Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

Analyses of Dynamic Behavior of Vertical Axis Wind Turbine in Transient Regime

Analyses of Dynamic Behavior of Vertical Axis Wind Turbine in Transient Regime Hindawi Advances in Acoustics and Vibration Volume 2019, Article ID 7015262, 9 pages https://doi.org/10.1155/2019/7015262 Research Article Analyses of Dynamic Behavior of Vertical Axis Wind Turbine in Transient Regime Bacem Zghal , Imen Bel Mabrouk, Lassâad Walha, Kamel Abboudi, and Mohamed Haddar Laboratory of Mechanical Modeling and Production (LAMP), National School of Engineers of Sfax (ENIS), University of Sfax, BP , , Tunisia Correspondence should be addressed to Bacem Zghal; bacem.zghal@gmail.com Received 19 October 2018; Accepted 6 March 2019; Published 10 April 2019 Academic Editor: Emil Manoach Copyright ©  Bacem Zghal et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 sha has a significant effect on the aerodynamic torque performance. 1. Introduction or vortex model [–] have been developed to optimize VAWTs performance. Generally, vertical axis wind turbines (VAWT) have a par- e (CFD) method has been widely used in developing ticular architecture compared with horizontal ones. ey are the characteristics of wind turbine (torque fluctuation, power composed of two main parts: the blade rotor in vertical output, and pressure distribution). Jiang et al. [] employed position and a mechanical gear transmission (bevel gear). a commercial CFD simulation for studying the effect of Vibrations of the aerodynamic part caused by the wind speed geometrical parameters and airfoil type on the performance variation are transmitted to the other part (gear transmission of the H-Darrieus turbine with fixed pitch angle. Also, a system) via sha , gears, and bearing. numerical analysis of H-rotor Darrieus turbine is introduced In literature, plenty of authors studied the aerodynamic by M.H Mohamed et al. []. e developing of torque fluctuation is a very important step for the reason that performance of Darrieus-type of VAWT. ere are two mainly approaches: momentum models and Computational the instantaneous torque produced by the rotor blade is Fluid Dynamics (CFD). e main benefit of momentum directly related to both the power generation and the gearbox models is that their time of resolution is quicker than the vibration. other approach. Although the Computational Fluid Dynam- Accordingly, there are many related literatures studying bevel gears transmission. Cai-Wan Chang-jian [] studied ics have been a useful design tool for studying the efficiency of wind turbine, the mesh generation in three-dimensional the nonlinear dynamic behavior of bevel gear system. Besides, analyses needs a lot of time for the simulation. Among Fujii et al. [] analyzed the dynamic vibration of straight bevel gear supported by angular bearings and tapered roller. analytical models are researches [–] based on Actuator Disk and Blade-Element Method (BEM) to predict the M. Li and H. Y. Hu [] studied the dynamic analysis of a aerodynamic torque of VAWT. In addition, computational spiral bevel-geared rotor-bearing system. Y. Wang et al. [] aerodynamics methods such as multiple stream tube method and J.F. Besseling [] developed a new approach based on  Advances in Acoustics and Vibration Wind V(t) Drag Lift 2 W(t) 6 =IM() W(t) Drag 6 MCH() Lift Drag W(t) F  Lift F : Flow velocities and forces in Darrieus wind turbine []. 2 2 finite element theory to model bevel gear systems. Moreover, () 𝑊= +𝑉𝑛 Driss Yassine et al. [] present the model of two-stage straight bevel gear system excited with only internal excitation which = 𝜔𝑅 + cos 𝜃 is the periodic fluctuations of the gear meshes’ stiffness. = 𝑎𝑉 sin 𝜃 () In this paper, we discuss the impact of some design parameters including number of blades, turbine radius, chord = 𝑉 𝑡 1−𝑎 ( )( ) length, blade length, and rotational speed on the aerody- namic torque of the H-Darrieus VAWT through analytical approach. e angle of attack is defined as the angle between the resul- e main objective of the present work is to predict the tant air velocity vector and the blade chord. It is expressed as dynamic behavior of the one-stage straight bevel gear system follows: used in vertical axis wind turbine and powered by two main sources of excitation which are the optimum aerodynamic 𝑉 (1−𝑎 ) sin 𝜃 −1 𝑛 𝛼 (𝜃 ) = tan ( ) = ( ) () torque selected through parametrical study and the periodic 𝑉 (1−𝑎 ) cos𝜃+𝜆 variation of the gear meshes’ stiffness. e value of the axial induction factor (a) can be introduced 2. Theoretical Modeling of VAWT Rotors by actuator disk theory. e resultant air velocity is dependent on the induced velocity and the tip speed ratio (TSR) defined In this section, analytical investigation of aerodynamic torque as of Darrieus wind turbine is established. e actuator disk the- ory is chosen for the aerodynamic study of the Darrieus-type wind turbine with straight blade. is theory characterizes 𝜆= () 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 , the relative flow velocity can be In this work, the induced velocity is modeled in deterministic obtained as follows: form, as a sum of several harmonics []: 𝜔𝑅 𝑉𝑎 𝑉𝑛 𝑉𝑎 𝑉𝑐 𝑉𝑐 Advances in Acoustics and Vibration Rotor (11)   11 1 Φ x1 y1 1 z1 Gear 12 K(t) Gear 21 2 2 Generator (22) Z K z2 Ψ y2 x2 F : Single-stage bevel gear model. 