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The dynamic response of many flexible multibody systems of practical interest is periodic. The investigation of such problems involves two intertwined tasks: first, the determination of the periodic response of the system and second, the analysis of the stability of this periodic solution. Starting from Hamilton’s principle, a unified solution procedure for continuous and discontinuous Galerkin methods is developed for these two tasks. In the proposed finite element formulation, the unknowns consist of the displacement and rotation components at the nodes, which are interpolated via the dual spherical linear interpolation technique. Periodic solutions are obtained by solving the discrete nonlinear equations resulting from continuous and discontinuous Galerkin methods. The monodromy matrix required for stability analysis is constructed directly from the Jacobian matrix of the solution process. Numerical examples are presented to validate the proposed approaches.

This paper proposes a novel spectral formulation for geometrically exact beams based on motion interpolation schemes. Motion interpolation schemes based on matrix, quaternion, and geodesic metrics ...

A novel time-discontinuous Galerkin (DG) method is introduced for the time integration of the differential-algebraic equations governing the dynamic response of flexible multibody systems. In contrast to traditional Galerkin methods, the rigid-body motion field is interpolated using the dual spherical linear scheme. Furthermore, the jumps inherent to time-DG methods are expressed in terms of a parameterization of the relative motion from one time-step to the next. The proposed scheme is third-order accurate for initial value problems of both rigid and flexible multibody dynamics.

In many applications, beams carry large axial forces, such as centrifugal forces, for instance, and the resulting axial stresses affect their torsional behavior. To capture this important coupling effect within the framework of beam models, a nonlinear theory must be developed that takes into account higher-order strain deformations. The proposed development starts with a three-dimensional elasticity model of beam-like structures. While strain components are assumed to remain small, the strain energy expression is expanded to retain strain components up to cubic order. This expansion of the three-dimensional model leads to material and geometric stiffness matrices. For linear problems, the three-dimensional warping fields induced by each of the six unit sectional strain components can be obtained. These warping fields are used to reduce the three-dimensional elasticity model to one-dimensional beam equations. Byproducts of this reduction process include the sectional stiffness matrix and six nonlinear stiffness matrices representing the geometric stiffness induced by unit sectional strains. It is shown that the nonlinear extension-twist coupling is captured by these geometric stiffness matrices. Comparison of the predictions of the proposed approach with analytical and experimental results demonstrate their effectiveness and accuracy.

Besides fusion structural materials themselves, research and development of small specimen test technologies (SSTTs) are also important for their joints for mechanical properties evaluation on resistance to neutron irradiation. Because the specimens usually fracture at the base metals (BM), not at the weld metal (WM), and the bonding properties of the joints cannot be obtained directly during tensile tests, small tensile specimens with different sizes of notches at the WM were utilized to try to extrapolate the bonding properties in the present study. A joint of reduced-activation ferritic/martensitic (RAFM) steel CLF-1 made by electron beam welding (EBW) after post-weld heat treatment (PWHT) were utilized to demonstrate the feasibility of this method. By combining tensile experiments, optical digital image correlation technology, and finite element method (FEM) simulation, effects of notch sizes on bonding properties such as strain, stress, and stress triaxiality were investigated. Finally, bonding properties for the joint were tentatively extrapolated from the results of the specimens with notches.

For beams undergoing large motions but small strains, the displacement field can be decomposed into an arbitrarily large rigid-section motion and a warping field. When applying beam theory to dynamic problems, it is customary to assume that all inertial effects associated with warping are negligible. This paper examines this assumption in details. It is shown that inertial forces affect the beam’s dynamic response in two manners: (1) warping motion induces inertial forces directly, and (2) secondary warping arises that alters the beam’s constitutive laws. Numerical examples demonstrate the range of validity of the proposed approach for beams made of both homogeneous isotropic and heterogeneous anisotropic materials. For low-frequency warping, it is shown that inertial forces associated with warping and secondary warping resulting from inertial forces are negligible. To examine the dynamic behavior of beams over a wider range of frequencies, their dispersion curves, natural vibration frequencies, and mode shapes are evaluated using both one- and three-dimensional models; good correlation is observed between the two models. Applications of the proposed beam theory to multibody problems are also presented; here again, good correlation is observed between the prediction of beam models and of full three-dimensional analysis.

