×

Dynamic analysis of a tapered cantilever beam under a travelling mass. (English) Zbl 1317.74056

Summary: 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.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)
74H45 Vibrations in dynamical problems in solid mechanics
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Fryba L (1972) Vibration of solids and structures under moving loads. Noordhoff International Publishing Company, Groningen · Zbl 0301.73015 · doi:10.1007/978-94-011-9685-7
[2] Wang PKC, Wei J (1987) Vibrations in a moving flexible robot arm. J. Sound Vib. 116:149-160 · doi:10.1016/S0022-460X(87)81326-3
[3] Dwivedy SK, Eberhard K (2006) Dynamic analysis of flexible manipulators, a literature review. Mech. Mach. Theory 41:749-777 · Zbl 1095.70005 · doi:10.1016/j.mechmachtheory.2006.01.014
[4] Ouyang H (2011) Moving-load dynamic problems: a tutorial (with a brief overview). Mech. Syst. Signal Proc. 25:2039-2060 · doi:10.1016/j.ymssp.2010.12.010
[5] Pesterev AV, Yang B, Bergman LA, Tan CA (2003) Revisiting the moving force problem. J. Sound Vib. 261:75-91 · doi:10.1016/S0022-460X(02)00942-2
[6] Sadiku S, Leipholz H (1987) On the dynamics of elastic systems with moving concentrated masses. Arch. Appl. Mech. 57:223-242 · Zbl 0606.73062
[7] Ting EC, Genin J, Ginsberg JH (1974) A general algorithm for moving mass problem. J. Sound Vib. 33:49-58 · Zbl 0276.73031 · doi:10.1016/S0022-460X(74)80072-6
[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 · doi:10.1007/BF02915961
[9] Golnaraghi MF (1991) Vibration suppression of flexible structures using internal resonance. Mech. Res. Commun. 18:135-143 · Zbl 0719.73504 · doi:10.1016/0093-6413(91)90042-U
[10] Golnaraghi MF (1991) Regulation of flexible structures via nonlinear coupling. Dyn Control 1:405-428 · doi:10.1007/BF02169768
[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 · doi:10.1007/BF00052456
[12] Siddiqui SAQ, Golnaraghi MF, Heppler GR (1998) Dynamics of a flexible cantilever beam carrying a moving mass. Nonlinear Dyn 15:137-154 · Zbl 0904.73031 · doi:10.1023/A:1008205904691
[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 · doi:10.1006/jsvi.1999.2449
[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 · Zbl 1348.74152 · doi:10.1016/S0020-7462(02)00084-7
[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 · doi:10.1006/jsvi.1999.2324
[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 · doi:10.1016/S0045-7949(03)00315-8
[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 · doi:10.1016/S0168-874X(03)00052-0
[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 · doi:10.1016/j.jsv.2004.01.013
[19] Goel RP (1976) Transverse vibrations of tapered beams. J. Sound Vib. 47:1-7 · Zbl 0332.73059 · doi:10.1016/0022-460X(76)90403-X
[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 · doi:10.1121/1.1914673
[21] De Rosa MD, Auciello NM (1996) Free vibrations of tapered beams with flexible ends. Comput. Struct. 60:197-202 · Zbl 0920.73141 · doi:10.1016/0045-7949(95)00397-5
[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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.