Welded Frame Finite Element Modelling and Experimental ...bth.diva-portal.org/smash/get/diva2:832056/FULLTEXT01.pdfIn welded frame finite element modeling, finite element method is

Master's Degree Thesis ISRN: BTH-AMT-EX--2012/D-19--SE

Supervisors: Ansel Berghuvud, BTH

Department of Mechanical Engineering Blekinge Institute of Technology

Karlskrona, Sweden

2012

Muralikrishna Koduri

Woodrow Wilson

Welded Frame Finite Element Modelling and Experimental

Modal Analysis

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This thesis has been submitted for the completion of Masters in Mechanical Engineering with Emphasis on Structural Mechanics at the Department of Mechanical engineering, Blekinge Institute of Technology, Karlskrona, Sweden.

ABSTRACT:

In this modern developing world finite element model is one of the main tools and they are becoming more and more advanced day by day in all mechanical fields. Before manufacturing a product the demands for finite element model is more between the manufacturers. The finite element method is considered as a powerful technique which is developed for solving the numerical solution of the complex problems in structural mechanics. In welded frame finite element modeling, finite element method is used and in for comparing the results, and its corresponding behavior, co-relation with the use of experimental modal analysis.

This thesis involves both analytical as well as experimental measurements. By means of using modal theory, analytical model is created so as to resemble the experimental data. The welded frame measurements are taken from college lab and the readings are observed by use of experimental modal analysis. The result of this thesis work includes updated model of the frame which is experimentally taken into consideration and there measurements are both analytically and experimentally co related. Here the initial finite element model of the welded frame showed a bit differences when compared to experimental data. There are four beams that are being investigated and compared with the experimental data.

KEY WORDS:

FRF (Frequency response function), Finite element model, Modal analysis, Experimental investigation, Modal updating.

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ACKNOWLEDGEMENT:

This thesis has been performed at the Department of Mechanical Engineering, Blekinge Institute of Technology Karlskrona, Sweden, under the supervision of Ansel Berghuvud (PhD, Mech Engg).

We would like to thank our supervisor Mr. Ansel Berghuvud for his support, guidance, patience and encouragement throughout our entire work. We have learnt many things from him during our two years study at b-th.

We would also like to thank colleagues and friends at the department for their support throughout our studies, without which this work will not be possible.

Karlskrona 2012

Muralikrishna Koduri

Woodrow Wilson

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CONTENT:

Abstract……………………………………………………………….2

Acknowledgement…………………………………………………... 3

Content………………………………………………………….…… 4

Notation…………………………………………………...…......….. 7

ABBREVATIONS ……………………………………………....… 8

1]. INTRODUCTION…………………………………………..... 12

1.1]. Background ………………………………………………..… 12

1.2]. Purpose and aim ……………………………………….….…. 13

1.3].Approach ………………………………………….…….……. 13

1.4]. Why Welded Frame ……………………………………….… 13

1.5]. Types of Vibrations ………………………………….………. 13

1.6]. Kinds of Modes ……………………………….…………….. 14

2]. THEORETICAL BACKGROUND…………………………15

2.1]. (FRF) Frequency response function ……………………........ 15

2.2]. Impact Testing …………………………………………..……15

2.3]. Mounting of test object ……………………………….…….. 15

2.4]. Excitation ………………………………………….…...…….16

2.5]. FE Analysis ………………………………………...……….. 16

2.6]. Bill of materials used for assembly …………………………. 17

2.7]. Simplification and limitations ………………………………..17

2.8]. Boundary conditions in FE analysis …………….……………17

2.9]. Results for Finite element analysis ……………….…………..18

2.10]. Discussion ……………………………………….…………..19

2.11]. How valid is Experimental investigation……………………..19

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2.12]. How valid is FE analysis …………………………………19

2.13]. Parameters used to update FE modal …………….…..….19

3]. EXPERIMENTAL TEST ...………….………………...….21

3.1]. Description of the work …………………………….……..21

3.2]. Measurement Preparation ……………………………….…22

3.3]. Point Selection ………………………………………...….. 22

3.4]. Reference Co ordinate System …………………………….22

3.5]. Position of Excitation (Input) ………………………………23

3.6]. Boundary conditions ……………………………………….24

3.6.1]. Types of Boundary condition investigated…………..24

3.7]. 3D Modeling of welded frame Assembly …………………..25

3.8]. Data Quality Assessment ……………….….....................…26

3.9]. Coherence …………………………………………….…....26

3.10]. Reciprocity …………………………………….…….....…26

3.11]. Selection of Reference point ………………………......… 26

3.12]. Testing Real Structures …………………………………....27

3.13]. Mode Indicator function …………………………………. 27

3.14]. Multivariate mode Indicator functions ………………….…28

3.15]. Linearity ……………………………………….…………. 28

3.16].List of equipments which are used during

The experimental setup…………………………………………… 28

3.17]. Mode Shapes …………………………….…….………….29

4]. RESULTS OF EXPERIMENTAL DATA……………...... 29

4.1]. Floor ……………………………………………….…...29

4.2]. Floor pillar readings analysis diagram ………………….33

4.3]. Rubber………………………………………...........……..36

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4.4]. Rubber pillar analysis diagram ..............................................38

4.5]. Wood ………………………………………..………………40

4.6]. Wood pillar analysis diagram ………………………………..42

5]. Comparison ……………………………………………………….44

5.1]. Fixed-Fixed Conditions ……………………….…………….44

5.2]. Experimental comparison 73 hertz (floor) …………..………47

5.3]. Experimental comparison 144 hertz (floor)…………………51

5.4]. Comparison Discussion …………………………….………52

5.5]. Experimental comparison diagram for 168 hertz (floor)….….55

5.6]. Comparison Discussion ………………………………….….56

5.7]. Free-Free Boundary conditions …………………….………56

5.8]. Comparison Discussion …………………………….……….60

5.9]. Experimental comparison for 17 hertz (rubber) …………….63

5.10]. Comparison Discussion ………………………………..….64

5.11]. Experimental comparison for 84 hertz (rubber) ………..….67

5.12]. Comparison Discussion ……………………………….….68

6]. CONCLUSION ………………………………………….….... 69

7]. FUTURE WORK ………………………………………………70

8]. REFERENCES ……………………………………….……...... 71

9]. Appendix ……………………………………...…………………72

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NOTATIONS:

k ------------ Stiffness N/m

L ----------- Length

M ----------- Mass

X ------------ Displacement

f ------------- Frequency in hertz (Hz)

x (t) ------------ Input signal time domain

Y (f) -------------- Response signal Frequency Domain

y (t) -------------- Response signal Time Domain

---------------- Pole

---------------- Residue

F ---------------- Force

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ABBREVATIONS:

FRF = Frequency response function

DOF = Degrees of Freedom

FEM = Finite Element Method

FE = Finite Element

SDOF = Single Degree Of Freedom

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List of Figures: Fig 2.9.1 Meshed diagram of the entire setup …………………….18

Fig3.1.1 Block diagram for the whole setup ………………………21

Fig 3.5.1 Position and origin of the whole setup ……………….….23

Fig 3.16.1 Data acquisition unit ……………………………….…..28

Fig 4.1.1 Mode plot for frequency 49 hertz (floor)………………..30

Fig 4.1.2 Mode plot for frequency 144 hertz (floor)………………..31

Fig 4.1.3 Mode plot for frequency 168 hertz (floor)………………32

Fig4.2.1 Mode plot for shaft at 23 hertz (floor)………………….…33

Fig4.2.2 Mode plot for shaft at 73 hertz (floor)…………………….34

Fig4.2.3 Mode plot for shaft at 167 hertz (floor)……………………35

Fig4.3.1 Mode plot for beam at 17 hertz (floor)……………………..36

Fig4.3.2 Mode plot for beam at 84 hertz (floor)……………………..37

Fig4.4.1 Mode plot for shaft at 10 hertz (rubber)………….………..38

Fig4.4.2 Mode plot for shaft at 167 hertz (rubber)…………….……..39

Fig 4.5.1 Mode plot of beam at 168 hertz (wood)…………………....40

Fig 4.5.2 mode plot of beam at 183 hertz (wood)…………………….41

Fig 4.6.1 mode plot for shaft at 16 hertz (wood)……………………..42

Fig 4.6.2 mode plot for shaft at 64 hertz (wood)……………………..43

FIXED-FIXED CONDITIONS (floor):

FIG 5.1.1 Finite element model for x displacement (72.95Hz)……….44

Fig 5.1.2 Finite element model for y displacement (72.95Hz)…………45

Fig 5.1.3 Finite element model for z displacement (72.95Hz)………….46

Fig 5.2.1 Experimental comparison for FE model (73 Hz)……………. 47

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Fig 5.2.2 Finite element model for x displacement (138Hz)…………….48

