Natural Frequencies of beams

Identification of Elastic Properties of Materials by Experimental Resonance Frequencies and using an Updating Methodology

Marco Dourado1, a, Jos Meireles1,b

1Mechanical Engineering Department, Azurm Campus, 4800-058 Guimares, Portugal

[email protected], [email protected]

ABSTRACT

The common method being used to determine elastic properties of materials, namely Yuongs modulus and Poisson ratio, is the tensile testing method. However, there are other methods for determining these properties, as for example the experimental vibrations testing. Such tests have the advantage of being applied to materials or components that cant be destroyed. The knowledge of resonance frequencies, allows, via analytical methods, to determine this elastic properties. However, there are components for which is difficult to establish a mathematical relationship between resonance frequencies and its physical properties. In this paper we propose a different methodology to determine Yuongs modulus and Poisson ratio values in anisotropic materials. For this purpose we use a finite element model updating methodology to estimate Youngs modulus and Poisson ratio of two different materials type, based on reference resonance frequencies. It is shown that Yuongs modulus and Poisson ratio values are obtained with high reliability. To validate the method efficiency, the Yuongs modulus and Poisson ratio values are also obtained by tensile testing and determined by analytical via using the theory of natural frequencies in beams.

Keywords: Elastic Properties, Finite Element Model Updating Methodology, Resonance Frequencies, Modal Analysis

1 Introduction

The knowledge of the elastic properties of materials, before their application, it is essential to ensure the intended mechanical behaviour of these materials under conditions of use. These properties are usually determined by tensile testing method. However, such tests has some disadvantages, namely: are destructives; in anisotropic materials must be measured more than one direction; expensive and time consuming to prepare samples; strain gauges become unusable after the test [1]. Non-destructive tests, as vibration testing, allow overcomes these disadvantages, because can be performed directly on the sample or structural components without destroying them. Some experimental works, for measurement of elastic properties based on kind of tests, were done by Caracciolo et al. [2, 3, 4, 5]. The authors present experimental methods for determining the Poisson ratio and the dynamic Youngs modulus in beams subject to external excitation at different temperatures in a broad frequency range. Indeed, resonance frequencies can be related with the elastic properties of materials by means of mathematical equations, through assumptions of Euler-Bernoulli beam [6], or some theories, as for example the high-order plate

element theory [7], Pickett theory [8], Rayleigh principle [9], and the torsional vibration for a beam of non-circular cross section theory [10]. However these equations can only be applied to simple geometries, such as beams or plates. For structures or components with complex geometry these equations can not be applied. In this paper we propose the application of a Finite Element Model updating methodology to overcome this limitation. The updating process is based on iterative indirect methods. By successive iterations, the elastic properties of material are estimated based on experimental resonance frequencies. Therefore, this automatized process is independent of any direct calculation, allowing be applied to any complex structure or component, since the dynamic

response of the system it is known.

An interesting work was developed by Zhou and Farquhar [11]. The authors developed a process to determine the mechanical properties of a living wheat stem. The mechanical properties were estimated by obtaining the analytical updated stiffness matrix of the structure. Should also be referred the work of Rahmani et al. [12]. The authors use the Regularized Model Updating method in alternative to the Finite Element Model Updating method, for accurate identification of mechanical properties of composite structures. A brief description of the methodology carried out in this study, is following presented. Samples with rectangular shape, of aluminium and steel material, are submitted to experimental modal analyses, in order to known its dynamic response (reference response) natural frequencies and mode shapes. Numerical models, representative of the rectangular samples, are modelled in finite element ANSYS code. An updating process is used to update the Youngs modulus and Poisson ratio of the numerical models. The updated Youngs modulus and Poisson ratio values are obtained when an objective function, explained in section 5, is minimized. It is means that the physical (reference samples with rectangular shape) and numerical models are correlated. In this study, is taken into account the anisotropy of the materials. Therefore, Youngs modulus in the parallel (Eyy) and perpendicular (Exx) direction to the forming process of the material are updated. Similarly Poisson ratio xy and yx are also updated. Note that:

xy is the Poisson ratio, the ratio between the strain obtained in the parallel direction to the forming process (y) and perpendicular direction to the forming process (x), when applying a stress in x direction.

yx is the Poisson ratio, the ratio between the strain obtained in the perpendicular direction to the forming process (x) and parallel direction to the forming process (y), when applying a stress in y direction.

