analysis of a wind turbine blade
TRANSCRIPT
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UNIVERSIDAD DE CONCEPCINFACULTADA DE INGENIERA AGRCOLA
NATURAL FRECUENCIES ANALYSIS OF A WIND TURBINE BLADE
RODRIGO FUENTES CCERES
CHILLAN CHILE
2009
TESIS PRESENTADA A LA ESCUELA DEGRADUADOS DE LA UNIVERSIDAD DECONCEPCION, PARA OPTAR AL GRADO DEMAGISTER EN INGENIERA AGRCOLA,MENCIN MECANIZACIN Y ENERGA.
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NATURAL FRECUENCIES ANALYSIS OF A WIND TURBINE BLADE
Aprobado por:
Gabriel Merino Coria
Licenciado en Fsica, Ph. D.Profesor Asociado Profesor Gua
Cristian Rodrguez GodoyIngeniero Civil Mecnico, DoctorProfesor Asistente Profesor Asesor Externo
Emilio Dufeu DelarzeIngeniero Civil Mecnico, DoctorProfesor Asociado Profesor Asesor Externo
Octavio Lagos RoaIngeniero Civil Agrcola, Ph. D.Profesor Asistente Director de Programa
Eduardo Holzapfel HocesIngeniero Agrnomo, Ph. D.Profesor Titular Decano
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INDEX
Page
Summary..................................................................................................
Resumen..................................................................................................
Introduction..............................................................................................
Finite Elements Model......................................................................
Experimental Modal Test..............................
Results............................................................................
Conclusions........................................................................
References...............................................................................................
Figures and Tables..................................................................................
1
3
5
8
9
11
13
14
16
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NATURAL FRECUENCIES ANALYSIS OF A WIND TURBINE BLADE
Keywords:Wind Turbine, FEM, Natural Frecuencies.
ABSTRACT
This paper studies the influence of the rotor angular speed on the natural
vibration frequencies of the blades for a low power wind turbine, due to the
stiffness effect produced by the centrifuge force. The wind turbine under
study corresponds to a 1 kW wind turbine, which consists of three blades of
compound material of fiber glass with epoxy resin, and whose range of
angular speed in operation fluctuates from 0 to 600 rpm.
The research has considered a finite elements model (FEM) of the blades
calculate their natural frequencies at several operational rotating speed, and
an experimental modal test for validating the numerical model at zero rotating
speed. For the FEM development, SAMCEF software was used, using
volume elements for the discretization of the blade. The impact test done on
the turbine blade was carried out reproducing the clampling condition of the
blade on the turbines rotor in the laboratory, using a Bruel & Kjaer, 8206
Model impact hammer, and a data collector-analysis system PXI, National
Instruments, 4472B Card was used to experimentally obtain the natural
vibration frequencies of the blade at a zero rotating speed.
Researches related to the natural frequencies about wind turbine blades have
considered a beam model to simulate the blades on the wind turbine. This
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research has focused on high power turbines. Results presented about the
stiffness effect of the blades on large turbines due to rotation speed, show a
non-significative stiffness effect, which means that the natural frequencies do
not significantly change in operation, regarding its resting position.
By using the FEM model developed in this study, graphs of the first, second
and third natural vibration frequencies of the blades in flapwise mode were
obtained, versus the turbines rotating speed where the blades stiffness
effect appears, due of the centrifugal force. Results indicate that the first
natural frequency, in flapwise bending, varies from 4 to 20 Hz, in a range of
speeds from 0 to 600 rpm, that is, an increase of five times the value of
natural frequency of the resting blade, regarding its value in operation at 600
rpm.
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ANALISIS DE LAS FRECUENCIAS NATURALES DE VIBRAR DE LAS
ASPAS DE UNA TURBINA EOLICA
Palabras ndice adicionales: Fotovoltaico, bombeo PV, bombeo de acopledirecto, monitoreo.
RESUMEN
En este trabajo se ha estudiado el grado de incidencia que produce la
velocidad de rotacin del rotor sobre las frecuencias naturales de vibrar delas aspas, en un aerogenerador de baja potencia, debido al efecto de
rigidizacin que produce la fuerza centrfuga. El modelo bajo estudio,
corresponde a una turbina de 1 kW de potencia, que consta de tres aspas de
material compuesto de fibra de vidrio con resina epxica y cuyo rango de
velocidad de rotacin, en operacin, flucta entre 0 y 600 rpm.
