^{1}

^{1}

^{1}

^{1}

^{1}

In this paper, the dynamic behavior of a one-stage bevel gear used in vertical axis wind turbine in transient regime is investigated. Linear dynamic model is simulated by fourteen degrees of freedom. Gear excitation is induced by external and internal sources which are, respectively, the aerodynamic torque caused by the fluctuation of input wind speed in transient regime and the variation of gear mesh stiffness. In this study, the differential equations governing the system motion are solved using an implicit Newmark algorithm. In fact, there are some design parameters, which influence the performance of vertical axis wind turbine. In order to get the appropriate aerodynamic torque, the effect of each parameter is studied in this work. It was found that the rotational speed of the rotor shaft has a significant effect on the aerodynamic torque performance.

Generally, vertical axis wind turbines (VAWT) have a particular architecture compared with horizontal ones. They are composed of two main parts: the blade rotor in vertical position and a mechanical gear transmission (bevel gear). Vibrations of the aerodynamic part caused by the wind speed variation are transmitted to the other part (gear transmission system) via shaft, gears, and bearing.

In literature, plenty of authors studied the aerodynamic performance of Darrieus-type of VAWT. There are two mainly approaches: momentum models and Computational Fluid Dynamics (CFD). The main benefit of momentum models is that their time of resolution is quicker than the other approach. Although the Computational Fluid Dynamics have been a useful design tool for studying the efficiency of wind turbine, the mesh generation in three-dimensional analyses needs a lot of time for the simulation. Among analytical models are researches [

The (CFD) method has been widely used in developing the characteristics of wind turbine (torque fluctuation, power output, and pressure distribution). Jiang et al. [

Accordingly, there are many related literatures studying bevel gears transmission. Cai-Wan Chang-jian [

In this paper, we discuss the impact of some design parameters including number of blades, turbine radius, chord length, blade length, and rotational speed on the aerodynamic torque of the H-Darrieus VAWT through analytical approach.

The main objective of the present work is to predict the dynamic behavior of the one-stage straight bevel gear system used in vertical axis wind turbine and powered by two main sources of excitation which are the optimum aerodynamic torque selected through parametrical study and the periodic variation of the gear meshes’ stiffness.

In this section, analytical investigation of aerodynamic torque of Darrieus wind turbine is established. The actuator disk theory is chosen for the aerodynamic study of the Darrieus-type wind turbine with straight blade. This theory characterizes the turbine as a disc with a discontinuity of pressure in the stream tube of air, which causes a deceleration of the wind speed. Referring to Figure

Flow velocities and forces in Darrieus wind turbine [

The tangential and normal forces as function of the azimuth angle

This part investigates the studying of the dynamic behavior of bevel gear system used in vertical axis wind turbine. The main excitations of the one-stage bevel gear system are the selected aerodynamic torque estimated through parametrical analysis in addition to the internal mesh stiffness excitation.

The dynamic model of single-stage bevel gear is presented by fourteen degrees of freedom (see Figure

Single-stage bevel gear model.

The second block includes the pinion

The proposed dynamic model is modeled by the generalized coordinate vector

The equations of motion describing the dynamic behavior of the model are established using the formalism of Lagrange for each degree of freedom of the system.

The tooth deflection following the line of action is defined by

Components of the tooth deflection.

c_{1} | |

c_{2} | |

c_{3} | |

c_{4} | |

c_{5} | |

c_{6} | |

c_{7} | |

c_{8} | |

c_{9} | |

c_{10} | |

c_{11} | |

c_{12} | |

The load vector can be written as

_{21} are the half-angle of the base circle of bevel gear (i) of the block (j) and the radius of the sphere which contains two bevel gears, _{12} = r_{21}).

The angles

The main purpose of this study is to investigate the effect of different design parameters (blade chord, number of blades, and radius turbine) on the aerodynamic torque evaluation of H-Darrieus turbine, using analytical approach.

