Evaluation of different support conditions when measuring eigenfrequencies of wooden boards

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Evaluation of different support conditions when measuring

eigenfrequencies of wooden boards

Växjö University 2009 Thesis no: TD 042/2009 Names:Hu Min, Abdul Qadir, Irawati Arvidsson School of Technology and Design School of Technology and Design, TD

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Organisation/ Organization Författare/Author(s)

School of Technology and Design Hu Min, Abdul Qadir, Irawati Arvidsson

Dokumenttyp/Type of document Handledare/tutor· Examinator/examiner Anders Olsson, Marie Johansson Anders Olsson

Titel och undertitel/Title and subtitle

Evaluation of different support conditions when measuring eigenfrequencies of wooden boards

Nyckelord

Abstract (in English)

The aim of this thesis is to evaluate how different support conditions influence on the possibilities of capturing the vibration content. The thesis comprises of two main parts: Laboratory work and finite element simulation.

In the laboratory part measurements are performed on ten boards of Norway spruce with different support conditions (elastic foundation and simple support) and for the finite element simulations a finite element model is created. The purpose is to compare laboratory results and simulation results.

The conclusion of this master thesis is that the elastic foundation boundary condition only affect the timber eigenfrequencies, in particular the first frequency, only slightly. Free-free boundary conditions can thus be replaced by an elastic foundation during the strength grading of timber. Simple support boundary condition, however, changes the mode shapes and the corresponding frequencies of the wooden boards substantially. Consequently simple support boundary condition would not be suggested by authors to apply for replacing free-free boundary condition when executing timber grading.

Key Words

Strength grading, timber, finite element, dynamic excitation, eigenfrequency, CALFEM

Utgivningsår/Year of issue Språk/Language Antal sidor/Number of pages 2009 English

Internet/WWW

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Abstract

The aim of this thesis is to evaluate how different support conditions influence on the possibilities of capturing the vibration content. The thesis comprises of two main parts:

Laboratory work and finite element simulation.

In the laboratory part measurements are performed on ten boards of Norway spruce with different support conditions (elastic foundation and simple support) and for the finite element simulations a finite element model is created. The purpose is to compare laboratory results and simulation results.

The conclusion of this master thesis is that the elastic foundation boundary condition only affect the timber eigenfrequencies, in particular the first frequency, only slightly. Free- free boundary conditions can thus be replaced by an elastic foundation during the strength grading of timber. Simple support boundary condition, however, changes the mode shapes and the corresponding frequencies of the wooden boards substantially.

Consequently simple support boundary condition would not be suggested by authors to apply for replacing free-free boundary condition when executing timber grading.

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Acknowledgement

We would sincerely like to thank our supervisors Professor Anders Olsson and Marie Johansson who have helped us during our master thesis work and also Bertil Enquist and Kirsi Jarneö who helped us during our laboratory experiments for collecting the data at the School of Technology and Design, Växjö University.

Finally, we would like to thank our parents for always supporting and encouraging us to make our studies possible.

--- --- --- Abdul Qadir Shaikh Hu Min Irawati Arvidsson

Email: ashvm08@student.vxu.se Email: humincom@hotmail.com Email: idjcq06@student.vxu.se

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1. Introduction

1.1 Background

Wood is a structural material that is used in many fields of industry. Normally we consider wood in both physical and mechanical properties. Physical properties of wood are moisture content, shrinkage, density etc. Mechanical properties of wood are elastic, strength and vibration characteristics. Generally wood is a material with strong orthotropic and elastic properties (Anon 2009a).

People have been going through a long history on understanding and gained excessive development of knowledge on timber engineering. As a very important application of those understanding and knowledge by people, the non-damage testing method- strength grading of timber has been developed for many years.

There are several ways to execute strength grading of timber. Dynamic excitation in the longitudinal direction of a board is a very commonly used method. Transducers like accelerometer, microphone, laser etc are used for detecting and capturing the vibrations. The frequency content of the vibrations is calculated using FFT (Fast Fourier Transformation) analyzer. Once the natural frequency has been found, it is easy to analyze and calculate the average stiffness of the board in longitudinal direction. The longitudinal stiffness correlates to bending strength of the board. Based on these procedures we may carry out a useful, though not perfect prediction of the bending strength of the board and correspondingly put the strength grading of timber into effect.

In recent years there are many researchers working on strength grading of timber based on dynamic excitation method. Previous research has shown that applying the dynamic excitation of the bending modes may give rise to a better prediction of the boards’ bending strength, correspondingly may assign the timber into a certain class of strength more reliable. Since it is still a new approach for timber grading, not used in industry, it remains many configurations and limitations to explore and master.

Based on what is mentioned above, our project will focus on the possibilities and limitations of using different support conditions on boards when executing the dynamic excitation test.

1.2 Expected Results

This project is mainly based on laboratory work. Measurements on ten boards using different support conditions are carried out. In addition to this some finite element calculations for complementing the results of the laboratory work are performed. The expected results after our research on this project is knowledge regarding the effects on vibration contents, i.e.

eigenfrequencies, depending on the choice of support conditions.