Tensile tests are performed in order to get Youngs modulus and Poisson ratio values of the two materials, in the parallel and perpendicular direction to the forming process. Theory of natural frequencies in beams and plates is explained in section 2 and applied in order to determine Eyy and Exx values based on reference resonance frequencies of the samples and its

physical properties. The analytical values are compared with the values obtained by updating process to confirm the reliability of the proposed methodology.

2 Theory of natural frequencies in beams

In this section we present the theories used in this work to calculate natural frequencies in beams and plates.

2.1 - Euler-Bernoulli beam

The dynamic response of a structure depends on its physical properties as, elastic properties, geometry and material density. The expression for calculating the natural frequency of bending modes of continuous systems based on assumptions of Euler-Bernoulli beam is given by [6],

32 lmIEK

f n

=

pi (1)

where, f is the natural frequency, Kn is a factor that depends of boundary conditions, E is the Youngs modulus related with parallel direction to the forming process (Eyy), I is the second area moment of inertia, m is the mass, and l is the length. The first five Kn values for a free-free beam conditions are given in table 1.

Table 1 First five Kn values for a free-free beam conditions.

Mode Kn value 1 22.3733 2 61.6728 3 120.9034 4 199.8594 5 298.5555

Transforming Equation (1), the Youngs modulus Eyy, can be calculated by,

I

lm

K

fE

n

yy

322

=

pi (2)

By other hand, the natural frequency of torsion modes are directly related with shear modulus G, which allows calculate the Youngs modulus in the perpendicular direction to the forming process Exx. This relationship is given by some theories as, high-order plate element theory, Pickett theory, Rayleigh principle, and torsional vibration for a beam of non-circular cross section theory, following, presented.

2.2 High-order plate element theory

The high-order plate element is an element of 4 nodes with four degrees of freedom (DOF) by each one: one lateral displacement w, two rotations x and y, and twist xy. The element of 16 DOF is shown in figure 1, adapted from [7].

Figure 1 High-order plate element with 16 DOF.

The stiffness coefficients kij of the element stiffness matrix Ke can be expressed in general form by [7]

( ) 65432

212

3

112

bl

b

l

l

b

bl

hEk ij

++

+

= (3)

where, l is the lenght, h is the thickness, b is the width, and constants 1, 2,..., 6 are given in table 2 for displacement i = 1 and node j = 1 [8]. Displacement i = 1 corresponds to the lateral displacement w in node j = 1. For torsion mode shape, the displacment w in node 1 can be also considered as bending in x-axis direction. Bending behaviour in x-axis direction can be relationed with Youngs modulus Exx.

Table 2 i Constants.

j i 1 2 3 4 5 6

1 1 156/35 156/35 72/25 0 0 0

Poisson ratio is given by,

yxxy = (4)

in that xy and yx values, are the values average obtained from tensile testing. By other hand, the stiffness constant kij can be expressed by,

mk ij =2

(5)

or

( ) mfk ij = 22pi

(6)

Replacing Equation (5) in Equation (3), the Youngs modulus Exx can be calculated by,

( )

( ) 65432

21

3

2

112

2

pi

bab

a

a

b

ba

h

mfE

yxxy

yxxy

xx

++

+

= (7)

2.3 Pickett theory

The relationship between shear modulus G and natural frequency of torsion mode is given in general form by Pickett equation [8],

mfBG = 2 (8)

where, B is given by,

Kan

IlB

p

= 2

4 (9)

where, Ip is the polar moment of inertia, n is the order of mode, a is the cross section and K is the shape factor for same cross section.

Knowing that,

xxyyxx

yyxx

xyEEE

EEG

++

=

2 (10)

and, substituting Equation (4) and Equation (8) in Equation (10), we can write Exx equation as,

( )mfIl

EKan

EE

p

yy

yxxy

yy

xx

+

=

2

2

421

(11)

2.4 Rayleigh principle

By Hearmon equations [13] we can make the relationship between the elastic constant D4 for orthotropic materials and shear modulus Gxy [9] by,

34xyG

D = (12)

However, Rayleigh principle give us the following relationship,

2

222

4274.0

h

blfD

(13)

Replacing Equation (13) and Equation (10) in Equation (12), we can write Exx equation as,

( ) 2222

274.0321

blf

Eh

EE

yy

yxxy

yy

xx

+

=

(14)

where, is the material density.

