finite element modeling of dynamic properties of power supply …831496/fulltext01.pdf · 2015. 6....

Master's Degree Thesis ISRN: BTH-AMT-EX--2015/D01--SE

Supervisors: Lars Håkansson, BTH

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Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden

2015

Mostafa Mohammadnejad Mahdi Ghazvini

Finite Element Modeling of Dynamic Properties of Power

Supply for an Industrial Application

Finite Element Modeling of Dynamic Properties of Power

Supply for an Industrial Application


Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden 2014

Thesis submitted for completion of the Master of Science in Mechanical Engineering with emphasis on Structural Mechanics in the Department of Mechanical Engineering, Blekinge Institute of Technology, Karlskrona, Sweden.

Abstract: In this thesis, the dynamic properties of the mechanic structure of Power Supply for an Industrial Application, an Alstom company product, are considered. A finite element model of the Power Supply mechanic structure have been generated with the aid of the MSC Marc software. Based on the FE model; modal analysis have been carried out and the eigenfrequencies and eigenmodes for the FE model have been calculated in a suitable frequency range. Relevant frequency response functions for the FE model have been produced using dynamic harmonic analysis. To validate and update the FE model, experimental modal analysis have been carried out on a Power Supply. For the experimental modal analysis the MIMO method the polyreference least-squares complex exponential method have been used. Based on the updated FE model some modified Power Supply designs are suggested with improved dynamic properties in an adequate frequency range.

Keywords: Finite element model, Dynamic properties, Experimental modal analysis, Eigenfrequency, Eigenmodes, Harmonic analysis

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Acknowledgements We are thankful to Allah the Almighty for helping us in all aspects of our life.

The thesis was carried out at the department of Mechanical engineering, Blekinge Institute of Technology (BTH), Karlskrona, Sweden and ALSTOM Company, Växjö, Sweden under supervision of Prof. Lars Håkansson.

We would like to express our sincere gratitude to Professor Lars Håkansson for his constant guidance, his support and valuable discussions during this thesis.

Also we wish to acknowledge financial support from ALSTOM Company.

In addition, special thanks should be dedicated to Dr. Ansel Berghuvud for his guidance and direction. Also, thanks for his help and support in case of unavailability of required equipment for experiment.

Last but not least, the most special thanks go to our respected and kind father and mother. Words alone cannot express the thanks we owe to them for their support and blessing encouragement.


Karlskrona, Sweden

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Contents

1 Notation 5 1.1 Abbreviations 6

2 Introduction 7

3 System Description and Mission statement 10 3.1 System Description 10

3.1.1 Geometry and material properties 13

3.2 Mission Statement 11

4 Methodology 12 4.1 Finite Element Method (FEM) 12

4.1.1 Steps of the modeling process 12

4.1.1.1 Creating different parts of the power supply in . MSC Marc 12

4.1.1.2 Repairing the geometry 13

4.1.1.3 Mesh definition and discretisation of the geometry 13

4.1.1.4 Selecting the material properties of elements 16

4.1.1.5 Define contacts between parts of the power supply 16

4.1.1.6 Applying boundary conditions 17

4.1.1.7 Dynamic analysis 18

4.1.1.7.1 Eigenvalue analysis 18

4.1.1.7.2 Harmonic response analysis 19

4.2 Experimental Modal Analysis (EMA) 20

4.2.1 Method for extracting modal parameters 21

4.2.2 Experimental setup 21

4.2.2.1 Suspension conditions of the power supply 21

4.2.2.2 Measurement equipment 22

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4.2.2.3 Select proper reference points 22

4.2.2.4 Select proper response points 23

4.2.3 The process of experiment 23

5 Results and Discussion 25 5.1 Results 25

5.1.1 Finite element model results 25

5.1.1.1 Eigenvalue analysis results 25

5.1.1.2 Harmonic analysis results 35

5.1.2 Experimental modal analysis results 43

5.2 Discussion 59

6 Modification, Suggestions and Conclusion 61 6.1 Modification of FE model and Suggestions to reduce the . vibration in the power supply 61

