Mechatronics, Electrical Power, and Vehicular Technology 05 (2014) 1-8 

 

Mechatronics, Electrical Power, and 

Vehicular Technology 
 

e-ISSN:2088-6985 

p-ISSN: 2087-3379 

Accreditation Number: 432/Akred-LIPI/P2MI-LIPI/04/2012 

 

 

www.mevjournal.com 
 

© 2014 RCEPM - LIPI All rights reserved 

doi: 10.14203/j.mev.2014.v5.1-8 

ROTOR-DYNAMIC CHARACTERISTIC EVALUATION OF INTERIOR 

PERMANENT MAGNET MOTOR USING FINITE ELEMENT METHOD 
 

Hilman Syaeful Alam 
a,
*, Pudji Irasari 

b
 

a
Technical Implementation Unit for Instrumentation Development, Indonesian Institute of Sciences, 

Jl. Sangkuriang Komplek LIPI Gedung 30 Bandung, 40135, Indonesia. 
b
Research Centre for Electrical Power and Mechatronics, Indonesian Institute of Sciences,  

Jl. Sangkuriang Komplek LIPI Gedung 20 Lantai 2 Bandung, 40135, Indonesia. 

 
Received 27 December 2013; received in revised form 12 February 2014; accepted 12 February 2014 

Published online 23 July 2014 

 

Abstract 
Dynamic characteristics of a critical speed of the rotor components at interior permanent magnet motor were evaluated using 

one-dimensional (1D) and three-dimensional (3D) finite element methods. Critical speed of the rotor wasinvestigated in the 

Campbell diagram, which shows the relationship between natural frequency and rotational velocity of the system when the motor 

is not in operation. The 1D finite element analysis shows that there are two modes which are close to the design frequency of 300 

Hz i.e. mode 1 and 2. However the critical rotational velocity in both modes are still far above the maximum velocity design of 
6,000 rpm. Validation using 3D finite element analysis demonstrated that all modes were still above the designed frequency and 

did not find any critical speed below 6,000 rpm. It can be concluded that the critical speed of the rotor of IPM motor is still 

outside the system resonance region, and can be operated safely. 

 
Keywords: natural frequency, campbell diagram, interior permanent magnet motor, finite element method. 

 

I. INTRODUCTION 
Interior permanent magnet (IPM) motor is one 

of electric motor type, which is widely applied in 

the compact system for its several advantages: 

lightweight, small size, simple mechanical 

construction, easy maintenance, high reliability 

as well as high energy to volume ratio [1-5]. One 

of the problems in designing the rotating 

machinery including IPM motors is the vibration, 

because the vibration is usually a direct cause of 

the component damage. Every spinning rotor has 

some vibration, at least a once-per-revolution 

frequency component (1st order) therefore it is 

impossible to make any rotor perfectly mass 

balanced. Rotor-dynamic analysis is essential for 

quantifying safe upper limits of allowable 

vibration levels by analyzing the critical speed of 

the system which is very useful. It can provide 

information about the resonance region of the 

system and can be used as a standard to monitor 

the possibility of harmful, due to damage of the 

components. In addition, it is very useful for 

designers in understanding the relationship 

between the selection of design schemes 

including the shaft size, bearing properties, 

housing stiffness and machine stability [6].  

There are two methods which are often 

adopted to deal with rotor dynamic problem. One 

is the transfer matrix method, such as Riccati 

transfer matrix method and the other is the finite 

element method (FEM), which has higher 

numerical stability but need more storage space 

of the computer. Both methods are widely used to 

solve the rotor dynamic problem. The former 

method divides the system into several parts after 

it gets the lumped mass model, such as the disk, 

the shaft and the bearing. Then it establishes the 

relation of the state vectors between the both 

ends of the cross-section and use the continuity 

conditions to obtain the relation between the state 

vectors in any cross section and the initial one. 

