Dynamic characteristic analysis of a roller bearing rotor system

2.1. Bearing

The bearing parameters are listed in Table 1.

Bearing stiffness can be expressed with Eq. (1) and Eq. (2) [4].

For angular contact ball bearing:

where $d$ represents the roller diameter (mm), $n$ denotes the roller number, $\beta$ represents the contact angle, $F_r$ represents the radial load (N).

Fig. 13-D model of the rotor system

For cylindrical roller bearing:

where $L$ represents the effective length of roller (mm), $n$ represents the number of rollers, $\beta$ represents the contact angle, $F_r$ represents the radial load (N).

It can be seen from Eq. (1) and Eq. (2) that the radial external load has great influence on the stiffness of the bearing. Thus, the external loads exerted by the rotor system must be analyzed to obtain the stiffness of the bearings. Weight distribution of the rotor system is shown in Fig. 2. The weight of the centrifugal impeller is $B1=$27.59 N. The weight of the gas turbine rotor assembly is $B3=$27.43 N, and the weights of other components are $B2=$24.42 N, $B4=$5.25 N and $B5=$5.03 N, respectively. Based on the principal of force equilibrium and the principal of moment equilibrium, the radial load of the bearing in support 1 is $F{r}_1=$37.11 N, and the radial load of the bearing in support 2 is $F{r}_2=$49.41 N.

Based on Eq. (1) and Eq. (2), the stiffness of the bearings for support 1 and support 2 are listed in Table 2.

Fig. 2Rotor system bearing loads distribution

2.2. Bearing pedestal

2.2.1. Bearing pedestal of support 1

The finite element model of the pedestal is shown in Fig. 3. In this model, all DOFs of the nodes on the flange are constrained. A central node and a rigid region are created at the axial location of the bearing. A unit force is imposed on the central node and the displacement is obtained. The displacement of the central node is $X=$1.6663×10^-9m, and the stiffness of the bearing pedestal is:

3

$K=\frac{f}{X}=\frac{1}{1.6663\times {10}^{-9}}=6\times {10}^8\text{N/m}.$

Fig. 3The finite element model of the bearing pedestal in support 1

2.3. Support

In this paper, the stiffness of the support should be considered as a combination of two springs in series. Thus, the support stiffness is:

where $K_s$ is support stiffness, $K_b$ is bearing stiffness and $K_{bp}$ is bearing pedestal stiffness.

According to the results of Section 2.1 and Section 2.2, the support stiffness calculated with Eq. (5) is shown in Table 4.

2.4. Squeeze film damper

The parameters of squeeze film damper are listed in Table 2.5. The squeeze film damper is considered as a liner damper and the equivalent damping is [6]:

where $\mu$ represents the dynamic viscosity of the oil film, $L$ represents the width of film, $R$ represents the radius of the oil film ring, $c$ represents the radius of the oil film clearance, and $\epsilon$ represents eccentricity.

According to Table 5, the damping of the squeeze film damper is 195.89 N·s/m.

2.5. The roller bearing rotor system

With the previously obtained support stiffness and damping, the finite element model of the roller-bearing rotor system is established, which is shown in Fig. 4. To simplify the modeling process and reduce the computational effort, blades of the turbine are modeled with one MASS21 element [7]. The supports are modeled with COMBI214 element. For the rest parts of the rotor system, the SOLID95 element is adopted. In total, the whole model is divided into 178417 elements and 72220 nodes.

Fig. 4The FE model of roller-bearing rotor system

3.1. Critical speed

Based on the FE model established in Section 2.5, critical speeds of the rotor system are obtained. The operating speed range of the rotor system is 31800 r/min-55320 r/min. And the rotational speed range 0-126000 r/min, which covers the operating speed range, is applied to the finite element model. The Campbell diagram is shown in Fig. 5. The first three critical speeds are listed in Table 6.

From Table 6 it can be seen that the distinction between first forward procession and backward procession is not so clear. The second critical speed is 20.6 % lower than the lower margin of the speed range (31800 r/min). The third critical speed is 105 % higher than the largest operating speed (55320 r/min). From the theory in reference [8], it is quite obvious that there is an enough safety margin between the critical speeds and the operating speed range. Also, there is no critical speed in the operating speed range, which ensures the safe and smooth operation of the rotor system.

Fig. 5Campbell diagram of rotor system

3.2. Unbalance response

Although the critical speeds of the rotor system have enough safety margins, the unbalance response of the rotor system in operating speed range needs to be considered for avoiding rubbing between the rotor and the case.

The residual imbalance of the rotor system is about 0.5 g·cm and is distributed at the front and rear end of the rotor system. In the operating speed range 31800 r/min-55320 r/min, the unbalance response of the rotor system is obtained with previously established FE model.

In the actual engine, the rubbing is prone to the turbine disk and the centrifugal impeller. Therefore, the radial displacements of the turbine disk and the centrifugal impeller under different rotational speeds are shown in Fig. 6.

Fig. 6Unbalance responses of turbine disk and centrifugal impeller

In Fig. 6, the amplitudes of the turbine disk and the centrifugal impeller have opposite trends. The clearance between the turbine and the case is 9×10^-4m and the clearance between the centrifugal impeller and the case is 7×10^-4m. It can be seen from Fig. 6 that the maximum amplitudes of the turbine disk and the centrifugal impeller are about 1.9×10^-6m and 2.1×10^-6m, respectively. Therefore, the rotor system can operate safely and smoothly in the operating speed range.