7 References 69

Appendix A 72

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1 Notation

a System parameter

C Damping

f Frequency (Hz)

F Load

h Impulse response function

H Transfer function

j

K Stiffness matrix

M Mass matrix

s Frequency domain

t Time, time domain

ω Eigenvalues (eigenfrequencies)

φ Eigenvectors

e Element

m Number of elements

i Numerical counters

T Matrix transpose

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1.1 Abbreviations

EMA Experimental Modal Analysis

ESP Electrostatic Precipitator

FE Finite Element

FEM Finite Element Method

FRF Frequency Response Function

HFPS High Frequency Power Supply

Im Imaginary part

IRF Impulse Response Function

MIMO Multi input – Multi output

Real Real part

T/R Transformer/high voltage rectifier

3-D Three Dimensional

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2 Introduction

In many industrial components and devices, damage and fatigue can occur because of vibration. Vibration causes dysfunction, mechanical fatigue and failure, etc. of e.g. civil constructions, mechanical devices and electrical instruments. This everyday existence of vibration in a board range of fields encourages to more focus on vibration, not only to protect our products and ourselves, but also because of its advantages and useful applications [1, 2].

Discomfort, mechanical fatigue of industrial components, regulation, etc. motivate companies and factories to find solutions that e.g. decrease and control the vibration of their products. To provide products with desirable dynamic properties require knowledge of dynamic properties of a product. Such knowledge is generally obtained via FE models and experimental modal analysis, etc. Based on modal analysis, etc., the behavior of mechanical structure or object, which is known as dynamic properties, can be determined [3-5].

In this thesis, the dynamic properties of the mechanic structure of a Power Supply for an Industrial Application are considered.

The ALSTOM Company is known as a global leader in the world of power transmission, power generation and rail infrastructure. This company represents and provide unique integrated power plant solutions. ALSTOM also provides some associated services for a wide different kind of energy sources such as hydro, nuclear, gas, coal, solar and wind. A wide range of solutions for power transmission, with emphasized smart grids, are offered by ALSTOM [6]. This company have branches and offices all over the world. The ALSTOM Company branch in Växjö, Sweden, produces equipment for air pollution control that is suitable for power plants and the industry in general. The company provides advanced systems and products for flue gas cleaning and capturing of carbon dioxide [7].

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ALSTOM’s advanced equipment for air pollution control has several electrical and mechanical parts. One of electrical parts is known as the power supply. The power supply consists of a number of parts. Parts in the power supply that are of particular interest in this work are; electromagnetic coils, an electrical circuit board and the electrical wiring between the coils and the circuit board. More details of the power supply is presented in appendix A. In this thesis, the structural dynamic properties of a power supply is investigated.

The dynamic properties of a structure may be approximately determined with the aid of the finite element method (FEM) in combination with experimental modal analysis (EMA). With a sufficient finite element model of a structure its spatial dynamic properties such as natural frequencies, mode shapes, etc. may be produced [3, 8, 9].

Modal analysis is done based on both modeling and experiments. The finite element method (FEM) can be implemented in order to obtain a good model of the dynamic properties of the system. By using a FE model of the dynamic properties of a structural system the behavior of the system under different types of dynamic loads may be predicted [10, 11]. Also, experimental measurements in terms of experimental modal analysis (EMA) provide frequency response functions, eigenfrequencies, and eigenmodes and the eigenmodes relative damping. Thus, enabling to update and validate the FE model. A correctly updated FE model is a reliable model, and it may be used for a variety of static and dynamic load simulations [12-15].

This thesis report consists of seven parts. The main body of the thesis which includes chapter 3, 4, 5 and 6 are explained as follows.

Chapter 3 – System Description and Mission statement: In this chapter, the power supply and the material properties of its structure are presented. Also, a geometry model of the power supply is shown. Details of the problem statement and the goal of this thesis are explained.