The latter method, namely FEM, was not adopted 

to analyze rotor dynamic problem until 

1970s.The key idea of the FEM is to transform 

the infinite DOF (Degrees of Freedom) problem 

into a finite number of DOF, and then solve it. As 

the computer technology develops, the FEM 

become very popular to analyze mechanical 

problems, not just the rotor dynamic [7]. 
*Corresponding Author. Tel.: +62-81394297528 
E-mail: alam_hilman@yahoo.com 

http://dx.doi.org/10.14203/j.mev.2014.v5.1-8


H. S. Alam and P. Irasari / Mechatronics, Electrical Power, and Vehicular Technology 05 (2014) 1-8 

 

 

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The FEM has several advantages such as 

reduction of time when solving complex equation 

system. In addition, it can be applied by the 

software and can be widely applied in solving 

problems in engineering field for high accuracy 

and flexibility [8]. Some previous research have 

been used finite element to evaluate the dynamic 

characteristics of the rotor shaft system for some 

applications, such as: main shaft system in a 

hydro-turbine [7], gas turbine rotor [8], rotating 

structures [9], rotor-shaft system for evaluating 

hydro-dynamic action in journal bearings [10], 

and induction motors [11], and the rotor 

deflection of permanent magnet (PM) generator 

[12]. In this study, the dynamic characteristic of 

the rotor is evaluated using finite element 

method, in order to determine the existence of a 

critical speed and safe upper limits of allowable 

vibration levels of IPM motors, so it can be used 

as a reference for dynamic analysis and a 

foundation for the design or improvement. 

 

II. MATERIALS AND METHODS 
The dynamic characteristic of the rotor was 

evaluated using finite element method by 

analyzing the critical speed based on the 

Campbell diagram which represents the 

relationship between the natural frequencies and 

the rotational velocity of the system. 1D finite 

element analysis is conducted numerically based 

on the development of the Lagrange equations 

and the effects of unbalanced mass and bearing 

influence are neglected. Then, the results of 

analysis are validated using ANSYS, a 

commercially software based on three 3D finite 

element analysis. The main components of rotor 

are the shaft, bearing and permanent magnet 

along with the crank which is modeled simply as 

a disk, shown in Figure 1. 

To determine the dynamic characteristics of 

the rotor, the calculation of the kinetic energy   
is required to characterize disk, shaft and 

unbalanced mass. In addition, the strain energy   
is employed to characterize the shaft. Force on 

the bearing as the action force on the shaft is used 

to calculate the virtual work    due to an 
external force. The general equation to evaluate 

the dynamic characteristics of the rotor can be 

modeled using Lagrange equation as follows 

[15]: 

 

  
(
  

   
) 

  

   
 
  

   
     (1) 

where  (     ) is the number degree of 
freedom,    are generalized independent 
coordinates,     are generalized forces, and 
denotes differentiation with respect to time  . 

 

A. 1D Finite Element 
In 1D finite element, shaft is modeled as a 

beam which has a constant circular cross-section. 

If the element has two nodal, it will form the 8-

order matrix. Each of the nodal has 4 degrees of 

freedom: 2 displacements and 2 slopes in both x-

y plane and y-z planes (Figure 2), therefore nodal 

displacement vector can be written in the 

equation [15]: 

  [                       ]
  (2) 

The relationship between displacement and 

slope are: 

  
  

  
 (3) 

   
  

  
 (4) 

Displacements    and   in accordance with 
movements in the X and Z directions written by 

the equation: 

   [           ]
  (5) 

   [           ]
  (6) 

Displacements in the finite element are 

formed from: 

    ( )   (7) 

    ( )   (8) 

  ( ) and   ( ) are the function of 
displacements for the beam that are subjected to 

bending loads: 

 
Figure 1. Rotor components of IPM motors 

 

 
Figure 2. Modeling of one dimensional (1D) rotor [15] 



 H. S. Alam and P. Irasari / Mechatronics, Electrical Power, and Vehicular Technology 05 (2014) 1-8 

 

 

3 

  ( )  *  
   

  
 
   

  
    

   

 
 

  

  
 
   

  
 
   

  
 
  

 
 
  

  
+ (9) 

  ( )  *  
   

  
 
   

  
   

   

 
 
  

  
 
   

  
 

   

  
  

  

 
 
  

  
+ (10) 

Kinetic energy of the shaft can be calculated as: 

  
  

 
∫ [  ̇   

 
    ̇    ̇

   
 
    ̇]   

 

 

  

 
∫ [  ̇ 

   
 

  

   

  
  ̇  

 

 

  ̇ 
   

 

  

   

  
  ̇]   

    ∫   ̇ 
   

 

  

   

  
          

 

 
 (11) 

where   is the mass per unit volume,   is the 
cross sectional area of the beam, and   is the area 

moment of inertia of the beam cross section 

about the neutral axis,   is the length of element 
and   is the angular velocity.  

