The complicated dynamics of heavy rigid bodies attached to deformable rods
Authors:
Stuart S. Antman, Randall S. Marlow and Constantine P. Vlahacos
Journal:
Quart. Appl. Math. 56 (1998), 431-460
MSC:
Primary 73K12; Secondary 70E15, 70K05, 73F15, 73H10
DOI:
https://doi.org/10.1090/qam/1637036
MathSciNet review:
MR1637036
Full-text PDF Free Access
Abstract |
References |
Similar Articles |
Additional Information
Abstract: We study the motion in space of light nonlinearly elastic and viscoelastic rods with heavy rigid attachments. The rods, which can suffer flexure, extension, torsion, and shear, are described by a general geometrically exact theory. We pay particular attention to the leading term of the asymptotic expansion of the governing equations as the inertia of the rod goes to zero. When the rods are elastic and weightless, and when they have appropriate initial conditions, they move irregularly through a family of equilibrium states parametrized by time; the motion of the rigid body is governed by an interesting family of multivalued ordinary differential equations. These ordinary differential equations for a heavy mass point attached to an elastica undergoing planar motion are explicitly treated. These problems illuminate such phenomena as snap-buckling. On the other hand, when the rods are viscoelastic and weightless, the rigid body is typically not governed by ordinary differential equations, but, as we show, the motion of the system is well-defined for arbitrary initial conditions. This analysis relies critically on the careful use of our properly invariant constitutive hypotheses.
- Stuart S. Antman, The paradoxical asymptotic status of massless springs, SIAM J. Appl. Math. 48 (1988), no. 6, 1319–1334. MR 968832, DOI https://doi.org/10.1137/0148081
- Stuart S. Antman, Nonlinear problems of elasticity, Applied Mathematical Sciences, vol. 107, Springer-Verlag, New York, 1995. MR 1323857
- Stuart S. Antman and Thomas I. Seidman, Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity, J. Differential Equations 124 (1996), no. 1, 132–185. MR 1368064, DOI https://doi.org/10.1006/jdeq.1996.0005
S. S. Antman and T. I. Seidman, Parabolic-hyperbolic systems governing the spatial motion of nonlinearly viscoelastic rods (in preparation)
- Paul F. Byrd and Morris D. Friedman, Handbook of elliptic integrals for engineers and scientists, Die Grundlehren der mathematischen Wissenschaften, Band 67, Springer-Verlag, New York-Heidelberg, 1971. Second edition, revised. MR 0277773
- R. L. Fosdick and R. D. James, The elastica and the problem of the pure bending for a nonconvex stored energy function, J. Elasticity 11 (1981), no. 2, 165–186. MR 614373, DOI https://doi.org/10.1007/BF00043858
R. Frisch-Fay, Flexible Bars, Butterworths, 1962
- Yu. S. Il′yashenko, Weakly contracting systems and attractors of Galerkin approximations of Navier-Stokes equations on the two-dimensional torus [translation of Adv. in Mech. 5 (1982), no. 1-2, 31–63; MR0719407 (85g:35099)], Selecta Math. Soviet. 11 (1992), no. 3, 203–239. Selected translations. MR 1181262
- Richard D. James, The equilibrium and post-buckling behavior of an elastic curve governed by a nonconvex energy, J. Elasticity 11 (1981), no. 3, 239–269. MR 625952, DOI https://doi.org/10.1007/BF00041939
- A. E. H. Love, A treatise on the Mathematical Theory of Elasticity, Dover Publications, New York, 1944. Fourth Ed. MR 0010851
- Robert H. Martin Jr., Nonlinear operators and differential equations in Banach spaces, Wiley-Interscience [John Wiley & Sons], New York-London-Sydney, 1976. Pure and Applied Mathematics. MR 0492671
E. L. Reiss, Column buckling—an elementary example of bifurcation, in Bifurcation Theory and Nonlinear Eigenvalue Problems, J. B. Keller and S. Antman, eds., Benjamin, 1969, pp. 1–16
- Edward L. Reiss and Bernard J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column, Quart. Appl. Math. 29 (1971), 245–260. MR 286344, DOI https://doi.org/10.1090/S0033-569X-1971-0286344-8
D. Shilkrut, A paradox of the classical approach in the theory of vibrations of geometrically nonlinear elastic systems, Ben Gurion University of the Negev, 1989
- Roger Temam, Infinite-dimensional dynamical systems in mechanics and physics, 2nd ed., Applied Mathematical Sciences, vol. 68, Springer-Verlag, New York, 1997. MR 1441312
- A. H. Zemanian, Distribution theory and transform analysis. An introduction to generalized functions, with applications, McGraw-Hill Book Co., New York-Toronto-London-Sydney, 1965. MR 0177293
S. S. Antman, The paradoxical asymptotic status of massless springs, SIAM J. Appl. Math. 48, 1319–1334 (1988)
S. S. Antman, Nonlinear Problems of Elasticity, Springer-Verlag, New York, 1995
S. S. Antman and T. I. Seidman, Quasilinear hyperbolic-parabolic equations of one-dimensional viscoelasticity, J. Differential Equations 124, 132–185 (1996)
S. S. Antman and T. I. Seidman, Parabolic-hyperbolic systems governing the spatial motion of nonlinearly viscoelastic rods (in preparation)
P. F. Byrd and M. D. Friedman, Handbook of Elliptic Integrals for Engineers and Scientists, 2nd edition, Springer-Verlag, New York, 1971
R. L. Fosdick and R. D. James, The elastica and the problem of pure bending for a non-convex stored energy function, J. Elasticity 11, 165–186 (1981).
R. Frisch-Fay, Flexible Bars, Butterworths, 1962
Yu. S. Ilyashenko, Weakly attracting systems and attractors of Galerkin approximations of Navier-Stokes equations on the two-dimensional torus, Selecta Math. Sov. 11, 203–239 (1992)
R. D. James, The equilibrium and post-buckling behavior of an elastic curve governed by a non-convex energy, J. Elasticity 11, 239–269 (1981)
A. E. H. Love, A Treatise on the Mathematical Theory of Elasticity, 4th edition, Dover Publications, New York, 1944
R. H. Martin, Nonlinear Operators and Differential Equations in Banach Spaces, Wiley-Interscience, New York-London-Sydney, 1976
E. L. Reiss, Column buckling—an elementary example of bifurcation, in Bifurcation Theory and Nonlinear Eigenvalue Problems, J. B. Keller and S. Antman, eds., Benjamin, 1969, pp. 1–16
E. L. Reiss and B. J. Matkowsky, Nonlinear dynamic buckling of a compressed elastic column, Quart. Appl. Math. 29, 245–260 (1971)
D. Shilkrut, A paradox of the classical approach in the theory of vibrations of geometrically nonlinear elastic systems, Ben Gurion University of the Negev, 1989
R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, 2nd edition, Springer-Verlag, New York, 1997
A. H. Zemanian, Distribution Theory and Transform Analysis, McGraw-Hill, New York, 1965
Similar Articles
Retrieve articles in Quarterly of Applied Mathematics
with MSC:
73K12,
70E15,
70K05,
73F15,
73H10
Retrieve articles in all journals
with MSC:
73K12,
70E15,
70K05,
73F15,
73H10
Additional Information
Article copyright:
© Copyright 1998
American Mathematical Society