An accurate and locking-free geometric exact beam formulation on the special orthogonal group SO(3)

IF 2.8 3区 工程技术 Q2 MECHANICS
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Abstract

An accurate and locking-free geometric exact beam formulation (GEBF) on the special orthogonal group SO(3) is developed for slender beams with large deformations and large rotations. Due to the nonlinear nature of the spatial rotations, the classical GEBFs usually have a non-constant mass matrix and complex expressions of inertia forces; meanwhile, the singularity and locking issues are also key matters of concern. Aiming at resolving these drawbacks, two main contributions are achieved in the present work. First, the angular velocity is independently interpolated, resulting in a constant mass matrix and explicitly representable inertial forces, which can offer significant advantages for efficient dynamic simulations. Second, inspired by the assumed natural strain method in shell theory, the stretch-shear strain is interpolated to obtain simplified and locking-free elastic forces, which is a new attempt to alleviate the locking problems for the GEBFs. In addition, the special orthogonal group SO(3) is utilized for updating incremental rotation vectors to eliminate singularities, and objective strain measurement is achieved by employing relative rotation vector interpolation. The effectiveness and superiority of the developed low-order and high-order elements are demonstrated through numerical simulations of standard benchmark examples. The present work contributes to the advancement of accurate and reliable formulation for slender beams; the constant mass matrix, the locking-free characteristics, and the elegant form of the formulation make it particularly suitable for multibody dynamic analysis.
特殊正交群 SO(3) 上精确且无锁定的几何精确波束公式
针对具有大变形和大旋转的细长梁,在特殊正交群 SO(3) 上开发了一种精确且无锁定的几何精确梁公式(GEBF)。由于空间旋转的非线性性质,经典的 GEBF 通常具有非恒定的质量矩阵和复杂的惯性力表达式;同时,奇异性和锁定问题也是关注的重点。为了解决这些问题,本研究做出了两大贡献。首先,对角速度进行独立插值,从而得到恒定的质量矩阵和可明确表示的惯性力,这为高效的动态模拟提供了显著优势。其次,受壳理论中假定自然应变方法的启发,对拉伸剪切应变进行内插,从而得到简化的无锁定弹性力,这是缓解 GEBFs 锁定问题的新尝试。此外,利用特殊正交群 SO(3) 更新增量旋转矢量以消除奇异性,并通过相对旋转矢量插值实现客观应变测量。通过对标准基准实例进行数值模拟,证明了所开发的低阶和高阶元素的有效性和优越性。本研究为细长梁精确可靠的计算方法的发展做出了贡献;恒定的质量矩阵、无锁定特性和优雅的计算方法使其特别适用于多体动力学分析。
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来源期刊
CiteScore
5.50
自引率
9.40%
发文量
192
审稿时长
67 days
期刊介绍: The International Journal of Non-Linear Mechanics provides a specific medium for dissemination of high-quality research results in the various areas of theoretical, applied, and experimental mechanics of solids, fluids, structures, and systems where the phenomena are inherently non-linear. The journal brings together original results in non-linear problems in elasticity, plasticity, dynamics, vibrations, wave-propagation, rheology, fluid-structure interaction systems, stability, biomechanics, micro- and nano-structures, materials, metamaterials, and in other diverse areas. Papers may be analytical, computational or experimental in nature. Treatments of non-linear differential equations wherein solutions and properties of solutions are emphasized but physical aspects are not adequately relevant, will not be considered for possible publication. Both deterministic and stochastic approaches are fostered. Contributions pertaining to both established and emerging fields are encouraged.
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