Rosenbrock-Nystrom Integrator for SSODE of Mechanical Systems

2003 ◽  
Author(s):  
Adrian Sandu ◽  
Dan Negrut ◽  
Corina Sandu ◽  
Edward J. Haug ◽  
Florian A. Potra
Keyword(s):  
Mechanical System ◽  
Time Evolution ◽  
Degrees Of Freedom ◽  
State Of The Art ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Generalized Coordinates ◽  
System A ◽  
Analytical Derivatives

When performing dynamic analysis of a constrained mechanical system, a set of index 3 Differential-Algebraic Equations (DAE) describes the time evolution of the system. The paper presents a state-space based method for the numerical solution of the resulting DAE. A subset of so called independent generalized coordinates, equal in number to the number of degrees of freedom of the mechanical system, is used to express the time evolution of the mechanical system. The second order statespace ordinary differential equations (SSODE) that describe the time variation of independent coordinates are numerically integrated using a Rosenbrock type formula. For stiff mechanical systems, the proposed algorithm is shown to significantly reduce simulation times when compared to state of the art existent algorithms. The better efficiency is due to the use of an L-stable integrator and a rigorous and general approach to providing analytical derivatives required by it.

scholarly journals A Review on the Control of Second Order Underactuated Mechanical Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-17 ◽  
Author(s):  
Soukaina Krafes ◽  
Zakaria Chalh ◽  
Abdelmjid Saka
Keyword(s):  
Mathematical Modeling ◽  
Mechanical System ◽  
Degrees Of Freedom ◽  
State Of The Art ◽  
Control Strategies ◽  
Mechanical Systems ◽  
Future Research ◽  
Underactuated Mechanical Systems ◽  
Underactuated Mechanical System ◽  
Year 2000

This paper describes some important classes of two degrees of freedom of underactuated mechanical system and also surveys review of the recent state-of-the-art concerning the mathematical modeling of these systems, their classification, and all the control strategies (linear, nonlinear, and intelligent) that have been made so far (i.e., from the year 2000 to date) to control these systems. Future research and challenges concerning the improvement, the effectiveness, and robustness of the proposed controllers for underactuated mechanical systems are presented.

State-Space Based Implicit Integration of the Differential-Algebraic Equations of Multibody Dynamics

1999 ◽  
Author(s):  
Dan Negrut ◽  
Edward J. Haug
Keyword(s):  
State Space ◽  
Mechanical System ◽  
Multibody Dynamics ◽  
Reference Frames ◽  
Differential Algebraic Equations ◽  
The State ◽  
Algebraic Equations ◽  
Implicit Integration ◽  
Euler Parameters ◽  
Generalized Coordinates

Abstract Three methods for the state-space based implicit integration of differential-algebraic equations of multibody dynamics are summarized and numerically compared. In the state-space approach, the time evolution of a mechanical system is characterized using a number of generalized coordinates equal with the number of degrees of freedom of the system. In this paper these independent generalized coordinates are a subset of the Cartesian position coordinates and orientation Euler parameters of body centroidal reference frames. Depending on the method, the independent generalized coordinates are implicitly integrated and dependent quantities (including Lagrange multipliers) are determined to satisfy constraint equations at position, velocity, and acceleration levels. Five computational algorithms based on the proposed methods are used to simulate the motion of a stiff 14-body vehicle model. Results show that the proposed methods deal effectively with challenges posed by stiff mechanical system simulation. A comparison with a state-space based explicit algorithm for the simulation of the same model indicates a speed-up of approximately two orders of magnitude.

Building Gaussian Process Based Metamodels Using Variable-Fidelity Experiments for Dynamic Analysis of Mechanical Systems

2007 ◽  
Author(s):  
Dan Negrut ◽  
Z. G. Peter Qian ◽  
Naresh Khude
Keyword(s):  
Time Evolution ◽  
Multibody Systems ◽  
Random Function ◽  
Mechanical Systems ◽  
Time Domain Analysis ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Highly Nonlinear ◽  
Variable Fidelity ◽  
Open Question

Given the highly nonlinear attribute of the underlying dynamics associated with the time evolution of multibody systems, an open question in mechanical system simulation is how one can reliably replace a model whose simulation is time consuming with a more expeditious one. Pushing this idea to the limit one can all together eliminate the dynamics of the problem using a set of simulations that train a predictor that is later used to provide the time evolution of the dynamic system. This paper investigates a Gaussian Random Function (GRF) based approach that attempts to address these questions. It relies on a framework recently proposed in the Statistical Analysis community that largely deals with the issues of model validation, calibration, and data integration. The approach investigated has several steps that are illustrated with a slider-crank mechanical systems whose time evolution is governed by a nonlinear set of index 3 Differential Algebraic Equations (DAEs). The paper concludes with a set of remarks on the potential of GRFs in the context of time domain analysis of mechanical systems.

scholarly journals Assessment of Linearization Approaches for Multibody Dynamics Formulations

2017 ◽  
Vol 12 (4) ◽  
Author(s):  
Francisco González ◽  
Pierangelo Masarati ◽  
Javier Cuadrado ◽  
Miguel A. Naya
Keyword(s):  
Mechanical System ◽  
Multibody Dynamics ◽  
Equations Of Motion ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Practical Applications ◽  
Linearization Methods ◽  
Highly Nonlinear ◽  
Wide Range ◽  
The Way