1.3 Limitations

In our thesis we will only consider softwood (Norway spruce) as the test material. Only 10 wooden boards of the dimension 45×145×3600mm will be tested in the laboratory. There are different methods used for the strength grading of wood but we will only focus on using the dynamic excitation test to measure the natural frequency of the wood.

We will consider two types of modes of vibration, longitudinal or axial modes and edge wise bending mode. Flat wise bending modes and twisting modes are not included in our research.

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1.4 Description of Material

The material for this research is Norway spruce or in Latin words called Picea abies.

Norway spruce is one of the most popular trees in Scandinavia. It is comparatively fast growing in this region and the wood is often used for construction.

The dimensions of wooden boards are 45×145×3600mm, and for this research we will test 10 boards of the same dimensions. These boards will be stored in a climate chamber with a temperature of 20 degrees Celsius and a relative humidity of 65%.

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2. Literature Review

2.1 Properties of Wood

2.1.1 Physical Properties

The purpose here is to describe the basis for strength grading of timber. To develop the accuracy of strength grading of timber one have to understand how the strength and stiffness properties of timber correlates with the clear wood properties (Carl-Johan Johansson, 2003).

In this case, we focus on introducing the properties of European species used for construction Directional properties

Wood is a material with strong orthotropic properties. The orientation of the wood fibres is determined of the way tree grows. Properties vary along three mutually perpendicular axes - longitudinal, radial and tangential (Figure 3.1). The longitudinal direction is parallel to the grain direction; the tangential direction is perpendicular to the longitudinal direction and tangent to the annual rings; the radial axis is perpendicular to the other two directions and normal to the annual rings. (Jerrold, 1994)

Figure 2.1 The three main directions of wood, longitudinal, tangential and radial direction (Jerrold, 1994)

Moisture Content

The weight of water in wood is the moisture content which is given as a percentage over the dry wood weight as

%

×100

= −

dryweight

dryweight ight

moisturewe MC

(2.1) The moisture content of wood generally correlates with the environment. It absorbs moisture from a humid environment and release moisture in a dry environment. In another words, the

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Density

Density of materials is defined as the mass per unit volume. In the case of wood, the density depends on the moisture content. So the density of wood must be given with respect to a particular moisture condition (Jerrold, 1994).

2.1.2 Mechanical Properties

Mechanical properties represent the material response to the external loads. The material stiffness may be described as the relation between stresses and strains. Wood is usually assumed to behave as an elastic material for most cases of engineering applications. When the load is removed, part of deformation from external loads would not be disappeared immediately; however, the remained deformation may be recovered in a period of time.

Strength properties give the ultimate resistance of a material to the external loads. As mentioned above, wood is an orthotropic material and the mechanical properties differ a lot in the three different directions. Strength values in longitudinal direction are far higher than those in the other two directions (Jerrold, 1994). Even for same species of wood, the strength may have a large variation among different specimens which is very different from other types of engineering materials, like for example steel.

2.2 Timber Grading

Strength grading of timber have been done by Visual Strength Grading and Machine Strength Grading. Dynamic excitation methods belong to the last category. In our work we elaborate on the dynamic excitation method which is the most commonly used method nowadays.

2.2.1 Visual Strength Grading

The traditional method used for determining the strength grade is visual strength grading. This method is used in Europe & North-America and is also widely used in Australia for making the structural products (Anon 2009b).

In visual strength grading each and every piece of timber is checked by an experienced person (with naked eyes), so that the effect of defects on strength and stiffness can be accessed properly.

It has been proved, from previous research, that defects such as knots and slope of grains reduce the strength of the timber. Relations between the strength of timber and information obtained from visual inspection have been found which gives a basis for rules for visual grading. For hardwoods and softwoods, the grading standards have thus already been defined so when the visual inspection is carried out, the position of the knots and cracks are being compared with the grading standards.

Most of the structural hardwood and some of the softwoods thicker than 45mm are assessed by means of visual strength grading.

2.2.2 Machine Strength Grading

In machine strength grading all pieces of wooden boards pass through a machine which measure one or several factors non-destructively. Strength and stiffness are predicted on the basis of these measurements.

One grading principle is based on measuring the flatwise bending stiffness of the wood. The flatwise bending stiffness/MOE is relatively easy to measure. The timber is passed through the machine between the load and roller supports, see Figure 2.2. Knowing the force and the deflection of the board the bending stiffness/MOE can be calculated.

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Figure 2.2 Bending type machine (Anon 2009b)

The second common principle of machine strength grading is measuring the density by means of x-ray or gamma rays (see the Figure2.3). It can determine the density distribution both across and along the timber. The accuracy of the measurements is related to the numbers of sensors (Johansson, CS.2003).

Figure 2.3 Measurement of density by means of radiant technique

Exploring the MOE (Modulus of Elasticity) by dynamic excitation method is also one of the widely used principles of machine strength grading. In recent years, this method has lead to more practical applications because of much development of dynamic excitation technique and use of Fast Fourier Transform (FFT).

