Estimation of speed-dependent bearing dynamic parameters in rigid rotor systems levitated by electromagnetic bearings

Mechanism and Machine Theory 92 (2015) 100–112

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Estimation of speed-dependent bearing dynamic parameters in rigid rotor systems levitated by electromagnetic bearings Rajiv Tiwari ⁎, Viswanadh Talatam Department of Mechanical Engineering, Indian Institute of Technology Guwahati, Guwahati, Assam 781039, India

a r t i c l e

i n f o

Article history: Received 14 August 2014 Received in revised form 12 March 2015 Accepted 8 May 2015 Available online xxxx Keywords: Active magnetic bearing Displacement and current stiffness Identification Unbalance

a b s t r a c t In the present work, an identification algorithm to estimate dynamic parameters (i.e., displacement stiffness and current stiffness) of active magnetic bearings (AMBs) and residual unbalances in a rigid rotor system levitated on AMBs has been developed. AMB dynamic parameters are considered to be dependent on the rotor speed and the run-up data could be used effectively in the present identification algorithm. To test the proposed algorithm, displacements and currents are generated with the help of a numerical simulation. For this purpose, a comprehensive four-degree-of-freedom model with a rigid rotor levitated on AMBs has been developed, which is also used for development of the identification algorithm. Then the estimation of parameters is performed based on leastsquares fit technique in frequency domain. To check the robustness of the identification algorithm, modelling errors and measurement noises have been incorporated during the parameter estimation and deviation in estimated parameters is found to be reasonable. © 2015 Elsevier Ltd. All rights reserved.

1. Introduction The most prominent problem in rotor dynamics is the dynamic balancing of rotors as even a small unbalance will cause excessive vibrations at high speeds, which may lead to catastrophic failures. Overall it reduces the efficiency of rotor-bearing systems and if this residual unbalance exceeds certain level will produce large amount of unbalance forces. This leads in vibrations that would prove to be fatal and causing failure to the entire rotor system [1–3]. Now-a-days there is a large usage of high-speed rotating machines; hence we require a control system integrated with the rotor system. Active magnetic bearings (AMBs) are such a choice, which are very much suitable to above requirements and also impart the contactless motion that helps in reducing the wear and tear of the rotor system. It is to be noted that the design of AMBs is very complex when compared to conventional bearings, as it includes the design of controllers, actuators and amplifiers, individually and simultaneously [4]. Vibrations in rotor systems can be reduced in principle actively and instantaneously by AMBs based on measured vibration responses from the system with the help of controllers, amplifiers and magnetic actuators. The several research efforts on AMBs have been performed on the control engineering. In synthesis of controllers, diverse control principles have been implemented to magnetic bearings from the traditional analogue controller to the modern digital ones [5, 6]. The modelling, simulation and control of the AMB system were studied by Binder et al. [7] with the PID and state space controllers. AMBs do not aim in calculating residual unbalances in the system and try to remove it by applying additional magnetic force opposite to unbalance forces. However, it will be advantageous, if we reduce residual unbalances to minimum by physically balancing the rotor so that it will reduce the continuous effort of the controller and the power consumption in rotor systems that are levitated on AMBs. Zhou and Shi [8] studied the active control and balancing of rotating systems, and they concluded that active balancing

⁎ Corresponding author. Tel.: +91 361 2582667; fax: +91 361 2690762. E-mail addresses: [email protected] (R. Tiwari), [email protected] (V. Talatam).

http://dx.doi.org/10.1016/j.mechmachtheory.2015.05.007 0094-114X/© 2015 Elsevier Ltd. All rights reserved.

