MODAL ANALYSIS OF ELECTROMAGNETIC ACTUATOR FOR NON-

CONTACT MODAL TESTING

MOHAMAD ASYRAF BIN AHMAD BUSTAMANG

Report submitted in partial fulfillment of the requirements

for the award of the degree of Bachelor of Mechanical Engineering

Faculty of Mechanical Engineering

UNIVERSITI MALAYSIA PAHANG

JUNE 2013

vii

ABSTRACT

Low surface finish, low accuracy and reduction in tool life were happening towards

the machining product and cutting tools during high speedmachining (HSM) process.

The defects were happening because of chatter effect or resonance of the whole

machine system. A study was conducted to study and optimizing between tool-life

and machining speed operation for controlling these chatter effect by using an

electromagnetic actuator (EMA) anon-contacting excitation force. The objective of

this project is to determine the modal analysis of electromagnetic actuator for non-

contact modal testing. The method use for determining modal analysis of

electromagnetic actuator by modal testing was by electromagnetic shaker. The modal

analysis of electromagnetic actuator was determined by comparison result of

computational and experimental modal analysis methods. The electromagnetic

actuator was analysed in two types that were structured with and without

electromagnetic core for each of modal analysis methods. ANSYS 14.0 was used as

the finite element analysis (FEA) software for generating modal analysis of the

structure and ME‟Scope software used for determining the modal analysis based on

the experimental method. The result of these modal analyses was supported by result

of operational deflection shape (ODS) for each type of assembling the structure. The

result of modal analysis of electromagnetic actuator was determined based on the

similarity data collected between computational and experimental methods. The

operational deflection shape determines the deflection of the structure under certain

operation condition. Based on modal analysis and operational deflection shape, the

structure of electromagnetic actuator can be used for the experiment with high

frequency excitation.

viii

ABSTRAK

Kemasan permukaan yang rendah, ketepatan yang rendah dan pengurangan jangka

hayat mata alat telah berlaku terhadap produk mesin dan mata alat pemotong semasa

proses pemesinan kelajuan tinggi (HSM). Kecacatan yang telah berlaku kerana kesan

chater atau resonans keseluruhan sistem mesin. Satu kajian telah dijalankan untuk

mengkaji dan mengoptimumkan antara jangka hayat mata alat dan kelajuan operasi

pemesinan untuk mengawal kesan chater dengan menggunakan aktuator

elektromagnet (EMA) sebagai daya perangsang tidak bersentuhan. Objektif projek

ini adalah untuk menentukan analisis modal aktuator elektromagnet untuk ujian

modal tidak bersentuhan.Kaedah yang digunakan untuk menentukan analisis modal

actuator electromagnet adalah ujian modal dengan penggoncang

electromagnet.Analisis modal untuk aktuator elektromagnet telah ditentukan oleh

hasil perbandingan kaedah analisis modal pengiraan dan eksperimen.Analisis modal

aktuator elektromagnet telah ditentukan oleh hasil perbandingan pengiraan dan

eksperimen kaedah analisis mod. Aktuator elektromagnet dianalisis dalam dua jenis

yang merupakan struktur dengan dan tanpa teras elektromagnet bagi setiap kaedah

analisis mod. ANSYS 14.0 telah digunakan sebagai perisian analisis unsur terhingga

(FEA) untuk menghasilkan analisis modal struktur dan perisian ME'Scope digunakan

untuk menentukan analisis modal berdasarkan eksperimen. Hasil analisis ini modal

disokong dengan hasil daripada bentuk pesongan operasi (ODS) bagi setiap jenis

struktur. Hasil analisis modal aktuator elektromagnet telah ditentukan berdasarkan

data yang dikumpul persamaan antara kaedah pengiraan dan eksperimen. Bentuk

pesongan operasi menentukan pesongan struktur di bawah keadaan operasi tertentu.

