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(1. College of Engineering, Northeast Agricultural University, Harbin，Heilongjiang 150030, China; 2. Heilongjiang Provincial Key Laboratory of Technology and Equipment for Utilization of Agricultural Renewable Resources in Cold Region, Harbin，Heilongjiang 150030, China; 3. College of Science, Northeast Agricultural University, Harbin,Heilongjiang 150030, China; 4. College of Energy and Power Engineering, Changsha University of Science and Technology, Changsha，Hunan 410114, China)

Abstract: The finite element analysis was carried out for a composite vertical axis wind turbine with lift-drag combined starting structures to ensure the structure safety of a vertical axis wind turbine (VAWT). The static and modal analysis of rotor of a composite vertical axis wind turbine was conducted by using ANSYS software. The relevant contour sketch of stress and deformation was obtained. The analysis was made for static structural mechanics, modal analysis of rotor and the total deformation and vibration profile to evaluate the influence on the working capability of the rotor. The analysis results show that the various structure parameters lie in the safety range of structural mechanics in the relative standards. The analysis showing the design safe to operate the rotor of a vertical axis wind turbine. The methods used in this study can be used as a good reference for the structural mechanics′ analysis of VAWTs.

Key words: vertical axis wind turbine;finite element analysis;static structural mechanics;lift-drag combined starting structure;model analysis

1 Introduction

Nowadays, the vertical axis wind turbine (VAWT) becomes a hot spot for investigators in view of the advantages including simple structure, conve-nient installation and maintenance. As for the remote areas such as islands and pasture lands, small-scale VAWT is the major application model of off-grid power systems[1].

In recent years, the application prospect of straight-bladed VAWT (SB-VAWT) has considered more promising. However, the poor starting characte-ristics are one of the main factors restricting the deve-lopment of SB-VAWT[2-4]. Therefore, many researc-hers installed auxiliary devices around[5]or above and below[6]the rotor, or developed new wind turbine structures[7]to enhance the starting characteristics of SB-VAWT. After the improvement of the starting cha-racteristics, the aerodynamic performance and structural mechanical characteristics of SB-VAWT will become different. At present, many researchers mainly evaluate the aerodynamic performance change of modified wind turbines by CFD[8]and wind tunnel test[9]. However, works about the structural mechanical characteristics of improved wind turbines are quite a few[10], which mean that there is little analysis on structural mechanics of original straight-bladed wind turbines[11]. According to the theory of related hydromechanics theory and small-scale VAWT design standard， a lift-drag combined starting structure was brought up, which aims to improve the starting characteristics of the SB-VAWT, and the aerodynamic performance analysis shows that the structure works effectively[12]. However, VAWT is a kind of rotating machinery: on the basis of enhancing aerodynamic characteristics, the reliability of the wind turbine structure is of great importance to its operation. Research about whether modification of SB-VAWT, by using the lift-drag combined starting structure, is reliable in structure and operation security has been necessary. In order to ensure the reliability and rationality of the structure and design of the wind turbine, the static analysis and modal analysis of the rotor were carried out, and whether this structure met the design standards was determined by the results.

2 Model design

Basing on the wind power situation demand, the LDCS-CVAWT aerodynamic performance was studied, and the relationship between the power of the wind turbine satisfying the demand and rotational speed is shown in Fig.1.

Fig.1 Power curve of LDCS-CVAWT

When the rated wind speed reaches 10 m/s, the power of the wind turbine is about 3 102 W, the utilization coefficient of the wind turbine is 0.3.

Fig.2 shows the structural model of the wind turbine. It mainly constitutes of lift-type blades, lift-drag combined structure blades, main axis, beam, flange and pillar. The parameters of the main components are shown in Tab.1. Among them, the lift-drag combined structure blades are the main improved components of this wind turbine, which is different from the traditional SB-VAWT. It can effectively provide the starting force moment of the SB-VAWT[12].

Fig.2 Three dimensional diagram of LDCS-CVAWT

Tab.1 Parameters of LDCS-CVAWT

3 Finite element analysis

3.1 Static structural mechanics analysis

The finite element method (FEM) was applied to analyze the design and mechanics structure of the wind turbine, and then to judge the feasibility of structural design[13]. Before the finite element analysis, the wind turbine structure was discretized into elements, and the original structural model was instead of the element collection. The displacement function was selected to calculate the relationship between node displacement and any point displacement among the element. The expression is[14]

u=Nδe，

(1)

Where:uis the displacement vector of any point in the element;Nis the shape function (a function of position coordinate);eis the node displacement vector of the element.

So the strain and stress in the element are[15]

ε=Bδe，

(2)

σ=Dε=DBδe=Sδe，

(3)

WhereBandSare strain matrix and stress matrix respectively.

Then the mechanical properties of the element can be analyzed. According to the elastic theory and the principle of minimum potential energy, it is obtained[16]

Keδe=fe，

(4)

WhereKeis the element stiffness matrix;feis the equivalent nodal force.

The stiffness matrix of each element is integrated into the stiffness matrix of the complete structure. Then the equivalent nodal force vector acting on each element is combined into the total load vector. So the equation of equilibrium for the complete structure is[17]

Kδ=f.

(5)

According to Equation (5), the boundary conditions of the wind turbine structure are set. After the proper modification of Equation (5), the displace-ment,δ, of the structure can be solved. The element stress or internal force can then be calculated by using the element characteristics.

