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.

The nodal-based floating frame of reference formulation with modal reduction

In a recent paper of the authors, a novel nodal-based floating frame of reference formulation (FFRF) for solid finite elements has been proposed. The nodal-based approach bypasses the unhandy inertia shape integrals ab initio, i.e. they neither arise in the derivation nor in the final equations of motion, leading to a surprisingly simple derivation and computer implementation without a lumped mass approximation, which is conventionally employed within commercial multibody codes. However, the nodal-based FFRF has so far been presented without modal reduction, which is usually required for efficient simulations. Hence, the aim of this follow-up paper is to bring the nodal-based FFRF into a suitable form, which allows the incorporation of modal reduction techniques to reduce the overall system size down to the number of modes included in the reduction basis, which further reduces the computational complexity significantly. Moreover, this exhibits a way to calculate the so-called FFRF invariants, which are constant “ingredients” required to set up the FFRF mass matrix and quadratic velocity vector, without integrals and without a lumped mass approximation.

Consistent and Inertia-Shape-Integral-Free Invariants of the Floating Frame of Reference Formulation

The conventional continuum-mechanics-based floating frame of reference formulation involves unhandy so-called inertia-shape-integrals in the equations of motion, which is why, commercial multibody software codes resort to a lumped mass approximation to avoid the evaluation of these integrals in their computer implementations. This paper recaps the conventional continuum mechanics floating frame of reference formulation and addresses its drawbacks by summarizing recent developments of the so-called nodal-based floating frame of reference formulation, which avoids inertia shape integrals ab initio, does not rely on a lumped mass approximation, and exhibits a way to calculate the so-called invariants, which are constant “ingredients” required to set up the equations of motion, in a consistent way.

Multibody Dynamics of Nonsymmetric Planar 3PRR Parallel Manipulator with Fully Flexible Links

This paper presents the implementation of the floating frame of reference formulation to model the flexible multibody dynamics of a nonsymmetric planar 3PRR parallel manipulator. All of the links, including the moving platform, of the manipulator under study are assumed flexible whereas the joints are assumed rigid. Using the Euler-Bernoulli beam, the flexibility of the links is modeled by using the Rayleigh-Ritz and finite element approximations. In both approximations, fixed-free boundary conditions are applied to the elastic coordinates of the links. These boundary conditions enable the evaluation of the elastic displacement at a link tip coincident with the end-effector of the manipulator which is of interest in the high precision robotics application. Both the approximations were compared by applying two different types of loads to the manipulator. It is shown that the elastic displacements obtained by using both the approximations have an agreement with a slight difference in the magnitude. In addition, the sensitivity analysis shows that the rigidity of the manipulator is much affected by the in-plane depth of the manipulator links’ cross section.