Combining the measurements of several characteristics gives better prediction accuracy. The company EuroGrecomat, for example, has produced a type of grading machine using flatwise bending for measuring the load-deflection relation, i.e. the stiffness, and X-rays for determination of density (Johansson, CS.2003).

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3. Methodology and Theory

3.1 Description of Methodology

The work comprises measurements on ten boards in the laboratory using different support conditions. The further work comprises finite element calculations, using the software MATLAB and CALFEM, on natural frequencies of boards as functions of stiffness.

The beam specimen is suspended via different supports to simulate different boundary conditions. The forced vibration test is carried out by a hammer impact test. The hammer is equipped with a piezoelectric force transducer, and the resulting vibration is measured by a small piezoelectric accelerometer which is fitted to the specimen with the use of a layer of wax. The registration and subsequent analysis is made by means of a two-channel FFT analyzer.

3.2 Euler-Bernoulli Beam Theory

Euler-Bernoulli beam theory neglects the influence of shear strain. Consequently it under- predicts deflections and over-predicts natural frequencies. For long and slender beams (beam length to thickness ratios of the order 20 or more) these effects are negligible. However these effects can be significant for thick beams. Euler-Bernoulli beam theory tells that the free flexural vibration of a prismatic element is governed by the following differential equation:

0

2 2 4

4

+

^∂_∂

=

∂

t

x

A

EI

^ν

ρ

^ν (3.1) Where,

E - Modulus of elasticity I - Moment of inertia

ν- Displacement in the direction transverse to the direction of the beam x - The coordinate along the beam

ρ - Mass density A- Cross section area t - Time

The eigenfrequencies for the free-free condition are:

mL4

F_B₋_n =γ_n EI

(3.2) From (3.2) we can get the Modulus of elasticity (MOE) as:

I mL E F

n n B n B

2 4 2

⎟⎟

⎠

⎞

⎜⎜

⎝

=⎛ ⁻

− γ π (3.3)

Where

m- Mass distribution along the beam [kg/m];

n- Mode number;

2

2 1⎥⎦⎤

⎢⎣⎡ +

n = n

γ (3.4)

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3.3 Timoshenko Beam Theory

The Timoshenko beam theory takes shear deformations into account, making it suitable for describing the behaviour of short beams.

For timber beams with a length vs. depth ratio of 20 or more difference in terms of estimated stiffness is negligible when calculating the first natural frequency but for higher modes or for lower length vs. depth ratios the effects are substantial. The finite element calculations carried out in this work and presented in Chapter 5 utilize Timoshenko beam theory.

The differential equation for Timoshenko beam theory is:

GA q EI M dx

d dx d dx

d ν ν_b ν_s α_s

−

= +

= ² ₂ ² ₂

2 2

(3.5)

Where

α_s - shear shape factor (1/1.2 = 0.833 for rectangular prismatic beams) ν – The total deflection of the neutral line of the beam

ν_b – Deflection due to bending ν_s– Deflection due to shear M - Bending moment G - Shear modulus 3.4 Dynamic Excitation

Dynamic Excitation is one of the most commonly used methods for strength grading of timber.

For research purposes a wooden beam may be hanged in two soft rubber supports. On one side of the beam an accelerometer is fixed with the help of wax and on the other side the beam is dynamically excited by means of the hammer. The striking of the hammer causes the vibrations to be produced in the beam and in turn be detected by the accelerometer. The frequency content of the vibration is calculated by the Fast Fourier Transformation (FFT) analyzer. Once the natural frequencies of the board have been determined it is easy to evaluate the mean stiffness of those boards. The measured stiffness relates with the strength of the boards and hence a particular grade can be assigned to the boards (see Figure 3.2).

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4. Laboratory Work and Results

In the laboratory work we are going to measure eigenfrequencies of ten boards using dynamic excitation method. We will apply different support conditions on boards to simulate different boundary conditions in order to investigate the influence of different boundary conditions on measuring eigenfrequencies of wooden boards. The boundary conditions we are going to implement are free-free, elastic foundation and simple support which are easily to be built up in the laboratory. This approach should give useful information regarding support conditions that could be developed further towards application in an industrial process.

4.1 Introduction of Measuring Test

The following equipments are used in the tests - FFT analyzer with power cord and network cable; accelerometer with cable (white); accelerometer with clip; mounting wax; impulse hammer with cable (black) and extension cord (power supply) etc.

Setting up the equipments follows these procedures - The impulse hammer (Dytran 5800B2) and accelerometer (Kistler 8772A50T) connects to channel one and two on the FFT-analyzer, use wax to stick an accelerometer on the board; the wax works same as glue; the accelerometer can be attached to different positions on beam in order to detect the vibrations in longitudinal, bending and flat-wise directions. After connecting the cables with channel one and two of FFT analyzer, and then connect the computer network with FFT analyzer; switch on the FFT analyzer, first a short peep and after a while a long one will be heard and then software can be started.

