Some Applications of Automatic Differentiation to Rigid, Flexible, and Constrained Multibody Dynamics

2005 ◽  
Author(s):  
D. Todd Griffith ◽  
James D. Turner ◽  
John L. Junkins
Keyword(s):  
Dynamical Systems ◽  
Multibody Dynamics ◽  
Multibody Systems ◽  
Automatic Differentiation ◽  
Equations Of Motion ◽  
Numerical Results ◽  
Flexible Multibody Systems ◽  
New Approach ◽  
Constrained Dynamical Systems ◽  
Constrained Multibody Dynamics

In this paper, we discuss several applications of automatic differentiation to multibody dynamics. The scope of this paper covers the rigid, flexible, and constrained dynamical systems. Particular emphasis is placed on the development of methods for automating the generation of equations of motion and the simulation of response using automatic differentiation. We also present a new approach for generating exact dynamical representations of flexible multibody systems in a numerical sense using automatic differentiation. Numerical results will be presented to detail the efficiency of the proposed methods.

Sensivity Analysis of Flexible Multibody Systems

2005 ◽  
Author(s):  
M. A. Neto ◽  
R. P. Leal ◽  
J. Ambro´sio
Keyword(s):  
Multibody Systems ◽  
Automatic Differentiation ◽  
Direct Method ◽  
Equations Of Motion ◽  
Flexible Multibody Systems ◽  
Element Analysis ◽  
Design Variables ◽  
Finite Differences Method ◽  
Flexible Multibody ◽  
Analytical Sensitivities

In this work a general formulation for the computation of the first order analytical sensitivities based on the direct method is presented. The direct method for sensitivity calculation is obtained by differentiating the equations that define the response of the flexible system with respect to the design variables. The design variables used here are the ply orientations of the laminated. The analytical sensitivities are compared with the numerical results obtained by using the finite differences method. For the beam composite material elements, the section properties and their sensitivities are found using an asymptotic procedure that involves a two-dimensional finite element analysis of their cross-section. The equations of the sensitivities are obtained by automatic differentiation and integrated in time simultaneously with the equations of motion of the multibody systems. The equations of motion and sensitivities of the flexible multibody system are solved and the accelerations and velocities and sensitivities of accelerations and velocities are integrated. Through the application of the methodology to a single flexible multibody systems the difficulties and benefices of the procedure are discussed.

The Motion Formalism for Flexible Multibody Systems

2021 ◽  
Author(s):  
Olivier Bauchau ◽  
Valentin Sonneville
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Solution Process ◽  
Flexible Multibody Systems ◽  
Structural Elements ◽  
Governing Equations ◽  
Euclidean Group ◽  
Reduced Order ◽  
Flexible Multibody ◽  
Element Approach

Abstract This paper describes a finite element approach to the analysis of flexible multibody systems. It is based on the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) recognizes that these are members of the Special Euclidean group thereby coupling their displacement and rotation components, and (3) resolves all tensors components in local frames. The goal of this review paper is not to provide an in-depth derivation of all the elements found in typical multibody codes but rather to demonstrate how the motion formalism (1) provides a theoretical framework that unifies the formulation of all structural elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, and (4) prevents the occurrence of singularities.

Recursive Method for the Dynamic Analysis of Open-Loop Flexible Multibody Systems

2003 ◽  
Author(s):  
Yunn-Lin Hwang
Keyword(s):  
Dynamic Analysis ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
Open Loop ◽  
Flexible Multibody Systems ◽  
Recursive Method ◽  
Flexible Bodies ◽  
System Equations ◽  
Generalized Newton ◽  
Time Invariant

The main objective of this paper is to develop a recursive method for the dynamic analysis of open-loop flexible multibody systems. The nonlinear generalized Newton-Euler equations are used for flexible bodies that undergo large translational and rotational displacements. These equations are formulated in terms of a set of time invariant scalars, vectors and matrices that depend on the spatial coordinates as well as the assumed displacement fields, and these time invariant quantities represent the dynamic coupling between the rigid body motion and elastic deformation. The method to solve for the equations of motion for open-loop systems consisting of interconnected rigid and flexible bodies is presented in this investigation. This method applies recursive method with the generalized Newton-Euler method for flexible bodies to obtain a large, loosely coupled system equations of motion. The solution techniques used to solve for the system equations of motion can be more efficiently implemented in the vector or digital computer systems. The algorithms presented in this investigation are illustrated by using cylindrical joints that can be easily extended to revolute, slider and rigid joints. The basic recursive formulations developed in this paper are demonstrated by two numerical examples.

