4.5 Numerical Experiments and Identification
4.5.1 Simulated Responses and Dynamic Effect of Residually Misaligned AMBs
Figure 4.6 Simulink diagram for FEM modelling response generation.
A Simulink model (shown in Figure 4.6) has been built to solve the global equation of motion (i.e., Equation (4.12)) of the unbalanced and misaligned flexible rotor-AMB system in the time domain, for residual and additional trial misalignments. The solutions would give the rotor displacement responses at all nodes and controlling current response at AMB locations for both the cases of AMB misalignment.
Table 4.1 The assumed data of the rotor and unbalance fault for the simulation purpose.
Table 4.2 The values of properties of AMBs and PID controller.
Rotor and unbalance fault parameters Assumed values
Shaft diameter, d and length, l 15.6 mm and 500 mm
Modulus of elasticity of shaft, E 2.1×1011 N/m2
Density of shaft material, ρ 7850 kg/m3
Disc 1 mass: md1 0.69 kg
Disc 2 and Disc 3 mass: md2, md3 0.85 kg, 1.5 kg Disc 4 and Disc 5 mass: md4, md5 1.5 kg, 0.85 kg Disc 1, Disc 3, Disc 4 eccentricities: e1, e2, e3 50, 80 and 100µm Disc 1, Disc 3, Disc 4 eccentricity phase: β1, β2, β3 10, 30 and 50deg.
Diametral mass moment of inertia of the disc 1, Id1 4.1×10-4 kg-m2 Diametral mass moment of inertia of the disc 2 and
disc 3, Id2, Id3
5.04×10-4 and 1.4×10-3 kg-m2 Diametral mass moment of inertia of the disc 4 and
disc 5, Id4, Id5
1.4×10-3 and 5.04×10-4 kg-m2 Polar mass moment of inertia of the disc 1, Ip1 8.2×10-4 kg-m2
Polar mass moment of inertia of the disc 2 and disc 3, Ip2, Ip3
10.08×10-4 and 2.8×10-3 kg-m2 Polar mass moment of inertia of the disc 4 and disc 5,
Ip4, Ip5
2.8×10-3 and 10.08×10-4 kg-m2 Distance between each disc, l1, l2, l3, l4, l5 70, 130, 120, 110, 70 (in mm)
AMB parameters Assumed Values Controller parameters
Assumed Values AMB 1 force-displacement
stiffness in x direction, ksx1
174150 N/m Proportional, kP 6000 A/m AMB 1 force-displacement
stiffness in y direction, ksy1
195000 N/m Derivative, kD 3 A-s/m AMB 2 force-displacement
stiffness in x direction, ksx2
365710 N/m Integral, kI 8000 A/m-s
AMB 2 force-displacement stiffness in y direction, ksy2
383000 N/m Air gap between rotor and AMB 1, s0
0.400 mm AMB 1 force-current
stiffness in x direction, kix1
34.83 N/A Air gap between rotor and AMB 2, s0
0.400 mm AMB 1 force-current
stiffness in y direction, kiy1
36.20 N/A Bias current, i0 2 A
AMB 2 force-current stiffness in x direction, kix2
73.14 N/A Number of poles 8
AMB 2 force-current stiffness in y direction, kiy2
75.40 N/A Angle between two
adjacent poles, α 45 deg
The numerical data of rotor and unbalance fault utilized for the numerical simulation is depicted in Table 4.1. Similarly, Table 4.2 presents the values of AMB and PID controller parameters. The assumed values of residual misalignments (δx1, δy1,δx2 and δy2) and trial misalignments (∆x1, ∆x1, ∆x1 and ∆x1) of AMBs have been taken from Table 3.1. The values of gains of PID controller (i.e., kP, kI and kD) in Table 4.2 have been selected on the basis of tuning from Routh-Hurwitz stability criteria (refer Appendix D), to have a stable vibration and controlling current responses of the flexible rotor-AMB system. For the assumed rotor model and AMB parameters, the first five natural frequencies of the rotor system using the free-free support boundary conditions are 0 Hz, 0 Hz, 53.88 Hz, 113.80 Hz and 178.48 Hz, respectively.
