Vibrations of rigid rotor systems with misalignment on squirrel cage supports

2.1. Description of the rotor system

A rotor test rig is designed to study the vibrations of the misalignment rotor system, as shown in Fig. 1. It consists of a hollow shaft and two bearings on squirrel cages. A steel disk is installed on the middle of shaft. Parameters of the rotor system are listed in Table 1.

Fig. 1The rigid rotor rig on two supports of squirrel cages

a) Photo of the rotor test rig

b) Sketch of the rotor test rig

2.2. The equations of motion

Schematic of a rotor system in test rig is shown in Fig. 2, which is considered as a rigid rotor on elastic supports with isotropic stiffness $k$. The mass center $C$ is located at distances $l_1$ and $l_2$ from the bearings. The polar moment of inertia is defined as $I_z$ and the transversal moment of inertia is defined as $I_x$.

Fig. 2Schematic of the rotor system

The kinematics is described by 4 coordinates, which are, transverse displacements of the mass center along $x$ and $y$ direction: $u$ and $v$, rotations about the $x$ and $y$: $\theta$ and $\varphi$. With these 4 coordinates, the displacements at the supports can be expressed as:

The strain energy of the elastic supports is:

Substituting Eq. (1) into Eq. (2) obtains:

For the rotor test rig shown in Fig. 1, $l_2=l_1$ and the last term in Eq. (3) disappears.

The kinematic energy of the rotor system is the sum of the translational and the rotational kinetic energy. The total kinetic energy is:

The four Lagrange equations are:

Introducing the complex coordinates:

Then Eq. (5) can be written as:

where $K_{\varphi }=2k{a}^2$.

The former equation of Eq. (7a) represents the cylindrical mode and the latter represents the concial mode. By solving the characteristic Eq. (8), frequency of the conical mode can be obtained:

The solutions consists of a Forward Whirl at $s=j{\omega }_1$:

and the Backward Whirl $s=-j{\omega }_2$:

Eq. (9) reveals that, the forward whirl frequency of the conical mode ${\omega }_1>\mathrm{\Omega }$ when $I_z>I_x$, which indicates the inexistence of forward whirl critical speed.

For the test rig shown in Fig. 1, parameters for simulations are listed in Table 2.

3.1. Modeling of the component

Bolt hole, chamfer and other tiny structures are overlooked and the main components are modelled, as shown in Table 3 and Fig. 3.

3.2. Connecting relations and boundary conditions

Mounting seat, connection pad, squirrel cage and bearing housing are rigid-connected together. A master node is established at the center of the hollow shaft and coupled with the corresponding nodes on shaft as rigid region. Bearing is simplified as four Combin14 elements with stiffness of 1×10⁸ N/m. All DOFs of nodes at the bottom of mounting seat are constrained. Finite element model of the rotor system is shown in Fig. 4, which has 10070 elements and 24812 nodes in all.

Fig. 3Finite element model of some components

a) Squirrel cage

b) Bearing

c) Bearing Housing

d) Assembly

Fig. 4Finite element model of the rotor system

4.1. Natural frequencies and modal shapes

Frequencies and modal shapes of the first three orders obtained by 3D finite element model are shown in Fig. 5. The results reveal that, for the 1st and 2nd mode, deformations mainly occur on the squirrel cages. Difference value between the 2nd and the 3rd order frequencies is over 500 Hz. When the running speed is below the 3rd mode, the rotor system can be considered as a rigid rotor on elastic supports.

Hammering method is used to obtain the natural frequencies of the static system and the FRF curves are shown in Fig. 6. Two peaks appear in the FRF, whose frequencies are 75.49 Hz and 139.98 Hz, respectively.

Fig. 5The first three modes of the rotor system

a) 75.85 Hz

b) 125.44 Hz

c) 648.33 Hz

The difference ratio of the 2nd order frequency between calculated and experimental results is relatively large, which results from the inconsistent of mass and moment of inertia because of the overlook of the bearing mass in modeling process. Model updating is adopted based on the measured modes and the results are listed in Table 4.