The solidity

The parameters of the bevel gear transmission are presented in Table

Parameters of the studied bevel gear system.

Teeth number | 18 / 45 |

module(m) | 0.004 |

Bearing stiffness (N/m) | |

_{ z1 }= _{z2 }= 4.10^{8} | |

Torsional stiffness(N/rd/m) | ^{8} |

Pressure angle | |

Contact ratio | _{α}=1.56 |

Average mesh stiffness(N/m) | _{ moy }=410^{8} |

Density (42CrMo4) | 7860kg/m^{3} |

Wind turbine specification.

Type | Straight blade Darrieus |

Airfoil profile | NACA0018 |

Airfoil chord(mm) | 480 |

Blade length (m) | 3.66 |

Turbine diameter (m) | 4 |

Blade number | 3 |

Speed of rotor (tr/min) | 50 |

The number of blades (n) is an important factor that influences the torque produced. It is well known that a bigger number of blades give rise to the solidity and produce a global torque with small fluctuation. However, increasing the number of blades lead to increase in the turbine drag by increasing the number of connecting shafts. Figure

Effect of blades number on the torque.

In Figure

Effect of blade chord on the torque.

As can be assumed from the torque equation (

Effect of radius turbine on the torque.

Effect of blade height on the torque.

Figure

Effect of rotation speed on the torque.

As shown in Figures

Effect of wind speed on the torque (N=180tr/min).

Effect of wind speed on the torque (N=50tr/min).

We clearly see only positive fluctuation of the aerodynamic torque at N=180 tr/min; however, this torque presents negative oscillation at small rotational speed (N=50 tr/min) at fixed wind speed of 14 m/s. Consequently, we conclude the most favorable speed excitation of the considerable Darrieus rotor that corresponds to wind velocity of 14m/s and rotational speed of 180 tr/min which respects the condition of positive torque evolution.

This paper presents a three-dimensional model of one-stage straight bevel gear system used in vertical axis wind turbine. Aerodynamic torque fluctuation and periodic oscillation of the gear meshes’ stiffness are the main sources of excitation for the bevel gear system.

Influence of geometrical parameters of Darrieus rotor has been done in order to select the appropriate parameters. The optimization process has been carried out on the effect of each parameter on the aerodynamic torque produced.

It was found that the aerodynamic torque increases when the chord, the radius, and the height of VAWT rise. However, the best performance is detected for 3-bladed VAWT. Finally, the most significant parameter that affects the aerodynamic torque is the rotation speed of the rotor shaft.

Axial induction factor

Turbine swept area

Chord (m)

Proportional damping matrix

Tangential force coefficient

Normal force coefficient

Average torque coefficient

Power coefficient

Lift and drag coefficients

Tangential force

Normal force

External excitation force

Blade height (m)

Moment of inertia of the wheel

Diametrical moment of wheel

Gear mesh of stiffness

Time stiffness matrix of the gear mesh fluctuation

Stiffness matrix of the average structure

Shafts torsional stiffness

Bending and traction-compression bearing stiffness

Mass matrix

Mass of block i

Vector of geometric parameters of the dynamic model

Blade number

Rotational speed of the rotor (tr/min)

Generalized coordinate vector

Radius of the wind turbine (m)

Resisting torque

Aerodynamic torque

Wind free stream velocity (m/sec)

Induced velocity

Chordal velocity component and the normal velocity component, respectively

Relative flow velocity

Azimuth angle

Angular velocity (rad/sec)

Air density[kg/m^{3}]

Tip speed ratio

Relative displacement of the contact point along the line of action

Geometric angle of bevel gear

Bearing displacements in each blocs (i=1:2)

Angular displacement of the bearing following X and Y, respectively

Angular displacement of wheel and gear following Z direction (i=1:2)

Solidity

Angle of attack.

All results are obtained by simulation with Matlab.

The authors declare that they have no conflicts of interest.