4.1.1 Method of Testing One

A beam with size 45×145×3600mm is hanged in two soft ribbons in order to resemble free- free boundary condition, see Figure 4.1. To conduct the forced vibration test an accelerometer is fastened in axial (x-direction) at one end of the board. Then the board is hit with an impulse hammer at the opposite end of the board. The striking of the hammer should be hard enough but not to hard in order to obtain a smooth frequency curve. The striking of the hammer gives different frequency modes of that particular board and than the values of those different frequency peaks are noted from the program. The process is performed both for the axial direction, i.e. for axial modes of vibration, and for edge-wise bending modes.

Figure 4.1 Method of testing one

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4.1.2 Method of Testing Two

In this method we use brushes which are shown in Figure 4.3 to simulate the elastic foundation (different number of brushes represents different stiffness of elastic foundation) and use the same equipment (Impulse hammer, FFT-analyzer, accelerometer) and measurement techniques like in previous test method. The differences is of only boundary conditions, here the board lies on a number of brushes. Initially four brushes at equal distance from each other have been used to conduct the test and then the number of the brushes has been increased to ten to conduct the same test for both in axial and longitudinal direction. The results from dynamic excitation and vibration analysis measurement by using these elastic supports will then be compared with method one.

Figure 4.2 is method of testing two with ten brushes put together and Figure 4.4 Stiffness tests, in this test we use.

Figure 4.2 Method of testing two with ten brushes

Figure 4.3 PVC (Polyvinyl chloride) brush

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(a) Digital displacement meter (b) Dynamometer Figure 4.4 Stiffness measuring

To measure the stiffness kx and ky of the elastic foundation, we put the beam on ten brushes and use digital displacement meter to measure the displacement of the beam when a certain force is applied, see Figure 4.4 (a). We use a dynamometer at the other side of the beam to measure the applied force, see Figure 4.4(b). The brushes are not perfectly elastic so the displacement cannot be recovered to zero after removal of the force which gives rise to the uncertainty of the measuring results. In other words this result is not accurate because of difficulty in measurements. The calculated stiffness values of the elastic foundation based on the test results are approximately kx≈4900 N/m², and ky ≈2043 N/m².

4.1.3 Method of Testing Three

The only difference between this test and the previously described is again the boundary condition. Now two steel rollers of diameter 50mm each are used to conduct the dynamic excitation test. The two rollers were set at 120 and 200 cm apart from each other. The boards were placed on the two roller supports as shown in the Figure 4.5. The position of the boards either vertically or horizontally also affects the results obtained from this test.

Figure 4.5 Method of testing three

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The position of the boards horizontally and vertically is shown in the Figure 4.6.

Horizontal board Vertical board Figure 4.6 Position of the boards

The roller supports which are used in the testing are shown in Figure 4.7.

Figure 4.7 Roller supports

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4.2 Experimental Results

The results in terms of eigenfrequencies using the different boundary conditions in the laboratory tests are shown in Table 4.1 and 4.2. Table 4.3 gives the comparison among the free-free, elastic foundation, and simple support boundary conditions for axial and edge-wise bending modes.

Precision of all the values in Table 4.1 and Table 4.2 is within ± 1.2% result in the accuracy of instruments.

Table 4.1 Axial frequency modes of all ten boards at different boundary conditions.

Axial mode

Frequency

Mode Board # Free-Free 4 B* 10 B 1200** mm

1 716.8 718.4 717.6 727.0

2 695.3 698.8 697.3 685.2

3 706.6 709.4 709.0 689.5

4 705.1 707.4 707.8 689.5

5 746.5 747.3 749.2 740.2

6 669.5 670.3 671.5 653.9

7 693.8 696.1 698.4 688.3

8 743.0 744.1 745.7 736.3

9 729.3 732.4 732.0 728.5

1^st (f1)

10 715.2 717.2 717.2 700.0

1 1443 1452 1450 1457

2 1377 1385 1384 1393

3 1413 1419 1420 1433

4 1388 1394 1393 1410

5 1486 1489 1492 1504

6 1321 1322 1325 1333

7 1401 1405 1409 1434

8 1480 1479 1482 1499

9 1446 1453 1454 1466

2^nd (f2)

10 1427 1431 1430 1450

1 2158 2171 2166 2164

2 2029 2038 2036 2046

3 2108 2118 2118 2125

4 2098 2106 2105 2115

5 2217 2220 2226 2243

6 2000 2002 2006 2011

7 2099 2105 2109 2096

8 2215 2216 2222 2239

9 2148 2160 2161 2250

3^rd (f3)

10 2125 2133 2132 2139

* Four brushes (elastic supports) at equal distance.

**1200mm is the distance between two roller supports.

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For these boards and modes, peak values are not available in the FRF graphs.

