Numerical simulations on static and dynamic response of full-scale mast structures for H-Darrieus wind turbine

Abstract

Mast structure is one of the most important parts of a vertical-axis wind turbine which supports generator and rotor and represents one-third of the overall costs in the production of a standard wind turbine (approximately 30%). All this may cause significant economic and physical losses when it is damaged or collapsed. The purpose of this research is to investigate numerically the static strength and structural dynamic responses of 10-kW vertical-axis wind turbine masts subjected to the aerodynamic and gravity loadings (according to the IEC 61400-2:2006 and EN 1991-1-4:2005 standards) using the SolidWorks finite element software. Mast structures with four different heights (12, 14, 16, and 18 m) and three various outer diameters (0.6, 0.7, and 0.8 m), in each height configuration, were evaluated. These analyses were performed to identify the stiffness, resistance, reliability, and natural frequency stiffness requirements within the mast structures, in order to save manufacturing cost. Based on static analysis, no structural failure is predicted for all masts during wind turbine operation according to maximum von Mises stresses at the bottom of the mast and maximum total deflections on the top of the mast. In addition, the dynamic parameters of these 12 models of masts have been studied to obtain the natural frequencies and corresponding mode shapes. Finally, the recommendations to avoid resonance and design strategy for each mast model are discussed.

Keywords

Wind turbine mast static strength analysis dynamic modal analysis natural frequency finite element analysis

Introduction

Wind power is now considered to be the most cost-effective and fastest growing of all the exploited renewable energy sources. According to the Renewable Energy Policy Network for the 21st century report (REN21), wind generation at the end of 2017 reached the total global installed capacity of 539 GW, an increase of 10.6% compared to 487 GW in 2016. This energy had its third strongest year ever, with 52 GW added (which is a 9.6% increase) (Danao et al., 2014; REN21, 2018). The modern wind turbines (WTs) are broadly classified into horizontal-axis wind turbine (HAWT) and vertical-axis wind turbine (VAWT), based on their axis of rotation. The interest for VAWTs has increased in the recent years because they claim many advantages. They operate independently of the wind direction (without yawing), and they also operate efficiently in turbulent wind conditions. In addition, the advantages also include manufacturing simplicity, less noise, and easy installation. Recently, most of the VAWTs installed are of smaller size (with the power rating of 50 kW or less) (Adaramola, 2014; Ferroudji et al., 2017; Kumar et al., 2018).

WTs are among those engineering structures that undergo the most extreme loading throughout their service life. Subjected to nature’s forces (winds, cold or extreme heat, sand, and earthquakes), gravitational loading varies with time as the turbine operates. To enable WT structures to operate in any external conditions and to ensure their expected service lifetime, each component of the WT has to be designed adequately, safely operated, and manufactured with suitable quality (Ferroudji et al., 2018; Söker, 2013). There are a number of guidelines that show the existing state of the art related to the design requirements of WTs and their key components such as the International Electrotechnical Commission standard IEC 61400-x Series.

VAWTs are usually attached to supporting steel masts that support the rotor, generator, and other components to increase their efficiency. During the time of development of WTs, several types of masts and towers have been tested such as tubular or conic steel structures, lattice (or truss) masts, concrete masts, and guyed masts (Schaffarczyk, 2014; Tong, 2010). One of the most popular choice design options for WT masts is tubular steel structure, because it is economical, has low aerodynamic resistance, and its maintenance and mounting are simple. These types of WT masts are manufactured in several segments, for technical conditions of mounting and easier transportation and erection (Hu et al., 2014). Taking into consideration that the cost of mast structure constitutes approximately one-third of the total cost of the WT, optimal design of the mast with satisfactory performance becomes a crucial step in the WT mast construction (Stavridou et al., 2019).

