Abstract: In this paper, both flap-wise and edge-wise natural frequencies of vibration of a variable blade length wind turbine are calculated. A stepped beam with two portions has been used to approximate the blade. The two portions of the blade have been approximated by a hollow beam and a solid beam with rectangular cross section. As the outboard portion of the blade can be slid in and out of the inboard portion, ten different configurations of the stepped beam have been investigated. A MATLAB program has been developed for a finite element analysis of a one dimensional model of the stepped beam. Subsequently a three dimensional model of the stepped beam has been developed in the finite element program Unigraphics NX4. The results found using the MATLAB program have been compared with those found with NX4 and satisfactory agreement between these results has been found. Additionally, the influence of varying the blade length on the natural frequencies has been investigated

Kilfoil, M.

Tartibu, Lagouge

Van der Merwe, A.J.

English

Name not available

 1 
Sixth South African Conference on Computational and Applied Mechanics 
SACAM08 
Cape Town,  26-28 March 2008 
SACAM 
 
ANALYSIS OF VIBRATION OF A VARIABLE BLADE LENGTH 
WIND TURBINE 
 
LAGOUGE TARTIBU 
 
KILFOIL, M. 
 
VAN DER MERWE, A.J. 
 
 Department of Mechanical Engineering, 
Cape Peninsula University of Technology, Box 652, Cape Town 8000, South Africa. 
 
1. Abstract 
 
In this paper, both flap-wise and edge-wise natural frequencies of vibration of a 
variable blade length wind turbine are calculated.  A stepped beam with two portions 
has been used to approximate the blade. The two portions of the blade have been 
approximated by a hollow beam and a solid beam with rectangular cross section. As 
the outboard portion of the blade can be slid in and out of the inboard portion, ten 
different configurations of the stepped beam have been investigated. A MATLAB 
program has been developed for a finite element analysis of a one dimensional model 
of the stepped beam. Subsequently a three dimensional model of the stepped beam has 
been developed in the finite element program Unigraphics NX4. The results found 
using the MATLAB program have been compared with those found with NX4 and 
satisfactory agreement between these results has been found. Additionally, the 
influence of varying the blade length on the natural frequencies has been investigated. 
 
2. Introduction 
 
In order to attain the highest possible power output in conditions of widely varying 
wind speed, a variable blade length is proposed. The basic concept of this variable 
blade length wind turbine is to attain higher energy capture in low wind conditions by 
increasing the blade length and to minimize mechanical loads in high wind conditions 
by decreasing the blade length [1]. The wind turbine blade consists of a fixed portion 
and a moveable blade portion which can be slid inside the fixed portion. 
 
Structural vibration problems present a major hazard and design limitation for a very 
wide range of engineering products. Vibration is important in wind turbines, because 
they are partially elastic structures, and they operate in an unsteady environment that 
tends to result in a vibrating response. The amplitude of the generated vibrations of a 
wind turbine blade depends on its stiffness [2]. One of the issues the variable blade 
length design presents to blade designers is that of structural dynamics. A wind 
turbine blade has certain characteristic natural frequencies and mode shapes which 
can be excited by mechanical or aerodynamic forces. This variable blade length 
design presents additional challenges because the stiffness and mass distribution 
change as the moveable blade portion slides in and out of the fixed blade portion.  
 
 2 
Manufacturers of wind turbines are interested in studying and verifying both edge-
wise and flap-wise vibrations (see Fig.1 below) of the turbine blade. The most visible 
and present source of excitation in a wind turbine system is the rotor.  
 The constant rotational speed is the first excitation frequency, mostly referred 
to as 1P. 
 The second excitation frequency is the rotor blade passing frequency: NbP in 
which Nb is the number of rotor blades: 2P for a turbine equipped with two 
rotor blades, 3P for a three bladed rotor, etc 
The structure should be designed such that its natural frequencies do not coincide with 
either 1P or NbP. Otherwise a resonance may occur in the whole structure of the 
turbine, leading to vibrations with increasing amplitude which may eventually destroy 
the whole wind turbine [3].  
 
 
 
Fig.1 Edge-wise and flap-wise vibrations of the blade [4] 
 
Dynamic analysis can be used to analyse and avoid the resonant vibration behaviour. 
Generally, research on the turbine blades focus on vibration frequencies and mode 
shapes. For simplification, a cantilevered beam can be used to replace the turbine 
blade [5]. Knowing the geometric shape and the material properties of the blade, the 
natural frequencies can be estimated using finite element analysis. 
 