𝑉 =14 + 2 sin (𝜔𝑡 )−1.75 sin (3𝜔𝑡 )+1.5 sin (5𝜔𝑡 ) ∞ e averagetorqueproduced by therotor (n blades) is generated from the average tangential force acting on one −1.25 sin 10𝜔𝑡 + sin 30𝜔𝑡 +0.5 sin 50𝜔𝑡 ( ) ( ) ( ) () blade: +0.25 sin (100𝜔𝑡 ) 2𝜋 𝑇 𝜃 =𝑛 ∫ 𝐹 𝜃 𝑅 () ( ) ( ) 2𝜋 e resulting aerodynamic forces in the blade can be founded by the interpolation of the li and drag coefficients relative to e average torque coefficient is calculated by the symmetrical airfoil used (NACA), the angle of attack, and the given Reynolds number. 𝑇 (𝜃 ) 𝐶 = () e tangential and normal forces as function of the 0.5𝜌𝑉 azimuth angle 𝜃 can be calculated using the blade-element Finally, the power coefficient Cp is estimated from the average theory []. torque coefficient: 𝐹 (𝜃 ) = 𝜌𝑐 𝑊 ℎ𝑐 𝑡 𝑡 = 𝜆𝐶 () () 𝐹 (𝜃 ) = 𝜌𝑐 𝑊 ℎ𝑐 3. Theoretical Modeling of the One-Stage Bevel 𝑛 𝑛 Gear System where C and C are the normal and tangential coefficients, n t is part investigates the studying of the dynamic behavior of respectively, calculated from the li and drag coefficients (C bevel gear system used in vertical axis wind turbine. e main and C ) using the same theory (blade-element momentum excitations of the one-stage bevel gear system are the selected theory), given by following expressions: aerodynamic torque estimated through parametrical analysis 𝐶 =𝐶 sin𝛼− 𝐶 cos 𝛼 in addition to the internal mesh stiffness excitation. 𝑡 𝐿 𝐷 () e dynamic model of single-stage bevel gear is presented 𝐶 =𝐶 cos𝛼+ 𝐶 sin 𝛼 𝑛 𝐿 𝐷 by fourteen degrees of freedom (see Figure ). is model 𝐶𝑝 𝐴𝑅 𝑑𝜃  Advances in Acoustics and Vibration includes two blocks where the first block is constituted by the e mesh stiffness matrix can be defined by Darrieus rotor modeled by the mass (11) linked to the wheel () (12) via a first sha with torsional rigidity K𝜃1 ;this block is [𝐾 (𝑡 )] =𝑘 (𝑡 )⟨𝐿 ⟩ . ⟨𝐿 ⟩ supported by a bearing. [k(t)] is the total meshing stiffness of the gear pair. Sha Wei et e second block includes the pinion (21),the second al. [] haveexpressed thetotal meshing stiffnessofthe gear sha with torsional rigidity K𝜃2 , and the generator modeled pair by a periodic excitation decomposed on Fourier series. by a mass (22). It is also supported by a bearing. e wheel e tooth deflection following the line of action is defined (12) is connected to the pinion (21) via teeth mesh stiffness. by e bearings are modeled by linear springs acting on the lines of action. e mesh stiffness characterizes the elastic 𝛿 𝑡 = 𝐿 .{𝑞 𝑡 } ( ) ⟨ ⟩ ( ) () deformations managing the relative positions of the two gear wheels; it can modeled by the mesh stiffness 𝑘(𝑡). {𝐿 } ={𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,0,𝑐 ,𝑐 ,0} () 1 2 3 4 5 6 7 8 10 11 9 12 e proposed dynamic model is modeled by the general- ized coordinate vector {𝑞}: e components of the tooth deflection are presented in Table . {𝑞 (𝑡 )} e load vector can be written as () =[𝑥 ,𝑦 ,𝑧 ,𝑥 ,𝑦 ,𝑧 ,𝜙 ,𝜓 ,𝜙 ,𝜓 ,𝜃 ,𝜃 ,𝜃 ,𝜃 ] 1 1, 1 2 2 2 1 1 2 2 11 12 21 22 {𝐹 (𝑡 )} () {𝑥𝑖, 𝑦𝑖, 𝑧𝑖} are the linear displacements of bearing in each 00 000 000 00𝑇 (𝑡 ) 00 −𝑅 (𝑡 ) ={ } block. {𝜙𝑖, 𝑖𝜓, 𝜃 ,𝜃 } are the angular displacement of bearing 1𝑖 2𝑖 T(t) and R (t) arethe aerodynamic torqueproduced by and wheel (i=:). the rotor and the resisting torque, respectively. [C] is the e equations of motion describing the dynamic behavior proportional damping matrix of the model are established using the formalism of Lagrange −5 for each degree of freedom of the system. ̃ [𝐶 ] =0.05 [𝑀 ] +10 [𝐾] () ̈ ̇ [𝑀 ] {𝑞} + [𝐶 ] {𝑞} + ([𝐾 ]+ [𝐾 (𝑡 )]){𝑞} = {𝐹 (𝑡 )} () where 𝑎 = sin(𝛿 ), 𝑏 = cos(𝛿 ), 𝑎 = sin(𝛿 ),and 12 𝑏12 12 𝑏12 21 𝑏21 𝑏 = cos(𝛿 ). 21 𝑏21 e total mass matrix is defined by 𝛿 and r are the half-angle of the base circle of bevel gear (i) of the block (j) and the radius of the sphere which [𝑀 ]0 [𝑀 ] =[ ] () contains two bevel gears, (12) and (21), respectively (r = 0[𝑀 ] r ). e angles 𝛿 and u describe the different parameters of [𝑀 ]= diag [𝑚 ,𝑚 ,𝑚 ,𝑚 ,𝑚 ,𝑚 ] () 𝐿 1 1 1 2 2 2 the bevel gear []. [𝑀 ]= diag [𝐼 +𝐼 ,𝐼 +𝐼 ,𝐼 +𝐼 ,𝐼 𝐴 11𝑥 12𝑥 11𝑦 12𝑦 21𝑥 22𝑥 21𝑦 () 4. Influence of Rotor Configuration on +𝐼 ,𝐼 ,𝐼 ,𝐼 ,𝐼 ] 22𝑦 11 12 21 22 Turbine Performance [𝐾 ] is the average stiffness matrix of the structure e main purpose of this study is to investigate the effect of different design parameters (blade chord, number of blades, [𝐾 ]0 𝑝 and radius turbine) on the aerodynamic torque evaluation of [𝐾 ]=[ ] () H-Darrieus turbine, using analytical approach. 0[𝐾 ] e solidity 𝜎 is defined as an important nondimensional parameter, which influences the self-starting abilities and where: [𝐾 ]= diag [𝑘 ,𝑘 ,𝑘 ,𝑘 ,𝑘 ,𝑘 ] () 𝑝 𝑥1 𝑦1 𝑧1 𝑥2 𝑦2 𝑧2 determines the applicability of the momentum models. e [𝐾 ] turbine is able to self-start for high solidity (𝜎≥ .) []. For straight bladed VAWTs, the solidity is calculated by 𝑘 0 0 0 00 00 𝜙1 𝑛𝑐 [ ] [ ] 𝜎= () 0𝑘 0 0 00 00 𝜓1 [ ] [ ] [ 00 𝑘 00 0 0 0 ] 𝜙2 [ ] In order to study the dynamic behavior of bevel gear system [ ] () [ 000 𝑘 00 00 ] used on vertical axis wind turbine and powered by the 𝜓2 [ ] [ ] aerodynamic torque in transient regime in addition to the 0000 𝑘 −𝑘 00 [ ] 𝜃1 𝜃1 [ ] periodic variations of mesh stiffness, we used numerical [ ] 0000 −𝑘 𝑘 00 [ 𝜃1 𝜃1 ] simulation based on implicit method of Newmark. [ ] [ ] e parameters of the bevel gear transmission are pre- 0000 0 0 𝑘 −𝑘 𝜃2 𝜃2 [ ] sented in Table . Specifications of the vertical axis wind 0000 0 0 −𝑘 𝑘 𝜃2 𝜃2 [ ] turbine are shown in Table . 