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101

suppresses vibrations due to unbalances as they eliminate residual unbalances, which is the cause for the vibration. Edwards et al. [9] used the finite element method to identify both residual unbalances and flexible support parameters of a rotor-foundation system. Sudhakar and Shekhar [10] have identified rotor unbalance while investigating model-based fault identification using vibration minimization methods. Tiwari et al. [11] identified inherent unbalances in a flexible rotor levitated on AMBs using an extended influence coefficient method, which correlates magnetic forces and applied unbalance forces with the help of trial unbalances. The majority of work is related to the reduction of residual unbalance responses in the system and a very little effort is contributed for development of methods for the experimental estimation of AMB dynamic parameters without which one cannot analyse the rotor bearing-system properly. However, for conventional bearings several methodologies for experimental estimation of bearing dynamic parameters have been reported [12]. Qiu and Tieu [13] developed a method to determine dynamic coefficients of journal bearings from impulse responses using least-squares estimation and verified their work using the experimentally identified coefficients. Tiwari [14] proposed an identification algorithm for a four-degree of freedom conventional rigid-rotor flexible-bearing system and in addition to that he proposed different methods to improve the condition of proposed algorithm. Tiwari and Chakravarthy [15] developed an identification algorithm for the simultaneous estimation of speed-dependent bearing parameters and residual unbalances by using the impulse response measurement for a multi-degree of freedom flexible conventional rotor-bearing system. The majority of works on the estimation of magnetic forces and stiffness parameters of AMBs have been performed by the FEM formulation of the bearing magnetic field. To evaluate the force and the stiffness of radial AMBs, the linear magnetic circuit theory was utilized [16]. In a canned motor pump, the conformity of measurement was observed to be sensible at small eccentricities. Using finite element techniques a magnetic bearing actuator force was studied [17]. A nonlinear finite element technique was utilized to establish the force of a radial magnetic bearing [18] through Maxwell's stress tensor technique. A non-linear two-dimensional finite element method is performed to calculate the effectiveness of radial magnetic bearings [19]. Linearized variables for the dynamic model of AMBs at diverse process conditions were obtained. Due to actual test conditions at bearing locations (magnetic flux leakage losses, effect of non-uniform temperature on magnetic material properties, etc.) are difficult to access practically, hence; even when theoretical analysis procedure is accurate, but does not estimate acceptable values. This encourages estimating these variables by experimentation in genuine test situation. On the evaluation of dynamic parameters of active magnetic bearings advances of the experimental method are exceptional. The AMB force magnitude on a rotor levitated by AMBs could be predictable indirectly by either measuring the magnetic flux density with imbedded Hall sensors into the air gap or by measuring actuator winding currents and shaft displacements [20, 4]. By application of calibrated forces on the rotor the effectiveness of these methods was compared by Aenis et al. [20]. They observed that the flux based procedure is more precise in comparison to the current–displacement method. However, accommodating Hall sensors in the AMB air gap is a real practical problem. To estimate the forces applied by an AMB through the FEM formulation an effort was made in [21]. A multi-point technique for the force estimation in AMBs was proposed by Kasarda et al. [22], in which they concluded that it was a more precise method than traditional force measuring methods. Tiwari and Chougale [23] developed an identification algorithm for the simultaneous estimation of speed-independent AMB parameters and residual unbalances for a rigid rotor system fully levitated on AMBs. Tiwari and Chougale [24] also developed the identification algorithm for the speed-independent AMB parameters using flexible rotor fully levitated on AMBs and validated their work with experimental data. Wang et al. [25] concluded in their work that both stiffness and damping of AMB system are complex functions of the frequency, and they also provided relations of the stiffness and damping performances with reference to the frequency. Hence, our focus lies on the development of methodology for the estimation AMB dynamic parameters along with residual unbalances in which AMB dynamic parameters have been considered to be dependent on the rotor speed, especially methodology suitable to be used with the help of measurements during the run-up data of a rotor system. This gives difficulty because the number of AMB parameters increases with the number of speeds of interest. So identification algorithm deviates as compared to the previous works on AMBs.

Fig. 1. Rigid rotor fully levitated on AMBs.

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In the present work, a methodology for simultaneously estimating the speed-dependent AMB dynamic parameters and residual unbalances has been proposed for a rigid-rotor system levitated on AMBs. To accomplish this, the measurement of controlling currents and unbalance responses at AMB locations has to be performed at different speeds or during run-up of the rotor system. To illustrate this methodology a simple rigid-rotor system levitated on two AMBs at two ends has been considered. Numerically simulated unbalance responses and controlling currents at discrete speeds have been utilized for estimating speed-dependent AMB parameters

Fig. 2. Flow chart for identification algorithm.

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Table 1 Physical properties of the rotor. Parameter

Numerical value

Total mass of the shaft, m Total length of the shaft (l1 + l2) Distance of C.G. from AMB 1, l1 Distance of C.G. from AMB 2, l2 Distance of C.G. from Disc 1, r1 Distance of C.G. from Disc 2, r2 Transverse moment of inertia of the system, it Polar moment of inertia of the system, ip Residual unbalance with angular position at Disc 1, (m1e1, ϕ1) Residual unbalance with angular position at Disc 2, (m2e2, ϕ2) Trail mass unbalance with angular position at Disc 1, (mt1et1, ϕt1) Trail mass unbalance with angular position at Disc 2, (mt2et2, ϕt2)

4 kg 400 mm 195 mm 205 mm 120 mm 180 mm 0.0786 kg-m2 0.0046 kg-m2 0.008 kg-m, 30° 0.004 kg-m, 60° 0.012 kg-m, 80° 0.020 kg-m, 110°

using the algorithm that is based on least-square fit. These estimates are checked in the presence of the addition of measurement noises and modelling errors. 2. Model descriptions and development of identification algorithms A description of the rotor system that has been considered for the present study is shown in Fig. 1. Distances mentioned in Fig. 1 are measured from the centre of gravity as a reference. l1 and l2 are distances of AMBs from the reference point, and r1 and r2 are distances of rigid discs, which are balancing planes also. In this model we have considered unbalance forces, inertia forces, gyroscopic effects and AMB forces. Equations of motion are given as follows [23] € −ωG η¼ f unb −Ks ðωÞη−Ki ðωÞic Mη


ð1Þ

where M, G and funb represent the mass and gyroscopic matrices, and the unbalance force vector, respectively; Ks(ω) and Ki(ω) represent the speed-dependent displacement and current stiffness matrices of AMBs, respectively; and η and ic represent the displacement and controlling current vectors, respectively. All matrices and vectors are provided in Appendix A. Now Eq. (1) will be transformed into frequency domain using a solution as η ¼ ηe jωt , ic = īc ejωt and f unb ¼ f unb e jωt . Here η, īc and f unb represent the complex displacement, controlling current and unbalance force vectors, respectively. On using these solutions, Eq. (1) could be transformed as
 2 2 ω M þ jω G η ¼ − f unb þ Ks ðωÞη þ Ki ðωÞic :

Fig. 3. A SIMULINK model for the rotor system.

