Analysis of Electromagnetic Excitations in an Integrated Centrifugal Pump and Permanent Magnet Synchronous Motor

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and Permanent Magnet Synchronous Motor

Kim Heesoo, Posa Atte, Heikkinen Janne, Nerg Janne, Sopanen Jussi

Kim, H., Posa, A., Heikkinen, J., Nerg, J., Sopanen, J. (2019). Analysis of Electromagnetic Excitations in an Integrated Centrifugal Pump and Permanent Magnet Synchronous Motor. IEEE Transactions on Energy Conversion. DOI: 10.1109/TEC.2019.2935785

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Analysis of Electromagnetic Excitations in an Integrated Centrifugal Pump and Permanent

Magnet Synchronous Motor

1

2

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Heesoo Kim , Atte Posa, Janne Nerg , Senior Member, IEEE, Janne Heikkinen, and Jussi T. Sopanen , Member, IEEE

4 5

Abstract—The effect of unbalanced magnetic pull (UMP) caused

6

by air gap eccentricity on the vibration of a permanent magnet

7

synchronous motor (PMSM) is investigated. The force model is

8

established analytically by the Maxwell stress method. For accu-

9

rate consideration of the eccentricity condition, mixed eccentricity,

10

axial-varying eccentricity, and eccentricity caused by motor frame

11

vibration are modeled and combined. The model of the rotor–

12

bearing system, which includes the UMP model, is developed with

13

two different methods. In the first method, UMP is added as a

14

linear negative spring to the rotor model, whereas in the second

15

method, UMP is included as an external force. The rotor system of

16

a centrifugal pump driven by an integrated PMSM is modeled using

17

beam elements, and the two distinct modeling approaches for UMP

18

are applied. From the results, the UMP effect on vibration and the

19

difference between the two modeling methods are investigated. To

20

verify the results of the analysis, experimental work is done with a

21

pump test rig, and results of frequency spectra are obtained. Based

22

on the analyses and experimental work, the negative stiffness effect

23

and additional vibration excitations caused by UMP are examined.

24

Index Terms—Axial-varying eccentricity, Eccentricity by frame

25

vibration, Permanent magnet synchronous motor, Mixed eccentric-

26

ity, Unbalanced magnetic pull.

27

I. INTRODUCTION 28

E

LECTROMECHANICAL interaction in rotating electrical

29

machines is a significant factor in the generation of nonlin-

30

ear dynamic behavior of a system. In machines with a small air

31

gap, such nonlinear dynamic behavior can be dangerous for the

32

rotor system, and therefore, many studies have been conducted

33

on unbalanced magnetic pull (UMP) caused by electromechan-

34

ical interaction.

35

The topic of UMP has been addressed in numerous studies

36

covering various factors such as asymmetry of rotor and stator,

37

rotor eccentricity, and magnetic saturation. Ortega et al. [1]

38

Manuscript received July 6, 2018; revised May 28, 2019; accepted July 15, 2019. This work was supported by LUT Doctoral School Funding. Paper no.

TEC-00728-2018. (Corresponding author: Heesoo Kim.)

H. Kim, J. Heikkinen, and J. T. Sopanen are with the Department of Me- chanical Engineering, LUT University, 53850 Lappeenranta, Finland (e-mail:

heesoo.kim@lut.fi; janne.heikkinen@lut.fi; jussi.sopanen@lut.fi).

A. Posa is with the Sulzer Pumps Finland Oy, 48600 Kotka, Finland (e-mail:

atte.posa@sulzer.com).

J. Nerg is with the Department of Electrical Engineering, LUT University, 53850 Lappeenranta, Finland (e-mail: janne.nerg@lut.fi).

Color versions of one or more of the figures in this article are available online at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TEC.2019.2935785

conducted experiments and a finite element analysis (FEA) 39

to study the effects of asymmetries caused by manufacturing 40

tolerances of the stator and rotor magnets on the performance 41

of a PMSM. Zhu et al. [2] developed a general analytical model 42

to predict UMP in a permanent magnet brushless AC and DC 43

machines having a diametrically asymmetric disposition of slots 44

and phase windings. Liu et al. [3] developed an analytical model 45

including the effect of interaction between the pole transitions 46

and the slot openings to analyze permanent magnet motors with 47

a slotted stator core. The rotor eccentricity, in particular, has been 48

widely investigated as a factor contributing to the UMP. Donat 49

[4] calculated UMP caused by air gap eccentricity based on an ⁵⁰ electromagnetic-coupled field analysis in the Ansys software. 51