The purpose of the present study is to calculate both flap-wise and edge-wise natural 
frequencies for ten configurations of a stepped beam which represent the blade with 
its outboard portion in ten different positions. In this way, the influence of varying 
blade length on the natural frequencies has been investigated. 
 
3. Methods 
 
The fixed portion and the moveable portion of the variable blade length wind turbine 
have been approximated respectively by a hollow beam and a solid beam which can 
be slid in and out as shown in Fig.2. 
 3 
 
Fig.2 Stepped beam 
 
The ten different configurations depending on the position of the second portion of 
the stepped beam are represented below. 
 
Fig.3 Ten configurations of stepped beam 
 
Finite Element Analysis 
 
In free vibration analysis, no loads are applied. The goal of the analysis is to 
determine at what frequencies a structure will vibrate if it is excited by a load that is 
applied suddenly and then removed. These frequencies are called natural frequencies 
and they are dependent on the fundamental characteristics of the structure, such as 
geometry, density and stiffness. These same characteristics can be included in a finite 
element model of a structural component. The finite element model can be used to 
determine the natural modes of vibration and corresponding frequencies. Once the 
geometry, density, and elastic material models have been defined for the finite 
element model, in the absence of damping, the dynamic character of the model can be 
expressed in matrix form as [6]: 
[K] {u i } =  
2
i [M] {u i }  (1) 
Here [K] is the stiffness matrix, [M] is the mass matrix, i is the natural frequency of 
vibration for a given mode and {u i } is the mode vector that expresses the 
corresponding mode shape. A finite element program uses iterative techniques to 
determine a set of frequencies and shapes that satisfy the finite element matrix 
equation. 
Portion1 
Portion2 
Configuration 5 
 
Configuration 1 
Configuration 2 
Configuration 3 
Configuration 4 
 
Configuration 6 
 
Configuration 7 
 
Configuration 8 
 
Configuration 9 
 
Configuration 10 
 
 4 
 
MATLAB 
 
A MATLAB program has been developed for a one dimensional model for the 
stepped beam. The geometry, material properties, vibration modes (flap-wise or edge-
wise), number of elements and configuration of the stepped beam have been made as 
selectable parameters which allow us to analyse blades with different sizes and 
properties. Both flap-wise and edge-wise natural frequencies have been calculated for 
the ten different configurations. 
 
NX4 
 
A three dimensional model of a stepped beam has been developed in the commercial 
finite element analysis program Unigraphics NX4. This model is designed to capture 
three-dimensional behaviour. The blade has been modelled as a cantilever, therefore it 
is fully constrained at the end of the inboard portion (where it is attached to the 
turbine shaft/hub). The outputs of the simulation are the natural frequencies of 
vibration: the flap-wise, edge-wise and torsional natural frequencies as well as their 
mode shapes. 
 
4. Results 
 
Values of the material and geometric properties of the two portion stepped beam 
under investigation are given in Table 1. 
 
Table 1 Material and geometric properties of the stepped beam 
 
 
 
Geometric properties 
Material properties 
(Carbon fiber composite) [7] 
 L(mm) W(mm) T(mm) Wh(mm) E(mN/mm^2)  (kg/mm^3) 
Portion1 1000 60 20 5 230 610  1.8 610  
Portion2 1000 50 10 N/A 230 610  1.8 610  
 
L: length 
W: width 
T: thickness 
Wh: wall thickness 
E: Young’s modulus 
 : density 
 
 
 NX4 results 
 
This section contains examples of the results obtained with NX4 for three different 
configurations of the stepped beam. The flap-wise, edge-wise and torsional 
deflections are represented. 
 
 
 
 
 
 
 5 
 
Mode 3 (Flap-wise)    Mode 5 (Edge-wise) 
    
 
Mode 6 (Torsional) 
 
Fig.4 Flap-wise, edge-wise and torsional deflection for configuration1 
 
Mode 4 (Flap-wise)    Mode 5 (Edge-wise) 
  
 
Mode 9 (Torsional) 
 
Fig.5 Flap-wise, edge-wise and torsional deflection for configuration5 
 
 
 
 6 
Mode 3 (Flap-wise)    Mode 8 (Edge-wise) 
  
 
Mode 10 (Torsional) 
 
Fig.6 Flap-wise, Edge-wise and Torsional deflection for configuration10 
 
 NX4 & MATLAB results comparison 
 
In this section, we compare the results found using the MATLAB program and NX4. 
The first nine natural frequencies (flap-wise and edge-wise) of the stepped beam are 
calculated successively for three different configurations. Torsional natural 
frequencies obtained with NX4 have been ignored due to the fact that the MATLAB 
program can only calculate flap-wise and edge-wise natural frequencies.  
 
