×

Nonlinear dynamics of elastic rods using the Cosserat theory: modelling and simulation. (English) Zbl 1167.74467

Summary: The method of Cosserat dynamics is employed to explore the nonplanar nonlinear dynamics of elastic rods. The rod, which is assumed to undergo flexure about two principal axes, extension, shear and torsion, are described by a general geometrically exact theory. Based on the Cosserat theory, a set of governing partial differential equations of motion with arbitrary boundary conditions is formulated in terms of the displacements and angular variables, thus the dynamical analysis of elastic rods can be carried out rather simply. The case of doubly symmetric cross-section of the rod is considered and the Kirchoff constitutive relations are adopted to provide an adequate description of elastic properties in terms of a few elastic moduli. A cantilever is given as a simple example to demonstrate the use of the formulation developed. The nonlinear dynamic model with the corresponding boundary and initial conditions are numerically solved using the Femlab/Matlab software packages. The corresponding nonlinear dynamical responses of the cantilever under external harmonic excitations are presented through numerical simulations.

MSC:

74K10 Rods (beams, columns, shafts, arches, rings, etc.)

Software:

FEMLAB; Matlab; COMSOL
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Antman, S. S.: Nonlinear problems of elasticity, Applied mathematical sciences (1991) · Zbl 0820.73002
[2] Antman, S. S.; Marlow, R. S.; Vlahacos, C. P.: The complicated dynamics of heavy rigid bodies attached to deformable rods, Quarterly of applied mathematics 86, 431-460 (1998) · Zbl 0960.74028
[3] Bishop, R. E. D.; Price, W. G.: Coupled bending and twisting of a Timoshenko beam, Journal of sound and vibration 50, 469-477 (1977) · Zbl 0367.73060 · doi:10.1016/0022-460X(77)90497-7
[4] Bolotin, V. V.: The dynamic stability of elastic systems, (1964) · Zbl 0125.15301
[5] Jr., J. C. Bruch; Mitchell, T. P.: Vibrations of mass-loaded clamped-free Timoshenko beam, Mitchell journal of sound and vibration 114, No. 2, 341-345 (1987) · Zbl 1235.74159
[6] Cao, D. Q.; Liu, D.; Wang, C. H. -T.: Nonlinear dynamic modelling for MEMS components via the Cosserat rod element approach, Journal of micromechanics and microengineering 15, 1334-1343 (2005)
[7] Cao, D. Q.; Liu, D.; Wang, C. H. -T.: Three-dimensional nonlinear dynamics of slender structures: Cosserat rod element approach, International journal of solids and structures 43, 760-783 (2006) · Zbl 1119.74594 · doi:10.1016/j.ijsolstr.2005.03.059
[8] Cartmell, M. P.: The equations of motion for a parametrically excited cantilever beam, Journal of sound and vibration 143, No. 3, 395-406 (1990)
[9] Comsol, Ab.: Femlab 3.0: Matlab interface guide, (2004)
[10] Da Silva, M. R. M. Crespo: Nonlinear flexural-flexural-torsional-extensional dynamics of beams – I. Formulation, International journal of solids and structures 24, No. 12, 1225-1234 (1988) · Zbl 0676.73030 · doi:10.1016/0020-7683(88)90087-X
[11] Da Silva, M. R. M. Crespo: Nonlinear flexural-flexural-torsional-extensional dynamics of beams – II. Response analysis, International journal of solids and structures 24, No. 12, 1235-1242 (1988) · Zbl 0676.73031 · doi:10.1016/0020-7683(88)90088-1
[12] Da Silva, M. R. M. Crespo; Glynn, C. C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams, I: Equations of motion, Journal of structural mechanics 6, No. 4, 437-448 (1978)
[13] Da Silva, M. R. M. Crespo; Glynn, C. C.: Nonlinear flexural-flexural-torsional dynamics of inextensional beams, II: Forced motions, Journal of structural mechanics 6, No. 4, 449-461 (1978)
[14] Cull, S. J.; Tucker, R. W.; Tung, R. S.; Hartley, D. H.: On parametrically excited flexural motion of an extensible and shearable rod with a heavy attachment, Technische mechanik 20, 147-158 (2000)
[15] Esmailzadeh, E.; Jalili, N.: Parametric response of cantilever Timoshenko beams with tip mass under harmonic support motion, International journal of non-linear mechanics 33, 765-781 (1998) · Zbl 1131.74317 · doi:10.1016/S0020-7462(97)00049-8