Dorrell et al. [5] considered a combination of static and dynamic 52

eccentricities and [6] studied a method for calculating UMP 53

in cage induction motors; the model includes magnetic satura- 54

tion and axial variation with static or dynamic eccentricity. Li 55

et al. [7] modeled axial-varying eccentricity by a superposition 56

method and verified the model by comparing its results with 57

3D FEA results. Tenhunen et al. [8] investigated UMP in an 58

induction motor when the rotor is in whirling motion by using 59

a method based on the principle of virtual work and measured 60

it for a test motor supported by active magnetic bearings. Guo 61

et al. [9] obtained analytical expressions of UMP by air gap 62

eccentricity for any pole pair number. Di et al. [10] modeled the 63

curved dynamic eccentricity caused by a bent rotor. 64

The effect of UMP has been investigated by studying the ⁶⁵ dynamic behavior of a rotor system. Chen et al. [11] studied 66

the analytical UMP calculation method considering the magne- 67

tomotive force (MMF) of the rotor and the stator in a PMSM. 68

They discussed the stability of the steady response by using an 69

eigenvalue analysis for the Jeffcott rotor. Xiang et al. [12] studied 70

the stiffness characteristics and nonlinear dynamic behavior of 71

the Jeffcott rotor system of a PMSM affected by UMP. Losak 72

et al. [13] modeled UMP as a spring element and investigated the 73

rotor deflection and critical speed. Xu et al. [14] examined a rotor 74

model considering both static and dynamic eccentricity. They 75

compared the results with a case including dynamic eccentricity 76

only and found that vibration displacement is increased and the 77

rotor shaft orbit is no longer centrosymmetric and is only ax- 78

isymmetric in the direction of the static eccentricity. Pennacchi 79

[15] studied a UMP model based on the actual position of the ⁸⁰ rotor not limited to circular orbits and validated the proposed 81 0885-8969 © 2019 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

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model by measuring the dynamical behavior of a steam turbo

82

generator.

83

The rotor eccentricity is an important cause of UMP and

84

simultaneously, a result of various factors, such as manufac-

85

turing tolerances, faults, and rotor whirling motion. Therefore,

86

accurate modeling of eccentricity is a prerequisite for a UMP

87

study. In most previous studies, the factors causing the rotor

88

eccentricity have been studied individually and the effect of

89

UMP has been investigated for a Jeffcott rotor system. In ac-

90

tual systems, however, different eccentricity-generating factors

91

occur simultaneously and the total eccentricity is a result of

92

complex interactions. Therefore, in this study, a combined model

93

with mixed eccentricity, axial-varying eccentricity, and eccen-

94

tricity caused by motor frame vibration is first developed and

95

the UMP model with combined eccentricity is then applied to a

96

rotor–bearing model for an actual prototype electrical machine

97

with two different approaches, and its effects are studied. The

98

results are verified by a comparison with experimental results

99

for a pump test rig.

100

II. ECCENTRICITYMODELING 101

In the conventional analysis, two special cases of whirling

102

motion, i.e., static eccentricity and dynamic eccentricity, are

103

typically studied as air gap eccentricity. In static eccentricity,

104

the whirling frequency is zero and the eccentricity results from

105

manufacturing tolerances, wear, and misalignment of bearings.

106

In dynamic eccentricity, the whirling frequency is equal to the

107

rotation speed of the rotor, and the center axis of whirling motion

108

is the same as the center axis of the stator. Such eccentricity

109

typically results from a bent shaft or unbalance mass of the rotor.

110

In a real system, eccentricity is a result of a combination of

111

both static and dynamic eccentricities and not identical in the

112

axial direction of an eccentric rotor. Moreover, eccentricity is

113

affected by frame vibration. To demonstrate this condition, these

114

eccentricities are modeled individually and combined.

115

A. Mixed Eccentricity

116

Under the assumption that the rotor and the stator are ideal

117

cylinders and the stator is rigid and does not vibrate, air gap

118

eccentricity can be defined in the form of mixed eccentricity

119

by combining the static and dynamic eccentricities. Mixed ec-

120

centricity can be explained as a condition in which the rotor

121

rotates with a certain whirling amplitude about the eccentric

122

axis, which is displaced from the center axis of the stator bore,

123

i.e., eccentricity between the stator bore and the bearing bore

124

center is given as a static condition, and the rotor rotates about

125

the bearing bore center with a whirling amplitude. Therefore,

126

mixed eccentricity can be defined as a displacement vector of

127

the rotor center with the stator bore center as the initial point. It is

128

dependent on time and expressed as magnitude e0and direction

129

angleθ0.

130

To model this condition, two reference coordinate systems

131

are defined as in Fig. 1. The origin O1of the x1-y1coordinate

132

system is the geometric center of the stator, and the origin O2of

133

the x2-y2coordinate system is the center of rotor whirling and

134

can also be regarded as the geometric center of the bearings.