Fig 5.2.2 Finite element model for y displacement (138Hz)……………49

Fig 5.2.3 Finite element model for z displacement (138Hz)…………….50

Fig 5.3.1 Experimental comparison for FE model (144 Hz)……………..51

Fig 5.4.1 Finite element model for x displacement (168.75Hz)…………..52

Fig 5.4.2 Finite element model for y displacement (168.75Hz)…………..53

Fig 5.4.3 Finite element model for z displacement (168.75Hz)………….54

Fig 5.5.1 Experimental comparison for FE model (168 Hz)………………55

Free-free (rubber) :

Fig 5.7.1 Finite element model for x displacement (11.9Hz)………………….56

Fig 5.7.2 Finite element model for y displacement (11.9Hz)…………………57

Fig 5.7.3 Finite element model for z displacement (11.9Hz)………………….58

Fig 5.7.4 Experimental comparison for FE model (10 Hz)……………………59

Fig 5.8.1 Finite element model for x displacement (17.7Hz)………………….60

Fig 5.8.2 Finite element model for y displacement (17.7Hz)………………….61

Fig 5.8.3 Finite element model for z displacement (17.7 Hz)………………….62

Fig 5.9.1 Experimental comparison for FE model (17 hz)…………………….63

Fig 5.10.1 Finite element model for x displacement (83.68 Hz)……………….. 64

Fig 5.10.2 Finite element model for y displacement (83.68 Hz)………………..65

Fig 5.10.3 Finite element model for z displacement (83.68 Hz)…………………66

Fig 5.11.1 Experimental comparison for FE model (84 Hz)……………………..67

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List of tables:

2.6.1 Bill of materials used …………………………….. 17

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1 INTRODUCTION:

In general the dynamic behavior study of the structure can be divided by two different divisions, namely (a).Experimental route (b).Theoretical route. The Finite element method is the theoretical route which is being used most widely. In the Finite element method the analysis can be done in order to characterize dynamic structures whereas by means of constructing numerical model where the structure can be simulated with the help of computer. While talking of modal analysis it has become the best method for what it’s more accurate known as the modal survey or (EMA) Experimental modal analysis. In the Experimental modal analysis the structures dynamic behavior in terms of modes of vibration can be obtained by means of testing its physical structure. Each mode’s is defined by its respective mode shape, modal damping and modal frequency.

In our work we have shown that how the analytical mode shape which we obtained from Finite element analysis (FEA) is getting co-related with the experimentally gained mode shape results.

1.1 BACKGROUND:

The dynamic analysis of frames and their designs are nowadays performed with a natural frequency analysis methodology. The frame design is meshed with the help of shell elements. Mat-lab is used to process the data of modal analysis, which we get by securing its experimental data of the system in the lab for gaining its dynamic characteristics. The welded frame design is experimented and the required data is collected through mat lab or I-DEAS. The data thus collected is processed using MATLAB to get required plots for analysis of data.

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1.2 PURPOSE AND AIM:

Here our task is to investigate the dynamic properties of a welded frame with the help of finite element modeling and checking the correlation of simulation results with experimental modal analysis results.

1.3 APPROACH:

The first problem is that how the real structures measurement can be taken into consideration and how it should be compared with our analytical finite element model. While we plan of comparing with experimental data some of the questions will be in our mind.

Is the real structure good enough?

Is this real structure linear or non-linear?

What will be error form before updating?

So such type of doubt must be analyzed first. There are some varieties of tools like FRF, coherence, reciprocity, comparison of modes which are used for this sake [1, 2, 3]. When good experimental data is collected, it’s ok to check up with data quality assessment.

1.4 Why welded frame?

Welded frame finds out its application in wide areas like bicycles, cars, metallic structures like bridges etc. A typical welded frame consists of many weld joints that contribute significantly to structures stiffness and dynamic characteristics. Modeling and characterization of dynamic properties of structures under vibrations is ongoing popular research topic.

1.7 Types of Vibration:

Generally all the vibration is a combination of both resonant and forced vibration.

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Resonant vibration:

This type of vibration occurs when one or more of resonances or natural mode of vibration of structure is excited. This vibrations somehow amplifies the responses of vibration some far beyond the deflection level, strain, stress which is caused due to the static loading.

Forced vibration:

It occurs when the motion or an alternating force is applied to a mechanical system.

1.8 Modes:

They are the inherent properties of structure. Modes or resonances can be easily determined by means of damping properties and mass stiffness.

The modes are used as an efficient and simple means for characterizing the resonant vibrations. Almost the majority of the structures can be made to resonate (which means), the structure can be made to vibrate with excessive oscillatory motion under proper conditions [3].

Kinds of modes:

Modes are of characterized as flexible bodies or rigid body modes. Generally all the bodies or structures can have six rigid body modes, three rotational modes and also 3three translational modes.

Rigid body mode: If the body somehow bounces on some soft springs its motion will be related to the rigid body mode.

The higher frequency mode shapes will be of more complex in their shape and appearance and hence not given a specific name to identify.

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2 Theoretical Backgrounds

2.1 FRF Measurements:

FRF denotes frequency response function which is one of the basic fundamental measurements which isolate the dynamic properties of a mechanical body or structure. The experimental mode parameters such as mode shape, damping, and frequency are gained only with the help of FRF measurements. Frequency response function generally defined as ratio of Fourier transform of the output response divided by the Fourier transform of input response that which causes the output.

2.2 IMPACT TESTING:

This is done with the help of roving hammer [4]. Here the output gets fixed, and the frequency response functions are being measured for the multiple inputs which corresponds to the elements measured from single row of frequency response function matrix.

IMPACT HAMMER:

It is attached with a load cell to its head in order to measure the input force.

ACCELEROMETER:

It is used to measure the acceleration response at a fixed point as well as fixed direction [4].

2.3 MOUNTING OF TEST OBJECT:

This type of mounting the test object can be done with the help of many possible ways, (i.e.) the structure can be tested by use of modal analysis in its operating conditions or without its operating condition, but somewhat in a nearby similar stage. The general condition used is the free-free condition which will be used to simulate free objects without considering any of the boundary conditions.

Here in our thesis the whole welded frame structure is placed on a rubber, as to consider it as a free-free boundary condition. Then the measurements are marked in respect to the structure and the readings are taken into consideration.

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When making a modal analysis, the ultimate purpose and aim of it was to correlate the results with the analytical test, hence as a result free-free condition is the mostly preferred one [2].

2.4 EXCITATION:

While planned of starting a modal test, the testable structure has to be given some form of vibration. In order to make the test structure vibrate, excitation force is used. One of the force transducer is used to monitor the accelerometer and the input signal by means of monitoring the response signal. The Force transducer and accelerometer generates time signal which are being used in the analysis. In welded frame design structure we are taking measurements without the use of shaker. Here we are only using accelerometer, data acquisition unit, roving hammer, and a system. The measurements are done in a simple way without the help of shaker. In this thesis the vibrations are generated in the test object by using impact test which is done with the help of hammer. The hammer can be attached to the force transducer together known as impulse hammer, or else the force transducer can be separately placed on the testing structure.

There are 2 forms of excitation (a) Impact excitation, (b) Shaker excitation.

Impact testing is done by use of roving hammer and accelerometers, and is usually performed where one or more fixed response points are chosen. The (FRF) frequency response function is measured by use of impact hammer around the structure of which we measure around all degrees of freedom which is to be measured one by one. While compared to shaker excitation, impact excitation is quite easy process [2] & [4].

Advantages of Impact excitation:

The hammer blow causes no extra loads on the structure you measure, and the measurements are so fast. The excitation points can be changed easily. The impact testing gets complicated only if the structure measured by you are non linear. In order to get good readings we need to be consistent in hitting the hammer blow over the structure.

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2.5 FE Analysis:

A three dimensional model of welded frame assembly was modeled to conduct finite element analysis and study the behavior of the system. A welded frame is selected to study the behavior of system under different boundary conditions with the help of experimental modal analysis. The frame assembly was modeled almost as real welded frame structure but with certain limitations to make the structure simple for finite element analysis purpose

2.6 Bill of materials for the assembly:

S.No Name of the part Material No: of parts

1 Welded Frame Hot rolled steel tubes of square cross-section

1

2 Flywheel Cast iron 2 3 Pillow block bearing Cast steel 8 4 Shaft Mild steel 4 5 Belt steel 1

Table no2.6.1 Bill of materials used.

2.7 Simplifications and limitations:

The welded frame assembly was modeled with some simplifications in its parts. The pillow block bearing that originally exists are considered and modeled as a single block. Since the bearing counts only for its weight it is modeled as single block so that the complications in meshing of the bearing can be avoided.

Bolts and nuts and their corresponding holes on the welded frame were totally removed and the weight is compensated with the bearing blocks. They are avoided to avoid complications in meshing.

2.8 Boundary conditions in finite element analysis:

In order to correlate the experimental modal analysis and the finite element analysis results we need to consider the boundary conditions that are

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considered in experimental modal analysis. The free-free boundary condition was implemented while doing finite element analysis. Here in this condition no loads were applied and this condition resembles the free-free condition (structure on rubber platform) of experimental modal analysis.

