Nonlinear Torsional Vibrations of Axially Loaded Pretwisted Beam with Primary Resonance Excitations

Document Type : Research Article

Authors

1 صنعتی شاهرود-مهندسی مکانیک

2 Dept. of Aerospace Engineering, Sharif Univ. of Tech.

Abstract

Frequently used thin walled beams have low torsional stiffness and their torsional deformations may be of such magnitudes that it is not adequate to treat the angles of cross section rotation as small. In this paper, nonlinear torsional vibrations of thin walled beams will be investigated. The method of multiple scales will be implemented as a solution method and different nonlinear phenomena will be studied. The obtained results are compared with the available results in the literature which reveals an excellent agreement between different solution methodologies. The outcomes of this study show that beam nonlinear torsional dynamics and the related phenomena could influence the linear torsional dynamic of beams under axial load, e.g. rotating beams. Forced torsional vibrations of a beam with the excitation in the form of primary resonance of the first and second modes have been investigated. It has been demonstrated that in the case of the beam with two ends clamped boundary conditions, three-to-one internal resonance will appear. The primary resonance of the first and second modes has been solved in two sets of boundary conditions, torsionally clamped-fixed and torsionally fixed-fixed. Nonlinear response, amplitude-phase equations, fixed points, and their stability have been studied.