2.5 Beam of non-circular cross section theory

The natural frequency of torsion mode can be also calculated considering beam of non-circular cross section theory. By [10] we know that, waves velocity, or also called by phase velocity is given by,

t

Tk

c

= (15)

where, kt for a free-free beam boundary conditions is,

l

nkT

pi= , with n = 1 (16)

So, Equation (15) can write as follows,

pi

=

n

lcT (17)

Being f= pi 2 , equation (17) is now,

pipi

=

n

lfcT

2 (18)

By other hand the torsional phase velocity is write as,

p

TI

Gc

=

(19)

where, is the torsional constant of the cross section.

Replacing Equation (18) in Equation (19), the shear modulus comes,

pi

pi22

=n

lfI

Gp

(20)

Replacing Equation (20) in Equation (10), we express Exx equation as,

( ) ( )( )22

221

lfI

En

EE

p

yy

yxxy

yy

xx

+

=

pipi

(21)

3 Experimental procedure

In this section it is explained the samples for experimental tests, and the experimental procedures are described. Two test types were performed: tensile testing and experimental modal analysis. Tensile testing are performed to determine the Youngs modulus and Poisson ratio values through parallel and perpendicular direction to the forming process of the material, to take into account the material anisotropy. Experimental modal analysis is carried out to identify the dynamic response of the rectangular plate samples. The samples for tensile testing and experimental modal analysis were obtained from aluminium and steel sheets by laser cutting process. In table 3 are presented the geometrical properties of experimental modal analysis samples.

Table 3 Geometrical properties of experimental modal analysis samples. Property Symbol Unit Material Value

Thickness t

m

Aluminium 1.97x10-3 Steel 3.91x10-3

Length l Aluminium 297x10-3 Steel Width b Aluminium 47.7x10-3 Steel

Cross section area a m2 Aluminium 9.397x10-4

Steel 1.865x10-4

Second area moment of inertia I

m4

Aluminium 3.039x10-11 Steel 2.376x10-11

Polar moment of inertia Ip Aluminium 1.785x10-8

Steel 3.560x10-8

Torsional constant of the cross section Aluminium 1.184x10-10

Steel 9.014x10-10

Shape factor K Aluminium 1.216x10-10

Steel 9.503x10-10

3.1 Tensile Test

The tensile testing sample, with standard dimensions, is shown in figure 2a. For each material type are performed tensile tests on six samples. Three samples in the parallel direction to the forming process and three samples in the perpendicular direction to the forming process. The tests were performed at room temperature, about 20 C, on a servo-hydraulic tensile testing machine.

In tensile testing the sample is subjected to an increasing tensile stress, suffering a progressive deformation. At the same time, force and displacement values are registered by the equipment software. Strain values are read using strain gauges applied directly in the sample, as shown in figure 2b, and registered by a data acquisition system.

Figure 2 (a) Sample for tensile testing and (b) sample with strain gauge submitted to tensile test.

3.2 Experimental Modal Analysis

The experimental modal analysis samples have rectangular shape and dimensions as shown in figure 3a. For each material type are performed experimental tests on three samples. The tests were performed at room temperature, about 20 C, using a frequency spectrum analyzer equipment. The samples are tested in free-free boundary conditions, suspending them in two points by a nylon yarn of sufficient length (350 mm) so as not to cause interference in the test, as shown in figure 3b. The tests are performed using an impact hammer to input the impact force in point P1, and the response measured with laser Doppler interferometer in eight points, P1 to P8, as shown in figure 3c.

Figure 3 (a) Sample schematic representation; (b) sample subject to experimental modal analysis; (c) location of the measured points.

The selected eight points are the minimum to represent the first eight mode shapes of the sample. The data is collected in the time domain (amplitude vs. time) and processed in the LMS modal analysis software to convert to the Frequency Response Function (FRF) domain.

4 Numerical models

Numerical models to update are built using the commercial finite element ANSYS code, with same geometrical properties (presented in table 3) of the experimental samples. The initial elastic properties and material density are presented in table 4. The initial Youngs modulus and Poisson ratio values are based on normal values for the respective materials. The density values are based on mass and dimensions of the samples. The rectangular plates are modeled with shell (shell 63) elements.

Table 4 Elastic properties and density of the numerical models to update. Material Property Symbol Units Model 1 Model 2 Model 3

Aluminium

Youngs modulus Exx Pa 70.0x109 70.0x109 70.0x109 Youngs modulus Eyy Pa 70.0x109 70.0x109 70.0x109

Poisson Ratio xy - 0.31 0.31 0.31 yx - 0.31 0.31 0.31

Density kg/m3 2712 2702 2709

Steel

Youngs modulus Exx Pa 200.0x109 200.0x109 200.0x109 Youngs modulus Eyy Pa 200.0x109 200.0x109 200.0x109


Density kg/m3 7826 7812 7772

Table 5 presents the parameters to update with their initial values, and lower and upper bounds.

