American Institute of Mathematical Sciences

June  2020, 28(2): 691-719. doi: 10.3934/era.2020036

The well-posedness and regularity of a rotating blades equation

 College of Applied Science, Beijing University of Technology, Beijing 100124, China

* Corresponding author: wangshu@bjut.edu.cn

Received  January 2020 Revised  March 2020 Published  April 2020

Fund Project: The author is supported by NSFC grant 11831003, 11771031, 11531010, 11726625 and 2017-ZJ-908 of Qinghai Province

In this paper, a rotating blades equation is considered. The arbitrary pre-twisted angle, arbitrary pre-setting angle and arbitrary rotating speed are taken into account when establishing the rotating blades model. The nonlinear PDEs of motion and two types of boundary conditions are derived by the extended Hamilton principle and the first-order piston theory. The well-posedness of weak solution (global in time) for the rotating blades equation with Clamped-Clamped (C-C) boundary conditions can be proved by compactness method and energy method. Strong energy estimates are derived under additional assumptions on the initial data. In addition, the existence and regularity of weak solutions (global in time) for the rotating blades equation with Clamped-Free (C-F) boundary conditions are proved as well.

Citation: Lin Shen, Shu Wang, Yongxin Wang. The well-posedness and regularity of a rotating blades equation. Electronic Research Archive, 2020, 28 (2) : 691-719. doi: 10.3934/era.2020036
References:
 [1] G. L. Anderson, On the extensional and flexural vibrations of rotating bars, International Journal of Non-Linear Mechanics, 10 (1975), 223-236.  doi: 10.1016/0020-7462(75)90014-1.  Google Scholar [2] H. Arvin and F. Bakhtiari-Nejad, Nonlinear free vibration analysis of rotating composite Timoshenko beams, Composite Structures, 96 (2013), 29-43.  doi: 10.1016/j.compstruct.2012.09.009.  Google Scholar [3] K. V. Avramov, O. S. Galas, O. K. Morachkovskii and C. Pierre, Analysis of flexural-flexural-torsional nonlinear vibrations of twisted rotating beams with cross-sectional deplanation, Strength of Materials, 41 (2009), 200-208.  doi: 10.1007/s11223-009-9111-x.  Google Scholar [4] A. Burchard and L. E. Thomas, On the cauchy problem for a dynamical Euler's Elastica, Communications in Partial Differential Equations, 28 (2013), 271-300.  doi: 10.1081/PDE-120019382.  Google Scholar [5] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Communications in Contemporary Mathematics, 6 (2004), 705-731.  doi: 10.1142/S0219199704001483.  Google Scholar [6] G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams, SIAM Journal on Control and Optimization, 25 (1987), 526-546.  doi: 10.1137/0325029.  Google Scholar [7] L. W. Chen, W. K. Peng, L. W. Chen and W. K. Peng, Dynamic stability of rotating blades with geometric non-linearity, Journal of Sound and Vibration, 187 (1995), 421-433.   Google Scholar [8] S.-C. Choi, J. S. Park and J. H. Kim, Vibration control of pre-twisted rotating composite thin-walled beams with piezoelectric fiber composites, Journal of Sound and Vibration, 300 (2007), 176-196.  doi: 10.1016/j.jsv.2006.07.051.  