 
Fig.7 MATLAB and NX4 results comparison 
 
 
 7 
 Influence of blade length variation 
 
The influence of varying the blade length has been studied and the results are shown 
in Fig.8 for the first five natural frequencies related to the configurations of the 
stepped beam. Table 2 provides values of these five first natural frequencies 
calculated for each configuration of the stepped beam shown in Fig. 3.  
 
Table 2. Computed natural frequencies (NX4)  
Configuration 
number 
Computed natural frequencies (Hz) 
Mode1 Mode2 Mode3 Mode4 Mode5 
1 36.5 109 228 638 675 
2 32.8 93.8 203 556 586 
3 29.4 82.1 174 421 510 
4 26.4 72.6 140 301 435 
5 23.6 64.6 107 249 364 
6 21.0 57.7 83.5 224 305 
7 18.6 51.7 67.6 205 259 
8 16.4 46.5 57.0 184 224 
9 14.5 41.8 49.9 162 198 
10 12.7 37.7 45.1 141 178 
 
 
Fig.8 Natural frequencies 
 
5. Discussion 
 
During design of a wind turbine blade, the 1st flap-wise, 2nd flap-
wise, 1st edge-wise and the 1st torsional natural frequencies shall be 
determined as a minimum [8]. It can be seen (Fig.7) that there is good agreement 
between the MATLAB and NX4 results for the first nine natural frequencies 
 in configuration1 : natural frequency 3000 Hz 
 in configuration5  : natural frequency 1300 Hz 
 in configuration10 : natural frequency 650 Hz 
It should be noted that only the frequency range between 0.5 Hz and 30 Hz is of 
relevance to wind turbine blades. In that range, the MATLAB and NX4 give the same 
results. Torsional natural frequencies have been calculated using NX4. The lowest 
torsional natural frequency (configuration10) determined is 677 Hz.  
 
 8 
The study of influence of blade length on natural frequencies represented in Fig.8 has 
shown that with increasing the blade length, the natural frequency decreases. This is 
probably due to the fact that the blade becomes more flexible as its length increases.  
The excitation loads are concentrated in the interval 0.5 Hz-30 Hz.  The natural 
frequencies included in the interval 12.7 Hz-36.5 Hz may coincide with these 
excitation frequencies. Therefore the first mode may be subjected to excitation. 
 
Finite element analysis has been used as a preliminary computational tool to precede 
an experimental modal analysis. As future work, experimental modal analysis will be 
performed on the blades being designed by a colleague. The experimental results will 
be compared with those obtained from the finite element analysis.  
 
6. Conclusions 
 
 Good agreement between results obtained with one dimensional model 
(MATLAB) and three dimensional model (NX4) has been found at lower 
frequencies, especially in the band of interest. 
 Results found in table 2 have shown that natural frequencies are function of 
configuration number.  
 Increasing blade length reduces the natural frequencies.   
 The excitation energy is concentrated in the frequency band of interest, 
typically in a range between 0.5 Hz to 30 Hz. Therefore, the first mode (flap-
wise natural frequency) may coincide with a forcing frequency.  
 The model includes some approximation. The result shows that further 
research with a more accurate model is needed. 
 
 
7. References 
[1] Pasupulati S., Wallace J. & Dawson M. 2005. Variable Length Blades Wind 
Turbine. IEEE Power Engineering Society General Meeting, Conference article 
(CA), Jun 12-16 2005, Pages 2097-2100. 
 
[2] Jureczko, M., Pawlak M. & Mężyk, A. 2005. Optimisation of wind turbine blades. 
Materials Processing Technology, 167(3-5): 463-471. August 30. 
 
[3] Burton, T., Sharpe, D., Jenkins, N. & Bossanyi, E. 2004. Wind Energy Handbook. 
Chichester. Wiley. 
 
[4] Grabau, P. & Petersensvej, H.C. 1999. Wind turbine with stress indicator. World 
International Property Organization, WO 99/57435 A1: 1-26. November 11. 
 