135

Or is the geometric center of the rotor. The x2-y2 coordinate

136

Fig. 1. Cross-sectional view of an eccentric rotor. (G: rotor mass center).

Fig. 2. Finite element model representation of an axial-varying static eccentric rotor (dotted line: real rotor condition).

system is translated parallel to the x1-y1 coordinate system as 137

the same degree as the static eccentricity. Therefore, the rotor 138

displacement can be defined as x2 and y2 coordinates of the 139

rotor center Or, and the mixed eccentricity can be defined by ¹⁴⁰ combining static eccentricity and rotor displacement. Therefore, 141

the magnitude and direction angle of mixed eccentricity are 142

defined as 143

e(t) =

(estcosθst+x2(t))²+ (estsinθst+y2(t))² (1) θ(t) = tan⁻¹

estsinθst+y2(t) estcosθst+x2(t)

(2) where estandθst are the amplitude and the direction angle 144

of static eccentricity, respectively. The coordinates x2 and y2 145

denote the instantaneous displacements of the rotor center. 146

B. Axial-Varying Eccentricity 147

In an actual system, air gap eccentricity is not consistent 148

with the axial direction because of manufacturing tolerances 149

and assembly misalignment. On the other hand, for calculation 150

of UMP, it must be assumed that the air gap is consistent 151

with the axial direction. Because the magnitude of the UMP 152

is proportional to the axial length of the electrical active rotor, 153

the rotor can be modeled as divided elements having axially 154

consistent and individual eccentricity, as in Fig. 2. To define 155

the mixed eccentricity in this finite element model, the static 156

(4)

Fig. 3. View (A) of the finite element model with axial-varying static eccentricity.

eccentricities (ei,st,θi,st) for all elements of the rotor have to

157

be determined. If the rotor is not bent, the static eccentricity

158

of an arbitrary intermediate rotor element can be calculated

159

from the eccentricities of the initial and final side using the

160

geometric relationship presented in Fig. 3. Static eccentricity

161

for an arbitrary ith element is defined as in (3) and (4).

162

ei,st=

x²_i,st+y_i,²st (3) θi,st= tan⁻¹

yi,st

xi,st

,when−π

2 < θ < π

2 (4)

in which

163

xi,st =eini,stcosθini,st

+2i−1

2n (eﬁn,stcosθﬁn,st−eini,stcosθini,st) (5) yi,st =eini,stsinθini,st

+2i−1

2n (eﬁn,stsinθﬁn,st−eini,stsinθini,st) (6) wherexi,standyi,st are the coordinates of the center of the ith

164

rotor element with respect to the x1-y1coordinate system. The

165

number of divided rotor elements is n, and therefore, i=1, 2,

166

…, n. The static eccentricity of the initial side of the rotor is

167

given by eini,standθini,st, whereas eﬁn,standθﬁn,stpresent the

168

static eccentricity of the final side of the rotor. Consequently, the

169

axial-varying eccentricity condition is modeled by using divided

170

elements that have individual eccentricity. Moreover, this model

171

can be easily applied to the rotor simulation model using finite

172

elements.

173

C. Eccentricity by Motor Frame Vibration

174

The UMP excites both the stator and the rotor. At the same

175

time, the vibration of the motor frame changes the air gap

176

Fig. 4. Motor frame vibration model of the machine under study.

eccentricity. Therefore, it is necessary to predict the vibration ¹⁷⁷ behavior of the motor frame and consider its effect on the eccen- 178

tricity. In this study, a simple model for the whole frame structure 179

is developed to be easily applied to the air gap eccentricity 180

calculation process. First, for the frame structure of the machine 181

under study, a modal analysis was performed with ANSYS, and 182

as a result, two frame-dominated modes were found [16]. Based 183

on these mode shapes, it is assumed that the motor frame and the 184

stator are rigid and considered one body, and the bearing housing 185

is rigid and connected rigidly to the ground. This approach is 186

taken as the focus of interest is on the effect of relative rotational 187

vibration between the motor frame and the bearing housing. 188

Based on these assumptions, the frame structure consisting of the 189

motor frame and the bearing housing and excited by the UMP is 190

modeled as a two-degree-of-freedom system as in Fig. 4. In this 191

model, the motor frame is connected with the bearing housing ¹⁹² by a rotational spring and a damper and rotated about a fixed 193

origin Oframe. The developed model is limited to the machine 194

structure of the study because the above assumptions are valid 195

for this structure only. Detailed information about the machine 196

structure is presented in Section V. 197

The rotational motion equation of the motor frame is 198

formulated as 199

I¨θ+C ˙θ+Kθ=T (7)

