Formulation of Shell Elements Based on the Motion Formalism

This paper presents a finite element implementation of plates and shells for the analysis of flexible multibody systems. The developments are set within the framework of the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) couples their displacement and rotation components by recognizing that configuration and motion are members of the Special Euclidean group, and (3) resolves all tensors components in local frames. The formulation based on the motion formalism (1) provides a theoretical framework that streamlines the formulation of shell elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, (4) circumvents the shear locking phenomenon that plagues shell formulations based on classical kinematic descriptions, and (5) prevents the occurrence of singularities in the treatment of finite rotation. Numerical examples are presented to illustrate the advantageous features of the proposed formulation.

An Approximation-Based Design Optimization Approach to Eigenfrequency Assignment for Flexible Multibody Systems

The use of flexible multibody simulation has increased significantly over recent years due to the increasingly lightweight nature of mechanical systems. The prominence of lightweight engineering design in mechanical systems is driven by the desire to require less energy in operation and to reach higher speeds. However, flexible lightweight systems are prone to vibration, which can affect reliability and overall system performance. Whether such issues are critical depends largely on the system eigenfrequencies, which should be correctly assigned by the proper choice of the inertial and elastic properties of the system. In this paper, an eigenfrequency assignment method for flexible multibody systems is proposed. This relies on a parametric modal model which is a Taylor expansion approximation of the eigenfrequencies in the neighborhood of a configuration of choice. Eigenfrequency assignment is recast as a quadratic programming problem which can be solved with low computational effort. The method is validated by assigning the lowest eigenfrequency of a two-bar linkage by properly adding point masses. The obtained results indicate that the proposed method can effectively assign the desired eigenfrequency.

The Motion Formalism for Flexible Multibody Systems

This paper describes a finite element approach to the analysis of flexible multibody systems. It is based on the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) recognizes that these are members of the Special Euclidean group thereby coupling their displacement and rotation components, and (3) resolves all tensors components in local frames. The goal of this review paper is not to provide an in-depth derivation of all the elements found in typical multibody codes but rather to demonstrate how the motion formalism (1) provides a theoretical framework that unifies the formulation of all structural elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, and (4) prevents the occurrence of singularities.

On design sensitivities in the structural analysis and optimization of flexible multibody systems

The structural analysis and optimization of flexible multibody systems become more and more popular due to the ability to efficiently compute gradients using sophisticated approaches such as the adjoint variable method and the adoption of powerful methods from static structural optimization. To drive the improvement of the optimization process, this work addresses the computation of design sensitivities for multibody systems with arbitrarily parameterized rigid and flexible bodies that are modeled using the floating frame of reference formulation. It is shown that it is useful to augment the body describing standard input data files by their design derivatives. In this way, a clear separation can be achieved between the body modeling and parameterization and the system simulation and analysis.

Review of the exponential and Cayley map on SE(3) as relevant for Lie group integration of the generalized Poisson equation and flexible multibody systems

The exponential and Cayley maps on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe–Kaas and generalized- α schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed-form relations along with the relevant proofs, including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized- α scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.

Reduced Isogeometric Analysis Models for Impact Simulations

Detailed impact simulations in flexible multibody systems are usually based on isoparametric finite element models. For modeling the dynamics of an impact, a precise representation of the geometry is essential. However, isoparametric finite elements involve the discretization of the geometry. This work tests the isogeometric analysis (IGA) as an alternative approach in flexible multibody systems. The IGA enables the exact representation of the geometry by using non-uniform rational basis splines (NURBS) as element shape functions. In the context of an efficient impact simulation a model reduction and a possible inclusion of the floating frame of reference formulation is beneficial. The degrees of freedom of the flexible bodies are reduced using component mode synthesis to save computation time in the multibody simulation. For the precise description of deformations and stresses in the contact area as well as elastodynamic effects, a large number of global shape functions is required. As testing examples, the impact of two elastic spheres and a multibody multicontact problem including wave propagation in a long elastic rod are simulated and compared to reference solutions.