[16] Femlab homepage: <http://www.femlab.com>.
[17] Forehand, D. I. M.; Cartmell, M. P.: On the derivation of the equations of motion for a parametrically excited cantilever beam, Journal of sound and vibration 245, No. 1, 165-177 (2001)
[18] Georgiou, I.: Advanced proper orthobonal decomposition tools: using reduced order models to identify normal modes of vibration and slow invariant manifolds in the dynamics of planar nonlinear rods, Nonlinear dynamics 41, 69-110 (2005) · Zbl 1142.74337 · doi:10.1007/s11071-005-2793-0
[19] Grant, D. A.: The effect of rotary inertia and shear deformation on the frequency and normal mode equations of uniform beams carrying a concentrated mass, Journal of sound and vibration 57, No. 3, 357-365 (1978) · Zbl 0375.70011 · doi:10.1016/0022-460X(78)90316-4
[20] Gratus, J.; Tucker, R. W.: The dynamics of Cosserat nets, Journal of applied mathematics, No. 4, 187-226 (2003) · Zbl 1205.74124 · doi:10.1155/S1110757X03110224
[21] Huang, T. C.: The effect of rotatory inertia and of shear deformation on the frequency and normal mode equations of uniform beams with simple end conditions, Transactions ASME journal of applied mechanics 28, 579-584 (1961) · Zbl 0102.19005 · doi:10.1115/1.3641787
[22] Kar, R. C.; Dwivedy, S. K.: Nonlinear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances, International journal of non-linear mechanics 34, 515-529 (1999) · Zbl 1342.74096
[23] Nayfeh, A. H.; Mook, D. T.: Nonlinear oscillations, (1979) · Zbl 0418.70001
[24] Rega, G.; Troger, H.: Dimension reduction of dynamical systems: methods, models, applications, Nonlinear dynamics 41, 1-15 (2005) · Zbl 1142.37320 · doi:10.1007/s11071-005-2790-3
[25] Rubin, M. B.: Cosserat theories: shells, rods and points, (2000) · Zbl 0984.74003
[26] Rubin, M. B.: Numerical solution procedures for nonlinear elastic rods using the theory of a Cosserat point, International journal of solids and structures 38, 4395-4437 (2001) · Zbl 1040.74030 · doi:10.1016/S0020-7683(00)00271-7
[27] Rubin, M. B.; Tufekci, E.: Three-dimensional free vibration of a circular arch using the theory of a Cosserat point, Journal of sound and vibration 286, 799-816 (2005)
[28] Saito, H.; Koizumi, N.: Parametric vibrations of a horizontal beam with a concentrated mass at one end, International journal of mechanical sciences 24, 755-761 (1982) · Zbl 0498.73062 · doi:10.1016/0020-7403(82)90026-1
[29] Simo, J. C.; Vu-Quoc, L.: On the dynamics in space of rods undergoing large motions – a geometrically exact approach, Computer methods in applied mechanics and engineering 66, 125-161 (1988) · Zbl 0618.73100 · doi:10.1016/0045-7825(88)90073-4
[30] Simo, J. C.; Vu-Quoc, L.: A geometrically-exact beam model incorporating shear and torsion warping deformation, International journal of solids and structures 27, 371-393 (1991) · Zbl 0731.73029 · doi:10.1016/0020-7683(91)90089-X
[31] Tucker, R. W.; Wang, C.: An integrated model for drill-string dynamics, Journal of sound and vibration 224, 123-165 (1999)
[32] Vu-Quoc, L.; Deng, H.: Galerkin projection for geometrically-exact sandwich beams allowing for ply drop-off, ASME journal of applied mechanics 62, 479-488 (1995) · Zbl 0833.73066 · doi:10.1115/1.2895955
[33] Vu-Quoc, L.; Deng, H.: Dynamics of geometrically-exact sandwich beams: computational aspects, Computer methods in applied mechanics and engineering 146, 135-172 (1997) · Zbl 0897.73071 · doi:10.1016/S0045-7825(96)01226-1
[34] Vu-Quoc, L.; Ebcioglu, I. K.: Dynamic formulation for geometrically-exact sandwich beams and 1-D plates, ASME journal of applied mechanics 62, 756-763 (1995) · Zbl 0836.73045 · doi:10.1115/1.2897011
[35] Vu-Quoc, L.; Li, S.: Dynamics of sliding geometrically-exact beams: large angle maneuver and parametric resonance, Computer methods in applied mechanics and engineering 120, 65-118 (1995) · Zbl 0852.73030 · doi:10.1016/0045-7825(94)00051-N
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.