La investigacin ha contemplado la generacin de un modelo numrico en
elementos finitos de las aspas, para calcular sus frecuencias naturales de
vibrar a distintas velocidades angulares de operacin, previo un ensayo
modal experimental para validar el modelo numrico, a velocidad de rotacin
cero. Para el desarrollo del modelo numrico, se utiliza el programa
SAMCEF, para modelacin por elementos finitos, utilizando elementos devolumen para la discretizacin del aspa. El ensayo de impacto realizado al
aspa de la turbina, se ejecuta replicando en banco de pruebas, la condicin
mecnica de empotramiento del aspa montada sobre el rotor de la turbina y
se utiliza un martillo de impacto Bruel & Kjaer, Modelo 8206 y un sistema de
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Adquisicin de datos PXI, National Instruments, Tarjeta 4472B para obtener
en forma experimental las frecuencias naturales de vibrar del aspa, a
velocidad de rotacin cero.
Investigaciones relacionadas a frecuencias naturales sobre aspas de
turbinas, han considerado un modelo de viga para simular las aspas en la
turbina, aunque los trabajos desarrollados se centran en turbinas de grandes
potencias. Resultados presentados sobre el efecto rigidizador de las aspas
en grandes turbinas, debido a la velocidad de rotacin, muestran un no gran
efecto rigidizador, esto significa que las frecuencias naturales no varan
significativamente, en operacin, con respecto a su condicin de reposo.
Al utilizar el modelo de elementos finitos desarrollado en este estudio, se
obtienen grficos de las tres primeras frecuencias naturales de vibrar de las
aspas, en modo flapwise, versus la velocidad de rotacin del rotor de la
turbina, en los cuales se aprecia el efecto de rigidizamiento de las aspas,
producto de la fuerza centrfuga. Se puede apreciar que la primera frecuencia
natural, en flexin flapwise, vara desde 4 Hz hasta 20 Hz, en un rango de
velocidades desde 0 rpm hasta 600 rpm, esto es, un aumento de cinco veces
el valor de la frecuencia natural del aspa en reposo, con respecto a su valor
en operacin a 600 rpm.
.
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INTRODUCTION
Due to the Chilean growing energy demand, which increases with the
countrys economic development, in addition to the exponential increase in
prices of the oil, to obtain electric energy from non-conventional renewable
sources is becoming more and more important. Wind power is a good
alternative source of non-contaminating renewable energy. Its low power
demand is growing, due to the increase of generation costs based on diesel
in isolated areas. The big barriers for using wind power have been the high
costs of the turbines and the lack of general knowledge in terms of the
selection, installation, performance and maintenance of the equipment.
Based on this fact, the objective of this study is to contribute with
indispensable knowledge for the development of this type of technology at a
national level.
Wind turbines can be classified as small and large, related to their power.
According to [12], we refer to small turbines as those with power less than 10
kW. Those with greater power are considered large wind turbines. According
to [9], currently, the majority of the machines that are installed present
nominal powers between 0.6 and 5 MW.
The objective of small wind turbines is to supply electric energy to residents,
farms, or small rural groups, isolated from the electric supply network. The
power of these machines usually varies between 0.1 and 10 kW [14]. These
wind turbines are much more simple than those used for large scale
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generation. In terms of its operating conditions, low power wind turbines have
operation speeds that can reach 1100 rpm, according to [14], and its blades
length between 1 and 4.5 m, compared with high power turbines, in which the
rotating speed does not exceed 60 rpm, according to [11] and [13], and
whose blades measure length between 10 and 55 m. These operational
characteristics produce a greater rigidity effect of the blades in the small
turbines, due to the angular velocity. In the large turbines, a greater rigidity
effect of the blades occurs due to their mass and radius.
In general, researches done about wind turbines related to the natural
vibration frequencies focuses on high power turbines, because of the
prioritary economic interest on the electric energy conversion. Tests done on
high power turbines blades show a minimal blades rigidity effect due to the
centrifugal force (see Figure 2).
In [4], the vibration response of a turbine tower and blades assembling under
the action of the stationary wind force was studied. In this case, the blades
are modeled as beams fitted to a rotating mass and only consider the
bending of one of its axis.
Studies done about beams fitted to a rotating mass, as in [1] and [2], where
the equations of motion of the beam are represented, considered as a Euler
Bernoulli beam, under large rotation displacements (rigid body motion) and
with small elastic deformations, like in [5], considering three variables:
bending in the most flexible direction (flapwise), bending in the most rigid
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direction (edgewise) and the lengthening of the beam (stretching). This way,
three partial differential equations about the beam motion are obtained.
In [2], these equations of motion are solved by the finite elements method and
an analysis of the dimensionless natural vibration frequencies versus the
beams dimensionless rotation speed was done on the three previously
mentioned variables. As they are dimensionless, these results are applied to
beams of any size.
