In order to predict the rotordynamics of a high speed induction motor in bending, an optimization procedure is proposed for identifying the constitutive properties especially those of the magnetic core. Modal parameters predicted by a finite element model based mainly on beam elements, and measured on an induction motor are included in modal error functions contained in a functional. The minimization of this functional by using the Levenberg-Marquardt algorithm permits extracting the constitutive properties along the magnetic core.

Figure 5. Experimental Setup.

Table 1 First four bending measured and predicted natural frequencies obtained with 7 sub-domains.

0.640.472.681.05Error (%)

231517051263847Predicted (Hz)

233017131230856Measured (Hz)

4th3rd2nd1st

Figure 8. Normalized Young’s modulus distributions. Figure 9. Normalized Coulomb’s modulus distributions.

Figure 7. Evolution of the global error norm versus the number of sub-domains.

Figure 6. The dashed and solid line represent measured and predicted mode shapes respectively. Predicted mode shapes were obtained with 7 sub-domains.

Figure 2. Short-circuiting rod in bending - Kinematic assumption.

ϕ ω = …, , 1, , 4.k k k

G. Mogenier(1-2), R. Dufour(1), T. Baranger(1), G. Ferraris Besso(1)(1) Université de Lyon, LaMCoS, INSA-Lyon, CNRS UMR5259, F-69621, France

(2) CONVERTEAM SAS, 54250, Champigneulles, France

Optimization Procedure for Identifying Constitutive Properties of High Speed Induction Motor

Optimization procedure tested on an industrial rotor with complex design. Constitutive property distributions depend on the number of sub-domainsused in the finite element (FE) modeling of the magnetic core. Distributions tend toward a particular shape versus the number of sub-domains: Constant values in the middle (55% of the magnetic core length) and Low values at its ends. The low level permits probably to consider the shear effect located close to the ends. This procedure is very useful for establishing an FE model based mainly on beam elements and therefore containing few degrees of freedom.

Screw

Short-Circuiting Rod Neutral Axis

Magnetic Core Neutral Axis

A B

A0 B0

λϕ λ ϕ

ϕ ϕ ϕ ϕ=≠

∂ ∂ ∂= − ∂ ∂ ∂

∂ = + ≡ + ∂

∑1

.

Tkk kk

p p p

mk

j k kj k k kjpj k

K M

x x x

c c V cx

( )

λ ϕλ ω

ϕ ϕ

− = = ==

ɶ …2

0, , 1, , ,

1

kk

k kT

k k

K Mk m

M

( ) ( )( ) ( ) ( ) ( ) ( )

( )

λ

λ

ρ

−

+

= − + ⋅∇ ∇ =

= +

∂= = =

∂… …

00

1

1

, given

, with .

where , 1, , , 1, , .

T Ti i i i i Tot ii i

i ii i

Tot is

sp

p

x

d J x J x I f x f x J x F x

x x d

F xJ p n s q

x

( ) ( )ϕϕ ϕ= 1- , ,

D iikk kF x MAC

( ) ( ) ( )ϕϕ ϕ ϕ ϕ

= ≠

= −∑i

1,

x , , ,ED

mi

jk k j kj j k

F MAC MAC

( )( )( )( )

( )( )( )

ϕϕ ϕ

ϕϕ=

= −∑i

1

x .maxmax

EC

i

krkj j

ki

j k lk ll l

F

( ) ( ) ( )=

= = = × ∑2

2

1

1 1, 4 .

2 2

qi Tot i Tot i

kk

f x F x F x q m

( ) ϕ ϕω ω ϕ ϕ ϕ ϕα α α α = , , , ,Ec EcD D HD HDTTot iF x F F F F

( )α =

0

1.

F x

Analytical Eigen-Derivatives

( )

ω ωω

= −ix 1,

ik

k

k

F

Multi-Objective Function

Efficient Convergence.

Less CPU-time Consuming.

Figure 4. Finite element model. Blue points represent measurement points.

Constitutive Property Distributions

− = β⋅ ⋅ ⋅ −

2

0 0 0 0

0 1 0 1.

0 0 0 0

0 1 0 1

TI TITI TI

TI

E SK e

L

Figure 1. Diagram of a squirrel cage induction motor.

Assembly Structure:

Tie rods, Short-circuiting rods, Laminated stack.

Prestressed Structure.

Laminations without central hole.

Short-circuiting rods are modeled as beams, of diameter DCC (Fig. 3), whose neutral axes coincide with the neutral axis of the magnetic core.

Tie rods can be seen as external tendons.

Laminated stack is modeled as a distributed orthotropic material.

Young’s and Coulomb’s modulus distributions tend to stabilize if h > 7 sub-domains.

Distributions have constant values in the middle (55 % of the magnetic core length) of the magnetic core and decrease at its ends. The global error norm decreases with the

number of sub-domains.

The global error norm obtained with h = 7is height times lower (Fig. 6) than this one obtained with an isotropic assumption.

The proposed method is more accuratethan classical identification methods considering restrictive assumptions such as homogenous or isotropic material.

Kinetic Assumptions

Nelson’s Method

( ) υ= − −

= = …3 2 ,3 1,3

, , , 1, , .p r r rp r r r

x E G r h

Tie rods stiffness matrix at clamping nodes A0 and B0 (Fig. 2):

The kth eigenvector derivative requires only the kth eigenvector.

where K and M are the stiffness and mass matrices respectively.

Distributions , considered by splitting the stack into h sub-domains such as

Figure 3. Cross section of the magnetic core.

Optimization strategy consists in minimizing the difference between the measured and predicted modaldata, at each iteration i:

Tie Rod

Short-Circuiting Rod

Stack of Steel

Laminations

Magnetic Core

Short-Circuiting Ring

Shaft End

Shaft End

The difference between measured and predicted natural frequencies and mode shapes is quantified by four modal error functions, for each mode k, k = 1…m:

Modal Testing

Weight coefficients based on the initial mean value, i = 0.

Ωr

∈ℜ = ∈ℕ, 3 ,nx n h h

The difference is defined as a non-linear least squares functional, a global error norm:

Shaker

Flat Sling

Sandow

Bolt-on Attachment Swivel

0.5 m

Force Sensor

Crane

Tie Rod

Short-Circuiting Rod

TID

TIeCCD

Great advantage for predicting the rotordynamics, i.e.unbalance responses and transient responses.

P1P62

( )?G x( )?E x ( )υ ?x

Objectives

Beam Finite Element ModelSquirrel Cage Rotor Magnetic Core Modeling

Sub-Domain Dependence

Conclusions

Levenberg-Marquardt Algorithm Undamped Eigenvalue ProblemModal Error Functions

Industrial Application Method Accuracy

Université de Lyon, LaMCoS, CNRS, INSA-Lyon UMR5259, 18-20 rue des Sciences - F69621 Villeurbanne Cedex