ð2Þ

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Ksx1

Ksx2

Ksy1

Ksy2

160000 140000 120000 100000 80000 60000 40000 20000 0 5 Hz

10 Hz

15 Hz

Fig. 4. Percentage deviation in AMB parameters with the addition of noise for spin speed of 5 Hz.

Now, terms in the above equation are arranged in such a way that all unknown quantities (i.e. related to residual unbalances and AMB dynamic parameters along with associated known coefficients) in the left-hand side and remaining known terms in right-hand side, so that we get
 2 2 Ks ðωÞη þ Ki ðωÞic −f unb ¼ ω M þ jω G η:

ð3Þ

Right-hand side matrices and vectors in Eq. (3) are then transformed in such a way that all known quantities (i.e. model parameters of the system, measured current information and unbalance responses) are in regression matrices and remaining unknown terms are in vectors. This form of equations is shown below ½ A1 ðωÞ A2 ðωÞ

9 8 < ks ðωÞ = A3 ðωÞ  ki ðωÞ ¼ b1 ðωÞ ; : uc

ð4Þ

with 2

X1 6 0 6 A1 ¼ 4 0 0

0 Y1 0 0

0 0 X2 0

3 0 0 7 7 ; 0 5 Y 2 44

2

Ix1 6 0 6 A2 ¼ 4 0 0

0 Iy1 0 0

0 0 Ix2 0

I y2

3 2 2 −ω ðl2 þ r 1 Þ −ω ðl2 −r 2 Þ 7 6 2 2 jω ðl2 −r 2 Þ 7 6 jω ðl2 þ r 1 Þ A3 ¼ 6 7 4 −ω2 ðl1 −r 1 Þ −ω2 ðl1 þ r 2 Þ 5 2 2 jω ðl1 −r 1 Þ − jω ðl1 þ r2 Þ 42 2

3 0 0 7 7 0 5

;

44

Kix1

Kix2

Kiy1

100 90 80 70 60 50 40 30 20 10 0 5 Hz

10 Hz

15 Hz

Fig. 5. AMB displacement stiffness parameters (N/m) for different speeds.

Kiy2

ð5Þ

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105

Table 2 Displacement responses for different speeds. Displacement responses (m)

Speed—5 Hz (1.32 (0.80 (0.74 (0.87 (1.71 (3.48 (−0.39 (5.36

X1 Y1 X2 Y2 X1t Y1t X2t Y2t

+ − + − + − + +

Speed—10 Hz

j0.71) j1.47) j0.85) j0.75) j3.09) j1.90) j5.26) j0.41)

× × × × × × × ×

−5

10 10−5 10−5 10−5 10−5 10−5 10−5 10−5

(0.32 (0.16 (0.14 (0.16 (0.39 (0.67 (−0.12 (0.94

+ − + − + − + +

j0.19) j0.27) j0.18) j0.13) j0.77) j0.33) j1.07) j0.11)

Speed—15 Hz × × × × × × × ×

−5

10 10−5 10−5 10−5 10−5 10−5 10−5 10−5

(0.25 + j0.15) (0.12 − j0.21) (0.11 + j0.14) (0.12 − j0.097) (0.31 + j0.60) (0.52 − j0.26) (−0.09 + j0.83) (0.72 + j0.082)

× × × × × × × ×


 2 2 b1 ðωÞ ¼ ω M þ jω G η

10−5 10−5 10−5 10−5 10−5 10−5 10−5 10−5

ð6Þ

and 8 9 ksx ðωÞ > > > > 1 > > > < ks ðωÞ > = y1 ks ðωÞ ¼ ; ð ω Þ k > > s > > x > > > 2 > : ; ksy ðωÞ

8 9 kix ðωÞ > > > > 1 > > > < ki ðωÞ > = y1 ki ðωÞ ¼ ; ð ω Þ k > > i > > x > > > 2 > : ; kiy ðωÞ

2

 uc ¼



U1 U2

ð7Þ

2

where X and Y are complex displacements, Ix and Iy are complex currents, subscripts 1 and 2 represent disc locations, ks and ki represent the displacement stiffness and current stiffness coefficients of the AMB, U represents the unbalance in a complex number form, and subscripts x and y represent orthogonal transverse directions. It should be noted that matrices on the left hand side and the vector on the right hand side in Eq. (4) are function of the spin speed, ω, so we get  ½ A4 ðωÞ A3 ðωÞ410

kðωÞ uc

 101

¼ b1 ðωÞ41

ð8Þ

with A4 ðωÞ ¼ ½ A1 ðωÞ

⌊

A2 ðωÞ  ;