Axial mode

Frequency Mode

Board # Free- Free 4 B 10 B 1200 mm

1 2931 2956 2941 2948

2 2732 2740 2742 2736

3 2809 2825 2822 2843

4 2768 2781 2781 2798

5 3001 3007 3014 3021

6 2640 2638 2643 2655

7 2832 2837 2848 2854

8 2952 2955 2964 2979

9 2953 2896 2890 2977

4^th (f4)

10 2942 2955 2914 2871

1 3395 3462 3846 3480

2 3466 3516 3520 3457

3 3464 3490 3496 3469

4 3513 3543 3542 3417

5 3748 3736 3759 3713

6 3392 3393

7 3542 3559 3570 3560

8 3769 3757 3773 3784

9 3671 3686 3688 3641

5^th (f5)

10 3595 3580 3581 3614

1 4248 4114

2 4071 4012 4075

3 4247 4295

4 4232 4261 4260 4305

5 4482 4490 4500 4519

6 4620

7 4177 4219 4201 4253

8 4463 4466 4476 4486

9 4342 4390 4368 4389

6^th (f6)

10 4397 4437 4414 4393

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For example the Figure 4.8 is showing the frequency modes in axial direction for board No.2, using ten brushes as boundary condition. It can be seen from the graph that there is no distinct sixth peak. Also some more graphs of the boards during dynamic excitation tests in axial direction are shown in the appendix where the higher frequency modes can not be read easily.

Figure 4.8 FRF graph in axial direction for board No.2

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Table 4.2 Bending modes of all ten boards at different boundary conditions.

Bending Mode

Frequency Mode

Board

#

Free

Free 4 B 10 B

2 Rollers at 1200 mm apart (vertical board)

2 Rollers at 1200 mm apart (Horizantal board)

1 60.00 61.25 60.63 58.75 57.50 59.38 60.00

2 58.13 58.13 58.13 58.75 54.38 58.13 58.13

3 56.88 56.25 57.50 58.75 54.38 56.88 56.88

4 61.88 61.88 61.88 62.50 57.50 61.88 61.88

5 61.88 63.13 61.88 59.38 58.75 61.88 61.88

6 59.38 61.88 60.00 60.00 56.25 60.00 59.38

7 55.63 55.63 56.25 57.50 53.75 56.25 56.88

8 60.00 62.50 60.00 60.63 57.50 60.00 60.00

9 58.75 57.50 58.75 59.38 57.50 58.75 58.75

1^st (f1)

10 59.38 56.88 58.75 60.00 56.88 59.38 59.38

1 157.5 157.5 158.8 158,8 167.5 143.8

2 155.6 155.0 156.3 140.0 131.9 132.5/ 176,3 141.9

3 151.9 151.9 152.5 189.4 123.1 /176,3 131.9

4 158.1 157.5 158.1 173.1 168.1 144.4

5 161.3 161.9 161.9 155.6 136.3 129.4 /156,3 148.1

6 153.1 153.1 153.8 173.1 127.5 126.9 138.1

7 152.5 152.5 152.5 126.3 140.6

8 156.3 156.9 157.5 161.3 135.6 128.8 / 170,0 144.4

9 151.3 152.5 152.5 163.8 132.5 145.6 135.0

2^nd (f2)

10 155.0 155.6 155.6 176.9 126.3 /167,5 143.8

1 293.1 293.1 294.4 301.9 295.6 306.3 296.9

2 276.9 276.3 277.5 286.8 276.9 316.9 306.3

3 278.1 279.4 279.4 276.3 279.4 314.4 303.1

4 291.3 291.3 291.3 295.0 295.6 320.0 303.1

5 290.0 290.0 291.9 294.4 293.8 282.5 306.3

6 280.6 280.0 281.3 285.0 281.9 327.5 286.3

7 280.6 280.6 280.6 274.4 282.5 275.6 /316,9 273.8/311,9

8 284.4 285.0 286.3 285.0 288.8 252.5 295.6

9 279.4 280.0 280.6 278.1 285.0 266.3 /307,5 293.1

3^rd (f3)

10 279.4 279.4 280.6 276.9 281.9 318.1 272.5

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Bending Mode

Frequency

Mode Board

#

Free

Free 4 B 10 B

1 448.8 448.1 450.6 450.0 454.4 428.8 456.9

2 422.5 422.5 424.4 428.8 432.5 427.5 430.0

3 430.0 432.5 431.9 441.3 437.5 431.9 434.4

4 437.5 438.8 440.0 443.1 443.8 438.1 448.1

5 435.0 440.0 440.6 440.6 446.9 437.5 446.3

6 428.8 431.9 436.3 435.6 435.0 437.5 440.6

7 431.3 431.3 433.1 433.1 437.5 456.3 439.4

8 436.9 436.9 437.5 433.1 440.6 438.1 441.3

9 431.9 431.9 433.8 431.3 441.9 433.8 436.9

4^th (f4)

10 434.4 435.0 435.6 417.5 442.5 435.0 439.4

1 606.3 606.9 608.8 630.0 620.6 608.1 608.8

2 585.0 577.5 586.9 614.4 610.6 589.4 578.1

3 616.0 579.4 616.3 633.1 608.1 623.8 578.1/623,1

4 614.4 615.6 616.3 643.1 615.6 613.1 613.8

5 599.4 600.6 602.5 605.6 628.8 613.1 601.9

6 629.4 627.5 630.6 664.4 596.9 636.3 627.5

7 605.0 600.6 601.9 644.4 616.9 601.9 599.4

8 601.9 601.9 603.8 646.3 593.8 602.5 605.0

9 594.4 593.8 595.6 629.4 627.5 598.8 596.3

5^th (f5)

