**1. Introduction**

Lately, as engineers have recognized the importance of having a high-quality working place, the effect of the noise and vibration emitted by inverter-fed induction machines has become a subject to study. Economic considerations force to use less active material. Since the encasing is less stiff, the machine becomes more sensitive to vibrations and noise. Less use of iron in the stator not only yields to a weaker structure but also higher field levels, thus causing higher magnetic forces, which yields to increased vibrations. Then the first aim of this work is to reach a wide knowledge how the levels of noise and vibration generated by the induction motor vary under different working conditions.

Electromagnetic noise is generated when the natural frequencies of vibration of induction motors match or are close to the frequencies present in the electromagnetic force spectrum. In order to avoid such noise and vibration, it is necessary to estimate the amplitude of the radial electromagnetic forces as well as the natural frequencies of the structure. For this reason, several papers have been published to analyze the natural frequencies, electromagnetic force, vibration and acoustic noise. For the analysis of the natural frequencies, a lot of papers have analyzed the stator core without winding. However, it is known that it is difficult to estimate the Young's modulus of winding. For the analysis of the radial force, vibration and acoustic noise, several papers have been published (Ishibashi et al., 2003, Shiohata et al., 1998, Munoz et al., 2003). They gave the amplitudes as well as the frequencies of the radial electromagnetic force. However, they mainly treated the case when the slip was 0. Ishibashi et al. did not consider the rotor current (Ishibashi et al., 2003), and Munoz et al. specified stator currents calculated by MATLAB/Simulink as input data not stator voltages (Munoz et al., 2003).

© 2012 Ishikawa, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Ishikawa, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

This paper investigates the vibration of induction motors fed by a Pulse Width Modulation (PWM) inverter. First we analyze the natural frequencies of the stator by considering the stator coil, and compare with the measured ones. Next, we analyze the radial electromagnetic force by using two-dimensional (2D) non-linear finite element method (FEM) which is considering the rotor current and is coupled with voltage equations, and discuss the calculated result with the measured vibration velocity. We clarify the influence of slip, the distributed stator winding and the PWM inverter on the radial force. Moreover, it is well known that a random PWM reduces the acoustic noise emitted from an inverter drive motor (Trzynadlowski et al., 1994). Then, we investigate the radial force of the motor fed by two types of random PWM method, namely, a randomized pulse position PWM and a randomized switching frequency PWM.

## **2. Natural frequencies**

### **2.1. Analysis method of natural frequencies**

The mechanical equation for the stator model with the free boundary condition is expressed as

$$\mathbf{[M]}\{\ddot{\mathbf{x}}\} + \mathbf{[K]}\{\mathbf{x}\} = \mathbf{[0]} \tag{1}$$

Analysis of Natural Frequency, Radial Force and Vibration of Induction Motors Fed by PWM Inverter 227

*<sup>i</sup>* . We solve (4) by discretizing the stator into a finite element mesh and using an

9 2 *E S* (0.0319 1.05) 10 N / m (5)

U U U

where, { } *<sup>i</sup>* is eigenvector representing the mode shape of the *i*-th natural angular

In the calculation of natural frequencies using FEM, the most important but unknown constant is Young's modulus of winding which is composed of the enameled wires, insulation films and vanish. Itori et al. has given the equivalent Young's modulus of

This chapter investigates the vibration characteristics of two motors, hereafter K-model and M-model, whose properties and characteristics are as follows. For M-model, 1.5 kW, 200 volt, 50 Hz, 6.8 A, 4 poles, number of stator slots: 36, number of rotor slots: 44, and one slotpitch skewing, see Fig.1 (Mori et al., 2005, 2005). For K-model, 100 volt, 50 Hz, 4 poles, number of stator slots: 24, number of rotor slots: 34, stator winding: 66 turns, rotor bar:


Natural frequencies are obtained by measuring the transfer function of the stator core. Fig. 3 shows an experimental setup to measure the natural frequencies. A piezoelectric accelerometer PV08A is placed at the top of the stator and is connected to one channel of a



eigenvalue subroutine utilized in International Mathematics and Statistics Library (IMSL).

winding in slot by the experimental investigation (Itori et al., 2002)

frequency

where, *S* is the space factor of winding.