Chapter 4 – Methodology: This chapter includes all the details of the processing method which is implemented in this thesis. The two main parts which concerns the Finite Element Method (FEM) and the Experimental Modal Analysis (EMA) are explained. The steps required to carry out FE modeling and Experimental Modal Analysis of the rectifier structure are presented in this chapter. Also, some brief descriptions concerning the used softwares, I-DEAS, I-MAT toolbox, MATLAB and MSC Marc, are given.

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Chapter 5 – Results and Discussion: In this part of this thesis, the results obtained with the aid of the developed FE model and the Experimental Modal Analysis are presented. Results from the FE model and the Experimental Modal Analysis are compared. The MAC matrix for the results is also provided.

Chapter 6 – Modification, Suggestions and Conclusion: As the final step of this thesis, some design modifications of the power supply are suggested with purpose of improve its structural dynamic properties are presented in this chapter. Also, the corresponding modified FE models are provided.

Figure 2.4 shows the summarized framework of this thesis.

Figure 2.4. Summarized framework of this thesis work and steps.

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3 System Description and Mission statement 3.1 System Description

The system that is modeled and analyzed in this thesis is a part of a Power Supply for an Industrial Application. This part is shown in the figure 3.1 in appendix A.

The power supply consists coils which are connected to a circuit board via solid wires. Also, the frame structure of the power supply is structurally coupled to the circuit board and the windings. In a complete power supply this part is submerged in mineral oil and contained in a stainless steel vessel.

In this thesis, the power supply structure is modeled without considering the effect of oil on its dynamic properties. After creating a proper model, the effect of oil on the vibration of the system can be considered in further projects and analysis. The weight of the rectifier without stainless steel vessel and oil is 25kg.

3.1.1 Geometry and material properties

There are several types of material used in the power supply. The presented figures, 3.2 and 3.3 in appendix A, show the geometry of power supply structure and the different materials it is made of, respectively.

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3.2 Mission Statement

The ultimate goal of this thesis is creating a validated FE model of the power supply for an industrial application. By having a validated FE model, it will be possible to calculate and model the system’s response to external vibration excitation.

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4 Methodology

In order to achieve the final objective of this thesis, the methodology includes three different parts. In the first part, a finite element model is generated using MSC Marc software. Also, the required analysis based on the model (such as eigenmodes and harmonic analysis) which are related to the dynamic properties of the system, are carried out with the aid of MSC Marc. The next part concerns an experimental modal analysis, in order to obtain estimates of the power supply structure's actual eigenmodes and frequency response functions (FRFs). In the final part, the results of the measurements are compared with the results of the FE model.

4.1 Finite Element Method (FEM)

4.1.1 The steps of the modeling process

4.1.1.1 Creating models of the different parts of the power supply structure in MSC Marc

In order to generate a finite element model of the power supply structure, a modeling and analyzing software is required. The software that is used in this thesis is the MSC Marc version 2013.1. The first step to generate the model is to create all different structural parts of the power supply in the software in terms of solids. In order to avoid changing the dynamic properties of the power supply, every part is created by all details in this thesis.

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Tetrahedral (most commonly) Hexahedral

4.1.1.2 Repairing the geometry

After creating solids in the software, the next step is “to repair” the geometry of solids, in order to make them suitable for finite element modeling. For example, the bolt threads are neglected in the modeling, and they can be removed from the geometry. Also, there is a great number of small electrical resistances attached to a circuit board of the power supply. These parts are neglected and removed, consider to their “very small” dimensions in comparison with the power supply. In this step, it is important to make sure that the geometry of the other solids is not changed, although all the free or broken curves should be removed.

4.1.1.3 Mesh definition and discretisation of the geometry

The mesh is known as the process of changing a physical problem into a discrete object or geometry. In order to discretize a physical problem and obtain a meshed geometry, some types of elements can be used. Based on the interpolation functions and the geometry shape, each element can be specified. Figure 4.2 shows the most common geometric shapes for the three – dimensional elements.