𝑏𝑗 Advances in Acoustics and Vibration T : Components of the tooth deflection. c 𝑏 sin(𝑎 𝑢 ) 1 12 12 12 c 𝑎 cos 𝑢 sin(𝑎 𝑢 )− sin 𝑢 cos(𝑎 𝑢 ) 2 12 12 12 12 12 12 12 c 𝑎 sin 𝑢 sin(𝑎 𝑢 )+ cos 𝑢 cos(𝑎 𝑢 ) 3 12 12 12 12 12 12 12 c 𝑏 sin(𝑎 𝑢 ) 4 21 21 21 c 𝑎 cos 𝑢 sin(𝑎 𝑢 )− sin 𝑢 cos(𝑎 𝑢 ) 5 21 21 21 21 21 21 21 c 𝑎 sin 𝑢 sin(𝑎 𝑢 )+ cos 𝑢 cos(𝑎 𝑢 ) 6 21 21 21 21 21 21 21 c 𝑐 𝑟 𝑏 cos(𝑎 𝑢 )− 𝑐 𝑟 (𝑎 cos 𝑢 cos (𝑎 𝑢 )+ sin 𝑢 sin(𝑎 𝑢 )) 7 2 12 12 12 12 1 12 12 12 12 12 12 12 12 c 𝑐 𝑟 (𝑎 sin 𝑢 cos (𝑎 𝑢 )− cos 𝑢 sin (𝑎 𝑢 )) − 𝑐 𝑟 𝑏 cos(𝑎 𝑢 ) 8 1 12 12 12 12 12 12 12 12 3 12 12 12 12 c 𝑐 𝑟 (𝑎 cos 𝑢 cos (𝑎 𝑢 )+ sin 𝑢 sin (𝑎 𝑢 )) − 𝑐 𝑟 (𝑎 sin 𝑢 cos (𝑎 𝑢 )+ cos 𝑢 sin(𝑎 𝑢 )) 9 3 12 12 12 12 12 12 12 12 2 12 12 12 12 12 12 12 12 c 𝑐 𝑟 (𝑎 cos 𝑢 cos (𝑎 𝑢 )+ sin 𝑢 sin (𝑎 𝑢 )) − 𝑐 𝑟 𝑏 cos(𝑎 𝑢 ) 10 4 21 21 21 21 21 21 21 21 5 21 21 21 21 c 𝑐 𝑟 (𝑎 sin 𝑢 cos (𝑎 𝑢 )− cos 𝑢 sin (𝑎 𝑢 )) − 𝑐 𝑟 𝑏 cos(𝑎 𝑢 ) 11 4 21 21 21 21 21 21 21 21 6 21 21 21 21 c −𝑐 𝑟 (𝑎 cos 𝑢 cos (𝑎 𝑢 )+ sin 𝑢 sin (𝑎 𝑢 )) + 𝑐 𝑟 (𝑎 sin 𝑢 cos (𝑎 𝑢 )− cos 𝑢 sin(𝑎 𝑢 )) 12 6 21 21 21 21 21 21 21 21 5 21 21 21 21 21 21 21 21 T : Parameters of the studied bevel gear system. Teeth number  / module(m) . 𝑘 =𝑘 =𝑘 =𝑘 =2 ⋅ 10 𝑥1 𝑦1 𝑥2 𝑦2 Bearing stiffness (N/m) k = k =. z z Torsional stiffness(N/rd/m) 𝑘 = 𝑘 = 𝜃1 𝜃2 Pressure angle 𝛼= 20 Contact ratio 𝜀 =. Average mesh stiffness(N/m) K = moy Density (CrMo) kg/m T : Wind turbine specification. Type Straight blade Darrieus Airfoil profile NACA Airfoil chord(mm) Blade length (m) . Turbine diameter (m) Blade number Speed of rotor (tr/min) e number of blades (n) is an important factor that thetorquebehavior ofVAWTturbine.Anincrease inradius influences the torque produced. It is well known that a bigger turbine and blade height advances the instantaneous torque number of blades give rise to the solidity and produce a as shown in Figures  and . global torque with small fluctuation. However, increasing the Figure  shows the total torque evolution versus azimuth angle generated for a complete revolution with differ- number of blades lead to increase in the turbine drag by increasing the number of connecting sha s. Figure  shows ent rotation speed. It can be observed that the torque the effect of the blade number on the torque in full revolution increases considerably with the increase of the rotation for a case where the radius and blade chord are maintained speed. Also, the torque fluctuation becomes positive when constant while the number of blades changes. As can be the rotation speed reaches  rev/min. However, for a deduced, -bladed VAWT perform more efficient than  and case of  rev/min and  rev/min the instantaneous bladed turbines. torque presents some fluctuation with negative magni- In Figure  the effect of blade chord on the torque tude which can explain the no self-starting of NACA fluctuation is presented. We remark that torque magnitude airfoil. increases when the chord length increases. Also, blades with As shown in Figures  and  rising wind speed from smaller chords require a bigger tip speed ratio to achieve a  to  m/s significantly affects the torque which increases considerably. e cyclic distribution of total torque at higher maximum torque so the selection of chord length affects the self-starting behavior of Darrieus turbine. rotational speed (N= rev/min) and at small rotational As can be assumed from the torque equation (), the rotor speed (N= rev/min) is observed at the same wind speed radius and the blade length have a major contribution in of  m/s.  Advances in Acoustics and Vibration −500 −1000 −1500 0 50 100 150 200 250 300 350 Azimuth angle 3-blade 2-blade 4-blade F : Effect of blades number on the torque. −500 −1000 0 50 100 150 200 250 300 350 Azimuth angle c=0.48 c=0.42 c=0.3 c=0.25 F : Effect of blade chord on the torque. −500 −1000 0 50 100 150 200 250 300 350 Azimuth angle R=3 R=2.5 R=1.8 R=1.4 F : Effect of radius turbine on the torque. Torque (N.m) Torque (N.m) Torque (N.m) Advances in Acoustics and Vibration −500 −1000 −1500 0 50 100 150 200 250 300 350 Azimuth angle h=3.66 h=2.5 h=1 F : Effect of blade height on the torque. −1000 −2000 0 50 100 150 200 250 300 350 Azimuth angle N=178 N=177 N=50 F : Effect of rotation speed on the torque. We clearly see only positive fluctuation of the aerody- Influence of geometrical parameters of Darrieus rotor has namic torque at N= tr/min; however, this torque presents been done in order to select the appropriate parameters. e negative oscillation at small rotational speed (N= tr/min) optimization process has been carried out on the effect of each at fixed wind speed of  m/s. Consequently, we conclude parameter on the aerodynamic torque produced. the most favorable speed excitation of the considerable It was found that the aerodynamic torque increases Darrieus rotor that corresponds to wind velocity of m/s and when the chord, the radius, and the height of VAWT rise. rotational speed of  tr/min which respects the condition of However, the best performance is detected for -bladed positive torque evolution. VAWT. Finally, the most significant parameter that affects the aerodynamic torque is the rotation speed of the rotor sha . 