Fundamental natural frequencies investigation for a typical 5-MW wind turbine blade

Abstract

Continuous structures such as beams, rods and plates can be modelled by discrete mass and stiffness parameters and analysed as multi-degree-of-freedom systems. The analysis of structural vibration is necessary to obtain the natural frequencies of a structure and the response to the external excitation. In this way, it can be determined whether a particular structure will fulfil its intended function and, in addition, the results of the dynamic loadings acting on a structure can be predicted. The lack of a sober analytical research about the vibrational behaviour of the 5-MW wind turbine blade pushed us to investigate about this crucial issue, however, most of the discreet researches are concerned with the aerodynamic effects rather than structural analysis. In this article, Rayleigh–Ritz method was implemented for a typical 5-MW wind turbine blade. MATLAB codes were developed and natural frequencies were obtained for both flapwise and edgewise vibrational behaviour. A good agreement was observed between the analytical results and the manufacturer results.

Keywords

Rayleigh–Ritz method vibration multi-degrees-of-freedom systems natural frequency modal analysis wind turbine

Introduction

The greater challenge for the future generations is the clean climate; one of the top promising alternatives is the use of pollution-free sources like the wind energy. The 5-MW wind turbine is considered one of the most commonly used, reliable and secure huge machine and has been on the market since 2005 for both onshore and offshore power generation source.¹ It is extremely important to investigate about the vibrational behaviour of the wind turbine blade. The fundamental natural frequencies for a 5-MW wind turbine blade were extracted and the results compared by a numerical solver FAST showing good agreement.² A novel computational tool was developed for the aeroelastic analysis of a National Renewable Energy Laboratory (NREL) 5-MW wind turbine blades with an alternative blade configurations to mitigate vibration and improve fatigue performance.³ The influence of vibrations on the aerodynamic loading on the blade of a 5-MW wind turbine was investigated using the blade element-momentum and the Beddoes–Leishman (B–L) dynamic stall model.⁴ A new approach to study and analyse the stresses and deformations of a 5-MW wind turbine blade under the steady-state condition has been provided, the three-dimensional model of blade was created using SOLIDWORKS and simulated numerically by ANSYS/Workbench.⁵ The effect of rotor position and weight adjustment on the vibration behaviour of the drive-train system has been addressed for a 5-MW direct-drive wind turbine by considering the unbalanced magnetic pull force.⁶

A variable length wind turbine blade natural frequencies were investigated and identified to avoid resonance both numerically and analytically.⁷ Rayleigh’s method has been implemented to track the resonance zone⁸ and the authors also studied the higher modes by finite element (FE) method. A study conducted by Ulriksen et al.⁹ proposed modal and wavelet analysis–based damage identification method to a 34-m wind turbine blade as a trailing-edge deboning was introduced. A modelling study on ultrasonic guided waves propagating in composite blades was used to determine the optimal frequency and location of the transducers for ensuring wave propagation, causing the required level of energy concentration and resulting shear stress across the leading edge of the turbine’s blade.¹⁰ An experimental analysis of the LM19-m blade has been compared with results from a FE-modelling of the same blade. For some of the higher modes, substantial discrepancies between the natural frequencies originating from the FE-modelling and the modal analysis, respectively, are observed. The aeroelastic response and performance of a 5-MW class rotating composite blade model were examined by numerically advanced coupled method based on computational fluid dynamics (CFD) and the computational flexible multi-body dynamics (CFMBD).¹¹ The basic dynamic characteristics of the 5-MW offshore wind energy plants (OWEP) and its environment by means of various measured signals had been discussed in Kraemer and Fritzen.¹² In this article, a further understanding for the implementation of Rayleigh–Ritz analytical method to extract the fundamental natural frequencies for both flapping and edgewise directions for a typical 5-MW wind turbine blade.

Description of the model

The wind turbine blade was modelled using ANSYS. The same blade will be analysed analytically by Rayleigh–Ritz method. A 5-MW NREL wind turbine blade was selected for this study, due to availability of required geometric design parameters and experimental data for verification. The blade is 61.5 m long, the geometric design parameters like mass, flexural rigidity, flapwise and edgewise stiffness with other properties are given in Appendix A.¹³ The x-axis of the blade is coinciding with the longitudinal axis of the blade as shown in Figure 1. In this work, axial strains are negligible because any change in length will be a small fraction of the original length. This will imply that the beam is inextensible. Also, it implies that the Bernoulli–Euler theorem is valid, which states that the curvature of the beam is proportional to the bending moment.¹⁴

Figure 1.