10 581.3 578.8 581.3 616.9 596.3 587.5 576.9

1 777.5 778.1 781.3 782.5 780.6 780.6 781.9

2 772.5 772.5 775.6 775.6 775.6 780.6 775.0

3 786.9 791.9 790.6 795.0 792.5 790.6 792.5

4 786.9 787.0 786.0 787.5 788.1 795.6 792.5

5 768.1 769.4 770.6 771.9 773.1 776.9 770.6

6 790.6 793.8 796.3 799.4 799.4 796.9 793.8

7 777.5 776.9 780.6 771.9 783.1 785.0 786.3

8 780.0 781.3 784.4 784.4 786.9 787.5 788.8

9 772.5 774.4 776.9 775.6 779.0 782.5 779.4

6^th (f6)

10 758.1 758.8 760.0 758.8 762.5 764.4 761.9

1 966.3 965.6 969.8 979.4 975.0 980.6 976.9

2 926.9 926.3 932.5 932.5 934.4 941.3 936.9

3 966.9 974.4 973.1 981.6 977.5 986.9 977.5

4 960.6 961.9 965.0 978.1 968.1 976.3 971.9

5 938.1 940.6 952.5 951.3 970.6 958.8

6 951.3 950.6 954.4 961.9 960.6 969.4 962.5

7 959.4 963.1 974.4 971.3 976.9 978.8

8 953.8 958.1 965.6 971.3 982.5 976.3

9 950.6 950.6 955.0 960.0 964.4 968.8 961.3

7th (f7)

10 917.5 919.4 921.9 927.5 922.5 935.0 935.0

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Precision of all the percentage values in Table 4.3 is also within ± 1.2% result in the accuracy of instruments.

Table 4.3 The comparison among the free-free, elastic foundation, and simple support boundary conditions for axial and edgewise bending modes.

1st Mode Axial Mode Bending Mode

Board # 4 B 10 B 1200 mm

(VB) 4 B* 10 B* ^{1200 mm}_(VB)** ^{1200 mm}_(HB)*** ^{2000 mm}_(VB) ^{2000 mm}_(HB)

1 0,2 % 0,1 % 1,4 % 2,1 % 1,0 % – 2,1 % – 4,2 % – 1,0 % 0.0 % 2 0,5 % 0,3 % – 1,4 % 0.0 % 0.0 % 1,1 % – 6,4 % 0.0 % 0.0 % 3 0,4 % 0,3 % – 2,4 % – 1,1 % 1,1 % 3,3 % – 4,4 % 0.0 % 0.0 % 4 0,3 % 0,4 % – 2,2 % 0.0 % 0.0 % 1,0 % – 7,1 % 0.0 % 0.0 % 5 0,1 % 0,4 % – 0,8 % 2,0 % 0.0 % – 4,0 % – 5,1 % 0.0 % 0.0 % 6 0,1 % 0,3 % – 2,3 % 4,2 % 1,0 % 1,0 % – 5,3 % 1,0 % 0.0 % 7 0,3 % 0,7 % – 0,8 % 0.0 % 1,1 % 3,4 % – 3,4 % 1,1 % 2,2 % 8 0,2 % 0,4 % – 0,9 % 4,2 % 0.0 % 1,0 % – 4,2 % 0.0 % 0.0 % 9 0,4 % 0,4 % – 0,1 % – 2,1 % 0.0 % 1,1 % – 2,1 % 0.0 % 0.0 % f1

10 0,3 % 0,3 % – 2,1 % – 4,2 % – 1,1 % 1,0 % – 4,2 % 0.0 % 0.0 %

* 4 B: Elastic foundation (four brushes); 10 B: Elastic foundation (ten brushes).

** VB: (Vertical Board) is like the board stands on the rollers as shown in Figure 4.6.

*** HB: (Horizontal Board) is like board lying on the rollers as shown in Figure 4.6.

1200mm or 2000mm is the distance between the two roller supports on which the board is lying.

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2nd Mode Axial Mode Bending Mode

Board # 4 B 10 B ^{1200 mm}_(VB) 4 B 10 B ^{1200 mm}_(VB) ^{1200 mm}_(HB) 2000 mm (VB)

2000 mm (HB)

1 0.6 % 0.5 % 1.0 % 0.0 % 0.8% 0.8 % 6.4 % – 8.7 % 2 0.6 % 0.5 % 1.2 % – 0.4 % 0.4 % ‐10.0% ‐15.2% ‐14.9% / 13.3% – 8.8 % 3 0.4 % 0.5 % 1.4 % 0.0 % 0.4% 24.7% ‐19.0% / 16.1% ‐13.2 % 4 0.4 % 0.4 % 1.6 % – 0.4 % 0.0 % 9.5% 6.3% – 8.7 % 5 0.2 % 0.4 % 1.2 % 0.4% 0.4% – 3.5 % ‐15.5% ‐19.8% / ‐3.1% – 8.2 % 6 0.1 % 0.3 % 1.0 % 0.0 % 0.4 % 13.1% ‐16.7% ‐17.1% – 9.8 %