The fixed boundary conditions are modeled for the finite element analysis as such the bottom or base of the welded frame structure was constrained as fixed. This consideration resembles to the fixed (structure on the floor platform) condition of experimental modal analysis.

2.9 Results for finite element analysis:

Figure 2.9.1 Meshed Diagram of the entire structure

FE model of the welded frame assembly was created based on the geometric details and material properties of the real structure. The assembly was modeled with Autodesk Inventor professional 2012 and FE analysis was done in the FE environment of Autodesk Inventor. The FE models for the welded frame

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assembly were created using three dimensional triangular shells FEs to model the whole structure. The assembly consists of 327316 total nodes and 166373 total elements in the whole structure.

The modal analysis of the meshed structure was done in the modal analysis in the frequency range 0 Hz- 200 Hz. The first 20 mode shapes were acquired in the frequency range mentioned and this mode shapes were compared with the experimental modal analysis results to observe the correlation of both the results.

2.10 Discussion:

The selected method for study namely impulse excitation is for cases where the measurement quality is of not main importance , such as for example for cases of trouble shooting impact excitation is preferred[2]. Impact excitation is preferred especially in the cases like when the system is to be studied under free-free bounday condition. This method is very fast compared to shaker excitation and the hammer blow doesn’t impose extra loading on the structure.

2.11 How valid is the experimental investigation?

The experimental result data obtained can be conducted for validity of the data with various data quality assessment techniques[2]. Here in our case the data is checked for its validity with Maxwells reciprocity and the FRFs obtained by changing the input and output positions viceversa .

2.12 How valid is FEA?

The FE modeling was done almost near to actual dimensions. The FEA results thus obtained will be valid enough to correlate with the experimental results. The results obtained in FEA will have some differences when correlated with the experimental results for the reasons as stated below.

1.Weight differences in the design model and real model

2.Joints of the welded frame : the frame model was designed as a single frame but in real case the frame was constructed with tubes connected by welded joints.

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3.Belt was cosidered as steel material in the design but it is not the case with the real belt

4.Pillow block bearing is considered as a single metal block instead of a bearing

2.13 Parameters used to update FE modal:

In order to update the Finite Element (FE) analytical modal the frequency response function is used. The ultimate aim of that was to make fit of both the analytical and the experimental natural frequencies. The boundary conditions have the capacity to change the natural frequency for both the experimental as well in the analytical part. In our thesis we have tested for both fixed-fixed and also for free-free boundary conditions, then whereas the comparisons are being done. When the testing structure is placed on the rubber it simulates free-free boundary conditions, and when the real testing structure is placed on the floor it comes under fixed-fixed boundary condition. The FE modal can be updated by changing its stiffness, mass. The natural frequencies depend on related to mass and stiffness, so if there will be a change the modal results will be changing. The mass can be updated by weighting the modal size, then by changing the structures density in the welds of the modal structure. The update can also be easily done by means of making changes in the welding properties.

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3 EXPERIMENTAL TESTS:

3.1 Description of work:

Figure 3.1.1 Block diagram for the whole setup of the structure

The welded frame horizontal beams on one side were divided into 10 equal parts each and marked as 9 points as shown in the figure 3.1.1. Nine points on each horizontal beam are used to make excitations while doing the experiment. The rubber platform, wooden platform and the floor was made ready to do experiment on all the three cases. The accelerometer and impulse hammer were connected to the data acquisition unit and the data acquisition unit was

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connected to the computer with mat lab. The accelerometer is meant to collect the response data and the impulse hammer is meant to collect excitation data. Both the data from transducers i.e. accelerometer and force transducer gives the data as time signals. The data is collected with both the devices using mat lab and stored as mat files. The data collected is checked with the help of some data quality assessment methods as mentioned. The mat files are used for further process to study the behavior of the system. Thus the data was collected on all the three platforms like rubber, wood and floor. The data collected on all the three case were correlated with the FE results.

3.2 Measurement preparation:

The important boundary condition which we used to concentrate is the free-free boundary condition, and then is the fixed-fixed condition. So in order to get the free-free boundary conditions we are placing our welded frame real testing structure get placed over the rubber. Then to change boundary condition, we place our testing structure on the floor and also in wood in order to identify its deformations based on different natural frequencies and different boundary conditions.

3.3 Point Selection:

In testing our real structure (welded frame) a fixed accelerometer and roving hammer were used. We have taken measurements by placing out structure over the floor, wood, and also over rubber. The measurements were taken for all the three conditions in all the three directions like x, y, and z directions. So as a result it will be easy for our task which is to be getting correlated with our analytical FE modal.

3.4 Reference coordinate system:

The reference co-ordinate system used is Cartesian coordinate system. In 3D three perpendicular planes and three coordinate points are the signed distances to each of planes. This can be generalized to create n coordinates for any point in n dimensional [6].

The bottom left corner at which pillow block bearing exists is considered as origin.

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3.5 Position of excitation (Input) and response (output):

Figure 3.5.1 Position and origin of the whole setup

Input: The four horizontal beams on one side are divided into nine equal points to make as excitation points. i.e. beam1 is divided in to 9 equal points, beam 2 is divided into 9 equal points and so on all the beams.

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Output: beam 2, position 5 was selected as response point. Position 5 covers all the mode shapes and is selected after examining various points to include all the mode shapes.

Position of accelerometer (Output Response):

0.018 m in X-direction

0.5345 m in Y-direction

0.715 m in Z-direction

3.6 Boundary conditions:

The welded frame has four bolt and nuts at the bottom four corners of the frame. The four bolts and nuts balance the whole structure and acts as boundary conditions.

3.6.1 Types of boundary conditions investigated:

We have tried for three different platforms on which the structure balances and more or less they are considered to resemble free-free boundary conditions and fixed boundary conditions.

Case1: Rubber platform

Here whole structure stands on the rubber platform and experimental modal analysis was carried out to get required mode shapes. This rubber platform is considered as free-free boundary condition and the results obtained here were compared with the finite element modeled structure under free-free conditions.

Case2: Floor platform

Here whole structure stands on the rubber platform and experimental modal analysis was carried out to get required mode shapes. This floor platform is considered as fixed condition and the results obtained here were compared with the finite element modeled structure under fixed conditions.

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Case3: Wooden platform

Here whole structure stands on the rubber platform and experimental modal analysis was carried out to get required mode shapes. The results obtained here with wooden

Platform lies in between the results obtained in case1 and case2. As the wooden platform is not much softer as the rubber platform and not as harder as floor, we hope that the results obtained in this case will lie in between case1 and case2.

\

3.7 3D modeling of welded frame assembly:

The 3D model comprises of following parts in the assembly

1. Welded frame 2. Flywheel3. shaft4. Pillow block bearing 5. Belt

Welded frame was designed with the frame generator in Autodesk inventor 2012. The material of the frame was selected as hot rolled steel tubes of square cross-section while designing the welded frame with frame generator. Welded frame was designed according to actual dimensions with the help of frame generator. Originally welded frame is constructed by joining different beams with welded joints. But in the design of welded frame it was designed as a single frame.

Flywheel was designed approximately near to actual dimensions. The flywheel is constrained with mate and flush options with shaft and the belt so that the flywheel is restricted to move. The movement of flywheel causes some non-linear problems and since the system is considered linear, the flywheel is restricted to move.

Shaft is designed selecting the material as mild steel. The shaft is inserted into the pillow block bearings using in the insert options in assemble section in Autodesk inventor. The shaft is also restricted to move with suitable constraints.

Pillow block bearing was designed considering as a single metal block instead of designing it as a bearing to remove the complications of meshing while

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meshing the structure. The pillow block metal structure was constrained with the mate options in all the three Cartesian coordinates. The design of pillow block as a single metal block instead of a bearing doesn’t make much difference as it is considered only weight considerations.

Belt is designed with sweep command in Autodesk inventor. The belt is in contact with the both the flywheels nearly with half of their circumferences. The lateral surface of the belt is constrained to the flywheel circumference. The belt side surfaces are also constrained with the flywheel outer ring side surface with mate option so that it doesn’t move in that direction too.

3.8 DATA QUALITY ASSESSMENT:

Before acquiring data some data quality assessments should be done to check the setup, for its correctness.

The data quality assessments most predominantly used is stated below [2].

1. The driving point frequency response should be checked.2. The linearity should be checked 3. Maxwell’s reciprocity should be checked. 4. Coherence function for some arbitrary frequency responses, in all possible directions should be checked.

3.9 COHERENCE: The anti resonances of the signal to noise ratio often gets so small, which is one of the cause for dip in some of the coherence at anti resonances and should be neglected. In our welded frame structure it is not an easy task to obtain coherence because it’s a real structure and the measurements are extracted from the structure without using shaker. Only impact testing method with the help of hammer is done. So coherence will not be gained in this structure.

3.10 RECIPROCITY:

It is tested by switching input as well as output points. The test is done with the help of single axis accelerometer and hammer.