Configuration of the modelled wind turbine blade.

Boundary conditions

The wind turbine blade is treated as a cantilever beam with following boundary conditions.

At fixed (clamped) end, both deflection and slope are zero

y (0) = \frac{d y (0)}{d x} = 0

(1)

At its free end, bending moment and shear force are zero

E I \frac{d^{2} y}{d x^{2}} (L) = E I \frac{d^{3} y}{d x^{3}} (L) = 0

(2)

Rayleigh–Ritz method

The Rayleigh–Ritz method enables the researcher to reduce an infinite number of degrees-of-freedom (DOFs) of a system into a finite number, which makes analysis possible and easier. The method relies on the approximation of the possible deformation shapes of the system, following the basic idea of Rayleigh’s principle which approximates a continuous system by an equivalent single degree-of-freedom (SDOF) system via assuming a single deformation shape. The number of DOF is equal to the number of Rayleigh–Ritz modes chosen. The method can be used to determine the natural frequencies and dynamic response of beams. The configuration of the modelled wind turbine blade is illustrated in Figure 1.

Consideration of the energy in a dynamic system together with the use of the Lagrange equation is a very powerful method of analysis for certain physically complex systems.¹⁵ This is an energy method that allows the equations of motion to be written in terms of any set of generalized coordinates. Generalized coordinates are a set of independent parameters that completely specify the system location and they are independent of each other. The fundamental equation of motion for a multi-DOF can be derived from Lagrange’s equation¹⁶ and can be written in terms of the generalized coordinates that describes the position of the system $q_{i}$ as follows

\frac{d}{d t} (\frac{\partial (T)}{\partial {\dot{q}}_{i}}) - \frac{\partial (T)}{\partial q_{i}} + \frac{\partial (U)}{\partial q_{i}} + \frac{\partial (D E)}{\partial {\dot{q}}_{i}} = Q_{i}

(3)

where $T$ is the total kinetic energy of the system, $U$ is the total strain energy of the system, $D E$ is the energy dissipating function and $Q_{i}$ is the generalized external force acting on the system. The subscript $i$ varies from 1 to $n$ denoting $n$ equations for a $n$ degree of freedom system, so that the Lagrange equation yields as many equations of motion as the number of degrees of freedom.

For a conservative system, $Q_{i}$ and $D E$ are both zero, so that

\frac{d}{d t} (\frac{\partial (T)}{\partial {\dot{q}}_{i}}) - \frac{\partial (T)}{\partial q_{i}} + \frac{\partial (U)}{\partial q_{i}} = 0

(4)

When dealing with free vibration under small displacements, it may be that $\partial (T) / \partial q_{i} = 0$ . The general Lagrange equation for free vibration can be written as

\frac{d}{d t} (\frac{\partial (T)}{\partial {\dot{q}}_{i}}) + \frac{\partial (U)}{\partial q_{i}} = 0 (i = 1, 2, 3, \dots, n)

(5)

where $q_{i}$ is the generalized coordinate of the system which has the same number of DOFs of the system.

Rotary inertia and shear effects must be taken into account in the analysis of high-frequency vibration of all beams.¹⁵ For this study, rotary inertia effects can be ignored due to low rotational velocity of the wind turbine as it is rated as 12 r/min.¹³ Therefore, the kinetic energy of the beam is given by

T = \frac{1}{2} \int_{0}^{L} m (x) \cdot {\dot{v}}^{2} (x, t) d x

(6)

By neglecting shear deformation, strain energy of the blade will be

U = \frac{1}{2} \int_{0}^{L} E I (x) . {(\frac{d^{2} v (x, t)}{d x^{2}})}^{2} d x

(7)

where $L$ is the length of the blade, $E I (x)$ is the flexural rigidity and $v (x, t)$ is the displacement function of the blade that satisfies the boundary conditions and can be written as

v (x, t) = \sum_{i = 1}^{n} V_{i} (x) q_{i} (t)

(8)

where $V_{i} (x)$ is the displacement function that satisfies the geometric boundary conditions. By introducing equation (8) into equations (6) and (7), following quadratic forms for energy can be obtained

T = \frac{1}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} m_{i j} \cdot {\dot{q}}_{i} . \cdot {\dot{q}}_{j}

(9)

U = \frac{1}{2} \sum_{i = 1}^{n} \sum_{j = 1}^{n} k_{i j} \cdot q_{i} \cdot q_{j}

