1 1
2 2
( ) ( )
( ) ( )
( n) ( n)
ω ω
ω ω
ω ω
=
A b
A b
x
A b
⋮ ⋮ (2.66)
and can be solved as Eqn. (2.65). The results obtained on the estimation of parameters covered in the identification algorithm are presented in the next section.
Figure 2-11 (a) x-displacement response during ramp up (b) Envelope of x-displacement response during ramp up
The 1× resonance is observed at 3684 rpm with peak displacement of 8.723×10-5 m. The 1
2× and 1
3
× resonances are observed at 1841 rpm and 1222 rpm respectively, with corresponding peak displacements of 2.289×10-5 m and 1.283×10-5 m, respectively. Vibration displacements at 5527 rpm is observed at 1.275×10-5 m and higher spin speeds (limited up to 10000 rpm) register vibration displacements lower than it. For testing the algorithm, two speed ranges are selected – the lower speed range below 1617 rpm (vibration displacement below 1.194×10-5 m) and the higher speed range above 5527 rpm (vibration displacement below 1.008×10-5 m). Spin speeds used for identification are presented in Table 2-4.
0 2000 4000 6000 8000
-1 -0.5 0 0.5
1x 10
-4
Spin speed (rpm) (a)
Displacement (m)
0 2000 4000 6000 8000
0 2 4 6 8x 10
-5
Spin speed (rpm) (b)
Displacement (m)
Table 2-4 Spin speeds used for identification
Speed range Spin speeds (rpm)
Lower 1140, 1200, 1260, 1320, 1380, 1440, 1500, 1560, 1620 (19 -27 Hz) Upper 5520, 5580, 5640, 5700, 5760, 5820, 5880, 5940, 6000 (92 -100 Hz)
To simulate real measurement conditions and to ascertain robustness of the identification procedure against instrument and measurement errors, random noise of 1%, 2%, and 5% are added serially to the generated response. Here, the Gaussian white noise is used for the simulation. It is defined as a statistical noise that has its probability density function equal to that of the normal distribution. It is a random signal with a flat power spectral density, uncorrelated and normally distributed with a mean zero and unit variance. The noising response signal could be obtained as
( ) ( ) ( ) (
0.5)
100
and p
noise
R N
r t r t r t −
= +
(2.67)
Here, r(t) is the vibration displacement signal as in Eqn. (2.27), Rand is a random scalar value with mean 0 and standard deviation 1 and Np is the noise %. Noisy response for AMB current is obtained similarly.
The random noise is generated with 4 different seeds and channeled to the 4 responses, viz. the x and y direction displacements and the x and y direction currents. Results of the estimation based on two speed ranges noted in Table 2-4, with clean and noise corrupted response signal are summarized in Table 2-5 and Table 2-6. It may be noted here that harmonics of vibration
displacements and AMB control currents for this estimation has been obtained from the FFT based full spectrum method.
Table 2-5 Noise sensitivity of parameters estimated in lower speed range Parameter Assumed
value
Estimated value at various noise
0% 1% 2% 5%
c 76 (Ns m-1) 75.3 75.32 74.72 74.69
% error -0.92 -0.88 -1.67 -1.71
∆k22 3×105 (N/m) 299490 299190 299770 299010
% error -0.16 -0.26 -0.8 -0.33
e 24 (µm) 23.95 23.92 23.97 24.27
% error -0.21 -0.32 -0.14 1.13
β 30º (deg.)
raradian)
30.002 30.007 30.002 30.001
% error 0.008 0.023 0.006 0.002
ks 105210 (N/m) 106410 106510 106520 106960
% error 1.14 1.24 1.25 1.66
kI 42.1 (N/A) 42.12 42.12 42.13 42.11
% error 0.04 0.04 0.05 0.03
It is noticeable from Table 2-5 that the estimation in lower speed range is quite robust against instrument errors. With up to 5% noise, the error introduced by noise is not severe for any of the parameters but the AMB constants ks and within limits acceptable for the identification purpose. It may be seen that ks shows the maximum estimation error with clean signal also. Estimates of additive crack stiffness (∆k22) and unbalance (e and β) are particularly less affected by noise. The results of identification in higher speed range is presented in Table 2-6.
Table 2-6 Noise sensitivity of parameters estimated in higher speed range Parameter Assumed
value
Estimated value at various noise
0% 1% 2% 5%
c 76 (Ns m-1) 75.86 75.71 75.57 78.52
% error -0.18 -0.37 -0.56 3.31
∆k22 3×105 (N/m) 299820 299930 299050 300210
% error -0.06 -0.02 -0.32 0.07
e 24 (µm) 23.95 23.96 23.94 24.14
% error -0.28 -0.16 -0.24 0.59
β 30º (deg.)
raradian)
30.04 30.04 29.83 30.06
% error 0.13 0.13 -0.57 0.2
ks 105210 (N/m) 107490 108470 102660 100960
% error 2.17 3.09 -2.42 -4.03
kI 42.1 (N/A) 42.157 42.253 41.77 41.9
% error 0.14 0.36 -0.77 -0.45
Comparison of results in Table 2-4(a) and Table 2-4(b) indicates more error of estimation of ks
(AMB force-displacement factor) in higher speed range. With clean response signal, the error in estimation of ks is 2.17% in higher speed range as compared to 1.14% in lower speed range. Other parameters are estimated with similar accuracy in both the speed ranges. It is notable that the noise corruption of response signals has greater effects in higher speed range as compared to the lower speed range, which is particularly pronounced in estimation of ks and c (viscous damping).
Due to multiple operational constraints, measurement of model parameters (known parameters in the grey box modelling) may contain errors which in turn would introduce error in the estimation process. Such errors are commonly known as the modelling error or bias. To simulate the modelling error and its impact on the identification algorithm, random errors (in form of noise) in range of 1%, 2% and 5% of the correct values are introduced in numerical model parameters, viz.
mass (m), intact shaft stiffness (k0) and shaft static deflection (δx). The response is generated in
accordance to Eqn. (2.31) with correct model parameters and then the error is added in model parameters in Eqn. (2.66) used for estimation in identification algorithm. Thus, correct response is used in a model containing error and the estimates obtained are compared with the assumed values to quantify estimation error due to bias error. Effect of bias error magnitude on results of estimation in lower speed range is summarized in Table 2-7. The effect of bias error is similar in higher speed range also and thus not included here.
Table 2-7 Sensitivity of estimated parameters to the modelling error Parameter Assumed
value
Estimated value at various bias error level
0% 1% 2% 5%
c 76 (Ns m-1) 75.3 76.43 76.92 79.01
% error -0.92 0.56 1.21 3.96
∆k22 3×105 (N/m) 299490 299820 299180 308120
% error -0.16 -0.06 -0.27 2.7
e 24 (µm) 23.95 23.96 23.96 23.96
% error -0.21 -0.13 -0.13 -0.13
β 30º (deg)
raradian)
30.002 30.002 30.002 30.002
% error 0.008 0.008 0.008 0.008
ks 105210
(N/m)
106410 108020 112780 114720
% error 1.14 2.67 7.23 9.51
kI 42.1(N/Amp) 42.12 42.14 42.34 44.07
% error 0.04 0.09 0.57 4.68
The developed identification algorithm found to be robust against bias errors. From Table 2-7, it is observed that rotor parameters are the least affected by bias errors with the additive crack stiffness (∆k22) deviating a maximum by 2.7% and viscous damping (c) by 3.96% at 5% bias error. AMB constants ks and kI are most vulnerable with 9.51% and 3.61% deviation, respectively, at 5% bias error.