⌋:

kðωÞ ¼ ksx ðωÞksy ðωÞksx ðωÞksy ðωÞkix ðωÞkiy ðωÞkix ðωÞkiy ðωÞ 1

1

2

2

1

1

2

2

ð9Þ

T

Matrix A4 is given in Appendix B. On splitting the real and imaginary parts in Eq. (8), we get h

r

i

r

i

A4 ðωÞ þ jA4 ðωÞ A3 ðωÞ þ jA3 ðωÞ



i 410

kðωÞ i u þ ju



r


 r i ¼ b1 ðωÞ þ jb1 ðωÞ

101

ð10Þ

41

where superscripts r and i denote the real and imaginary parts, respectively. On expanding Eq. (10) and rearranging as follows, "

r

r

i

A4 ðωÞ A3 ðωÞ ‐A3 ðωÞ i i r A4 ðωÞ A3 ðωÞ A3 ðωÞ

9 8 " # r < kðωÞ = b1 ðωÞ r ¼ u i : i ; b1 ðωÞ 81 u 820 201

#

ð11Þ

Table 3 Current responses for different speeds. Current responses (Amp)

Speed—5 Hz

Speed—10 Hz

Speed—15 Hz

Ix1 Iy1 Ix2 Iy2 Ix1t Iy1t Ix2t Iy2t

0.0508 + j0.0317 0.0358 − j0.0569 0.0274 + j0.0357 0.0365 − j0.0278 0.0607 + j0.1281 0.1439 − j0.0673 −0.0287 + j0.2096 0.2134 + j0.0300

0.0115 + j0.0093 0.0080 − j0.0099 0.0047 + j0.0080 0.0070 − j0.0041 0.0111 + j0.0332 0.0287 − j0.0094 −0.011 + j0.0042 0.0369 + j0.0098

0.0085 + j0.0081 0.0069 − j0.0073 0.0032 + j0.0066 0.0057 − j0.0028 0.0067 + j0.0270 0.0230 − j0.0056 −0.0113 + j0.0324 0.0283 + j0.0099

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Table 4 Comparison of the assumed and estimated bearing parameters for the rotor spin speed of 5 Hz. Parameter

Assumed value

Estimated value

5

% Deviation

5

0.0365

ksx1 (N/m)

1.0500 × 10

ksx2 (N/m)

1.3000 × 105

1.3000 × 105

0.0254

ksy1 (N/m)

1.2500 × 105

1.2500 × 105

0.0273

ksy2 (N/m)

1.5000 × 105

1.5000 × 105

0.0226

kix1 (N/A)

85.00

85.00

0.0021

kix2 (N/A)

80.00

80.00

0.0023

kiy1 (N/A)

68.00

68.00

0.0030

kiy2 (N/A)

73.00

73.00

0.0029

1.0500 × 10

with 

r

u ¼

(

r  U1 ; r U2

i

)

U1 i U2

i

u ¼

:

Now we will write the above equation in a matrix form to get a single regression equation  ½ A5 ðωÞ A6 ðωÞ 812

kðωÞ u

 121

¼ b2 ðωÞ81

ð12Þ

with " A5 ¼

 u¼

r

A4 ðωÞ i A4 ðωÞ

r

u i u

 ;

#

" ;

A6 ¼

88

( b2 ðωÞ ¼

r

b1 ðωÞ i b1 ðωÞ

r

i

A3 ðωÞ ‐A3 ðωÞ i r A3 ðωÞ A3 ðωÞ

# ; 84

) :

ð13Þ

81

Matrix A5 contains information related to the displacement and current responses, Matrix A6 represents the information related to unbalances and vector b2 contains information that includes rotor model. Matrices A5, A6 and b2(ω) are given in Appendix B. From the above form of estimation equation, it could be seen that the number of unknowns (i.e., twelve) are more than number of equations (i.e., eight), which is a case of under-determinate system of linear simultaneous equations. It could be made determinate equations of system and the condition of regression matrix can be improved by taking measurements with different sets of trial masses (minimum one without trial mass and the other with a trial mass; it is assumed that the rotor has always some amount of residual unbalance) at the same rotational speed of the rotor. Now if we consider known trial masses, equations of motion can be written as follows t

€ −ωG η¼ f unb þ f unb −Ks η−Ki ic : Mη


ð14Þ

Table 5 Comparison of the assumed and estimated bearing parameters for the rotor spin speed of 10 Hz. Parameter

Assumed value

Estimated value

% Deviation

ksx1 (N/m)

67.020 × 105

67.020 × 105

0.0074

ksx2 (N/m)

82.980 × 105

82.980 × 105

0.0063

ksy1 (N/m)

79.790 × 105

79.780 × 105

0.0065

ksy2 (N/m)

95.750 × 105

95.740 × 105

0.0061

kix1 (N/A)

138.41

138.41

0.0037

kix2 (N/A)

130.26

130.26

0.0048

kiy1 (N/A)

110.73

110.73

0.0009

kiy2 (N/A)

118.87

118.87

0.0007

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107

Table 6 Comparison of the assumed and estimated bearing parameters for the rotor spin speed of 15 Hz. Parameter

Assumed value

Estimated value

5

% deviation

5

0.0095

ksx1 (N/m)

198.97 × 10

ksx2 (N/m)

246.34 × 105

246.33 × 105

0.0079

ksy1 (N/m)

236.87 × 105

236.85 × 105

0.0074

ksy2 (N/m)

284.24 × 105

284.22 × 105

0.0069

kix1 (N/A)

245.22

245.27

0.0220

kix2 (N/A)

