Nonlinear dynamics of rotating pretwisted cylindrical panels under 1:2 internal resonances. (English) Zbl 1453.70009
Summary: This paper investigates the nonlinear vibrations of the rotating pretwisted cylindrical panel under higher-frequency primary resonance and lower-frequency primary resonance for the case of 1:2 internal resonances. An accurate strain-displacement relationship is derived by the Green strain tensor. First-order shear deformation theory and Hamilton principle are utilized to establish the partial differential governing equation of the rotating cylindrical panel. Galerkin approach is employed to obtain the two-degree-of-freedom nonlinear system, which contains coupling between linear stiffness terms of the two transverse modes. The method of multiple scales is used to obtain the modulation equations for the amplitudes and phases. Numerical simulations are performed to show amplitude-frequency responses and bifurcation behaviors of the system. Two types of numerical methods are compared to describe the amplitude-frequency responses of the system. The results show the accuracy of our proposed method. The effects of the detuning parameter, the damping coefficient and the excitation amplitude on amplitude-frequency responses and bifurcation behaviors are fully discussed.
MSC:
| 70K28 | Parametric resonances for nonlinear problems in mechanics |
| 74H45 | Vibrations in dynamical problems in solid mechanics |
Keywords:
rotating pretwisted blade; rotating cylindrical panel; internal resonance; amplitude-frequency response; bifurcation behaviorReferences:
| [1] | Arvin, H. & Nejad, F. B. [2013] “ Nonlinear free vibration analysis of rotating composite Timoshenko beams,” Compos. Struct.96, 29-43. |
| [2] | Bai, B., Li, H., Zhang, W. & Cui, Y. C. [2020] “ Application of extremum response surface method-based improved substructure component modal synthesis in mistuned turbine bladed disk,” J. Sound Vibr.472, 115210. |
| [3] | Cao, D. X., Liu, B. Y., Yao, M. H. & Zhang, W. [2017] “ Free vibration analysis of a pre-twisted sandwich blade with thermal barrier coatings layers,” Sci. China-Technol. Sci.60, 1747-1761. |
| [4] | Chandiramani, N. K., Librescu, L. & Shete, C. D. [2002] “ On the free-vibration of rotating composite beams using a higher-order shear formulation,” Aerosp. Sci. Technol.6, 545-561. · Zbl 1018.74505 |
| [5] | Chern, Y. C. & Chao, C. C. [2000] “ Comparison of natural frequencies of laminates by 3D theory, Part II: Curved panels,” J. Sound Vibr.230, 1009-1030. |
| [6] | Chen, Y. G., Zhai, J. Y. & Han, Q. K. [2016] “ Vibration and damping analysis of the bladed disk with damping hard coating on blades,” Aerosp. Sci. Technol.58, 248-257. |
| [7] | Crespo da Silva, M. R. M. & Hodges, D. H. [1986] “ Nonlinear flexure and torsion of rotating beams with application to helicopter rotor blades-II: Response and stability results,” Vertica10, 171-186. |
| [8] | Dey, S. & Karmakar, A. [2012] “ Natural frequencies of delaminated composite rotating conical shells — A finite element approach,” Finite Elem. Anal. Des.56, 41-51. |
| [9] | Du, C. C. & Li, Y. H. [2013] “ Nonlinear resonance behavior of functionally graded cylindrical shells in thermal environments,” Compos. Struct.102, 164-174. |
| [10] | Farhadi, S. & Hashemi, S. H. [2011] “ Aeroelastic behavior of cantilevered rotating rectangular plates,” Int. J. Mech. Sci.53, 316-328. |
| [11] | Fazelzadeh, S. A., Malekzadeh, P., Zahedinejad, P. & Hosseini, M. [2007] “ Vibration analysis of functionally graded thin-walled rotating blades under high temperature supersonic flow using the differential quadrature method,” J. Sound Vibr.306, 333-348. |
| [12] | Hao, Y. X., Zhang, W. & Yang, J. [2013] “ Nonlinear dynamics of cantilever FGM cylindrical shell under 1:2 internal resonance relations,” Adv. Mater. Struct.20, 819-833. |