Table 5 Parameters to update.

Material Property Variable Units Initial Value Lower bound

Upper bound

Aluminium Youngs modulus Exx Pa 70.0x109 60.0x109 80.0x109 Youngs modulus Eyy Pa 70.0x109 60.0x109 80.0x109


Steel Youngs modulus Exx Pa 200.0x109 160.0x109 240.0x109 Youngs modulus Eyy Pa 200.0x109 160.0x109 240.0x109


The aim is to find the optimal value of the referred parameters. These values are found when resonance frequencies and mode shapes of numerical and experimental model are correlated.

5 Updating process

The finite element model updating methodology use an optimization technique explained in [14, 15, 167] to find the Youngs modulus and Poisson ratio value. The optimization problem consists in minimization of an objective function defined by a sum of three specific functions as described below,

( ) ( ) ( ) ( )xxxx ffff UC ++= (22)

The Cf function represents the quantification of the difference between numerical and

reference correlated mode pairs. CN is the number of correlated mode pairs values, of the

diagonal MAC matrix, to sum. Cf is given by,

( ) ( )( )

=

=

=C

C

N

i ii

N

i ii

C

MAC

MACf

10

1

x

xx (23)

where,

( )( )( )( )( )( )NumjTNumjRefiTRefi

Numj

TRefi

ijMAC

2

= (24)

where, Refi is the thi reference mode shape and Numj is the thj numerical mode shape [17].

The Uf function represents the quantification of the difference between numerical and reference uncorrelated mode pairs. UN is the number of uncorrelated mode pairs values, outside of the diagonal MAC matrix, to sum. Uf is given by,

( )( )( )

=

=

=

=

=

U U

U U

N

j

N

ji ij

N

j

N

ji ij

U

U

MAC

MAC

Nf

111

0

1111

x

x

x (25)

The f function represents the quantification of the difference between numerical and reference

frequencies. N is the number of eigenvalues corresponding to the correlated mode pairs.

f is given by,

( )( )( )( )( )

=

=

=

=

=

N

ji ji

N

ji ji

f

11

20

11

2

x

x

x (26)

where,

pi 2Ref

i = (27)

is the reference frequency and,

pi 2Num

j = (28)

is the numerical frequency. Ref is the reference eigenvalue and Num is the numerical eigenvalue. Quadratic term is used to accelerate the convergence of Equation (26) and to obtain only positive differences between the frequencies of the two models. The denominator is used to obtain the normalized difference. x is the vector with the updating Youngs modulus and

Poisson ratio parameters used in the numerical model updating. Numerical mode shapes NUM

and numerical eigenvalues NUM are function of these updating parameters, and can be expressed as, ( ) ( )pNUMNUM xxxxf ,...,,,, 321= (29)

where, p is the number of updating parameters. 0x is the vector with the initial updating Youngs modulus and Poisson ratio parameters. Updating parameters x are subject to lower and upper bounds inequality constraints defined as,

UBLB xxx (30)

The updated Youngs modulus and Poisson ratio value are obtained when objective function f is minimized. It is means that the modes are correlated. However, the minimal objective function value is different for all updated models, and therefore cant be considered as direct

reference to evaluate the reliability of the updated values. The minimal objective function value only indicates that the Youngs modulus and Poisson ratio optimal value was found. Then, the reliability evaluation is made by the average difference defined as,

( )100Difference Average 1

1

=

=

=

i

N

ji

finalji

N

(31)

where, finalj is the numerical final frequency obtained after updating. Multiply by 100 to obtain the average percentage difference.

The updating routine is built in MATLAB, using some tools from your Toolbox. The routine is prepared to interact with the finite element ANSYS program. The main steps of the updating process are the following:

1. Starts the ANSYS program with a given numerical model input file to update, with

updating parameters assigned in 0x ; 2. Reads the output file of the ANSYS program and processes it in order to build the

objective function f and constraints, defined as UBLB xxx , used for the optimization process;

3. Stops the calculation process if an optimal value on the updating process has been achieved, or goes to the next step on the updating process;

4. Obtains the new parameters x defined by the optimization algorithm, through MATLAB;

5. Modifies the finite element model input file with the new parameters x ; 6. Starts a new analysis by going to Step 1 with the new input file.

