The TC rotor usually consists of a radial outflow compressor wheel and a radial inflow turbine wheel on a single shaft. Bearings are mounted inboard, with the compressor and turbine overhung, as shown in Figure 1. FRB consists of two hydrodynamic fluid film bearings in series with the ring as an additional degrees-of-freedom (DOF). The inner film has two rotating surfaces, i.e., journal and ring with rotational speeds of *Ω*_{j} and *Ω*_{r}, and the outer film has only one rotating surface, i.e., ring with rotational speed of *Ω*_{r}. This arrangement can only be used for high speed and light weight applications, such as in the automotive industry.

### 2.1 FRB Model

Unlike conventional plain journal bearing, the floating ring rotates with the sum of speed of inner and outer hydrodynamic oil films, which is a key parameter to the analysis of TC rotor’s dynamic characteristics. However, the rotational speed of floating ring is very difficult to monitor in the actual working condition. For the purpose of simplicity, the bearing feeding holes are not included and the oil feeding conditions are not considered in the FRB model. In addition, the isothermal fluid flow condition is assumed for the proposed model. The Reynolds equations for the inner and outer oil films can be written as follows:

$$ \frac{1}{{R_{j}^{2} }}\frac{\partial }{{\partial \theta_{i} }}\left( {\frac{{h_{i}^{3} }}{{12\mu_{i} }}\frac{{\partial p_{i} }}{{\partial \theta_{i} }}} \right) + \frac{\partial }{{\partial Z_{i} }}\left( {\frac{{h_{i}^{3} }}{{12\mu_{i} }}\frac{{\partial p_{i} }}{{\partial Z_{i} }}} \right) = \frac{{\varOmega_{j} + \varOmega_{r} }}{2}\frac{{\partial h_{i} }}{{\partial \theta_{i} }} + \frac{{\partial h_{i} }}{\partial t}, $$

(1)

$$ \frac{1}{{R_{o}^{2} }}\frac{\partial }{{\partial \theta_{o} }}\left( {\frac{{h_{o}^{3} }}{{12\mu_{o} }}\frac{{\partial p_{o} }}{{\partial \theta_{o} }}} \right) + \frac{\partial }{{\partial Z_{o} }}\left( {\frac{{h_{o}^{3} }}{{12\mu_{o} }}\frac{{\partial p_{o} }}{{\partial Z_{o} }}} \right) = \frac{{\varOmega_{r} }}{2} + \frac{{\partial h_{o} }}{{\partial \theta_{o} }} + \frac{{\partial h_{o} }}{\partial t}, $$

(2)

where subscripts *i* and *o* identify the parameters of the inner oil film and outer oil film, respectively. Subscripts *j* and *r* distinguish the parameters between journal and floating rings. *p* is the oil film pressure, and *μ* denotes the lubricating oil film viscosity. *R*_{j} and *R*_{o} correspond to the radius of journal and outer floating ring, respectively. *θ* is the angular coordinate for the inner and outer oil films. *Z*_{i} and *Z*_{o} denote the axial coordinates of the inner and outer films, respectively. *h*_{i} and *h*_{o} represent the oil film thicknesses of inner and outer, respectively. *Ω*_{j} and *Ω*_{r} are the angular velocity of journal and floating ring, respectively.

The equilibrium positions are determined by an iterative procedure, the linearized bearing coefficients of inner film and outer film can be obtained from Eqs. (1) and (2). Actually, the FRB is usually used for very high-speed applications and the floating ring is designed to have high ring speed to ensure the proper hydrodynamic lubrication. At very high speed, the eccentricity ratios are very small and the journal, ring, and bearing are nearly concentric. When the eccentricity ratios are zero, the journal torque *T*_{j} and ring torque *T*_{r} balance becomes

$$ T_{j} = 2\uppi\mu_{i} R_{i}^{3} (\varOmega_{i} - \varOmega_{o} )L_{i} /C_{i} = T_{r} = 2\uppi\mu_{o} R_{o}^{3} \varOmega_{o} L_{o} /C_{o} . $$

(3)

Thus, the ring speed *Ω*_{r} can be determined from the above Eq. (3). Once the ring speed *Ω*_{r} is known, the two Reynolds equations for the inner and outer films can be solved separately and the bearing coefficients can be determined just like the conventional plain cylindrical bearing for linear analysis. The maximum possible ring speed *Ω*_{r} can be conveniently estimated using the torque balance with concentric journal, ring, and bearing [21]:

$$ \varOmega_{r} = \frac{{\varOmega_{j} }}{{1 + \left[ {(\mu_{o} /\mu_{i} )(R_{o} /R_{i} )^{3} (L_{o} /L_{i} )(C_{i} /C_{o} )} \right]}}, $$