In [3], the blade is modeled considering its complete geometry, but only for
zero angular speed, for which the influence of the centrifugal force due to
rotational speed on the behavior of the blades natural vibration frequency
was not considered. In this case, a mathematical model is presented
regarding a high power turbine blade (19 m in length) mounted on a rigid
mass (See Figure 1). The equations of motion describe small rotations of the
mass (hub), small flapwise and edgewise bending, and considers the warp
movement of the blade (warping). The mathematical model is solved as a
eigenvalues problem and then compared with an experimental modal
analysis, with excellent results.
As previously mentioned, the stiffness increase that is result of the rotating
speed of blades on large size turbines is not considerable. According to [11],
it is worth asking what the influence of the rotating speed of the blades on
small size turbines is, since the rotation speed are greater, considering its
geometry and the material parameters it is made of.
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The principal objective of this study is to investigate the degree of influence of
the rotors rotating speed on the natural vibration frequencies of the blades,
due to the stiffness increase that the centrifugal force produces on a low
power wind turbine (1 kW). For this purpose, a numerical model of the
turbines blade will be generated, with the SAMCEF software, for modeling
by the finite elements method. In addition, the natural frequencies of the
turbines resting blade will be determined through an experimental modal test.
Then, the numerical model at rotation speed zero will be validated, comparing
it with the natural frequencies experimentally obtained, to then determine
through the numerical model, the natural frequencies of the wind turbines
blade for different operation angular speeds.
FINITE ELEMENTS MODEL (FEM)
The modal dynamical analysis, through the finite elements method, was done
with the SAMCEF program. In this study the blade is discretized using volume
elements, pentahedrons and hexahedrons of first order [6], using a total of
4646 nodes of 3 degrees of freedom, and 2600 elements, with a total of 200
pentahedron elements (7.7%) and 2400 hexahedrons (92.3%). The model
was made considering the blade clamped in one of its extremes, as shown in
Figure 4, and replicating the real profile of the blade, as observed in Figures 3
and 5. The blade is built of fiber glass with epoxy resin, for which, according
to [10] and [8], an elasticity module (Young) of 38600 MPa and a Poissons
model of 0.26 were considered. On the other hand, the measured density is
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1.9 X 10-9 ton/mm3. This data was used in the simplified construction of the
model, where the non-linear elastic orthotropic properties that compound
materials generally present were not considered, but the isotropic properties
with behavior within a linear elastic range of the materials were, as was
estimated in [3], achieving good results.
With the model in question, the modal study of the blade was carried out, at a
rotating speed zero, where the natural vibration frequencies were obtained,
which were then compared with those obtained in the experimental modal
test in Table 1.
Next, the results obtained from the numeric model are presented, in terms of
the three first flapwise bending modes (regarding the Y axis) of the blades
vibration. The first vibration mode has a natural frequency of 4.4 Hz and
posseses a node on the extreme of the blades clamping, see Figure 6. The
second vibration mode has a frequency of 27.7 Hz and posseses two nodes,
as observed in Figure 7. The third vibration mode is observed in Figure 8,
which has a frequency of 77.4 Hz, with its corresponding three nodes.
EXPERIMENTAL MODAL TEST
In the impact test, a Bruel & Kjaer, 8206 Model impact hammer, a Bruel &
Kjaer, 4513-001 model accelerometer sensor and a PXI, National
Instruments, card 4472B data adquisition card were used, as well as
LabView software.
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The blade was clamped to a base in the laboratory, as obserevd in Figure 9,
to simulate its behavior when mounted on the turbines rotor. The modal
impact test consisted of hitting the blade, which is measured with the impact
hammer, and then the vibratory amplitude of the blades vibration is measured
by the accelerometer. This signal is captured by the data acquisition card,
and with the LabView software, the frequencies answer function of the
vibration was calculated, where the natural vibration frequencies of the blade
were obtained [7].
In Figure 10, that answer in the time of the amplitude of the blades vibratory
acceleration is observed, due to the impact given in its free extremity. The
accelerometer was installed in the clamped side of the blade. The impact
given and the blades response of the free - damped vibration can be seen in
the waveform, which decreases in amplitude as time passes. In Figure 11,
the spectrum in frequencies of the acceleration amplitude of the vibration is
observed, in a range of frequencies from 0 to 800 Hz. The several
frequencies of which the vibratory wave is composed of can be seen, among
which the blades natural vibration frequencies are found.
The following natural vibration frequencies were obtained by this impact test,
which are compared with those obtained by the numerical model by finite
elements, in Table 1.