Berdasarkan analisis modal dan bentuk pesongan operasi, struktur penggerak

elektromagnet boleh digunakan untuk eksperimen dengan pengujaan frekuensi

tinggi.

ix

TABLE OF CONTENTS

Page

EXAMINER’S DECLARATION ii

SUPERVISOR’S DECLARATION iii

STUDENT’S DECLARATION iv

DEDICATION v

ACKNOWLEDGEMENTS vi

ABSTRACT vii

ABSTRAK viii

TABLE OF CONTENTS ix

LIST OF TABLES xii

LIST OF FIGURES xiii

LIST OF SYMBOLS xv

LIST OF ABBREVIATIONS xvi

CHAPTER 1 INTRODUCTION

1.1 Background Study 1

1.2 Problem Statement 2

1.3 Project Objective 3

1.4 Project Scope 3

1.5 Project Flow Chart 4

1.6 Organization of Thesis 6

CHAPTER 2 LITERATURE REVIEW

2.1 Introduction 7

2.2 High Speed Machining 7

2.3 Chatter Effect 8

2.4 Modal Testing 9

2.4.1 Impact Hammer Modal Testing 9

2.4.2 Electromagnetic Shaker Modal Testing 11

x

2.5 Non-Contact Modal Testing 12

2.5.1 Acoustic Excitation Methods 12

2.5.2 Pulsed Air Excitation 13

2.5.3 Non-Contacting Electromagnetic Excitation 13

2.5.4 Eddy Current Excitation 14

2.5.5 Laser Pulse Excitation 14

2.6 Experimental Modal Analysis 15

2.6.1 Natural Frequency 15

2.6.2 Mode Shape 15

2.6.3 Damping 15

2.7 Signal Analysis 16

2.7.1 Time Domain 16

2.7.2 Frequency Domain 16

2.7.3 Fast Fourier Transform 16

2.7.4 Frequency Response Function 17

2.8 Finite Element Analysis 18

2.9 Operational Deflection Shape 19

CHAPTER 3 METHODOLOGY

3.1 Introduction 20

3.2 Computational Modal Analysis Procedure 23

3.2.1 Structure Modeling with CAD Software 23

3.3

3.2.2 Modal Analysis with FEA Software 24

Experiment Procedure 28

3.3.1 Modal Testing with Electromagnetic Shaker 28

3.3.2 ODS Experiment with Motor Excitation 33

3.4 Modal Analysis Validation Procedure 35

3.5 Mathematical Validation Procedure 35

3.6 Summary 35

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CHAPTER 4 RESULT AND DISCUSSION