Most recent research studies available on VAWT and HAWT are concentrated on the analysis of the aerodynamics and performance of VAWTs through computational fluid dynamic (CFD) simulations (Amano et al., 2013; Bianchini et al., 2017; Kono et al., 2017) and experimental testing (Araya, 2016; Li et al., 2016; Peng et al., 2016; Wang et al., 2018). Concerning system design optimization, the majority of the studies are focused on the WT itself (power output, economics, and cost optimization; Prowell et al., 2009; Vatanchian and Shooshtari, 2016; Wang et al., 2017), and a few studies are found to deal with the problem of structural design optimization. Negm and Maalawi (2000) developed and tested five optimization strategies in order to optimize WT tower structure. In all five strategies, the wall thickness and height of each segment were chosen as the key design variables. Bazeos et al. (2002) and Lavassas et al. (2003) investigated the stability behavior for prototype steel WT tower with power ratings of 450 kW and 1 MW, respectively. Both studies used finite element analyses (FEAs) and other simplified models with the rotor and nacelle mass lumped at the top of the tower. Uys et al. (2007) established a procedure to calculate the least cost design (wall thickness) of a conical steel tower. Dimopoulos and Gantes (2014) investigated an experimental and numerical study of the effect of door opening and stiffening rings on the strength of cantilever cylindrical shells. To permit analytic approximations of tower stresses, a design process based on modeling the tubular lattice towers as either tripods or quadrapods was performed by Adhikari et al. (2014). Hu et al. (2014) analyzed the structural behavior of WT towers with different design configurations (with three different heights) under wind loads, by means of finite element (FE) method modeling. In their study, Stratton et al. (2016) selected the optimal tower design using a multilevel decision-making procedure, after the nine towers were analyzed with different geometries and materials using FEA.

In this research, a 10-kW VAWT mast structure with four different heights (12, 14, 16, and 18 m) is developed by FE method using SolidWorks (2016) simulation computer program. In each height case, masts have three different outer diameters: 0.6 m (model 1), 0.711 m (model 2), and 0.8 m (model 3). In addition, the static analysis of all mast configurations is used to determine the von Mises stresses, deflections, and factor of security caused by loads, including the aerodynamic (extreme wind and thrust) and gravity (rotor, generator, and mast itself) loadings. Afterwards, the dynamic analysis of these mast structures is performed; this analysis is used to determine the vibration characteristics (natural frequencies and mode shapes), and suggestions to avoid resonance for each mast configuration under the wind loads are proposed in detail.

The remainder of the article is structured as follows: section “Design of the WT mast structure” presents the design of WT mast structures. Section “Load characteristics” presents the application of the IEC 61400-2:2006 and EN 1991-1-4:2005 standards to determine the load characteristics, which are used in the FEAs. Section “FE development” presents the FE development model, and section “Results and discussions” presents the numerical results. Finally, section “Conclusion” presents the main concluding remarks.

Design of the WT mast structure

This article describes the design of a mast structure that supports a 10-kW H-Darrieus VAWT. The main specifications of the WT are listed in Table 1. The mast is divided into two cross-sectional segments along the vertical direction transportation and erection purposes that are bolted together by bolts and each segment has a constant linear slope. The mast is fully fixed at the bottom (joined to a soil–foundation) and freely supported at the top. The different variations of mast height are considered (L = 12, 14, 16, and 18 m) and the total heights of the WT including the rotor and the blades are 17.5, 19.5, 21.5, and 23.5 m. The outer diameters are 0.6 m (model 1), 0.711 m (model 2), and 0.8 m (model 3), and the uniform thickness for all masts is 10 mm. All of the 12 three-dimensional (3D) solid models of the masts are realized using the SolidWorks (version 2016) computer-aided software tool. The four masts for model 3 are shown in Figure 1.

Table 1.

Main specifications of the 10-kW wind turbine.

Parameter	Unit	Value
Rated power	kW	10
Rotor diameter	m	8.9
Blade length	m	7.6
Cut-in wind speed	m/s	3
Rated wind speed	m/s	10
Cut-out wind speed	m/s	20
Wind speed extreme	m/s	55
Swept area	m²	68
Rotor diameter	m	8.9
Rotor height	m	7.6
Generator mass	ton	1
Rotor weight (with blades)	ton	0.8

Figure 1.

