Abstract
The problem of an elastic beam under the periodic loading of successive moving masses is investigated as a pragmatic case for studying dynamic stability of linear time-varying systems. This model serves to highlight the odds of multi-solutions coexistence, a form of hidden instability which reveals dangerous as it may be precipitated by the slightest disturbance or variation in the model. Since no engineering model perfectly represents a physical system, such situations for which Floquet theory naively predicts stability are potentially inevitable. The harmonic balancing method is used in order to thoroughly explore the stability diagrams for detecting these instability gaps. Although this phenomenon has also been described in other physical systems, it has not been addressed for beam–moving mass systems. This result may find particular importance in applications involving self-induced vibrations of elastic structures and hence also appears of practical relevance.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Yang Y.B., Yau J.D., Wu Y.S.: Vehicle Bridge Interaction Dynamics: With Applications to High Speed Railways. World Scientific Publishing Company, Singapore (2004)
Gerdemeli I., Esen I., Özer D.: Dynamic response of an overhead crane beam due to a moving mass using moving finite element approximation. Key Eng. Mater. 450, 99–102 (2010)
Yau J.D., Yang Y.B.: Vibration of a suspension bridge installed with a water pipeline and subjected to moving trains. Eng. Struct. 30, 632–642 (2008)
Shiau T.N., Huang K.H., Wang F.C., Hsu W.C.: Dynamic response of a rotating multi-span shaft with general boundary conditions subjected to a moving load. J. Sound Vib. 323, 1045 (2009)
Cojocaru E.C., Foo J., Irschik H.: Quasi-static response of a Timoshenko beam loaded by an elastically supported moving rigid beam. Tech. Mech. 24, 79–90 (2004)
Pan L., Qiao N., Lin W., Liang Y.: Stability and local bifurcation in a simply-supported beam carrying a moving mass. Acta Mech. Solida Sin. 20, 123–129 (2007)
Mazilu T.: Instability of a train of oscillators moving along a beam on a viscoelastic foundation. J. Sound Vib. 332, 4597–4619 (2013)
Ju S.H.: Nonlinear analysis of high-speed trains moving on bridges during earthquakes. Nonlinear Dyn. 69, 173–183 (2012)
Walker, W.H., Veletsos, A.S.: Response of simple span highway bridges to moving vehicles. University of Illinois—Engineering Experiment Station—Bulletin, pp. 69 (1966)
Michaltsos G.T., Sophianopoulos D., Kounadis A.N.: The effect of moving mass and other parameters on the dynamic response of a simply supported beam. J. Sound Vib. 191, 357–362 (1996)
Cojocaru E.C., Irschik H.: Dynamic response of an elastic bridge loaded by a moving elastic beam with a finite length Interact. Multiscale Mech. 3, 343–363 (2010)
Newland D.E.: Instability of an elastically supported beam under a travelling inertia load. J. Mech. Eng. Sci. 12, 373–374 (1970)
Karimpour H., Eftekhari M.: Exploiting internal resonance for vibration suppression and energy harvesting from structures using an inner-mounted oscillator. Nonlinear Dyn. 77, 699–727 (2014)
Sultan A., Siddiqui Q.: Nonlinear Beam Behaviour with a Moving Mass. Univ. of Waterloo, Waterloo (1998)
Pirmoradian M., Keshmiri M., Karimpour H.: On the parametric excitation of a Timoshenko beam due to intermittent passage of moving masses: instability and resonance analysis. Acta Mech. 226, 1241–1253 (2015)
Lee S.H., Jeong W.B.: Steady-state vibration analysis of modal beam model under parametric excitation. Int. J. Precis. Eng. Manuf. 13, 927–933 (2012)
Verichev S.N., Metrikine A.V.: Instability of vibrations of mass that moves uniformly along a beam on a periodically inhomogeneous foundation. J. Sound Vib. 260, 901–925 (2003)
Fossen T.I., Nijmeije H.: Parametric Resonance in Dynamical System. Springer, New York (2012)
Mackertich S.: Dynamic stability of a beam excited by a sequence of moving mass particles. Acoust. Soc. Am. 115, 1416–1419 (2004)
Jianjun, P.T., Jin, W.: Some results in Floquet theory, with application to periodic epidemic models, Appl. Anal. doi:10.1080/00036811.2014.918606 (2014)
Hartono, H., van der Burgh, A.H.P.: A linear differential equation with a time-periodic damping coefficient: stability diagram and an application. J. Eng. Math. 49, 99–112 (2004)
Nayfeh A.H., Mook D.T.: Nonlinear Oscillations. Wiley, New York (1979)
Benedetti G.A.: Dynamic stability of a beam loaded by a sequence of moving mass particles. J. Appl. Mech. 41, 1069–1071 (1974)
Ng L., Rand R.: Nonlinear effects on coexistence phenomenon in parametric excitation. Nonlinear Dyn. 31, 73–89 (2003)
Recktenwald G.D.: The Stability of Parametrically Excited Systems: Coexistence and Trigonometrification. Cornell University, Ithaca (2006)
Brown, B.M., Eastham, M.S.P., Schmidt, K.M.: Periodic differential operators, operator theory: advances and applications Floquet theory 230, 1–29 (2013)
Recktenwald G., Rand R.: Coexistence phenomenon in autoparametric excitation of two degree of freedom systems. Int. J. Non-Linear Mech. 40, 1160–1170 (2005)
Núñez D., Torres P.J.: On the motion of an oscillator with a periodically time-varying mass. Nonlinear Anal. RealWorld Appl. 10, 1976–1983 (2009)
Pak C.H., Rand R.H., Moon F.C.: Free vibrations of a thin elastica by normal modes. Nonlinear Dyn. 3, 347–364 (1992)
Rand, R.H.: Lecture Notes On Nonlinear Vibrations, Published On-Line by the Internet-First University Press, http://dspace.library.cornell.edu/handle/1813/79 (2004)
Doedel E.J., Aronson D.G., Othmer H.G.: The dynamics of coupled current-biased Josephson junctions: Part 1. IEEE Trans. Circuits Syst. 35, 810–817 (1988)
Rand R.H., Tseng S.F.: On the stability of a differential equation with application to the vibrations of a particle in the plane. J. Appl. Mech. 36, 311–313 (1969)
Broer H., Puig J., Simó C.: Resonance tongues and instability pockets in the Quasi–Periodic Hill–Schrödinger Equation. Commun. Math. Phys. 241, 467–503 (2003)
Pirmoradian, M., Keshmiri, M., Karimpour, H.: Instability and resonance analysis of a beam subjected to moving mass loading via incremental harmonic balance method. J. Vibroeng. (2014)
Ghomeshi Bozorg M., Keshmiri M.: Stability analysis of nonlinear time varying system of beam-moving mass considering friction interaction. Indian J. Sci. Tech. 6, 54–59 (2013)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Karimpour, H., Pirmoradian, M. & Keshmiri, M. Instance of hidden instability traps in intermittent transition of moving masses along a flexible beam. Acta Mech 227, 1213–1224 (2016). https://doi.org/10.1007/s00707-015-1551-8
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00707-015-1551-8