Figure 4.2. Most common 3-D geometry shapes.

On the other hand, interpolation functions for each element give a specific property to it. Two common interpolation functions of each element are shown in figures 4.3.

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Figure 4.3. Two common interpolation functions for each element.

It should be noted that interpolation functions are expressed with respect to the area coordinate system. The stiffness matrix of calculations can be integrated considered to a single integrated point at the centroid. Thus, by using a single integration point at the centroid of the face, the distributed load on the face can be integrated [16].

The geometric shape and the interpolation function of the element are tetrahedral and linear, respectively (Element type: 134). Figure 4.4 shows the shape of element type 134 consisting four nodes with linear interpolation function.

Figure 4.4. Shape of the element type 134 with four nodes.

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This type of element is known as a linear isoperimetric three-dimensional tetrahedron element. Since the interpolation function of the element type 134 is linear, therefore, the strains are constant throughout the element. Also, one point at the centroid of the element is used in order to be integrated numerically. For achieving an accurate solution using this element, it is required to define a fine mesh. In order to solve more complex problems such as nonlinear problem, the higher-order element 127 can e.g. be used [17, 18].

The convention for the ordering of the connectivity array is explained as follows. As can be seen in figure 4.5, nodes with a number of 1, 2 and 3 should be the corners of the first face, given in counterclockwise order when viewed from inside the element. Also, node 4 is located on the opposing vertex. It should be noted that normally the elements and their connectivity are constructed automatically via a preprocessor of the software such as CAD program.

Figure 4.5. Arrangement of nodes for elements connectivity.

Figure 4.6 shows the discretized geometry in MSC Marc using element type 134.

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Figure 4.6. Meshed object using element type 134.

4.1.1.4 Selecting the material properties of the elements

In this step, the material properties of elements are selected. Table 4.1 in appendix A, presents material properties of different parts of the power supply including density, Young’s modulus and poison’s ratio.

4.1.1.5 Define contacts between parts of the power supply

The power supply consists of several parts connected via joints (figure 4.7 in appendix A shows joints of power supply). Joints usually introduce non-linearities in a system. In this thesis, the finite element model is generated assuming rigid connections at the joints. By applying such assumption the following advantages may be obtained [19, 20]:

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- Making the assumption of a rigid connection at the joints will provide an “almost” linear model of the power supply structure, and the system will become simpler. Also, the differences in the results compared to the experimental results may be addressed in a more consistent and symmetric way.

4.1.1.6 Applying boundary conditions

When the model is meshed properly, the boundary conditions are applied on the model and they are selected to adequately model the actual boundary conditions of the power supply

In this thesis, the object is suspended to approximate its actual operating conditions (this will be discussed in part 4.2). Thus, the boundary conditions applied to the finite element is shown in figure 4.8.

Figure 4.8. Boundary conditions used in the FE model of the power supply.

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)3.4(02nn yKM

)1.4(tFtuKtuCtuM

)2.4(02 UKM

4.1.1.7 Dynamic analysis

When the finite element model of the power supply is a linear system, two different calculations can be processed by using MSC Marc’s dynamic analysis capability. First, calculations concerning eigenvalue analysis and secondly; harmonic response analysis.

4.1.1.7.1 Eigenvalue analysis

By defining the equation of motion for the FE model and based on Newton’s second law, the general and compact N degrees-of-freedom governing equation is:

where {u(t)}={u1(t), u2(t), …, uN(t)}T is the displacement vector, un(t), n Î{1,2, …, N}, are the displacement coordinates, [K] is the stiffness matrix, [M] is the mass matrix and [C] is the damping matrix. By looking at the free vibrations (F(t)=0) and an undamped system ([C]=0) the Fourier transform of the governing equation can be summarized as follow:

Hence, the eigenvalues ω2n and mode shapes {y}n corresponding to the

undamped linear N degrees-of-freedom is obtained via the eigenvalue problem:

where [K] is the stiffness matrix, [M] is the mass matrix, ω2n is an

eigenvalue (squared angular eigenfrequency) and {y}n is the corresponding mode shape vector [21].