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3.11 SELECTION OF REFERENCE POINTS:

If you decide to take measurements from your structure by use of one reference point, then for sure you should be aware that all the modes of interest should be included in the reference point of which you select. (i.e.) your reference point should not be placed near by the nodal line for any mode. The reference point should not be changed until the entire measurements are being obtained from the structure. In order to choose a proper exact reference point you can use Finite Element model structure if available. If you have finite element model you can skip different testing choices. The general strategy is that you may choose a point whereas the point should maximizes the smallest resonance. When you are selecting a reference point for a three dimensional objects or structure the mode shape directions must also be taken into consideration. In order to get a clear crystal results for three dimensional structures, you can try to find a skewed reference point which makes the force gets excited in all the three directions.

3.12 TESTING REAL STRUCTURES:

Real structures are quite tricky to test. A real continuous structure has an infinite number of modes as well as infinite number of degree of freedom. From testing aspect a real continuous structure can be sampled as we wish (i.e.) choosing degree of freedom as we like. There will be no limit to the number of unique degree of freedom between which the frequency response function (FRF) is to be made. By analyzing the time and cost of structure in our mind we will be only measuring the small subset of the frequency response function which is to be measured on the structure. By obtaining the small subset of the frequency response function we can quite clearly define the mode shapes that are within certain frequencies of measurements. By means of taking more measurements of the structure, the more definitions will be given to the mode shapes.

3.13 MODE INDICATOR FUNCTION (MIF):

The phase resonance criterion is frequently checked by using the mode indicator function (MIF) in the linear structures. In parametric estimation process, that of finding poles, a crucial point is to find out the best assumption for the modal order. Hence so mode indicator function (MIF) is used. In general mode indicator function (MIF) is the sum of all the frequency response function (or) by summing the absolute value of imaginary part of frequency response function. The common idea of mode indicator function is that it

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shows peaks at frequencies, whereas the global modes where most measured degree of freedoms gets moved and less important modes will be suppressed.

3.14 MULTIVARIATE MODE INDICATOR FUNCTION:

This is one of the most sophisticated MIF. It denotes the fact that, at resonance frequencies phase relationship should be exact to 90 degrees in between force and displacement.

3.15 LINEARITY:

When the frequency response does not get changed in amplitude the testing structure can be approximated as linear [2]. The first basic assumption is that the structure is to be considered to be linear. (i.e.) the response of the real testing physical structure to any of the combination of forces which is simultaneously applied is to be the sum of individual responses to each of the forces which is to be acting alone. In general for large and complicated structures this is being considered as a very good assumption. So when the structure is linear its corresponding behavior can be easily characterized and controlled by use of excitation experiments.

3.16 List of Equipments used during the measurement setup:

Figure 3.16.1 Data Acquisition Unit

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(1). IMPULSE HAMMER

(2).DATA ACQUISTION UNIT

(3). IMPULSE HAMMER

(3). COMPUTER WITH MAT-LAB SOFTWARE

(4). WELDED FRAME REAL TESTING STRUCTURE

(5). ACCELEROMETERS.

3.17 MODE SHAPES:

They are called shapes the reason they don’t have value, but they are unique in shape. When we update the analytical model it is important to check the mode shapes of the analytical against the experimental mode shape. In our welded frame thesis the mode shapes are compared with the help of inventor software. The experimental mode shapes are obtained with the help of mat-lab software.

Here when visualizing the modes strange behaviors’, unrealistic pattern and flaws, can also be detected.

[4]. Results of Experimental data :

4.1 FLOOR:

The welded frame assembly was standing on the floor and it is considered as fixed condition. The accelerometer was placed on the second beam of the welded frame at position 4. The frame was excited at different points and the FRFs were plotted. At the frequencies 49 Hz, 144 Hz, 168 Hz we get reasonable data. The following plots were the mode shapes at the above mentioned frequencies noted in the figure 4.1.1, figure 4.1.2 and figure 4.1.3

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Figure 4.1.1 Mode plot for frequency 49 hertz (floor)

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Figure 4.1.2 Mode plot for frequency 144 hertz floor)

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Figure 4.1.3 Mode plot for frequency 168 hertz (floor)

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4.2 FLOOR PILLAR READINGS ANALYSIS DIAGRAMS:

Here the accelerometer was positioned on the middle short beam looking from the right hand side. The accelerometer was placed exactly on the center of the beam. The frame is excited at different points and FRFs were plotted. We get reasonable data at frequencies 23 Hz, 73 Hz, 167 Hz. The following are the mode shapes for the above mentioned frequencies mentioned in the figure 4.2.1, figure 4.2.2 and figure 4.2.3

Figure 4.2.1 Mode shape for shaft at 23 hertz (floor)

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4.3 RUBBER:

The welded frame assembly was standing on the rubber platform and it is considered as free-free condition. The accelerometer was placed on the second beam of the welded frame at position 4. The frame was excited at different points and the FRFs were plotted. At the frequencies 17 Hz, 84 Hz, we get reasonable data. The following plots were the mode shapes at the above mentioned frequencies are noted in the figure 4.3.1 and 4.3.2

Figure 4.3.1 Mode plot for beam at 17 hertz (rubber)

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Figure 4.3.2 Mode plot for beam at 84 hertz (rubber)

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4.4 RUBBER PILLAR READINGS ANALYSIS DIAGRAMS:

Here the accelerometer was positioned on the middle short beam looking from the right hand side. The accelerometer was placed exactly on the center of the beam. The frame is excited at different points and FRFs were plotted. We get reasonable data at frequencies 10 Hz, 167 Hz. The following are the mode shapes for the above mentioned frequencies are mentioned in the figure 4.4.1 and 4.4.2

Figure 4.4.1 Mode shape for shaft at 10 hertz (rubber)

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Figure 4.4.2 Mode shape for shaft at 167 hertz (rubber)

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4.5 WOOD:

The welded frame assembly was standing on the wooden platform and it is considered as fixed condition. The accelerometer was placed on the second beam of the welded frame at position 4. The frame was excited at different points and the FRFs were plotted. At the frequencies 168 Hz, 183 Hz we get reasonable data. The following plots were the mode shapes at the above mentioned frequencies are noted in figure number 4.5.1 and 4.5.2

Figure 4.5.1 Mode shape for beam at 168 hertz (wood)

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Figure 4.5.2 Mode shape for beam at 183 hertz (wood)

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4.6 WOOD PILLAR READINGS:

Here the accelerometer was positioned on the middle short beam looking from the right hand side. The accelerometer was placed exactly on the center of the beam. The frame is excited at different points and FRFs were plotted. We get reasonable data at frequencies 16 Hz, 64 Hz. The following are the mode shapes for the above mentioned frequencies are mentioned clearly in figure 4.6.1 and 4.6.2

Figure 4.6.1 Mode shape for shaft at 16 hertz (wood)

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Figure 4.6.2 Mode shape for shaft at 64 hertz (wood)

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5 Comparisons of Results between Experimental and Analytical Modal:

5.1 FIXED-FIXED CONDITIONS:

In this comparison part we have clearly mentioned the displacement of the finite element modal for the entire three axis, whereas the comparison with the experimental modal is made quite easy. When we take a clear look at the analytical diagram and the experimental results gained graph, the deflection of the structure and the beam deformations can be noted clearly.

Figure 5.1.1 Finite element model for 72.95 hertz(x displacement)

The above figure 5.1.1 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 72.95 Hz. The displacement of various parts in the x-direction at frequency 72.95 Hz is shown above. Here we can see clearly that the upper left corners of the frame and one of the two flywheels gets displaced. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.1.2 Finite element model for 72.95 HERTZ (Y- DISPLACEMENT):

The above figure 5.1.2 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 72.95 Hz. The displacement of various parts in the Y-direction at frequency 72.95 Hz is shown above. Here we can see the displacement of flywheels and the belt. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.1.3 Finite element model for 72.95 HERTZ (Z- DISPLACEMENT):

The above figure 5.1.3 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 72.95 Hz. The displacement of various parts in the Z-direction at frequency 72.95 Hz is shown above. Here we can see the displacement of flywheels and one of the eight pillow block bearing. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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5.2 EXPERIMENTAL COMPARISON FOR THE ABOVE FE-MODEL (73HZ)

Figure 5.2.1 Experimental comparison for the above FE-MODEL (73HZ)

The above figure 5.2.1 is the mode shapes at different parts in the welded frame assembly at frequency 73 Hz. For beam1 at position 2 and at position 3 the figure shows that it crosses the nodal lines since at these positions the mode

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Shape crosses zero. Zero accelerance at certain position says that the position is nodal position.

For beam2 it shows the nodal position occurs at position3.the nodal position at 3 means that we cannot acquire any displacements of the beam at this position.

When FE-results get compared with experimental results there are some similarities in deflections in flywheel and shaft in the Z-direction.

Figure 5.2.2 Finite element model for 138hertz (X- DISPLACEMENT)

The above figure 5.2.2 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 138 Hz. The displacement of various parts in the X-direction at frequency 138 Hz is shown above. Here we can see the displacement of the shaft beneath the upper flywheel. The displacement of various parts here is meant to compare with the experimental

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modal analysis results at corresponding frequency and study the amount of correlation.

