Access the full text.
Sign up today, get DeepDyve free for 14 days.
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 shas. 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 : Shas 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
Advances in Acoustics and Vibration – Hindawi Publishing Corporation
Published: Apr 10, 2019
You can share this free article with as many people as you like with the url below! We hope you enjoy this feature!
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
Copy and paste the desired citation format or use the link below to download a file formatted for EndNote
Access the full text.
Sign up today, get DeepDyve free for 14 days.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.