Effect of Spinning Speed Fluctuation along with the Twist Angle on the Nonlinear Vibration and Stability of an Asymmetrical Twisted Slender Beam

Document Type : Research Article

Authors

Faculty of Mechanical Engineering, Shahrood University of Technology

Abstract

In this study, the effect of spinning speed fluctuations along with the twist angle, on the stability and bifurcation of spinning slender twisted beams, with linear twist angle and large transverse deflections, near the primary and parametric resonances have been analyzed using the Euler–Bernoulli model. The spinning speed fluctuation along with the twist angle, asymmetry and imbalance, play an important role on the frequency response of the twisted beam. The equations of motion, in the case of pure single mode motion, are analyzed by using the multiple scales method after discretization by the Galerkin's procedure. The instability of the twisted and untwisted beams is investigated and cases and domains are determined in which bifurcation could occur. Effects of the speed fluctuations, twist angle, damping ratio, asymmetry, eccentricity and mass moment of inertia about the longitudinal axis on the frequency response of the twisted beam are investigated. This is explained that the spinning speed fluctuation effect is weak in lower modes and smaller twist angles while asymmetry effect is dominant. By ascending the mode number and twist angle, spinning speed fluctuation effect amplifies the amplitude of system. The results are compared and validated with the results obtained from Runge-Kutta numerical method in steady state, and confirmed with some previous researches.

