A Detailed Derivation of the Velocity-Dependent Inertia Forces in the Floating Frame of Reference Formulation

Abstract

In the case of complex multibody systems, an efficient and time-saving computation of the equations of motion is essential; in particular, concerning the inertia forces. When using the floating frame of reference formulation for modeling a multibody system, the inertia forces, which include velocity-dependent forces, depend nonlinearly on the system state and, therefore, have to be updated in each time step of the dynamic simulation. Since the emphasis of the present investigation is on the efficient computation of the velocity-dependent inertia forces as along with a fast simulation of multibody systems, a detailed derivation of the latter forces for the case of a general rotational parameterization is given. It has to be emphasized that the present investigations revealed a simpler representation of the velocity-dependent inertia forces compared to results presented in the literature. In contrast to the formulas presented in the literature, the presented formulas do not depend on the type of utilized rotational parameterization or on any associated assumptions.

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

“…Yet, essential elements of its implementation, such as the representation of inertia or Coriolis, centrifugal and gyroscopic forces that result when kinetic energy is differentiated, are usually not described in sufficient detail. As a result, complex and error-prone derivations are often required, for example as in [7] and [8]. Moreover, most of the derivations presented in the literature are specific to a unique set of rotational coordinates and might not meet the requirements of a particular analysis.…”

“…Appropriate formulas can be easily found in the literature, for example, in the papers of Yoo and Haug [10] and, more recently, Lugrís et al [4] and Sherif and Nachbagauer [8] or, for a detailed derivation, in the well-known book by Shabana [7]. The present paper differs from its predecessors in the way it presents a detailed and clearly elaborated derivation of the velocity-dependent inertia forces, with a detailed and explicit use of the inertia shape integrals, and with the derivation accomplished using angular velocity and angular acceleration vectors.…”

On the relevance of inertia related terms in the equations of motion of a flexible body in the floating frame of reference formulation

“…In order to avoid redundancy, an extensive derivation of the equations of motion is omitted and only relevant equations are reviewed. Details can be found in Shabana [1] and the work of Sherif and Nachbagauer [2]. For better readability with respect to the mentioned publications, the same notation is used in this work.…”

“…In contrast to the referenced equation, the constraints for a nonminimal set of rotational coordinates (e.g., Euler parameters) are omitted here, since they are not relevant for the considerations in this work. According to [2,Eqs. (25), (54) and (66)], the so-called (n × 1) quadratic velocity vector Q v can be written as…”

A complete strategy for efficient and accurate multibody dynamics of flexible structures with large lap joints considering contact and friction

“…with Q v being the quadratic velocity vector, also known as the velocity inertia vector [29], Q e including the generalized external forces, and Q nl,f containing the generalized contact and friction forces and the generalized flexible forces. For body i in the multibody system, the equation of motion can be written in a partitioned matrix form as…”