In order to process the calculations and extract eigenvalues and eigenvectors of the model, either the inverse power sweep method or the Lanczos method is used in MSC Marc software.

In the inverse Power Sweep, the initial trial vector is created by MSC Marc and this initial vector is multiplied by the mass matrix and the inverse (factorized) stiffness matrix in order to obtain the new vector. This process is repeated until convergence is reached. After MSC Marc has calculated the first

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eigenvalue, by using the Gram-Schmidt orthogonalization procedure, the trial vector at each iteration is orthogonalized to previously extracted vectors. In this method, eigenvalue extraction is controlled by maximum number of iterations per mode. An eigenvalue has converged when the difference between the eigenvalues in two consecutive sweeps divided by the eigenvalue is less than the tolerance [21].

In the Lanczos method, the Lanczos algorithm converts the original eigenvalue problem into the determination of the eigenvalues of a tridiagonal matrix. It should be noted that, this method can be used either for the determination of all modes or the calculation of a small number of modes. In the Lanczos iteration method, eigenvalue extraction is controlled by the maximum number of iterations for all modes. The Lanczos iteration method has converged when the normalized difference between all eigenvalues satisfies the tolerance [21].

4.1.1.7.2 Harmonic response analysis

In a harmonic response analysis of MSC Marc, structures vibrating are analyzed around an equilibrium state which can be unstressed or statically pre-stressed. In the most of practical applications, objects or components are dynamically excited which are often harmonic and cause only small amplitude vibrations. By linearizing the problem around the equilibrium state, the vibration problem can be solved as a linear problem using complex arithmetic in MSC Marc [21].

This analytical procedure consists four steps. In the first step, MSC Marc calculates the response of the structure to a static preload (which can be nonlinear) based on the constitutive equation for the material response. In this portion of the analysis, the program ignores inertial effects.

In the second step, the complex-valued amplitudes of the superimposed response for each given frequency and amplitude of the boundary tractions or displacements, is calculated. In this part of the analysis, the program considers both material behavior and inertial effects.

As the third step, different loads with different frequencies or changes the static preload at discretion can be applied. It should be noted that, during the calculation of the complex response, all data relevant to the static response is stored.

In the final step, in order to obtain harmonic response analysis, the

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)4.4(1

*

*N

r r

NNr

r

NNrNN j

AjA

H

)5.4(1 2rrrrr j

)6.4(Trrrr QA

HARMONIC parameter should be used. By using HARMONIC history definition option, the excitation frequency can be determined. In order to define incremental data only for the harmonic excitation, it is required to enter the HARMONIC history definition option with a set of incremental data. This can also be done for applying boundary conditions, as well as loads [21-23].

4.2 Experimental Modal Analysis (EMA) The process to determine the dynamic properties of a structural system

experimentally, is known as Experimental modal Analysis (EMA). In EMA dynamic properties of the system such as eigenfrequencies, mode shapes and damping ratios can be estimated. Such results can be obtained based on experimental simultaneous measurements of suitable forces exciting a system and the vibration responses of the system [22, 24]. In experimental modal analysis the excitation force signals and the vibration response signals from the sensors attached on a system, can be recorded by data acquisition system connected to a PC with adequate software [25]. Modern modal analysis are usually based on a PC with the proper software for experimental modal analysis (such as I-DEAS). The collected data can be utilized to determine the frequency response function of the system.

Based on modal analysis theory the governing equation for modal model can be written as follow [26]:

In equation 4.4, H(ω) is frequency response function that consists sum of modes of system. Each mode can be determined by its residues [Ar] and poles λr that are shown in equation 4.5 and 4.6, respectively.

where ωr is undamped resonance frequency and ζr is the relative damping for mode r. It should be noted that poles are known as eigenvalues.

