The consideration of Finite Element (FE) structures in multibody simulation (MBS) has become an established method, especially when the number of force application points remains moderate. In recent years, a trend can be observed in which distributed loads are considered as well, such as those arising from the contact of two elastic bodies. For finely-meshed FE structures, this results in a large number of possible force application points. In such a case, conventional methods fail, leading to exorbitantly high computation times.
In the last decade, approaches for the reduced computation of deformations inside distributed load application areas were introduced. Special approach vectors are used, called "local modes" here. These local modes lead to a reduction in the involved equations by several orders of magnitude. However, for very large potential load application areas, a large number of local modes is still required—for example, several thousand. Since each local mode leads to a differential equation, a fast numerical time integration is not possible with common methods. In this work, two methodological improvements are proposed for a fast and accurate time integration of such systems.
Inhaltsverzeichnis (Table of Contents)
- Kurzfassung
- Abstract
- Table of contents
- List of abbreviation
- Introduction
- Motivation
- Goal
- Outline
- Brief review on model reduction of finite element structures via projection
- Linear FE models
- Introduction
- General equations
- Component Mode Synthesis
- Moment matching
- Balanced truncation
- A brief comparison of CMS, MM and BT
- Nonlinear FE models
- Introduction
- Literature on MOR for geometric nonlinearities
- MOR for nonlinearities caused by contact and friction
- Data based reduction methods
- Challenge of many modes in case of distributed nonlinear surface forces
- Detailed analysis of the equations of motion of a free flexible body in the floating frame of reference formulation
- Introduction
- Brief review of the equations of motion
- Assumptions and simplifications
- Euler parameter
- Linearized mean-axis conditions
- Use of (pseudo) free surface modes together with an FF origin fixed to the center of gravity of the undeformed body
- Use of mass normalized modes
- Use of central axis of inertia
- Final equation of motion
- Assumption of small deformations
- Illustrative Examples used in the section
- Small elastic deformations with respect to the body's dimension
- Magnitude of modes and modal coordinates
- Magnitude of entries in invariants W₁, W2 and W3
- Magnitude of entries in matrix W₁
- Magnitude of entries in matrices W2 and W3
- Investigations on the relevance of the single terms in the equations of motion
- Rotational inertia of the deformed body
- Inertia coupling
- Quadratic velocity vector
- "Set of guidlines" for practical use
- Benefit
- Separate time integration for flexible body's with a large number of modes
- Introduction
- Theory
- Brief review of equations of motions of a multibody system
- Properties of local modes and its implications
- Separate consideration of a flexible body's non-stiff and stiff equations
- Separate time integration of a flexible body's stiff equations via a fixed point iteration based on the HHT method
- Numerical examples
- 2 dof Oscillator
- Planar crank drive with elastic piston and elastohydrodynamic oil film model
- Discussion and Conclusion
- Approximation of surface loads via stress trial vectors
- General concept
- Determination of a proper stress mode base
- Simulation-free generation of potential stress space
- Simulation-based generation of a potential stress space
Zielsetzung und Themenschwerpunkte (Objectives and Key Themes)
This work aims to address the challenges posed by the inclusion of finite element (FE) structures in multibody simulation (MBS), particularly when considering distributed loads that arise from the contact of two elastic bodies. These loads lead to a large number of possible force application points, posing significant challenges to traditional methods due to exorbitantly high computation times. The work proposes two methodological advancements to achieve fast and accurate time integration of such systems.
- Reduced Computation of Deformations
- Efficient Time Integration with Many Modes
- Hyper-Reduction for Distributed Forces
- Contact Mechanics in Multibody Simulation
- Applications in Engineering Problems
Zusammenfassung der Kapitel (Chapter Summaries)
- Introduction: The chapter introduces the motivation behind the work, outlining the challenges and opportunities associated with incorporating FE structures into multibody simulations. It highlights the specific issues arising from distributed loads and the need for efficient computational methods.
- Brief review on model reduction of finite element structures via projection: This chapter delves into model reduction techniques used for FE structures, reviewing existing methods like Component Mode Synthesis (CMS), Moment Matching (MM), and Balanced Truncation (BT). It discusses the limitations of these methods when dealing with nonlinearities caused by contact and friction, and introduces the concept of local modes as a potential solution.
- Detailed analysis of the equations of motion of a free flexible body in the floating frame of reference formulation: The chapter provides an in-depth analysis of the equations of motion for a flexible body in the floating frame of reference formulation. It examines the assumptions and simplifications made, and analyzes the various terms contributing to the final equation of motion. The chapter aims to provide a theoretical foundation for the proposed methods.
- Separate time integration for flexible body's with a large number of modes: This chapter presents the first proposed method to improve efficiency, focusing on separate time integration for stiff and non-stiff equations arising from the presence of many local modes. It describes the theory behind this approach, including the use of a fixed-point iteration based on the HHT method, and provides numerical examples to illustrate its effectiveness.
- Approximation of surface loads via stress trial vectors: The chapter introduces the second proposed method for enhancing efficiency, which involves approximating surface loads using stress trial vectors. It outlines the general concept and discusses different strategies for determining a proper stress mode base, including both simulation-free and simulation-based approaches.
Schlüsselwörter (Keywords)
The central keywords and focus topics of this work include: flexible multibody dynamics, floating frame of reference formulation, numerical time integration, model reduction, hyper-reduction, local modes, contact mechanics, distributed loads, and efficient computation.
- Quote paper
- Wolfgang Witteveen (Author), 2021, Computational efficient flexible multibody dynamics with nonlinear loads and many modes, Munich, GRIN Verlag, https://www.grin.com/document/1190639