Comparing the blades natural frequencies obtained through the experimental
test with those obtained through the numerical model (on SAMCEF), in Table
1, a minimum difference can be seen (less than 8%), in the results of both
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methods regarding the three first natural frequencies. The dispersion of the
values of high frequencies could be caused by a lack of refinement of the
mesh (4646 nodes), and by the using elements of first order. However, this
does not detriment the study objectives, as they require analyzing the
dynamic behavior of the blades for a range of the turbines operational speed,
for which, the high frequencies are very outside the range. Therefore, it is not
necessary to make any adjustments to the numerical model.
With this, the numerical model is validated at zero angular speed and will be
used to simulate the behavior of the blades natural frequencies for the
turbines several operational speeds, and study its stiffness effect on the
blade and the influence on the behavior of its natural frequencies under
operational conditions.
RESULTS OF NATURAL VIBRATION FREQUENCIES OF THE BLADES
VERSUS ROTATING SPEED
The range of rotating speed in normal operation of the turbine is from 0 to 600
rpm. The modal analysis was simulated for rotating speeds of 10, 20, 30, 40,
50 and 60 rad/s, that is, between 95 and 668 rpm. With this data, a graph was
created with the behavior of the three first natural vibration frequencies for
flapwise mode, versus the rotating speed in rpm (see Figure 12). In this first
graph, it can be seen that the behavior of the second and third modes natural
frequency are out of the range of turbines frecuencies, 1X, 2X and 3X.
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For better observation about if there is at some time a resonant zone of one
of the two first natural vibration frequencies, in flapwise bending, a graph
was made of these two first natural frequencies (in Hz) versus the rotating
speed of the turbine (in Hz) in its range of operational speed. It can be
observed that the rotating frequency 1X, is out of the range of the second
natural vibration frequency. In terms of the first natural vibration frecuency,
which at a zero rotating speed is of 4.92 Hz, it can be observed that as the
rotating speed increases, it generates an increase of the value of the natural
frequency, due to the stiffness effect, unable the 1X to reach this first natural
frequency, for which there are no problems of resonance.
In terms of the blades frequency, it means 3X, there is a problem around the
2 Hz (120 rpm) of rotation, as the 3X = 6 Hz coincide with the first natural
frequency, for which there is a posible problem of structural resonance at this
speed (see Figure 13). It must be considered that this condition is given for
the blades frequency, when passing in front of the tower tube, for which the
excitation force should be transmitted from the tower to the turbines rotor and
then to the blades, therefore it is not a direct excitation, which is not very
significative.
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CONCLUSIONS
The experimented by [3] is proven in this paper; by developing a modal
dynamical model of the blades, the non-linear elastic orthotropic condition of
the compound materials can be approximated by a simplified linear elastic
isotropic model, with good results, at a zero rotating speed.
Under the conditions of the simplifed model previously indicated, it is possible
to simulate the behavior of the blades of a wind turbine with the method of
finite elements, acheiving a very good approximation by comparing the
obtained natural vibration frequencies with the finite elements model (FEM),
versus those obtained through an experimental modal test, at a zero
rotational speed.
In this paper, it can be appreciated that the stiffness effect of the blades due
to rotating speed, for small sized turbines, is considerably significative.
Graphs are presented of the blades three first natural vibration frequencies,
in flapwise mode, versus the turbines rotors rotating speed, in which it can
be seen that the first natural flapwise frequency varies from 4 Hz to 20 Hz, in
a range of speed from 0 to 600 rpm, that is, an increase of up five times the
blades natural frequency while resting, versus its operational value at 600
rpm. Therefore, for the design of low power turbine blades, where rotational
speed is high, it is not sufficient to only calculate the natural frequencies with
an impact test of the resting blade, rather it also requires a study of their
dynamical behavior, to avoid resonances and prolong its lifespan.
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REFERENCES.
[1] H. H. Yoo, R. R. Ryan and R. A. Scott 1995, Journal of Sound and
Vibration 181, 261 278. DYNAMICS of FLEXIBLE BEAMS UNDERGOING
OVERALL MOTIONS.
[2] J. Chung and H. H. Yoo 2002, Journal of Sound and Vibration 249, 147
164. DYNAMICS ANALYSIS of a ROTATING CANTILEVER BEAM by
USING the FINITE ELEMENT METHOD.
[3] A. Baumgart 2002, Journal of Sound and Vibration 251, 1 12. A
MATHEMATICAL MODEL FOR WIND TURBINE BLADES.
[4] P.J. Murtagh, B. Basu, B.M. Broderick 2005, Engineering Structures 27,
1209 1219. ALONG WIND RESPONSE OF A WIND TURBINE TOWER
WITH BLADE COUPLING SUBJECTED TO ROTATIONALLY SAMPLED
WIND LOADING.