5. Conclusion is paper presents a three-dimensional model of one-stage Nomenclature straight bevel gear system used in vertical axis wind turbine. Aerodynamic torque fluctuation and periodic oscillation of a: Axial induction factor the gear meshes’ stiffness are the main sources of excitation A: Turbine swept area for the bevel gear system. c: Chord (m) Torque (N.M) Torque (N.m)  Advances in Acoustics and Vibration −1000 0 50 100 150 200 250 300 350 Azimuth angle UR=14 UR=12 UR=10 UR=8 F : Effect of wind speed on the torque (N=tr/min). −500 −1000 0 50 100 150 200 250 300 350 Azimuth angle UR=14 UR=12 UR=10 UR=8 F : Effect of wind speed on the torque (N=tr/min). [C]: Proportional damping matrix C : Tangential force coefficient [𝐾 ]: Stiffness matrix of the average structure C : Normal force coefficient k : Sha s torsional stiffness 𝜃 i C : Average torque coefficient T 𝐾 , 𝑘 , 𝑘 , 𝑥𝑖 𝑦𝑖 𝑧𝑖 Cp: Power coefficient 𝐾 and 𝐾 : Bending and traction-compression Φ𝑖 𝜓𝑖 𝐶 , 𝐶 : Li and drag coefficients 𝐿 𝐷 bearing stiffness F:Tangentialforce t [𝑀]:Massmatrix F : Normal force n m : Mass of block i {𝐹(𝑡)} :Externalexcitationforce ⟨𝐿⟩ : Vector of geometric parameters of the h: Blade height (m) dynamic model I : Moment of inertia of the wheel j of block i ij n: Blade number I ,I : Diametrical moment of wheel j of block i N: Rotational speed of the rotor (tr/min) ijx ijy respectively following the X- and Y-axes {𝑞(𝑡)} : Generalized coordinate vector 𝑘(𝑡): Gear mesh of stiffness R: Radius of the wind turbine (m) [𝐾(𝑡)] :Timestiffnessmatrixofthegearmesh 𝑅 (𝑡): Resisting torque fluctuation T(t):Aerodynamictorque Torque (N.m) Torque (N.m) Advances in Acoustics and Vibration Engineering Science and Technology, an International Journal, vol. , no. , pp. –, . V(t): Wind free stream velocity (m/sec) [] C.-W. Chang-Jian, “Nonlinear dynamic analysis for bevel-gear Va: Induced velocity system under nonlinear suspension-bifurcation and chaos,” Vc, Vn: Chordal velocity component and the Applied Mathematical Modelling, vol., no., pp. –, normal velocity component, respectively W: Relative flow velocity [] M. Fujii, Y. Nagasaki, and M. Nohara Trauchi, “Effect of bearing 𝜃:Azimuthangle on dynamic behavior of straight bevel gear train,” Transactions 𝜔:Angularvelocity(rad/sec) of the Japan Society of Mechanical Engineers A,vol., pp. – 𝜌:Airdensity[kg/m ] , . 𝜆 : Tip speed ratio [] M. Li and H. Y. Hu, “Dynamic analysis of a spiral bevel-geared 𝛿(𝑡) : Relative displacement of the contact point rotor-bearing system,” Journal of Sound and Vibration,vol. , along the line of action no. , pp. –, . 𝛿 , 𝑢 : Geometric angle of bevel gear 𝑗𝑏𝑖 [] Y. Wang, W. Zhang, and H. Cheung, “A finite element approach {𝑥𝑖, 𝑦𝑖, 𝑧𝑖} : Bearing displacements in each blocs to dynamic modeling of flexible spatial compound bar–gear (i=:) systems,” Mechanism and Machine eory, vol., no.,pp. {𝜙𝑖, 𝑖𝜓}: Angular displacement of the bearing –, . following X and Y, respectively [] J. F. Besseling, “Derivatives of deformation parameters for bar {𝜃 ,𝜃 }: Angular displacement of wheel and gear 1𝑖 2𝑖 elements and their use in bucking and postbucking analysis,” following Z direction (i=:) Computer Methods in Applied Mechanics and Engineering,vol. 𝜎:Solidity ,pp.–,. 𝛼(𝜃) :Angleofattack. [] D.Yassine,H. Ahmed,W.Lassaad, and H.Mohamed, “Effects of gear mesh fluctuation and defaults on the dynamic behavior of two-stage straight bevel system,” Mechanism and Machine Data Availability eory, vol., pp.–, . [] M. Islam, D. Ting, and A. Fartaj, “Aerodynamic models for All results are obtained by simulation with Matlab. Darrieus-type straight-bladed vertical axis wind turbines,” Renewable & Sustainable Energy Reviews,vol.,no.,pp.– Conflicts of Interest , . [] A. Mirecki, Comparative study of energy conversion system e authors declare that they have no conflicts of interest. dedicated to a small wind turbine [Ph.D. thesis],Polytechnic National Institute, Toulouse, France, . References [] S. Wei, J. Zhao, Q. Han, and F. Chu, “Dynamic response analysis on torsional vibrations of wind turbine geared transmission [] R. E. Wilson, “Wind-turbine aerodynamic,” Journal of Industrial system with uncertainty,” Journal of Renewable Energy,vol. , Aerodynamics,pp. –, . pp. –, . [] C. Javier, Small-scale vertical axis wind turbine design. Bach- elors esis, Aeronautical Engineering, Tampere University of Applied Sciences, . [] D. Prathamesh and C. L. Xian, “Numerical study of giromill- type wind turbines with symmetrical and non-symmetrical airfoils,” Journal of Science and Technology,. [] J. H. Strickland, “e Darrieus turbine: a performance pre- diction model using multiple streamtube,” Sandia Laboratories Report. SAND, pp. –, , United States of America. [] I. paraschivoiu, “Double-multiple stream tube model for study- ing vertical-axis wind turbines,” Journal of Propulsion and Power, vol.,no.,pp.–, . [] H. Beri and Y. Yao, “Double multiple stream tube model and numerical analysis of vertical axis wind turbine,” International Journal of Energy and Power Engineering, vol.,no. ,pp.– , . [] L. Wang, L. Zhang, and N. Zeng, “A potential flow -D vortex panel model: applications to vertical axis straight blade tidal turbine,” Energy Conversion and Management,vol.,no. ,pp. –, . [] Z.-C. Jiang, Y. Doi, and S.-Y. Zhang, “Numerical investigation on the flow and power of small-sized multi-bladed straight Darrieus wind turbine,” Journal of Zhejiang University SCIENCE A,vol.,no.,pp. –, . [] M.H. Mohamed,A. M.Ali, and A.A. Hafiz,“CFD analysis for H-rotor Darrieus turbine as a low speed wind energy converter,” 𝑖𝑗 International Journal of Advances in Rotating Machinery Multimedia Journal of The Scientific Journal of Engineering World Journal Sensors Hindawi Hindawi Publishing Corporation Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 http://www www.hindawi.com .hindawi.com V Volume 2018 olume 2013 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 Submit your manuscripts at www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Hindawi Hindawi Hindawi Volume 2018 Volume 2018 Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com www.hindawi.com www.hindawi.com Volume 2018 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Acoustics and Vibration Hindawi Publishing Corporation

Analyses of Dynamic Behavior of Vertical Axis Wind Turbine in Transient Regime

Loading next page...