Fig. 6FRF curves obtained by hammering method

4.2. Critical speed

Campbell diagram of the rotor system when the rotational frequency varies from 0 to 300 Hz, as shown in Fig. 7. Campbell diagram reveals that natural frequency of the 1st order is not affected by rotational speed, while the 2nd order shows obvious difference between the forward and backward whirling motions. Critical speed of the first two orders are listed in Table 5.

Fig. 7Campbell diagram of the rotor system

a) FE results

b) Analytical results

The rotor starts up from 0 Hz up to 90 Hz rapidly and then freely decays to stationary state. Electrical vortex sensor is used to detect the vibration of the supporting point. Spectrum cascade obtained in run-up and free decay test is shown in Fig. 8. Peak of the vibration amplitude appears at 74.63 Hz, which corresponds to the 1st order critical speed.

4.3. Imbalance response

Damping parameters should be determined when the imbalance response is calculated. Using of the amplitude-frequency diagram and Half-power Bandwidth Method:

Damping ratio is obtained as $\xi =\mathrm{}$0.0356.

The amount of unbalance 50 g∙cm, combined with the damping ratio $\xi =\mathrm{}$0.0356 are introduced into the finite element model. Transverse responses of the node near the end far from motor are shown in Fig. 9, in which the simulated and measured results agree well with each other.

Results of the natural characteristics, critical speed and imbalance response all validate the effectiveness of the 3D finite element model.

Fig. 8Spectrogram obtained by run-up and free decay test

Fig. 9Comparision of the simulated and measured results

5.1. Description of the bearing misalignment

For the rotor system with two bearings as shown in Fig. 1, bearing misalignment often appears on the bearing far from the motor end (see Fig. 10). When the mounting seat has a bias of $\delta \mathrm{\text{'}}$ and thus bearing $A$ has an eccentricity of $\delta$, the deformation mainly appears on the squirrel cage elastic supports and an additional bending moment arises.

In the FEM of rotor system, a bias $\delta \mathrm{\text{'}}$ is imposed the mounting seat to simulate the bearing misalignment. Response of the gyroscopic effect and large rotation effects is considered in the calculation. Then vibrations of the shaft and the stress on squirrel cage elastic supports are investigated.

Fig. 10Rigid rotor on elastic supports with bearing misalignment

5.2. Vibration of the shaft

Three cases are selected for simulation, that is below the 1st order critical speed 65 Hz, at the critical speed 75 Hz and over the 1st order critical speed 85 Hz. Simulation results are shown in Table 6 and Fig. 11.

Fig. 11 shows the transverse vibrations and rotor orbits at different rotational speeds under the eccentricity $\delta \text{'}=$ 2 mm. No obvious features appear in the frequency spectra and orbits. Measured results show the same feature, as shown in Fig. 12.

Fig. 11Simulation results when eccentricity δ'= 2 mm

a)$\mathrm{\Omega }=\mathrm{}$65 Hz

b)$\mathrm{\Omega }=\mathrm{}$75 Hz

c)$\mathrm{\Omega }=\mathrm{}$85 Hz

Fig. 12Measured results when δ'= 2 mm at the frequency of 65 Hz

5.3. Stress on the squirrel cage elastic supports

Stress is calculated and measured to study the influence of the bearing misalignment on the dynamic stresses of squirrel cage elastic supports. The location of strain gauge is shown on Fig. 13.

Fig. 13Location of the strain gauge

Fig. 14Stress of simulation on the squirrel cage elastic supports

a) Distribution of the stress

b)$\delta \text{'}=0$

c)${\delta }^{\text{'}}=\mathrm{}$1 mm

d)${\delta }^{\text{'}}=\mathrm{}$2 mm

Fig. 14(a) shows the stress distribution on squirrel cage elastic supports, and Fig. 14(b)-14(d) illustrate the stress under different eccentricities, which are obtained by simulation. It can be seen that, the stress values increase with the increase of eccentricity $\delta \mathrm{\text{'}}$. However, the peak-to-peak value is consistent.

Fig. 15 is the measured results of stress on the squirrel cage, the same features as simulation is obtained.

Fig. 15Stress of the measured results at 65 Hz when δ'= 2 mm

a) Time domain wave

b) Frequency spectrum