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References

[1] S. Sarkar, H. Bijl, Nonlinear aeroelastic behavior of an oscillating airfoil during stall-induced vibration, Journal of Fluids and Structures, 24(6) (2008) 757-777.
[2] C. Tran, D. Petot, Semi-empirical model for the dynamic stall of airfoils in view of the application to the calculation of responses of a helicopter blade in forward flight,  (1980).
[3] J. Marshall, M. Imregun, A review of aeroelasticity methods with emphasis on turbomachinery applications, Journal of fluids and structures, 10(3) (1996) 237-267.
[4] D. Whitehead, Torsional flutter of unstalled cascade blades at zero deflection. R&M-3429, British ARC,  (1966).
[5] M. Ohtsuka, Untwist of rotating blades,  (1975).
[6] D.H. Hodges, Torsion of pretwisted beams due to axial loading,  (1980).
[7] O. Bauchau, R. Loewy, P. Bryan, Approach to ideal twist distribution in tilt rotor VTOL blade designs, Rensselaer Polytechnic Institute, Troy, NY, RTC Report No. D-86-2,  (1986).
[8] R. LAKE, M. NIXON, M. WILBUR, J. SINGLETON, P. MIRICK, A demonstration of passive blade twist control using extension-twistcoupling, in:  33rd Structures, Structural Dynamics and Materials Conference, 1992, pp. 2468.
[9] P.V. Bayly, S.A. Metzler, A.J. Schaut, K.A. Young, Theory of torsional chatter in twist drills: model, stability analysis and composition to test, J. Manuf. Sci. Eng., 123(4) (2001) 552-561.
[10] D. Wajchman, K.-C. Liu, J. Friend, L. Yeo, An ultrasonic piezoelectric motor utilizing axial-torsional coupling in a pretwisted non-circular cross-sectioned prismatic beam, ieee transactions on ultrasonics, ferroelectrics, and frequency control, 55(4) (2008) 832-840.
[11] D.H. Hodges, Nonlinear composite beam theory, American Institute of Aeronautics and Astronautics, 2006.
[12] V. Vlasov, Thin-walled elastic beams, national science foundation, Washington, DC, USA,  (1961).
[13] J. Buckley, The bifilar property of twisted strips, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 28(168) (1914) 778-787.
[14] H. Wagner, Torsion and buckling of open sections, 1936.
[15] J. Goodier, Elastic torsion in the presence of initial axial stress,  (1950).
[16] J.C. Houbolt, G.W. Brooks, Differential equations of motion for combined flapwise bending, chordwise bending, and torsion of twisted nonuniform rotor blades, National Advisory Committee for Aeronautics, 1957.
[17] K. Washizu, Some considerations on a naturally curved and twisted slender beam, Journal of Mathematics and Physics, 43(1-4) (1964) 111-116.
[18] A. Rosen, The effect of initial twist on the torsional rigidity of beams—another point of view,  (1980).
[19] A. Rosen, Theoretical and experimental investigation of the nonlinear torsion and extension of initially twisted bars,  (1983).
[20] D.H. Hodges, D. Harursampath, V.V. Volovoi, C.E. Cesnik, Non-classical effects in non-linear analysis of pretwisted anisotropic strips, International Journal of Non-Linear Mechanics, 34(2) (1999) 259-277.
[21] D. Harursampath, D.H. Hodges, Asymptotic analysis of the non-linear behavior of long anisotropic tubes, International journal of non-linear mechanics, 34(6) (1999) 1003-1018.
[22] B. Popescu, D.H. Hodges, Asymptotic treatment of the trapeze effect in finite element cross-sectional analysis of composite beams, International Journal of Non-Linear Mechanics, 34(4) (1999) 709-721.
[23] M. Cullimore, The shortening effect, a nonlinear feature of pure torsion, Engineering structures,  (1949) 153-164.
[24] M. Gregory, Elastic Torsion of Members of Thin Open Cross Sections, Department of Civil Engineering, University of Tasmania, 1961.
[25] W. Tso, A. Ghobarah, Non-linear non-uniform torsion of thin-walled beams, International Journal of Mechanical Sciences, 13(12) (1971) 1039-1047.
[26] B. Rozmarynowski, C. Szymczak, Non-linear free torsional vibrations of thin-walled beams with bisymmetric cross-section, Journal of Sound and Vibration, 97(1) (1984) 145-152.
[27] M.M. Attard, Nonlinear shortening and bending effect under pure torque of thin-walled open beams, Thin-walled structures, 4(3) (1986) 165-177.
[28] M.M. Attard, Nonlinear theory of non-uniform torsion of thin-walled open beams, Thin-walled structures, 4(2) (1986) 101-134.
[29] N.S. Trahair, Non-Linear Elastic Non-Uniform Torsion (No. R828),  (2003).
[30] G. Şakar, M. Sabuncu, Dynamic stability analysis of pretwisted aerofoil cross-section blade packets under rotating conditions, International Journal of Mechanical Sciences, 50(1) (2008) 1-13.
[31] W.-R. Chen, Parametric instability of spinning twisted Timoshenko beams under compressive axial pulsating loads, International journal of mechanical sciences, 52(9) (2010) 1167-1175.
[32] S.K. Sinha, K.E. Turner, Natural frequencies of a pre-twisted blade in a centrifugal force field, Journal of Sound and Vibration, 330(11) (2011) 2655-2681.
[33] C.M. Saravia, S.P. Machado, V.H. Cortínez, Free vibration and dynamic stability of rotating thin-walled composite beams, European Journal of Mechanics-A/Solids, 30(3) (2011) 432-441.
[34] B. Deepak, R. Ganguli, S. Gopalakrishnan, Dynamics of rotating composite beams: A comparative study between CNT reinforced polymer composite beams and laminated composite beams using spectral finite elements, International Journal of Mechanical Sciences, 64(1) (2012) 110-126.
[35] M. Yao, Y. Chen, W. Zhang, Nonlinear vibrations of blade with varying rotating speed, Nonlinear Dynamics, 68(4) (2012) 487-504.
[36] H.A. Salim, J.F. Davalos, Torsion of open and closed thin-walled laminated composite sections, Journal of composite materials, 39(6) (2005) 497-524.
[37] F. Mohri, N. Damil, M.P. Ferry, Large torsion finite element model for thin-walled beams, Computers & structures, 86(7-8) (2008) 671-683.
[38] N.-I. Kim, D.K. Shin, Torsional analysis of thin-walled composite beams with single-and double-celled sections, Engineering structures, 31(7) (2009) 1509-1521.
[39] K.-C. Liu, J. Friend, L. Yeo, The axial–torsional vibration of pretwisted beams, Journal of Sound and Vibration, 321(1-2) (2009) 115-136.
[40] E. Sapountzakis, V. Tsipiras, Nonlinear nonuniform torsional vibrations of bars by the boundary element method, Journal of Sound and Vibration, 329(10) (2010) 1853-1874.
[41] P. Prasad, D. Harursampath, Closed-form nonlinear sectional analysis of pretwisted anisotropic beam with multiple delaminations,  (2012).
[42] S. Sina, H. Haddadpour, H. Navazi, Nonlinear free vibrations of thin-walled beams in torsion, Acta Mechanica, 223(10) (2012) 2135-2151.
[43] S. Sina, H. Haddadpour, Axial–torsional vibrations of rotating pretwisted thin walled composite beams, International Journal of Mechanical Sciences, 80 (2014) 93-101.
[44] J. Sicard, J. Sirohi, Modeling of the large torsional deformation of an extremely flexible rotor in hover, AIAA journal, 52(8) (2014) 1604-1615.
[45] J.F. Sicard, J. Sirohi, An analytical investigation of the trapeze effect acting on a thin flexible ribbon, Journal of Applied Mechanics, 81(12) (2014) 121007.
[46] S. Han, O.A. Bauchau, On the nonlinear extension-twist coupling of beams, European Journal of Mechanics-A/Solids, 72 (2018) 111-119.
[47] D. Hoffmeyer, J. Høgsberg, Damping of torsional vibrations in thin-walled beams by viscous bimoments, Mechanics of Advanced Materials and Structures, 28(3) (2021) 308-320.
[48] A.H. Nayfeh, B. Balachandran, Modal interactions in dynamical and structural systems,  (1989).
[49] A.H. Nayfeh, D.T. Mook, Nonlinear oscillations, John Wiley & Sons, 2008.
[50] D. Sado, Energy transfer in two-degree-of-freedom vibrating systems–a survey, Journal of Theoretical and Applied Mechanics, 31(1) (1993) 151-173.
[51] M. Ruijgrok, Studies in parametric and autoparametric resonance, Universiteit Utrecht, Faculteit Wiskunde en Informatica, 1995.
[52] E. Sapountzakis, V. Tsipiras, Shear deformable bars of doubly symmetrical cross section under nonlinear nonuniform torsional vibrations—application to torsional postbuckling configurations and primary resonance excitations, Nonlinear Dynamics, 62(4) (2010) 967-987.
[53] H.D. Al-Budairi, Design and analysis of ultrasonic horns operating in longitudinal and torsional vibration, University of Glasgow, 2012.
Amirkabir Journal of Mechanical Engineering
Volume 54, Issue 6
September 2022
Pages 1271-1302

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