230.80

230.77

0.0280

kiy1 (N/A)

196.18

196.23

0.0104

kiy2 (N/A)

210.60

210.58

0.0087

198.95 × 10

A trial mass unbalance force vector is given by

t t t 2 j f unb ¼ m1 e1 ω e ð

t ωtþϕ1

8 8 9 9 ðl þ r 1 Þ > > > > > ðl2 −r 2 Þ > > > < 2 = = t < Þ − jðl2 þ r1 Þ þ mt et ω2 e jðωtþϕ2 Þ − jðl2 −r 2 Þ 2 2 ðl þ r 2 Þ > > > > ðl1 −r 1 Þ > > > > : : 1 ; ; − jðl1 −r 1 Þ − jðl1 þ r 2 Þ

ð15Þ

where mt1 and mt2 are trial masses at discs 1 and 2, respectively; et1 and et2 are eccentricity of discs 1 and 2, respectively; and ϕt1 and ϕt2 are trial mass angular locations at discs 1 and 2, respectively. It should be noted that corresponding to Eq. (14), displacements and currents also are expressed with superscript t. Following the procedure to find the identification algorithm as described above we get the estimation equation with the inclusion of trial mass as follows, 

t

t

A5 ðωÞ A6 ðωÞ



 812



kðωÞ u

t

121

¼ b2 ðωÞ81 :

ð16Þ t

The superscript t in above denotes the inclusion of trial mass in the rotor-bearing system. Elements of At5 ; At6 andb2 ðωÞ are given in Appendix B. Combining Eqs. (12) and (16), we get a matrix form as follows  ½ A7 ðωÞ A8 ðωÞ 1612

kðωÞ u

 121

¼ b3 ðωÞ161

ð17Þ

with A5 ðωÞ ; t A5 ðωÞ

 A8 ðωÞ ¼

A6 ðωÞ ; t A6 ðωÞ

 b3 ðωÞ ¼

 b2 ðωÞ : t b2 ðωÞ

ð18Þ

1% Noise 2% Noise 5% Noise

10

Percentage deviation

 A7 ðωÞ ¼

8 6 4 2 0

ksx1

kix1

ksy1

kiy1

ksx2

kix2

Fig. 6. AMB current stiffness parameters (N/Amp) for different speeds.

ksy2

kiy2

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Percentage deviation

10 1% Noise 2% Noise 5% Noise

8 6 4 2 0

ksx1

kix1

ksy1

kiy1

ksx2

kix2

ksy2

kiy2

Fig. 7. Percentage deviation in AMB parameters with the addition of noise for spin speed of 10 Hz.

For the consistent estimation, the rotor is run at n different speeds. Hence, number of equations would be equal to (16 × n) and the number of unknowns will be equal to (8 × n + 4), where n represents the number of speeds used for the estimation. The estimation equation would take the following form 2

A7 ðω1 Þ ½0168 ½0168 6 ½0168 A7 ðω2 Þ ½0168 6 4 ⋮ ⋮ ⋱ ½0168 ½0168 …

8 9 3 3 2 kðω1 Þ > > > > … A8 ðω1 Þ b3 ðω1 Þ > > > > ð Þ k ω < = 2 6 b3 ðω2 Þ 7 … A8 ðω2 Þ 7 7 7 ¼6 : ⋮ 5 5 4 ⋮ ⋮ ⋮ > > > > k ω ð Þ > > n > A7 ðωn Þ A8 ðωn Þ ðð16nÞð8nþ4ÞÞ > : ; b3 ðωn Þ ðð16nÞ1Þ u ðð8nþ4Þ1Þ

ð19Þ

From above equation all speed-dependent AMB dynamic parameters and residual unbalances would be estimated in a single run. In the next section, Eq. (19) is used for the simultaneous estimation of bearing parameters and unbalances with the help of numerically generated displacements and current responses. 3. Numerical simulation The procedure for testing the proposed identification algorithm is illustrated with the help of a flowchart in Fig. 2. The proposed identification algorithm is tested with a 4-DOF rigid rotor system using a numerical simulation and physical properties of which are shown in Table 1. The PID controller has been chosen with following properties: KP = 4200 Amp/m, KI = 2000 Amp/m-s and KD = 10.0 Amp-s/m. The numerical model is made with the help of SIMULINK to get rotor displacements and AMB currents in two orthogonal directions at two AMB locations, and is shown in Fig. 3. To generate responses using the numerical simulation, we have assumed both AMB dynamic parameters for discrete speeds (shown in Figs. 4 and 5) and residual unbalances (which are independent of speed and are given in Table 1). For the present case, the rotor is given spin speeds of 5 Hz, 10 Hz and 15 Hz, and assumed trial mass unbalances at discs 1 and 2 are given in Table 1. Numerically generated displacements and currents for different speeds are tabulated in Tables 2 and 3, respectively. In these tables, subscripts 1 and 2 indicate disc locations and subscript t represents inclusion of trial masses. The complex form

Percentage deviation

10

1% Noise 2% Noise 5% Noise

8 6 4 2 0

ksx1

kix1

ksy1

kiy1

ksx2

kix2

ksy2

Fig. 8. Percentage deviation in AMB parameters with the addition of noise for spin speed of 15 Hz.