Keywords

Main Subjects

References

[1]  Washizu, K., Some Consideration on a Naturally Curved and Twisted Slender Beam, Journal of Mathematics and Physics, 43(2) (1964)111-116.
[2] Slyper, H.A., Coupled bending vibrations of pret twisted cantilever beams, Journal of Mechanical Engineering Science, 4(4) (1962)365-379.
[3]   Dawson, B., Carnegie, W., Modal curves of pre-twisted beams of rectangular cross-section, Journal of Mechanical Engineering Science, 11(1) (1969)1-13.
[4] Imregun, T., Numerical derivation of FRF matrices for curved, twisted and rotating beams, Journal of Mechanical Engineering Sciences, 203(6) (1989)393-401.
[5]  Oh, S.Y., Song, O., Librescu, L., 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.
[6] Banerjee, J.R., Development of an exact dynamic stiffness matrix for free vibration analysis of a twisted Timoshenko beam, Journal of Sound and Vibration, 270(1-2) (2004) 379–401.
[7] Yardimoglu, B., Yildirim, T., Finite element model for vibration analysis of pre- twisted Timoshenko beam, Journal of Sound and Vibration, 273(4-5) (2004)741–54.
[8] Subuncu, M., Evran, K., Dynamic stability of a rotating pre-twisted asymmetric cross-section Timoshenko beam subjected to an axial periodic force, International Journal of Mechanical Science, 48(6) (2006)579–90.
[9] Lin, S.M., Lee, J.F., Lee, S.Y., Wang, W.R., Prediction of rotating damped beams with arbitrary pre-twist, International Journal of Mechanical Science, 48(12) (2006)1494–504.
[10] Avramov, K.V., Pierre, C., Flexural-flexural-torsional Nonlinear Vibrations of Pre-twisted Rotating Beams with Asymmetric Cross-sections, Journal of Vibration and Control, 13(4) (2007)329–364.
[11]  Verichev, N.N., Verichev, S.N., Erofeyev V.I., Damping lateral vibrations in rotary machinery using motor speed modulation, Journal of Sound and Vibration, 329(1) (2010)13–20.
[12] Yao, M.H.,  Chen, Y.P.,  Zhang, W., Nonlinear vibrations of blade with varying rotating speed. Nonlinear Dynamics, 68(4) (2012)487–504.
[13] Ebrahimi, F., Mokhtari, M., Vibration analysis of spinning exponentially functionally graded Timoshenko beams based on differential transform method, Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering, 229(14) (2015) 2559-2571.
[14]   Bekir, B., Romero, L., Burak, O., Three dimensional dynamics of rotating structures under mixed boundary conditions, Journal of Sound and Vibration, 358(2015)176–191.
[15]  Adair, D.,  Jaeger, M., Vibration analysis of a uniform pre-twisted rotating Euler–Bernoulli beam using the modified Adomian decomposition method, Mathematics and Mechanics of Solids, 23(9) (2017)1345–1363.
[16]Tekinalp, O., Ulsoy, A.G., Modeling and finite element analysis of drill bit vibrations, ASME Journal of Vibration and Acoust, 111(2) (1989)148–55.
[17] Tekinalp, O., Ulsoy, A.G., Effects of geometric and process parameters on drill transverse vibrations, ASME Journal of Manufacturing Science and Engineering, 112(2)) 1990) 189–94.
[18] Huang, B.W., Dynamic characteristics of a drill in the drilling process, Proceeding Instruments Mechanical Engineers Part B, 217(2003).
[19] Young, T.H., Gau, C.Y., Dynamic stability of pre-twisted beams with non-constant spin rates under axial random force International Journal of Solids and Structures, 40(18) (2003) 4675-4698
[20] Gurgeoze, M., On the dynamic stability of a pre-twisted beam subject to a pulsating axial load, Journal of Sound and Vibration, 102(3) (1985) 415-422.
[21] Lee, H.P., Dynamic stability of spinning pre-twisted beams, International Journal of Solids and Structures, 31(18) (1994) 2509–17.
[22] Lee, H.P., Dynamic stability of spinning pre-twisted beams subject to axial pulsating loads, Computer Methods in Applied Mechanics and Engineering, 127(1-4) (1995)115–26.
[23]        Tan, T.H., Lee, H.P., Leng, G.S.B., Parametric instability of spinning pre-twisted beams subjected to sinusoidal compressive axial loads, Computers and Structures, 66(6) (1998)745–64.
[24]        Chen, W.R., On the vibration and stability of spinning axially loaded pre-twisted Timoshenko beams, Finite Elements in Analysis and Design, 46(11) (2010)1037–47.
[25] Chen, C.H., Huang, Y., Shan, J., The finite element model research of the pre- twisted beam, Applied Mechanics and Materials, 188 (2012)31–6.
[26] Chen, W.R., Hsin, S.W., Chu, T.H., Vibration analysis of twisted Timoshenko beams with internal Kelvin–Voigt damping, Procedia Engineering, 67 (2013)525–32.
[27]Chen, W.R., Effect of local Kelvin-Voigt damping on eigenfrequencies of cantilevered twisted Timoshenko beams, Procedia Engineering, 79 (2014)160-165.
[28] Chen, W.R., Parametric studies on bending vibration of axially-loaded twisted Timoshenko beams with locally distributed Kelvin–Voigt damping, International Journal of Mechanical Sciences, 88 (2014)61-70.
[29] Li, X., Li, Y.H., Qin, Y., Free vibration characteristics of a spinning composite thin-walled beam under hygrothermal environment, International Journal of Mechanical Sciences, 119 (2016)253-265.
[30] Li, X., Qin, Y., Li, Y.H., Zhao, X., The coupled vibration characteristics of a spinning and axially moving composite thin-walled beam, Mechanics of Advanced Materials and Structures, 25(9) (2108) 722-731.
[31] Zhu, K., Chung, J., Dynamic Modelling and analysis of a spinningRayleigh beam under deployment. International Journal of Mechanical Sciences, 115 (2016) 392-405.
[32]  Zhu, K., Chung, J., Vibration and stability analysis of a simply-supported Rayleigh beam with spinning and axial motions, Applied Mathematical Modelling, 66 (2019) 362-382.
[33]  Thomas, O., Senechal, A., Deu, J.F., Hardening/softening behavior and reduced order modeling of nonlinear vibrations of rotating cantilever beams, Nonlinear Dynamics, 86(2) (2016) 1293-1318.
[34] Sheng, G.G., Wang, X., The geometrically nonlinear dynamic responses of simply supported beams under moving loads, Applied Mathematical Modelling, 48 (2017)183-195.
[35] Farsadi, T., Ozgun, S., Kayran, A., Free vibration analysis of uniform and asymmetric composite pretwisted rotating thin walled beam, (2017)1-22.
[36] Mirtalaie, S. H., Hajabasi, M. A., Nonlinear axial-lateral-torsional free vibrations analysis of Rayleigh rotating shaft, Archive of Applied Mechanics, 87(2017)1465-1494.
[37] Qaderi, M.S., Hosseini, S.A.A., Zamanian, M., Combination parametric resonance of nonlinear unbalanced rotating shafts, 13(11) (2018)1-8.
[38] Xinwei, W., Zhang, X.Y., Three-dimensional vibration analysis of curved and twisted beams with irregular shapes of cross-sections by sub-parametric quadrature element method, Computers and Mathematics with Applications,76(6) (2018)1486-1499.
[39] Karimi Nobandegani, A., Fazelzadeh, S.A., Ghavanloo, E., Non-conservative stability of spinning pre-twisted cantilever beams, Journal of Sound and Vibration, 412 (2018)130-147.
[40] Tajik, M., Karami Mohammadi, A., Nonlinear vibration, stability, and bifurcation analysis of unbalanced spinning pre-twisted beam, Mathematics and Mechanics of Solids, 24(11) (2019)3514-3536.
[41] Nayfeh A.H., Pai P.F., Linear and nonlinear structural mechanics, Edition1ed, Wiley Interscience, New York, 2004.
[42]  Nayfeh, A.H., Lacarbonara, W., On the discretization of spatially continuous systems with quadratic and cubic nonlinearities, JSME International Journal Series C, 41(3) (1998)510–531.
[43]Nayfeh, A.H., Mook, D.T., Nonlinear oscillations, Edition 1ed, Wiley Interscience, New York, 1995.
[44]  Nayfeh, A.H., Balachandran, B., Applied nonlinear dynamics: analytical, computational, and experimental methods, Edition 1ed, Wiley, New York, 1995.
[45] Shahgholi, M., Khadem, S.E., Resonance analysis of gyroscopic nonlinear spinning shafts with parametric excitations and speed fluctuations, International Journal of Mechanical Sciences, 64(1) (2012)94–109.
AUT Journal of Mechanical Engineering
Volume 4, Issue 3
September 2020
Pages 363-380

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