Mast prototype configurations.

Load characteristics

The design of the WT mast structure has been made according to the IEC 61400-2:2006 and EN 1991-1-4:2005 standards. The mast design is based primarily on types of load acting on the mast. These loads include gravitational loads (i.e. dead loads), aerodynamic loads on the rotor, and wind loads on the mast itself. All the 12 masts were subjected to the extreme wind speed model (EWM; IEC 61400-2:2006).

Aerodynamic loading

The main external consideration for structural integrity is wind condition. Wind load acting on WT structure is divided into two components: the first is a concentrated force (wind thrust) which is acting on the top of the mast, and the second component contains a distributed wind which is acting along the height of the mast (DNV/Risø, 2002). The wind loads are introduced in the following sections.

Extreme wind loads

The design wind velocity, $V_{e 50} (z)$ , is distributed along the mast according to IEC (IEC 61400-2:2006; Hau, 2013) and is obtained from

V_{e 50} (z) = 1.4 \cdot V_{ref} {(\frac{z}{z_{hub}})}^{0.11}

(1)

where $V_{ref}$ is the reference wind velocity (10 min average of extreme wind speed with a 50-year return period). For the site Adrar, Algeria (Wind Zone 2), $V_{ref} = 28 m / s$ (Règlement Neige et Vent (RNV), 1999). z and $z_{hub}$ are the height above ground level and hub height, respectively. According to EN 1991-1-4:2005, the relationship between the static wind load pressure $P_{W, j}$ acting on a structure or a structural component (segment j) and the design wind velocity is given by the following equation

P_{W, j} = \frac{1}{2} C_{e} (z_{j}) \cdot C_{f, j} \cdot C_{s} C_{d} \cdot ρ_{air} \cdot [V_{e 50} (z)]^{2}, j = 1, \dots, 18

(2)

where $ρ_{air}$ is the mass density of air (1.225 kg/m³), $C_{s} C_{d} = 1$ is the structural coefficient (EN 1991-1-4:2005), $C_{f, j}$ is the force coefficient for the segment j, and $C_{e} (z_{j})$ is the exposure factor at height z, which is given by (EN 1991-1-4:2005)

C_{e} (z) = C_{r}^{2} \cdot C_{t}^{2} \cdot [1 + 2 k_{p} I_{v}]

(3)

where $k_{p}$ is the peak factor, $k_{p} = 3.5$ (Karpat, 2013); $C_{t}$ is the topography coefficient; $C_{r}$ is the roughness coefficient which is given by equation (4); and $I_{v}$ is the intensity of the turbulence

C_{r} (z) = k_{T} \cdot \ln (\frac{z}{z_{0}}), z_{\min} \leq z \leq 200 m

(4)

\begin{matrix} C_{r} (z) = k_{T} \cdot \ln (\frac{z_{\min}}{z_{0}}), z < z_{\min} = 4 m \\ I_{v} (z) = \frac{1}{C_{r} C_{t}} \end{matrix}

(5)

where $k_{T} = 0.19$ is the terrain factor, $z_{0} = 0.05$ is the roughness length, and $C_{t} = 1$ . From tables and graphs in EN 1991-1-4:2005 and RNV (1999), the values of $k_{T}$ , $z_{0}$ , and $C_{t}$ are obtained for sea or level area.

