added damping, which greatly affect the characteristics of the structures. The relation between the obtained hydrodynamic forces and the displacements (and accelerations) of the driveshaft, gives the equivalent coefficients. The mass of the viscous fluid is significant since the amount of fluid present between the driveshaft and the stator is considerable for the system to re-establish the equilibrium conditions at each rub-impact. The resulting viscous forces \((F_{X_{1}},F_{Y_{1}})\) about the driveshaft centre in the two-dimensional \(X_{1}\) and \(Y_{1}\) direction flow, acting on the whirling driveshaft in an unsteady flow are expressed as  
\[F_{X_{1}}(r,\theta_{1},t)=-m_{0}\Upsilon_{o}(\pi-\Upsilon_{1})\frac{d^{2}X_{1 }}{dt^{2}}-m_{0}\Omega\Upsilon_{o}\Upsilon_{1}\frac{dX_{1}}{dt}+F_{fIX_{1}} \tag{30}\]  
\[F_{Y_{1}}(r,\theta_{1},t)=-m_{0}\Upsilon_{o}\Upsilon_{1}\frac{d^{2}\Upsilon_ {1}}{dt^{2}}+m_{0}\Omega\Upsilon_{o}(\pi-\Upsilon_{1})\frac{d\Upsilon_{1}}{dt }+F_{fY_{1}} \tag{31}\]  
where the fluid mass and damping coefficients per unit fluid depth represent the inherent flexibility and damping of the viscous film. By applying a spring-damper concept the fluid parameters interacting with the driveshaft are given by the following expressions  
\[