Google Scholar [9] S. A. Fazelzadeh and M. Hosseini, Aerothermoelastic behavior of supersonic rotating thin-walled beams made of functionally graded materials, Pressure Vessels and Piping Conference, (2006), 227–236. doi: 10.1115/PVP2006-ICPVT-11-93624.  Google Scholar [10] S. A. Fazelzadeh, P. Malekzadeh, P. Zahedinejad and M. Josseini, Vibration analysis of functionally graded thin-walled rotating blades under high temperature supersonic flow using the differential quadrature method, Journal of Sound and Vibration, 306 (2007), 333-348.  doi: 10.1016/j.jsv.2007.05.011.  Google Scholar [11] F. Georgiades, J. Latalski and J. Warminski, Equations of motion of rotating composite beam with a nonconstant rotation speed and an arbitrary preset angle, Meccanica, 49 (2014), 1833-1858.  doi: 10.1007/s11012-014-9926-9.  Google Scholar [12] B. Z. Guo and J.-M. Wang, The well-posedness and stability of a beam equation with conjugate variables assigned at the same boundary point, IEEE Transactions on Automatic Control, 50 (2005), 2087-2093.  doi: 10.1109/TAC.2005.860275.  Google Scholar [13] A. Hasanov and O. Baysal, Identification of unknown temporal and spatial load distributions in a vibrating Euler-Bernoulli beam from Dirichlet boundary measured data, Automatica J. IFAC, 71 (2016), 106-117.  doi: 10.1016/j.automatica.2016.04.034.  Google Scholar [14] G. Hegarty and S. Taylor, Classical solutions of nonlinear beam equations: Existence and stabilization, SIAM Journal on Control and Optimization, 50 (2012), 703-719.  doi: 10.1137/100793864.  Google Scholar [15] X. X. Hu, T. Sakiyama, H. Matsuda and C. Morita, Fundamental vibration of rotating cantilever blades with pre-twist, Journal of Sound and Vibration, 271 (2004), 47-66.  doi: 10.1016/S0022-460X(03)00262-1.  Google Scholar [16] L. Hugo and Z. Barrios, Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations, Journal of Mathematical Analysis and Applications, 285 (2003), 191-211.  doi: 10.1016/S0022-247X(03)00401-3.  Google Scholar [17] B. Kundu and R. Ganguli, Analysis of weak solution of Euler-Bernoulli beam with axial force, Applied Mathematics and Computation, 298 (2017), 247-260.  doi: 10.1016/j.amc.2016.11.019.  Google Scholar [18] J. E. Lagnese, Boundary controllability of nonlinear beams to bounded states, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 63-72.  doi: 10.1017/S0308210500028316.  Google Scholar [19] J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, Journal of Differential Equations, 91 (1991), 355-388.  doi: 10.1016/0022-0396(91)90145-Y.  Google Scholar [20] L. Librescu, S.-Y. Oh and O. Song, Thin-walled beams made of functionally graded materials and operating in a high temperature environment: Vibration and stability, Journal of Thermal Stresses, 28 (2005), 649-712.  doi: 10.1080/01495730590934038.  Google Scholar [21] L. Librescu and O. Song, Thin-walled Composite Beams: Theory and Application, Springer, Dordrecht, 2006. Google Scholar [22] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [23] L. Nirenberg, On Elliptic Partial Differential Equations. IL Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali, Springer, Berlin, 1959. Google Scholar [24] A. Holt and Z. Garabed, Piston theory, a new aerodynamic tool for the aeroelastican, Journal of the Aeronautical Sciences, 23 (1956), 1109-1118. Google Scholar [25] S. Y. Oh, O. Song and L. Librescu, Effects of pretwist and presetting on coupled bending vibrations of rotating thin-walled composite beams, International Journal of Solids and Structures, 40 (2003), 1203-1224.  