In structural analysis, many components are approximated as plates. More often that not, classical plate theories, such as Kirchhoff or Reissner-Mindlin plate theories, form the basis of the analytical developments. The advantage of these approaches is that they leads to simple kinematic descriptions of the problem: the plate's normal material line is assumed to remain straight and its displacement field is fully defined by three displacement and two rotation components. While such approach is capable of capturing the kinetic energy of the system accurately, it cannot represent the strain energy adequately. For instance, it is well known from three-dimensional elasticity theory that the normal material line will warp under load for laminated composite plates, leading to three-dimensional deformations that generate complex stress states. To overcome this problem, several layer-wise plate theories have been proposed. While these approaches work well for some cases, they often lead to inefficient formulations because they introduce numerous additional variables. This paper presents a novel, single-layer theory using local-global Approach: based on a finite element semi-discretization of the normal material line, the two-dimensional plate equations are derived from three-dimensional elasticity using a rigorous dimensional reduction procedure. Three-dimensional stresses through the plate's thickness can be recovered accurately from the plate's stress resultants.

The Hamiltonian approach is generalized to beams consisting of thin-walled panels.The proposed approach can handle curved sectional geometries.Closed-form central and extremity solutions are found.Correct boundary conditions based on the weak form formulation are derived.The beam's 6×6 sectional stiffness matrix is a by-product of the analysis. In this paper, the Hamiltonian approach developed for beam with solid cross-section is generalized to deal with beams consisting of thin-walled panels. The governing equations of plates and cylindrical shells for the panels are cast into Hamiltonian canonical equations and closed-form central and extremity solutions are found. Typically, the end-effect zones for thin-walled beams are much larger than those for beams with solid cross-sections. Consequently, extremity solutions affect the solution significantly. Correct boundary conditions based on the weak form formulation are derived. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are found to be in good agreement with those of plate and shell FEM analysis.

The manipulation of motion and ancillary operations are important tasks in kinematics, robotics, and rigid and flexible multibody dynamics. Motion can be described in purely geometric terms, based on Chasles’ theorem. Representations and parameterizations of motion are also available, such as Euler motion parameters and the vectorial parameterization, respectively. Typical operations to be performed on motion involve the selection of local or global parameterizations and the derivation of the associated expressions for the motion tensor, velocity or curvature vector, composition of motions, and tangent tensors. Many of these tasks involve arduous, error-prone algebra. The use of dual entities has been shown to ease the manipulation of motion, yet this concept has received little attention outside of the fields of kinematics and robotics. This paper presents a comprehensive treatment of the topic using a notation that eliminates the bookkeeping parameter typically used in dual number algebra, thereby recasting all operations within the framework of linear algebra and streamlining the process. The manipulation of geometric entities is recast within this formalism, paving the way for the manipulation of motion. All developments are presented within the framework of dual numbers directly; the principle of transference is never invoked: The manipulation of rotation is a particular case of that of motion, as should be. The problem of interpolation of motion, a thorny issue in finite element applications, is also addressed.

Based on the symplectic transfer-matrix method, this paper develops a novel approach for the analysis of beams presenting periodic heterogeneities along their span. The approach, rooted in the Hamiltonian formalism, generalizes developments presented earlier by the authors for spanwise uniform beams. Starting from the kinematics of a unit cell, the approach proceeds through a set of structure-preserving symplectic transformations and decomposes the solution into its central and extremity components. The geometric configuration and material properties of the unit cell may be arbitrarily complex as long as the cell's two end cross sections are identical. The proposed approach identifies an equivalent, homogenized beam with uniform curvatures and sectional stiffness characteristics along its span. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are found to be in excellent agreement with those obtained by full finite-element analysis.

This paper proposes a novel solution strategy for Saint-Venant's problem based on Hamilton's formalism. Saint-Venant's problem focuses on helicoidal beams and its solution hinges upon the determination of the subspace of the system's Hamiltonian matrix associated with its null and pure imaginary eigenvalues. A projection approach is proposed that reduces the system Hamiltonian matrix to a matrix of size 12 × 12, whose eigenvalues are identical to the null and purely imaginary eigenvalues of the original system, with the same Jordan structure. A fundamental theoretical result is established: Saint-Venant's solutions exist because rigid-body motions create no strains. Indeed, the solvability conditions for the governing equations of the problem are satisfied because a matrix identity holds, which expresses the fact that rigid-body motions create no strains. Because it avoids the identification of the Jordan structure of the original system, the implementation of the proposed projection for large, realistic problems is straightforward. Closed-form solutions of the reduced problem are found and three-dimensional stress and strain fields can be recovered from the closed-form solution. Numerical examples are presented to demonstrate the capabilities of the analysis. Predictions are compared to exact solutions of three-dimensional elasticity and three-dimensional FEM analysis.