مدل سازی ارتعاشات آزاد غیرخطی برج توربین باد

نوع مقاله : مقاله پژوهشی

نویسندگان

1 دانشکده فنی مهندسی، دانشگاه فردوسی مشهد، مشهد، ایران

2 دانشگاه فردوسی مشهد

چکیده

هدف این مقاله بررسی ارتعاشات آزاد دامنه بلند برج توربین باد می باشد که به صورت تیر مقطع متغیر با جرم خارج از مرکز مدل شده است. در این مدل سازی، تاثیر اعمال نیروی محوری متغیر بر تیر ناشی از میدان جاذبه در نظر گرفته شده است. معادلات دیفرانسیل حاکم بر ارتعاشات غیرخطی سیستم و شرایط مرزی مربوطه با استفاده از اصل همیلتون به همراه فرضیات اولر- برنولی استخراج شده است. سپس از یک روش تفاضل محدود برای بدست آوردن فرکانس های طبیعی و شکل مودهای سیستم استفاده شده است. در ادامه معادلات دیفرانسیل با مشتقات جزئی حاکم بر دینامیک سیستم با استفاده از روش گلرکین به معادلات دیفرانسیل معمولی بر حسب جابجایی های انتهایی کاهش یافته اند که به دلیل وجود خارج از مرکزی طولی به هم وابسته اند. این معادلات زمانی با استفاده از روش اغتشاشاتی مقیاس های زمانی چندگانه به صورت تحلیلی حل شده اند. نتایج بدست آمده از حل تحلیلی تطابق خوبی با نتایج شبیه سازی عددی دارد. نتایج این پژوهش می تواند به منظور بررسی اثرات وجود جرم نوک خارج از مرکز، مقطع متغیر و اثر جاذبه را بر ارتعاشات دامنه بلند برج توربین باد، با هدف بهبود رفتار دینامیکی آن مورد استفاده قرار گیرد.

کلیدواژه‌ها

موضوعات


عنوان مقاله [English]

Modeling Nonlinear Free Vibrations of Wind Turbine Tower

نویسنده [English]

  • Hasan Malaeke 1
1 Ferdowsi university of Mashhad
2 Room 214, School of Mechanical Engineering, Department of Eng, Ferdowsi University of Mashhad, Mashhad, Iran
چکیده [English]

The goal of this paper is to study the large amplitude free vibration of wind turbine tower which modeled as a variable cross-section beam with eccentric mass. The effects of variable axial force due to gravity is also taken into account. The nonlinear governing equations of motion and the corresponding boundary conditions of the system are obtained using the Hamilton's principle as well as Euler-Bernoulli’s assumptions. Then a numerical finite difference scheme is utilized to find the natural frequencies and the mode shapes of the system. Using Galerkin method, the partial differential equations governing dynamic of the system are reduced to ordinary differential equations in terms of the end displacements which are coupled due to the presence of the transverse eccentricity. These temporal coupled ODEs are then solved analytically using the multiple time scales perturbation technique. The obtained analytical results are compared with the numerical ones and excellent agreement is observed. The results of this research can be used to study the effect of the eccentric tip mass, variable cross-section and gravity on the large amplitude vibration of wind turbine tower for improved dynamic performance.

کلیدواژه‌ها [English]