kiy2

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109

Table 7 Comparison of the assumed and estimated unbalances for both magnitude and phase and its percentage deviation after the addition of noise in responses. Disc

1 2

Assumed me (kg-m)

Assumed ϕ (deg)

Estimate-d me (kg-m)

Estimated ϕ (deg)

Percentage deviation in me on addition of noise 1%

2%

5%

1%

2%

5%

0.008 0.004

30 60

0.008 0.004

30 60

0.001 0.001

0.001 0.002

0.001 0.001

0.007 0.004

0.001 0.002

0.004 0.006

Percentage deviation in ϕ on addition of noise

of displacements and currents is generated by considering phase information with respect to a fixed reference point of the shaft and these data will help in identifying the unbalance angular position also with respect to the shaft reference point. With the help of these currents and displacements, Eq. (19) is used to get estimated values of assumed AMB (for all speeds of interest) and residual unbalance parameters. The comparison of assumed and estimated bearing dynamic parameters is tabulated in Tables 4, 5 and 6 for spin speeds of 5 Hz, 10 Hz and 15 Hz, respectively. Comparison is excellent when no noise in responses is added. Up to the decimal point presented the data looks same but at higher decimal point some difference exists and that reflects in a minimal percent variation of the estimated and assumed parameters. In order to check the robustness of the algorithm, a random noise (1, 2 or 5%) is added to the current and displacement signals, and the estimates are obtained. From these tables we can infer that the algorithm is robust enough as there is minimal variation in the assumed and estimated bearing dynamic parameters. Figs. 6–8 show the percentage deviation of estimated bearing parameters after the addition of noise for the rotor spin speed of 5 Hz, 10 Hz and 15 Hz, respectively. In these figures, the abscissa represents bearing dynamic parameters and the ordinate the percentage deviation of estimated parameters from assumed values. It can be inferred from these figures that addition of the noise up to 2% gives less than 5% deviation on estimated parameters and for the noise of 5% gives deviation of less than 11% in estimates. The comparison of assumed and estimated unbalance magnitude and phase of discs 1 and 2 without the addition of noise and their percentage deviation after the addition of noise in responses is summarized in Table 7. It can be inferred that the addition of noise has hardly any effect on the estimates of residual unbalance magnitude and phase. Now modelling errors in the present model are introduced to see variations of estimates. Modelling errors are simply the error in values of model parameters, which may occur due to lack of knowledge of the system or measurement constraints. Random errors in the range of 1, 2 and 5% are added to system parameters, the like damping and stiffness coefficients, which are used in estimation model. AMB parameters and residual unbalances are then estimated with the present identification algorithm. Fig. 9 shows the percentage deviation of estimated bearing parameters after the addition of bias error in rotor parameters (geometrical and mass parameters in Table 1) for the rotor spin speed of 5 Hz. In this figure, the abscissa represents bearing parameters and the ordinate the percentage deviation of the estimated parameters from assumed values. A comparison of the assumed and estimated unbalance magnitudes and phases of discs 1 and 2 without the addition of modelling errors and their percentage deviation after the addition of modelling errors is summarized in Table 8. From Fig. 9 and Table 8, we can infer that as modelling errors increase the estimation of bearing parameters also increases in the same proportion, however, residual unbalances show minimal variations even for 5% modelling error.

4. Conclusions In the present work, an identification algorithm for the simultaneous estimation of speed-dependent AMB parameters and residual unbalances has been developed. The identification algorithm has been illustrated with the help of SIMULINK model by which responses are generated and these responses are used in the estimation of AMB parameters and residual unbalances. The estimates (AMB dynamic parameters and residual unbalances) are found to be in good agreement when compared them with their assumed

1% Error 2% Error 5% Error

Percentage deviation

10 8 6 4 2 0

ksx1

kix1

ksy1

kiy1

ksx2

kix2

ksy2

kiy2

Fig. 9. Percentage deviation in AMB parameters with the addition of modelling errors for speed of 5 Hz.

110

R. Tiwari, V. Talatam / Mechanism and Machine Theory 92 (2015) 100–112

Table 8 Comparison of the assumed and estimated unbalances for both magnitude and phase and its percentage deviation after the addition of modelling errors. Disc

1 2

Assumed me (kg-m)

Assumed ϕ (deg)

Estimated me (kg-m)

Estimated ϕ (deg)

0.008 0.004

30 60

0.008 0.004

30 60

Percentage deviation in me on addition of errors

Percentage deviation in ϕ on addition of errors

1%

2%

5%

1%

2%

5%

0.001 0.002

0.002 0.003

0.004 0.007

0.001 0.001

0.002 0.001

0.004 0.002

values. The identification algorithm has been checked against the addition of modelling errors and measurement noises and it is found to be robust against them. In future, a functional form for the variation of AMB dynamic parameters with respect to the speed can be developed. Also this work could be extended by considering the cross-coupled stiffness terms in developing the estimation algorithm. Appendix A. Rotor-AMB system matrices Detailed matrices of equations of motions Eq. (1) are given below [23] Mass matrix: 2

