Modal and harmonic analysis of three-dimensional wind turbine models

Abstract

In this article, the dynamic responses of wind turbine systems are analytically and numerically investigated. For this purpose, analytic differential equations of motion of wind turbine components subjected to vibration (the blades, the nacelle, and the tower) are solved. This allows determining their dynamic characteristics, mode shapes, and natural frequencies. Two models of two three-dimensional (3D) micro-turbine that are created by the finite element method are set up using the new version of the academic finite element analysis software ANSYS. The first wind turbine is a standard micro three-bladed turbine and the second one is a micro six-bladed Rutland 504. Their natural frequencies and mode shapes are identified based on the modal analysis principle to check the validity of designed models. Dynamic behaviors at several operating conditions of wind turbines are established. Then, spectrum graphs of the structures along x-, y- and z-axis are analyzed.

Keywords

Modal analysis harmonic analysis wind turbine three-dimensional model natural frequencies mode shapes spectral graphs

Introduction

It has been indicated (Baringo and Conejo, 2014; Carrasco et al., 2006) that the Global Wind Energy Council declares that more than 35 GW new wind power capacity was brought online in 2013. Many works (Fernando et al., 2006; Tsili and Papathanassiou, 2009) have confirmed that wind energy capacity is estimated to carry on being one of the most essential sources of energy around the world. It has been stated (Hameed et al., 2009; Sellami et al., 2014) that turbine designing and monitoring are still considered as important issues.

Nevertheless, as wind turbines are subjected to several dynamic loads such as aerodynamic loads, changes in wind direction, and gravitational forces, their structure vibrations have been stated (Sellami et al., 2016; Wu et al., 2015) important. In fact, Staino and Basu (2015) have argued that the challenge remains on better understanding their dynamic properties via analytical, numerical, and experimental means. Many of the vibration and noise related problems contribute to resonance. The resonant vibration is mostly caused by interactions between the elastic and inertial properties of the materials of the structure since small forces can cause important deformations, and possibly, serious damages. That is why many works (Hu et al., 2015; Wei et al., 2015) have indicated that the resonant and the natural frequencies of the wind turbine structure need to be quantified.

The most important part of dynamic finite element analysis is the modal analysis. Modal analysis is a well-known method of extracting the mode shapes of vibrations of the structure. It has been indicated (Iliopoulos et al., 2015; Juang and Pappa, 1985) that new or improved product prototypes always need structural dynamics modal testing to assess their real dynamic behaviors. Modal analysis techniques were described (Deyuan et al., 2004; Pandey et al., 1991) based on frequency domain consisting of identifying the dynamic characteristics of the wind turbine, mode shapes, and natural (modal or resonant) frequencies. At the natural frequency of a mode, the overall shape of the vibration will tend to be dominated by the mode shape of the resonance. Each mode shape is defined by a natural frequency, modal damping frequency calculated by the response function of the system. It has been stated (Allen et al., 2011; Hansen, 2003) that these parameters depend on the material properties of the structure (damping, mass, and stiffness) and the boundary conditions applied to it.

Wind turbine model

Wind turbine converts wind energy to electrical one. The basic configuration of the system is sketched in Figure 1. The wind turbine system is basically composed of blades, a tower, and a nacelle.

Figure 1.

Designed wind turbine model using ANSYS software.

The wind turbine aerodynamic captured power from the wind P_a can be written by the nonlinear expression

P_{a} = \frac{1}{2} ρ π R^{2} V^{3} C_{p} (λ, β)

(1)

where R is the rotor radius, V is the wind speed, ρ is the airflow density, C_p is the power coefficient defined as a function of the pitch angle β, and the tip speed ratio λ defined by $λ = R ω_{r} / V$ , $ω_{r}$ is the wind turbine rotor speed.

T_a is the rotor aerodynamic torque given by

T_{a} = \frac{P_{a}}{ω_{r}}

(2)

T_a is involved in the mechanical equation of the turbine defined by

J_{t} ω_{r} = T_{a} - K ω_{r} - T_{g}

(3)

where K is the turbine’s total external damping, T_g is the generator’s electromagnetic torque, and J_t is the turbine’s total inertia.