(10)

where $m_{i j}$ and $k_{i j}$ are the generalized mass and stiffness matrices of the system, respectively, and are given by

m_{i j} = \int_{0}^{L} m (x) \cdot V_{i} (x) \cdot V_{j} (x) d x

(11)

k_{i j} = \int_{0}^{L} E I (x) \cdot \frac{d^{2} V_{i} (x)}{d x^{2}} \cdot \frac{d^{2} V_{j} (x)}{d x^{2}} d x

(12)

Substituting equations (9) and (10) into Lagrange’s equation (5), one can obtain the total differential equation of the free vibration

\sum_{j = 1}^{n} m_{i j} \cdot {\overset{\cdot\cdot}{q}}_{j} + \sum_{j = 1}^{n} k_{i j} \cdot q_{j} = 0 (i = 1, 2, 3, \dots, n)

(13)

Assuming solution for equation in the form

q_{i} (t) = {\bar{q}}_{i} . s i n (ω t +) (i = 1, 2, 3, \dots, n)

(14)

where ${\bar{q}}_{i}$ is the amplitude of the displacement, $ω$ is the frequency and ά is the phase angle. Substituting equation (14) into equation (13) lead to a set of characteristic equations

\sum_{j = 1}^{n} (k_{i j} - m_{i j} . ω^{2}) \cdot {\bar{q}}_{j} = 0 (i = 1, 2, 3, \dots, n)

(15)

Equation (15) contains a set of characteristic equations and can be solved for $n$ discrete approximate values of $ω^{2}$ ; these approximate values $ω_{1}^{2}, ω_{2}^{2}, \dots, ω_{n}^{2}$ have the following property

λ_{1}^{2} \leq ω_{1}^{2}, λ_{2}^{2} \leq ω_{2}^{2}, \dots, λ_{n}^{2} \leq ω_{n}^{2}

(16)

where $λ_{1}^{2}, λ_{2}^{2}, \dots, λ_{n}^{2}$ , are the smallest exact Eigen values of the original problem.

Associated with each Eigen value is a set of values of ${\bar{q}}_{1}, {\bar{q}}_{2}, \dots, {\bar{q}}_{n}$ . These ${\bar{q}}_{i}$ values are not determined absolutely; only their ratio may be obtained and the rth mode shape is given by

Ø_{r} (x) = \sum_{i = 1}^{n} V (x) . {\bar{q}}_{i}^{(r)} (r = 1, 2, \dots, n)

(17)

The characteristic equations given by equation (17) can be easily put into a form suitable for numerical calculation, using matrix notation. If the beam is divided into $n$ spanwise stations, the mass and stiffness terms given by equations (13) and (14) can be written in the following matrix forms

[m_{i j}] = [V] [m] [W] {[V]}^{T}

(18)

where $[W]$ is a matrix of weighting coefficients introduced to transform the definite integrals such as equations (8)–(12) into the form of equations (18) and (19). It depends on the value of number $n$ and the numerical techniques used for the integration

[k_{i j}] = [\frac{d^{2} V}{d x^{2}}] [E I] [W] {[\frac{d^{2} V}{d x^{2}}]}^{T}

(19)

[V] = [\begin{matrix} V_{1} (x_{1}) & \dots & V_{1} (x_{n}) \\ ⋮ & ⋱ & ⋮ \\ V_{n} (x_{1}) & \dots & V_{n} (x_{n}) \end{matrix}]

(20)

and ${[V]}^{T}$ is the transpose matrix of $[V]$ , rewriting equation (15) in matrix form yields

ω^{2} [m_{i j}] {\bar{q}} = [k_{i j}] {\bar{q}}

(21)

By substituting equations (18) and (19) into equation (21) we get

\begin{array}{l} ω^{2} [V] [m] [W] {[V]}^{T} {\bar{q}} = [\frac{d^{2} V}{d x^{2}}] \\ [E I] [W] {[\frac{d^{2} V}{d x^{2}}]}^{T} {\bar{q}} \end{array}

(22)

The complete solution of the problem of free vibration of any system would require the determination of all the natural frequencies and of the mode shape associated with each. In practice, it is often necessary to know only a few of the natural frequencies, and sometimes only one. Usually the lowest frequencies are the most important.¹⁷

Since the wind turbine blade is a cantilever beam, the following polynomial expression will be used for the mode shapes functions

\begin{array}{l} V_{j} (x) = \frac{1}{6} (j + 2) (j + 3) {(\frac{x}{L})}^{j + 1} - \frac{1}{3} j (j + 3) {(\frac{x}{L})}^{j + 2} + \\ \frac{1}{6} j (j + 1) {(\frac{x}{L})}^{j + 3} (j = 1, 2, \dots, n) \end{array}

(23)

The 61.5-m long wind turbine blade is divided into 50 equal segments, 51 spanwise stations will be created with each of 1.23 m starting from the root to the tip of the blade as shown in Figure 2.