The force coefficient for mast with a cylindrical cross section (without free-end flow) is given by (EN 1991-1-4:2005)

C_{f, j} = C_{f 0, j} . ψ_{λ}

(6)

where $ψ_{λ}$ is the slenderness reduction or end-effect factor and $C_{f 0, j}$ is the basic force coefficient for segment j. $ψ_{λ}$ should be determined as a function of effective slenderness $λ_{e} = Min . (h / d; 70)$ and the effective area $φ = 1$ (for closed constructions) (h and d are the height and the outside diameter of the mast, respectively). $C_{f 0, j}$ is presented as a function of the Reynolds number $(R_{e} = d \cdot V (z_{e}) / γ)$ and the ratio $k / d$ , where $V (z_{e}) = C_{r} V_{ref}$ is the peak wind velocity at height $z_{e}$ , $γ$ is the kinematic viscosity of the air $(γ = 15 \times 10^{- 6} m / s)$ , and $k$ is the equivalent roughness, from Table 7.13 in EN 1991-1-4:2005 for the steel surface $k = 0.05$ . The values of $ψ_{λ}$ and $C_{f 0, j}$ are obtained from Figures 7.36 and 7.28, respectively, in EN 1991-1-4:2005. The mast is divided into segments which have identical length of $h = 1 m$ . All the results of extreme wind loads for the 18-m mast (model 2) are presented in Table 2.

Table 2.

Extreme wind pressure at different mast heights.

$z_{j} (m)$	$C_{r}$	$I_{v}$	$C_{e}$	$V_{e 50} (m / s)$	$R_{e}$	$ψ_{λ}$	$C_{f 0, j}$	$C_{f, j}$	$P_{W, j} (N / m^{2})$
1	0.569	1.757	3.985	30.43	$7.55 \times 10^{5}$	0.61	1.170	0.714	1613.75
2	0.701	1.426	5.396	32.84	$9.30 \times 10^{5}$	0.65	1.172	0.762	2716.06
3	0.778	1.285	6.050	34.34	$1.03 \times 10^{6}$	0.66	1.172	0.773	3377.86
4	0.832	1.202	6.516	35.44	$1.10 \times 10^{6}$	0.67	1.172	0.785	3934.99
5	0.875	1.143	6.891	36.32	$1.16 \times 10^{6}$	0.68	1.173	0.798	4443.07
6	0.909	1.100	7.189	37.06	$1.21 \times 10^{6}$	0.69	1.173	0.809	4892.53
7	0.939	1.065	7.455	37.69	$1.25 \times 10^{6}$	0.70	1.173	0.821	5325.36
8	0.964	1.037	7.675	38.25	$1.28 \times 10^{6}$	0.71	1.173	0.833	5729.18
9	0.986	1.014	7.873	38.75	$1.31 \times 10^{6}$	0.72	1.173	0.844	6111.28
10	1.006	0.994	8.054	39.20	$1.33 \times 10^{6}$	0.73	1.173	0.856	6488.79
11	1.025	0.975	8.221	39.61	$1.36 \times 10^{6}$	0.74	1.173	0.868	6857.41
12	1.041	0.961	8.374	39.99	$1.38 \times 10^{6}$	0.75	1.173	0.880	7218.13
13	1.056	0.947	8.507	40.35	$1.40 \times 10^{6}$	0.76	1.173	0.891	7558.70
14	1.070	0.934	8.630	40.68	$1.42 \times 10^{6}$	0.77	1.174	0.904	7907.64
15	1.084	0.922	8.759	40.99	$1.44 \times 10^{6}$	0.78	1.174	0.916	8256.80
16	1.096	0.912	8.870	41.28	$1.45 \times 10^{6}$	0.79	1.174	0.927	8582.00
17	1.107	0.903	8.971	41.55	$1.47 \times 10^{6}$	0.80	1.174	0.939	8907.47
18	1.118	0.894	9.072	41.82	$1.48 \times 10^{6}$	0.81	1.174	0.951	9241.82

Thrust load

Two additional loads act on the rotor (rotating blades) and the mast. The force thrust on the rotor eventually transmitted to the mast on the top $(F_{T})$ is given by (Bisoi and Haldar, 2017; IEC 61400-2:2006)

F_{T} = \frac{1}{2} C_{T} ρ_{air} A_{rotor} V_{e 50}^{2}

(7)

where A is the swept area by the rotor (68 m²) and $C_{T}$ is the thrust coefficient of the rotor (equal to 0.75). For the four masts, the thrust loads are as follows: $F_{T, 12 m} = 49.955 kN$ , $F_{T, 14 m} = 51.693 kN$ , $F_{T, 16 m} = 53.230 kN$ , and $F_{T, 18 m} = 54.632 kN$ .