Figure 5.2.3 Finite element model for 138hertz (Y- DISPLACEMENT)

The above figure 5.2.3 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 138 Hz. The displacement of various parts in the Y-direction at frequency 138 Hz is shown above. Here we can see the displacement of the shaft beneath the upper flywheel and its corresponding pillow block bearing. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.2.4 Finite element model for 138hertz (Z- DISPLACEMENT)

The above figure 5.2.4 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 138 Hz. The displacement of various parts in the X-direction at frequency 138 Hz is shown above. Here we can see the displacement of all the horizontal beams except the bottom horizontal beams and also we can see the displacement in the lower flywheel. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure5.3.1 Experimental comparison for the above FE model (144HZ)

The figure 5.3.1 is the representation of mode shapes at frequency 144 Hz plotted with experimental modal analysis.

Beam1: it shows that the displacements for beam1 can be acquired and it doesn’t show any nodal position.

Beam2: approximately at position 6 and 8 it crosses zero and it shows that the nodal position occurs at these positions.

Beam3: at positions 5 and 8 the beam has nodal position and we cannot acquire any displacement at these position.

Beam4: at position 8 it crosses zero and it shows the nodal position at this position.

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The above figure shows the displacements at different position of the beam and this result is compared with FE analysis results for any correlation.

5.4 Comparison discussion:

While comparing experimental results with FE- results at frequency 144 hertz there occurs some similarities of mode shapes in the horizontal beams of the structure particularly in the Z-direction. The three first horizontal beams deflect and it shows the deflections in experimental results too. But there is no deflection in the fourth beam of FE results.

Figure 5.4.1 Finite element models for 168.75 hertz (X-DISPLACEMENT)

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The figure 5.4.1 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 168 Hz. The displacement of various parts in the X-direction at frequency 168 Hz is shown above. Here we can see the displacement of the pillow block bearings beneath the upper flywheel and slight displacement of one of its beam on which this bearing seats. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

Figure 5.4.2 Finite element model for 168.75hertz (Y-DISPLACEMENT)

The figure 5.4.2 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 168 Hz. The displacement of various parts in the Y-direction at frequency 168 Hz is shown above. Here we can see the displacement of the top horizontal beam, lower flywheel and one of the third horizontal beams. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure5.4.3 Finite element model for168.75 hertz (Z-DISPLACEMENT)

The figure 5.4.3 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 168 Hz. The displacement of various parts in the Z-direction at frequency 168 Hz is shown above. Here we can see the displacement of the top horizontal beam and the two pillow block bearings one on the third horizontal beam and other on the bottom horizontal beam. Lower flywheel and one of the third horizontal beams, the displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.5.1 Experimental comparison for the above FE-MODEL (168HZ)


Beam1: it shows that position 1 and 9 are nodal positions. We can see the displacement of the beam in-between these two points.

Beam2: approximately at position 5 it crosses zero and it shows that the nodal position occurs at these positions.

Beam3: the beam from position 4 to position5 shows zero accelerance and the beam at this position is nodal and we cannot acquire any displacement at these position.

Beam4: it shows that position 1 and 9 are nodal positions. We can see the displacement of the beam in-between these two points. The above figure shows

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the displacements at different position of the beam and this result is compared with FE analysis results for any correlation.

5.6 Comparison discussion: Observing FE- results with the experimental results at 168hertz in y and z directions some similarities in mode shapes for first and third horizontal beams of the structure exists. In both Y and Z direction FE results the top beam bends and it bends similarly in the experimental results.

5.7 FREE-FREE BOUNDARY CONDITIONS:

Figure 5.7.1 Finite element model for 11.9 hertz (X- DISPLACEMENT)

The figure 5.7.1 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 11.9 Hz. The displacement of various parts in the X-direction at frequency 11.9 Hz is shown above. Here we can see the displacement only in the belt. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.7.2 Finite element model for 11.9 hertz (Y- DISPLACEMENT)

The above figure 5.7.2 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 11.9 Hz. The displacement of various parts in the Y-direction at frequency 11.9 Hz is shown above. Here we can see the displacement only in the belt. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.7.3 Finite element model for 11.9 hertz (Z- DISPLACEMENT)

The above figure 5.7.3 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 11.9 Hz. The displacement of various parts in the Z-direction at frequency 11.9 Hz is shown above. Here we can see the displacement only in the belt. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.7.4 Experimental model diagram for frequency 10 hertz (rubber)



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Beam2: we cannot see any nodal points in this beam and we can acquire displacements at all the positions.

Shaft12 and Wheel: we cannot see any nodal points in this beam and we can acquire displacements at all the positions.


5.8 Comparison discussion

Here in all the FE results we can see the deflection in belts but not in any other part. The experimental results doesn’t compares here with FE results.

Figure 5.8.1 Finite element model for 17.71 hertz (X-DISPLACEMENT)

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The figure 5.8.1 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 17.71 Hz. The displacement of various parts in the X-direction at frequency 17.71 Hz is shown above. Here we can see the displacement in the vertical beams and a slight displacement in the belt. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

Figure 5.8.2 Finite element model for 17.71 hertz (Y-DISPLACEMENT)

The figure 5.8.2 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 17.71 Hz. The displacement of various parts in the Y-direction at frequency 17.71 Hz is shown above. Here we can see the displacement in the top and bottom horizontal and also in the vertical beams at their top and bottom positions. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.8.3 Finite element model for 17.71 hertz (Z-DISPLACEMENT)

The figure 5.8.3 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 17.7 Hz. The displacement of various parts in the Z-direction at frequency 17.71 Hz is shown above. Here we can see the displacement in the top and bottom horizontal and also in the vertical beams at their top and bottom positions. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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5.9 Experimental comparison for the above diagram (17hertz):

Figure 5.9.1 Experimental comparison for the above diagram (17hertz)

The above figure 5.9.1 is the representation of mode shapes at frequency 17 Hz plotted with experimental modal analysis.


Beam2: at position 3,5,7,9 it crosses zero and it shows that the nodal position occurs at these positions.



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We found some similarities of mode shapes in the horizontal beams of the structure as well as in the vertical beams in the Z-axis direction. All the horizontal beams in the Z axis gets deflected somewhat like zigzag shape exactly as it shows for all the beams in experimental results.

Figure 5.10.1 Finite element model for 83.68 hertz (X-DISPLACEMENT)

The figure 5.10.1 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 83.68 Hz. The displacement of various parts in the X-direction at frequency 83.68 Hz is shown above. Here we can see the displacement in the upper flywheel and also some tilt in the flywheel. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.10.2 Finite element model for 83.68 hertz (Y-DISPLACEMENT)

The figure 5.10.2 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 83.68 Hz. The displacement of various parts in the Y-direction at frequency 83.68 Hz is shown above. Here we can see some tilt in the flywheel and minor displacements in the belt and also minor displacement in the pillow block bearing seating on the third horizontal beam. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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Figure 5.10.3 Finite element model for 83.68 hertz (Z-DISPLACEMENT)

The figure 5.10.3 shows the modal analysis of the welded frame assembly in Autodesk inventor professional 2012 at frequency 83.68 Hz. The displacement of various parts in the Z-direction at frequency 83.68 Hz is shown above. Here we can see displacement in the second horizontal beams. The displacement of various parts here is meant to compare with the experimental modal analysis results at corresponding frequency and study the amount of correlation.

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5.11 Experimental comparison for the above diagram (84hertz):

Figure 5.11.1 Experimental comparison for the above diagram (84hertz)

The above figure 5.11.1 is the representation of mode shapes at frequency 84 Hz plotted with experimental modal analysis.

Beam1: it shows that approximately at position4 since the accelerance crosses zero the position 4 is nodal position. We can see the displacement of the beam in-between these two points.

Beam2: at position 3 and 7 it crosses zero and it shows that the nodal position occurs at these positions.


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Beam4: it shows that approximately at position4 since the accelerance crosses zero the position 4 is nodal position. We can see the displacement of the beam in-between these two points.



We found some similarities in the horizontal beam 2 at Z-axis. We can see clearly the horizontal beam2 bends throughout the beam in FE results as it also happens in experimental results for beam2.

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6 CONCLUSIONS:

The aim of the work is to study about the dynamic behavior of the welded frame.

Mode shapes at frequencies 17 Hz and 84 Hz for free condition and 144 Hz and 168 Hz for fixed conditions were obtained with experimental modal analysis. These mode shapes were compared with FE modal analysis results for correlation.

Finite element modeling was done and it was simulated for mode shapes in the frequency range 0 Hz- 200 Hz. The mode shapes obtained in FE simulation was compared with the experimentally obtained mode shapes. At certain areas both the results gets correlated. Both the results don’t get correlated at certain areas too. The data was correlated partially at some frequencies and it doesn’t correlates for some frequencies.

In order to match both the results we found reasons as explained in section 2.5. The experimental results showed that the finite element modal should be updated in order to match the experimental results. It was found that the structure resonates at frequencies 17 and 84 Hz for free-free conditions and at 144 and 168Hz for fixed conditions. There are similarities between the experimental and FE results at these frequencies. The dissimilarities at certain positions may be due to some change in parameters that explained in section [2.5].