4.1 Introduction 36

4.2 Data of Computational Modal Analysis 37

4.2.1 Computational Analysis of Structure with EMA core 37

4.3

4.2.2 Computational Analysis of Structure without EMA core 38

Data of Experimental Modal Analysis 39

4.3.1 Modal Analysis of Structure with EMA Core 39

4.3.2 Modal Analysis of Structure without EMA Core 40

4.4 Validation of Modal Analysis Data 41

4.4.1 Validation of EMA structure with EMA Core 41

4.4.2 Validation of EMA structure without EMA Core 44

4.5 Data of ODS Setup 46

4.5.1 Experimental ODS of EMA structure with EMA Core 46

4.5.2 Experimental ODS of EMA structure without EMA Core 47

4.6 Mathematical Analysis 48

4.6.1 ODS of Structure with EMA Core 49

4.6.2 ODS of Structure without EMA Core 52

4.7 Summary 55

CHAPTER 5 CONCLUSION AND RECOMMENDATION

5.1 Conclusion 56

5.2 Recommendation 57

5.2.1 Computational Analysis 57

5.2.2 Experimental Modal Testing 57

5.2.3 Experimental ODS with Motor Excitation 58

5.2.4 Future Work 58

References 59

Appendix A 61

Appendix B 64

xii

LIST OF TABLES

Table No. Title Page

4.1 Computed Modal Analysis of structure with EMA core 37

4.2 Computed Modal Analysis of structure without EMA core 38

4.3 Percentage of Error for structure with EMA core 42

4.4 Modal Analysis Data of structure with EMA core 43

4.5 Percentage of Error for structure without EMA core 45

4.6 Modal Analysis Data of structure without EMA core 46

4.7 Signal Analysis of structure with EMA core 46

4.8 Signal Analysis of structure without EMA core 47

xiii

LIST OF FIGURES

Figure No. Title Page

1.1 Project Flow Chart 4

2.1 Impact Hammer Testing Setup 10

2.2 Electromagnetic Modal Testing Setup 11

2.3 Block Diagram of an FRF 17

3.1 Modal Testing and Modal Analysis Procedure 21

3.2 ODS Setup and Validation Procedure 22

3.3 Structure with EMA Core 23

3.4 Structure without EMA Core 24

3.5 Selecting Analysis Type 25

3.6 Defining Structure Material 25

3.7 Defining Material Type 26

3.8 Importing the Geometry 26

3.9 Modal Testing with Electromagnetic Shaker 28

3.10 Module of Signal Generation 29

3.11 Electromagnetic Shaker 29

3.12 Stinger and Force Transducer 30

3.13 Module for Collecting and Data Analysis 31

3.14 ODS Experiment with Motor Excitation 33

3.15 Sensor Setup 34

3.16 Placement of Accelerometer Sensor 34

4.1 Experimental Modal Analysis of Structure with EMA core 39

4.2 Experimental Modal Analysis of Structure without EMA core 40

4.3 Data Validation of Structure with EMA Core 41

4.4 Data Validation of Structure without EMA Core 44

4.5 ODS at First Deflection Shape 49

4.6 ODS at Second Deflection Shape 50

4.7 ODS at Third Deflection Shape 50

4.8 Dominant Deflection Shape of ODS 51

4.9 ODS at First Deflection Shape 52

4.10 ODS at Second Deflection Shape 53

xiv

4.11 ODS at Third Deflection Shape 53

4.12 Dominant Deflection Shape of ODS 54

xv

LIST OF SYMBOLS

Fourier transform

Continuous function (in time)

A Amplitude ratio

Magnitude of transfer function

Force acting on modal mode

Response for spatial mode

Response in modal

Matrix

Force

xvi

LIST OF ABBREVIATIONS

HSM High Speed Machining

EMA Electromagnetic Actuator

ODS Operational Deflection Shape

FYP I Final Year Project 1

FYP II Final Year Project 2

FEA Finite Element Analysis

FRF Frequency Response Function

3D Three Dimensions

YAG Neodymium-doped Yttrium Aluminium Garnet

FFT Fast Fourier Transform

DOF Degree of Freedom

FEM Finite Element Method

CAD Computer Aided Design

DASYLab Data Acquisition System Laboratory

DAQ Data Acquisition

NI-MAX National Instrument Measurement and Automation

Explorer

USB Universal Serial Bus

RPM Revolution per Minutes

CHAPTER 1

INTRODUCTION

1.1 BACKGROUND STUDY

High speed machining (HSM) machine is the modern technology used in

manufacturing industries. The advantage of HSM machine was capable to increase

the efficiency of the machining process. It also can increase the accuracy and the

quality of the machining product together with reducing the production. This

machine was also used for processing material in a short time compared to the

conventional machine. HSM is referring the machining process by light milling

passes to achieve a high metal removal rate by high spindle speed and feed rate.

The application of this HSM machine can be effective at machining the core

and cavity geometries such as machining the mold. It also effective in machining the

large component structure of the aircraft or fighter jet from a block of material. For

manufacturing companies of aircraft component, this technology can reduce the cost

and time for producing it. The total assembly time of single aircraft can up till one

year depending on the size of the aircraft. This problem was happening due to

machining process of every single component that requiring on time and quality. By

HSM technology, this problem with time can quality of the machining process can

solve and together it can reduce the production cost.

2

1.2 PROBLEM STATEMENT

Some scholar defines HSM as the machining process at or near the resonant

frequency of the machine. This statement is due to the high frequency that be

produced by the machine when its speed is increased. When the speed is increased,

the frequency has also increased. When machining with high speed, at certain

frequencies that produce can cause the resonance towards the machine. This

resonance frequency does not cause towards the machine itself but it affect toward

the whole machining system that related. The resonance frequency knows as the

chatter effect. The chatter effect can cause to low surface finish, low product

accuracy and reducing tool life.

The cutting tool and the tool holder were included together with machining

system that can affect to chatter effect. It was resulting to the large displacement of

the cutting tool and the work piece. Because of that, it can be the cause to the low

surface finish and low product accuracy.

From this case, a study conducted to determine the dynamic properties of the

cutting tool under dynamic conditions. This study will determine the dynamic

characteristic of the cutting tool for reducing the effect of chatter. The research was

using basic modal testing to determine the dynamic characteristic of a component.

For non-contact modal testing, the Electromagnet Actuator (EMA) has used

as the experimental tool. EMA designed for studying and optimizing between tool-

life and speed of machining operation for controlling these chatter effect. EMA used

to produce magnetic flux and forces as the excitation force and capturing the force

that has exerted. Since this EMA structure mounted on the machining structure, it

also can be effected to the chatter effect. The dynamic properties of the EMA

structure also need to identify before the experiment started.