[5] Hansen, M.O.L., Sørensen, J.N., Voutsinas, S., Sørensen, N., Madsen, H.Aa. 
2006. State of the art in wind turbine aerodynamics and aeroelasticity. Aerospace 
sciences, 42: 285-330. December. 
 
[6] Mckittrick, L.R., Cairns, D.S., MandelI, J., Combs, D.C., Rabem, D.A. & 
VanLuchene, R.D. 2001. Analysis of a Composite Blade Design for the AOC 15/50 
Wind Turbine using a Finite Element Model.  SAND2001-1441. Sandia National 
Laboratories Contractor Report. May 2001. 
 
[7] Zweben, C.; Introduction to Mechanical Behavior and Properties of Composites 
Materials; DCDE, Volume 1; 1989. 
 
[8] Larsen, G.C., Hansen, M.H., Baumgart, A., Carlen, I. 2002. Modal Analysis of 
Wind Turbine Blades. Technical Report Risø-R-1181. Risø National Laboratory. 


Analysis of vibration of a variable blade length wind turbine

University of Johannesburg Institutional Repository

 1 Sixth South African Conference on Computational and Applied Mechanics SACAM08 Cape Town,  26-28 March 2008 SACAM  ANALYSIS OF VIBRATION OF A VARIABLE BLADE LENGTH WIND TURBINE  LAGOUGE TARTIBU  KILFOIL, M.  VAN DER MERWE, A.J.   Department of Mechanical Engineering, Cape Peninsula University of Technology, Box 652, Cape Town 8000, South Africa.  1. Abstract  In this paper, both flap-wise and edge-wise natural frequencies of vibration of a variable blade length wind turbine are calculated.  A stepped beam with two portions has been used to approximate the blade. The two portions of the blade have been approximated by a hollow beam and a solid beam with rectangular cross section. As the outboard portion of the blade can be slid in and out of the inboard portion, ten different configurations of the stepped beam have been investigated. A MATLAB program has been developed for a finite element analysis of a one dimensional model of the stepped beam. Subsequently a three dimensional model of the stepped beam has been developed in the finite element program Unigraphics NX4. The results found using the MATLAB program have been compared with those found with NX4 and satisfactory agreement between these results has been found. Additionally, the influence of varying the blade length on the natural frequencies has been investigated.  2. Introduction  In order to attain the highest possible power output in conditions of widely varying wind speed, a variable blade length is proposed. The basic concept of this variable blade length wind turbine is to attain higher energy capture in low wind conditions by increasing the blade length and to minimize mechanical loads in high wind conditions by decreasing the blade length [1]. The wind turbine blade consists of a fixed portion and a moveable blade portion which can be slid inside the fixed portion.  Structural vibration problems present a major hazard and design limitation for a very wide range of engineering products. Vibration is important in wind turbines, because they are partially elastic structures, and they operate in an unsteady environment that tends to result in a vibrating response. The amplitude of the generated vibrations of a wind turbine blade depends on its stiffness [2]. One of the issues the variable blade length design presents to blade designers is that of structural dynamics. A wind turbine blade has certain characteristic natural frequencies and mode shapes which can be excited by mechanical or aerodynamic forces. This variable blade length design presents additional challenges because the stiffness and mass distribution change as the moveable blade portion slides in and out of the fixed blade portion.    2 Manufacturers of wind turbines are interested in studying and verifying both edge-wise and flap-wise vibrations (see Fig.1 below) of the turbine blade. The most visible and present source of excitation in a wind turbine system is the rotor.   The constant rotational speed is the first excitation frequency, mostly referred to as 1P.  The second excitation frequency is the rotor blade passing frequency: NbP in which Nb is the number of rotor blades: 2P for a turbine equipped with two rotor blades, 3P for a three bladed rotor, etc The structure should be designed such that its natural frequencies do not coincide with either 1P or NbP. Otherwise a resonance may occur in the whole structure of the turbine, leading to vibrations with increasing amplitude which may eventually destroy the whole wind turbine [3].     Fig.1 Edge-wise and flap-wise vibrations of the blade [4]  Dynamic analysis can be used to analyse and avoid the resonant vibration behaviour. Generally, research on the turbine blades focus on vibration frequencies and mode shapes. For simplification, a cantilevered beam can be used to replace the turbine blade [5]. Knowing the geometric shape and the material properties of the blade, the natural frequencies can be estimated using finite element analysis.  