Substitution of equation (9) and (10) into 

equation (11), then: 

  
 

 
  ̇     ̇  

 

 
  ̇     ̇  

 

 
  ̇     ̇  

 

 
  ̇     ̇     ̇

      

      (12) 

   and    are the classical mass matrix,    and 
   are generated due to a secondary effect of 
rotor inertia, and    is generated due to 
gyroscopic. By applying the Lagrange equation, 

then: 

 

  
(
  

  ̇
) 

  

  
 (    ) ̈    ̇ (13) 

where   and    are obtained from 
            and   from   , thus: 

 

  
   

   

[
 
 
 
 
 
 
 
                
                
                 

                   

                
                
                   
                 ]

 
 
 
 
 
 
 

 (14) 

   
  

   

[
 
 
 
 
 
 
 
               
             
               
               

             
               
                
               ]

 
 
 
 
 
 
 

 (15) 

  
   

   

[
 
 
 
 
 
 
 
               
             
               
               

             
               
                
               ]

 
 
 
 
 
 
 

 (16) 

Strain energy on the shaft can be calculated by the equation: 

  
  

 
∫[   

    
 

   
    
   

      
    

 

   
    
   

  ]  

 

 

 

 
  

 
∫ *   

   
 

  

   

  
      

   
 

  

   

  
  +  

 

 
 (17) 

Then, after integration: 

  
 

 
        

 

 
        

 

 
        

 

 
        (18) 



H. S. Alam and P. Irasari / Mechatronics, Electrical Power, and Vehicular Technology 05 (2014) 1-8 

 

 

4 

where    and    are the classical stiffness matrix (  ),    and    are the matrix due to the axial force 
(  ).  

The effect of shear force can be calculated by the equation: 

  
    

    
 
 (19) 

with shear modulus: 

  
 

 (   )
 (20) 

where   is the Poissons ratio and E is Young’s modulus of the material. Then by applying Eq. (17) to the 
Lagrange equation: 

  

  
    (21) 

        (22) 

where    dan   , can be calculated 

   
  

(   )  

[
 
 
 
 
 
 
 
 
               
             
   (   )       (   )   

     (   )      (   )  

             
               
   (   )       (   )   

     (   )      (   )  ]
 
 
 
 
 
 
 
 

 (23) 

   
 

   

[
 
 
 
 
 
 
 
               
             
           (    

               

             
               
               
               ]

 
 
 
 
 
 
 

 (24) 

 

Furthermore the equations for the components 

of permanent magnet and the hollow crank are 

modeled as a disk, which only characterized by 

its kinetic energy. The node of the rotor has four 

degrees of freedom: two displacements   and   
and two slopes about the x-y and y-z planes, 

which are   and   respectively. Then if the nodal 
displacement vector δ of center of the disk is 

  [       ]  (25) 

By applying the Lagrange equations, the equation 

for the disk: 

 

  
(
  

  ̇
) 

  

  
 [

     
     
      
      

][

 ̈
 ̈
 ̈
 ̈

] 

 

[
 
 
 
    
    
       
      ]

 
 
 
[

 ̇
 ̇
 ̇
 ̇

] (26) 

where the first matrix is the classical stiffness 

matrix and the second one is the matrix due to 

gyroscopic effects. 

In this study, the effects of unbalanced mass 

and bearing influence are neglected so that the 

general equation of rotor dynamics is 

  ̈      ( ) (27) 

where  is all nodal displacement vectors,  is the 
mass matrix,  is the stiffness matrix and  ( )is 
the force vector.  