[5] J. C. Simo 1985, Computer Methods in Applied Mechanics and
Engineering 49, 55 70. A FINITE STRAIN BEAM FORMULATION. THE
THREE DIMENSIONAL DYNAMIC PROBLEM. PART I.
[6] T. Chandrupatla, A. Belegundu 1999, INTRUDUCCION AL ESTUDIO DEL
ELEMENTO FINITO EN INGENIERIA. Prentice Hall.
[7] Peter Avitabile 2001, Sound and Vibration 35, 20 31. EXPERIMENTAL
MODAL ANALYSIS. A SIMPLE NON MATHEMATICAL PRESENTATION.
[8] C.Kong, J. Bang, Y. Sugiyama 2005, Energy 30, 2101 2114.
STRUCTURAL INVESTIGATION OF COMPOSITE WIND TURBINE BLADE
CONSIDERING VARIOUS LOAD CASES AND FATIGUE LIFE.
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[9] J. W. Larsen 2005, Tesis for the degree of Doctor of Philosophy.
Department of Civil Engineering, Faculty of Engineering and Science, Aalborg
University. NONLINEAR DYNAMICS OF WIND TURBINE WINGS.
[10] R. F. Gibson 1994: PRINCIPLE OF COMPOSITE MATERIALS
MECHANICS. Mc Graw-Hill Ed. p. 1.
[11] R. Gasch, J.Twele 2002, WIND POWER PLANTS. Fundamentals,
Design, Construction and Operation.
[12] J.F. Sanz 2001. GUIA DE LAS ENERGIAS RENOVABLES APLICADAS
A LAS PYMES.
[13] J. Wiley and Sons Ltda. 2001, WIND ENERGY HANDBOOK.
[14] M. Sagrillo 2002, APPLES AND ORANGES CATALOGUE
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Figure 1. Motion Sequence of the Blade in its Second Flapwise Mode. Three
times in the same figure are shown.
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Figure 2. Graph of the Primary Natural Vibration Frequencies versus Rotation
Velocity of the Blade of a Turbine. Turbine Test DEBRA (DLR) [15].
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Figure 3. Profile of the Bergey XL1 turbine blade, of 1 kW. Its clamped
condition to the rotor of the turbine can be seen.
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Figure 4. Model of turbine blade done on SAMCEF. The fitted condition of the
model is marked the red oval.
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Figure 5. Profile of the Bergey XL1 turbine blade, model done on SAMCEF.
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FIGURE 6. First vibration mode of the blade in flapwise bending.
nodo
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FIGURE 7. Second vibration mode of the blade in flapwise bending.
nodo 1
nodo 2
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FIGURE 8. Third vibration mode of the blade in flapwise bending.
nodo 1
nodo 2
nodo 3
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Figure 9. Impact test with the blade bolted to its base, such as the turbines
rotor.
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Figure 10. Response of the vibratory acceleration amplitude (g) in time, of theturbines blade, when is hit with the hammer.
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Figure 11. Response in frequencies (from 0 to 800 Hz) of the vibratory
acceleration amplitude (g), of the turbines blade, when is hit with the
hammer.
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0
20
40
60
80
100
120
0 100 200 300 400 500 600 700 800
velocidad de rotacin, rpm.
fnatural,Hz.
fn 1er modo flapwise
fn 2o modo flapwise
fn 3er modo flapwise
1X
2X
3X
Figure 12. Graph of the three primary natural vibratuon frequencies, in
flapwise mode versus rotation velocity in rpm of the turbines blade.
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0
5
10
15
20
25
30
35
40
45
0,00 2,00 4,00 6,00 8,00 10,00 12,00
velocidad de rotacin, Hz.
fnatural,Hz. fn 1er modo flapwise
fn 2o modo flapwise
1X
2X
3X
Figure 13. Graph of the two primary natural vibration frequencies,
inflapwisemode, versus rotating speed, in Hz, of the turbines blade
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Modal Impact Test
FRF
Numeric Model
FEM Deviation
Flapwise
Bending Mode Frequency (Hz) Frequency (Hz)
Percentage
%
1 4.2 4.4 4.8
2 26.2 27.7 5.7
3 72.0 77.4 7.5
4 138.4 151.1 9.2
5 214.6 248.4 15.8
6 315.6 368.6 16.8
7 402.4 510.6 26.9
Table 1. Comparison of the natural frequencies of the first seven modes of
flapwise bending, obtained with the numerical model FEM and the modal test
FRF.