## On the dynamics in space of rods undergoing large motions - A geometrically exact approach.(English)Zbl 0618.73100

The dynamics of a fully nonlinear rod model, capable of undergoing finite bending, shearing and extension, is considered in detail. Unlike traditional nonlinear structural dynamics formulation, due to the effect of finite rotations the deformation map takes values in $${\mathbb{R}}^ 3\times SO(3)$$, which is a differentiable manifold and not a linear space. An implicit time stepping algorithm that furnishes a canonical extension of the classical Newmark algorithm to the rotation group (SO(3) is developed. In addition to second-order accuracy, the proposed algorithm reduces exactly to the plane formulation. Moreover, the exact linearization of the algorithm and associated configuration update is obtained in closed form, leading to a configuration-dependent nonsymmetric tangent inertia matrix. As a result, quadratic rate of convergence is attained in a Newton-Raphson iterative solution strategy. The generality of the proposed formulation is demonstrated through several numerical examples that include finite vibration, centrifugal stiffening of a fast rotating beam, dynamic instability and snap-through, and large overall motions of a free-free flexible beam. Complete details on implementation are given in three appendices.

### MSC:

 74S30 Other numerical methods in solid mechanics (MSC2010) 74H45 Vibrations in dynamical problems in solid mechanics 74B20 Nonlinear elasticity 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74G60 Bifurcation and buckling
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### References:

 [1] Antman, S.S., The theory of rods, () · Zbl 0188.57501 [2] Antman, S.S., Kirchhoff problem for nonlinearly elastic rods, Quart. J. appl. math., 23, 3, 221-240, (1974) · Zbl 0302.73031 [3] Antman, S.S.; Jordan, K.B., Qualitative aspects of the spatial deformation of non-linearly elastic rods, (), 85-105, 5 · Zbl 0351.73076 [4] Antman, S.S.; Liu, T.P., Traveling waves in hyperelastic rods, Quart. J. appl. math., 377-399, (1979) · Zbl 0408.73043 [5] Antman, S.S.; Kenney, C.S., Large buckled states of nonlinear elastic rods under torsion, thrust and gravity, Arch. rat. mech. anal., 76, 4, 289-337, (1981) · Zbl 0472.73036 [6] Argyris, J.H.; Balmer, H.; St. Doltsinis, J.; Dunne, P.C.; Haase, M.; Kleiber, M.; Malejannakis, G.A.; Mlejenek, J.P.; Muller, M.; Scharpf, D.W., Finite element method-the natural approach, Comput. meths. appl. mech. engrg., 17/18, 1-106, (1979) · Zbl 0407.73058 [7] Argyris, J.H.; Symeonides, Sp., Nonlinear finite element analysis of elastic systems under nonconservative loading-natural formulation. part I. quasistatic problems, Comput. meths. appl. mech. engrg., 26, 75-123, (1981) · Zbl 0463.73073 [8] Argyris, J.H.; Symeonides, Sp., A sequel to: nonlinear finite element analysis of elastic systems under nonconservative loading-natural formulation: part I. quasistatic problems, Comput. meths. appl. mech. engrg., 26, 377-383, (1981) · Zbl 0484.73059 [9] Argyris, J.H.; Symeonides, Sp., Nonlinear finite element analysis of elastic systems under nonconservative loading-natural formulation. part II. dynamic problems, Comput. meths. appl. mech. engrg., 28, 241-258, (1981) · Zbl 0489.73070 [10] Argyris, J.H., An excursion into large rotations, Comput. meths. appl. mech. engrg., 32, 85-155, (1982) · Zbl 0505.73064 [11] Belytschko, T.; Hughes, T.J.R., Computational methods for transient analysis, (1983), North-Holland Amsterdam · Zbl 0533.73002 [12] Canavin, J.R.; Likins, P.W., Floating reference frames for flexible spacecrafts, J. spacecraft, 14, 12, 724-732, (1977) [13] Choquet-Bruhat, Y.; DeWitt-Morette, C., Analysis, manifolds and physics, (1982), North-Holland Amsterdam · Zbl 0492.58001 [14] Hughes, T.J.R., Stability, convergence and growth and decay of energy of the average acceleration method in nonlinear structural dynamics, Comput. & structures, 6, 313-324, (1976) · Zbl 0351.73102 [15] Hughes, T.J.R.; Winget, J., Finite rotation effects in numerical integration of rate constitutive equations arising in large-deformation analysis, Internat. J. numer. meths. engrg., 15, (1980) · Zbl 0463.73081 [16] Hughes, T.J.R., Numerical implementation of constitutive models: rate-independent deviatoric plasticity, (), 29-63 [17] Kane, T.R.; Ryan, R.R.; Banerjee, A.K., Dynamics of a beam attached to a moving base, (), 12-15 [18] Laskin, R.A.; Likins, P.W.; Longman, R.W., Dynamical equations of a free-free beam subject to large overall motions, J. astronaut. sci., 31, 4, 507-528, (1983) [19] Love, A.E.H., The mathematical theory of elasticity, (1944), Dover New York · Zbl 0063.03651 [20] Mital, N.K.; King, A.I., Computation of rigid-body rotation in three-dimensional space from body-fixed linear acceleration measurements, J. appl. mech., 46, 925-930, (1979) [21] Nordgren, R.P., On computation of the motion of elastic rods, J. appl. mech., 777-780, (1974) · Zbl 0292.73022 [22] Parker, D.F., The role of saint Venant’s solutions in rods and beam theories, J. appl. mech., 46, 861-866, (1979) · Zbl 0418.73044 [23] Reissner, E., On a one-dimensional finite strain beam theory: the plane problem, J. appl. math. phys., 23, 795-804, (1972) · Zbl 0248.73022 [24] Reissner, E., On a one-dimensional large-displacement finite-strain beam theory, Stud. appl. math., 52, 87-95, (1973) · Zbl 0267.73032 [25] Reissner, E., On finite deformations of space-curved beams, Z. angew. math. phys., 32, 734-744, (1981) · Zbl 0467.73048 [26] Reissner, E., Some remarks on the problem of column buckling, Ing. arch., 52, 115-119, (1982) · Zbl 0487.73047 [27] Simo, J.C.; Hjelmstad, K.H.; Taylor, R.L., Numerical formulations for the elasto-viscoplastic response of beams accounting for the effect of shear, Comput. meths. appl. mech. engrg., 42, 301-330, (1984) · Zbl 0517.73074 [28] Simo, J.C., A finite strain beam formulation. part I, Comput. meths. appl. mech. engrg., 49, 55-70, (1985) · Zbl 0583.73037 [29] J.C. Simo and L. Vu-Quoc, Three-dimensional finite-strain rod model. Part II: Computational aspects, Comput. Meths. Appl. Mech. Engrg. (to appear). · Zbl 0608.73070 [30] Simo, J.C.; Vu-Quoc, L., On the dynamics of flexible beams under large overall motions-the plane case, () · Zbl 0607.73058 [31] Simo, J.C.; Marsden, J.E.; Krishnaprassad, P.S., The Hamiltonian structure of nonlinear elasticity. the convected representation of solids, rods and plates, Arch. rat. mech. anal., (1987), (to appear) [32] Stanley, G., Continuum based shell elements, () [33] R.L. Taylor, Private communication, 1985. [34] Vu-Quoc, L., ()
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