Numerical analysis of a vertical rotor rub system partially immersed in inviscid incompressible fluid

1.1. Vertical rotor system description and governing equations

The proposed model of the coupling between lateral and torsional vibrations of the isotropic vertical rotor is found quite often in literature as a horizontal rotor [3, 5]. In this work, the model of vertical Jeffcott rotor of lumped-disc-shaft submerged in an inviscid incompressible fluid is presented for fault analysis, as shown in Fig. 1(a).

Fig. 1(a) is an illustration of the schematic diagram of elastic rotor-stator model partially immersed in an incompressible inviscid fluid medium, taking into account the effect of an axial force. The governing equations of the fully coupled rotor lateral-torsional vibrations are determined under various assumptions and based on the factor such as unbalance as formulated in [3], with consideration of the rotor-stator rub and axial force effects. The obtained nonstationary 4-Degree of Freedom (DOF) responses consists of two orthogonal bending deflections of the shaft-disc of fixed coordinate centre $\left(X,Y,Z\right)$, one axial displacement along $Z$-axis with axial force$\mathrm{P}$and finally one vertical shaft rotation $\theta$. The general coordinates applied to derive the system equations are displayed in Fig. 1(b), where $\left(X,Y,Z\right)$ is the inertial coordinate frame; $\left(x_m{y}_m\right)$ is associated with the motor gearbox system. The unbalance mass, $m_u$, is located by the eccentricity distance $e$ from the reference linked to the rotating disc $O_1$.

Fig. 1a) Full vertical rotor-stator model, b) rotor deformation in fixed coordinate

a)

b)

1.2. Energy principle expression

Based on the geometry and rotor deformation shown in Fig. 1(b), and considering the modal damping, the kinetic dissipative energy of the motor-disc-shaft system is written as:

where $J_M$ represents the mass moment of inertia of the motor, $M$ is the mass of the rotor, $J_D$ is the disc mass moment of inertia and $m_u$ is the mass unbalance.

1.3. Expression of potential energy

The expression of the potential energy of the rotor system comprises the lateral shaft deflection, shaft strain energy and is given by:

where $K_{XX}$, $K_{YY}$, are the laterals shaft stiffness and $K_{ZZ}$ is the axial stiffness coefficients associated with the system DOF. A rotor’s high-speed level generates axial vibration in the bottom of the vertical rotor, which excessively creates an axial force that influences the rotor stiffness as shown in [6] and expressed as follows:

where $L$, $K_0$, and $P\left(t\right)$ respectively are, the shaft’s length, the initial stiffness for first mode deflection and the rotor axial compressive force. In this study, $P$, which is assumed to fluctuate harmonically about a constant mean value $P_0$ is written as:

where $P_f$, $P_0$ and $\theta$ are amplitudes and shaft rotational angle respectively. Considering the effect of the viscous modal damping coefficient, the total dissipated energy of the system can be expressed as:

where, $C_{XX}$, $C_{YY}$ and $C_{ZZ}$ are the respective laterals and axial damping coefficient.

1.4. Mathematical expression of rub-contact

Upon applying the method in [5], into Fig. 2, the geometrical location of the shaft centre and the contact forces between the rotor and stator in the Cartesian coordinate frame are:

where, $K_S$ and $\mu$ are stator radial stiffness and coefficient of dry friction, respectively.

Fig. 2a) Schematic of the contact system, b) 2D contact force model idealization

a)

b)

1.5. Hydrodynamic excitation force expressions

In this section, the hydrodynamic forces expressions coupled with the oscillating vertical rotor under lateral excitation are established. The equations of fluid motion can be found with more details in [3] and will not be repeated herein. Considering the basic mechanism of the filled rectangular tank, the lateral vibration of the vertical shaft modifies the stream flow at the interface with the fluid which is subjected to sinusoidal excitation displacement along $XY$-direction expressed as:

where, $X_0$, $Y_0$ are the excitation amplitude, and $\mathrm{\Omega }$ is a cyclical frequency in cycles per unit of time much lower than its fundamental frequency ($\omega \gg \mathrm{\Omega }$). In the assumption that the disc has approximately the same radius with the shaft ($r_{shaft}\approx r_{disc}$), the hydrostatic pressure ($\rho gz$) is neglected. Considering the total velocity potential function $\mathrm{\Phi }$, the hydrodynamic pressure interfacing at any point along with the rotor is determined by the pressure Eq. (8):

where $p_0=\rho X_0{\mathrm{\Omega }}^2\mathrm{c}\mathrm{o}\mathrm{s}\theta \mathrm{c}\mathrm{o}\mathrm{s}\mathrm{\Omega }t$. The acting force along the $X$-axis can systematically be obtained by resolving the hydrodynamic pressure at the first mode $n=$ 1 and $\theta =0$ as shown in [3]:

where the overall mass of the fluid is defined by $m_f=\rho \pi h{W}^2$ and ${\chi }_1=\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{h}(h{\xi }_1/W)/h{\xi }_1$. The force acting on the bottom of the container and along the $y$-axis are nil:

The pressure is symmetrically distributed at $XY$-direction along the tank wall to cancel the integration over it. The fluid force acting on the shaft can then be differentially simplified to:

For incompressible inviscid fluid flow, the obtained extra fluid mass is added and coupled into the inertial force of the shaft as [3]:

1.6. The general equations of motion

The Lagrangian equation of a system in $q_n$ generalized coordinate frame is:

Upon substitution of Eqs. (1), (2), (5), (6) and (11) into Eq. (13), performing the requisite differentiation and algebraic manipulation, the final rotor system expression obtained in a matrix form reads:

where the first element of Eq. (17) represents the inertia rotor mass matrix, which displays coupling between different generalized coordinate of the rotor system [5]. $\left\{Q_n\right\}$ represents the nonlinear terms of Coriolis components and $\left\{F_n\right\}$ the vector component of the external forces associated with the with each DOF. The damping ratio and natural frequency of the first asymmetric mode of a cylindrical rotor are ${\xi }_1=$1.841 and ${\omega }_1=\sqrt{g{\xi }_1/W}$, where $W$ is the container width, $g$ is the acceleration due to gravity [3].