The partial safety factors (load factors) for the extreme static loads are given by IEC 61400-2:2006. The factors for dead loads, wind loads, and WT loads are equal to 1.2, 1.4, and 1.35, respectively (Ma and Meng, 2014).

Thrust load

The loading effect of gravity has been considered. In a structural model, the static load due to gravity consists of the weight of the mast and is estimated directly using the FEA software SolidWorks Simulation, as a function of the dimensions of the mast and unit mass of the material density. The weight of the rotor and the generator will depend on a vertical force (18 kN) that acts on the top of the mast.

FE development

All the 12 detailed 3D FE models of the WT mast are developed using the FE commercial software SolidWorks Simulation (2016). For all mast structures, the steel materials used have to fulfill the requirements regarding strength, toughness, cold deformability, and suitability for welding (Baniotopoulos et al., 2011). The material used in the FEA was selected from SolidWorks Simulation library and is similar to the one of the mast structure. The material is steel (AINS 1020 Steel, cold rolled) with Young’s modulus (E) of $20.5 \times 10^{10} N / m^{2}$ , Poisson’s ratio (v) of 0.29, and density (ρ) of $7870 kg / m^{3}$ .

The boundary conditions at the bottom of the mast models were fixed for all degrees of freedom (DoFs). The mast was divided through an axial plane to create two halves. This permitted the wind load and thrust load on the mast to be applied in the same direction and to act normal to the mast surface. All boundary conditions and the applied load for 18 m mast (model 2) are shown in Figure 2. Generating a high-quality mesh is crucial to the best balanced accuracy and efficiency in numerical simulations. The models were meshed with parabolic tetrahedral solid elements (10 nodes per element). The resultant mesh models after the sweep and different localized refinements made in areas of interest are shown in Figure 2. The summary of mesh sizes and DoF numbers for models of the various mast structures are provided in Table 3. The software is run on a workstation at the Renewable Energy Research Unit in Saharan Medium (URER/MS), using 12 processors (Intel^® Core™ i7-4770K CPU, RAM 16.0 Go) at a 3.5-GHz clock frequency computer.

Table 3.

Meshing summary and DoF numbers.

Height of mast (m)	Models	Minimum–maximum element size (mm)	Degrees of freedom	No. of nodes	No. of elements
12	Model 1	13.04–65.24	862,683	289,340	147,374
	Model 2	13.05–65.24	926,907	310,659	157,845
	Model 3	13.04–65.24	935,100	313,297	159,308
14	Model 1	13.71–68.56	952,311	319,163	162,077
	Model 2	13.57–67.85	952,311	319,163	162,077
	Model 3	13.71–68.56	1,007,076	337,306	172,629
16	Model 1	14.06–70.30	948,060	317,698	161,339
	Model 2	14.74–73.73	948,060	317,698	161,339
	Model 3	14.06–70.30	1,060,659	355,163	183,911
18	Model 1	13.13–70.65	989,943	331,640	168,418
	Model 2	14.74–73.73	995,046	333,192	170,363
	Model 3	14.13–70.65	1,316,892	440,553	223,174

Figure 2.

Illustration of 3D FE 14-m mast structure (model 3), load application, and boundary condition. Red, pink, and blue represent the extreme wind, thrust, and gravity loadings, respectively, and green area is assumed as fixed as boundary condition.