Vibration analysis of the wind turbine model

The turbine system can be composed of several parts, where each one includes a mass m, a stiffness spring k, and a viscous damping coefficient c. The vectors u, $\overset{\cdot}{u}$ , and $\overset{\cdot\cdot}{u}$ represent the displacement, velocity, and acceleration of the various masses, respectively. Every part produces a resistive force “ $k u$ ,” inertia force, “ $m \overset{\cdot\cdot}{u}$ ” and a damping mechanism dissipating the energy “ $c \overset{\cdot}{u}$ .”

The degrees of freedom number of a vibrating system can be defined by “the minimum number of displacement components required to describe the configuration of the system during vibration” as been indicated in (Jhonson and Kienholz, 1982). The wind turbine system can be modeled with several complicated levels.

Modal dynamics of the tower

The dynamics of the tower vibrations can be considered as a single degree of freedom system because its mass can only move along the vertical z-axis. Authors Chiang et al. (2015), He and Ge (2015), and Jianbing et al. (2015) argued that the nacelle and the blades could be considered as one concentrated tip mass.

The equation of motion for the vibration of a viscously damped tower is given by

m \overset{\cdot\cdot}{u} (t) + c \overset{\cdot}{u} (t) + k u (t) = F (t)

(4)

where F is the excitation to the system.

Dividing equation (4) by the total mass m, we obtain

\overset{\cdot\cdot}{u} (t) + 2 ζ ω_{n} \overset{\cdot}{u} (t) + ω_{n}^{2} u (t) = F (t) / m

(5)

where the commonly used parameters for determining the model performance are the natural pulsation ω_n, the damping ratio ζ, and the viscous damping factor, which are defined by

ζ = \frac{c}{2 m ω_{n}}, ω_{n} = \sqrt{\frac{k}{m}}

F(t) presents the force that acts on the mass resulting from pressure loading of upstream and downstream disturbances.

Based on the modal analysis method, the equation of motion tower in the case of free vibration is

\overset{\cdot\cdot}{u} (t) + 2 ζ ω_{n} \overset{\cdot}{u} (t) + ω_{n}^{2} u (t) = 0

(6)

For damped second-order systems, the solution of the previous equation is given by

u (t) = A_{1} e^{s_{1} t} + A_{2} e^{s_{2} t}

(7)

where $s_{1, 2} = - ζ \pm ω_{n} \sqrt{ζ^{2} - 1}$ , and A₁ and A₂ are the two constants of integration. Their values depend on the initial displacement $u_{0} = u_{(t = 0)}$ and initial velocity ${\overset{\cdot}{u}}_{0} = d u_{(t = 0)} / d t$ of the system.

For simplification, damping of the tower can be neglected. Thus, the equation of motion tower (6) becomes

\overset{\cdot\cdot}{u} (t) + ω_{n}^{2} u (t) = 0

(8)

Solving the previous equation, we obtain the displacement u(t) of the mass m over time t given by an exponential form

u (t) = A e^{s t}

(9)

Introducing equation (9) into equation (8) and dividing through by $A e^{s t}$ , we obtain the characteristic equation

s^{2} + ω_{n}^{2} = 0

(10)

Then

s_{1, 2} = \pm i ω_{n}

(11)

Inserting s₁ and s₂ into equation (9), the general solution of equation (8) is written as

u (t) = A c o s (ω_{n} t - ϕ)

(12)

This solution indicates that the system executes, with the natural frequency $f_{n}$ , a harmonic oscillation where the amplitude A and the phase angle $ϕ$ are

A = \sqrt{u_{0}^{2} + {(\frac{{\overset{\cdot}{u}}_{0}}{ω_{n}})}^{2}}, ϕ = \tan^{- 1} \frac{{\overset{\cdot}{u}}_{0}}{u_{0} ω_{n}}

Then, the n^th mode shape undamped natural frequency expression of the tower system is given by

f_{n} = \frac{ω_{n}}{2 π} = (\frac{1}{2 π}) \sqrt{\frac{k}{m}}

(13)

In general, the damped natural frequency expression is given by

f_{d} = f_{n} \sqrt{1 - ξ^{2}}

(14)

Modal dynamics of the blades

Several works (Basu et al., 2015; Jureczko et al., 2005) have stated that the dynamics of a blade vibration, as a complex system, could be considered as a multi-degree of freedom system. The vectors {u}, { $\overset{\cdot}{u}$ }, and { $\overset{\cdot\cdot}{u}$ } represent the displacement, velocity, and acceleration vectors of the various masses, respectively. Under an external force action, the differential equation of motion is given by

[M] {\overset{\cdot\cdot}{u} (t)} + [C] {\overset{\cdot}{u} (t)} + [K] {u (t)} = {F (t)}

(15)

where [M] is the generalized mass matrix, [C] is the generalized damping matrix, [K] is the generalized stiffness matrix, {F(t)} is an external matrix, and u represents the displacement in lateral direction.