 
/lp/hindawi-publishing-corporation/analyses-of-dynamic-behavior-of-vertical-axis-wind-turbine-in-j05YUZcXSy
Publisher
Hindawi Publishing Corporation
Copyright
Copyright © 2019 Bacem Zghal et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ISSN
1687-6261
eISSN
1687-627X
DOI
10.1155/2019/7015262
Publisher site
See Article on Publisher Site

Abstract

Hindawi Advances in Acoustics and Vibration Volume 2019, Article ID 7015262, 9 pages https://doi.org/10.1155/2019/7015262 Research Article Analyses of Dynamic Behavior of Vertical Axis Wind Turbine in Transient Regime Bacem Zghal , Imen Bel Mabrouk, Lassâad Walha, Kamel Abboudi, and Mohamed Haddar Laboratory of Mechanical Modeling and Production (LAMP), National School of Engineers of Sfax (ENIS), University of Sfax, BP , , Tunisia Correspondence should be addressed to Bacem Zghal; bacem.zghal@gmail.com Received 19 October 2018; Accepted 6 March 2019; Published 10 April 2019 Academic Editor: Emil Manoach Copyright ©  Bacem Zghal et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 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 sha has a significant effect on the aerodynamic torque performance. 1. Introduction or vortex model [–] have been developed to optimize VAWTs performance. Generally, vertical axis wind turbines (VAWT) have a par- e (CFD) method has been widely used in developing ticular architecture compared with horizontal ones. ey are the characteristics of wind turbine (torque fluctuation, power composed of two main parts: the blade rotor in vertical output, and pressure distribution). Jiang et al. [] employed position and a mechanical gear transmission (bevel gear). a commercial CFD simulation for studying the effect of Vibrations of the aerodynamic part caused by the wind speed geometrical parameters and airfoil type on the performance variation are transmitted to the other part (gear transmission of the H-Darrieus turbine with fixed pitch angle. Also, a system) via sha , gears, and bearing. numerical analysis of H-rotor Darrieus turbine is introduced In literature, plenty of authors studied the aerodynamic by M.H Mohamed et al. []. e developing of torque fluctuation is a very important step for the reason that performance of Darrieus-type of VAWT. ere are two mainly approaches: momentum models and Computational the instantaneous torque produced by the rotor blade is Fluid Dynamics (CFD). e main benefit of momentum directly related to both the power generation and the gearbox models is that their time of resolution is quicker than the vibration. other approach. Although the Computational Fluid Dynam- Accordingly, there are many related literatures studying bevel gears transmission. Cai-Wan Chang-jian [] studied ics have been a useful design tool for studying the efficiency of wind turbine, the mesh generation in three-dimensional the nonlinear dynamic behavior of bevel gear system. Besides, analyses needs a lot of time for the simulation. Among Fujii et al. [] analyzed the dynamic vibration of straight bevel gear supported by angular bearings and tapered roller. analytical models are researches [–] based on Actuator Disk and Blade-Element Method (BEM) to predict the M. Li and H. Y. Hu [] studied the dynamic analysis of a aerodynamic torque of VAWT. In addition, computational spiral bevel-geared rotor-bearing system. Y. Wang et al. [] aerodynamics methods such as multiple stream tube method and J.F. Besseling [] developed a new approach based on  Advances in Acoustics and Vibration Wind V(t) Drag Lift 2 W(t) 6 =IM() W(t) Drag 6 MCH() Lift Drag W(t) F  Lift F : Flow velocities and forces in Darrieus wind turbine []. 2 2 finite element theory to model bevel gear systems. Moreover, () 𝑊= +𝑉𝑛 Driss Yassine et al. [] present the model of two-stage straight bevel gear system excited with only internal excitation which = 𝜔𝑅 + cos 𝜃 is the periodic fluctuations of the gear meshes’ stiffness. = 𝑎𝑉 sin 𝜃 () In this paper, we discuss the impact of some design parameters including number of blades, turbine radius, chord = 𝑉 𝑡 1−𝑎 ( )( ) length, blade length, and rotational speed on the aerody- namic torque of the H-Darrieus VAWT through analytical approach. e angle of attack is defined as the angle between the resul- e main objective of the present work is to predict the tant air velocity vector and the blade chord. It is expressed as dynamic behavior of the one-stage straight bevel gear system follows: used in vertical axis wind turbine and powered by two main sources of excitation which are the optimum aerodynamic 𝑉 (1−𝑎 ) sin 𝜃 −1 𝑛 𝛼 (𝜃 ) = tan ( ) = ( ) () torque selected through parametrical study and the periodic 𝑉 (1−𝑎 ) cos𝜃+𝜆 variation of the gear meshes’ stiffness. e value of the axial induction factor (a) can be introduced 2. Theoretical Modeling of VAWT Rotors by actuator disk theory. e resultant air velocity is dependent on the induced velocity and the tip speed ratio (TSR) defined In this section, analytical investigation of aerodynamic torque as of Darrieus wind turbine is established. e actuator disk the- ory is chosen for the aerodynamic study of the Darrieus-type wind turbine with straight blade. is theory characterizes 𝜆= () 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 , the relative flow velocity can be In this work, the induced velocity is modeled in deterministic obtained as follows: form, as a sum of several harmonics []: 𝜔𝑅 𝑉𝑎 𝑉𝑛 𝑉𝑎 𝑉𝑐 𝑉𝑐 Advances in Acoustics and Vibration Rotor (11)   11 1 Φ x1 y1 1 z1 Gear 12 K(t) Gear 21 2 2 Generator (22) Z K z2 Ψ y2 x2 F : Single-stage bevel gear model. 