scholarly journals Formulation of Shell Elements Based on the Motion Formalism

2021 ◽  
Vol 2 (4) ◽  
pp. 1009-1036
Author(s):  
Olivier Bauchau ◽  
Valentin Sonneville
Keyword(s):  
Multibody Systems ◽  
Equations Of Motion ◽  
Solution Process ◽  
Flexible Multibody Systems ◽  
Governing Equations ◽  
Euclidean Group ◽  
Shell Elements ◽  
Reduced Order ◽  
Flexible Multibody ◽  
Plates And Shells

This paper presents a finite element implementation of plates and shells for the analysis of flexible multibody systems. The developments are set within the framework of the motion formalism that (1) uses configuration and motion to describe the kinematics of flexible multibody systems, (2) couples their displacement and rotation components by recognizing that configuration and motion are members of the Special Euclidean group, and (3) resolves all tensors components in local frames. The formulation based on the motion formalism (1) provides a theoretical framework that streamlines the formulation of shell elements, (2) leads to governing equations of motion that are objective, intrinsic, and present a reduced order of nonlinearity, (3) improves the efficiency of the solution process, (4) circumvents the shear locking phenomenon that plagues shell formulations based on classical kinematic descriptions, and (5) prevents the occurrence of singularities in the treatment of finite rotation. Numerical examples are presented to illustrate the advantageous features of the proposed formulation.

Comparison of Kane’s and Lagrange’s Methods in Analysis of Constrained Dynamical Systems

Robotica ◽  
2020 ◽  
Vol 38 (12) ◽  
pp. 2138-2150
Author(s):  
Amin Talaeizadeh ◽  
Mahmoodreza Forootan ◽  
Mehdi Zabihi ◽  
Hossein Nejat Pishkenari
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Constrained Systems ◽  
Robot Dynamics ◽  
Energy Error ◽  
Kane’S Method ◽  
Kane's Method ◽  
Constrained Dynamical Systems ◽  
Error Energy

SUMMARYDynamic modeling is a fundamental step in analyzing the movement of any mechanical system. Methods for dynamical modeling of constrained systems have been widely developed to improve the accuracy and minimize computational cost during simulations. The necessity to satisfy constraint equations as well as the equations of motion makes it more critical to use numerical techniques that are successful in decreasing the number of computational operations and numerical errors for complex dynamical systems. In this study, performance of a variant of Kane’s method compared to six different techniques based on the Lagrange’s equations is shown. To evaluate the performance of the mentioned methods, snake-like robot dynamics is considered and different aspects such as the number of the most time-consuming computational operations, constraint error, energy error, and CPU time assigned to each method are compared. The simulation results demonstrate the superiority of the variant of Kane’s method concerning the other ones.

Automatic Differentiation in Automatic Generation of the Linearized Equations of Motion

2021 ◽  
Author(s):  
Bruce Minaker ◽  
Francisco González
Keyword(s):  
Multibody Systems ◽  
Automatic Differentiation ◽  
Equations Of Motion ◽  
Computational Cost ◽  
Order Reduction ◽  
Automatic Generation ◽  
Analysis Model ◽  
Direct Evaluation ◽  
Base Function ◽  
Linearized Equations

