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**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 |

### Keywords:

covariant implicit time stepping algorithm; proper invariance requirements under superposed rigid body motions; dynamics; fully nonlinear rod model; finite bending; shearing; extension; implicit time stepping algorithm; extension of the classical Newmark algorithm; rotation group; second-order accuracy; exact linearization; configuration update; configuration-dependent nonsymmetric tangent inertia matrix; quadratic rate of convergence; Newton-Raphson iterative solution strategy; finite vibration; centrifugal stiffening; fast rotating beam; dynamic instability; snap-through; large overall motions; free-free flexible beam
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\textit{J. C. Simo} and \textit{L. Vu-Quoc}, Comput. Methods Appl. Mech. Eng. 66, No. 2, 125--161 (1988; Zbl 0618.73100)

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### References:

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