𝑉 =14 + 2 sin (𝜔𝑡 )−1.75 sin (3𝜔𝑡 )+1.5 sin (5𝜔𝑡 ) ∞ e averagetorqueproduced by therotor (n blades) is generated from the average tangential force acting on one −1.25 sin 10𝜔𝑡 + sin 30𝜔𝑡 +0.5 sin 50𝜔𝑡 ( ) ( ) ( ) () blade: +0.25 sin (100𝜔𝑡 ) 2𝜋 𝑇 𝜃 =𝑛 ∫ 𝐹 𝜃 𝑅 () ( ) ( ) 2𝜋 e resulting aerodynamic forces in the blade can be founded by the interpolation of the li and drag coefficients relative to e average torque coefficient is calculated by the symmetrical airfoil used (NACA), the angle of attack, and the given Reynolds number. 𝑇 (𝜃 ) 𝐶 = () e tangential and normal forces as function of the 0.5𝜌𝑉 azimuth angle 𝜃 can be calculated using the blade-element Finally, the power coefficient Cp is estimated from the average theory []. torque coefficient: 𝐹 (𝜃 ) = 𝜌𝑐 𝑊 ℎ𝑐 𝑡 𝑡 = 𝜆𝐶 () () 𝐹 (𝜃 ) = 𝜌𝑐 𝑊 ℎ𝑐 3. Theoretical Modeling of the One-Stage Bevel 𝑛 𝑛 Gear System where C and C are the normal and tangential coefficients, n t is part investigates the studying of the dynamic behavior of respectively, calculated from the li and drag coefficients (C bevel gear system used in vertical axis wind turbine. e main and C ) using the same theory (blade-element momentum excitations of the one-stage bevel gear system are the selected theory), given by following expressions: aerodynamic torque estimated through parametrical analysis 𝐶 =𝐶 sin𝛼− 𝐶 cos 𝛼 in addition to the internal mesh stiffness excitation. 𝑡 𝐿 𝐷 () e dynamic model of single-stage bevel gear is presented 𝐶 =𝐶 cos𝛼+ 𝐶 sin 𝛼 𝑛 𝐿 𝐷 by fourteen degrees of freedom (see Figure ). is model 𝐶𝑝 𝐴𝑅 𝑑𝜃  Advances in Acoustics and Vibration includes two blocks where the first block is constituted by the e mesh stiffness matrix can be defined by Darrieus rotor modeled by the mass (11) linked to the wheel () (12) via a first sha with torsional rigidity K𝜃1 ;this block is [𝐾 (𝑡 )] =𝑘 (𝑡 )⟨𝐿 ⟩ . ⟨𝐿 ⟩ supported by a bearing. [k(t)] is the total meshing stiffness of the gear pair. Sha Wei et e second block includes the pinion (21),the second al. [] haveexpressed thetotal meshing stiffnessofthe gear sha with torsional rigidity K𝜃2 , and the generator modeled pair by a periodic excitation decomposed on Fourier series. by a mass (22). It is also supported by a bearing. e wheel e tooth deflection following the line of action is defined (12) is connected to the pinion (21) via teeth mesh stiffness. by e bearings are modeled by linear springs acting on the lines of action. e mesh stiffness characterizes the elastic 𝛿 𝑡 = 𝐿 .{𝑞 𝑡 } ( ) ⟨ ⟩ ( ) () deformations managing the relative positions of the two gear wheels; it can modeled by the mesh stiffness 𝑘(𝑡). {𝐿 } ={𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,𝑐 ,0,𝑐 ,𝑐 ,0} () 1 2 3 4 5 6 7 8 10 11 9 12 e proposed dynamic model is modeled by the general- ized coordinate vector {𝑞}: e components of the tooth deflection are presented in Table . {𝑞 (𝑡 )} e load vector can be written as () =[𝑥 ,𝑦 ,𝑧 ,𝑥 ,𝑦 ,𝑧 ,𝜙 ,𝜓 ,𝜙 ,𝜓 ,𝜃 ,𝜃 ,𝜃 ,𝜃 ] 1 1, 1 2 2 2 1 1 2 2 11 12 21 22 {𝐹 (𝑡 )} () {𝑥𝑖, 𝑦𝑖, 𝑧𝑖} are the linear displacements of bearing in each 00 000 000 00𝑇 (𝑡 ) 00 −𝑅 (𝑡 ) ={ } block. {𝜙𝑖, 𝑖𝜓, 𝜃 ,𝜃 } are the angular displacement of bearing 1𝑖 2𝑖 T(t) and R (t) arethe aerodynamic torqueproduced by and wheel (i=:). the rotor and the resisting torque, respectively. [C] is the e equations of motion describing the dynamic behavior proportional damping matrix of the model are established using the formalism of Lagrange −5 for each degree of freedom of the system. ̃ [𝐶 ] =0.05 [𝑀 ] +10 [𝐾] () ̈ ̇ [𝑀 ] {𝑞} + [𝐶 ] {𝑞} + ([𝐾 ]+ [𝐾 (𝑡 )]){𝑞} = {𝐹 (𝑡 )} () where 𝑎 = sin(𝛿 ), 𝑏 = cos(𝛿 ), 𝑎 = sin(𝛿 ),and 12 𝑏12 12 𝑏12 21 𝑏21 𝑏 = cos(𝛿 ). 21 𝑏21 e total mass matrix is defined by 𝛿 and r are the half-angle of the base circle of bevel gear (i) of the block (j) and the radius of the sphere which [𝑀 ]0 [𝑀 ] =[ ] () contains two bevel gears, (12) and (21), respectively (r = 0[𝑀 ] r ). e angles 𝛿 and u describe the different parameters of [𝑀 ]= diag [𝑚 ,𝑚 ,𝑚 ,𝑚 ,𝑚 ,𝑚 ] () 𝐿 1 1 1 2 2 2 the bevel gear []. [𝑀 ]= diag [𝐼 +𝐼 ,𝐼 +𝐼 ,𝐼 +𝐼 ,𝐼 𝐴 11𝑥 12𝑥 11𝑦 12𝑦 21𝑥 22𝑥 21𝑦 () 4. Influence of Rotor Configuration on +𝐼 ,𝐼 ,𝐼 ,𝐼 ,𝐼 ] 22𝑦 11 12 21 22 Turbine Performance [𝐾 ] is the average stiffness matrix of the structure e main purpose of this study is to investigate the effect of different design parameters (blade chord, number of blades, [𝐾 ]0 𝑝 and radius turbine) on the aerodynamic torque evaluation of [𝐾 ]=[ ] () H-Darrieus turbine, using analytical approach. 0[𝐾 ] e solidity 𝜎 is defined as an important nondimensional parameter, which influences the self-starting abilities and where: [𝐾 ]= diag [𝑘 ,𝑘 ,𝑘 ,𝑘 ,𝑘 ,𝑘 ] () 𝑝 𝑥1 𝑦1 𝑧1 𝑥2 𝑦2 𝑧2 determines the applicability of the momentum models. e [𝐾 ] turbine is able to self-start for high solidity (𝜎≥ .) []. For straight bladed VAWTs, the solidity is calculated by 𝑘 0 0 0 00 00 𝜙1 𝑛𝑐 [ ] [ ] 𝜎= () 0𝑘 0 0 00 00 𝜓1 [ ] [ ] [ 00 𝑘 00 0 0 0 ] 𝜙2 [ ] In order to study the dynamic behavior of bevel gear system [ ] () [ 000 𝑘 00 00 ] used on vertical axis wind turbine and powered by the 𝜓2 [ ] [ ] aerodynamic torque in transient regime in addition to the 0000 𝑘 −𝑘 00 [ ] 𝜃1 𝜃1 [ ] periodic variations of mesh stiffness, we used numerical [ ] 0000 −𝑘 𝑘 00 [ 𝜃1 𝜃1 ] simulation based on implicit method of Newmark. [ ] [ ] e parameters of the bevel gear transmission are pre- 0000 0 0 𝑘 −𝑘 𝜃2 𝜃2 [ ] sented in Table . Specifications of the vertical axis wind 0000 0 0 −𝑘 𝑘 𝜃2 𝜃2 [ ] turbine are shown in Table . 