Inertia forces and shape integrals in the floating frame of reference formulation

2019

Multibody Syst Dyn

The Floating Frame of Reference Formulation (FFRF) is one of the most widely used methods to analyze flexible multibody systems subjected to large rigid-body motion but small strains and deformations. The FFRF is conventionally derived via a continuum mechanics approach. This tedious and circuitous approach, which still attracts attention among researchers, yields so-called inertia shape integrals. These unhandy volume integrals, arising in the FFRF mass matrix and quadratic velocity vector, depend not only on the degrees of freedom, but also on the finite element shape functions. That is why conventional computer implementations of the FFRF are laborious and error prone; they require access to the algorithmic level of the underlying finite element code or are restricted to a lumped mass approximation. This contribution presents a nodal-based treatment of the FFRF to bypass these integrals. Each flexible body is considered in its spatially discretized state ab initio, wherefore the integrals are replaced by multiplications by a constant finite element mass matrix. Besides that, this approach leads to a simpler and concise but rigorous derivation of the equations of motion. The steps to obtain the inertia-integral-free equations of motion (in 2D and 3D spaces) are presented in a clear and comprehensive way; the final result provides ready-to-implement equations of motion without a lumped mass approximation, in contrast to the conventional formulation. Keywords Floating frame of reference formulation • Flexible multibody dynamics • Inertia shape integrals • Quadratic velocity vector • Finite element method 1 Introduction An increasing focus on the optimization of products to higher levels of reliability and efficiency in combination with the growing complexity of our technology-dominated world B A. Zwölfer

mentioning

confidence: 99%

mentioning

confidence: 99%

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A concise nodal-based derivation of the floating frame of reference formulation for displacement-based solid finite elements

Zwölfer

2019

Multibody Syst Dyn

“…The computation of the inertia terms is usually time consuming, and the implementation is laborious. Related formulas involving inertia integrals can be found in [15,19,21]. An alternative approach avoiding the necessity of evaluating integrals has been given in [12], with reference to [18].…”

Section: Co-rotational Formulationmentioning

confidence: 99%

A projection-based approach for the derivation of the floating frame of reference formulation for multibody systems

Winkler

2018

Acta Mech

The reduction in the number of coordinates for flexible multibody systems is necessary in order to achieve acceptable simulation times of real-life structures and machines. The conventional model order reduction technique for flexible multibody systems is based on the floating frame of reference formulation (FFRF), using a rigid body frame and superimposed small flexible deformations. The FFRF leads to strongly coupled terms in rigid body and flexible coordinates as well as to a non-constant mass matrix. As an alternative to the FFRF, a formulation based on absolute coordinates has been proposed which uses a co-rotational strain. In this way, a constant mass matrix and a co-rotational stiffness matrix are obtained. In order to perform a reduction in the number of coordinates, by means of the component mode synthesis, e.g., the number of modes needs to be increased, such that all modes are represented in every possible rotated configuration. This approach leads to the method of generalized component mode synthesis (GCMS). The present paper shows in detail how the equations of motion of the FFRF evolve from the ones of the GCMS by considering rigid body constraint conditions and subsequently eliminating them via an appropriate null-space projection. This approach allows a straightforward, term-by-term interpretation of the FFRF mass matrix and of the generalized gyroscopic forces, which, to the same extent, cannot be deduced from former publications on the FFRF. From a practical point of view, the resulting expressions allow to calculate all inertia coefficients from the constant finite element mass matrix together with standard input data of the finite element model in the course of a preprocessing step. Then, the repeated updates of the FFRF mass matrix and of the gyroscopic forces in the course of time integration involve only simple vector matrix operations of low dimensions. In contrast to previous implementations of the FFRF, no evaluations of extra inertia integrals are required. Consequently, the present formulation can be implemented entirely independent of the related finite element code.

“…There are several ways to model flexible multibody systems [19,24]. Among them the absolute coordinate formulation known as generalized component mode synthesis [16], which is a promising alternative to well-established flexible multibody formalisms, such as the floating frame of reference formulation (FFRF) [15,20,21], since the so-called generalized component modes not only describe rigid body motion, but can also represent the deformation modes in any possible orientation, which preserves a linear relationship between the global displacement field and the DOFs of the considered domain; this yields a constant mass matrix, a constant but co-rotated stiffness matrix with one co-rotational frame for each system body only, a trivial quadratic velocity vector and a simple structure of the governing equations. The mass and stiffness matrix are simply the standard linear FE system matrices pre-and post-multiplied with a constant reduction matrix.…”

Section: Introductionmentioning

confidence: 99%

Preconditioning strategies for linear dependent generalized component modes in 3D flexible multibody dynamics

Zwölfer

2019

Multibody Syst Dyn

The trend away from physical towards virtual prototyping as well as increasing industrial demands require advanced simulation tools for dynamical systems. Virtually all engineering systems are assemblies and are associated with stresses, noise and vibrations; therefore, flexible multibody simulations are inevitable for accurate predictions. However, real-world finite element models contain millions of degrees of freedom that cannot be reasonably handled without model reduction techniques. Generalized component mode synthesis is a promising tool for flexible 3D multibody systems, since the generalized component modes not only represent the deformation modes in any possible orientation, but also rigid body motion, which preserves a linear configuration space, yielding a constant mass matrix, a co-rotated but constant stiffness matrix, no quadratic velocity vector and a simple structure of the equations of motion. In this novel framework, the displacement is approximated by a linear combination of generalized component modes generated from vibration eigenmodes, undeformed nodal coordinates and the Cartesian base vectors. The emerging system matrices may be ill-conditioned and may introduce significant numerical errors, because of linearly dependent generalized component modes and due to different orders of magnitude of their Euclidean norm. However, this issue has not received much attention in the open literature despite its importance. Hence, the current contribution sheds light on this problem and derives preprocessing procedures to convert ill-conditioned into well-conditioned problems, which shall improve the formulation's applicability. The new findings are illustrated by numerical experiments of simple bodies and a crankshaft. Keywords Flexible multibody systems • Generalized component mode synthesis • Absolute coordinate formulation • Finite element method • Modal reduction • Model order reduction B A. Zwölfer