Abstract
The nonlinear free vibration of an axially translating viscoelastic beam with an arbitrarily varying length and axial velocity is investigated. Based on the linear viscoelastic differential constitutive law, the extended Hamilton’s principle is utilized to derive the generalized third-order equations of motion for the axially translating viscoelastic Bernoulli–Euler beam. The coupling effects between the axial motion and transverse vibration are assessed under various prescribed time-varying velocity fields. The inertia force arising from the longitudinal acceleration emerges, rendering the coupling terms between the axial beam acceleration and the beam flexure. Semi-analytical solutions for the governing PDE are obtained through the separation of variables and the assumed modes method. The modified Galerkin’s method and the fourth-order Runge–Kutta method are employed to numerically analyze the resulting equations. Further, dynamic stabilization is examined from the system energy standpoint for beam extension and retraction. Extensive numerical simulations are presented to illustrate the influences of varying translating velocities and viscoelastic parameters on the underlying dynamic responses. The material viscosity always dissipates energy and helps stabilize the transverse vibration.
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References
Mote C.D. Jr: Dynamic stability of axially moving materials. Shock Vib. Dig. 4, 2–11 (1972)
Ulsoy A.G., Mote C.D. Jr: Band saw vibration and stability. Shock Vib. Dig. 10, 3–15 (1978)
Wickert J.A., Mote C.D. Jr: Current research on the vibration and stability of axially moving materials. Shock Vib. Dig. 20, 3–13 (1988)
Wickert J.A., Mote C.D. Jr: On the energetics of axially moving continua. J. Acoust. Soc. Am. 85, 1365–1368 (1989)
Wickert J.A., Mote C.D. Jr.: Classical vibration analysis of axially moving continua. ASME J. Appl. Mech. 57, 738–744 (1990)
Rao G.V.: Linear dynamics and active control of an elastically supported traveling string. Comput. Struct. 43, 1041–1049 (1992)
Pakdemirli M., Ozkaya E.: Group-theoretic approach to axially accelerating beam problem. Acta Mech. 155, 111–123 (2002)
Tabarrok B., Leech C.M., Kim Y.I.: On the dynamics of an axially moving beam. J. Frankl. Inst. 297, 201–220 (1974)
Zajaczkowski J., Lipinski J.: Instability of the motion of a beam of periodically varying length. J. Sound Vib. 63, 9–18 (1979)
Wang P.K.C., Wei J.: Vibration in a moving flexible robot arm. J. Sound Vib. 116, 149–160 (1987)
Buffinton K.W.: Dynamics of elastic manipulators with prismatic joints. ASME J. Dyn. Syst. Meas. Control 113, 34–40 (1992)
Yamamoto T., Yasuda K., Koto M.: Vibration of a string with time-variable length. Bull. Jpn. Soc. Mech. Eng. 21, 1677–1684 (1978)
Terumichi Y., Ohtsuka M., Yoshizawa M., Fukawa Y., Tsujioka Y.: Nonstationary vibrations of a string with time-varying length and a mass-spring system attached at the lower end. Nonlinear Dyn. 12, 39–55 (1997)
Zhu W.D., Ni J.: Energetic and stability of translating media with an arbitrarily varying length. J. Vib. Acoust. 122, 295–304 (2000)
Stylianou M., Tabarrok B.: Finite element analysis of an axially moving beam. Part I: time integration. J. Sound Vib. 178, 433–453 (1994)
Stylianou M., Tabarrok B.: Finite element analysis of an axially moving beam. Part II: stability analysis. J. Sound Vib. 178, 455–481 (1994)
Tadikonda S.S.K., Baruh H.: Dynamics and control of a translating flexible beam with a prismatic joint. ASME J. Dyn. Syst. Meas. Control 114, 422–427 (1992)
Wang L.H., Hu Z.D., Zhong Z., Ju J.W.: Hamiltonian dynamic analysis of an axially translating beam featuring time-variant velocity. Acta Mech. 206, 149–161 (2009)
Theodore R.J., Arakeri J.H., Ghosal A.: The modeling of axially translating flexible beams. J. Sound Vib. 191, 363–376 (1996)
Pakdemirli M., Ulsoy A.G.: Stability analysis of an axially accelerating string. J. Sound Vib. 203, 815–832 (1997)
Fung R.F., Huang J.S., Chen Y.C., Yao C.M.: Nonlinear dynamics analysis of the viscoelastic string with a harmonically varying transport speed. Comput. Struct. 66, 777–784 (1998)
Yang X.D., Chen L.Q.: Nonlinear forced vibration of axially moving viscoelastic beams. Acta Mech. Solida Sin. 19, 365–373 (2006)
Zhang L.X., Zu J.W.: Nonlinear vibrations of viscoelastic moving belts. Part I: free vibration analysis. J. Sound Vib. 216, 75–91 (1998)
Zhang L.X., Zu J.W.: Nonlinear vibrations of viscoelastic moving belts. Part II: forced vibration analysis. J. Sound Vib. 216, 93–105 (1998)
Oz 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)
Pelicano F., Vestroni F.: Non-linear dynamics and bifurcations of an axially moving beam. ASME J. Vib. Acoust. 122, 21–30 (2000)
Pelicano F., Fregolent A., Bertuzzi A., Vestroni F.: Primary and parametric non-linear resonances of a power transmission belt. J. Sound Vib. 244, 669–684 (2001)
Lee U., Oh H.: Dynamics of an axially moving viscoelastic beam subject to axial tension. Int. J. Solids Struct. 42, 2381–2398 (2005)
Chen L.Q., Yang X.D.: Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation. Chaos Solitons Fractals 27, 748–757 (2006)
Marnowski K., Kapitaniak T.: Zener internal damping in modeling of axially moving viscoelastic beam with time-dependent tension. Int. J. Non-linear Mech. 42, 118–131 (2007)
Hou Z.C., Zu J.W.: Non-linear free oscillations of a moving viscoelastic belt. Mech. Mach. Theory 37, 925–940 (2002)
Koivurova H.: The numerical study of the nonlinear dynamics of a light, axially moving string. J. Sound Vib. 320, 373–385 (2009)
Cooper J.: Asymptotic behavior for the vibrating string with a moving boundary. J. Math. Anal. Appl. 174, 67–87 (1993)
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This work was sponsored by the National Natural Science Foundation of China (NSFC) under Project No. 10572104.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Wang, L.H., Hu, Z.D., Zhong, Z. et al. Dynamic analysis of an axially translating viscoelastic beam with an arbitrarily varying length. Acta Mech 214, 225–244 (2010). https://doi.org/10.1007/s00707-010-0287-8
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DOI: https://doi.org/10.1007/s00707-010-0287-8