7 0.3 % 0.6 % 2.4 % 0.0 % 0.0 % ‐17.2% – 7.8 %

8 – 0.1 % 0.1 % 1.3 % 0.4% 0.8% 3.2% ‐13.2% ‐17.6% / 8.8% – 7.6 % 9 0.5 % 0.6 % 1.4 % 0.8% 0.8% 8.3% ‐12.4% – 3.8 % ‐10.8%

f2

10 0.3 % 0.2 % 1.6 % 0.4% 0.4% 14.1% ‐18.5% / 8.1% – 7.2 %

3rd Mode Axial Mode Bending Mode

Board # 4 B 10 B ^{1200 mm}_(VB) 4 B 10 B ^{1200 mm}_(VB) ^{1200 mm}_(HB) ^{2000 mm}_(VB) ^{2000 mm}_(HB)

1 0.6% 0.4% 0.3% 0.0 % 0.4% 3.0% 0.8% 4.5% 1.3%

2 0.4% 0.3% 0.8% – 0.2 % 0.2% 3.6% 0.0 % 14,4% 10,6%

3 0.5% 0.5% 0.8% 0.5 % 0.5% – 0.6% 0.5% 13,1% 9.0%

4 0.4% 0.3% 0.8% 0.0 % 0 1.3% 1.5% 9.8% 4.1%

5 0.1% 0.4% 1.2% 0.0 % 0.7% 1.5% 1.3% ‐2.6% 5.6%

6 0.1% 0.3% 0.6% – 0.2 % 0.2% 1.6% 0.5% 16,7% 2.0%

7 0.3% 0.5% – 0.1 % 0.0 % 0.0 % – 2.2% 0.7% – 1.8 % / 12,9%

– 2.4%/

11,2%

8 0.1% 0.3% 1.1% 0.2% 0.7% 0.2% 1.6% ‐11,2% 3.9%

9 0.6% 0.6% 4.8% 0.2% 0.4% – 0.5% 2.0% – 4.7% /

10,1% 4.9%

f3

10 0.4% 0.3% 0.7% 0.0 % 0.4% – 0.9% 0.9% 13,8% – 2.5%

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4th Mode Axial Mode Bending Mode

1 0.8% 0.3% 0.6% – 0.2% 0.4% 0.3% 1.2% – 4.5% 1.8%

2 0.3% 0.4% 0.2% 0.0% 0.4% 1.5% 2.4% 1.2% 1.8%

3 0.6% 0.5% 1.2% 0.6% 0.4% 2.6% 1.7% 0.4% 1.0%

4 0.5% 0.5% 1.1% 0.3% 0.6% 1.3% 1.4% 0.1% 2.4%

5 0.2% 0.4% 0.7% 1.2% 1.3% 1.3% 2.7% 0.6% 2.6%

6 – 0.1% 0.1% 0.6% 0.7% 1.8% 1.6% 1.4% 2.0% 2.8%

7 0.2% 0.6% 0.8% 0.0 % 0.4% 0.4% 1.4% 5.8% 1.9%

8 0.1% 0.4% 0.9% 0.0 % 0.1% – 0.9% 0.8% 0.3% 1.0%

9 – 1.9% – 2.1% 0.8% 0.0 % 0.4% – 0.1% 2.3% 0.4% 1.2%

f4

10 0.4% – 1.0% – 2.4% 0.1% 0.3% – 3.9% 1.9% 0.1% 1.2%

1 1.8% 13,3% 2.5% 0.1% 0.4% 3.9% 2.4% 0.3% 0.4%

2 1.4% 1.6% – 0.3% – 1.3% 0.3 % 5.0% 4.4% 0.8% – 1.2%

3 0.8% 0.9% 0.1% – 5.9% 0.1% 2.8% – 1.3% 1.3% – 6.2%

/ 1.2%

4 0.8% 0.8% – 2.7% 0.2% 0.3% 4.7% 0.2% – 0.2% – 0.1%

5 – 0.3% 0.3% – 0.9% 0.2% 0.5% 1.0% 4.9% 2.3 % 0.4%

6 0.0% – 0.3% 0.2% 5.6% – 5.2% 1.1% – 0.3%

7 0.5% 0.8% 0.5% – 0.7% – 0.5% 6.5% 2.0% – 0.5% – 0.9%

8 – 0.3% 0.1% 0.4% 0.0 % 0.3% 7.4% – 1.4% 0.1% 0.5%

9 0.4% 0.5% – 0.8% – 0.1% 0.2% 5.9% 5.6% 0.7% 0.3%

f5

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1 – 3.2% 0.1% 0.5% 0.6% 0.4% 0.4% 0.6%