I=

Iframe, x 0 0 Iframe, y

,C=

cframe, x 0 0 cframe, y

,

K=

kframe, x 0 0 k^frame, y

,T=

Tx

Ty

,

θ=

θstator, x

θstator, y

,θ˙=

θ˙stator, x

θ˙stator, y

,θ¨=

θ¨stator, x

θ¨stator, y

(8) where I is the mass moment of inertia of the motor frame 200

and C and K are the rotational damping and stiffness of the 201

motor frame, respectively. Rotational displacement, velocity, 202

and acceleration vectors of the frame are denoted byθ,θ, and˙ θ.¨ ²⁰³

(5)

The moment caused by the UMP (Fx,Fy) acting on the frame

204

can be calculated as

205

Tx= − ⁿ

i=1

Fi, yl^ump, i (9)

Ty= − ⁿ

i=1

Fi, xlump, i (10) When the motor frame is vibrated by the moment, the dis-

206

placements of the stator bore at the ith node where the UMP is

207

applied are given by

208

xstator, i=lump, itanθy, ystator, i=lump, itanθx (11) wherelump,iis the axial length between the stator rotation center

209

and the node where the eccentricity is defined. Eccentricity gen-

210

erated by the displacement of the stator bore can be calculated

211

by

212

estator, i(t) =

(xstator, i(t))²+ (ystator, i(t))² (12) θ^stator,i(t) = tan⁻¹

ystator, i(t) xstator, i(t)

(13) D. Combined Eccentricity

213

From previous eccentricity models, the combined eccentricity

214

of the ith element of an electrically active part of the rotor is

215

defined as in (14) and (15).

216

ecomb,i(t) =

(ei,stcosθi,st+x2(t) +xstator, i(t))² +(e_i,stsinθ_i,st+y2(t) +ystator, i(t))²

1/2

(14) θ^comb,i(t) = tan⁻¹

ei,stsinθi,st+y2(t) +ystator, i(t) ei,stcosθ_i,st+x2(t) +xstator, i(t)

(15) III. ANALYTICALCALCULATION OF THEUMPBY 217

ECCENTRICITY ATPMSM

218

In this section, an analytical UMP model for a permanent

219

magnet synchronous motor is described. The basic concept is

220

to determine the air gap flux by modulating the fundamental

221

component of the magnetomotive force (MMF) wave consider-

222

ing the air gap permeance and calculate the corresponding force

223

components by the Maxwell stress tensor method.

224

In the model, the fundamental component of the air gap MMF

225

is taken from the study by Chen et al. [11] using the same

226

assumptions but with a slight modification, i.e., also the relative

227

permeability of the permanent magnet is considered to calculate

228

the rotor MMF as in (17). The amplitude of the fundamental

229

MMF of the air gap is written as

230

Fm=

Fsm² +Frm² −2FsmFrmsinϕ (16) Frm= 4Brhm

πμr,PMμ0sinαpπ 2

, Fsm=

√2mN kw

πp I (17)

TABLE I

PARAMETERS OF THEELECTRICALMACHINEUNDERSTUDY

where Fsm and Frm are the amplitudes of the fundamental 231

MMF waves for the stator and the permanent magnet rotor, 232

respectively. The variables are explained in Table I. 233

The resulting force of the UMP can be obtained by direct ²³⁴ integration using the Maxwell stress over the rotor surface as 235

Fx= 2π

0 σRrlcosαdα (18)

Fy= 2π

0 σRrlsinαdα (19)

Here,σis the Maxwell stress normal to the iron and air bound- 236

ary and αis a variable for defining air-gap’s circumferential 237

location on the rotor surface as in Fig. 1. 238

IV. MODELING OF AROTOR-BEARINGSYSTEMWITHUMP 239

In this section, the rotor-bearing model with UMP is estab- 240

lished using two modeling approaches. In the first approach, the 241

UMP is applied to the rotor model as a linear spring element. In 242

the second approach, the UMP is applied as a nonlinear force. 243

For the simulation using the first approach, a linearized UMP 244

stiffness is first defined employing the UMP model developed 245

by Guo et al. [9] with several assumptions. In the UMP model, 246

if oscillating terms except the first constant term are ignored 247

and the power series having only the first two terms as in (20) 248

are used under the assumptionε²1, the UMP model can be ²⁴⁹

simplified as in (21) and (22). 250

1

1−ε² ≈1 +ε²,

1−ε²≈1−ε²

2 (20)

Fx(ε, θ) =Rlπμ0

2δ0² Fm²

ε+5

4ε³+ 5 16ε⁵+ 1

16ε⁷

cosθ (21)

(6)