(4)

where *µ* is the oil dynamic viscosity, *L* is the axial length of the bearing, and *C* is the bearing radial clearance. Subscripts *i* and *o* are for the inner and outer film properties. Since the inner film has much higher surface velocity and smaller clearance, the lubricant operating temperature is higher than that of the outer film. Typical ratio of *µ*_{o}/*µ*_{i} ranges from 1.2 to 2.0. The clearance ratio *C*_{o}/*C*_{i} is about 1.5 to 4.0. For FRB, the clearance ratio is probably the most important parameter, the optimal value depends on the operating conditions and ratios of *R*_{o}/*R*_{i} and *L*_{o}/*L*_{i}. Smaller clearance ratio will decrease the ring speed which is not desirable from the lubrication point of view, although it may increase the rotor stability. Large clearance ratio increases the ring speed, however, it is not desirable from rotor vibration and stability point of view. However, the nonlinear fluid film forces for the inner and outer films are dependent on the motions of the journal and ring.

### 2.2 TC Rotor with Induced Unbalance

Considering a high-speed TC rotor, since the partial differential equations given by modeling of the rotor shaft in a continuous system are difficult to be tackled, discretizing the continuum to a discretized system by the FE method. The motion governing equation for the investigated TC rotor-FRB system is derived as follows:

$$ \varvec{M\ddot{x}} + \varvec{C}_{SG} \dot{\varvec{x}} + \varvec{Kx} = \varvec{F}(t), $$

(5)

where *M* represents the mass matrix contained the mass and inertia moments of the rotor with *n* DOFs. *C*_{SG} is the damping coefficients and gyroscopic matrix. *K* is the system stiffness coefficients matrix included diagonal and cross coupled stiffness. *x* is the response vector included two translational and two rotational displacements at each station in the horizontal direction *X* and the vertical direction *Y*. *F*(*t*) consists of unbalance force *F*_{ub}(*Ω*, *t*), static gravity force *F*_{s} in *Y* direction and nonlinear bearing force *F*_{i}(\( x,\;\dot{x},\;t \)). Thus, the *F*(*t*) can be rewritten as

$$ {\varvec{F}}(t) = {\varvec{F}}_{i} (x,\;\dot{x},\;t) + {\varvec{F}}_{ub} (\varOmega ,\;t) + {\varvec{F}}_{s} . $$

(6)

Considering the practical structure and the locations probability of unbalance on gasoline engine TC, the *F*_{ub} (*Ω*, *t*) usually can be decomposed \( {\varvec{F}}_{ub}^{c} \) and \( {\varvec{F}}_{ub}^{t} , \) which exists at the end of compressor and turbine impellers, as shown in Eqs. (7) and (8). *m*_{c} and *m*_{t} denote the mass of the compressor and turbine impellers, respectively. *e* is the unbalance displacement. *ϕ* is the rotor rotating angle around the *Z* axis, and therefore, \( \dot{\phi } \) and \( \ddot{\phi } \) represent the angular speed and acceleration, respectively.

$$ {\varvec{F}}_{ub}^{c} = \left( \begin{aligned} F_{ub}^{{x\text{c}}} \hfill \\ F_{ub}^{yc} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} m_{c} e\dot{\phi }^{2} \cos \phi + m_{c} e\ddot{\phi }\sin \phi \hfill \\ m_{c} e\dot{\phi }^{2} \sin \phi - m_{c} e\ddot{\phi }\cos \phi \hfill \\ \end{aligned} \right), $$

(7)

$$ {\varvec{F}}_{ub}^{t} = \left( \begin{aligned} F_{ub}^{xt} \hfill \\ F_{ub}^{yt} \hfill \\ \end{aligned} \right) = \left( \begin{aligned} m_{t} e\dot{\phi }^{2} \cos \phi + m_{t} e\ddot{\phi }\sin \phi \hfill \\ m_{t} e\dot{\phi }^{2} \sin \phi - m_{t} e\ddot{\phi }\cos \phi \hfill \\ \end{aligned} \right). $$

(8)

The residual mass unbalance of a rotating assembly is usually determined by using the multi-plane balancing machines. However, with the development of dynamic balancing technology, the balancing precision is very high, it is urgent to know how to control the unbalance magnitude for small vibration and stable operation in the design process. It is inevitably that the unbalance existed in the rotation, especially for the high-speed rotating machine. These mass unbalance locates at different locations of two impellers with variable magnitude of *me*. As expressed in unbalance force Eqs. (7), (8) and the governing Eq. (6), it is obvious seen that the unbalance *F*_{ub} directly affects the motion of TC rotor and bearing system. The rotor motion may change with the variable magnitude of unbalance in turbine or compressor wheel. Thus, it is necessary to investigate the relationship between unbalance magnitude and dynamic characteristics for the small-sized and high-speed TC rotor with FRB.