| [13] | Hashemi, S. H., Farhadi, S. & Carra, S. [2009] “ Free vibration analysis of rotating thick plates,” J. Sound Vibr.323, 366-384. |
| [14] | Hu, X. X. & Tsuiji, T. [1999] “ Free vibration analysis of curved and twisted cylindrical thin panels,” J. Sound Vibr.219, 63-88. |
| [15] | Hu, H. Y. [2000] Applied Nonlinear Dynamics (Aviation Industry, Beijing). |
| [16] | Hu, X. X., Sakiyama, T., Xiong, Y., Matsuda, H. & Morita, C. [2004a] “ Vibration analysis of twisted plates using first order shear deformation theory,” J. Sound Vibr.277, 205-222. |
| [17] | Hu, X. X., Sakiyama, T., Matsuda, H. & Morita, C. [2004b] “ Fundamental vibration of rotating cantilever blades with pre-twist,” J. Sound Vibr.271, 47-66. |
| [18] | Ji, J. C. [2014a] “ Secondary resonances of a quadratic nonlinear oscillator following two-to-one resonant Hopf bifurcations,” Nonlin. Dyn.78, 2161-2184. |
| [19] | Ji, J. C. [2014b] “ Design of a nonlinear vibration absorber using three-to-one internal resonances,” Mech. Syst. Sign. Proc.42, 236-246. |
| [20] | Ji, J. C. [2015] “ Two families of super-harmonic resonances in a time-delayed nonlinear oscillator,” J. Sound Vibr.349, 299-314. |
| [21] | Ji, J. C. & Brown, T. [2017] “ Periodic and chaotic motion of a time-delayed nonlinear system under two coexisting families of additive resonances,” Int. J. Bifurcation and Chaos27, 1750066-1-15. · Zbl 1367.70045 |
| [22] | Ji, J. C. & Zhou, J. [2017] “ Coexistence of two families of sub-harmonic resonances in a time-delayed nonlinear system at different forcing frequencies,” Mech. Syst. Sign. Proc.93, 151-163. |
| [23] | Kobayashi, Y. & Leissa, A. W. [1995] “ Large amplitude free vibration of thick shallow shells supported by shear diaphragms,” Int. J. Non-Lin. Mech.30, 57-66. · Zbl 0819.73036 |
| [24] | Kou, H. J. & Yuan, H. Q. [2014] “ Rub-induced non-linear vibrations of a rotating large deflection plate,” Int. J. Non-Lin. Mech.58, 283-294. |
| [25] | Leung, A. Y. T. & Fan, J. [2010] “ Natural vibration of pre-twisted shear deformable beam systems subjected to multiple kinds of initial stresses,” J. Sound Vibr.329, 1901-1923. |
| [26] | Liew, K. M., Lim, C. W. & Ong, L. S. [1994] “ Vibration of pretwisted cantilever shallow conical shells,” Int. J. Solids Struct.31, 2463-2476. · Zbl 0943.74515 |
| [27] | Li, S. & Caracoglia, L. [2019] “ Surrogate model monte carlo simulation for stochastic flutter analysis of wind turbine blades,” J. Wind Eng. Ind. Aerodyn.188, 43-60. |
| [28] | Mao, X. Y., Ding, H. & Chen, L. Q. [2017] “ Dynamics of a super-critically axially moving beam with parametric and forced resonance,” Nonlin. Dyn.89, 1475-1487. |
| [29] | Niu, Y., Zhang, W. & Guo, X. Y. [2019] “ Free vibration of rotating pretwisted functionally graded composite cylindrical panel reinforced with graphene platelets,” Eur. J. Mech. A — Solids77, 103798. · Zbl 1475.74057 |
| [30] | Nosier, A. & Reddy, J. N. [1991] “ A study of non-linear dynamic equations of higher-order deformation plate theories,” Int. J. Nonlin. Mech.26, 233-249. · Zbl 0734.73041 |
| [31] | Qatu, M. S. & Leissa, A. W. [1991] “ Vibration studies for laminated composite twisted cantilever plates,” Int. J. Mech. Sci.33, 927-940. |
| [32] | Rao, J. S. & Gupta, K. [1987] “ Free vibrations of rotating small aspect ratio pretwisted blades,” Mech. Mach. Th.22, 159-167. |
| [33] | Reddy, J. N. [2004] Mechanics of Laminated Composite Plates and Shells: Theory and Analysis (CRC Press, NY). · Zbl 1075.74001 |
| [34] | Reddy, R. M. R., Karunasena, W. & Lokuge, W. [2018] “ Free vibration of functionally graded-GPL reinforced composite plates with different boundary conditions,” Aerosp. Sci. Technol.78, 147-156. |
| [35] | Rostami, H., Ranji, A. R. & Nejad, F. B. [2016] “ Free in-plane vibration analysis of rotating rectangular orthotropic cantilever plates,” Int. J. Mech. Sci.115-116, 438-456. |