Fy(ε, θ) =Rlπμ0

2δ0² Fm²

ε+5

4ε³+ 5

16ε⁵+ 1 16ε⁷

sinθ

(22) Moreover, if it is assumed that the eccentricity has purely

251

horizontal or vertical direction components, i.e., the coupled

252

terms between the x and y directions are ignored, the UMP

253

stiffness (kump,x, kump,y) can be obtained as follows. When

254

θis zero (pure x-direction eccentricity, x=δ0ε),

255

Fx=Rlπμ0

2δ0³ Fm²

x+5x³

4δ0² + 5x⁵ 16δ0⁴ + x⁷

16δ0⁶

(23) Whenθisπ/2 (pure y-direction eccentricity, y=δ⁰ε),

256

Fy= Rlπμ⁰ 2δ³₀ Fm²

y+5y³

4δ₀² + 5y⁵ 16δ₀⁴ + y⁷

16δ₀⁶

(24) Above, the UMP force is linearized as a derivative in a static

257

eccentricity (xst, yst), and the UMP stiffness is presented as

258

kump,x≈ dFx(x) dx

xst

, kump,y≈ dFy(y) dy

yst

(25) In conclusion, the equation of motion with UMP can be

259

established by using the UMP stiffness vector Kumplinearized

260

at a given static eccentricity as

261

M¨q+ (C+ ΩG)(˙q+ (K−Kump)q=Fub+Fg (26) where q is the displacement vector and M, C, G, and K are

262

the mass, damping, gyroscopic, and stiffness matrices, respec-

263

tively. TermΩis the rotor angular velocity. Correspondingly,

264

Fub andFg denote the unbalance force and the gravity force,

265

respectively.

266

In the second method, UMP is added as an external force

267

(Fump) directly to the equation of motion as follows

268

M¨q+ (C+ ΩG)˙q+Kq=Fub+Fg+Fump (27) V. SIMULATION

269

In this section, simulations using the rotor–bearing system

270

models in Section IV are presented. The simulations were con-

271

ducted for a centrifugal pump with an integrated PMSM. Results

272

for several cases were obtained and compared to identify the

273

effects of UMP on the vibration of the machine.

274

A. Electrical Machine Under Study

275

The machine under study consists of a motor, a bearing unit,

276

and a volute case. It has an overhang structure in which the rotor

277

is not supported on the motor rear side; further, the impeller is

278

not considered. The structure of the machine is shown in Fig. 5,

279

and the machine parameters are given in Table I. The mass

280

unbalance of the rotor is located at the face of the nonpump

281

side, and its magnitude is 110.25 g·mm corresponding to the

282

balancing grade G2.5. Two angular contact ball bearings support

283

the rotor, and their stiffness is estimated by using a simple

284

method proposed by Gargiulo [17]. According to this method,

285

the stiffness is calculated by considering the ball diameter, the

286

ball numbers, and the bearing radial load. Here, the bearing load

287

is estimated on the assumption of 20% static eccentricity and a

288

Fig. 5. Structure of the electrical machine under study.

20μm (0-peak) rotor whirling vibration condition, because the 289

result of the 20% eccentricity case will be compared with the 290

experimental result. In conclusion, each bearing has constant 291

stiffness (kbearing=1.4·10⁸N/m) and damping (3.5·10³N·s/m) ²⁹² in both the horizontal and vertical directions. Here, damping is ²⁹³ estimated as 2.5·10⁻⁵·kbearingbased on the suggestion presented 294

in [18]. 295

B. Simulation Method 296

For the rotordynamic analysis, the rotor is modeled with 297

beam finite elements that have four degrees of freedom per 298

node. This model assumes that there is no displacement in 299

the axial direction and no rotation around the rotor axis. To 300

consider axial-varying static eccentricity, the electrically active ³⁰¹ part of the rotor is divided into four parts and individual static 302

eccentricity is applied to each part. In this study, seven cases 303

in total are simulated; without UMP, with UMP in three condi- 304

tions of eccentricity (mixed and axial-varying eccentricity with 305

0–0%, 10–0% and 20–0% static eccentricities), and three cases 306

with UMP including frame vibration. The axial-varying static 307

eccentricity is expressed as initial side eccentricity and final side 308

eccentricity. The direction angle of the static eccentricity is set 309

to zero in all cases. The static eccentricities of the divided rotor 310

parts are calculated using (3) to (6). 311

For the simulation using the linear spring model of the UMP, 312

the UMP stiffness is calculated using the process presented in 313

Section IV. To consider axial-varying static eccentricity, the 314

individual UMP stiffness is calculated at the obtained static 315

eccentricity of each rotor part and applied to the node of the rotor ³¹⁶ part. Total UMP stiffness values in the horizontal direction for 317

the entire rotor are calculated as 1.399·10⁶for 0–0%, 1.403·10⁶ 318

for 10–0%, and 1.414·10⁶ N/m for 20–0% static eccentricity. 319

The vertical direction stiffness is the same 1.399·10⁶N/m for all 320

cases because there is no eccentricity in the vertical direction. 321

The motion equation is defined as an eigenvalue problem and the 322

critical speeds are calculated by solving this problem. However, 323

the motor frame vibration model cannot be included in this 324

method. 325

In the simulation applying the external force model of the 326

UMP, a time transient analysis using numerical integration is 327

(7)