3

1.3 PROJECT OBJECTIVE

Modal analysis is the study of dynamic properties of structures under

vibrational excitation. This method is used to measure and analyses the dynamic

response of the structure or fluid when being excited by an input. The objective of

this study is to determine modal analysis of electromagnetic actuator (EMA) for non-

contact modal testing. This EMA will undergo modal testing experiment and the data

for modal analysis verified by computational analysis using Finite Element Analysis

(FEA).

1.4 PROJECT SCOPE

For achieving the objective of this research, the scope that relates to the

project that's been identified were:

i. Determining dynamic properties of Electromagnetic Actuator (EMA) by

experimental, computational and mathematical approach.

ii. Conducting an experimental modal testing with vibration shaker setup.

iii. Validation of experimental modal analysis with computational modal

analysis of FEA software.

iv. Conducting an experimental Operational Deflection Shape (ODS) with

motor excitation.

v. Validation of experimental ODS with mathematical calculation.

4

1.5 PROJECT FLOW CHART

Figure 1.1 Project Flow Chart

Start

Defining the Background of

Study

Identifying Problem Statement,

Objective and Scope of Study

Literature Review

Methodology

Computational, Experimental and

Mathematical Setup

Data

Validation

Data Result

Data Analysis and Discussion

Presentation and

Documentation

End

FYP I

FYP II

YES

NO

5

Figure 1.1 shows the flow chart for this project. For Final Year Project (FYP)

1, this research project started with defining the background of study that relates to

the topic that by identify the real situation that relating with the research. Next, the

process followed by identifying problem statement, objective and scope of the study.

Under this section, the main problem identified based on the real situation that has

the connection between it. Scope of study also been defined as the limitation of the

research.

All journals, articles, reference books and reliable source reviewed as the

project references under literature review section. Under methodology section, the

computational modal analysis of the structure identified by FEA software simulation.

Experimental setup for modal testing constructed together with suitable software for

modal analysis of modal testing data in FYP 2.

The second section of FYP will be continuing with setup the experimental of

modal testing. Under methodology, the experimental setup for Operational

Deflection Shape (ODS) also been constructed together with .Under data result, data

collected form modal testing and ODS been analysed by signal analysis software.

Under data analysis and discussion, the analysed data were validated by

computational modal analysis for modal data analysis and calculation approach. The

validation data for each experiment has been set up has discussed for any changes.

Under presentation and documentation, the data collected presented and documented

in both experiments setups.

The project progress for FYP 1 and FYP 2 can reviewed through the Gantt

chart in the Appendix A.

6

1.6 ORGANIZATION OF THESIS

Chapter 1 introduces the background of the study. The problem statement that

based on the project background has discussed in this chapter. The objective of the

research, the limitation of research through scope of study and project flow chart also

have been detailed in.

In chapter 2, the detail research about the non-contact modal testing,

contacting modal testing, modal analysis, Electromagnetic Actuator (EMA), and

Finite Element Analysis (FEA) will discuss. It together with the Operational

Deflection Shape (ODS) and mathematical calculation had been through. Anything

that relates to this research also discuss in this chapter.

Chapter 3 will explain the experimental method used and relate to study of

modal analysis of EMA structure. This is including the setup for experimental modal

testing and modal analysis, computational modal analysis by FEA software and

validation of modal analysis. The experimental ODS and validation by mathematical

approach included. The setup for software used in this project determine clearly. The

validation process of both experiments setups also defined.

In chapter 4, the data being collected and validate between experimental

modal analysis and computational modal analysis for identifying the modal analysis

of structure. And for ODS, data collected is analysed and validate by mathematical

methods. Based on validation data, the discussion was drawn for both experiments.

The validation data will identify the suitable and reliable data as the result of this

data analysis.

Through chapter 5, the conclusion was drawn based on data collected, data

validation and discussion. Any suggestion for data result improvement also been

stated clearly in this chapter under the recommendation and future works.

CHAPTER 2

LITERATURE REVIEW

2.1 INTRODUCTION

This chapter was dealing with the related research paper, journal, articles, and

reference book that can use in determining the dynamic properties of EMA structure.