1D finite element analysis in this study uses 

the discretization of 4 elements and 5 nodal. 

Simplification scheme of the rotor with 

dimension in mm is shown in Figure 3. 

To produce the equality throughout the 

element, the local element matrix of each element 

is arranged into a global matrix. Every element, 

which has the same number of nodal and that of 

degrees of freedom, is placed on the same row 

and column and this applies to the mass matrix, 

stiffness matrix and damping matrix [14]. The 



 H. S. Alam and P. Irasari / Mechatronics, Electrical Power, and Vehicular Technology 05 (2014) 1-8 

 

 

5 

illustrative example of the element assembly 

from local elements into global element for two 

local elements can be seen in Figure 4. 

Globalizing matrix in this way is the fastest and 

easy, even for a lot of elements and nodal degrees 

of freedom. In addition to the dimensions of each 

component of the rotor, the mechanical properties 

of the component are also required as the input 

data of the calculation. In this study, the material 

of the shaft is JIS S45C steel, with Poissons ratio 

  = 0.27 to 0.30, tensile strength = 569 MPa, 
yield strength = 343MPa, Young's modulus, E = 

190 to 210 GPa and density = 7,700 to 8,030 

kg/m
3 

[16]. Meanwhile, the disk consists of two 

components, namely the crank is made of 

steelST37 and the permanent magnet type is 

NdFeB. Both the density of the material is almost 

the same which is about 8,000 kg/m
3
. 

B. 3D Finite Element  
The evaluation procedures of 3D finite 

element analysis using ANSYS to analyze rotor-

dynamic characteristics consist of geometric 

modeling, material definition, meshing 

(discretization), the determination of boundary 

conditions, calculation, and displaying the results 

[6][7]. In the geometry modeling stage, the 

geometry of every rotor component is made in 

3D and its material type is defined from 

specification. Meshing stage is conducted by 

dividing the components into small elements with 

a finite number and then doing analysis according 

to the material properties, boundary conditions 

and loading. In this study, the tetrahedral 3D 

element type on auto-mode is used, meaning that 

the discretization step is performed automatically 

by ANSYS program, including the selection of 

the elements number. After the processes of 

iteration and calculation are solved, then the final 

stage is to display the results of the simulation. 

 

III. RESULTS AND DISCUSSION 

A. 1D Finite Element 
Campbell diagram showing the output results 

of numerical simulations using the finite element 

method represents a natural frequency as a 

function of rotational speed. Figure 5 shows the 

dynamic characteristics of the rotor using 1D 

finite element that are visualized by Campbell 

diagram, where the X axis is the rotation in rpm 

and the Y axis is the natural frequency of the 

system. The black solid line indicates the 

frequency of the rotor which equals to the 

frequency of rotation, while the black dashed line 

indicates the frequency of the rotor which equals 

half of the frequency of the rotation.  

 

Figure 4. Element assembly from local elements into global 

element for two elements [15][16] 
 

Figure 5.Campbell diagram of 1D finite element analysis 

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0

500

1000

1500

2000

2500

3000

N (rpm)

F
 (

H
z
)

CAMPBELL DIAGRAM

 

 

F=N/60

F=0.5N/60

mode 1 BW

mode 1 FW

mode 2 BW

mode 2 FW

mode 3 BW

mode 3 FW

mode 4 BW

mode 4 FW

 

Figure 3. Simplification scheme of the discretization in 1D 
finite element analysis for the rotor 



H. S. Alam and P. Irasari / Mechatronics, Electrical Power, and Vehicular Technology 05 (2014) 1-8 

 

 

6 

The simulation results illustrate only the first 

four modes in the diagram because the value of 

the other modes is far above the operating speed 

of the motor. Each mode has two lines of 

response in the opposite direction those are FW 

(solid line) and BW (dashed line). When the 

response lines equal to the rotational speed or 

showing intersections in the diagram then these 

intersections are called critical speed. The 

maximum speed of the IPM motor refers to the 

maximum frequency of the speed control used 

that is 300 Hz. Having 6 poles number, the 

maximum speed of the motor equals to 120 

multiplied by the frequency and divided by the 

number of poles yields 6,000 rpm. According to 

the simulation results, the possibilities of rotor 

critical speeds are in mode 1 and 2, therefore the 

analysis is focused on both frequencies. 

