-
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]
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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
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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.
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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)
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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
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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.
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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.
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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.
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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
Poisson Ratio xy - 0.26 0.26 0.26 yx - 0.26 0.26 0.26
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.
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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
Poisson Ratio xy - 0.31 0.30 0.32 yx - 0.31 0.30 0.32
Steel Youngs modulus Exx Pa 200.0x109 160.0x109 240.0x109 Youngs
modulus Eyy Pa 200.0x109 160.0x109 240.0x109
Poisson Ratio xy - 0.26 0.23 0.29 yx - 0.26 0.23 0.29
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.
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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.
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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
Poisson Ratio xy - 0.31 0.31 0.31 yx - 0.31 0.31 0.31
Steel Youngs modulus Exx Pa 191.3x109 198.9x109 200.6x109 Youngs
modulus Eyy Pa 227.1x109 229.8x109 234.5x109
Poisson Ratio xy - 0.26 0.24 0.25 yx - 0.27 0.27 0.27
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
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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
Poisson Ratio xy - 0.31 0.31 0.31 yx - 0.31 0.31 0.31
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
Table 9 Dynamic behavior evolution of aluminium material
numerical model 2.
Mode Ref. Freq. (Hz) Num.
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
Table 10 Dynamic behavior evolution of aluminium material
numerical model 3.
Mode Ref. Freq. (Hz) Num.
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.
Mode Ref. Freq. (Hz) Num.
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
Table 13 Dynamic behavior evolution of steel material numerical
model 2.
Mode Ref. Freq. (Hz) Num.
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
Table 14 Dynamic behavior evolution of steel material numerical
model 3.
Mode Ref. Freq. (Hz) Num.
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.
8 REFERENCES
[1] D. V. Boeri, Caraterizao de materiais compostos por
ultra-som, Master Thesis, Escola Politcnica da Universidade de So
Paulo, So Paulo, 2006.
[2] R. Caracciolo, M. Giovagnoni, Frequency dependence of
Poisson's ratio using the method of reduced variables, Mechanics of
Materials, 24 (1996) 75-85.
[3] R. Caracciolo, A. Gaspararetto, M. Giovagnoni, Measurement
of the isotropic dynamic Youngs modulus in a seismically excited
cantilever beam using a laser sensor, Journal of Sound and
Vibration, 231(5) (2000) 1339-1353.
[4] R. Caracciolo, A. Gaspararetto, M. Giovagnoni, Application
of causality check and of the reduced variables method for
experimental determination of Youngs modulus of a viscoelastic
material, Mechanics of Materials 33 (2001) 693-703.
[5] R. Caracciolo, A. Gaspararetto, M. Giovagnoni, An
experimental technique for complete dynamic characterization of a
viscoelastic material, Journal of Sound and Vibration, 272 (2004)
1013-1032.
[6] S. F. Almeida, J. B. Hanai, Anlise Dinmica Experimental da
Rigidez de Elementos de Concreto Submetidos Danificao Progressiva
at a Ruptura, Cadernos de Engenharia de Estruturas, So Carlos,
10(44) (2008) 49-66.
[7] R. Szilard, Theories and Applications of Plate Analysis:
Classical, Numerical and Engineering Methods, John Wiley &
Sons, Inc., New Jersey, 2004.
-
[8] S. Spinner, R. C. Valore, Jr., Comparison of Theoretical and
Empirical Relations Between the Shear Modulus and Torsional
Resonance Frequencies for Bars of Rectangular Cross Section,
Journal of Research of the National Bureau of Standards, 60(5) 1958
459-464.
[9] M. E. Mcintyre, J. Woodhouse, On Measuring the Elastic and
Damping Constants of Orthotropic Sheet Materials, Acta metal, 36(6)
(1988) 1397-1416.
[10] M. Caresta, Vibrations of a free-free beam, Cambridge
University.
[11] J. Zhou, T. Farquhar, Wheat stem moduli in vivo via
reference basis model updating, Journal of Sound and Vibration, 285
(2005) 1109-1122.
[12] B. Rahmani, F. Mortazavi, I. Villemure, M. Levesque, A new
approach to inverse identification of mechanical properties of
composite materials: Regularized model updating, Composite
Structures, 105 (2013) 1109-1122.
[13] R. F. S. Hearmon, Introduction to Applied Anisotropic
Elasticity, Oxford Univ. Press, Oxford, 1961.
[14] J. Meireles, Anlise Dinmica de Estruturas por Modelos de
Elementos Finitos Identificados Experimentalmente, Master Thesis,
University of Minho, Guimares, 2007.
[15] J. Meireles, J. Ambrsio, J. Montalvo e Silva, A. Pinho,
Structural Dynamic Analysis by Finite Element Models Experimentally
Identified: An Approach Using Modal Data, in: Proc. Experimental
Vibration Analysis for Civil Engineering Structures, Porto,
2007.
[16] M. Dourado, J. Meireles, A. M. A. C. Rocha, Structural
Dynamic Updating Using a Global Optimization Methodology, in Proc.
5th Int. Operational Modal Analysis Conference, Guimares, 2013, pp.
1-8.
[17] R. J. Allemang, D. L. Brown, A Correlation Coefficient for
Modal Vector Analysis, in Proc. 1st Int. Conference & Exhibit
Modal Analysis, Florida, 1982, pp. 110-116.