Ce papier présente un modèle électromécanique pour simuler le comportement dynamique d'une machine asynchrone / arbre du rotor. Le modèle décrit dans ce travail combine un modèle de réseau de perméances d'un moteur à induction à cage d'écureuil et un modèle dynamique simple de son arbre de rotor. Un échantillonnage angulaire est utilisé pour permettre d'investiguer les variations de vitesse angulaire instantanée et de couple. Le système rotatif est décrit en se référant à la position angulaire de l'arbre. La méthodologie de couplage électromécanique est bien détaillée. En se basant sur un certain nombre de simulations, on montre l'influence des variations de couple provenant de la machine à 'induction sur le comportement dynamique du système mécanique

Bourdon, Adeline

Chaari, Fakher

Feki, Nabih

Fourati, Aroua

Haddar, Mohamed

Remond, Didier

I-Revues

22ème Congrès Français de Mécanique                                               Lyon, 24 au 28 Août 2015 
 
 
An electromagnetic-mechanical modeling of an 
induction motor and rotor shaft coupled to angular 
approach 
 
Aroua. FOURATIab, Adeline. BOURDONa, Didier. REMONDa,           
Nabih. FEKIb, Fakher. CHAARIb, Mohamed. HADDARb 
 
a. LaMCoS UMR5259, CNRS, INSA-Lyon, University of Lyon, France 
{aroua.fourati,adeline.bourdon,didier.remond}@insa-lyon.fr 
b. La2MP, National School of Engineers of Sfax – BP 1173-3038, Tunisia  
{arwafourati,fekinabih,fakher.chaari}@gmail.com 
mohamed.haddar@enis.rnu.tn 
Résumé:  
Ce papier présente un modèle électromécanique pour simuler le comportement dynamique d’une 
machine asynchrone / arbre du rotor. Le modèle décrit dans ce travail combine un modèle de réseau de 
perméances d'un moteur à induction à cage d'écureuil et un modèle dynamique simple de son arbre de 
rotor. Un échantillonnage angulaire est utilisé pour permettre d’investiguer les variations de vitesse 
angulaire instantanée et de couple. Le système rotatif est décrit en se référant à la position angulaire 
de l'arbre. La méthodologie de couplage électromécanique est bien détaillée. En se basant  sur un 
certain nombre de simulations, on montre l'influence des variations de couple provenant de la machine 
à 'induction sur le comportement dynamique du système mécanique. 
 
Abstract:  
This paper present an electromechanical model to simulate the dynamic behavior of induction motor/ 
rotor shaft. The model detailed in this work combines a permeance network model of a squirrel cage 
induction motor and a simple dynamic model of its rotor shaft. Angular sampling is used to allow 
investigation of instantaneous angular speed and torque variations. The rotating system is described 
referring to the angular position of the rotating shaft. The methodology for electro-mechanical coupling 
is well detailed. Based on a number of simulations, it is shown the influence of torque variations derived 
from the induction machine on the dynamic behavior of the mechanical system. 
 
Mots clefs: Induction machine, electro-mechanical modeling, Instantaneous 
Angular Speed, condition monitoring 
 
1 Introduction 
Induction machines connected to mechanical devices are frequently integrated in industrial applications. 
In these conditions, diagnosis is crucial in terms of fault monitoring to ensure optimal performance. 
However, these systems are operating under non-stationary conditions. This reality has been the catalyst 
to the development of surveillance techniques for machines under variable operating conditions. In 
numerical domain, several sophisticated models were proposed in the literature for modeling industrial 
components [1], [2]. In the other hand, the angular approach is developed to reproduce varying operating 
22ème Congrès Français de Mécanique                                               Lyon, 24 au 28 Août 2015 
 
conditions [3]. In this paper, an electromechanical model of an induction machine coupled to its rotor 
shaft in angular domain is proposed in order to extend modeling to varying angular speed. Coupling 
between the electrical and the mechanical subsystems is dealt by dual interaction via torque, angular 
speed and air-gap. A number of numerical results are presented to illustrate electrical-magnetic and 
mechanical subsystems interactions in order to highlight the importance of angular sampling for 
diagnosis of rotating machines in non-stationary conditions. 
 
2 Angular background                       
In the framework of rotating machinery modeling, angular approach is well developed to investigate in 
modeling of rotating machinery in the case non-stationary rotational speed. It consists on sampling 
variables in reference to an angular step rather than a time step. 
A system of n degrees of freedom is represented in angular domain via angular sampling by the 
following system [1]:  
         R
R R
R
R R R t R2n2n 2n 2n 2n 2n
R R2n
dt 1
d ( )
dQ( ) 1
A(t, ) . Q( ) B(t, ) . U (t) U ( )
d ( )

  

   
    

 
   
 
          (1) 
Where the variable used for the integration is the rotational degree of freedom 
R  and time is resulting 
from the relationship between angle and time through the instantaneous angular speed function. This 
angular approach allows to highlight angular periodicity of a rotating system. 
 
3 Electro-mechanical coupling 
3.1    Electrical modeling: permeance network model 
The permeance network model offers a detailed representation of an induction machine. It makes 
possible to consider winding distribution, stator and rotor slotting and rotor eccentricity. It consists in 
discretizing the induction machine on a finite number of nodes. The induction machine is represented 
by a permeance network model (Fig.1) and a classic electrical circuit (Fig.2). 
 
