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Dynamic analysis of a tapered cantilever beam under a travelling mass

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Abstract

We study the vibration of a tapered cantilever (Euler–Bernoulli) beam carrying a moving mass. The tapering is assumed to be parabolic. Using the Galerkin method we find approximate solutions in an energy formulation that takes into account dynamic mass-beam coupling due to inertial, Coriolis and centrifugal effects. The approximate solutions are expanded in terms of the mode shapes of the free tapered beam, which can be obtained analytically. We then study the effect the tapering as well as the magnitude and velocity of the mass have on the tip deflections of the beam. We consider two different initial conditions, one where the mass starts moving from a statically deformed beam and one where the beam is initially triggered to vibrate. We find that tip deflections are more irregular for strongly tapered beams. Our results are of interest for barreled launch systems where tip deflections may adversely affect projectile motion.

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References

  1. Fryba L (1972) Vibration of solids and structures under moving loads. Noordhoff International Publishing Company, Groningen

    Book  MATH  Google Scholar 

  2. Wang PKC, Wei J (1987) Vibrations in a moving flexible robot arm. J. Sound Vib. 116:149–160

    Article  ADS  Google Scholar 

  3. Dwivedy SK, Eberhard K (2006) Dynamic analysis of flexible manipulators, a literature review. Mech. Mach. Theory 41:749–777

    Article  MATH  MathSciNet  Google Scholar 

  4. Ouyang H (2011) Moving-load dynamic problems: a tutorial (with a brief overview). Mech. Syst. Signal Proc. 25:2039–2060

    Article  ADS  Google Scholar 

  5. Pesterev AV, Yang B, Bergman LA, Tan CA (2003) Revisiting the moving force problem. J. Sound Vib. 261:75–91

    Article  ADS  Google Scholar 

  6. Sadiku S, Leipholz H (1987) On the dynamics of elastic systems with moving concentrated masses. Arch. Appl. Mech. 57:223–242

    MATH  Google Scholar 

  7. Ting EC, Genin J, Ginsberg JH (1974) A general algorithm for moving mass problem. J. Sound Vib. 33:49–58

    Article  ADS  MATH  Google Scholar 

  8. Ryu BJ, Lee JW, Yim KB, Yoon YS (2006) Dynamic behaviors of an elastically restrained beam carrying a moving mass. J. Mech. Sci. Technol. 20:1382–1389

    Article  Google Scholar 

  9. Golnaraghi MF (1991) Vibration suppression of flexible structures using internal resonance. Mech. Res. Commun. 18:135–143

    Article  MATH  Google Scholar 

  10. Golnaraghi MF (1991) Regulation of flexible structures via nonlinear coupling. Dyn Control 1:405–428

    Article  MathSciNet  Google Scholar 

  11. Khalily F, Golnaraghi MF, Heppler GR (1994) On the dynamic behaviour of a flexible beam carrying a moving mass. Nonlinear Dyn 5:493–513

    Article  Google Scholar 

  12. Siddiqui SAQ, Golnaraghi MF, Heppler GR (1998) Dynamics of a flexible cantilever beam carrying a moving mass. Nonlinear Dyn 15:137–154

    Article  MATH  Google Scholar 

  13. Siddiqui SAQ, Golnaraghi MF, Heppler GR (2000) Dynamics of a flexible beam carrying a moving mass using perturbation, numerical and time-frequency analysis techniques. J. Sound Vib. 229:1023–1055

    Article  ADS  Google Scholar 

  14. Siddiqui SAQ, Golnaraghi MF, Heppler GR (2003) Large free vibrations of a beam carrying a moving mass. Int. J. Nonlinear Mech. 38:1481–1493

    Article  MATH  Google Scholar 

  15. Wu JJ, Whittaker AR (1999) The natural frequencies and mode shapes of a uniform cantilever beam with multiple two-dof spring-mass systems. J. Sound Vib. 227:361–381

    Article  ADS  Google Scholar 

  16. Wu JJ (2003) Use of effective stiffness matrix for the free vibration analyses of a non-uniform cantilever beam carrying multiple two degree-of-freedom spring-damper-mass systems. Comput. Struct. 81:2319–2330

    Article  Google Scholar 

  17. Wu JJ (2004) Free vibration analysis of beams carrying a number of two-degree-of-freedom spring-damper-mass systems. Finite Elem. Anal. Des. 40:363–381

    Article  Google Scholar 

  18. Wu JJ (2005) Use of equivalent-damper method for free vibration analysis of a beam carrying multiple two degree-of-freedom spring-damper-mass systems. J. Sound Vib. 281:275–293

    Article  ADS  Google Scholar 

  19. Goel RP (1976) Transverse vibrations of tapered beams. J. Sound Vib. 47:1–7

    Article  ADS  MATH  Google Scholar 

  20. Mabie HH, Rogers CB (1974) Transverse vibrations of double-tapered cantilever beams with end support and with end mass. J. Acoust. Soc. Am. 55:986–991

    Article  ADS  Google Scholar 

  21. De Rosa MD, Auciello NM (1996) Free vibrations of tapered beams with flexible ends. Comput. Struct. 60:197–202

    Article  MATH  Google Scholar 

  22. Zhou D (1996) The exact analytical solution of transverse free vibration of a type of beams with variable cross-sections. J. Vib. Shock 15:12–15

    Google Scholar 

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Acknowledgments

The financial support of China Scholarship Council (CSC) under the Grant No. 201306260082 is gratefully acknowledged.

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Correspondence to Zhen Dong Hu.

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Zhao, X.W., Hu, Z.D. & van der Heijden, G.H.M. Dynamic analysis of a tapered cantilever beam under a travelling mass. Meccanica 50, 1419–1429 (2015). https://doi.org/10.1007/s11012-015-0112-5

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  • DOI: https://doi.org/10.1007/s11012-015-0112-5

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