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)7.4()(2

1

' t

NN

N

rrNN

r

io

ioeAth

And governing equation of residues is:

where Qr is scaling constant for mode r and {ψ} is mode shape vector. By looking at the impulse response matrix [h(t)] for N degree of freedom and assume that poles are not complex conjugate pairs, the governing equation for multiple input (i) and multiple output (o) is [26]:

4.2.1 Method for extracting modal parameters

In this thesis, the method for extracting modal parameters from measured accelerations and forces on the power supply structure is a multi input – multi output (MIMO) method. In MIMO method, when several shakers are used the excitation energy can be more distributed. Thus, non-linearities may be handled in a more consistent way. Some advantages of MIMO methods are [24, 26]:

- All modes in different directions can be excited at the same time. - Usually, for symmetric structures, this method with several

accelerometers gives better results. - The reciprocity can be checked directly. - More data can be acquired simultaneously. Thus, the curve fitting

results will be improved.

4.2.2 Experimental setup

4.2.2.1 Suspension conditions of the power supply

The first step to take in the preparation for the experiment is to suspend the structure with “free-free” boundary condition or its operating condition. Considering the dimensions of the electrical power supply, the free-free boundary conditions may not be an alternative. The suspension condition of the object for the experimental test in this thesis is presented in figure 4.9 in

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appendix A. In this suspension, power supply is attached with bolts. Hence, the suspension of the power supply is stationary during the measurements and the boundary conditions remain constant for the entire test.

4.2.2.2 Measurement equipment

The equipment used to carry out the experimental modal analysis is presented in table 4.2.

Table 4.2. Model and number of measurement equipment.

Equipment Model Number

Accelerometer PCB 333A32 16

LDS Shaker v201 2

Impedance head Brüel & Kjær 8001 2

Data acquisition unit HP VXI E1432 front-end

1

PC with IDEAS version 6 1

4.2.2.3 Select proper reference points

Selecting proper reference positions with adequate measurement direction on a structure for a subsequent experimental modal analysis of it is crucial for the final result [26].

The reference positions should not be near to nodal lines of mode shapes, and they should be chosen in the positions with a direction that all the required modes can be excited [26]. To select appropriate reference positions for an experimental modal analysis a finite element model that is already created can be helpful. In this thesis, two positions on the power supply structure are selected based on the FE model, (Figure 4.10 in appendix A). By scrutinizing the mode shapes estimated with the aid of the FE model in chapter

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5 (Eigenvalue analysis results) it follows that these positions are excited for all modes of interest in this work.

4.2.2.4 Select proper response points

The response positions should be selected to enable a good separation of modes. The MAC matrix can be used to see the correlation between modes [26]. Figure 4.11 in appendix A shows some of the installed accelerometers on the selected response points for the experiments.

4.2.3 The process of experiment

After determining the reference and response positions and directions, the shakers are suspended with approx. free boundary conditions in the frequency range of concern, see Figure 4.12 and also figure 4.13 in appendix A.

Figure 4.12. Suspension of shakers for the experimental setup (Note: picture of the device is

not presented clearly due to confidentiality).

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Also, suitable stingers are selected in order to excite the reference positions in adequate direction. In the experiments, two impedance heads are used to measure the force and acceleration in the reference positions simultaneously. A burst random signal is selected as the excitation signal. The accelerometers are mounted on the response positions. A data acquisition system which is connected to a PC is used in order to collect the signal from the transducers. The collected data are analyzed by the I-DEAS software on the PC. With this software, it is possible to estimate FRFs, coherence functions, mode indicator function, parameters to modal model, eigenmodes, the MAC matrix, etc. immediately after doing measurements. All the calculated results can be imported to MATLAB by using I-MAT software for further analysis.