The results with wood as boundary condition are likely to stay in between the cases floor and rubber.

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7 Future work:

We would like to suggest looking over the following areas for future investigations.

(1). Difference of weight between FE-modal and Original structure.

(2). Difference of Stiffness between FE-modal and Original structure.

(3). the experiment can be conducted with shaker excitation for better results.

(4). the influence of weld joints on the modal properties of the structure.

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8 REFERENCES:

(1). Vibrations : Analytical and Experimental modal analysis, Dr. Randall J. Allemang, Professor, Structural Dynamics Research Laboratory Department of Mechanical, Industrial and Nuclear Engineering.

(2). Experimental Modal Analysis in Practice, Kjell Ahlin and Anders Brandt. Saven Edu Tech AB, Sweden.

(3). D.J. Ewins, Modal testing theory Practice and application, Second edition.

(4). Experimental Modal Analysis, By Brian J. Schwarz & Mark H. Richardson.

(5). Finite element and Boundary element analysis of beams and thin walled structures. Jaroslav Mackerle.

(6). www.wikipedia.org

(7). www.mathworks.com

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9. Appendix Wood

clcclear allclose all

% load beam1ex1.mat% [A1,fr]=fantrans(a,fs);% [F1,fr]=fantrans(f,fs);% H1=A1./F1;% plot(fr,imag(H1))







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% load beam2ex9.mat% [A10,fr]=fantrans(a,fs);% [F10,fr]=fantrans(f,fs);% H10=A10./F10;% plot(fr,imag(H10))%% load beam3ex1.mat% [A11,fr]=fantrans(a,fs);% [F11,fr]=fantrans(f,fs);% H11=A11./F11;% plot(fr,imag(H11))

% load beam3ex3(topbearing).mat% [A12,fr]=fantrans(a,fs);% [F12,fr]=fantrans(f,fs);% H12=A12./F12;% plot(fr,imag(H12))



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%% load beam4ex5.mat% [A17,fr]=fantrans(a,fs);% [F17,fr]=fantrans(f,fs);% H17=A17./F17;% plot(fr,imag(H17))%% load beam4ex7.mat% [A18,fr]=fantrans(a,fs);% [F18,fr]=fantrans(f,fs);% H18=A18./F18;% plot(fr,imag(H18))%% load beam4ex9(topbearing).mat% [A19,fr]=fantrans(a,fs);% [F19,fr]=fantrans(f,fs);% H19=A19./F19;% plot(fr,imag(H19))

%H=1/16*(H1+H2+H3+H4+H5+H6+H7+H8+H9+H10+H11+H12+H13+H14+H15+H16);% start=0;% stop=200;%% [Ht,ft] = frftrunc(H,fr,start,stop)% semilogy(ft,abs(Ht))% figure% plot3(ft,real(Ht),imag(Ht))

x=[1 2 3 4 5 6 7 8 9 ];b1=[-0.008898 0.5836 0.2891 0.3836 0.4057 0.3813 0.08009 0.4778 0.04069]b2=[0.006378 0 0.01068 0 0 0 -0.002408 0 0]b3=[0.01223 0 0 0 0 0 0 0 0];b4=[0 -0.1321 0 -0.467 0 -0.5288 -0.17166 0 0.1386]subplot(4,1,1);plot(x,b1)xlabel('position of beam')ylabel('Accelerance')title('mode shape for beam1 at frequency 168(WOOD)')subplot(4,1,2);plot(x,b2)xlabel('position of beam')ylabel('Accelerance')title('mode shape for beam2 at frequency 168(WOOD)')subplot(4,1,3);plot(x,b3)xlabel('position of beam')ylabel('Accelerance')title('mode shape for beam3 at frequency 168(WOOD)')subplot(4,1,4);plot(x,b4)

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xlabel('position of beam')ylabel('Accelerance')title('mode shape for beam4 at frequency 168(WOOD)')figure

% % mode shapes for frequency 183x=[1 2 3 4 5 6 7 8 9 ];b21=[0.03909 -0.06073 1.258 -0.05231 1.584 -0.06073 0.3039 -0.05533 0.2512]b22=[-0.003004 0 -0.0377 0 0 0 -0.002728 0 0];b23=[0.01964 0 0 0 0 0 0 0 0];b24=[0 0.07251 0 0.01223 0 0.1002 -1.346 0 -0.4091]subplot(4,1,1);plot(x,b21)xlabel('position of beam')ylabel('Accelerance')title('mode shape for beam1 at frequency 183(WOOD)')subplot(4,1,2);plot(x,b22)xlabel('position of beam')ylabel('Accelerance')title('mode shape for beam2 at frequency 183(WOOD)')subplot(4,1,3);plot(x,b23)xlabel('position of beam')ylabel('Accelerance')title('mode shape for beam3 at frequency 183(WOOD)')subplot(4,1,4);plot(x,b24)xlabel('position of beam')ylabel('Accelerance')title('mode shape for beam4 at frequency 183(WOOD)')

% figure% plot3(x,y2,'b')% title('Mode shape for mode2(183)');% xlabel('Length of the beam (unit)')% ylabel('Imaginary part of accelerance(Nm/s^2)')%

Rubber


% load beam1ex1.mat% [A1,fr]=fantrans(a,fs);

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% [F1,fr]=fantrans(f,fs);% H1=A1./F1;% plot(fr,imag(H1))

% load beam1ex2.mat% [A2,fr]=fantrans(a,fs);% [F2,fr]=fantrans(f,fs);% H2=A2./F2;% plot(fr,imag(H2))%% load beam1ex3.mat% [A3,fr]=fantrans(a,fs);% [F3,fr]=fantrans(f,fs);% H3=A3./F3;% plot(fr,imag(H3))%% load beam1ex4.mat% [A4,fr]=fantrans(a,fs);% [F4,fr]=fantrans(f,fs);% H4=A4./F4;% plot(fr,imag(H4))%% load beam1ex5.mat% [A5,fr]=fantrans(a,fs);% [F5,fr]=fantrans(f,fs);% H5=A5./F5;% plot(fr,imag(H5))%% load beam1ex6.mat% [A6,fr]=fantrans(a,fs);% [F6,fr]=fantrans(f,fs);% H6=A6./F6;% plot(fr,imag(H6))%% load beam1ex7.mat% [A7,fr]=fantrans(a,fs);% [F7,fr]=fantrans(f,fs);% H7=A7./F7;% plot(fr,imag(H7))%% load beam1ex8.mat% [A8,fr]=fantrans(a,fs);% [F8,fr]=fantrans(f,fs);% H8=A8./F8;% plot(fr,imag(H8))%% load beam1ex9.mat% [A9,fr]=fantrans(a,fs);% [F9,fr]=fantrans(f,fs);% H9=A9./F9;% plot(fr,imag(H9))%% load beam2ex1.mat% [A10,fr]=fantrans(a,fs);

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% [F10,fr]=fantrans(f,fs);% H10=A10./F10;% plot(fr,imag(H10))%% load beam2ex2.mat% [A11,fr]=fantrans(a,fs);% [F11,fr]=fantrans(f,fs);% H11=A11./F11;% plot(fr,imag(H11))%% load beam2ex3.mat% [A12,fr]=fantrans(a,fs);% [F12,fr]=fantrans(f,fs);% H12=A12./F12;% plot(fr,imag(H12))%% load beam2ex4.mat% [A13,fr]=fantrans(a,fs);% [F13,fr]=fantrans(f,fs);% H13=A13./F13;% plot(fr,imag(H13))%% load beam2ex6.mat% [A14,fr]=fantrans(a,fs);% [F14,fr]=fantrans(f,fs);% H14=A14./F14;% plot(fr,imag(H14))%% load beam2ex7.mat% [A15,fr]=fantrans(a,fs);% [F15,fr]=fantrans(f,fs);% H15=A15./F15;% plot(fr,imag(H15))%% load beam2ex8.mat% [A16,fr]=fantrans(a,fs);% [F16,fr]=fantrans(f,fs);% H16=A16./F16;% plot(fr,imag(H16))%% load beam2ex9.mat% [A17,fr]=fantrans(a,fs);% [F17,fr]=fantrans(f,fs);% H17=A17./F17;% plot(fr,imag(H17))% % % load beam3ex1.mat% [A18,fr]=fantrans(a,fs);% [F18,fr]=fantrans(f,fs);% H18=A18./F18;% plot(fr,imag(H18))%% load beam3ex2.mat% [A19,fr]=fantrans(a,fs);

