Dynamic response of axially moving Timoshenko beams: integral transform solution

doi:10.1007/s10483-014-1879-7

Abstract

Abstract:

The generalized integral transform technique (GITT) is used to find a semianalytical numerical solution for dynamic response of an axially moving Timoshenko beam with clamped-clamped and simply-supported boundary conditions, respectively. The implementation of GITT approach for analyzing the forced vibration equation eliminates the space variable and leads to systems of second-order ordinary differential equations (ODEs) in time. The MATHEMATICA built-in function, NDSolve, is used to numerically solve the resulting transformed ODE system. The good convergence behavior of the suggested eigenfunction expansions is demonstrated for calculating the transverse deflection and the angle of rotation of the beam cross-section. Moreover, parametric studies are performed to analyze the effects of the axially moving speed, the axial tension, and the amplitude of external distributed force on the vibration amplitude of axially moving Timoshenko beams.

Key words: hybrid solution, integral transform, axially moving Timoshenko beam, transverse vibration

2010 MSC Number:

O242

Chen AN;Jian SU . Dynamic response of axially moving Timoshenko beams: integral transform solution. Applied Mathematics and Mechanics (English Edition), 2014, 35(11): 1421-1436.

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References

[1] Pakdemirli, H. R., Öz, M., and Boyaci, H. Non-linear vibrations and stability of an axially movingbeam with time-dependent velocity. International Journal of Non-Linear Mechanics, 36(1), 107-115 (2001)
[2] Yang, X. D. and Chen, L. Q. Dynamic stability of axially moving viscoelastic beams with pulsatingspeed. Appl. Math. Mech. -Engl. Ed., 26(8), 989-995 (2005) DOI 10.1007/BF02466411
[3] Lee, U. and Oh, H. Dynamics of an axially moving viscoelastic beam subject to axial tension.International Journal of Solids and Structures, 42(8), 2381-2398 (2005)
[4] Liu, K. F. and Deng, L. Y. Identification of pseudo-natural frequencies of an axially movingcantilever beam using a subspace-based algorithm. Mechanical Systems and Signal Processing,20(1), 94-113 (2006)
[5] Chen, L. Q. and Yang, X. D. Nonlinear free transverse vibration of an axially moving beam:comparison of two models. Journal of Sound and Vibration, 299(1-2), 348-354 (2007)
[6] Jakši?, N. Numerical algorithm for natural frequencies computation of an axially moving beammodel. Meccanica, 44(6), 687-695 (2009)
[7] Ponomareva, S. V. and van Horssen, W. T. On the transversal vibrations of an axially movingcontinuum with a time-varying velocity: transient from string to beam behavior. Journal of Soundand Vibration, 325(4-5), 959-973 (2009)
[8] Chang, J. R., Lin, W. J., Huang, C. J., and Choi, S. T. Vibration and stability of an axiallymoving rayleigh beam. Applied Mathematical Modelling, 34(6), 1482-1497 (2010)
[9] Huang, J. L., Su, R. K. L., Li, W. H., and Chen, S. H. Stability and bifurcation of an axiallymoving beam tuned to three-to-one internal resonances. Journal of Sound and Vibration, 330(3),471-485 (2011)
[10] Wang, B. Asymptotic analysis on weakly forced vibration of axially moving viscoelastic beamconstituted by standard linear solid model. Appl. Math. Mech. -Engl. Ed., 33(6), 817-828 (2012)DOI 10.1007/s10483-012-1588-8
[11] Marynowski, K. Dynamic analysis of an axially moving sandwich beam with viscoelastic core.Composite Structures, 94(9), 2931-2936 (2012)
[12] Lee, U., Kim, J., and Oh, H. Spectral analysis for the transverse vibration of an axially movingTimoshenko beam. Journal of Sound and Vibration, 271(3-5), 685-703 (2004)
[13] Cojocaru, E. C., Irschik, H., and Schlacher, K. Concentrations of pressure between an elasticallysupported beam and a moving Timoshenko-beam. Journal of Engineering Mechanics-ASCE,129(9), 1076-1082 (2003)
[14] Yang, X. D., Tang, Y. Q., Chen, L. Q., and Lim, C. W. Dynamic stability of axially acceleratingTimoshenko beam: averaging method. European Journal of Mechanics A-Solids, 29(1), 81-90(2010)
[15] Tang, Y. Q., Chen, L. Q., and Yang, X. D. Natural frequencies, modes and critical speeds ofaxially moving Timoshenko beams with different boundary conditions. International Journal ofMechanical Sciences, 50(10-11), 1448-1458 (2008)
[16] Tang, Y. Q., Chen, L. Q., and Yang, X. D. Parametric resonance of axially moving Timoshenkobeams with time-dependent speed. Nonlinear Dynamics, 58(4), 715-724 (2009)
[17] Tang, Y. Q., Chen, L. Q., and Yang, X. D. Nonlinear vibrations of axially moving Timoshenkobeams under weak and strong external excitations. Journal of Sound and Vibration, 320(4-5),1078-1099 (2009)
[18] Li, B., Tang, Y. Q., and Chen, L. Q. Nonlinear free transverse vibrations of axially movingTimoshenko beams with two free ends. Science China-Technological Sciences, 54(8), 1966-1976(2011)
[19] Tang, Y. Q., Chen, L. Q., Zhang, H. J., and Yang, S. P. Stability of axially accelerating viscoelasticTimoshenko beams: recognition of longitudinally varying tensions. Mechanism and MachineTheory, 62, 31-50 (2013)
[20] Ghayesh, M. H. and Balar, S. Non-linear parametric vibration and stability analysis for twodynamic models of axially moving Timoshenko beams. Applied Mathematical Modelling, 34(10),2850-2859 (2010)
[21] Ghayesh, M. H. and Amabili, M. Three-dimensional nonlinear planar dynamics of an axiallymoving Timoshenko beam. Archive of Applied Mechanics, 83(4), 591-604 (2013)
[22] Cotta, R. M. Integral Transforms in Computational Heat and Fluid Flow, CRC Press, Boca Raton(1993)
[23] Cotta, R. M. and Mikhailov, M. D. Heat Conduction—Lumped Analysis, Integral Transforms,Symbolic Computation, John Wiley & Sons, Chichester, England (1997)
[24] Cotta, R. M. The Integral Transform Method in Thermal and Fluids Science and Engineering,Begell House, New York (1998)
[25] An, C., Gu, J. J., and Su, J. Integral transform solution of bending problem of clamped orthotropicrectangular plates. International Conference on Mathematics and Computational Methods Appliedto Nuclear Science and Engineering, Rio de Janeiro, Brazil (2011)
[26] Ma, J. K., Su, J., Lu, C. H., and Li, J. M. Integral transform solution of the transverse vibration ofan axial moving string. Journal of Vibration, Measurement & Diagnosis, 26(117), 104-107 (2006)
[27] An, C. and Su, J. Dynamic response of clamped axially moving beams: integral transform solution.Applied Mathematics and Computation, 218(2), 249-259 (2011)
[28] Gu, J. J., An, C., Duan, M. L., Levi, C., and Su, J. Integral transform solutions of dynamicresponse of a clamped-clamped pipe conveying fluid. Nuclear Engineering and Design, 254, 237-245 (2013)
[29] Gu, J. J., An, C., Levi, C., and Su, J. Prediction of vortex-induced vibration of long flexiblecylinders modeled by a coupled nonlinear oscillator: integral transform solution. Journal of Hydrodynamics,24(6), 888-898 (2012)
[30] Wolfram, S. The Mathematica Book, 5th ed., Wolfram Media/Cambridge University Press, Champaign(2003)