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5 Results and Discussion In this chapter, results are presented. In part 5.1.1.1, the result of the finite element model which is created using MSC Marc are provided. These results include the eigenfrequencies and eigenmodes which are obtained based on eigenvalue analysis. The steps of generating finite element model and doing eigenvalue analysis are discussed in the methodology chapter.

In part 5.1.1.2, the results of harmonic analysis of the generated model in MSC Marc are displayed in a number of different figures. As discussed in the methodology chapter, the frequency response functions between different positions on the model can be obtained by carrying harmonic analysis of the FE model.

Part 5.1.2 presents the results of experimental modal analysis followed by each frequency response function estimate, the coherence plots for the measurements are provided indicating e.g. the quality of the measurements.

In the last part of this chapter, results produced using FEM and EMA are compared with each other in tables and graphs.

5.1 Results

5.1.1 Finite element model results

5.1.1.1 Eigenvalue analysis results

The results of eigenvalue analysis on the FE model, including eigenfrequencies and eigenmodes are presented. Table 5.1 presents eighteen eigenfrequencies that are obtained in the frequency range of interest (5-200 Hz).

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Table 5.1. Obtained eigenfrequencies from the FE Model. Eigen-

frequencies (Hz)

30.2 38.4 39.6 48.6 77.5 84.5 102.4 106.7 107.7

113.8 114.5 131.2 137.2 156.5 172.2 176.9 180.8 194.7

Also, the mode shapes of power supply for each eigenfrequency are shown in figures 5.1 to 5.18.

Figure 5.1. Mode Shape 1 from the simulation of the FE model of the power supply in MSC Marc.

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28



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30



31



32



33



34



35

.


5.1.1.2 Harmonic analysis results The frequency response function for FE model based on the harmonic

analysis of it are shown in section. The produced FRFs for the finite element model with the aid of the harmonic analysis are presented in figures 5.19 to 5.34.

Figure 5.19. FRF based on Simulation of the FE model for position 1 in z-direction.

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Figure 5.20. FRF based on Simulation of the FE model for point 2 in y-direction.


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Figure 5.22. FRF based on Simulation of the FE model for point 4 in z-direction.


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39



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Figure 5.29. FRF based on Simulation of the FE model for point 11 in x-direction.

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Figure 5.31. FRF based on Simulation of the FE model for point 13 in x-direction.

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43


5.1.2 Experimental modal analysis results

The frequency response function estimates produced in the experimental modal analysis of the power supply and the corresponding coherence functions are shown in figures 5.35 to 5.50.

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b)

Figure 5.35. a) Frequency response function between force at driving points and acceleration at point 1 in z-direction. b) Multiple-Coherence function for acceleration at point 1.

a)

45

b)


a)

46

b)


a)

47

b)


a)

48

b)

Figure 5.39. a) Frequency response function between force at driving points and acceleration

at point 5 in z-direction. b) Multiple-Coherence function for acceleration at point 5.

a)

49

b)


a)

50

b)


a)

51

b)


a)

52

b)


a)

53

b)


a)

54

b)



a)

55

b)


a)

56

b)



a)

57

b)


a)

58

b)


a)

59

b)


a)

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5.2 Discussion The obtained eigenfrequencies from the FE model and experimental

modal analysis of the power supply are presented in table 5.2.

Table 5.2. Eigenfrequencies from the FE model and experimental modal analysis of the power supply.

Simulation Eigen-frequency (Hz)

Experimental Eigen-frequency (Hz)

30.2 28

38.4 35

39.6 40

48.6 45

77.5 72

84.5 87

102.4 105

106.7 108

107.7 110

113.8 113

114.5 118

131.2 130

137.2 141

156.5 155

172.2 174

176.9 179

180.8 184

194.7 197

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In order to determine the relevance of the experimental setup, the MAC matrix is calculated. As can be seen in figure 5.51, the obtained matrix is diagonal-dominant, and the off-diagonal elements have substantially lower values as compared to the off-diagonal elements

Figure 5.51. MAC matrix of mode shapes produced using EMA.