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% [F19,fr]=fantrans(f,fs);% H19=A19./F19;% plot(fr,imag(H19))%% load beam3ex3.mat% [A20,fr]=fantrans(a,fs);% [F20,fr]=fantrans(f,fs);% H20=A20./F20;% plot(fr,imag(H20))%% load beam3ex4.mat% [A21,fr]=fantrans(a,fs);% [F21,fr]=fantrans(f,fs);% H21=A21./F21;% plot(fr,imag(H21))%% load beam3ex5.mat% [A22,fr]=fantrans(a,fs);% [F22,fr]=fantrans(f,fs);% H22=A22./F22;% plot(fr,imag(H22))%% load beam3ex6.mat% [A23,fr]=fantrans(a,fs);% [F23,fr]=fantrans(f,fs);% H23=A23./F23;% plot(fr,imag(H23))%% load beam3ex7.mat% [A24,fr]=fantrans(a,fs);% [F24,fr]=fantrans(f,fs);% H24=A24./F24;% plot(fr,imag(H24))%% load beam3ex8.mat% [A25,fr]=fantrans(a,fs);% [F25,fr]=fantrans(f,fs);% H25=A25./F25;% plot(fr,imag(H25))%load beam3ex9.mat[A26,fr]=fantrans(a,fs);[F26,fr]=fantrans(f,fs);H26=A26./F26; plot(fr,imag(H26))%% load beam4ex1.mat% [A27,fr]=fantrans(a,fs);% [F27,fr]=fantrans(f,fs);% H27=A27./F27;% plot(fr,imag(H27))%% load beam4ex2.mat% [A28,fr]=fantrans(a,fs);

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% [F28,fr]=fantrans(f,fs);% H28=A28./F28;% plot(fr,imag(H28))%% load beam4ex3.mat% [A29,fr]=fantrans(a,fs);% [F29,fr]=fantrans(f,fs);% H29=A29./F29;% plot(fr,imag(H29))%% load beam4ex4.mat% [A30,fr]=fantrans(a,fs);% [F30,fr]=fantrans(f,fs);% H30=A30./F30;% plot(fr,imag(H30))%% load beam4ex5.mat% [A31,fr]=fantrans(a,fs);% [F31,fr]=fantrans(f,fs);% H31=A31./F31;% plot(fr,imag(H31))%% load beam4ex6.mat% [A32,fr]=fantrans(a,fs);% [F32,fr]=fantrans(f,fs);% H32=A32./F32;% plot(fr,imag(H32))%% load beam4ex7.mat% [A33,fr]=fantrans(a,fs);% [F33,fr]=fantrans(f,fs);% H33=A33./F33;% plot(fr,imag(H33))%% load beam4ex8.mat% [A34,fr]=fantrans(a,fs);% [F34,fr]=fantrans(f,fs);% H34=A34./F34;% plot(fr,imag(H34))%% load beam4ex9.mat% [A35,fr]=fantrans(a,fs);% [F35,fr]=fantrans(f,fs);% H35=A35./F35;% plot(fr,imag(H35))%%%%%H=1/35*(H1+H2+H3+H4+H5+H6+H7+H8+H9+H10+H11+H12+H13+H14+H15+H16+H17+H18+H19+H20+H21+H22+H23+H24+H25+H26+H27+H28+H29+H30+H31+H32+H33+H34+H35);% % M=H./(2*pi*fr*(-1)^0.5);

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%% semilogy(fr,abs(H),'b');% xlabel('Freequency in hertz');% ylabel('accelerance in db(Nm/s^2)');% title('frf for rubber readings');%% plot3(fr,real(H),imag(H))% e1=[1 2 3 4 5 6 7 8 9];% b1=[0 -0.1322 -0.09733 -0.1304 -0.1487 0 -0.1079 -0.1305 -0.1149];% b2=[-0.1127 -0.1198 0 -0.1396 0 -0.1199 0 -0.1299 0];% b3=[-0.1283 -0.1347 -0.9706 0 0 0 0 0 0];% b4=[-0.1297 -0.1126 0 -0.1197 -0.1293 -0.1227 -0.1298 0 -0.1306];% subplot(4,1,1);% plot(e1,b1)% xlabel('position of the beam')% ylabel('Accelerance')% title('Mode plot for beam1 at frequency 17 Hz(RUBBER)')%%% subplot(4,1,2);% plot(e1,b2)% xlabel('position of the beam')% ylabel('Accelerance')% title('Mode plot for beam2 at frequency 17 Hz(RUBBER)')%%% subplot(4,1,3);% plot(e1,b3)% xlabel('position of the beam')% ylabel('Accelerance')% title('Mode plot for beam3 at frequency 17 Hz(RUBBER)')%%% subplot(4,1,4);% plot(e1,b4)% xlabel('position of the beam')% ylabel('Accelerance')% title('Mode plot for beam4 at frequency 17 Hz(RUBBER)')%% figure%%%%% e2=[1 2 3 4 5 6 7 8 9];% b21=[0.4497 0.3396 0.158 -0.06238 -0.2414 -0.4435 -0.5586 -0.6912 -0.7291];% b22=[0.6477 0.7829 0 0.82 0 0.4769 0 -0.2588 -0.5949];% b23=[0.4523 0.1653 0 0 0 0 -0.8729 -0.9079 0];% b24=[0.5087 0.3882 0.2285 0.07594 -0.1085 -0.2521 -0.4346 -0.5829 -0.6646];% subplot(4,1,1);

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% plot(e2,b21)% xlabel('position of the beam')% ylabel('Accelerance')% title('Mode plot for beam1 at frequency 84 Hz(RUBBER)')%%% subplot(4,1,2);% plot(e2,b22)% xlabel('position of the beam')% ylabel('Accelerance')% title('Mode plot for beam2 at frequency 84 Hz(RUBBER)')%%% subplot(4,1,3);% plot(e2,b23)% xlabel('position of the beam')% ylabel('Accelerance')% title('Mode plot for beam3 at frequency 84 Hz(RUBBER)')%%% subplot(4,1,4);% plot(e2,b24)% xlabel('position of the beam')% ylabel('Accelerance')% title('Mode plot for beam4 at frequency 84 Hz(RUBBER)')%%%%%%

Rubber pillar


load pillar2ex1[A1,fr]=fantrans(a,fs);[F1,fr]=fantrans(f,fs);H1=A1./F1;% plot(fr,imag(H1))


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load pillarex1[A5,fr]=fantrans(a,fs);[F5,fr]=fantrans(f,fs);H5=A5./F5;% plot(fr,imag(H5))





load pillarex6[A10,fr]=fantrans(a,fs);[F10,fr]=fantrans(f,fs);H10=A10./F10;

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% plot(fr,imag(H10))


load pillarex8[A12,fr]=fantrans(a,fs);[F12,fr]=fantrans(f,fs);H12=A12./F12;% % plot(fr,imag(H12))

load shaft1[A13,fr]=fantrans(a,fs);[F13,fr]=fantrans(f,fs);H13=A13./F13;% plot(fr,imag(H13))

load shaft2[A14,fr]=fantrans(a,fs);[F14,fr]=fantrans(f,fs);H14=A14./F14;% plot(fr,imag(H14))

load wheel[A15,fr]=fantrans(a,fs);[F15,fr]=fantrans(f,fs);H15=A15./F15;% plot(fr,imag(H15))

H=(1/15)*(H1+H2+H3+H4+H5+H6+H7+H8+H9+H10+H11+H12+H13+H14+H15);% M=H./(2*pi*j*fr);

semilogy(fr,(H),'b'); xlabel('frequency in hertz'); ylabel('accelerance in db(Nm/s^2'); title('Frf for rubber pillar');

% figure% plot(real(M),imag(M))

Floor

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clcclear allclose all%% load beam1ex1.mat% [A1,fr]=fantrans(a,fs);% [F1,fr]=fantrans(f,fs);% H1=A1./F1;% plot(fr,imag(H1))

% load beam1ex2.mat% [A2,fr]=fantrans(a,fs);% [F2,fr]=fantrans(f,fs);% H2=A2./F2;% plot(fr,imag(H2))% % % load beam1ex3.mat% [A3,fr]=fantrans(a,fs);% [F3,fr]=fantrans(f,fs);% H3=A3./F3;% plot(fr,imag(H3))%% load beam1ex4.mat% [A4,fr]=fantrans(a,fs);% [F4,fr]=fantrans(f,fs);% H4=A4./F4;% plot(fr,imag(H4))

% load beam1ex5.mat% [A5,fr]=fantrans(a,fs);% [F5,fr]=fantrans(f,fs);% H5=A5./F5;% plot(fr,imag(H5))%% load beam1ex6.mat% [A6,fr]=fantrans(a,fs);% [F6,fr]=fantrans(f,fs);% H6=A6./F6;% plot(fr,imag(H6))%% load beam1ex7.mat% [A7,fr]=fantrans(a,fs);% [F7,fr]=fantrans(f,fs);% H7=A7./F7;% plot(fr,imag(H7))% % % load beam1ex8.mat% [A8,fr]=fantrans(a,fs);% [F8,fr]=fantrans(f,fs);% H8=A8./F8;% plot(fr,imag(H8))% % % load beam1ex9.mat