doi: 10.1016/S0020-7683(02)00605-4.  Google Scholar [26] P. Pei, M. A. Rammaha and D. Toundykov, Well-posedness of mindlin-timoshenko plate with nonlinear boundary damping and sources, Applied Mathematics and Optimization, 76 (2017), 429-464.  doi: 10.1007/s00245-016-9357-1.  Google Scholar [27] E. Pesheck, C. Pierre and S. W. Shaw, Accurate reduced-order models for a simple rotor blade model using nonlinear normal modes, Mathematical and Computer Modelling, 33 (2001), 1085-1097.  doi: 10.1016/S0895-7177(00)00301-0.  Google Scholar [28] Z. M. Qin and L. Librescu, On a shear-deformable theory of anisotropic thin-walled beams: Further contribution and validations, Composite Structures, 56 (2002), 345-358.  doi: 10.1016/S0263-8223(02)00019-3.  Google Scholar [29] O. Rand and S. M. Barkai, A refined nonlinear analysis of pre-twisted composite blades, Composite Structures, 39 (1997), 39-54. Google Scholar [30] J. Simon, Compact sets in the space $L^p(0, T; B)$, Annali Di Matematica Pura Ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [31] S. K. Sinha, Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end, International Journal of Non-Linear Mechanics, 40 (2005), 113-149.  doi: 10.1016/j.ijnonlinmec.2004.05.019.  Google Scholar [32] O. Song and L. Librescu, Free vibration of anisotropic composite thin-walled beams of closed cross-section contour, Journal of Sound and Vibration, 167 (1993), 129-147.  doi: 10.1006/jsvi.1993.1325.  Google Scholar [33] O. Song and L. Librescu, Structural modeling and free vibration analysis of rotating composite thin-walled beams, Journal of the American Helicopter Society, 42 (1997), 358-369.  doi: 10.4050/JAHS.42.358.  Google Scholar [34] D. M. Tang and E. H. Dowell, Nonlinear response of a non-rotating rotor blade to a periodic gust, Journal of Fluids and Structures, 10 (1996), 721-742.  doi: 10.1006/jfls.1996.0050.  Google Scholar [35] L. Tebou, Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping, Nonlinear Anal., 71 (2009), e2288–e2297. doi: 10.1016/j.na.2009.05.026.  Google Scholar [36] D. Thakkar and R. Ganguli, Induced shear actuation of helicopter rotor blade for active twist control, Thin-Walled Structures, 45 (2007), 111-121.  doi: 10.1016/j.tws.2006.11.001.  Google Scholar [37] G. Turkalj, J. Brnic and J. Prpic-Orsic, Large rotation analysis of elastic thin-walled beam-type structures using ESA approach, Computers and Structures, 81 (2003), 1851-1864.  doi: 10.1016/S0045-7949(03)00206-2.  Google Scholar [38] Z. J. Yang, On an extensible beam equation with nonlinear damping and source terms, Journal of Differential Equations, 254 (2013), 3903-3927.  doi: 10.1016/j.jde.2013.02.008.  Google Scholar [39] M. H. Yao, Y. P. Chen and W. Zhang, Nonlinear vibrations of blade with varying rotating speed, Nonlinear Dynamics, 68 (2012), 487-504.  doi: 10.1007/s11071-011-0231-z.  Google Scholar