Figure 2.

The modelled blade divided into 50 equal segments.

Flapwise and edgewise vibrations

Two main types of vibration occur in wind turbine blades, flapwise and edgewise. Figure 3 shows flapwise direction (F–F) and edgewise direction (E–E). The first flapwise natural bending frequency of the blade can resonate with the wind turbine tower first bending frequency whereas the second natural bending frequency of the blade is in most cases so high that it no longer interferes.¹⁸ The second natural frequency of interest is therefore the first edgewise natural frequency.

Figure 3.

Flapwise (F–F) and edgewise (E–E) vibration directions of a typical wind turbine blade.

By assuming two mode shape functions, the lateral displacement matrix equation is given by

{Ø} = {V_{1}} {\bar{q}}_{1} + {V_{2}} {\bar{q}}_{2}

(24)

where ${\bar{q}}_{1}$ and ${\bar{q}}_{2}$ . are generalized coordinates representing the contributions of each of the functions, ${Ø}$ , while ${V_{1}}$ and ${V_{2}}$ are the column matrices with 51 elements represents values of $V_{1} (x)$ and $V_{2} (x)$ at 51 spanwise stations, respectively

V_{1} (x) = 2 {(\frac{x}{L})}^{2} - \frac{4}{3} {(\frac{x}{L})}^{3} + \frac{1}{3} {(\frac{x}{L})}^{4}

(25)

V_{2} (x) = \frac{10}{3} {(\frac{x}{L})}^{3} - \frac{10}{3} {(\frac{x}{L})}^{4} + {(\frac{x}{L})}^{5}

(26)

where $V_{1} (x)$ and $V_{2} (x)$ are the displacement functions that satisfy the geometrical boundary conditions for a cantilever beam. They are from equation (23) for j values of 1 and 2.

The displacement second-derivative matrix $[d^{2} V / d x^{2}]$ is obtained by differentiating equations (25) and (26) twice with respect to x and evaluating the results for 51 spanwise stations. Results of differentiating are given by

[\frac{d^{2} V_{1} (x)}{d x^{2}}] = \frac{1}{L^{2}} (4 - 8 (\frac{x}{L}) + 4 {(\frac{x}{L})}^{2})

(27)

[\frac{d^{2} V_{2} (x)}{d x^{2}}] = \frac{1}{L^{2}} (20 (\frac{x}{L}) - 40 {(\frac{x}{L})}^{2} + 20 {(\frac{x}{L})}^{3})

(28)

Mass density of the wind turbine blade and both the flapwise and edgewise flexural rigidity are given in Appendix A.¹³ The matrix of weighting coefficients $[W],$ depends on the value of number $n$ which is 50 equal segments, according to Simpson’s rule

\int_{a}^{b} f (x) d x \approx \frac{d}{3} (\begin{matrix} f_{o} + 4 f_{1} + 2 f_{2} + 4 f_{3} + 2 f_{4} + \dots \\ + 2 f_{n - 2} + 4 f_{n - 1} + f_{n} \end{matrix})

(29)

where (a–b) is the length of the blade which equals to 61.5 m with $d$ equals each interval so $d = 61.5 / 50 = 1.23 m$ and $d / 3 = 0.41 m$ .

Results

Using MATLAB, equation (22) was solved for first two flapwise and edgewise natural frequencies. Results are shown in Table 1.

Table 1.

Natural frequencies comparison for two modes.

Mode number	Rayleigh–Ritz results (Hz)	Flapwise results¹³ (Hz)	Relative error (flapwise; %)	Rayleigh–Ritz edgewise (Hz)	Edgewise results¹³ (Hz)	Relative error (edgewise; %)
1	1.05	0.70	50	1.33	1.09	22
2	5.83	1.93	202.1	4.72	2.02	133.7

Big error is noticed by comparing analytical results to manufacturer results,¹³ especially for the second flapping frequency. Less errors are found for the first two edgewise frequencies. By plugging the three mode shape functions and solving it again, results are improving. Same procedure was followed by assuming higher mode shapes until reaching sixth mode shapes given by

V_{6} (x) = 12 {(\frac{x}{L})}^{7} - 18 {(\frac{x}{L})}^{8} + 7 {(\frac{x}{L})}^{9}

(30)

And its second derivatives with respect to $x$ are given as

[\frac{d^{2} V_{6} (x)}{d x^{2}}] = \frac{1}{L^{2}} (\begin{array}{l} 504 {(\frac{x}{L})}^{5} - 1008 {(\frac{x}{L})}^{6} + \\ 504 {(\frac{x}{L})}^{7} \end{array})

(31)

Natural frequencies converge towards the manufacturer results and errors have been decreased considerably. Results for natural frequencies after applying sixth mode shape are given in Table 2.