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% [A9,fr]=fantrans(a,fs);% [F9,fr]=fantrans(f,fs);% H9=A9./F9;% plot(fr,imag(H9))%% load beam2ex1.mat% [A10,fr]=fantrans(a,fs);% [F10,fr]=fantrans(f,fs);% H10=A10./F10;% plot(fr,imag(H10))%% load beam2ex2.mat% [A11,fr]=fantrans(a,fs);% [F11,fr]=fantrans(f,fs);% H11=A11./F11;% plot(fr,imag(H11))%% load beam2ex3.mat% [A12,fr]=fantrans(a,fs);% [F12,fr]=fantrans(f,fs);% H12=A12./F12;% plot(fr,imag(H12))% % % load beam2ex4.mat% [A13,fr]=fantrans(a,fs);% [F13,fr]=fantrans(f,fs);% H13=A13./F13;% plot(fr,imag(H13))%% load beam2ex5.mat% [A14,fr]=fantrans(a,fs);% [F14,fr]=fantrans(f,fs);% H14=A14./F14;% plot(fr,imag(H14))%% load beam2ex6.mat% [A15,fr]=fantrans(a,fs);% [F15,fr]=fantrans(f,fs);% H15=A15./F15;% plot(fr,imag(H15))% % % load beam2ex7.mat% [A16,fr]=fantrans(a,fs);% [F16,fr]=fantrans(f,fs);% H16=A16./F16;% plot(fr,imag(H16))%% load beam2ex8.mat% [A17,fr]=fantrans(a,fs);% [F17,fr]=fantrans(f,fs);% H17=A17./F17;% plot(fr,imag(H17))%% load beam2ex9.mat

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% [A18,fr]=fantrans(a,fs);% [F18,fr]=fantrans(f,fs);% H18=A18./F18;% plot(fr,imag(H18))%% load beam3ex1.mat% [A19,fr]=fantrans(a,fs);% [F19,fr]=fantrans(f,fs);% H19=A19./F19;% plot(fr,imag(H19))% % % load beam3ex2.mat% [A20,fr]=fantrans(a,fs);% [F20,fr]=fantrans(f,fs);% H20=A20./F20;% plot(fr,imag(H20))%% load beam3ex3.mat% [A21,fr]=fantrans(a,fs);% [F21,fr]=fantrans(f,fs);% H21=A21./F21;% plot(fr,imag(H21))%% load beam3ex4.mat% [A22,fr]=fantrans(a,fs);% [F22,fr]=fantrans(f,fs);% H22=A22./F22;% plot(fr,imag(H22))%% load beam3ex5.mat% [A23,fr]=fantrans(a,fs);% [F23,fr]=fantrans(f,fs);% H23=A23./F23;% plot(fr,imag(H23))

% load beam3ex6.mat% [A24,fr]=fantrans(a,fs);% [F24,fr]=fantrans(f,fs);% H24=A24./F24;% plot(fr,imag(H24))%% load beam3ex7.mat% [A25,fr]=fantrans(a,fs);% [F25,fr]=fantrans(f,fs);% H25=A25./F25;% plot(fr,imag(H25))% % % load beam3ex8.mat% [A26,fr]=fantrans(a,fs);% [F26,fr]=fantrans(f,fs);% H26=A26./F26;% plot(fr,imag(H26))%% load beam3ex9.mat

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% [A27,fr]=fantrans(a,fs);% [F27,fr]=fantrans(f,fs);% H27=A27./F27;% plot(fr,imag(H27))% % % load beam4ex1.mat% [A28,fr]=fantrans(a,fs);% [F28,fr]=fantrans(f,fs);% H28=A28./F28;% plot(fr,imag(H28))

% load beam4ex2.mat% [A29,fr]=fantrans(a,fs);% [F29,fr]=fantrans(f,fs);% H29=A29./F29;% plot(fr,imag(H29))% % % load beam4ex3.mat% [A30,fr]=fantrans(a,fs);% [F30,fr]=fantrans(f,fs);% H30=A30./F30;% plot(fr,imag(H30))%% load beam4ex4.mat% [A31,fr]=fantrans(a,fs);% [F31,fr]=fantrans(f,fs);% H31=A31./F31;% plot(fr,imag(H31))%% load beam4ex5.mat% [A32,fr]=fantrans(a,fs);% [F32,fr]=fantrans(f,fs);% H32=A32./F32;% plot(fr,imag(H32))%% load beam4ex6.mat% [A33,fr]=fantrans(a,fs);% [F33,fr]=fantrans(f,fs);% H33=A33./F33;% plot(fr,imag(H33))%% load beam4ex7.mat% [A34,fr]=fantrans(a,fs);% [F34,fr]=fantrans(f,fs);% H34=A34./F34;% plot(fr,imag(H34))%% load beam4ex8.mat% [A35,fr]=fantrans(a,fs);% [F35,fr]=fantrans(f,fs);% H35=A35./F35;%% plot(fr,imag(H35))load beam4ex9.mat

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[A36,fr]=fantrans(a,fs);[F36,fr]=fantrans(f,fs);H36=A36./F36;plot(fr,imag(H36))%% % % % % % %H=1/35*(H1+H2+H3+H4+H5+H6+H7+H8+H9+H10+H11+H12+H13+H14+H15+H16+H17+H18+H19+H20+H21+H22+H23+H24+H25+H26+H27+H28+H29+H30+H31+H32+H33+H34+H35);% % % M=H./(2*pi*fr*(-1)^0.5);% % % semilogy(fr,abs(H),'b');% xlabel('frequency in hertz');% ylabel('accelerance in db(Nm/s^2)');% title('Frf for floor readings');% % plot3(fr,real(H),imag(H))%% % x1=[1 2 3 4 5 6 7 8 9];% % % % y1=[-0.6228 -0.7138 -0.5606 -0.4897 -0.6204 -0.5689 -0.7395 -0.693 -0.647];% % y2=[-0.5662 -0.05852 -0.6477 -1.26 -0.9533 -0.9529 -0.9167 -1.105 -0.719];% % y3=[-0.7879 -1.056 -1.301 -1.521 -0.5168 -1.125 -0.7289 -1.048 0];% % y4=[-0.5639 -0.5338 -0.4504 -0.4946 -0.5912 -0.529 -0.5578 -0.6285 -0.7403];% % subplot(4,1,1);% % plot(x1,y1)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam1 at frequency 49 Hz(FLOOR)')% % % % subplot(4,1,2);% % plot(x1,y2)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam2 at frequency 49 Hz(FLOOR)')% % % % subplot(4,1,3);% % plot(x1,y3)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam3 at frequency 49 Hz(FLOOR)')% % % % subplot(4,1,4);% % plot(x1,y4)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam4 at frequency 49 Hz(FLOOR)')% %

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% % figure% % % % % %% % x2=[1 2 3 4 5 6 7 8 9];% % % % y21=[2.032 2.63 2.06 2.346 2.251 1.602 1.242 0 -0.4898];% % y22=[1.699 1.672 1.363 1.178 0.688 0.236 -0.1488 -0.5955 -0.9154];% % y23=[0.9911 0.1808 -1.097 -1.705 0 -2.771 -2.026 0 -1.312];% % y24=[1.921 2.346 1.955 1.929 1.684 1.168 0.6633 0.22430.4962];% % subplot(4,1,1);% % plot(x2,y21)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam1 at frequency 144 Hz(FLOOR)')% % % % % % subplot(4,1,2);% % plot(x2,y22)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam2 at frequency 144 Hz(FLOOR)')% % % % subplot(4,1,3);% % plot(x2,y23)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam3 at frequency 144 Hz(FLOOR)')% % % % % % subplot(4,1,4);% % plot(x2,y24)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam4 at frequency 144 Hz(FLOOR)')% % % % figure% % % % x3=[1 2 3 4 5 6 7 8 9];% % % y=[0 0 0 0 0 0 0 0 0];% % y31=[-0.7043 -2.363 -3.774 -4.67 -5.208 -4.894 -4.224 -2.903 -0.215];% % y32=[0.4024 0.4446 0.3608 0.1788 0.06046 -0.05326 -0.1129 -0.126 -0.1157];% % y33=[0.2229 0.06267 -0.04701 0 0 -0.2448 -0.2767 -0.2959 -0.2354];% % y34=[1.387 2.785 3.925 4.745 5.158 4.775 3.3633 2.164 0.9925];% % % % subplot(4,1,1);% % plot(x3,y31)% % xlabel('position of the beam')

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% % ylabel('Accelerance')% % title('Mode plot for beam1 at frequency 168 Hz(FLOOR)')% % % % subplot(4,1,2);% % plot(x3,y32)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam2 at frequency 168 Hz(FLOOR)')% % % % subplot(4,1,3);% % plot(x3,y33)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam3 at frequency 168 Hz(FLOOR)')% % % % subplot(4,1,4);% % plot(x3,y34)% % xlabel('position of the beam')% % ylabel('Accelerance')% % title('Mode plot for beam4 at frequency 168 Hz(FLOOR)')% % % % % %

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

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Welded Frame Finite Element Modelling and Experimental ...bth.diva-portal.org/smash/get/diva2:832056/FULLTEXT01.pdfIn welded frame finite element modeling, finite element method is