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References:
 [1] G. L. Anderson, On the extensional and flexural vibrations of rotating bars, International Journal of Non-Linear Mechanics, 10 (1975), 223-236.  doi: 10.1016/0020-7462(75)90014-1.  Google Scholar [2] H. Arvin and F. Bakhtiari-Nejad, Nonlinear free vibration analysis of rotating composite Timoshenko beams, Composite Structures, 96 (2013), 29-43.  doi: 10.1016/j.compstruct.2012.09.009.  Google Scholar [3] K. V. Avramov, O. S. Galas, O. K. Morachkovskii and C. Pierre, Analysis of flexural-flexural-torsional nonlinear vibrations of twisted rotating beams with cross-sectional deplanation, Strength of Materials, 41 (2009), 200-208.  doi: 10.1007/s11223-009-9111-x.  Google Scholar [4] A. Burchard and L. E. Thomas, On the cauchy problem for a dynamical Euler's Elastica, Communications in Partial Differential Equations, 28 (2013), 271-300.  doi: 10.1081/PDE-120019382.  Google Scholar [5] M. M. Cavalcanti, V. N. Domingos Cavalcanti and J. A. soriano, Global existence and asymptotic stability for the nonlinear and generalized damped extensible plate equation, Communications in Contemporary Mathematics, 6 (2004), 705-731.  doi: 10.1142/S0219199704001483.  Google Scholar [6] G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams, SIAM Journal on Control and Optimization, 25 (1987), 526-546.  doi: 10.1137/0325029.  Google Scholar [7] L. W. Chen, W. K. Peng, L. W. Chen and W. K. Peng, Dynamic stability of rotating blades with geometric non-linearity, Journal of Sound and Vibration, 187 (1995), 421-433.   Google Scholar [8] S.-C. Choi, J. S. Park and J. H. Kim, Vibration control of pre-twisted rotating composite thin-walled beams with piezoelectric fiber composites, Journal of Sound and Vibration, 300 (2007), 176-196.  doi: 10.1016/j.jsv.2006.07.051.  Google Scholar [9] S. A. Fazelzadeh and M. Hosseini, Aerothermoelastic behavior of supersonic rotating thin-walled beams made of functionally graded materials, Pressure Vessels and Piping Conference, (2006), 227–236. doi: 10.1115/PVP2006-ICPVT-11-93624.  Google Scholar [10] S. A. Fazelzadeh, P. Malekzadeh, P. Zahedinejad and M. Josseini, Vibration analysis of functionally graded thin-walled rotating blades under high temperature supersonic flow using the differential quadrature method, Journal of Sound and Vibration, 306 (2007), 333-348.  doi: 10.1016/j.jsv.2007.05.011.  Google Scholar [11] F. Georgiades, J. Latalski and J. Warminski, Equations of motion of rotating composite beam with a nonconstant rotation speed and an arbitrary preset angle, Meccanica, 49 (2014), 1833-1858.  doi: 10.1007/s11012-014-9926-9.  Google Scholar [12] B. Z. Guo and J.-M. Wang, The well-posedness and stability of a beam equation with conjugate variables assigned at the same boundary point, IEEE Transactions on Automatic Control, 50 (2005), 2087-2093.  doi: 10.1109/TAC.2005.860275.  Google Scholar [13] A. Hasanov and O. Baysal, Identification of unknown temporal and spatial load distributions in a vibrating Euler-Bernoulli beam from Dirichlet boundary measured data, Automatica J. IFAC, 71 (2016), 106-117.  doi: 10.1016/j.automatica.2016.04.034.  Google Scholar [14] G. Hegarty and S. Taylor, Classical solutions of nonlinear beam equations: Existence and stabilization, SIAM Journal on Control and Optimization, 50 (2012), 703-719.  doi: 10.1137/100793864.  Google Scholar [15] X. X. Hu, T. Sakiyama, H. Matsuda and C. Morita, Fundamental vibration of rotating cantilever blades with pre-twist, Journal of Sound and Vibration, 271 (2004), 47-66.  doi: 10.1016/S0022-460X(03)00262-1.  Google Scholar [16] L. Hugo and Z. Barrios, Existence, stability and smoothness of a bounded solution for nonlinear time-varying thermoelastic plate equations, Journal of Mathematical Analysis and Applications, 285 (2003), 191-211.  doi: 10.1016/S0022-247X(03)00401-3.  Google Scholar [17] B. Kundu and R. Ganguli, Analysis of weak solution of Euler-Bernoulli beam with axial force, Applied Mathematics and Computation, 298 (2017), 247-260.  doi: 10.1016/j.amc.2016.11.019.  Google Scholar [18] J. E. Lagnese, Boundary controllability of nonlinear beams to bounded states, Proc. Roy. Soc. Edinburgh Sect. A, 119 (1991), 63-72.  doi: 10.1017/S0308210500028316.  