Results and discussions

Static strength analysis

The main aim of a static strength analysis is to give information on the capability of the structure to withstand the external static loading, strains, and structural properties. This information enables to know the strength, stiffness, and stability of the structure. In this analysis, the inertial and damping effects are ignored because they are not important. It is used when all the loads acting on the structure are applied very slowly and gradually (Khelifi and Ferroudji, 2016). Structural behavior of the mast structure is examined in terms of maximum shear stress (according to the von Mises yield criterion) and the mast-top deflection. The maximum von Mises stress and maximum deflection are plotted for each mast structure for the three models. The resulting von Mises stress distributions (MPa) and the deflection distributions (mm) of static strength analysis are, respectively, shown in Figures 3 –6. Table 4 shows the analysis results of maximum von Mises stress, maximum deflection, and factor of safety in all the mast models.

Table 4.

Static results of all mast models (maximum values).

Height of mast (m)	Models	Variables
		von Mises stress (MPa)	Mast deflection (mm)	Factor of safety
12	Model 1	112.66	11.074	3.107
	Model	116.19	7.6320	3.012
	Model 3	118.04	5.5920	2.965
14	Model 1	153.00	20.300	2.288
	Model 2	157.02	14.041	2.229
	Model 3	161.70	10.974	2.164
16	Model 1	163.00	34.362	2.149
	Model 2	178.00	23.722	1.968
	Model 3	178.00	18.538	1.970
18	Model 1	236.73	55.479	1.478
	Model 2	260.06	38.584	1.346
	Model 3	266.30	30.209	1.314

Figure 3.

von Mises stress and the deflection of the 12-m mast structure: (a) model 1, (b) model 2, and (c) model 3.

Figure 4.

von Mises stress and the deflection of the 14-m mast structure: (a) model 1, (b) model 2, and (c) model 3.

Figure 5.

von Mises stress and the deflection of the 16-m mast structure: (a) model 1, (b) model 2, and (c) model 3.

Figure 6.

von Mises stress and the deflection of the 18-m mast structure: (a) model 1, (b) model 2, and (c) model 3.

As expected, the analyses of the von Mises stress values revealed that maximum stress concentrations were located in the vicinity of the bottom of all the mast models (base), but these localized von Mises stress dissipates rapidly throughout the structure. The maximum stress values for all mast models are between 112.62 and 266.3 MPa (Table 4). Between all these models, it was noted that higher stresses occur within the 18-m mast for model 3 and is equal to 266.3 MPa. This value remains under the yield strength of the material which is 350 MPa.

However, maximum deflection values are determined in masts unlike von Mises stresses. It is obvious that the deflections along the mast are different and the maximum deflections occur at the mast top. Due to the pressure applied on the structure by the extreme wind load and rotor thrust force, the mast bottom is set up fully constrained. These loads resulted in a total deflection value between 5.592 and 55.479 mm at the top of all the mast models (Table 4). The higher deflection (55.479 mm) occurs within the 18-m mast for model 1. All the maximum deflection values are acceptable according to the Code for Design of high-rise structures, and the maximum deflection is limited to L/100 (Adhikari et al., 2014; Ma and Meng, 2014).

Based on the static strength results, it can be concluded that the structures of the masts will not fail during WT operation because the maximum stresses are all lower than the yield strength of the material (lowest factor of safety is equal to 1.3) and all related maximum total deformations meet material deflection.

Dynamic modal analysis

Vibration analysis consists of performing a modal analysis, which yields the natural frequencies (eigenvalues) and corresponding mode shapes (eigenvectors) for all masts excluding the mass of generator, rotor, and blades. These results are used to compute the structure response to vibration excitation. Results of the modal analysis inform the design decisions to avoid the resonance of mast structure under the wind loads. There are three different design strategies of the mast structure (soft–soft, soft–stiff, and stiff–stiff) for dynamic performance (Hu et al., 2014; Petrini et al., 2010). The design strategies include the following: (1) a soft–soft design refers to the case where $f_{nat}$ of the structure is placed below the operational frequency of the turbine (rotor frequency $f_{r}$ , 1P excitation)—this type of mast is very flexible (very slender) and fatigue effects are significant; (2) a soft–stiff design refers to the case where $f_{nat}$ is greater than $f_{r}$ and is less than the blade passing frequency $f_{b}$ ( $f_{r}$ times number of blades) for a three-bladed WT ( $f_{b} = 3 f_{r}$ , 3P excitation)—considering design economy (material saved) and safety rules, this type of design is the most economical for modern WT mast; and (3) a stiff–stiff design refers to the case where $f_{nat}$ is higher than $f_{b}$ (3P excitation)—this mast is uneconomical because it requires large quantities of material and in general, the fatigue effects are not significant.