2

ml2 þ it 6 0 6 M¼6 4 ml1 l2 −it 0

0 2 ml2 þ it 0 ml1 l2 −it

ml1 l2 −it 0 2 ml1 þ it 0

3 0 7 ml1 l2 −it 7 7 5 0 2 ml1 þ it

ðA:1Þ

Gyroscopic matrix: 2

−ip 0 ip 0

0 6 ip 6 G¼4 0 −ip

3 ip 0 7 7 −ip 5 0

0 −ip 0 ip

ðA:2Þ

Displacement vector: 8 9 8 9 x > X > > > > > < 1> < 1> = = y1 Y1 ηðt Þ ¼ and η ¼ x X > > > > > 2; > > : : 2> ; y2 Y2

ðA:3Þ

Displacements with trial masses including the above vector components would have t as superscript. Current vector:

iðt Þ ¼

9 8 i ðt Þ > > > = < ix1 ðt Þ > y1

i ðt Þ > > > ; : x2 > iy2 ðt Þ

and i ¼

8 9 I > > > = < I x1 > y1

I > > > ; : x2 > Iy2

ðA:4Þ

Currents with trial masses including the above vector components would have t as superscript.Unbalance force vector: 8 8 9 9 ðl þ r 1 Þ > ðl2 −r 2 Þ > > > > > > > < 2 < = = −jðl2 þ r 1 Þ − jðl2 −r2 Þ 2 jðωtþϕ1 Þ 2 jðωtþϕ2 Þ f unb ðt Þ ¼ m1 e1 ω e þ m2 e2 ω e ðl þ r 2 Þ > > ðl1 −r 1 Þ > > > > > > : : 1 ; ; −jðl1 −r 1 Þ − jðl1 þ r2 Þ

ðA:5Þ

Controlling force vector:

f c ðt Þ ¼

8 9 k x þ kix ix1 > > > > 1 > sx1 1 > > < ks y1 þ ki iy > = y y 1 1

1

ksx x2 þ kix ix2 > > 2 2 > > :k y þ k i sy

2

2

iy

2

y2

> > > > ;

ðA:6Þ

R. Tiwari, V. Talatam / Mechanism and Machine Theory 92 (2015) 100–112

111

Displacement stiffness matrix: 2

ksx 6 01 6 Ks ¼ 6 6 0 4 0

0 ksy 0 0

0

0 7 7 7 0 7 5 ksy

0 1

3

0

ksx 2 0

ðA:7Þ

2

Current stiffness matrix: 2

kix 6 01 6 Ki ¼ 6 6 0 4 0

0

0

kiy

0 1

0

0 7 7 7 0 7 5 kiy

kix

2

0

3

0

0

ðA:8Þ

2

Appendix B. Regression matrices and vectors Detailed matrices used in regression equations Eqs. (8), (12), and (16) are presented as follows 2

X1 6 0 6 A4 ¼ 4 0 0 2

0 Y1 0 0

r

X1 6 0 6 6 6 0 6 6 0 6 A5 ðωÞ ¼ 6 X i 6 1 6 6 0 6 6 0 4 0

0 0 X2 0

0 0 0 Y2

Ix1 0 0 0

0 Iy1 0 0

r

3 0 0 7 7 0 5 Iy2 48

0 0 Ix2 0

0 r Y1 0 0

0 0 r X2 0

0 0 0 r Y2

Ix1 0 0 0

0 i Y1

0 0

0 0

Ix1 0

i Iy1

0 0

0 0

X2 0

i

0 i Y2

0 0

0 0

Ix2 0

2

2

i

−ω ðl2 þ r 1 Þ 6 6 0 6 6 −ω2 ðl −r Þ 1 1 6 6 0 t 6 A6 ðωÞ ¼ A6 ðωÞ ¼ 6 6 6 2 0 6 ω ðl þ r Þ 6 2 1 6 4 0 2 ω ðl1 −r 1 Þ

0

r Iy1

0 0 0

0 0 r Ix2 0

i

2

ðB:1Þ

3 0 0 7 7 7 0 7 r 7 Iy2 7 7 0 7 7 7 0 7 7 0 7 5 i Iy2 88

−ω ðl2 −r 2 Þ 0 2 −ω ðl1 þ r 2 Þ 0 0 2 ω ðl2 −r 2 Þ 0 2 ω ðl1 þ r 2 Þ

ðB:2Þ

3 0 0 7 2 2 −ω ðl2 þ r 1 Þ −ω ðl2 −r 2 Þ 7 7 7 0 0 7 7 2 2 −ω ðl1 −r1 Þ −ω ðl1 þ r 2 Þ 7 7 2 2 −ω ðl2 þ r 1 Þ −ω ðl2 −r 2 Þ 7 7 7 0 0 7 7 2 2 −ω ðl1 −r1 Þ −ω ðl1 þ r 2 Þ 5 0 0 84




 9 8 2 2 r 2 r 2 i 2 i > −ω ml2 þ it X 1 −ω ðml1 l2 −it ÞX 2 þ ω ip Y 1 − ω ip Y 2 > > > > >


 > > > 2 2 r 2 r 2 i 2 i > > > > > −ω ml 2 þ it Y 1 −ω ðml1 l2 −it ÞY 2 − ω ip X 1 þ ω ip X 2 > > > > > >