Abstract In the ongoing search for mathematically efficient methods of predicting the motion of vehicle and other multibody systems, and presenting the associated results, one of the avenues of continued interest is the linearization of the equations of motion. While linearization can potentially result in reduced fidelity in the model, the benefits in computational speed often make it the pragmatic choice. Linearization techniques are also useful in modal and stability analysis, model order reduction, and state and input estimation. This paper explores the application of automatic differentiation to the generation of the linearized equations of motion. Automatic differentiation allows one to numerically evaluate the derivative of any function, with no prior knowledge of the differential relationship to other functions. It exploits the fact that every computer program must evaluate every function using only elementary arithmetic operations. Using automatic differentiation, derivatives of arbitrary order can be computed, accurately to working precision, with minimal additional computational cost over the evaluation of the base function. There are several freely available software libraries that implement automatic differentiation in modern computing languages. In the paper, several example multibody systems are analyzed, and the computation times of the stiffness matrix are compared using direct evaluation and automatic differentiation. The results show that automatic differentiation can be surprisingly competitive in terms of computational efficiency.

Semi-Symbolical Equations of Motion for Flexible Multibody Systems

1993 ◽  
Author(s):  
Frank Melzer
Keyword(s):  
Data Model ◽  
Multibody Systems ◽  
Equations Of Motion ◽  
General Purpose ◽  
Flexible Multibody Systems ◽  
Dynamical Analysis ◽  
Lightweight Structures ◽  
Elastic Bodies ◽  
Flexible Multibody ◽  
Increasing Demand

Abstract The need for computer aided engineering in the analysis of machines and mechanisms led to a wide variety of general purpose programs for the dynamical analysis of multibody systems. In the past few years the incorporation of flexible bodies in this methodology has evolved to one of the major research topics in the field of multibody dynamics, due to the use of more lightweight structures and an increasing demand for high-precision mechanisms such as robots. This paper presents a formalism for flexible multibody systems based on a minimum set of generalized coordinates and symbolic computation. A standardized object-oriented data model is used for the system matrices, describing the elastodynamic behaviour of the flexible body. Consequently, the equations of motion are derived in a form independent of the chosen modelling technique for the elastic bodies.

Incremental Finite Element Formulations and Flexible Multibody Dynamics

1999 ◽  
Author(s):  
Marcello Berzeri ◽  
Marcello Campanelli ◽  
A. A. Shabana
Keyword(s):  
Finite Element ◽  
Multibody Dynamics ◽  
Multibody Systems ◽  
Absolute Nodal Coordinate Formulation ◽  
Flexible Multibody Dynamics ◽  
Flexible Multibody Systems ◽  
Large Displacement ◽  
Absolute Nodal Coordinate ◽  
Flexible Multibody ◽  
Finite Element Formulations

Abstract In this investigation, the performance of two different large displacement finite element formulations in the analysis of flexible multibody systems is investigated. These are the incremental corotational procedure proposed by Rankin and Brogan [8] and the non-incremental absolute nodal coordinate formulation recently proposed [9]. It is demonstrated in this investigation that the limitation resulting from the use of the nodal rotations in the incremental corotational procedure can lead to simulation problems even when very simple flexible multibody applications are considered.

On First-Order Decoupling of Equations of Motion for Constrained Dynamical Systems

1993 ◽  
Author(s):  
Timothy A. Loduha ◽  
Bahram Ravani
Keyword(s):  
Dynamical Systems ◽  
Equations Of Motion ◽  
Relaxation Method ◽  
Nonholonomic Systems ◽  
Dynamical Equations ◽  
Generalized Coordinates ◽  
First Order ◽  
Constraint Relaxation ◽  
Constrained Dynamical Systems ◽  
Generalized Coordinate

Abstract In this paper we present a method for obtaining first-order decoupled equations of motion for multi-rigid body systems. The inherent flexibility in choosing generalized velocity components as a function of generalized coordinates is used to influence the structure of the resulting dynamical equations. Initially, we describe how a congruency transformation can be formed that represents the transformation between generalized velocity components and generalized coordinate derivatives. It is shown that the proper choice for the congruency transformation will insure generation of first-order decoupled equations of motion for holonomic systems. In the case of nonholonomic systems, or more complex dynamical systems, where the appropriate congruency transformation may be difficult to obtain, we present a constraint relaxation method based on the use of orthogonal complements. The results are illustrated using several examples. Finally, we discuss numerical implementation of congruency transformations to achieve first-order decoupled equations for simulation purposes.

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