The related aspect of modal testing, modal analysis, Finite Element Analysis (FEA)

and Operational Deflection Shape (ODS) have been through. The chapter shows the

experiment designs by the pervious researcher based on same research scope through

the research paper and journal. The reference book was used as the guided in

conducting the experiment with the proper setup.

2.2 HIGH SPEED MACHINING

High speed machining (HSM) is a type of material removal technique that

use by big manufacturing company in producing the product in a short period. This

technique for material removal change the manufacturing process of aircraft

components and it produces by machined monolithic components. The replacement

of the sheet metal assemblies‟ methods is reducing the cost of the manufacturing

process and improving the structure of the machined components (Davies et al.,

1998).

A lot of time require by the workers for processing of the smaller component

and assemble it into the aircraft. The application of this technique was helping the

aircraft manufacturing company reducing time by producing the aircraft component

only by one large workpiece. The advantage of a high speed machining process of

8

aluminium is reducing in machining period, improving workpiece surface finished,

reducing in thermal and mechanical stress on the workpiece and tool, and improving

dynamic stability (Halley et al., 1999).

2.3 CHATTER EFFECT

Either in conventional machining or high speed machining, the chatter was

the factor that can contribute to lowering the productivity of the material removal

machine. This effect was the cause of poor quality of a workpiece surface. The tool

also can damage when the high effort applied to accelerate tool wear. The chatter

was the uncontrolled and it was the unstable vibration of the cutting tool. This

unstable vibration behaviour was resulting in large amplitude relative displacement

between the cutter tool and the workpiece.

There were two main causes of chatter effect that were a mode of coupling

and regeneration of waviness. Mode coupling was two-dimensional phenomena that

occurred when the tool was experience force feedback at least in two directions that

was simultaneously. Both directions have the same values of frequency vibration.

This phase shift of the cutting tool was the cause of the unstable of elliptical motion.

The generation of waviness effect was the other cause of this chatter effect. This

waviness effect was the caused by the surface variation that associated with previous

cuts and undulating surface create when each tool flute pass over making cuts on the

material surface. This wavy surface encounter after the next flute passes and makes

another cut (Caulfield, 2002).

This chatter effect was developed of the excitation at the natural frequency of

the machine system that were either the cutting tool, workpiece, machine, table

machine or other parts of the machine. The excitation was caused by the modulated

of chip thickness that undergoes vibration. This chip modulated was creating the

dynamic cutting forces and the frequency develop was close to the natural frequency

that related inside the machine system (Kayhan et al., 2009).

9

2.4 MODAL TESTING

Modal testing can be found in two types, that were impacted hammer modal

testing and vibration shaker. Impact hammer and vibration shaker were the

conventional devices for the conventional modal testing method. This conventional

impact hammer modal testing method used to excite the target structure by force in

frequency vibration technique. A conventional impact hammer used for this

conventional modal testing method by applying the impact force with a certain

amplitude of frequency vibration.

It is slightly different with the function of shaker vibration that used to shake

the structure with certain types of frequency waves that is depending on the apply

frequency. The conventional method of modal testing is the frequency force applied

by excitation in contact between structure and the impact or vibration device. This

shows that the method used to in contact modal testing. The frequency response of

the target structure measured by the device that mounted on the target structure.

2.4.1 Impact Hammer Modal Testing

Impact hammer testing was the popular modal testing method since 1970.

This modal test method has the ability to compute frequency response function

(FRF) measurement in a Fast Fourier Transform (FRF) analyser. It was also the

popular technique for finding mode shape of the structure with fast, convenient and

in low cost. In order to perform testing with impact hammer, the required setup was

an impact hammer with a load cell attached to its head. This load cell functioning to

measure the input force that applied for force excitation.

A singe accelerometer used to measure the frequency response at a single

fixed point. For the procedure of impact testing, the accelerometer attached only at a

single point of the target structure. And the impact hammer was used to apply the

impact force at many points and many directions. This many points and directions

were required in order to determine its mode shape. FRF computed one at a time that

10

is between each impact point and the fixed response point. The modal parameters

were defined by the curve fitting the resulting set of FRF (Richardson, 1997).

Figure 2.1 Impact Hammer Modal Testing Setup

Sources: Richardson (1997)

The figure shows the measurement setup of impact hammer modal testing.