It has been stated (Bekhti et al., 2015) that cyclic loading is resulted from pressure loading of upstream and downstream disturbances.

Vibrations due to initial disturbance of the blade system present the case of modal analysis study, which are characterized by

[M] {\overset{\cdot\cdot}{u} (t)} + [C] {\overset{\cdot}{u} (t)} + [K] {u (t)} = \vec{0}

(16)

In the absence of damping, the equation of a free vibration equation (16) becomes

[M] {\overset{\cdot\cdot}{u} (t)} + [K] {u (t)} = \vec{0}

(17)

Assuming a solution given by

{u (t)} = {U} e^{j w_{n} t}

(18)

where {U} is the displacement amplitude vector.

Equation (18) leads to

[K] - ω_{n}^{2} [M] = \vec{0}

(19)

Equation (19) is solved for the eigenvalues under several boundary conditions of the blade of the turbine.

Simulation results of modal and harmonic analysis of a three-dimensional wind turbine model

Drive train vibration model

The wind turbine is considered as a hybrid system combining mechanical and structural parts; its basic components include the hub, high- and low-speed shaft, generator, nacelle, blades, and the tower. The blade and the tower can be considered as structural parts. However, the generator, shaft, and the nacelle present the mechanical ones. As for the mechanical parts, their strain and stress are not as significant as those of the blades and the tower; moreover, their structural characteristics can be considered as a rigid body system. The blade is completely rigid and has a low mass as it is subjected to strong vibrations during the operation.

Three-dimensional (3D) designs of two micro-turbines are made. The first one is a standard three-bladed micro-turbine and the second one is a six-bladed Rutland 504 micro-turbine. Their characteristics are presented in Appendix 1. For both wind turbine models, the towers are modeled as Euler–Bernoulli beam systems with distributed parameters, having flexible and damping properties (Farrar and James, 1997). The tower vibration is represented by the acceleration measurements from the tower.

Singiresu (2007) and Hearn and Testa (1991) have described the blade as system composed of a blade body and a blade root. Several sections were drawn. In this article, according to a profile airfoil data, data of blade element sections are obtained. Then, sections and lofting are generated using CATIA software as shown in Figure 2. The blade’s vibration is represented by the acceleration measurements from the blades.

Figure 2.

Airfoil profile drawn by CATIA software.

To simplify the analysis, the material of the blades is considered homogeneous isotropic. The blade’s mechanical behavior material is supposed to be elastic linear. The nacelle and the tower are considered rigid and fixed, and the connection between the different structures is rigid.

Wind turbine structure modal analysis for validating designed models

After specifying the materials, natural frequencies of the two created turbine models have to be calculated to check their validity. Thus, the modal analysis is carried out, and both the structure natural frequencies and the first mode shapes are acquired using ANSYS software.

Table 1 illustrates the first 10 natural frequencies of the first and second wind turbines. The natural frequencies relative to the first mode shape are simultaneously 5.4373 Hz and 18.018 Hz for the two micro-turbines. The frequencies keep increasing with the next mode shapes.

Table 1.

First and second wind turbines’ first 10 mode shapes and their corresponding natural frequencies.

Mode shapes	First turbine natural frequencies [Hz]	Second turbine natural frequencies [Hz]
1	5.4373	18.018
2	5.5416	25.097
3	5.8304	26.727
4	8.6265	34.766
5	14.095	35.664
6	14.191	49.853
7	19.886	51.89
8	26.472	52.002
9	28.884	63.717
10	32.983	66.021

These evaluated natural frequencies permit to verify the created model of the wind turbine by comparing the structure parameters to those obtained by the analytical resolution in previous section. The first natural frequency of the first turbine, whose radius is 1 m, is 5.4373 Hz. The first natural frequency of the second turbine, whose radius is 0.26 m, is 18.018 Hz. These results lead to deduce that small wind turbines have important natural frequency values, while huge turbines have low ones.