By enlarging the Campbell diagram near the 

maximum design frequency of 300Hz, modes 1 

and 2 are investigated as shown in Figure 6.As 

shown on Figure 6, there is intersecting point 

between the response line and mode 1 at the 

rotational speed of 7,000 rpm (116.4 Hz). 

However, that critical speed resulted from the 1D 

finite element analysis is above the motor 

maximum speed of 6,000 rpm and frequency of 

300 Hz so that the motor can operate safely. 

These results are then compared with the results 

obtained from 3D finite element analysis. 

 

B. 3D Finite Element 
Discretization or meshing in ANSYS uses 

tetrahedral 3D element in auto mode. The 

meshing process produces 18,077 elements and 

37,571 nodal, shown in Figure 7. If we compared 

with 1D finite element, that has very few 

elements and nodal, 3D finite element will 

generate a more detailed analysis on each 

segment of the rotor.  

After iteration and calculation process are 

solved by the computer, Campbell diagram and 

each natural frequency modes will be generated 

from the results of calculations. In this study, the 

first ten modes are examined and represented in 

Table 1. From the 10 modes generated by 3D 

finite element, the smallest natural frequency is 

well above the design frequency of 300 Hz. 

Campbell diagram in Figure 8 shows that all 

modes do not indicate critical speeds in the 

system for the rotational velocity 1,000 rad/s or 

9,554.14 rpm. Based on these results, the IPM 

motor designed with the maximum speed of 

6,000 rpm and the frequency of 300 Hz can be 

operated safely. The other simulation result is the 

modal map in different modes.  

The modal maps of the first six modes are 

shown in Figure 9. The first and the second 

modes appear to be bending on the disk to the 

shaft position, while the third and fourth modes 

there is a shift toward the disk position to the 

shaft axis. Then at the fifth and sixth modes, 

bending occurs on the shaft. 

 

IV. CONCLUSION 
Rotor-dynamic characteristics of the IPM 

motor in the form of the critical speed in the 

Campbell diagram has been evaluated using the 

1D and 3D finite element method. 1D finite 

 
Figure 6. Rotor critical speeds in mode 1 and 2 

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000
0

50

100

150

200

250

300

350

400

450

500

N (rpm)

F
 (

H
z
)

CAMPBELL DIAGRAM

 

 

F=N/60

F=0.5N/60

mode 1 BW

mode 1 FW

mode 2 BW

mode 2 FW

 

Figure 7. Discretization using tetrahedral elements 

Table 1. 
Frequencies at the first ten modes 

Mode Frequency (Hz) 

1 505.08 

2 552.16  

3 655.71  

4 992.08  

5 1,084.80 

6 1,087.40 

8 3,288.90 

9 3,684.30 

10 3,711.70 

 



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7 

 

 

Figure 8. Campbell diagram of 3D finite element analysis 

 

 

Figure 9. The model maps of the first six modes 



H. S. Alam and P. Irasari / Mechatronics, Electrical Power, and Vehicular Technology 05 (2014) 1-8 

 

 

8 

element analysis yields two modes (modes 1 and 

2) which are in the range of frequency design of 

300 Hz, however the critical rotational velocity in 

both modes are still far above the maximum 

velocity design of 6,000 rpm.  

Validation using 3D finite element analysis 

demonstrates that all modes are still above of the 

designed frequency and did not find any critical 

speeds below 6,000 rpm. It can be concluded that 

the critical speed of the rotor of IPM motor is still 

outside the system resonance region and could be 

operated safely. 
 

ACKNOWLEDGEMENT 
Authors would like to thank to Kegiatan 

Kompetitif LIPI 2013, which has funded this 

research. Moreover thanks to all the team 

members of electric machines in The Research 

Centre for Electrical Power and Mechatronics, 

for any assistance that has been given. 

 

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