**Figure 1.** Experimental motor, M-model.

**2.3. Measurement of natural frequencies** 


0

0.05

aluminium, and no skewing, see Fig. 2 (IEEJ, 2002).

U U U

**2.2. Experimental motors** 

where, { } **x** is the node displacement, [ ] **M** and [ ] **K** are the global mass matrix and stiffness matrix. In the two-dimensional (2D) case, the plate is assumed to have constant mass density ρ, area A, uniform thickness h, and motion restricted to the {x, y} plane. The element mass matrix is expressed as a 6 × 6 matrix

$$\mathbf{M}^{\epsilon} = \frac{\rho A h}{12} \begin{bmatrix} 2 & 0 & 1 & 0 & 1 & 0 \\ 0 & 2 & 0 & 1 & 0 & 1 \\ 1 & 0 & 2 & 0 & 1 & 0 \\ 0 & 1 & 0 & 2 & 0 & 1 \\ 1 & 0 & 1 & 0 & 2 & 0 \\ 0 & 1 & 0 & 1 & 0 & 2 \end{bmatrix} \tag{2}$$

The element stiffness matrix for plane strain is given by

$$\mathbf{K}^{\epsilon} = \frac{E}{(1+\nu)(1-2\nu)} \begin{vmatrix} 1-\nu & \nu & 0\\ \nu & 1-\nu & 0\\ 0 & 0 & (1-2\nu)/2 \end{vmatrix} \tag{3}$$

where *E* is Young's modulus and is Poisson's ratio.

Equation (1) leads to the eigenvalue problem,

$$\langle (\mathbf{K}) - \alpha\_i^2 [\mathbf{M}] \rangle \{ \mathbf{0} \}\_i = \{ \mathbf{0} \} \tag{4}$$

where, { } *<sup>i</sup>* is eigenvector representing the mode shape of the *i*-th natural angular frequency *<sup>i</sup>* . We solve (4) by discretizing the stator into a finite element mesh and using an eigenvalue subroutine utilized in International Mathematics and Statistics Library (IMSL).

In the calculation of natural frequencies using FEM, the most important but unknown constant is Young's modulus of winding which is composed of the enameled wires, insulation films and vanish. Itori et al. has given the equivalent Young's modulus of winding in slot by the experimental investigation (Itori et al., 2002)

$$E = (0.0319S - 1.05) \times 10^9 \left[ \text{N} / \text{m}^2 \right] \tag{5}$$

where, *S* is the space factor of winding.

### **2.2. Experimental motors**

226 Induction Motors – Modelling and Control

a randomized switching frequency PWM.

mass matrix is expressed as a 6 × 6 matrix

where *E* is Young's modulus and

Equation (1) leads to the eigenvalue problem,

*e Ah*

The element stiffness matrix for plane strain is given by

*<sup>e</sup> E*

**2.1. Analysis method of natural frequencies** 

**2. Natural frequencies** 

as

This paper investigates the vibration of induction motors fed by a Pulse Width Modulation (PWM) inverter. First we analyze the natural frequencies of the stator by considering the stator coil, and compare with the measured ones. Next, we analyze the radial electromagnetic force by using two-dimensional (2D) non-linear finite element method (FEM) which is considering the rotor current and is coupled with voltage equations, and discuss the calculated result with the measured vibration velocity. We clarify the influence of slip, the distributed stator winding and the PWM inverter on the radial force. Moreover, it is well known that a random PWM reduces the acoustic noise emitted from an inverter drive motor (Trzynadlowski et al., 1994). Then, we investigate the radial force of the motor fed by two types of random PWM method, namely, a randomized pulse position PWM and

The mechanical equation for the stator model with the free boundary condition is expressed

[ ]{ } [ ]{ } { } **M Kx 0**

where, { } **x** is the node displacement, [ ] **M** and [ ] **K** are the global mass matrix and stiffness matrix. In the two-dimensional (2D) case, the plate is assumed to have constant mass density ρ, area A, uniform thickness h, and motion restricted to the {x, y} plane. The element

> 201010 020101 102010 12 010201 101020 010102

> > 1 0 1 0

 

(1 )(1 2 ) 0 0 (1 2 ) / 2

 

is Poisson's ratio.

