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

2019 ◽  
Vol 46 (1) ◽  
pp. 77-105 ◽  
Author(s):  
Wolfgang Witteveen ◽  
Florian Pichler
Keyword(s):  
Equations Of Motion ◽  
Frame Of Reference ◽  
Flexible Body ◽  
Reference Formulation ◽  
Floating Frame Of Reference ◽  
Floating Frame

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

2020 ◽  
Author(s):  
Andreas Zwölfer ◽  
Johannes Gerstmayr
Keyword(s):  
Continuum Mechanics ◽  
Equations Of Motion ◽  
Frame Of Reference ◽  
Reference Formulation ◽  
Lumped Mass ◽  
Floating Frame Of Reference ◽  
Mass Approximation ◽  
Floating Frame ◽  
Software Codes ◽  
Set Up

Abstract 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.

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

2019 ◽  
Vol 49 (3) ◽  
pp. 291-313 ◽  
Author(s):  
Andreas Zwölfer ◽  
Johannes Gerstmayr
Keyword(s):  
Finite Element ◽  
Mass Matrix ◽  
Equations Of Motion ◽  
Frame Of Reference ◽  
Body Motion ◽  
Reference Formulation ◽  
Lumped Mass ◽  
Floating Frame Of Reference ◽  
Mass Approximation ◽  
Floating Frame

AbstractThe 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.

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

2020 ◽  
Author(s):  
Andreas Zwölfer ◽  
Johannes Gerstmayr
Keyword(s):  
Mass Matrix ◽  
Equations Of Motion ◽  
Frame Of Reference ◽  
Reference Formulation ◽  
Lumped Mass ◽  
Simple Derivation ◽  
Floating Frame Of Reference ◽  
Mass Approximation ◽  
Modal Reduction ◽  
Floating Frame

AbstractIn 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.

A Floating-Frame-of-Reference Formulation for Deformable Rotors Using the Properties of Free Elastic Vibration Modes

2009 ◽  
Author(s):  
H. Irschik ◽  
M. Nader ◽  
M. Stangl ◽  
H.-G. v. Garssen
Keyword(s):  
Rigid Body ◽  
Equations Of Motion ◽  
Linear Equations ◽  
Axial Rotation ◽  
Frame Of Reference ◽  
Reference Formulation ◽  
Floating Frame Of Reference ◽  
Non Linear ◽  
Floating Frame ◽  
Linear Equations Of Motion

Formulations in rotordynamics are usually based on the assumption that the displacements of the bearings of the rotor are small, such that, besides the axial rotation, no large rigid-body motions have to be taken into account. This results in linear equations of motion with gyroscopic terms. When the axial angular speed of a rotor is increased, however, as well as for rapidly changing transient conditions, a non-linear coupling between the large axial rotation and the small rigid body motion induced by the compliance of the bearings and the small elastic deformation of the rotor body itself is to be expected. It is the scope of the present contribution to present a rational strategy for dealing with this situation. First, we present a problem-oriented version of the floating-frame-of-reference formulation (FFRF). We use a co-rotating rigid rotor as reference configuration, which allows using linear modes of the non-rotating elastic rotor as Ritz approximations. The position vector of the origin of a body-fixed coordinate system and three suitable Bryant angles are used as rigid body coordinates, and free elastic modes of the rotor are considered as elastic Ritz approximations. The properties of the latter and their consequences upon simplifying the necessary spatial integrals in the FFRF are addressed in some detail. The free modes are obtained from a Finite Elements pre-processing of the elastic rotor body. The non-linear equations of motion of the rotor are obtained afterwards by means of symbolic computation This formulation leads to a set of relations, in which the rigid-body degrees of freedom need not to be small, and which is integrated using an implicit scheme. Results for a rotor with unbalance forces, accelerated by external forces and having linear visco-elastic bearings are successfully compared to a commercial multi-body dynamics code.

Development of Reduced Order Thermomechanical Model Using Floating Frame of Reference Formulation

2017 ◽  
Author(s):  
Hiroki Yamashita ◽  
Rohit Arora ◽  
Hiroyuki Kanazawa ◽  
Hiroyuki Sugiyama
Keyword(s):  
Multibody Systems ◽  
Thermal Deformation ◽  
Frame Of Reference ◽  
Flexible Multibody Systems ◽  
Flexible Body ◽  
Reference Formulation ◽  
Thermomechanical Model ◽  
Reduced Order ◽  
Floating Frame Of Reference ◽  
Floating Frame

In this paper, a reduced order thermomechanical model based on the Craig-Bampton component mode synthesis method is extended to the floating frame of reference formulation for the thermomechanical analysis of flexible multibody systems. To this end, coupled structural and thermal equations of finite element models are partitioned in terms of the internal and interface coordinates, each of which consists of the structural and thermal coordinates. Both deformation including the thermal effect and temperature in the internal region are then defined by a linear combination of the thermomechanical fixed-interface normal modes and thermomechanical constraint modes to account for structural and thermal modes associated with external forces and heat sources applied to the system. The final form of equations include equations of motion associated with a flexible body that incorporates thermal deformation and the reduced order heat equations that describe the transient change in the temperature over the flexible body. For this reason, the inertia coupling of the reference motion and the thermal deformation is automatically considered using the floating frame of reference formulation. Both equations are integrated forward in time simultaneously using general multibody dynamics computer algorithms to account for the coupled structural and thermal behavior of flexible multibody systems. Several numerical examples are presented to demonstrate the use of the numerical procedure developed in this study.

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

2017 ◽  
Vol 88 (3) ◽  
pp. 1953-1968 ◽  
Author(s):  
Grzegorz Orzechowski ◽  
Marko K. Matikainen ◽  
Aki M. Mikkola
Keyword(s):  
Frame Of Reference ◽  
Reference Formulation ◽  
Floating Frame Of Reference ◽  
Floating Frame ◽  
Inertia Forces
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