  • Wind turbine tower
  • Large amplitude vibration
  • Variable cross-section beam
  • Eccentric mass
  • Multiple time scales
[1] G.J. Herbert, S. Iniyan, E. Sreevalsan, S. Rajapandian, "A review of wind energy technologies," Renewable and sustainable energy Reviews", 2007, vol. 11, pp. 1117-1145.
[2] J.F. Manwell, J.G. McGowan, A.L. Rogers, "Wind energy explained: theory, design and application," John Wiley & Sons, 2010.
[3] A. Mostafaeipour, "Feasibility study of offshore wind turbine installation in Iran compared with the world," Renewable and Sustainable Energy Reviews, 2010, vol. 14, pp. 1722-1743.
[4] A. Mollahosseini, S.A. Hosseini, M. Jabbari, A. Figoli, A. Rahimpour, "Renewable energy management and market in Iran: A holistic review on current state and future demands," Renewable and Sustainable Energy Reviews, 2017, vol. 80, pp. 774-788.
[5] A. Dabbaghiyan, F. Fazelpour, M.D. Abnavi, M.A. Rosen, "Evaluation of wind energy potential in province of Bushehr, Iran," Renewable and Sustainable Energy Reviews, 2016, vol. 55, pp. 455-466.
[6] I. Lavassas, G. Nikolaidis, P. Zervas, E. Efthimiou, I. Doudoumis, C. Baniotopoulos, "Analysis and design of the prototype of a steel 1-MW wind turbine tower," Engineering structures, 2003, vol. 25, pp. 1097-1106.
[7] N. Bazeos, G. Hatzigeorgiou, I. Hondros, H. Karamaneas, D. Karabalis, D. Beskos, "Static, seismic and stability analyses of a prototype wind turbine steel tower," Engineering structures, 2002, vol. 24, pp. 1015-1025.
[8] J.-Y. Han, C.-H. Hong, J.-H. Jeong, B.-Y. Moon, "Dynamic Characteristics Analysis of Filament-wound Composite Towers for Large Scale Offshore Wind-Turbine," The KSFM Journal of Fluid Machinery, 2012, vol. 15, pp. 55-60.
[9] C. You-liang, Q. Jiang-man, X. Zhan-pu, J. Yan, "Dynamic Analysis of Wind Power Turbine's Tower under the Combined Action of Winds and Waves, International Journal of Plant Engineering and Management, 2017, vol. 22(3), pp. 140-149.
[10] P. Murtagh, B. Basu, B. Broderick, "Simple models for natural frequencies and mode shapes of towers supporting utilities," Computers & structures, 2004, vol. 82, pp. 1745-1750.
[11] J. Wang, D. Qin, T.C. Lim, "Dynamic analysis of horizontal axis wind turbine by thin-walled beam theory," Journal of Sound and Vibration, 2010, vol. 329, pp. 3565-3586.
[12] U. Lee, "Vibration analysis of one-dimensional structures using the spectral transfer matrix method," Engineering structures, 2000, vol. 22, pp. 681-690.
[13] S.-T. Choi, S.-Y. Mau, "Dynamic analysis of geared rotor-bearing systems by the transfer matrix method," Journal of mechanical design, 2001, vol. 123, pp. 562-568.
[14] X. Rui, B. He, Y. Lu, W. Lu, G. Wang, "Discrete time transfer matrix method for multibody system dynamics," Multibody System Dynamics, 2005, vol. 14, pp. 317-344.
[15] B. He, X. Rui, H. Zhang, "Transfer matrix method for natural vibration analysis of free system," Mathematical Problems in Engineering, 2012, vol. 19, pp. 123-131.
[16] S.-C. Hsieh, J.-H. Chen, A.-C. Lee, "A modified transfer matrix method for the coupling lateral and torsional vibrations of symmetric rotor-bearing systems," Journal of Sound and Vibration, vol. 289, pp. 294-333.
[17] M. Wang, Z. Wang, H. Zhao, "Analysis of Wind-Turbine Steel Tower by Transfer Matrix," in:  International Conference on Energy and Environment Technolog (ICEET), IEEE, 2009, pp. 526-529.
[18] W. Meng, W. Zhangqi, "The vibration frequencies of wind turbine steel tower by transfer matrix method," in:  Measuring Technology and Mechatronics Automation (ICMTMA), IEEE, 2011, pp. 995-998.
