Abstract
The thermo-mechanical nonlinear dynamics of a buckled axially moving beam is numerically investigated, with special consideration to the case with a three-to-one internal resonance between the first two modes. The equation of motion of the system traveling at a constant axial speed is obtained using Hamilton’s principle. A closed form solution is developed for the post-buckling configuration for the system with an axial speed beyond the first instability. The equation of motion over the buckled state is obtained for the forced system. The equation is reduced into a set of nonlinear ordinary differential equations via the Galerkin method. This set is solved using the pseudo-arclength continuation technique to examine the frequency response curves and direct-time integration to construct bifurcation diagrams of Poincaré maps. The vibration characteristics of the system at points of interest in the parameter space are presented in the form of time histories, phase-plane portraits, and Poincaré sections.
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References
Swope R.D., Ames W.F.: Vibrations of a moving threadline. J. Frank. Inst. 275, 36–55 (1963)
Mote C.D.J.: Dynamic stability of axially moving materials. Shock Vib. Dig. 4(4), 2–11 (1972)
Thurman A.L., Mote C.D.J.: Free, periodic, nonlinear oscillation of an axially moving strip. ASME J. Appl. Mech. 36, 83–91 (1969)
Shih L.Y.: Three-dimensional non-linear vibration of a traveling string. Int. J. Non-Linear Mech. 6, 427–434 (1971)
Yuh J., Young T.: Dynamic modeling of an axially moving beam in rotation: simulation and experiment. J. Dyn. Syst. Meas. Control 113, 34–40 (1991)
Treyssède F.: Prebending effects upon the vibrational modes of thermally prestressed planar beams. J. Sound Vib. 307, 295–311 (2007)
Pradeep V., Ganesan N., Bhaskar K.: Vibration and thermal buckling of composite sandwich beams with viscoelastic core. Compos. Struct. 81, 60–69 (2007)
Sharnappa G.N., Sethuraman R.: Dynamic modeling of active constrained layer damping of composite beam under thermal environment. J. Sound Vib. 305, 728–749 (2007)
Xiang H.J., Yang J.: Free and forced vibration of a laminated FGM Timoshenko beam of variable thickness under heat conduction. Compos. B Eng. 39, 292–303 (2008)
Wu G.: The analysis of dynamic instability and vibration motions of a pinned beam with transverse magnetic fields and thermal loads. J. Sound Vib. 284, 343–360 (2005)
Manoach E., Ribeiro P.: Coupled, thermoelastic, large amplitude vibrations of Timoshenko beams. Int. J. Mech. Sci. 46, 1589–1606 (2004)
Ghayesh, M.H., Kazemirad, S., Darabi, M.A., Woo, P.: Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system. Arch. Appl. Mech. 82, 317–331 (2012)
Chen L.-Q.: Analysis and control of transverse vibrations of axially moving strings. Appl. Mech. Rev. 58, 91–116 (2005)
Mote J.C.D.: On the nonlinear oscillation of an axially moving string. ASME J. Appl. Mech. 33, 463–464 (1966)
Pakdemirli M., Ulsoy A.G., Ceranoglu A.: Transverse vibration of an axially accelerating string. J. Sound Vib. 169, 179–196 (1994)
Pakdemirli M., Ulsoy A.G.: Stability analysis of an axially accelerating string. J. Sound Vib. 203, 815–832 (1997)
Öz H.R., Pakdemirli M., Özkaya E.: Transition behaviour from string to beam for an axially accelerating material. J. Sound Vib. 215, 571–576 (1998)
Pakdemirli M., Özkaya E.: Approximate boundary layer solution of a moving beam problem. Math. Comput. Appl. 3, 93–100 (1998)
Öz H.R., Pakdemirli M.: Vibrations of an axially moving beam with time-dependent velocity. J. Sound Vib. 227, 239–257 (1999)
ÖZkaya E., Pakdemirli M.: Vibrations of an axially accelerating beam with small flexural stiffness. J. Sound Vib. 234, 521–535 (2000)
Öz H.R., Pakdemirli M., Boyaci H.: Non-linear vibrations and stability of an axially moving beam with time-dependent velocity. Int. J. Non-Linear Mech. 36, 107–115 (2001)
Yang X.-D., Tang Y.-Q., Chen L.-Q., Lim C.W.: Dynamic stability of axially accelerating Timoshenko beam: Averaging method. Eur. J. Mech. A Solids 29, 81–90 (2010)
Chen L.-Q., Tang Y.-Q., Lim C.W.: Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams. J. Sound Vib. 329, 547–565 (2010)
Ding H., Chen L.-Q.: Galerkin methods for natural frequencies of high-speed axially moving beams. J. Sound Vib. 329, 3484–3494 (2010)
Chen L.H., Zhang W., Yang F.H.: Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations. J. Sound Vib. 329, 5321–5345 (2010)
Chen L.-Q., Chen H.: Asymptotic analysis on nonlinear vibration of axially accelerating viscoelastic strings with the standard linear solid model. J. Eng. Math. 67, 205–218 (2010)
Chen L.-Q., Ding H.: Steady-State transverse response in coupled planar vibration of axially moving viscoelastic beams. J. Vib. Acoust. 132, 011009 (2010)
Ding H., Zhang G.C., Chen L.-Q.: Supercritical equilibrium solutions of axially moving beams with hybrid boundary conditions. Mech. Res. Commun. 38, 52–56 (2011)
Marynowski K., Kapitaniak T.: Kelvin-Voigt versus Bürgers internal damping in modeling of axially moving viscoelastic web. Int. J. Non-Linear Mech. 37, 1147–1161 (2002)