The interaction algorithm flowchart between updating process in MATLAB and ANSYS is presented in figure 4.

Figure 4 Interaction flowchart between Matlab and Ansys.

6 Results and discussion

In this section the results and respective discussion are presented.

6.1 - Tensile tests results

In this section are presented the tensile tests results. Figure 5 and 6 present, respectively, the stress-strain graphs (-) for aluminium material samples in the parallel direction and perpendicular direction to the forming process. Figure 7 and 8 present, respectively, the stress-strain graphs (-) for steel material samples in the parallel direction and perpendicular direction to the forming process.

Figure 5 Stress-strain graph for aluminium material samples in the parallel direction to the forming process.

Figure 6 Stress-strain graph for aluminium material samples in the perpendicular direction to the forming process.

Figure 7 Stress-strain graph for steel material samples in the parallel direction to the forming process.

Figure 8 Stress-strain graph for steel material samples in the perpendicular direction to the forming process.

The compilation of the tensile testing results is presented in table 6. For aluminium material, the obtained Exx and Eyy values present a range of, respectively, 5x109 Pa and 2.4x109 Pa. For steel material, the obtained Exx and Eyy values present a range of, respectively, 9.3x109 Pa and 7.4x109 Pa. The obtained Poisson ratio (xy and yx) values, shown high consistency for both materials.

Table 6 - Elastic properties compilation from tensile tests. Material Property Symbol Units Sample 1 Sample 2 Sample 3

Aluminium Youngs modulus Exx Pa 67.5x109 63.4x109 68.4x109 Youngs modulus Eyy Pa 73.7x109 76.1x109 73.9x109


Steel Youngs modulus Exx Pa 191.3x109 198.9x109 200.6x109 Youngs modulus Eyy Pa 227.1x109 229.8x109 234.5x109


6.2 Updating results

In this section is shown the results after updating to the aluminium and steel material numerical models.

6.2.1 Aluminium material numerical models

Table 7 present the updated elastic properties for aluminium material numerical models.

Table 7 Updated elastic properties for aluminium material numerical models. Property Symbol Units Model 1 Model 2 Model 3

Youngs modulus Exx Pa 65.7x109 66.8x10e9 66.6x10e9 Youngs modulus Eyy Pa 71.2x109 70.4x109 71.2x109


For aluminium material, Exx and Eyy present a range of, respectively, 1.1x109 Pa and 0.8x109 Pa in the updated values. The Poisson ratio (xy and yx) values are equals to the values obtained by tensile testing method. Therefore, updating method reveals more consistency and lower dispersion in the results than tensile testing method. Table 8, 9 and 10 show, respectively, the dynamic behaviour evolution for aluminium material numerical model 1, 2 and 3.

Table 8 Dynamic behavior evolution of aluminium material numerical model 1.

Mode Ref. Freq. (Hz) Num.

initial Freq. (Hz)

Difference before

Updating (%)

Num. final Freq.

(Hz)

Difference after

Updating (%)

Initial MAC

Final MAC

1 117.758 116.817 0.799 117.768 0.008 0.992 0.992 2 326.148 323.244 0.890 325.828 0.098 0.982 0.982 3 435.699 439.066 0.773 435.703 0.001 0.983 0.983 4 641.552 636.489 0.789 641.438 0.018 0.997 0.997 5 888.119 893.742 0.633 887.208 0.103 0.996 0.996 6 1064.620 1056.671 0.747 1064.620 0.000 0.984 0.984 7 1366.728 1378.696 0.876 1369.362 0.193 0.992 0.992 8 1594.797 1584.378 0.653 1595.881 0.068 0.986 0.986



initial Freq. (Hz)

Difference before

Updating (%)

Num. final Freq.

(Hz)

Difference after

Updating (%)

Initial MAC

Final MAC

1 117.410 117.050 0.307 117.410 0.000 0.987 0.987 2 325.175 323.887 0.396 324.861 0.097 0.988 0.988 3 435.571 439.938 1.003 436.466 0.205 0.881 0.881 4 639.594 637.754 0.288 639.597 0.001 0.995 0.995 5 890.388 895.518 0.576 888.682 0.192 0.998 0.998 6 1061.718 1058.771 0.278 1061.681 0.003 0.995 0.995 7 1371.459 1381.436 0.727 1371.440 0.001 0.991 0.991 8 1590.901 1587.527 0.212 1591.648 0.047 0.929 0.929



initial Freq. (Hz)

Difference before

Updating (%)

Num. final Freq.