Table 2.

Natural frequencies comparison for six mode shapes.

Mode number	Rayleigh–Ritz flapwise results (Hz)	Flapwise results¹³ (Hz)	Relative error (flapwise; %)	Rayleigh–Ritz edgewise results (Hz)	Edgewise results¹³ (Hz)	Relative error (edgewise; %)
1	0.75	0.70	7.1	1.12	1.09	2.6
2	1.95	1.93	1	3.31	2.02	63.9
3	5.21	N/A	–	7.65	N/A	–
4	12.83	N/A	–	16.13	N/A	–
5	37.24	N/A	–	34.37	N/A	–
6	96.13	N/A		150.88

It is clear from the results of Rayleigh–Ritz method that convergence of the fundamental natural frequencies of the modelled wind turbine blade been achieved after the fifth mode shape function for both the flapwise and edgewise vibrations. It is noticed that the first natural frequency is converging steadily while this is not the case for higher mode natural frequencies. This is mainly because Rayleigh–Ritz method is an extension of Rayleigh’s principle which represents a technique for estimating the lowest Eigenvalue, that is, the first natural frequency.¹⁹ It is also noticed that the relative error has reduced significantly for the second flapping mode but not for the second edging mode this is mainly referred to the designed structural properties of the wind turbine blade stiffness which has much higher values for edgewise as compared with flapwise stiffness as shown in Appendix A, in other words, the deflection in the edgewise direction is limited as compared with that in the flapwise direction, moreover, the second edgewise mode is in fact the fourth natural frequency and as mentioned earlier the Rayleigh–Ritz method has superior accuracy for the lowest fundamental natural frequencies.

Discussion of the analytical analysis

The Rayleigh–Ritz method has been implemented to perform analytical analysis for the wind turbine blade. Properties of the blade were provided by NREL, a renowned wind turbine manufacturer. A MATLAB code was developed for defining both flapwise and edgewise natural frequencies for the mode shapes. Convergence of natural frequencies was achieved by six mode shapes. Relative Rayleigh–Ritz method errors as compared with the manufacturer results for clean blade were 7% and 1% for the first and second flapwise frequencies, respectively, and less than 3% for the first edgewise frequency.

Conclusion

This article started with a description of the wind turbine model. The Rayleigh–Ritz method was then introduced together with all the key equations and matrices. The solution procedure was started by two mode shapes. Mass and stiffness matrices were established along with other relevant matrices according to the properties provided by the blade manufacture. The matrices were then fed into MATLAB programme code developed for Rayleigh–Ritz method. The results of natural frequencies were compared with results obtained by the manufacturer. The procedure was repeated by increasing the number of the mode shapes until convergence has achieved by five mode shapes, which was for the clean blade. Results of analytical analysis for the three vibration modes agree well with the manufacturer results knowing that the Rayleigh–Ritz method is most suitable for the lower modes, namely the first mode.

Footnotes

Appendix

Appendix A.

Wind Turbine Blade Structural Properties [14].