Marynowski K.: Non-linear vibrations of an axially moving viscoelastic web with time-dependent tension. Chaos Solit. Fract. 21, 481–490 (2004)
Marynowski K., Kapitaniak T.: Zener internal damping in modelling of axially moving viscoelastic beam with time-dependent tension. Int. J. Non-Linear Mech. 42, 118–131 (2007)
Suweken G., Van Horssen W.T.: On the weakly nonlinear, transversal vibrations of a conveyor belt with a low and time-varying velocity. Nonlinear Dyn. 31, 197–223 (2003)
Suweken G., Van Horssen W.T.: On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part II: the beam-like case. J. Sound Vib. 267, 1007–1027 (2003)
Suweken G., Van Horssen W.T.: On the transversal vibrations of a conveyor belt with a low and time-varying velocity. Part I: the string-like case. J. Sound Vib. 264, 117–133 (2003)
Sze K.Y., Chen S.H., Huang J.L.: The incremental harmonic balance method for nonlinear vibration of axially moving beams. J. Sound Vib. 281, 611–626 (2005)
Huang J.L., Su R.K.L., Li W.H., Chen S.H.: Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances. J. Sound Vib. 330, 471–485 (2011)
Stylianou M., Tabarrok B.: Finite element analysis of an axially moving beam, part I: time integration. J. Sound Vib. 178, 433–453 (1994)
Riedel C.H., Tan C.A.: Coupled, forced response of an axially moving strip with internal resonance. Int. J. Non-Linear Mech. 37, 101–116 (2002)
Hwang S.J., Perkins N.C.: Supercritical stability of an axially moving beam part II: vibration and stability analyses. J. Sound Vib. 154, 397–409 (1992)
Tan C.A., Yang B., Mote J.C.D.: Dynamic response of an axially moving beam coupled to hydrodynamic bearings. J. Vib. Acoust. 115, 9–15 (1993)
Pellicano F., Zirilli F.: Boundary layers and non-linear vibrations in an axially moving beam. Int. J. Non-Linear Mech. 33, 691–711 (1998)
Ghayesh M.H., Balar S.: Non-linear parametric vibration and stability of axially moving visco-elastic Rayleigh beams. Int. J. Solids Struct. 45, 6451–6467 (2008)
Ghayesh M.H.: Nonlinear transversal vibration and stability of an axially moving viscoelastic string supported by a partial viscoelastic guide. J. Sound Vib. 314, 757–774 (2008)
Ghayesh M.H.: Stability characteristics of an axially accelerating string supported by an elastic foundation. Mech. Mach. Theory 44, 1964–1979 (2009)
Ghayesh M.H., Yourdkhani M., Balar S., Reid T.: Vibrations and stability of axially traveling laminated beams. Appl. Math. Comput. 217, 545–556 (2010)
Ghayesh M.H.: Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation. Int. J. Non-Linear Mech. 45, 382–394 (2010)
Ghayesh M.H., Païdoussis M.P.: Three-dimensional dynamics of a cantilevered pipe conveying fluid, additionally supported by an intermediate spring array. Int. J. Non-Linear Mech. 45, 507–524 (2010)
Ghayesh, M.H., Païdoussis, M.P.: Dynamics of a fluid-conveying cantilevered pipe with intermediate spring support. ASME Conf. Proc. (FEDSM 2010) 893–902 (2010)
Ghayesh M.H., Balar S.: Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams. Appl. Math. Model. 34, 2850–2859 (2010)
Sahebkar S.M., Ghazavi M.R., Khadem S.E., Ghayesh M.H.: Nonlinear vibration analysis of an axially moving drillstring system with time dependent axial load and axial velocity in inclined well. Mech. Mach. Theory 46, 743–760 (2011)
Ghayesh M.H., Païdoussis M.P., Modarres-Sadeghi Y.: Three-dimensional dynamics of a fluid-conveying cantilevered pipe fitted with an additional spring-support and an end-mass. J. Sound Vib. 330, 2869–2899 (2011)
Ghayesh M.H., Moradian N.: Nonlinear dynamic response of axially moving, stretched viscoelastic strings. Arch. Appl. Mech. 81, 781–799 (2011)
Ghayesh, M.H.: Stability and bifurcations of an axially moving beam with an intermediate spring support. Nonlinear Dyn. (2011, in press)
Ghayesh M.H.: On the natural frequencies, complex mode functions, and critical speeds of axially traveling laminated beams: parametric study. Acta Mech. Solida Sin. 24, 373–382 (2011)
Ghayesh M.H.: Nonlinear forced dynamics of an axially moving viscoelastic beam with an internal resonance. Int. J. Mech. Sci. 53, 1022–1037 (2011)
Ghayesh M.H., Kafiabad H.A., Reid T.: Sub- and super-critical nonlinear dynamics of a harmonically excited axially moving beam. Int. J. Solids Struct. 49, 227–243 (2012)
Ghayesh M.H., Kazemirad S., Amabili M.: Coupled longitudinal-transverse dynamics of an axially moving beam with an internal resonance. Mech. Mach. Theory 52, 18–34 (2012)
Ghayesh, M.H.: Subharmonic dynamics of an axially accelerating beam. Arch. Appl. Mech. accepted (2012)
Pellicano F., Vestroni F.: Nonlinear dynamics and bifurcations of an axially moving beam. J. Vib. Acoust. 122, 21–30 (2000)
Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.: AUTO 97: Continuation and Bifurcation Software for Ordinary Differential Equations (with HomCont). Concordia University, Montreal (1998)
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Kazemirad, S., Ghayesh, M.H. & Amabili, M. Thermo-mechanical nonlinear dynamics of a buckled axially moving beam. Arch Appl Mech 83, 25–42 (2013). https://doi.org/10.1007/s00419-012-0630-8
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DOI: https://doi.org/10.1007/s00419-012-0630-8