| [36] | Roy, P. A. & Meguid, S. A. [2018] “ Nonlinear transient dynamic response of a blade subject to a pulsating load in a decaying centrifugal force field,” Int. J. Mech. Mater. Des.14, 709-728. |
| [37] | Sahoo, B., Panda, L. N. & Pohit, G. [2017] “ Stability, bifurcation and chaos of a traveling viscoelastic beam tuned to 3:1 internal resonance and subjected to parametric excitation,” Int. J. Bifurcation and Chaos27, 1750017-1-20. · Zbl 1362.74021 |
| [38] | Shen, H. S. & Wang, H. [2014] “ Nonlinear vibration of shear deformable FGM cylindrical panels resting on elastic foundations in thermal environments,” Compos. Pt. B-Eng.60, 167-177. |
| [39] | Sinha, S. K. & Turner, K. E. [2011] “ Natural frequencies of a pre-twisted blade in a centrifugal force field,” J. Sound Vibr.330, 2655-2681. |
| [40] | Sreenivasamurthy, S. & Ramamurti, V. [1981] “ A parametric study of vibration of rotating pre-twisted and tapered low aspect ratio cantilever plates,” J. Sound Vibr.76, 311-328. |
| [41] | Sun, J., Kari, L. & Arteaga, I. L. [2013a] “ A dynamic rotating blade model at an arbitrary stagger angle based on classical plate theory and the Hamilton’s principle,” J. Sound Vibr.332, 1355-1371. |
| [42] | Sun, J., Arteaga, I. L. & Kari, L. [2013b] “ General shell model for a rotating pretwisted blade,” J. Sound Vibr.332, 5804-5820. |
| [43] | Tornabene, F., Viola, E. & Inman, D. J. [2009] “ 2D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures,” J. Sound Vibr.328, 259-290. |
| [44] | Wang, F. X. & Qu, Y. H. [2014] “ Period doubling motions of a nonlinear rotating beam at 1:1 resonance,” Int. J. Bifurcation and Chaos24, 1450159. · Zbl 1305.70041 |
| [45] | Wang, G. X., Ding, H. & Chen, L. Q. [2020] “ Dynamic effect of internal resonance caused by gravity on the nonlinear vibration of vertical cantilever beam,” J. Sound Vibr.474, 115265. |
| [46] | Xie, F. T., Ma, H., Cui, C. & Wen, B. C. [2017] “ Vibration response comparison of twisted shrouded blades using different impact models,” J. Sound Vibr.397, 171-191. |
| [47] | Yao, M. H., Chen, Y. P. & Zhang, W. [2012] “ Nonlinear vibrations of blade with varying rotating speed,” Nonlin. Dyn.68, 487-504. · Zbl 1348.74160 |
| [48] | Yao, M. H., Ma, L. & Zhang, W. [2018] “ Nonlinear dynamics of the high-speed rotating plate,” Int. J. Aerosp. Eng.2018, 5610915. |
| [49] | Yao, M. H., Niu, Y. & Hao, Y. X. [2019] “ Nonlinear dynamic responses of rotating pretwisted cylindrical shells,” Nonlin. Dyn.95, 151-174. · Zbl 1439.74182 |
| [50] | Yoo, H. H., Kwak, J. Y. & Chung, J. [2001] “ Vibration analysis of rotating pre-twisted blades with a concentrated mass,” J. Sound Vibr.240, 891-908. |
| [51] | Yoo, H. H. & Kim, S. K. [2002] “ Free vibration analysis of rotating cantilever plates,” AIAA J.40, 2188-2196. |
| [52] | Yoo, H. H. & Pierre, C. [2003] “ Modal characteristic of a rotating rectangular cantilever plate,” J. Sound Vibr.259, 81-96. |
| [53] | Yoo, H. H., Lee, S. H. & Shin, S. H. [2005] “ Flapwise bending vibration analysis of rotating multi-layered composite beams,” J. Sound Vibr.286, 745-761. · Zbl 1243.74063 |
| [54] | Yuan, J., Allegri, G., Scarpa, F., Patsias, S. & Rajasekaran, R. [2016] “ A novel hybrid Neumann expansion method for stochastic analysis of mistuned bladed discs,” Mech. Syst. Sign. Proc.72-73, 241-253. |
| [55] | Zhang, B., Ding, H. & Chen, L. Q. [2019] “ Super-harmonic resonances of a rotating pre-deformed blade subjected to gas pressure,” Nonlin. Dyn.98, 2531-2549. · Zbl 1430.74096 |
| [56] | Zhang, W., Niu, Y. & Behdinan, K. [2020] “ Vibration characteristics of rotating pretwisted composite tapered blade with graphene coating layers,” Aerosp. Sci. Technol.98, 105644. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