Fig.1.A part of the permeance network model              Fig.2.Rotor squirrel cage electrical circuit        
22ème Congrès Français de Mécanique                                               Lyon, 24 au 28 Août 2015 
 
Considering magnetic and electrical models, electromechanical behavior of an induction machine can 
be modelled using a differential equation in the angular domain: 
    

   
  
    RR R R
R R
R
d GXZ( )d I
L GXZ( ) . . R . . I V(t)
d d
d
(t)
dt

    
 


 
    
 
 

                         (2) 
Where 
R  the instantaneous angular position of the rotor shaft. I is the generalized stator and rotor 
currents vector,  V(t) is the power supply vector,  R and  L are respectively matrices of inductances 
and resistances of the stator and the rotor and  RGXZ( )  is the matrix describing electrical-magnetical 
coupling in the induction machine. 
It is important to notice that being modeled in angular domain, results of simulation of the induction 
motor can be extended to manage non-stationary rotational speed conditions. 
 
3.2 Mechanical modeling: simple rotor shaft model 
The mechanical model of the rotor shaft is simulated by a simple 3-nodes Timoshenko beam elements 
whereas bearings are introduced by additional lumped stiffness elements as shown in Fig.3. The 
mechanical system dynamic behavior is therefore accounted for by 3 nodes with 2 degrees-of-freedom 
per node: axial rotation and radial displacement. 
 
Fig.3. 3-nodes dynamic rotor shaft model  
 
The general equation of motion of the rotating shaft may be written in angular domain as following: 
        
RR R t R
R
dQ(t)
A( ) . Q(t) B( ) . U (t) U ( )
dt
d
(t)
dt
  


 
   
 

                                                          (3)   
Where R  is the rotational degree of freedom of the shaft.  Q(t) is displacement and velocity vector, 
 tU (t) and  R RU ( )  external forces vectors depending respectively from time and angle,  RA( )
and  RB( ) are matrices representing the dynamic characteristics of the shaft. 
 
3.3 Coupling electrical and mechanical models 
The electromechanical coupling is realized by defining a global state vector including electrical and 
mechanical outputs. In each iteration, magnetic, electric and mechanic models are solved while ensuring 
input/output relationships between subsystems. In order to take into account dynamic variations of the 
system, the rotor angular position, the electromagnetic torque, the rotor eccentricity, the rotational speed 
22ème Congrès Français de Mécanique                                               Lyon, 24 au 28 Août 2015 
 
and the time are up-dated at every step. Electromagnetic torque generated by the induction machine is 
applied as an external force at node 2 of the shaft. In another hand, instantaneous angular speed and 
radial displacement of the same node of the mechanical model are used to feed the electrical model of 
the rotational speed and the air-gap eccentricity as shown in Fig.4.  
 
Fig.4. Electric-magnetic-mechanical coupling   
 
4    Results  
The proposed electromechanical system is used to simulate the influence of the interaction between 
electrical and mechanical models on the dynamic of the whole system. Fig.5 illustrate angle variations 
of the first-phase stator current. Fig.6 shows air-gap eccentricity amplitude. In Fig.7, instantaneous 
angular speed response is represented. Torque variations are shown in Fig.8.  
 
 
Fig.5. First phase stator current  
 
Fig.6. Air-gap eccentricity amplitude   
22ème Congrès Français de Mécanique                                               Lyon, 24 au 28 Août 2015 
 
 
Fig.7. Instantaneous angular speed  
 
 
Fig.8. Motor torque  
 
It is important to notice that results presented in this paper are simulated for a constant macroscopic 
angular speed. However, analysis can be extended to varying speed. Influence of this variable on the 
other dynamic variables of this system (eccentricity, torque, currents …) may be investigated. 
Finally, to show the potential of the angular approach, a comparison between simulations of the model 
using time and angular sampling in the same conditions has shown that simulations in angular domain 
provide a gain of 70% of the computing time. 
 
5     Conclusion 
The proposed electromechanical model makes possible to simulate behavior of an induction machine 
and a simple rotating shaft dynamically connected. The electrical model, based on the permeance 
network model was modified by coupling with angular approach to be suitable to non-stationary 
operating conditions. A simple mechanical model of the rotor shaft was sampled in angular domain to 
make access to angular speed variations. Results presented in this paper shows the interconnection 
between subsystems. Further works will present the combination of this resulted system with more 
complex mechanical systems (bearing …) that add angular periodicities able to influence the dynamic 
of the whole system.  
 
Références  
[1] A. Bourdon, H. André, D. Rémond, Introducing angularly periodic disturbances in dynamic 
models of rotating systems under non-stationary conditions, Mechanical systems and signal processing 
44 (2014) 60-71 
[2] N. Feki, G. Clerc, P. Velex, Gear and motor fault modelling and detection based on motor current 
analysis, Electric power systems research, 95 (2013) 28-37  
[3] A. Bourdon, H. André, D. Rémond, A new way for writing motion equations in rotating machines 
by translation into the angular domain, Proceedings of the 8th IFToMM International Conference on 
Rotordynamics (2010), KIST, Seoul, Korea 


An electromagnetic-mechanical modeling of an induction motor and rotor shaft coupled to angular approach

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An electromagnetic-mechanical modeling of an induction motor and rotor shaft coupled to angular approach

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