<sup>2</sup> ([ ] [ ]){ } { } 

 

*x* (1)

**M** (2)

*i i* **KM 0** (4)

**K** (3)

This chapter investigates the vibration characteristics of two motors, hereafter K-model and M-model, whose properties and characteristics are as follows. For M-model, 1.5 kW, 200 volt, 50 Hz, 6.8 A, 4 poles, number of stator slots: 36, number of rotor slots: 44, and one slotpitch skewing, see Fig.1 (Mori et al., 2005, 2005). For K-model, 100 volt, 50 Hz, 4 poles, number of stator slots: 24, number of rotor slots: 34, stator winding: 66 turns, rotor bar: aluminium, and no skewing, see Fig. 2 (IEEJ, 2002).

**Figure 1.** Experimental motor, M-model.

### **2.3. Measurement of natural frequencies**

Natural frequencies are obtained by measuring the transfer function of the stator core. Fig. 3 shows an experimental setup to measure the natural frequencies. A piezoelectric accelerometer PV08A is placed at the top of the stator and is connected to one channel of a charge amplifier UV-06. An impulse hammer PH-51 is connected to the other channel. The charge amplifier is connected to a signal analyzer SA-01A4, and then to a PC where a software for SA-01A4 is installed.

Analysis of Natural Frequency, Radial Force and Vibration of Induction Motors Fed by PWM Inverter 229

Mode Frequency [Hz] 2 1,325 2 1,337 3 3,425 3 3,875

Next, the natural frequencies of the stator with winding of M-model are measured as shown in Fig. 4. The natural frequencies around 1,200Hz are generated from rotor. Three lowest

> Mode Frequency [Hz] 2 637 3 1,770 4 2,694

100 1000 10000

First, we calculate the natural frequencies for the stator core only of the M-model motor, whose mechanical properties include mass density of 7,850kg/m3, Young's modulus of 2.1× 1010N/m2 and Poisson's ratio of 0.3. Table 3 shows the comparison of the calculated natural frequencies with the measured ones. It shows a good agreement between the measured values and the calculated ones. In this calculation, we use 18,811 finite element nodes. If we calculate the natural frequencies with a rough mesh, they become higher values. Fig. 5 shows the modes of stator due to each harmonic. The natural frequencies of 1,369 and

**Table 1.** Measured natural frequencies of the stator core of M-model motor.

natural frequencies except around 1,200Hz are shown in Table 2.

**Figure 4.** Natural frequencies measured for the whole motor M-model.

**2.4. Calculation of natural frequencies** 

6.25


1,425Hz have mode 2, and 3,446 and 3,926Hz have mode 3.

**Table 2.** Three lowest natural frequencies of the M-model motor with stator winding.

The transfer function is measured by hammering the stator surface. First, the natural frequencies of the stator core only of M-model are measured. We have removed the stator windings from the stator. Table 1 shows the four lowest measured natural frequencies.

**Figure 2.** Experimental motor, K-model.

**Figure 3.** Experimental setup for measurement of natural frequencies.


**Table 1.** Measured natural frequencies of the stator core of M-model motor.

228 Induction Motors – Modelling and Control

software for SA-01A4 is installed.

**Figure 2.** Experimental motor, K-model.


0

0.05

**Figure 3.** Experimental setup for measurement of natural frequencies.

charge amplifier UV-06. An impulse hammer PH-51 is connected to the other channel. The charge amplifier is connected to a signal analyzer SA-01A4, and then to a PC where a

The transfer function is measured by hammering the stator surface. First, the natural frequencies of the stator core only of M-model are measured. We have removed the stator windings from the stator. Table 1 shows the four lowest measured natural frequencies.




SA-01A4 installed PC Signal analyzer SA-01A4

U U

U U

Charge Amp.

UV-06 installed PC

Pick-up PV08A

Target motor

Hammer PH-51

Next, the natural frequencies of the stator with winding of M-model are measured as shown in Fig. 4. The natural frequencies around 1,200Hz are generated from rotor. Three lowest natural frequencies except around 1,200Hz are shown in Table 2.

**Figure 4.** Natural frequencies measured for the whole motor M-model.


**Table 2.** Three lowest natural frequencies of the M-model motor with stator winding.