Radius	BlFract	AeroCent	StrcTwst	BMassDen	FlpStff	EdgStff	GJStff	EAStff	Alpha	FlpIner	EdgIner	PrecrvRef	PreswpRef	FlpcgOf	EdgcgOf	FlpEAOf	EdgEAOf
(m)	(-)	(-)	(°)	(kg/m)	(N•m2)	(N•m2)	(N•m2)	(N)	(-)	(kg•m)	(kg•m)	(m)	(m)	(m)	(m)	(m)	(m)
1.50	0.00000	0.25000	13.308	678.935	18110.00E+6	18113.60E+6	5564.40E+6	9729.48E+6	0.0	972.86	973.04	0.0	0.0	0.0	0.00017	0.0	0.0
1.70	0.00325	0.25000	13.308	678.935	18110.00E+6	18113.60E+6	5564.40E+6	9729.48E+6	0.0	972.86	973.04	0.0	0.0	0.0	0.00017	0.0	0.0
2.70	0.01951	0.24951	13.308	773.363	19424.90E+6	19558.60E+6	5431.59E+6	10789.50E+6	0.0	1091.52	1066.38	0.0	0.0	0.0	−0.02309	0.0	0.0
3.70	0.03577	0.24510	13.308	740.550	17455.90E+6	19497.80E+6	4993.98E+6	10067.23E+6	0.0	966.09	1047.36	0.0	0.0	0.0	0.00344	0.0	0.0
4.70	0.05203	0.23284	13.308	740.042	15287.40E+6	19788.80E+6	4666.59E+6	9867.78E+6	0.0	873.81	1099.75	0.0	0.0	0.0	0.04345	0.0	0.0
5.70	0.06829	0.22059	13.308	592.496	10782.40E+6	14858.50E+6	3474.71E+6	7607.86E+6	0.0	648.55	873.02	0.0	0.0	0.0	0.05893	0.0	0.0
6.70	0.08455	0.20833	13.308	450.275	7229.72E+6	10220.60E+6	2323.54E+6	5491.26E+6	0.0	456.76	641.49	0.0	0.0	0.0	0.06494	0.0	0.0
7.70	0.10081	0.19608	13.308	424.054	6309.54E+6	9144.70E+6	1907.87E+6	4971.30E+6	0.0	400.53	593.73	0.0	0.0	0.0	0.07718	0.0	0.0
8.70	0.11707	0.18382	13.308	400.638	5528.36E+6	8063.16E+6	1570.36E+6	4493.95E+6	0.0	351.61	547.18	0.0	0.0	0.0	0.08394	0.0	0.0
9.70	0.13335	0.17156	13.308	382.062	4980.06E+6	6884.44E+6	1158.26E+6	4034.80E+6	0.0	316.12	490.84	0.0	0.0	0.0	0.10174	0.0	0.0
10.70	0.14959	0.15931	13.308	399.655	4936.84E+6	7009.18E+6	1002.12E+6	4037.29E+6	0.0	303.60	503.86	0.0	0.0	0.0	0.10758	0.0	0.0
11.70	0.16585	0.14706	13.308	426.321	4691.66E+6	7167.68E+6	855.90E+6	4169.72E+6	0.0	289.24	544.70	0.0	0.0	0.0	0.15829	0.0	0.0
12.70	0.18211	0.13481	13.181	416.820	3949.46E+6	7271.66E+6	672.27E+6	4082.35E+6	0.0	246.57	569.90	0.0	0.0	0.0	0.22235	0.0	0.0
13.70	0.19837	0.12500	12.848	406.186	3386.52E+6	7081.70E+6	547.49E+6	4085.97E+6	0.0	215.91	601.28	0.0	0.0	0.0	0.30756	0.0	0.0
14.70	0.21465	0.12500	12.192	381.420	2933.74E+6	6244.53E+6	448.84E+6	3668.34E+6	0.0	187.11	546.56	0.0	0.0	0.0	0.30386	0.0	0.0
15.70	0.23089	0.12500	11.561	352.822	2568.96E+6	5048.96E+6	335.92E+6	3147.76E+6	0.0	160.84	468.71	0.0	0.0	0.0	0.26519	0.0	0.0
16.70	0.24715	0.12500	11.072	349.477	2388.65E+6	4948.49E+6	311.35E+6	3011.58E+6	0.0	148.56	453.76	0.0	0.0	0.0	0.25941	0.0	0.0
17.70	0.26341	0.12500	10.792	346.538	2271.99E+6	4808.02E+6	291.94E+6	2882.62E+6	0.0	140.30	436.22	0.0	0.0	0.0	0.25007	0.0	0.0
19.70	0.29595	0.12500	10.232	339.333	2050.05E+6	4501.40E+6	261.00E+6	2613.97E+6	0.0	124.61	398.18	0.0	0.0	0.0	0.23155	0.0	0.0
21.70	0.32846	0.12500	9.672	330.004	1828.25E+6	4244.07E+6	228.82E+6	2357.48E+6	0.0	109.42	362.08	0.0	0.0	0.0	0.20382	0.0	0.0
23.70	0.36098	0.12500	9.110	321.990	1588.71E+6	3995.28E+6	200.75E+6	2146.86E+6	0.0	94.36	335.01	0.0	0.0	0.0	0.19934	0.0	0.0
25.70	0.39350	0.12500	8.534	313.820	1361.93E+6	3750.76E+6	174.38E+6	1944.09E+6	0.0	80.24	308.57	0.0	0.0	0.0	0.19323	0.0	0.0