The purpose of the present study is to calculate both flap-wise and edge-wise natural frequencies for ten configurations of a stepped beam which represent the blade with its outboard portion in ten different positions. In this way, the influence of varying blade length on the natural frequencies has been investigated.  3. Methods  The fixed portion and the moveable portion of the variable blade length wind turbine have been approximated respectively by a hollow beam and a solid beam which can be slid in and out as shown in Fig.2.  3  Fig.2 Stepped beam  The ten different configurations depending on the position of the second portion of the stepped beam are represented below.  Fig.3 Ten configurations of stepped beam  Finite Element Analysis  In free vibration analysis, no loads are applied. The goal of the analysis is to determine at what frequencies a structure will vibrate if it is excited by a load that is applied suddenly and then removed. These frequencies are called natural frequencies and they are dependent on the fundamental characteristics of the structure, such as geometry, density and stiffness. These same characteristics can be included in a finite element model of a structural component. The finite element model can be used to determine the natural modes of vibration and corresponding frequencies. Once the geometry, density, and elastic material models have been defined for the finite element model, in the absence of damping, the dynamic character of the model can be expressed in matrix form as [6]: [K] {u i } =  2i [M] {u i }  (1) Here [K] is the stiffness matrix, [M] is the mass matrix, i is the natural frequency of vibration for a given mode and {u i } is the mode vector that expresses the corresponding mode shape. A finite element program uses iterative techniques to determine a set of frequencies and shapes that satisfy the finite element matrix equation. Portion1 Portion2 Configuration 5  Configuration 1 Configuration 2 Configuration 3 Configuration 4  Configuration 6  Configuration 7  Configuration 8  Configuration 9  Configuration 10   4  MATLAB  A MATLAB program has been developed for a one dimensional model for the stepped beam. The geometry, material properties, vibration modes (flap-wise or edge-wise), number of elements and configuration of the stepped beam have been made as selectable parameters which allow us to analyse blades with different sizes and properties. Both flap-wise and edge-wise natural frequencies have been calculated for the ten different configurations.  NX4  A three dimensional model of a stepped beam has been developed in the commercial finite element analysis program Unigraphics NX4. This model is designed to capture three-dimensional behaviour. The blade has been modelled as a cantilever, therefore it is fully constrained at the end of the inboard portion (where it is attached to the turbine shaft/hub). The outputs of the simulation are the natural frequencies of vibration: the flap-wise, edge-wise and torsional natural frequencies as well as their mode shapes.  4. Results  Values of the material and geometric properties of the two portion stepped beam under investigation are given in Table 1.  Table 1 Material and geometric properties of the stepped beam    Geometric properties Material properties (Carbon fiber composite) [7]  L(mm) W(mm) T(mm) Wh(mm) E(mN/mm^2)  (kg/mm^3) Portion1 1000 60 20 5 230 610  1.8 610  Portion2 1000 50 10 N/A 230 610  1.8 610   L: length W: width T: thickness Wh: wall thickness E: Young’s modulus  : density    NX4 results  This section contains examples of the results obtained with NX4 for three different configurations of the stepped beam. The flap-wise, edge-wise and torsional deflections are represented.        5  Mode 3 (Flap-wise)    Mode 5 (Edge-wise)      Mode 6 (Torsional)  Fig.4 Flap-wise, edge-wise and torsional deflection for configuration1  Mode 4 (Flap-wise)    Mode 5 (Edge-wise)    Mode 9 (Torsional)  Fig.5 Flap-wise, edge-wise and torsional deflection for configuration5     6 Mode 3 (Flap-wise)    Mode 8 (Edge-wise)    Mode 10 (Torsional)  Fig.6 Flap-wise, Edge-wise and Torsional deflection for configuration10   NX4 & MATLAB results comparison  In this section, we compare the results found using the MATLAB program and NX4. The first nine natural frequencies (flap-wise and edge-wise) of the stepped beam are calculated successively for three different configurations. Torsional natural frequencies obtained with NX4 have been ignored due to the fact that the MATLAB program can only calculate flap-wise and edge-wise natural frequencies.    