Fig. 6. 3D spectral map of horizontal direction displacement (µm) at the unbalance mass location to determine a change in critical speeds.

conducted to solve the motion equation. The numerical integra-

328

tion is performed using the ode15s function in MATLAB. In the

329

analysis process, the rotor location is updated at every time step

330

and the UMP is calculated using eccentricity from the updated

331

rotor location. For the simulation including frame vibration, the

332

inertia, stiffness, and damping of the frame presented in Table I

333

are used. These values were predicted by using two natural

334

frequencies (74 and 93 Hz) related to the frame’s rotational

335

vibration modes determined by experimental measurement [16].

336

To investigate the UMP effects, a ramp from 10 to 70 Hz in 60 s

337

is simulated for all cases.

338

C. Simulation Results

339

From the simulation using the linear spring model, critical

340

speeds of the rotor are obtained for four cases without frame

341

vibration. Again, 3 D spectral maps of the rotor vibration were

342

obtained from the simulation using the external force model.

343

Here, displacement vibration is measured at the rotor node

344

where the unbalance mass is located as in Fig. 5., because

345

rotor vibration can be only measured when not including the

346

frame model. For a comparison with the experimental results,

347

in the case including the frame model, also velocity is mea-

348

sured at the same frame location with the experimental mea-

349

surement, which is presented in Section IV on experimental

350

verification.

351

The 3 D spectral maps for the four cases: without UMP, with 352

UMP at 0–0% and 0–20% axial-varying static eccentricity, and 353

with UMP including frame vibration at 20–0% axial-varying 354

static eccentricity are presented in Figs. 6 and 7. From the 355

peak point of the 3D spectral map, the first backward/forward 356

critical speeds can be found. Changes in the critical speeds 357

caused by the UMP are detected and presented in Table II. 358

The negative stiffness effect was observed similarly in both 359

simulation methods. The critical speeds were decreased by the 360

UMP, and this effect was slightly amplified by frame vibration. ³⁶¹ The effect of static eccentricity on the negative stiffness was 362

small in comparison with the effect of dynamic eccentricity. This 363

finding is in agreement with the calculation result indicating that 364

the UMP stiffness change from the static eccentricity is small. 365

Moreover, the effect of static eccentricity on the negative 366

stiffness was shown with only a little difference between two 367

methods. In the method using the linear spring model of the 368

UMP, the effect by static eccentricity appeared to be smaller 369

than in the other method. Furthermore, it is shown that the 370

excitation of the backward whirling mode increases when the 371

static eccentricity increases as shown in Fig. 6. The backward 372

whirling mode is excited by the anisotropic support stiffness 373

[19], which is amplified by the anisotropic negative stiffness 374

caused by static eccentricity. 375

To determine the additional excitations from the UMP, 3D ³⁷⁶ spectral maps with a decreased range of the z axis are presented 377

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Fig. 7. 3D spectral map of horizontal direction displacement (µm) at unbalance mass location to determine higher frequency excitations.

TABLE II

CHANGE INCRITICALSPEEDS(SIMULATION1: LINEARSPRINGMODEL, SIMULATION2: NONLINEARFORCEMODEL)

∗)Backward mode (without UMP case) is not excited in Simulation 2, and therefore, the effect of UMP is found by comparing with the critical speed (54.1 Hz) in Simulation 1.

in Fig. 7. It is found that harmonics with 2 Ω, 3 Ω, 4 Ω,

378

5Ω, and 6Ω(2·line frequency) frequencies are generated, and

379

their amplitudes increase when the static eccentricity increases.

380

In particular, it is shown that the 6 Ω frequency component

381

occurs especially when static eccentricity is present. Finally,

382

TABLE III

VIBRATIONFREQUENCIES INSIMULATIONRESULTWITHUMPAS AN EXTERNALNONLINEARFORCE

∗)Line freq.=pΩ =3Ω, p: pole pair no.