27.70	0.42602	0.12500	7.932	294.734	1102.38E+6	3447.14E+6	144.47E+6	1632.70E+6	0.0	62.67	263.87	0.0	0.0	0.0	0.14994	0.0	0.0
29.70	0.45855	0.12500	7.321	287.120	875.80E+6	3139.07E+6	119.98E+6	1432.40E+6	0.0	49.42	237.06	0.0	0.0	0.0	0.15421	0.0	0.0
31.70	0.49106	0.12500	6.711	263.343	681.30E+6	2734.24E+6	81.19E+6	1168.76E+6	0.0	37.34	196.41	0.0	0.0	0.0	0.13252	0.0	0.0
33.70	0.52358	0.12500	6.122	253.207	534.72E+6	2554.87E+6	69.09E+6	1047.43E+6	0.0	29.14	180.34	0.0	0.0	0.0	0.13313	0.0	0.0
35.70	0.55610	0.12500	5.546	241.666	408.90E+6	2334.03E+6	57.45E+6	922.95E+6	0.0	22.16	162.43	0.0	0.0	0.0	0.14035	0.0	0.0
37.70	0.58862	0.12500	4.971	220.638	314.54E+6	1828.73E+6	45.92E+6	760.82E+6	0.0	17.33	134.83	0.0	0.0	0.0	0.13950	0.0	0.0
39.70	0.62115	0.12500	4.401	200.293	238.63E+6	1584.10E+6	35.98E+6	648.03E+6	0.0	13.30	116.30	0.0	0.0	0.0	0.15134	0.0	0.0
41.70	0.65366	0.12500	3.834	179.404	175.88E+6	1323.36E+6	27.44E+6	539.70E+6	0.0	9.96	97.98	0.0	0.0	0.0	0.17418	0.0	0.0
43.70	0.68618	0.12500	3.332	165.094	126.01E+6	1183.68E+6	20.90E+6	531.15E+6	0.0	7.30	98.93	0.0	0.0	0.0	0.24922	0.0	0.0
45.70	0.71870	0.12500	2.890	154.411	107.26E+6	1020.16E+6	18.54E+6	460.01E+6	0.0	6.22	85.78	0.0	0.0	0.0	0.26022	0.0	0.0
47.70	0.75122	0.12500	2.503	138.935	90.88E+6	797.81E+6	16.28E+6	375.75E+6	0.0	5.19	69.96	0.0	0.0	0.0	0.22554	0.0	0.0
49.70	0.78376	0.12500	2.116	129.555	76.31E+6	709.61E+6	14.53E+6	328.89E+6	0.0	4.36	61.41	0.0	0.0	0.0	0.22795	0.0	0.0
51.70	0.81626	0.12500	1.730	107.264	61.05E+6	518.19E+6	9.07E+6	244.04E+6	0.0	3.36	45.44	0.0	0.0	0.0	0.20600	0.0	0.0
53.70	0.84878	0.12500	1.342	98.776	49.48E+6	454.87E+6	8.06E+6	211.60E+6	0.0	2.75	39.57	0.0	0.0	0.0	0.21662	0.0	0.0
55.70	0.88130	0.12500	0.954	90.248	39.36E+6	395.12E+6	7.08E+6	181.52E+6	0.0	2.21	34.09	0.0	0.0	0.0	0.22784	0.0	0.0
56.70	0.89756	0.12500	0.760	83.001	34.67E+6	353.72E+6	6.09E+6	160.25E+6	0.0	1.93	30.12	0.0	0.0	0.0	0.23124	0.0	0.0
57.70	0.91382	0.12500	0.574	72.906	30.41E+6	304.73E+6	5.75E+6	109.23E+6	0.0	1.69	20.15	0.0	0.0	0.0	0.14826	0.0	0.0
58.70	0.93008	0.12500	0.404	68.772	26.52E+6	281.42E+6	5.33E+6	100.08E+6	0.0	1.49	18.53	0.0	0.0	0.0	0.15346	0.0	0.0
59.20	0.93821	0.12500	0.319	66.264	23.84E+6	261.71E+6	4.94E+6	92.24E+6	0.0	1.34	17.11	0.0	0.0	0.0	0.15382	0.0	0.0
59.70	0.94636	0.12500	0.253	59.340	19.63E+6	158.81E+6	4.24E+6	63.23E+6	0.0	1.10	11.55	0.0	0.0	0.0	0.09470	0.0	0.0
60.20	0.95447	0.12500	0.216	55.914	16.00E+6	137.88E+6	3.66E+6	53.32E+6	0.0	0.89	9.77	0.0	0.0	0.0	0.09018	0.0	0.0
60.70	0.96260	0.12500	0.178	52.484	12.83E+6	118.79E+6	3.13E+6	44.53E+6	0.0	0.71	8.19	0.0	0.0	0.0	0.08561	0.0	0.0
61.20	0.97073	0.12500	0.140	49.114	10.08E+6	101.63E+6	2.64E+6	36.90E+6	0.0	0.56	6.82	0.0	0.0	0.0	0.08035	0.0	0.0
61.70	0.97886	0.12500	0.101	45.818	7.55E+6	85.07E+6	2.17E+6	29.92E+6	0.0	0.42	5.57	0.0	0.0	0.0	0.07096	0.0	0.0
62.20	0.98699	0.12500	0.062	41.669	4.60E+6	64.26E+6	1.58E+6	21.31E+6	0.0	0.25	4.01	0.0	0.0	0.0	0.05424	0.0	0.0
62.70	0.99512	0.12500	0.023	11.453	0.25E+6	6.61E+6	0.25E+6	4.85E+6	0.0	0.04	0.94	0.0	0.0	0.0	0.05387	0.0	0.0
63.00	1.00000	0.12500	0.000	10.319	0.17E+6	5.01E+6	0.19E+6	3.53E+6	0.0	0.02	0.68	0.0	0.0	0.0	0.05181	0.0	0.0