The applied impact hammer modal test can use to determine the dynamic properties

of the target structure

Impact hammer modal testing cannot apply for all types of structures. The

structure with delicate surfaces was cannot be tested by impact hammer testing. It

can cause by its limited range of frequency or low density over a wide spectrum, so

the impacting force is not sufficient to excite the mode of interest. FRF measurement

must be applying an artificial excitation with one or more shaker that attached to the

structure if the impact hammer cannot use.

11

2.4.2 Electromagnetic Shaker Modal Testing

Electromagnetic and hydraulic shaker were the common types of shakers. A

stinger or long slender rod usually attached to the target structure. The shaker will

impart the force to the target structure by along the axis of the stinger. The stinger

was the axis of force measurement. In between the structure and the stinger, a load

cell attached in order to measure the excitation force. The tri-axial accelerometer and

3D motion of the structure measured at each test point can use for the analyser that

having four channels (Schwarz & Richardson, 1999).

Figure 2.2 Electromagnetic Modal Testing Setup

Sources: Schwarz & Richardson(1999)

12

2.5 NON-CONTACT MODAL TESTING

The other technique of modal testing also been developed this is non-contact

modal testing. This method is the different version of the contact modal testing

technique. The differences between contact and non-contact method are from the

experiment setup that is the technique of force excitation and frequency response

measurement. The setup for contact modal testing has defined clearly in the previous.

For non-contact modal testing, the force being excited to the structure target without

contacting between it with the force exciter. The sensor of frequency response

measurement also can connect either contact or non-contact with the target structure

depending type of measurement sensor to be used

.

2.5.1 Acoustic Excitation Methods

Acoustic excitation method is the method that exposes a sound field at the

same condition as the simple ways of causing a structure to vibrate. By applying

sound excitation towards the target structure, it is possible to use it as the excitation

force by non-contacting method. The excitation signal generated acoustically and this

method has been attracting the researcher on non-contact excitation.

Weaver and Dowdell (1984) discussed to use the speaker for modal testing by

applying acoustic excitation towards a plate. They conclude that the speaker

excitation can considered be multiple input excitations in the sense that acoustic

waves strike the plate over entire surface and not just in one place. Both of

researchers make no mention of the fact that the acoustic wave striking the plate was

identical frequency composition. The waves can considered as a multiple input

excitation, hey represent a series of correlated inputs that will make it impossible to

determine which force is responsible for the particular response.

Musson and Stevens (1985) conducted on the same test and attempted a fully

non-contacting excitation and the response test by using acoustical method in 1986.

The work concluded that the method was satisfactory when precise modal definitions

were not required. Huber et al. (2006) was conducted non-contact vibrational mode

13

of reed organ pipe by using radiation force to perform ultrasound stimulate

excitation. The radiation force was setup by using a pair of ultrasonic beams with

different frequency of audio-range. By the same persons, they also had a non-contact

modal testing hard-drive suspension with the same method of ultrasound radiation

force.

In general, then, acoustic sources and ultrasound stimulated can used to excite

structures. Both methods used by focusing the source into one point in order to

provide the excitation force.

2.5.2 Pulsed Air Excitation

The pulsed air excitation method was the experiment setup that applying

pulsed air used to excite a structure. Vanlanduit et al. (2006) was set up an

experimental modal testing by using pressurized air excitation to determine to excite

the structure. The purpose of the experiment was to determine the exerted force by

the pneumatic excitation system. The impinging jet set up by aiming at the desk. The

impact forces that exert on the disk measure by the influence of the supply pressure.

Farshidi et al. (2010) set up the experimental modal analysis by using air

excitation and microphone array. A simple aluminium 6061 cantilever beam use by

fixing at the end of the beam. The array of microphone sensor positioned

symmetrically around the air excitation sources. Based on the experiment set-up by

using air excitation force, it can demonstrate the effectiveness of this method based

on the accuracy of the experimental result. And these methods are cost effectively

measured structural dynamics by translational and rotational degrees of non-contact

excitation and sensor mechanism.

2.5.3 Non-Contacting Electromagnetic Excitation

Firrone and Berruti (2011) purposed a paper of the non-contact excitation

system based on electromagnets. The system aims at exciting cyclically symmetric

structure like bladed disk by generating typical engine order like travelling wave

excitation that bladed disk encounter during the service. Their paper proposed an