(Hz)

Difference after

Updating (%)

Initial MAC

Final MAC

1 117.886 116.895 0.841 117.923 0.032 0.993 0.993 2 326.272 323.458 0.862 326.239 0.010 0.990 0.990

3 437.973 439.356 0.316 437.926 0.011 0.988 0.988 4 642.520 636.910 0.873 642.208 0.049 0.977 0.977 5 891.611 894.333 0.305 891.610 0.000 0.995 0.995 6 1065.127 1057.370 0.728 1065.838 0.067 0.988 0.988 7 1375.429 1379.607 0.304 1375.862 0.031 0.989 0.989 8 1597.642 1585.425 0.765 1597.638 0.000 0.966 0.967

Dynamic behaviour evolution shows that elastic properties values are updated with high reliability. The mean percentage difference, obtained by application of Equation (31), between resonance frequencies of the numerical and experimental model is very closer to zero: 0.061% for model 1, 0.068% for model 2 and 0.025% for model 3. By other hand, the fact of initial and final MAC values are very close to 1, show that mode shapes of numerical and experimental model are correlated.

6.2.2 Steel material numerical models

Table 11 present the updated elastic properties for steel material numerical models.

Table 11 Updated elastic properties for aluminium material numerical models. Property Symbol Units Model 1 Model 2 Model 3

Youngs modulus Exx Pa 167.4x109 165.6x10e9 163.3x109 Youngs modulus Eyy Pa 215.6x10e9 214.7x109 211.5x109

Poisson Ratio xy - 0.25 0.25 0.25 Poisson Ratio yx - 0.27 0.27 0.27

For steel material, both Exx and Eyy present a range of 4.1x109 Pa in the updated values. The Poisson ratio (xy and yx) values are very similar to the values obtained in tensile testing method. Therefore, updating method reveals more consistency and lower dispersion in the results than tensile testing method. Table 12, 13 and 14 show, respectively, the dynamic behaviour evolution for steel material numerical model 1, 2 and 3.

Table 12 Dynamic behavior evolution of steel material numerical model 1.


initial Freq. (Hz)

Difference before

Updating (%)

Num. final Freq.

(Hz)

Difference after

Updating (%)

Initial MAC

Final MAC

1 240.194 230.599 3.995 239.399 0.331 0.997 0.997 2 662.650 637.285 3.828 661.362 0.194 0.999 0.999 3 870.4540 884.728 1.640 870.454 0.000 0.992 0.992 4 1299.687 1252.997 3.592 1299.68 0.001 0.997 0.997 5 1771.660 1798.512 1.516 1771.596 0.004 0.971 0.971 6 2149.142 2077.129 3.351 2153.294 0.193 0.985 0.985 7 2727.595 2768.903 1.514 2732.564 0.182 0.988 0.989 8 3207.203 3110.443 3.017 3222.688 0.483 0.991 0.991



initial Freq. Difference

before Num.

final Freq. Difference

after Initial MAC

Final MAC

(Hz) Updating (%)

(Hz) Updating (%)

1 239.867 230.812 3.775 239.116 0.313 1.000 1.000 2 661.742 637.874 3.607 660.575 0.176 0.999 0.999 3 868.362 885.546 1.979 868.149 0.025 0.993 0.993 4 1298.117 1254.155 3.387 1298.115 0.000 0.999 0.999 5 1766.966 1800.174 1.879 1766.966 0.000 0.994 0.994 6 2145.847 2079.048 3.113 2150.672 0.225 0.998 0.998 7 2720.872 2771.462 1.859 2725.573 0.173 0.993 0.992 8 3203.205 3113.317 2.806 3218.721 0.484 0.992 0.992



initial Freq. (Hz)

Difference before

Updating (%)

Num. final Freq.