𝑏𝑗 Advances in Acoustics and Vibration T : Components of the tooth deflection. c 𝑏 sin(𝑎 𝑢 ) 1 12 12 12 c 𝑎 cos 𝑢 sin(𝑎 𝑢 )− sin 𝑢 cos(𝑎 𝑢 ) 2 12 12 12 12 12 12 12 c 𝑎 sin 𝑢 sin(𝑎 𝑢 )+ cos 𝑢 cos(𝑎 𝑢 ) 3 12 12 12 12 12 12 12 c 𝑏 sin(𝑎 𝑢 ) 4 21 21 21 c 𝑎 cos 𝑢 sin(𝑎 𝑢 )− sin 𝑢 cos(𝑎 𝑢 ) 5 21 21 21 21 21 21 21 c 𝑎 sin 𝑢 sin(𝑎 𝑢 )+ cos 𝑢 cos(𝑎 𝑢 ) 6 21 21 21 21 21 21 21 c 𝑐 𝑟 𝑏 cos(𝑎 𝑢 )− 𝑐 𝑟 (𝑎 cos 𝑢 cos (𝑎 𝑢 )+ sin 𝑢 sin(𝑎 𝑢 )) 7 2 12 12 12 12 1 12 12 12 12 12 12 12 12 c 𝑐 𝑟 (𝑎 sin 𝑢 cos (𝑎 𝑢 )− cos 𝑢 sin (𝑎 𝑢 )) − 𝑐 𝑟 𝑏 cos(𝑎 𝑢 ) 8 1 12 12 12 12 12 12 12 12 3 12 12 12 12 c 𝑐 𝑟 (𝑎 cos 𝑢 cos (𝑎 𝑢 )+ sin 𝑢 sin (𝑎 𝑢 )) − 𝑐 𝑟 (𝑎 sin 𝑢 cos (𝑎 𝑢 )+ cos 𝑢 sin(𝑎 𝑢 )) 9 3 12 12 12 12 12 12 12 12 2 12 12 12 12 12 12 12 12 c 𝑐 𝑟 (𝑎 cos 𝑢 cos (𝑎 𝑢 )+ sin 𝑢 sin (𝑎 𝑢 )) − 𝑐 𝑟 𝑏 cos(𝑎 𝑢 ) 10 4 21 21 21 21 21 21 21 21 5 21 21 21 21 c 𝑐 𝑟 (𝑎 sin 𝑢 cos (𝑎 𝑢 )− cos 𝑢 sin (𝑎 𝑢 )) − 𝑐 𝑟 𝑏 cos(𝑎 𝑢 ) 11 4 21 21 21 21 21 21 21 21 6 21 21 21 21 c −𝑐 𝑟 (𝑎 cos 𝑢 cos (𝑎 𝑢 )+ sin 𝑢 sin (𝑎 𝑢 )) + 𝑐 𝑟 (𝑎 sin 𝑢 cos (𝑎 𝑢 )− cos 𝑢 sin(𝑎 𝑢 )) 12 6 21 21 21 21 21 21 21 21 5 21 21 21 21 21 21 21 21 T : Parameters of the studied bevel gear system. Teeth number  / module(m) . 𝑘 =𝑘 =𝑘 =𝑘 =2 ⋅ 10 𝑥1 𝑦1 𝑥2 𝑦2 Bearing stiffness (N/m) k = k =. z z Torsional stiffness(N/rd/m) 𝑘 = 𝑘 = 𝜃1 𝜃2 Pressure angle 𝛼= 20 Contact ratio 𝜀 =. Average mesh stiffness(N/m) K = moy Density (CrMo) kg/m T : Wind turbine specification. Type Straight blade Darrieus Airfoil profile NACA Airfoil chord(mm) Blade length (m) . Turbine diameter (m) Blade number Speed of rotor (tr/min) e number of blades (n) is an important factor that thetorquebehavior ofVAWTturbine.Anincrease inradius influences the torque produced. It is well known that a bigger turbine and blade height advances the instantaneous torque number of blades give rise to the solidity and produce a as shown in Figures  and . global torque with small fluctuation. However, increasing the Figure  shows the total torque evolution versus azimuth angle generated for a complete revolution with differ- number of blades lead to increase in the turbine drag by increasing the number of connecting sha s. Figure  shows ent rotation speed. It can be observed that the torque the effect of the blade number on the torque in full revolution increases considerably with the increase of the rotation for a case where the radius and blade chord are maintained speed. Also, the torque fluctuation becomes positive when constant while the number of blades changes. As can be the rotation speed reaches  rev/min. However, for a deduced, -bladed VAWT perform more efficient than  and case of  rev/min and  rev/min the instantaneous bladed turbines. torque presents some fluctuation with negative magni- In Figure  the effect of blade chord on the torque tude which can explain the no self-starting of NACA fluctuation is presented. We remark that torque magnitude airfoil. increases when the chord length increases. Also, blades with As shown in Figures  and  rising wind speed from smaller chords require a bigger tip speed ratio to achieve a  to  m/s significantly affects the torque which increases considerably. e cyclic distribution of total torque at higher maximum torque so the selection of chord length affects the self-starting behavior of Darrieus turbine. rotational speed (N= rev/min) and at small rotational As can be assumed from the torque equation (), the rotor speed (N= rev/min) is observed at the same wind speed radius and the blade length have a major contribution in of  m/s.  Advances in Acoustics and Vibration −500 −1000 −1500 0 50 100 150 200 250 300 350 Azimuth angle 3-blade 2-blade 4-blade F : Effect of blades number on the torque. −500 −1000 0 50 100 150 200 250 300 350 Azimuth angle c=0.48 c=0.42 c=0.3 c=0.25 F : Effect of blade chord on the torque. −500 −1000 0 50 100 150 200 250 300 350 Azimuth angle R=3 R=2.5 R=1.8 R=1.4 F : Effect of radius turbine on the torque. Torque (N.m) Torque (N.m) Torque (N.m) Advances in Acoustics and Vibration −500 −1000 −1500 0 50 100 150 200 250 300 350 Azimuth angle h=3.66 h=2.5 h=1 F : Effect of blade height on the torque. −1000 −2000 0 50 100 150 200 250 300 350 Azimuth angle N=178 N=177 N=50 F : Effect of rotation speed on the torque. We clearly see only positive fluctuation of the aerody- Influence of geometrical parameters of Darrieus rotor has namic torque at N= tr/min; however, this torque presents been done in order to select the appropriate parameters. e negative oscillation at small rotational speed (N= tr/min) optimization process has been carried out on the effect of each at fixed wind speed of  m/s. Consequently, we conclude parameter on the aerodynamic torque produced. the most favorable speed excitation of the considerable It was found that the aerodynamic torque increases Darrieus rotor that corresponds to wind velocity of m/s and when the chord, the radius, and the height of VAWT rise. rotational speed of  tr/min which respects the condition of However, the best performance is detected for -bladed positive torque evolution. VAWT. Finally, the most significant parameter that affects the aerodynamic torque is the rotation speed of the rotor sha . 5. Conclusion is paper presents a three-dimensional model of one-stage Nomenclature straight bevel gear system used in vertical axis wind turbine. Aerodynamic torque fluctuation and periodic oscillation of a: Axial induction factor the gear meshes’ stiffness are the main sources of excitation A: Turbine swept area for the bevel gear system. c: Chord (m) Torque (N.M) Torque (N.m)  Advances in Acoustics and Vibration −1000 0 50 100 150 200 250 300 350 Azimuth angle UR=14 UR=12 UR=10 UR=8 F : Effect of wind speed on the torque (N=tr/min). −500 −1000 0 50 100 150 200 250 300 350 Azimuth angle UR=14 UR=12 UR=10 UR=8 F : Effect of wind speed on the torque (N=tr/min). [C]: Proportional damping matrix C : Tangential force coefficient [𝐾 ]: Stiffness matrix of the average structure C : Normal force coefficient k : Sha s torsional stiffness 𝜃 i C : Average torque coefficient T 𝐾 , 𝑘 , 𝑘 , 𝑥𝑖 𝑦𝑖 𝑧𝑖 Cp: Power coefficient 𝐾 and 𝐾 : Bending and traction-compression Φ𝑖 𝜓𝑖 𝐶 , 𝐶 : Li and drag coefficients 𝐿 𝐷 bearing stiffness F:Tangentialforce t [𝑀]:Massmatrix F : Normal force n m : Mass of block i {𝐹(𝑡)} :Externalexcitationforce ⟨𝐿⟩ : Vector of geometric parameters of the h: Blade height (m) dynamic model I : Moment of inertia of the wheel j of block i ij n: Blade number I ,I : Diametrical moment of wheel j of block i N: Rotational speed of the rotor (tr/min) ijx ijy respectively following the X- and Y-axes {𝑞(𝑡)} : Generalized coordinate vector 𝑘(𝑡): Gear mesh of stiffness R: Radius of the wind turbine (m) [𝐾(𝑡)] :Timestiffnessmatrixofthegearmesh 𝑅 (𝑡): Resisting torque fluctuation T(t):Aerodynamictorque Torque (N.m) Torque (N.m) Advances in Acoustics and Vibration Engineering Science and Technology, an International Journal, vol. , no. , pp. –, . V(t): Wind free stream velocity (m/sec) [] C.