Fig. 8. Pump test rig, (left) two accelerometers for measurement, (right) axial- varying static eccentricity by shim.

it is observed that the UMP effect is slightly amplified by frame 383

vibration. The critical speeds are lower, and especially the ex- 384

citations that meet the frame natural frequency are significantly 385

amplified. 386

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Fig. 9. Comparison between the 3D spectral maps for the simulation and the experimental results of horizontal direction velocity (mm/s) (Ω: rotating speed).

VI. EXPERIMENTALVERIFICATION 387

To verify the simulation results, experiments were carried

388

out with a pump test rig. Axial-varying static eccentricity was

389

adjusted by placing a shim between the bearing housing and the

390

motor frame as in Fig. 8. The eccentricity was checked with a

391

customized feeler gauge in which the measurement limits were

392

set at 0.25 mm increments. For the vibration measurement, two

393

accelerometers (IMI VO-622) were attached to the nonpump

394

side of the motor frame in the x and y-directions as in Fig. 8.

395

A ramp from 0 to 4100 rpm in 70 s was driven and 3D spectral

396

maps for the velocity vibration of the frame were obtained as a

397

result. In this experiment, only the cases including UMP were

398

investigated because it is impossible to drive the rotor without

399

the effect of the UMP. Therefore, higher frequency excitations

400

from the UMP and the effect caused by the variation of the

401

static eccentricity can be seen from the experimental results,

402

but the negative stiffness effect from the presence of the UMP

403

cannot be accurately verified. 3D spectral maps for vibration

404

in horizontal direction for two cases with 0–0% and 20–0%

405

axial-varying static eccentricities are presented in Fig. 9 with

406

simulation results measured at the same location using the same

407

vibration unit.

408

The experimental results show that harmonic excitations oc-

409

cur similarly to the simulation result, and these components are

410

amplified at frame natural frequencies. As mentioned above in 411

the frame vibration modeling section, the whole frame has two 412

natural frequencies (73 and 93 Hz) within the operating speed 413

range. However, in simulation results presented in Fig. 9, the 414

amplification is shown at only first natural frequency, of which 415

the mode shape is related with vibration in horizontal direction. 416

Moreover, it is shown that the 2Ωand 6Ωfrequency components 417

are significantly amplified by the static eccentricity. These re- 418

sults are in agreement with the simulation results. However, the 419

amplitudes of these components are much higher than the ones 420

found in the simulation results. This discrepancy is probably due ⁴²¹ to the simplifications made in the modeling of frame vibration; 422

the topic, however, requires further study. 423

VII. CONCLUSION 424

An eccentricity model including mixed eccentricity and axial- 425

varying eccentricity considering frame vibration was developed. 426

Based on the eccentricity model, analyical unbalanced magnetic 427

pull (UMP) model was applied to the rotor simulation model 428

of a centrifugal pump with a PMSM. From simulation results 429

for several cases, the vibration effects caused by the UMP were 430

studied, and the results were verified by an experimental analysis 431

of a pump test rig. 432

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The key findings of the study on the effect of UMP on vibration

433

and conclusions about the simulations can be summarized as

434

follows.

435

1) The negative stiffness effect caused by the UMP decreases

436

rotor critical speeds. This effect is mainly due to dynamic

437

eccentricity and slightly amplified by static eccentricity and

438

frame vibration. Moreover, the anisotropy of static eccentric-

439

ity amplifies the excitation of the backward whirling mode.

440

2) Additional vibration components with frequencies of 2Ω,

441

6Ω(2·line frequency), and other speed multiple frequencies

442

are generated by the UMP. The vibration component of the

443

2·line frequency is mainly generated when static eccentricity

444

is present, and its amplitude is increased when the static

445

eccentricity increases.

446

3) The simulation method using a linear negative spring model

447

of UMP can be used to estimate the effect of negative stiffness

448

on rotor critical speeds. However, the method applying UMP

449

as an external force can only be used for prediction of higher

450

frequency excitations produced by UMP.

451

4) A comparison between the simulations and the experimental

452

results showed that the proposed simulation model can be

453

used to predict electromagnetic excitations caused by UMP

454

from air gap eccentricity.

455

5) In the experimental results, the amplitude of the electrome-

456

chanical excitation at the natural frequency of the frame was

457

significantly higher than in the simulations. This difference

458

is probably due to the assumptions used in the modeling of

459

the frame vibration. Hence, further studies on the modeling

460

of the frame vibration effects are required.

461

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Energy Convers., vol. 32, no. 3, pp. 1002–1011, Sep. 2017.

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method,” IEEE Trans. Magn., vol. 53, no. 11, Nov. 2017, Art. no. 1400404.

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magnetic forces of the cage rotor in conical whirling motion,” IEE Proc.- 486

Electr. Power Appl., vol. 150, no. 5, pp. 563–568, Sep. 2003.