Fig.7 MATLAB and NX4 results comparison    7  Influence of blade length variation  The influence of varying the blade length has been studied and the results are shown in Fig.8 for the first five natural frequencies related to the configurations of the stepped beam. Table 2 provides values of these five first natural frequencies calculated for each configuration of the stepped beam shown in Fig. 3.   Table 2. Computed natural frequencies (NX4)  Configuration number Computed natural frequencies (Hz) Mode1 Mode2 Mode3 Mode4 Mode5 1 36.5 109 228 638 675 2 32.8 93.8 203 556 586 3 29.4 82.1 174 421 510 4 26.4 72.6 140 301 435 5 23.6 64.6 107 249 364 6 21.0 57.7 83.5 224 305 7 18.6 51.7 67.6 205 259 8 16.4 46.5 57.0 184 224 9 14.5 41.8 49.9 162 198 10 12.7 37.7 45.1 141 178   Fig.8 Natural frequencies  5. Discussion  During design of a wind turbine blade, the 1st flap-wise, 2nd flap-wise, 1st edge-wise and the 1st torsional natural frequencies shall be determined as a minimum [8]. It can be seen (Fig.7) that there is good agreement between the MATLAB and NX4 results for the first nine natural frequencies  in configuration1 : natural frequency 3000 Hz  in configuration5  : natural frequency 1300 Hz  in configuration10 : natural frequency 650 Hz It should be noted that only the frequency range between 0.5 Hz and 30 Hz is of relevance to wind turbine blades. In that range, the MATLAB and NX4 give the same results. Torsional natural frequencies have been calculated using NX4. The lowest torsional natural frequency (configuration10) determined is 677 Hz.    8 The study of influence of blade length on natural frequencies represented in Fig.8 has shown that with increasing the blade length, the natural frequency decreases. This is probably due to the fact that the blade becomes more flexible as its length increases.  The excitation loads are concentrated in the interval 0.5 Hz-30 Hz.  The natural frequencies included in the interval 12.7 Hz-36.5 Hz may coincide with these excitation frequencies. Therefore the first mode may be subjected to excitation.  Finite element analysis has been used as a preliminary computational tool to precede an experimental modal analysis. As future work, experimental modal analysis will be performed on the blades being designed by a colleague. The experimental results will be compared with those obtained from the finite element analysis.   6. Conclusions   Good agreement between results obtained with one dimensional model (MATLAB) and three dimensional model (NX4) has been found at lower frequencies, especially in the band of interest.  Results found in table 2 have shown that natural frequencies are function of configuration number.   Increasing blade length reduces the natural frequencies.    The excitation energy is concentrated in the frequency band of interest, typically in a range between 0.5 Hz to 30 Hz. Therefore, the first mode (flap-wise natural frequency) may coincide with a forcing frequency.   The model includes some approximation. The result shows that further research with a more accurate model is needed.   7. References [1] Pasupulati S., Wallace J. & Dawson M. 2005. Variable Length Blades Wind Turbine. IEEE Power Engineering Society General Meeting, Conference article (CA), Jun 12-16 2005, Pages 2097-2100.  [2] Jureczko, M., Pawlak M. & Mężyk, A. 2005. Optimisation of wind turbine blades. Materials Processing Technology, 167(3-5): 463-471. August 30.  [3] Burton, T., Sharpe, D., Jenkins, N. & Bossanyi, E. 2004. Wind Energy Handbook. Chichester. Wiley.  [4] Grabau, P. & Petersensvej, H.C. 1999. Wind turbine with stress indicator. World International Property Organization, WO 99/57435 A1: 1-26. November 11.  [5] Hansen, M.O.L., Sørensen, J.N., Voutsinas, S., Sørensen, N., Madsen, H.Aa. 2006. State of the art in wind turbine aerodynamics and aeroelasticity. Aerospace sciences, 42: 285-330. December.  [6] Mckittrick, L.R., Cairns, D.S., MandelI, J., Combs, D.C., Rabem, D.A. & VanLuchene, R.D. 2001. Analysis of a Composite Blade Design for the AOC 15/50 Wind Turbine using a Finite Element Model.  SAND2001-1441. Sandia National Laboratories Contractor Report. May 2001.  [7] Zweben, C.; Introduction to Mechanical Behavior and Properties of Composites Materials; DCDE, Volume 1; 1989.  [8] Larsen, G.C., Hansen, M.H., Baumgart, A., Carlen, I. 2002. Modal Analysis of Wind Turbine Blades. Technical Report Risø-R-1181. Risø National Laboratory. 

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Analysis of vibration of a variable blade length wind turbine

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