The values of rotor speed (operational interval) for the 10-kW WT at cut-in and rated conditions are 18 and 60 r/min, respectively. This means that the corresponding operational frequency (obtained by dividing the rotor speed (r/min) by 60) lies between 0.3 and 1 Hz, and the blade passing frequency $(3 f_{r})$ between 0.9 and 3 Hz. In the WT industry, a ±10% safety distance should be applied to the natural frequency of the mast to take into account the calculation uncertainties and design assumptions (Von der Haar and Marx, 2015; Wang and Ran, 2014). So, the first natural frequency constraint of the mast is between 0.33 Hz ( $1.1 f_{r}$ of the cut-in wind speed) and 2.7 Hz ( $2.7 f_{r}$ of the rated wind speed), where $1.1 f_{r} \leq f_{nat} \leq 2.7 f_{r}$ .

The mast models are still considered with fixed boundary conditions at their bottoms. Similarly, all the other parameters (material property, mast dimensions, and types of elements) of the masts are identical to those of the previous corresponding models (static analysis) without the extreme loads (modal analysis does not require applying loads). The FFEPlus solver (Lanczos method) performance output was used to extract natural frequencies and mode shapes (SolidWorks, 2016).

Only odd vibration modes of the 12-, 14-, 16-, and 18-m masts for the three models are shown in Figures 7 –10, respectively, since modes n and n + 1 (with n = 1, 3, ...) have almost the same frequency but vibration occurs in orthogonal planes, due to the symmetry of the structure. The results show that, for all models of the 12- and 14-m masts (Figures 3 and 4, respectively), the first and the third mode shapes bend (fore-and-after) along the X-direction in the XY plane, the second and fourth mode shapes bend (side-to-side) along the Z-direction in the YZ plane, and the fifth mode shape is the first torsional mode around the Y-axis in the XZ plane. For the 16- and 18-m masts (Figures 5 and 6, respectively), the first and the third mode shapes bend (fore-and-after) along the Z-direction in the YZ-plane, the second and the fourth mode shapes bend (side-to-side) along the X-direction in the XY-plane, and the first torsional mode (Y-axis) is the seventh mode shape. The first 10 natural frequencies of the masts are presented in Table 5.

Table 5.

Natural frequency of the WT masts (Hz).

Modes	Model 1 (Hz)				Model 2 (Hz)				Model 3 (Hz)
	12 m	14 m	16 m	18 m	12 m	14 m	16 m	18 m	12 m	14 m	16 m	18 m
1	3.2456	2.4905	1.9636	1.5846	3.9550	3.0286	2.3836	1.9205	4.4986	3.4774	2.7347	2.1808
2	3.2458	2.4905	1.9636	1.5847	3.9554	3.0287	2.3838	1.9206	4.4996	3.4779	2.7348	2.1810
3	21.876	16.230	12.479	9.9358	25.998	19.403	14.959	11.921	29.199	22.034	16.999	13.442
4	21.877	16.231	12.479	9.9359	26.000	19.405	14.960	11.922	29.206	22.036	16.999	13.443
5	49.870	44.450	35.277	28.541	54.573	48.226	41.628	33.767	44.800	44.749	44.630	37.767
6	60.093	44.657	35.279	28.542	55.391	52.621	41.634	33.768	44.806	44.754	44.638	37.770
7	60.095	44.710	40.158	36.564	55.394	52.627	43.189	39.072	57.270	50.146	44.726	40.471
8	76.754	76.868	68.656	54.372	64.811	55.350	55.798	55.780	57.869	56.562	46.774	44.003
9	76.758	76.882	68.658	54.373	64.814	55.357	55.803	55.823	57.870	56.566	46.780	44.008
10	82.334	79.700	71.187	64.263	70.504	63.748	59.077	56.826	78.434	58.981	49.784	45.674