The first natural frequency ‘0 Hz’ corresponds to the translational motion in the x-direction at node 1, whereas the second natural frequency ‘0 Hz’ corresponds to the rotational motion in the x-z plane at the same node. The third natural frequency ’53.88 Hz’ corresponds to the translational motion in the x-direction at node 2, whereas the fourth natural frequency ’113.8 Hz’ corresponds to the rotational motion in the x-z plane at the same node, i.e., node 2. The fifth natural frequency ’178.48 Hz’ corresponds to the translational motion in the x-direction at node 3. There will be an identical set of natural frequencies for the translational motion in the y-direction and the rotational motion in the y-z plane at the corresponding nodes.
Here, the first two natural frequencies are found to be zero, which represent the translational and rotational rigid body mode shapes of the rotor. The rotor will have rigid body up and down motion corresponding to the translational rigid body mode and it will have rigid body rotation about the centre of gravity for the rotational rigid body mode.
The rotor displacements at all nodes and current responses at both AMB positions have been obtained using a fourth-order Runge-Kutta differential solver with 0.0001 s fixed time step size. The numerical simulation for the flexible rotor-AMB system was undertaken for 5 s using data given in Table 4.1, Table 4.2 and Table 3.1 (only the values of residual and trial
misalignments). To investigate the dynamic influence of AMB residual misalignment in a flexible rotor mounted on active magnetic bearings, the simulated results have been obtained at 120 Hz rotor angular frequency (i.e., rotor spin speed of 753.98 rad/s) for without and with misalignment. Figure 4.7 presents the orbit plots for the generated rotor displacement as well as controlling current output to show the effect of residual misalignment of AMB on the rotor system, at node 2 and node 5 of the shaft. Residual misalignment is the unknown misalignment between the axes of rotor and supported AMBs, which may be created due to initial setting errors, assembling errors and sensor measurement errors, etc. The nodes 2 and 5 are the locations of AMB 1 and AMB 2 in the considered flexible rotor-AMB system. At node 2, the peak values of displacement without and with misalignment are found to be at 2.86×10-5 m and 4.20×10-5 m, respectively. The absolute maximum values of current are 0.183 A and 0.893 A for without and with misalignment at the same node. Related to AMB perfect alignment at both locations, these responses are observed to be higher for the misalignment state. These increments in the percentage are 46.99% and 388% respectively, for the rotor displacement and AMB current.
Figure 4.7 Influence of AMB residual misalignment on the rotor system orbital response (a) Rotor displacement at AMB 1 (b) Controlling current at AMB 1 (c) Rotor displacement at AMB 2 (d) Controlling current at AMB 2.
Similarly, for the AMB misalignment state, the absolute peak values of vibration response and current output of PID controller at node 5 are 5.05×10-5 m and 0.982 A, respectively. However, the values are only 3.29×10-5 m and 0.211 A for the perfectly aligned case. Hence, the displacement and current responses at this node have been enhanced by 53.5%
and 365.4% due to severe misalignment fault. Growth in the response magnitudes and more power consumption motivates a researcher to identify quantitatively the residual misalignments between the rotor and AMB. One of important points can be observed from Figure 4.7(b) and 4.7(d) that the current orbit for AMB misalignment condition at nodes 2 and 5 has been moved away from the zero mean positon unlike the perfect alignment state. This is caused due to
-5 0 5
x 10-5 -5
0 5x 10-5
ux2-displacement (m) (a)
u y2-displacement (m)
-1 -0.5 0 0.5
-1 -0.5 0 0.5
ix1-current (A) (b) i y1-current (A)
-1 -0.5 0 0.5 1
x 10-4 -1
-0.5 0 0.5
1x 10-4
ux5-displacement (m) (c)
u y5-displacement (m)
-1 -0.5 0 0.5
-1 -0.5 0 0.5
ix2-current (A) (d) i y2-current (A)
With misalignment No misalignment
constant force of AMB, f1mq1 in the x- and y-directional misaligned AMB force (refer Equation (4.4)), which appears during present formulation of misalignment. This constant force also causes to displace the rotor displacement responses, but these response signals get compensated through the integral gain factor (kI) of PID controller. The integral factor shifts them to the mean position by introducing a non-zero mean biased controlling currents.
Figure 4.8 System responses in the time domain (a, b) x-displacement and current at AMB 1 (c, d) x-displacement and current at AMB 2 for residual misalignment case (e, f) x- displacement and current at AMB 1 (g, h) x-displacement and current at AMB 2 for additional trial misalignment case.