2 – 1.4% 0.1% 0.0 % 0.4 % 0.4% 0.4% 1.1% 0.3%

3 1.1% 0.6% 0.5% 1.0% 0.7% 0.5% 0.7%

4 0.7% 0.7% 1.7% 0.0% – 0.1% 0.1% 0.2% 1.1% 0.7%

5 0.2% 0.4% 0.8% 0.2% 0.3% 0.5% 0.6% 1.2% 0.3%

6 0.4% 0.7% 1.1% 1.1% 0.8% 0.4%

7 1.0% 0.6% 1.8% – 0.1% 0.4% – 0.7% 0.7% 1.0% 1.1%

8 0.1% 0.3% 0.5% 0.2% 0.6% 0.6% 0.9% 1.0% 1.1%

9 1.1% 0.6% 1.1% 0.2% 0.6% 0.4% 0.8% 1.3% 0.9%

f6

10 0.9% 0.4% – 0.1% 0.1% 0.2% 0.1% 0.6% 0.8% 0.5%

1 – 0.1% 0.4% 1.4% 0.9% 1.5% 1.1%

2 – 0.1% 0.6% 0.6% 0.8% 1.6% 1.1%

3 0.8% 0.6% 1.5% 1.1% 2.1% 1.1%

4 0.1% 0.5% 1.8% 0.8% 1.6% 1.2%

5 0.3% 1.5% 1.4% 3.5% 2.2%

6 – 0.1% 0.3% 1.1% 1.0% 1.9% 1.2%

7 0.4% 1.6% 1.2% 1.8% 2.0%

8 0.4% 1.2% 1.8% 3.0% 2.4%

9 0.0 % 0.5% 1.0% 1.4% 1.9% 1.1%

f7

10 0.2% 0.5% 1.1% 0.5% 1.9% 1.9%

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Green box indicates that there is no value found for corresponding frequency modes while testing the boards in axial and bending direction on four elastic brushes and that is why they can not be compared.

Same is the case with the pink boxes but this time the boards are being tested on the ten elastic supports in both axial and longitudinal direction.

This time the boards were being tested on the two rollers supports 1200mm apart from each other but again some higher modes for some boards were still unable to record in axial direction. Even in the longitudinal direction it was difficult to read the 2nd frequency mode for most of the boards.

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5. Finite Element Simulation and Results

5.1 Introduction of CALFEM

In this thesis work, we used CALFEM to execute the finite element simulation. CALFEM (Computer Aided Learning of Finite Element Method) is a popular computer program which has been developed since the late 1970s at the Division of Structural Mechanics, Lund University. CALFEM is a MATLAB toolbox for finite element applications which is commonly used to learn the finite element method. CALFEM can be used for different kinds of structural mechanics problems and field problems including certain dynamic and nonlinear analysis. Nowadays, many co-workers both former and present are still engaged in improvement and strengthening the functionalities of CALFEM in different stages.

In this project work, the most important calculation we are going to implement is dynamic analysis (the eigenvalue analysis).

5.2 Two-Dimensional Model Simulation in CALFEM

We are going to calculate a 2-dimensional beam structure with different boundary conditions such as “Free-Free”, “the Elastic foundation” and “two simple supports” etc. The details of model simulation with those different boundary conditions are described in the following chapters.

5.2.1 Free-free Boundary Condition Consider the two dimensional beam shown below.

Figure 5.1 Two dimensional beam model

The length of the beam is 3000mm. The dimensions of the cross-section are 145×45mm and the material parameters given in Table 5.1 are applied to the beam.

Table 5.1 Material data

The beam is initially divided into (n-1) elements. The numbering of the elements and the numbering of degrees of freedom are marked in Figure 5.1.

In the first step, we define the finite element model in CALFEM.

ITEM VALUE

Young’s modulus(N/ m²) 1.2×10¹⁰ Cross-section area(m²) 6.525×10^-3 Moment of inertia(m⁴) 1.143×10^-5 Shear modulus(N/ m²) 7.0×10⁸

Density(kg/ m³) 450

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Defining the finite element model comprises of following procedures:

¾ Input of the material data according to Table 5.1;

¾ Define the topology matrix as:

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

. . . . . . .

12 11 10 9 8 7 3

9 8 7 6 5 4 2

6 5 4 3 2 1 1

¾ Define the coordinates of elements, give the x,y coordinates to nodes

¾ Define the degree of freedom which gives relative freedoms to corresponding nodes

¾ Generate element matrices [K^e], [M^e], assemble in global matrices [K], [M] by using the CALFEM functions beam2t and beam2d.

After defining the model in CALFEM, we print the finite element mesh, see Figure 5.2.

Figure5.2 Beam model created in CALFEM

Now we execute the eigenvalue analysis. It is accomplished by the following set of commands.

[W, X]=eigen (K, M);

Freq=sqrt (W)/ (2*pi);

The results of these commands are the eigenvlues stored in “W” and eigen vectors stored in

“X”. The corresponding eigenfrequencies in Hz are calculated and stored in “Freq”.

We also plot the first eight eigen mode, described by the eigenvectors stored in “X”, see Figure 5.3. Calculated eigenfrequencies in Hz, using 15 elements, are:

Freq=[58,401 154,26 285,14 439,87 610,44 718,53 791,45 979,18 1170,4 1361,2…]^T,

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1 2 3

4 5 6

7 8

Figure 5.3 The first eight eigen modes

Evaluation of different support conditions when measuring eigenfrequencies of wooden boards