 > > > > 2 r 2 2 r 2 i 2 i > > > −ω ðml1 l2 −it ÞX 1 −ω ml1 þ it X 2 − ω ip Y 1 þ ω ip Y 2 > > > > >


 > > > > 2 r 2 2 r 2 i 2 i > < −ω ðml1 l2 −it ÞY 1 −ω ml1 þ it Y 2 þ ω ip X 1 − ω ip X 2 > =


 b2 ðωÞ ¼ 2 2 i 2 i 2 r 2 r > > > > −ω
ml2 þ it X 1 −ω ðml1 l2 −it ÞX 2 −
ω ip Y 1 þ
ω ip Y 2 > > > > > > > 2 2 i 2 i 2 r 2 r > > > > > ð ÞY −ω ml þ i −ω ml l −i þ ω i − ω i Y X X > > 2 1 2 1 2 t 1 2 t p p > > > >


 > > > 2 i 2 2 i 2 r 2 r > > > X Y Y −ω ð ml l −i ÞX −ω ml þ i þ ω i − ω i > 1 1 2 1 2 > 1 2 t t p p > > > >


 > > > > 2 i 2 2 i 2 r 2 r ; : −ω ðml1 l2 −it ÞY 1 −ω ml1 þ it Y 2 − ω ip X 1 þ ω ip X 2

81

ðB:3Þ

ðB:4Þ

112

R. Tiwari, V. Talatam / Mechanism and Machine Theory 92 (2015) 100–112

2

r

X 1t 6 0 6 6 6 0 6 6 0 t 6 A5 ðωÞ ¼ 6 X i 6 1t 6 6 0 6 6 0 4 0

r

0 r Y 1t 0 0

0 0 r X 2t 0

0 0 0 r Y 2t

I x1t 0 0 0

0 i Y 1t

0 0

0 0

I x1t 0

0 0

X 2t 0

i

0 i Y 2t

0 0

0

0 0

0 0

Ix2t 0

0 i Iy1t

0 0

0 0

Ix2t 0

r Iy1t

i

r

i

0 0 0

3

7 7 7 7 r 7 I y2t 7 7 0 7 7 7 0 7 7 0 7 5 i I y2t 88

ðB:5Þ






9 8 2 2 r 2 r 2 i 2 i t 2 t t 2 t > −ω ml2 þ it X 1t −ω ðml1 l2 −it ÞX 2t þ ω ip Y 1t − ω ip Y 2t þ m 1 ω ðl2 þ r 1 Þ cos ϕ1 þ m 2 ω ðl2 −r 2 Þ cos ϕ2 > > > > > >




> > > 2 2 r 2 r 2 i 2 i t 2 t t 2 t > > > > −ω Y Y X þ m ml þ i −ω ð ml l −i ÞY − ω i þ ω i þ m ω ð l þ r Þ sin ϕ ω ð l −r Þ sin ϕ > t 1 2 t p p 2 1 2 2 2 1t 2t 1t 2t 1 1 2 2 > > > >




> > > > 2 r 2 2 r 2 i 2 i t 2 t t 2 t > > > > −ω ðml1 l2 −it ÞX 1t −ω ml1 þ it X 2t − ω ip Y 1t þ ω ip Y 2t þ m 1 ω ðl1 −r 1 Þ cos ϕ1 þ m 2 ω ðl1 þ r 2 Þ cos ϕ2 > > > > >




 > > > > > = < −ω2 ðml1 l2 −it ÞY r1t −ω2 ml21 þ it Y r2t þ ω2 ip X i1t − ω2 ip X i2t þ mt 1 ω2 ðl1 −r 1 Þ sin ϕt1 þ mt 2 ω2 ðl1 þ r 2 Þ sin ϕt2 > t




 b2 ðωÞ ¼ 2 2 i 2 i 2 r 2 r t 2 t t 2 t > > > −ω ml2 þ it X 1t −ω ðml1 l2 −it ÞX 2t − ω ip Y 1t þ ω ip Y 2t þ m 1 ω ðl2 þ r 1 Þ sin ϕ1 þ m 2 ω ðl2 −r 2 Þ sin ϕ2 > > > >




 > > > > > > −ω2 ml 2 þ i Y i −ω2 ðml l −i ÞY i þ ω2 i X r − ω2 i X r −mt ω2 ðl þ r Þ cos ϕt −mt ω2 ðl −r Þ cos ϕt > > > > > t 1 2 t p p 2 1 2 2 2 1t 2t 1t 2t 1 1 2 2 > > >




> > > > > 2 i 2 2 i 2 r 2 r t 2 t t 2 t > −ω ðml1 l2 −it ÞX 1t −ω ml1 þ it X 2t þ ω ip Y 1t − ω ip Y 2t þ m 1 ω ðl1 −r 1 Þ sin ϕ1 þ m 2 ω ðl1 þ r 2 Þ sin ϕ2 > > > > > > >




 > > > 2 i 2 2 i 2 r 2 r t 2 t t 2 t ; : −ω ðml l −i ÞY −ω ml þ i Y − ω i X þ ω i X −m ω ðl −r Þ cos ϕ −m ω ðl þ r Þ cos ϕ > 1 2

t

1t

1

t

2t

p

1t

p

2t

1

1

1

1

2

1

2

2

81

ðB:6Þ

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