-W. Chang-Jian, “Nonlinear dynamic analysis for bevel-gear Va: Induced velocity system under nonlinear suspension-bifurcation and chaos,” Vc, Vn: Chordal velocity component and the Applied Mathematical Modelling, vol., no., pp. –, normal velocity component, respectively W: Relative flow velocity [] M. Fujii, Y. Nagasaki, and M. Nohara Trauchi, “Effect of bearing 𝜃:Azimuthangle on dynamic behavior of straight bevel gear train,” Transactions 𝜔:Angularvelocity(rad/sec) of the Japan Society of Mechanical Engineers A,vol., pp. – 𝜌:Airdensity[kg/m ] , . 𝜆 : Tip speed ratio [] M. Li and H. Y. Hu, “Dynamic analysis of a spiral bevel-geared 𝛿(𝑡) : Relative displacement of the contact point rotor-bearing system,” Journal of Sound and Vibration,vol. , along the line of action no. , pp. –, . 𝛿 , 𝑢 : Geometric angle of bevel gear 𝑗𝑏𝑖 [] Y. Wang, W. Zhang, and H. Cheung, “A finite element approach {𝑥𝑖, 𝑦𝑖, 𝑧𝑖} : Bearing displacements in each blocs to dynamic modeling of flexible spatial compound bar–gear (i=:) systems,” Mechanism and Machine eory, vol., no.,pp. {𝜙𝑖, 𝑖𝜓}: Angular displacement of the bearing –, . following X and Y, respectively [] J. F. Besseling, “Derivatives of deformation parameters for bar {𝜃 ,𝜃 }: Angular displacement of wheel and gear 1𝑖 2𝑖 elements and their use in bucking and postbucking analysis,” following Z direction (i=:) Computer Methods in Applied Mechanics and Engineering,vol. 𝜎:Solidity ,pp.–,. 𝛼(𝜃) :Angleofattack. [] D.Yassine,H. Ahmed,W.Lassaad, and H.Mohamed, “Effects of gear mesh fluctuation and defaults on the dynamic behavior of two-stage straight bevel system,” Mechanism and Machine Data Availability eory, vol., pp.–, . [] M. Islam, D. Ting, and A. Fartaj, “Aerodynamic models for All results are obtained by simulation with Matlab. Darrieus-type straight-bladed vertical axis wind turbines,” Renewable & Sustainable Energy Reviews,vol.,no.,pp.– Conflicts of Interest , . [] A. Mirecki, Comparative study of energy conversion system e authors declare that they have no conflicts of interest. dedicated to a small wind turbine [Ph.D. thesis],Polytechnic National Institute, Toulouse, France, . References [] S. Wei, J. Zhao, Q. Han, and F. Chu, “Dynamic response analysis on torsional vibrations of wind turbine geared transmission [] R. E. Wilson, “Wind-turbine aerodynamic,” Journal of Industrial system with uncertainty,” Journal of Renewable Energy,vol. , Aerodynamics,pp. –, . pp. –, . [] C. Javier, Small-scale vertical axis wind turbine design. Bach- elors esis, Aeronautical Engineering, Tampere University of Applied Sciences, . [] D. Prathamesh and C. L. Xian, “Numerical study of giromill- type wind turbines with symmetrical and non-symmetrical airfoils,” Journal of Science and Technology,. [] J. H. Strickland, “e Darrieus turbine: a performance pre- diction model using multiple streamtube,” Sandia Laboratories Report. SAND, pp. –, , United States of America. [] I. paraschivoiu, “Double-multiple stream tube model for study- ing vertical-axis wind turbines,” Journal of Propulsion and Power, vol.,no.,pp.–, . [] H. Beri and Y. Yao, “Double multiple stream tube model and numerical analysis of vertical axis wind turbine,” International Journal of Energy and Power Engineering, vol.,no. ,pp.– , . [] L. Wang, L. Zhang, and N. Zeng, “A potential flow -D vortex panel model: applications to vertical axis straight blade tidal turbine,” Energy Conversion and Management,vol.,no. ,pp. –, . [] Z.-C. Jiang, Y. Doi, and S.-Y. Zhang, “Numerical investigation on the flow and power of small-sized multi-bladed straight Darrieus wind turbine,” Journal of Zhejiang University SCIENCE A,vol.,no.,pp. –, . [] M.H. Mohamed,A. M.Ali, and A.A. Hafiz,“CFD analysis for H-rotor Darrieus turbine as a low speed wind energy converter,” 𝑖𝑗 International Journal of Advances in Rotating Machinery Multimedia Journal of The Scientific Journal of Engineering World Journal Sensors Hindawi Hindawi Publishing Corporation Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 http://www www.hindawi.com .hindawi.com V Volume 2018 olume 2013 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 Journal of Control Science and Engineering Advances in Civil Engineering Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 Submit your manuscripts at www.hindawi.com Journal of Journal of Electrical and Computer Robotics Engineering Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 VLSI Design Advances in OptoElectronics International Journal of Modelling & Aerospace International Journal of Simulation Navigation and in Engineering Engineering Observation Hindawi Hindawi Hindawi Hindawi Volume 2018 Volume 2018 Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com www.hindawi.com www.hindawi.com Volume 2018 International Journal of Active and Passive International Journal of Antennas and Advances in Chemical Engineering Propagation Electronic Components Shock and Vibration Acoustics and Vibration Hindawi Hindawi Hindawi Hindawi Hindawi www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018 www.hindawi.com Volume 2018

Journal

Advances in Acoustics and VibrationHindawi Publishing Corporation

Published: Apr 10, 2019

References