[19] M. Feyzollahzadeh, M. Mahmoodi, "Dynamic Analysis of Offshore Wind Turbine Towers with Fixed Monopile Platform Using the Transfer Matrix Method," Journal of Solid Mechanics, 2016, vol. 8, pp. 130-151.
[20] M. Feyzollahzadeh, M. Mahmoudi, "Free Vibration Analysis of Offshore Wind Turbine with Fixed Monopile Platform", Journal Of Marine Engineering, 2015, vol. 10, pp. 11-26, (in persian)
[21] G.C. Larsen, M.H. Hansen, A. Baumgart, I. Carlén, "Modal analysis of wind turbine blades", Risø National Laboratory, Roskilde, Denmark, , 2002, pp. 1-72.
[22] S. J. Hosseininia, K. Khalili, S. M. Emam, "Modal analysis of wind turbine blade using machine vision", Modares Mechanical Engineering, 2015, vol. 15(11), pp. 377-386, (in Persian)
[23] J. Chen, D. Jiang, "Modal analysis of wind turbine tower", World Non-Grid-Connected Wind Power and Energy Conference (WNWEC), 2010, IEEE, pp. 1-3.
[24] J. Prescott, Applied elasticity: Longmans, Green and Co., 1924.
[25] L. A. Pipes, L. R. Harvill, Applied mathematics for engineers and physicists, 1970.
[26] R. Goel, "Vibrations of a beam carrying a concentrated mass," Journal of Applied Mechanics, 1973, vol. 40, pp. 821-832.
[27] B. Bhat, H. Wagner, "Natural frequencies of a uniform cantilever with a tip mass slender in the axial direction," Journal of Sound and Vibration, 1976, vol. 45(2), pp. 304-307.
[28] C. W. S. To, "Vibration of a cantilever beam with a base excitation and tip mass," Journal of Sound and Vibration, 1982, vol. 83(4), pp. 445-460.
[29] N. M. Auciello, "Transverse vibrations of a linearly tapered cantilever beam with tip mass of rotary inertia and eccentricity," Journal of Sound and Vibration, 1996, vol. 194(1), pp. 25-34.
[30] H. Conway, J. Dubil, "Vibration frequencies of truncated-cone and wedge beams," Journal of Applied Mechanics, 1965, vol. 32(4), pp. 932-934.
[31] D. Sanger, "Transverse vibration of a class of non-uniform beams," Journal of Mechanical Engineering Science, 1968, vol. 10(2), pp. 111-120.
[32] H. Mabie, C. Rogers, "Transverse vibrations of double‐tapered cantilever beams with end support and with end mass," The Journal of the Acoustical Society of America, 1974, vol. 55(5), pp. 986-991.
[33] D. Caruntu, "On bending vibrations of some kinds of beams of variable cross-section using orthogonal polynomials," Revue Roumaine des Sciences Techniques - Série de Mécanique Appliquée, 1996, vol. 41, pp. 265-272.
[34] H. C. Wang", Generalized hypergeometric function solutions on the transverse vibration of a class of nonuniform beams," Journal of Applied Mechanics, 1967, vol. 34(3), pp. 702-708.
[35] S. Naguleswaran, "A direct solution for the transverse vibration of Euler-Bernoulli wedge and cone beams," Journal of Sound and Vibration, 1994, Vol. 172(3), pp. 289-304.
[36] A. H. Nayfeh, D. T. Mook, Nonlinear oscillations: John Wiley & Sons, 2008.
[37] L. Zavodney, A. Nayfeh, "The non-linear response of a slender beam carrying a lumped mass to a principal parametric excitation: theory and experiment," International journal of non-linear mechanics, 1989, vol. 24(2), pp. 105-125.
[38] M. N. Hamdan, N. H. Shabaneh, "On the large amplitude free vibrations of a restrained uniform beam carrying an intermediate lumped mass," Journal of Sound and Vibration, 1997, vol. 199(5), pp. 711-736.
[39] H. Moeenfard, B. Motakef Imani, M. Davoudi, A. Rahimzadeh, "Dynamic instability in tapered beams under wind excitation," Modares Mechanical Engineering, 2015, vol. 15(3), pp. 153-161, (in Persian)
[40] J.R. Banerjee, H. Su, D.R. Jackson, "Free vibration of rotating tapered beams using the dynamic stiffness method", Journal of Sound and Vibration, 2006, vol. 298, pp. 1034–1054.