Figure 5.52 shows the MAC matrix that is obtained from the comparison of mode shapes produced using EMA and mode shapes produced for the FE model. It indicates that the results have a fairly good correlation, particularly for the mode shapes that are related to the circuit board in the power supply.

Figure 5.52. MAC matrix for comparison of experimental and simulation results.

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6 Modification, Suggestions and Conclusion

6.1 Modification of FE model and Suggestions to reduce the vibration in the power supply

It may be observed from the figures showing the mode shapes for the power supply in chapter 5.1.1.1, that the maximum displacement of most of the modes (in operating vibration range) occurs in the circuit board that also is involved in the wire failures. One important reasons is likely to be the symmetry in the structure of the power supply. Symmetry of structures may poses vibration problem when excited with dynamic forces. One of the suggestions with purpose to reduce the vibration response of the system might be to change the structure of the power supply and making it non-symmetric. Such changes can be applied on the validated FE model.

Another reason is the stiffness of the board is likely to be low compared to the other parts of the power supply. Therefore, increasing the dynamic stiffness of the plate can be considered as another suggestion. The require increase in dynamic stiffness can be determined using the validated model provided in this work.

Consider to the above suggestions, some modifications are implemented on the verified FE model. Thus, as a result the mode displacement of the circuit board is decreased (see Figure 6.1). Figures 6.1 and 6.2, shows the modification that is carried out.

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Figure 6.1. The considered circuit board of the power supply.

In figure 6.2, the stiffening of the board with the aid of plates is illustrated.

Figure 6.2. The kind of modification for desire plate of the power supply.

The plate (circuit board)

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Based on the suggested modification of the circuit board, a new FE model of the power supply is obtained. By implementing the dynamic modal analysis, the eigenfrequencies and mode shapes are determined for the modified circuit board. Figures 6.3 to 6.9 show the resulting mode shapes of the power supply in the operating frequency range (5-200 Hz).

As can be seen, there are only seven eigenfrequencies for the modified model. Furthermore, the amplitude of the plate part of the mode shapes has decreased substantially.

Figure 6.3. Mode Shape 1 for modified FE model of the power supply in MSC Marc.

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66



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In this thesis, based on the created FE model which is validated by experiments, the dynamic properties of the power supply are modeled. According to the results, it is found that there are several eigenfrequencies in the interesting range of frequency of this power supply. The estimated eigenmodes show that the maximum amplitudes happen in a thin plate (circuit board) of the power supply.

Hence, this is a likely to be explanation to the failures of the wires connected to the plate in the power supply. In order to avoid the wire failures in the power supply, some modifications in the structure of the device are presented.

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7 References 1. Tedesco, J.W., W.G. McDougal, and C.A. Ross, Structural dynamics:

theory and applications. 1999: Addison-Wesley Montlo Park, California.

2. Meirovitch, L., Fundamentals of vibrations. International Edition, McGraw-Hill. 2001.

3. Allemang, D.R.J., VIBRATIONS: ANALYTICAL AND EXPERIMENTAL MODAL ANALYSIS. 1994, Cincinnati, Ohio: University of Cincinnati. 203.

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Crankshaft. IOSR Journal of Engineering, 2012. 2(4): p. 674-684.

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25. Schwarz, B.J. and M.H. Richardson, Experimental modal analysis. CSI Reliability week, 1999. 35(1): p. 1-12.

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27 - 30. These references are presented in appendix A

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Appendix A

This part is removed because of CONFIDENTIALITY OBLIGATION and agreement with ALSTOM

Company, Växjö, Sweden.

School of Engineering, Department of Mechanical Engineering Blekinge Institute of Technology SE-371 79 Karlskrona, SWEDEN

Telephone: E-mail:

+46 455-38 50 00 [email protected]

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