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship and/or publication of this article.

ORCID iD

Abdel Salam Y Alsabagh

References

Musial

Butterfield

Ram

. Energy from offshore wind. Golden, CO: National Renewable Energy Lab (NREL), 2006.

Sun

, et al. Modal analysis of three rotating blades of large wind turbine. Appl Mech Mater 2012; 157–158: 697–701.

Lago

Ponta

Otero

. Analysis of alternative adaptive geometrical configurations for the NREL-5 MW wind turbine blade. Renew Energ 2013; 59: 13–22.

Liu

Liang

, et al. Influence of the vibration of large-scale wind turbine blade on the aerodynamic load. Energy Procedia 2015; 75: 873–879.

Zuheir

Abdullah

Al-Maliki

. Stress and vibration analyses of the wind turbine blade (A NREL 5MW). J Mech Eng Res Dev 2019; 42(4): 14–19.

. The effect of the rotor adjustment on the vibration behaviour of the drive-train system for a 5 MW direct-drive wind turbine. P I Mech Eng C-J Mec 2018; 232(17): 3027–3044.

Tartibu

Kilfoil

Van Der Merwe

. Vibration analysis of a variable length blade wind turbine. Int J Adv Eng Technol 2012; 4(1): 630–639.

Gürsel

Çoban

Özdamar

. Vibration analysis of rotor blades of a farm wind-power plant. Math Comput Appl 2012; 17(2): 164–175.

Ulriksen

Tcherniak

Kirkegaard

, et al. Operational modal analysis and wavelet transformation for damage identification in wind turbine blades. Struct Health Monit 2016; 15(4): 381–388.

10.

Habibi

Cheng

Zheng

, et al. A dual de-icing system for wind turbine blades combining high-power ultrasonic guided waves and low-frequency forced vibrations. Renew Energ 2015; 83: 859–870.

11.

Kim

. Performance prediction of a 5MW wind turbine blade considering aeroelastic effect. World Acad Sci Eng Technol 2011; 81: 771–775.

12.

Kraemer

Fritzen

. Aspects of operational modal analysis for structures of offshore wind energy plants. In: Proceedings of the society for experimental mechanics, IMAC-XXVIII, Jacksonville, FL, 1–4 February 2010.

13.

Jonkman

Butterfield

Musial

, et al. Definition of a 5-MW reference wind turbine for offshore system development. Golden, CO: National Renewable Energy Lab, 2009.

14.

de Silva

. Vibration: fundamentals and practice. Boca Raton, FL: CRC Press, 2000.

15.

Beards

. Structural vibration: analysis and damping. London: Arnold; Halsted Press, 1996.

16.

Rao

. Vibration of continuous systems, vol. 464. Hoboken, NJ: John Wiley & Sons, 2007.

17.

Stokey

. Vibration of systems having distributed mass and elasticity. In: Harris

Crede

(eds) Shock and vibration handbook. New York: McGraw-Hill Book Company, 1961, pp. 7.1–7.50.

18.

Hau

. Wind turbines: fundamentals, technologies, application, economics. Berlin: Springer, 2010.

19.

Meirovitch

. Fundamentals of vibrations. 1st ed. New York: McGraw-Hill Higher Education, 2001.