Google Scholar [19] J. E. Lagnese and G. Leugering, Uniform stabilization of a nonlinear beam by nonlinear boundary feedback, Journal of Differential Equations, 91 (1991), 355-388.  doi: 10.1016/0022-0396(91)90145-Y.  Google Scholar [20] L. Librescu, S.-Y. Oh and O. Song, Thin-walled beams made of functionally graded materials and operating in a high temperature environment: Vibration and stability, Journal of Thermal Stresses, 28 (2005), 649-712.  doi: 10.1080/01495730590934038.  Google Scholar [21] L. Librescu and O. Song, Thin-walled Composite Beams: Theory and Application, Springer, Dordrecht, 2006. Google Scholar [22] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar [23] L. Nirenberg, On Elliptic Partial Differential Equations. IL Principio Di Minimo E Sue Applicazioni Alle Equazioni Funzionali, Springer, Berlin, 1959. Google Scholar [24] A. Holt and Z. Garabed, Piston theory, a new aerodynamic tool for the aeroelastican, Journal of the Aeronautical Sciences, 23 (1956), 1109-1118. Google Scholar [25] S. Y. Oh, O. Song and L. Librescu, Effects of pretwist and presetting on coupled bending vibrations of rotating thin-walled composite beams, International Journal of Solids and Structures, 40 (2003), 1203-1224.  doi: 10.1016/S0020-7683(02)00605-4.  Google Scholar [26] P. Pei, M. A. Rammaha and D. Toundykov, Well-posedness of mindlin-timoshenko plate with nonlinear boundary damping and sources, Applied Mathematics and Optimization, 76 (2017), 429-464.  doi: 10.1007/s00245-016-9357-1.  Google Scholar [27] E. Pesheck, C. Pierre and S. W. Shaw, Accurate reduced-order models for a simple rotor blade model using nonlinear normal modes, Mathematical and Computer Modelling, 33 (2001), 1085-1097.  doi: 10.1016/S0895-7177(00)00301-0.  Google Scholar [28] Z. M. Qin and L. Librescu, On a shear-deformable theory of anisotropic thin-walled beams: Further contribution and validations, Composite Structures, 56 (2002), 345-358.  doi: 10.1016/S0263-8223(02)00019-3.  Google Scholar [29] O. Rand and S. M. Barkai, A refined nonlinear analysis of pre-twisted composite blades, Composite Structures, 39 (1997), 39-54. Google Scholar [30] J. Simon, Compact sets in the space $L^p(0, T; B)$, Annali Di Matematica Pura Ed Applicata, 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar [31] S. K. Sinha, Non-linear dynamic response of a rotating radial Timoshenko beam with periodic pulse loading at the free-end, International Journal of Non-Linear Mechanics, 40 (2005), 113-149.  doi: 10.1016/j.ijnonlinmec.2004.05.019.  Google Scholar [32] O. Song and L. Librescu, Free vibration of anisotropic composite thin-walled beams of closed cross-section contour, Journal of Sound and Vibration, 167 (1993), 129-147.  doi: 10.1006/jsvi.1993.1325.  Google Scholar [33] O. Song and L. Librescu, Structural modeling and free vibration analysis of rotating composite thin-walled beams, Journal of the American Helicopter Society, 42 (1997), 358-369.  doi: 10.4050/JAHS.42.358.  Google Scholar [34] D. M. Tang and E. H. Dowell, Nonlinear response of a non-rotating rotor blade to a periodic gust, Journal of Fluids and Structures, 10 (1996), 721-742.  doi: 10.1006/jfls.1996.0050.  Google Scholar [35] L. Tebou, Well-posedness and stability of a hinged plate equation with a localized nonlinear structural damping, Nonlinear Anal., 71 (2009), e2288–e2297. doi: 10.1016/j.na.2009.05.026.  Google Scholar [36] D. Thakkar and R. Ganguli, Induced shear actuation of helicopter rotor blade for active twist control, Thin-Walled Structures, 45 (2007), 111-121.  doi: 10.1016/j.tws.2006.11.001.  Google Scholar [37] G. Turkalj, J. Brnic and J. Prpic-Orsic, Large rotation analysis of elastic thin-walled beam-type structures using ESA approach, Computers and Structures, 81 (2003), 1851-1864.  doi: 10.1016/S0045-7949(03)00206-2.  Google Scholar [38] Z. J. Yang, On an extensible beam equation with nonlinear damping and source terms, Journal of Differential Equations, 254 (2013), 3903-3927.  doi: 10.1016/j.jde.2013.02.008.  Google Scholar [39] M. H. Yao, Y. P. Chen and W. Zhang, Nonlinear vibrations of blade with varying rotating speed, Nonlinear Dynamics, 68 (2012), 487-504.  doi: 10.1007/s11071-011-0231-z.  Google Scholar
Rotating blades
The cross-section of rotating blades
Position vectors of ponit $A^*$ in reference frames
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