From the results plotted in Figure 7, it can be seen that the natural frequencies of 12 m mast (all models) are greater than the blade passing frequency (2.7 Hz), so the 12-m masts are of stiff–stiff design, and the resonance problem will not appear during operational WT. The 14 m (models 2 and 3) and 16 m (model 3) masts are similar to those of 12 m masts. The natural frequencies of the 14 m (model 1), 16 m (models 1 and 2), and 18 m (all models) masts all lie between 0.33 and 2.7 Hz, which satisfies the natural frequency stiffness requirements of $1.1 f_{r} \leq f_{nat} \leq 2.7 f_{r}$ and indicates that these masts are of soft–stiff design (see Figures 8 –10). Table 6 shows the design of each type of mast model.

Table 6.

Design of each type of mast model.

Height of mast (m)	Models	Mast design type
12	1	Stiff–stiff
	2	Stiff–stiff
	3	Stiff–stiff
14	1	Soft–stiff
	2	Stiff–stiff
	3	Stiff–stiff
16	1	Soft–stiff
	2	Soft–stiff
	3	Stiff–stiff
18	1	Soft–stiff
	2	Soft–stiff
	3	Soft–stiff

Figure 7.

Natural mode shapes for the 12-m mast structure: (a) model 1, (b) model 2, and (c) model 3.

Figure 8.

Natural mode shapes for the 14-m mast structure: (a) model 1, (b) model 2, and (c) model 3.

Figure 9.

Natural mode shapes for the 16-m mast structure: (a) model 1, (b) model 2, and (c) model 3.

Figure 10.

Natural mode shapes for the 18-m mast structure: (a) model 1, (b) model 2, and (c) model 3.

Conclusion

In this research article, the evaluations of masts’ structural performance for 10 kW VAWT were investigated using detailed FEAs. The simulations included the evaluation of WT masts with four different heights and three various outer diameters for each height case subjected to aerodynamic and gravity loadings (according to the IEC 61400-2:2006 and EN 1991-1-4:2005 standards). Static strength and structural dynamic responses were obtained to identify stiffness, resistance, reliability, and natural frequency stiffness requirements within the mast models, in order to save the manufacturing cost. Numerical results reveal that

The structure of the masts will not fail during WT operation because the maximum stresses are all lower than the yield strength of the material (lowest factor of safety is equal to 1.3) and all related maximum total deformations meet material deflection;

The natural frequencies of 12 m (all models), 14 m (models 2 and 3), and 16 m (model 3) masts are greater than the blade passing frequencies (3P excitation), so these masts are of stiff–stiff design (uneconomical) and will not experience resonance during operational WT. However, the natural frequencies of 14 m (model 1), 16 m (models 1 and 2), and 18 m (all models) masts satisfied the natural frequency stiffness requirements, which indicates that these masts are of soft–stiff design (economical).

Footnotes

Acknowledgements

The authors would like to thank the research financial mast provided by the Algerian Government Ministry of Higher Education and Scientific Research and the General Directorate for Scientific Research and Technological Development.

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) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is a part of the research project “Mechanical and Vibratory Advanced Design of a (4*25 kW) Vertical-Axis Wind Turbine with Direct Attack” funded by the Renewable Energy Research Unity in Saharan Medium (URER/MS) linked to the Renewable Energy Development Center (CDER).

ORCID iD

Fateh Ferroudji

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