Figure 3 shows that the first turbine’s total displacement at the frequency of 32.983 Hz (10th mode shape) is concentrated on the three blades. It reaches the maximum value of 28.877 mm. The second turbine’s total displacement at the frequency of 66.021 Hz (10th mode shape) is concentrated on the ring. It reaches the maximum value of 99.522 mm. The results prove that the blades or the rings are the most sensitive parts to vibration as they are facing more aerodynamic problems than the tower and the nacelle.

Figure 3.

First and second turbines’ total displacement at the natural frequency relative to the 10th mode shape.

Figure 4 presents the total displacement of the first wind turbine at different frequencies; the wind turbine’s structure total displacement at the frequency of 32.983 Hz (10th mode shape) is concentrated on the edge of the three blades. It reaches the maximum at the value of 28.877 mm. This total displacement at the frequency of 106.32 Hz (20th mode shape) is concentrated on one of the blades. It reaches the maximum at the value of 114.02 mm. At the frequency of 161.28 Hz (30th mode shape), the total displacement is still concentrated on the previous blade. But the maximum value reached has decreased to 82.876 mm.

Figure 4.

Wind turbine structure total displacement at the frequency of 32.983 Hz (10th mode shape), 106.32 Hz (20th mode shape) and 161.28 Hz (30th mode shape).

Wind turbine’s structure harmonic response

After accomplishing a validated numerical model of the wind turbine, many tests can be made to study the turbine dynamics. It has been stated (Adhikaria and Bhattacharya, 2012; Maalawi and Badr, 2003) that through exciting the structure, its acceleration responses in different positions are given by ANSYS, as the wind turbine is exposed to a pressure caused by the wind. A normal pressure, as indicated in “Norme NV65,” is applied perpendicularly to the tower along the x-axis to get the basic vibration spectral graphs. Frequency band studied is 0.50 Hz. Three accelerometers are placed in different positions of the wind turbine structure: the first one is placed on the tower, the second one on a blade, and the last one is on the head of the nacelle. Acceleration magnitudes of the three accelerometers are plotted simultaneously along the x-, y- and z-axis.

Figure 5 shows that exposed to the wind along the x-axis, the blade acceleration maximum magnitude along the x-axis reaches 2.152 10⁵ mm/s². While the tower and the nacelle acceleration maximum magnitudes are 8449.7 and 22761 mm/s², respectively. Thereby, the blades are most sensitive to vibrate as they have the highest magnitudes of accelerations compared to those of the tower and the nacelle. This is due to their particular geometries and materials.

Figure 5.

Acceleration spectral graphs along the x-axis of (a) the tower, (b) the blade, and (c) the nacelle.

Figures 6 and 7 demonstrate that along any accelerometer direction, mode shapes and its natural frequencies can be identified. However, the most important values of acceleration magnitudes are presented along the x-axis. Meanwhile, the study of wind turbine vibration, along the direction of the exciting harmonic force, is sufficient.

Figure 6.

Acceleration spectral graphs along the z-axis of (a) the tower, (b) the blade, and (c) the nacelle.

Figure 7.

Acceleration spectral graphs along the y-axis of (a) the tower, (b) the blade, and (c) the nacelle.

Conclusion

In this article, the vibration of the wind turbine structure is first studied by an analytical analysis to calculate the natural frequencies and the mode shapes of the structure. Then, developed 3D models of complete wind turbines are designed for both three-bladed and six-bladed micro-turbines. Natural frequencies are obtained by ANSYS to check the validity of designed models. It is deducted that micro wind turbines have important natural frequency values, while large turbines have low ones. The work proves that the blades (for the first wind turbine) or the ring (for the second wind turbine) are the most sensitive parts to vibrate because they are facing more aerodynamic problems than the tower and the nacelle. Consequently, structural damage can happen to any component of the wind turbine system; however, the most common type of damage is blade and ring damage. This article also asserts that the study of wind turbine vibrations along the direction of the harmonic exciting force is sufficient.

Footnotes

Appendix 1

The tower of the wind turbine is constructed from steel. The blades and the nacelle are constructed from complex materials.

Tables 2 and 3 present the dimensions of the two designed micro-turbines.

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.

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