(Hz)

Difference after

Updating (%)

Initial MAC

Final MAC

1 238.519 231.401 2.984 237.927 0.248 0.999 0.999 2 658.648 639.501 2.907 657.285 0.207 0.999 0.999 3 864.263 887.805 2.724 864.116 0.017 0.999 0.999 4 1291.660 1257.356 2.656 1291.64 0.001 0.998 0.998 5 1758.733 1804.768 2.618 1758.733 0.000 0.998 0.998 6 2135.699 2084.353 2.404 2139.929 0.198 0.993 0.993 7 2708.713 2778.534 2.578 2712.814 0.151 0.998 0.998 8 3187.893 3121.262 2.090 3202.623 0.462 0.997 0.997

Dynamic behaviour evolution shows that elastic properties values are updated with high reliability. The mean percentage difference, obtained by application of Equation (10), between resonance frequencies of the numerical and experimental model is very low: 0.174% for model 1, 0.175% for model 2 and 0.161% for model 3. By other hand, the fact of initial and final MAC values are very close to 1, show that mode shapes of numerical and experimental model are correlated.

6.3 Analytical method

In this section is shown the obtained results based on analytical theories presented in section 2.

6.3.1 Analytical results for Eyy

Considering the eight first mode shapes extracted from experimental modal analysis: modes 1, 2, 4, 6 and 8 are bending shapes; mode 3 is torsion shape; modes 5 and 7 are mixed shapes. Note that in table 1 (first five Kn values for a free-free beam conditions), mode 1 corresponds to the experimental mode 1, mode 2 corresponds to the experimental mode 2, mode 3 corresponds to the experimental mode 4, mode 4 corresponds to the experimental mode 6 and mode 5 corresponds to the experimental mode 8. Youngs modulus Eyy value can be calculated using any one of five modes. However, the lowest error in Eyy value is obtained using the first mode. Using Equation (2) and replacing f and m variables by reference experimental values, presented

in table 15 for respective sample, we calculate Eyy values of aluminium and steel material. See table 3 for the values of geometrical variables. Analytical Eyy values are presented in table 16.

Table 15 Values of f (bending mode shape) and m variables for aluminium and steel material. Variable Units Aluminium material Steel material Sample 1 Sample 2 Sample 3 Sample 1 Sample 2 Sample 3

f1

(Hz)

117.758 117.410 117.886 240.194 239.867 238.519 f2 326.148 325.175 326.272 662.650 661.742 658.648 f4 641.552 639.594 642.520 1299.687 1298.117 1291.660 f6 1064.620 1061.718 1065.127 2149.142 2145.847 2135.699 f8 1594.797 1590.901 1597.642 3207.203 3203.205 3187.893 m (Kg) 0.0757 0.0754 0.0756 0.4335 0.4327 0.4305

Table 16 Eyy Values using five experimental bending mode shapes. Mode Material Property Symbol Units Sample 1 Sample 2 Sample 3

f1

Aluminium Youngs modulus Eyy Pa

71.3x109 70.6x109 71.4x109 f2 72.0x109 71.3x109 71.9x109 f4 72.5x109 71.7x109 72.6x109 f6 73.0x109 72.3x109 73.0x109 f8 73.4x109 72.8x109 73.8x109 f1

Steel Youngs modulus Eyy Pa

217.3x109 216.3x109 212.8x109 f2 217.6x109 216.6x109 213.5x109 f4 217.8x109 216.9x109 213.7x109 f6 218.0x109 216.9x109 213.8x109 f8 217.5x109 216.6x109 213.4x109

Table 17 shows the comparison between tensile testing, updating and analytical method. For this comparison only are used Eyy values calculated from f1 mode.

Table 17 Comparison of Eyy values using the three methods.

Sample Material Property Symbol Units Updating Analytical Tensile Testing 1

Aluminium Youngs modulus Eyy Pa

71.2x109 71.3x109 73.7x109 2 70.4x109 70.6x109 76.1x109 3 71.2x109 71.4x109 73.9x109 1

Steel Youngs modulus Eyy Pa

215.6x10e9 217.3x109 227.1x109 2 214.7x109 216.3x109 229.8x109 3 211.5x109 212.8x109 234.5x109

The results show that analytical values for Youngs Modulus Eyy are very similar with the results obtained by updating method. Both methods allow obtain values more consistent and reliable than tensile testing method.

6.3.2 Analytical results for Exx

Using equation (7) we calculate Exx values based on high-order plate element theory. The xy and yx values for both materials, are presented in table 6. Replacing f and m variables by reference

experimental values, presented in table 18 for respective sample, we calculate Exx values for aluminium and steel material. See table 3 for the values of geometrical variables. The analytical Exx values are presented in table 19.