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[9] D. Guo, F. Chu, and D. Chen, “The unbalanced magnetic pull and its effects 488

on vibration in a three-phase generator with eccentric rotor,” J. Sound Vib., 489

vol. 254, no. 2, pp. 297–312, 2003.

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unbalanced magnetic pull in cage induction motors with curved dynamic 492

eccentricity,” IEEE Trans. Magn., vol. 51, no. 8, p. 8106507, Aug. 2015.

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[13] P. Lošák and R. Vlach, “Study of the response of the shaft loaded by unbal- 501 anced magnetic pull,” in Proc. 16th Int. Conf. Mechatronics, Mechatronika, 502

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[14] X. Xu, Q. Han, and F. Chu, “Nonlinear vibration of a generator rotor with 504 unbalanced magnetic pull considering both dynamic and static eccentric- 505 ities,” Arch. Appl. Mech., vol. 86, no. 8, pp. 1521–1536, 2016. 506 [15] P. Pennacchi, “Computational model for calculating the dynamical be- 507 haviour of generators caused by unbalanced magnetic pull and experi- 508 mental validation,” J. Sound Vib., vol. 312, no. 1–2, pp. 332–353, 2008. 509 [16] A. Posa, “Vibration behavior of a centrifugal pump with integrated perma- 510 nent magnet motor,” M.S. thesis, Dept. Mech. Eng., Lappeenranta Univ. 511

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Germany: Springer-Verlag GmbH, 1993. 516

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vol. 84-2, 1995, pp. 991–1000. 519

Heesoo Kim was born in Seoul, Korea, in 1979. He 520 received the B.S. and M.S. degrees in mechanical 521 engineering from Hanyang University, Seoul, Korea, 522 in 2005 and 2007, respectively. After graduation, 523 he worked as a Turbocharger Development Engi- 524 neer for marine and vehicle engines. From 2017, he 525 is working toward the Ph.D. degree at the Depart- 526 ment of Mechanical Engineering, LUT University, 527 Lappeenranta, Finland. His research interests include 528 rotordynamics for electrical machines, specifically 529 study of electromechanical interaction from air gap 530 eccentricity, stator deformation, and other geometric non-idealities. 531 532

Atte Posa received the M.Sc. degree in mechanical 533 engineering from LUT University, Finland, in 2016. 534 After the master’s studies, he started working as Prod- 535 uct Development Engineer in Sulzer Pumps Finland 536 Oy, Kotka, Finland. His work as Product Develop- 537 ment Engineer is focused on mechanical seals and 538

sealing systems. 539

540

Janne Nerg (M’99–SM’12) received the M.Sc. 541 degree in electrical engineering, the Licentiate of 542 Science (Technology) degree, and the D. Sc. (Tech- 543 nology) degree from LUT University, Lappeenranta, 544 Finland, in 1996, 1998, and 2000, respectively. He 545 is currently an Associate Professor with the Depart- 546 ment of Electrical Engineering at LUT University. 547 His research interests include electrical machines and 548 drives, especially electromagnetic and thermal mod- 549 eling and design of electromagnetic devices. 550 551

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Janne Heikkinen received the M.Sc. degree in me- 552

chanical engineering from LUT University, Lappeen- 553

ranta, Finland, in 2010. After the master’s studies, he 554

started his research career as a Ph.D. student at LUT 555

University. He successfully defended his Doctoral 556

Dissertation and received the D.Sc. degree from LUT 557

University in 2014. He is currently working as a 558

Postdoctoral Researcher in Laboratory of Machine 559

Dynamics at LUT University. His research interests 560

include rotating electric machines, especially high- 561

speed machinery. His main expertise is in rotordy- 562

namics, structural vibrations, and vibration measurements.

563 564

Jussi T. Sopanen (M’14) was born in 1974 in 565 Enonkoski, Finland. He received the M. Sc. degree in 566 mechanical engineering and the D. Sc. (Technology) 567 degree from LUT University, Lappeenranta, Finland, 568 in 1999 and 2004, respectively. He has been a Re- 569 searcher with the Department of Mechanical Engi- 570 neering at LUT University during 1999–2006. He has 571 also worked as a Product Development Engineer in 572 electric machine manufacturer, Rotatek Finland Ltd., 573 from 2004 to 2005. During 2006–2012, he worked 574 as Principal Lecturer in mechanical engineering and 575 Research Manager with the Faculty of Technology in Saimaa University of 576 Applied Sciences, Lappeenranta, Finland. He is currently serving as a Profes- 577 sor in Machine Dynamics Lab at LUT University. His research interests in- 578 clude rotordynamics, multi-body dynamics, and mechanical design of electrical 579

machines. 580

581

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