3.2 Modal analysis

The natural frequency and vibration mode of wind turbine structure can be determined by modal analysis. By analyzing the natural frequency and vibration mode, it can be confirmed that the vibration reaction of the structure under the action of the vibration source is within this frequency range. The structural dynamics model is established by using the finite element method of elastic mechanics, and its finite element dynamics can be expressed as Equation (6)[16]

(6)

When there is no external constraint or excitation force for the structure and the damping is ignored, Equation (7) can be changed into the undamped free vibration differential equation

(7)

The frequency equation can be obtained by solving non-zero solutions about Equation (8)[16]

|{K-ω2[M]}|=0,

(8)

Whereωis the natural frequency of the structure, the natural frequency of the system can be obtained by solving the differential equation, and then the natural vibration profile of each order can be obtained.

4 Results and analysis

4.1 Static structural mechanics analysis of rotor

The finite element structural model of the rotor is shown in Fig.3. At this time, only the wind pressure load and gravity load are considered.

LDCS-CVAWT is generally used in places with good wind speed, such as rural regions border regions and cold regions. Wind resources are relatively good. The rated working wind speed of the wind turbine can be designed forU=10 m/s, and the formula of wind pressure load is set in Equation (9)[17]

(9)

WherepZis the distribution of wind pressure load, N/m;ρis the density of air,ρ=1.225 kg/m3;Uis wind speed,U=10 m/s;Zis the height of the wind turbine,Z=12 m;H0is the based height,H0=10 m;U0is the wind speed of 10 m high;αqis ground roughness and ground wind shear coefficient,αq=0.12. It can be calculated thatpZ=63.98 N/m. When the finite element analysis and calculation are carried out, the full constraint is applied to the bottom of the axis, and the wind load and self-gravity load are introduced. The stress and strain contour plots under the working condition are shown in Fig.4 and Fig. 5.

Fig.3 Finite element model of rotor

Fig.4 shows that maximum stress of LDCS-CVAWT is 237.83 MPa. Fig.5 shows the maximum displacement of LDCS-CVAWT is 37.572 mm, maximum stress and maximum deformation occurs in central lift-type blades. According to article 4.7 of safety factor and material characteristic value criterion of GB/T 17646—2013, for plastic and non-plastic deformed materials, ultimate tensile strength (UTS) would be adopted as material characteristic value,Rchar, for fatigue and ultimate load.γis the safety factor, the value depends on the loading condition and material, the composite material safety factor is 1.25, the strength limit of the central lift-type blades is 320 MPa,σd=256 MPa. Fig.4 shows that the maximum stress of LDCS-CVAWT is 237.83 MPa, which is not overσd, so the design meets the safety requirements. According to article 2.0.8 of the high-rise structure design specification of GB50135—2019, the control condition of the limit state of the smooth operation of high-rise structure should conform to that, under the action of wind load, the horizontal displacement of any point in the high-rise structure should not be greater than 1/100 of the height above the ground. The center height of LDCS-CVAWT above the ground is 8.6 m. Fig.5 shows the maximum displacement of LDCS-CVAWT is 37.572 mm, which is lower than 86 mm, metting the design requirements.

Fig.4 Stress distributions of rotor

Fig.5 Total deformation of rotor

4.2 Modal analysis of rotor

It can be seen in Fig.6 that the vibration shape of LDCS-CVAWT is at the first order mode natural frequency, and the blade is shown as bending vibration, to be specific, waving vibration, the same as second order and third order.

At the fourth and fifth order, the blade vibration is mainly reflected as the complex vibration of swing and oscillating vibration. At the sixth order, torsional vibration becomes a major vibration of blades, accompanied by swing vibration and oscillating vibration. According to the vibration theory,the energy in the vibration process is mainly concentrated in the first and se-cond order. Therefore, waving vibration is the main vibration of LDCS-CVAWT.

Since swing vibration appears more frequently on the blades, which mainly occurs in the lift-type blade and lift-type blade tension belt, it can be indicated that the weak radial stiffness of blades needs to be strengthened. In addition, the tension belt should further enhance the strength and stiffness while production for avoiding fatigue damage, casualties and economic losses.

Fig.6 First six order mode vibration profile of rotor

The natural frequencies of the first six order mode of LDCS-CVAWT are as collected in Tab.2.

Tab.2 Natural frequency of the first six order mode of rotor

It can be seen from Tab.2 that the lowest natural frequency of the wind turbine isf1=2.931 8 Hz, which is calculated by the formula of the first critical speed of the blade (2)

n0=60f.

(10)

It can be calculated that the first order critical speed isn0= 175.9 r/min, while the working speed range of the wind turbine is about 5—100 r/min, which is far less than the first order critical speed. According to the requirements of article 5.2.5.3 of the national standard of small vertical axis wind turbine gene-rator set of GB/T 29494—2013, the support structures and towers of wind turbines should be evaluated for the resonance to avoid resonance. The calculation results show that the resonance phenomenon would not occur in the operation of the LDCS-CVAWT structure.

4.3 Prototype wind turbine for field test

According to the model design and analysis results of LDCS-CVAWT, a prototype was made and installed in a farm of Northeast Agricultural University of China for testing, which is shown in Fig.7.

Fig.7 Prototype of LDCS-CVAWT

Fig.8 shows the local wind speed distribution in a year, which indicates that the prototype can work safely and stably.

Fig.8 Local wind speed distribution in a year

5 Conclusions

A composite SB-VAWT with lift-drag combined starting structures was designed to develop a model for the study of the static structural mechanics analysis and modal analysis of rotor. The static and dynamic characteristic of the rotor was analyzed by using ANSYS software, in which the relevant contour plots of stress and deformation were developed. The static structural mechanics analysis was carried out to evaluate the strength and stiffness. The displacement deformation and stress distributions of the rotor show that the structural design of the wind turbine is within the safety range of structural mechanics. The natural frequencies obtained from the modal analysis show the resonance can be avoided and the wind turbine can operate reliably.