Table 18 Values of f (torsion mode shape) and m variables for aluminium and steel material. Variable Units Aluminium material Steel material Sample 1 Sample 2 Sample 3 Sample 1 Sample 2 Sample 3

f3 (HZ) 435.699 435.571 437.973 870.454 868.362 864.263 m (Kg) 0.0757 0.0754 0.0756 0.4335 0.4327 0.4305

Using equation (11) we calculate Exx values based on Picket theory. Replacing f and m variables by reference experimental values, presented in table 18 for respective sample, we calculate Exx values for aluminium and steel material. Eyy is given by values calculated in section 6.3.1 through first mode. See table 3 for the values of geometrical variables. Eyy is given by values calculated in section 6.3.1 through first mode. The analytical Exx values are presented in table 19.

Using equation (14) we calculate Exx values based on Rayleigh principle. Replacing f and m variables by reference experimental values, presented in table 18 for respective sample, we calculate Exx values for aluminium and steel material. See respectively, table 3 and 4 for the values of geometrical variables and values of material density. Eyy is given by values calculated in section 6.3.1 through first mode. The analytical Exx values are presented in table 19.

Using equation (21) we calculate Exx values based on torsional vibration for beam of non-circular cross section theory. Replacing f and m variables by reference experimental values, presented in table 18 for respective sample, we calculate Exx values for aluminium and steel material. See respectively, table 3 and 4 for the values of geometrical variables and values of material density. Eyy is given by values calculated in section 6.3.1 through first mode. The analytical Exx values are presented in table 19.

Table 19 Exx values obtained using plate theories. Theory Material Property Symbol Units Sample 1 Sample 2 Sample 3

High-order plate

element

Aluminium Youngs modulus Exx Pa

64.6x109 64.3x109 65.2x109

Picket 67.7x109 68.0x109 69.2x109 Rayleigh 43.5x109 43.6x109 44.3x109 Beam of

non-circular

cross section

72.5x109 72.8x109 74.2x109

High-order Steel Youngs Exx Pa 195.0x109 193.7x109 190.9x109

plate element

modulus

Picket 172.8x109 171.2x109 169.1x109 Rayleigh 115.5x109 114.6x109 113.1x109 Beam of

non-circular

cross section

194.9x109 193.1x109 190.7x109

With High-order plate element and Picket theory we obtain very similar Exx values for aluminium material, and are closer to the values obtained by updating method. The Exx values obtained by Rayleigh principle are lower than expected. The values obtained by torsional vibration for beam of no-circular cross section theory are higher than expected. For steel material the Picket theory gives Exx values closest to the obtained by updating method. The values obtained by Rayleigh principle are lower than expected. The values obtained by High-order plate element and torsional vibration for beam of no-circular cross section theory are higher than expected. The high-order plate element theory it is more effective than smaller is the thickness, when the width keeps constant. This justifies the fact of the Exx values obtained by high-order plate element theory, for steel material, are more distant of the values obtained by updating for the same material. In steel material samples the thickness is about 12 times smaller than the width dimension, while in aluminium material samples this ratio is approximately 24 times. Therefore Exx values for aluminium material obtained from this theory are very close to the values obtained by updating method. Table 20 shows the comparison between the results obtained by tensile testing, updating and analytical method for Exx values. The analytical Exx values used for aluminium material are the obtained by high-order plate element theory. The analytical Exx values used for steel material are the obtained by Picket theory.

Table 20 Comparison of Exx values using the three methods.

Sample Material Plate Property Symbol Units Updating Analytical Tensile Testing 1

Aluminium Youngs modulus Exx Pa

65.7x109 64.6x109 67.5x109 2 66.8x109 64.3x109 63.4x109 3 66.6x109 65.2x109 68.4x109 1

Steel Youngs modulus Exx Pa

167.4x109 173.0x109 191.3x109 2 165.6x109 171.2x109 198.9x109 3 163.3x109 169.1x109 205.0x109

Analytical Exx values are similar relatively to the updated Exx values. Both methods, analytical and updating, allow obtain higher accuracy, more consistency and lower dispersion in the results than tensile testing method.

7 CONCLUSIONS

This paper approaches a different way to estimate elastic properties of materials, namely Youngs Modulus and Poisson Ratio, usually obtained by tensile tests. A finite element model updating methodology is applied to practical cases and the process is validated. The estimated values from presented updating methodology, and validated by analytical theories, reveals to be efficient and reliable than tensile testing method. The use of resonance frequencies allow to the updating method to be very sensitive to the slight variations in elastic properties caused by forming process. Moreover have the advantage of be a non-destructive test and can be applied to more complex structures and components.

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