Dynamics Analysis of Parallel Mechanism With Flexible Moving Platform Based on Floating Frame of Reference Formulation

2019 ◽  
Vol 11 (4) ◽  
Author(s):  
Gengxiang Wang
Keyword(s):  
Parallel Mechanism ◽  
Reference Conditions ◽  
Frame Of Reference ◽  
Dynamic Responses ◽  
Dynamics Analysis ◽  
Dynamics Model ◽  
Reference Formulation ◽  
Floating Frame Of Reference ◽  
Floating Frame ◽  
Moving Platform

The dynamics model of 4-SPS/PS parallel mechanism with a flexible moving platform is formulated based on the equation of motion. Firstly, the dynamics model of flexible moving platform is formulated based on the floating frame of reference formulation. In order to avoid the wrong solutions caused by an inappropriate set of reference conditions, the fixed-fixed reference conditions are carefully selected according to the structure of parallel mechanism. Secondly, considering that the original Craig–Bampton (CB) method only represents the free-free modes. In order to use CB method to obtain fixed-fixed modes, the original CB method is improved by imposing the reference conditions prior to obtaining the static correction modes and fixed interface modes. In addition, the dynamics analysis of 4-SPS/PS parallel mechanism with flexible moving platform based on both free-free modes and fixed-fixed modes are implemented, respectively. Finally, the simulations show that the dynamic responses obtained using fixed-fixed modes are close to the ideal dynamic response, which proves the correctness of improved CB method. Moreover, the maximum percentage error of simulation results between using free-free modes and using fixed-fixed modes exceeds 100%, it is clear that the solutions based on free-free modes are not reasonable. Eventually, the conclusions prove that the deformation caused by high-speed and heavy-load should not be neglected in the parallel mechanism.

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

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.

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