Formulating the dynamics equations of a mechanical system following a multibody dynamics approach often leads to a set of highly nonlinear differential-algebraic equations (DAEs). While this form of the equations of motion is suitable for a wide range of practical applications, in some cases it is necessary to have access to the linearized system dynamics. This is the case when stability and modal analyses are to be carried out; the definition of plant and system models for certain control algorithms and state estimators also requires a linear expression of the dynamics. A number of methods for the linearization of multibody dynamics can be found in the literature. They differ in both the approach that they follow to handle the equations of motion and the way in which they deliver their results, which in turn are determined by the selection of the generalized coordinates used to describe the mechanical system. This selection is closely related to the way in which the kinematic constraints of the system are treated. Three major approaches can be distinguished and used to categorize most of the linearization methods published so far. In this work, we demonstrate the properties of each approach in the linearization of systems in static equilibrium, illustrating them with the study of two representative examples.

An Accurate Spatial Discretization and Substructural Method With Application to Moving Elevator Cable-Car Systems: Part I—Methodology

2011 ◽  
Author(s):  
H. Ren ◽  
W. D. Zhu
Keyword(s):  
Differential Algebraic Equations ◽  
Distributed Parameter ◽  
Algebraic Equations ◽  
Spatial Discretization ◽  
Matching Conditions ◽  
Generalized Coordinates ◽  
Axial Forces ◽  
Linear Algebraic Equations ◽  
Dependent Variables ◽  
Elevator Cable

A spatial discretization and substructure method is developed to calculate the dynamic responses of one-dimensional systems, which consist of length-variant distributed-parameter components such as strings, rods, and beams, and lumped-parameter components such as point masses and rigid bodies. The dependent variable, such as the displacement, of a distributed-parameter component is decomposed into boundary-induced terms and internal terms. The boundary-induced terms are interpolated from the boundary motions, and the internal terms are approximated by an expansion of trial functions that satisfy the corresponding homogeneous boundary conditions. All the matching conditions at the interfaces of the components are satisfied, and the expansions of the dependent variables of the distributed-parameter components absolutely and uniformly converge. The spatial derivatives of the dependent variables, which are related to the internal forces/moments, such as the axial forces, bending moments, and shear forces, can be accurately calculated. Assembling the component equations and the geometric matching conditions that arise from the continuity relations leads to a system of differential algebraic equations (DAEs). When some matching conditions are linear algebraic equations, some generalized coordinates can be represented by others so that the number of the generalized coordinates can be reduced. The methodology is applied to moving elevator cable-car systems in Part II of this work.

Trajectory Tracking of Underactuated Multibody Systems

2011 ◽  
Author(s):  
Stefan Reichl ◽  
Wolfgang Steiner
Keyword(s):  
Trajectory Tracking ◽  
Multibody Systems ◽  
Degrees Of Freedom ◽  
Inverse Dynamics ◽  
Feedforward Control ◽  
Equations Of Motion ◽  
Three Dimensional ◽  
Differential Algebraic Equations ◽  
State Variables ◽  
Algebraic Equations

This work presents three different approaches in inverse dynamics for the solution of trajectory tracking problems in underactuated multibody systems. Such systems are characterized by less control inputs than degrees of freedom. The first approach uses an extension of the equations of motion by geometric and control constraints. This results in index-five differential-algebraic equations. A projection method is used to reduce the systems index and the resulting equations are solved numerically. The second method is a flatness-based feedforward control design. Input and state variables can be parameterized by the flat outputs and their time derivatives up to a certain order. The third approach uses an optimal control algorithm which is based on the minimization of a cost functional including system outputs and desired trajectory. It has to be distinguished between direct and indirect methods. These specific methods are applied to an underactuated planar crane and a three-dimensional rotary crane.

scholarly journals Numerical solution for consistent initial conditions of constrained mechanical systems

2003 ◽  
Vol 25 (3) ◽  
pp. 170-185
Author(s):  
Dinh Van Phong
Keyword(s):  
Numerical Solution ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
System Of Equations ◽  
Algebraic Equations ◽  
Flexible Approach ◽  
Constrained Mechanical Systems ◽  
Consistent Initial Values

The article deals with the problem of consistent initial values of the system of equations of motion which has the form of the system of differential-algebraic equations. Direct treating the equations of mechanical systems with particular properties enables to study the system of DAE in a more flexible approach. Algorithms and examples are shown in order to illustrate the considered technique.

Well-Posed Formulations for Nonholonomic Mechanical System Dynamics

2020 ◽  
Vol 15 (10) ◽  
Author(s):  
Edward J. Haug
Keyword(s):  
System Dynamics ◽  
Mechanical System ◽  
Lagrange Multiplier ◽  
Variational Formulation ◽  
Direct Consequence ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
First Order ◽  
Well Posed ◽  
Problem Data

Abstract Four formulations of nonholonomic mechanical system dynamics, with both holonomic and differential constraints, are presented and shown to be well posed; i.e., solutions exist, are unique, and depend continuously on problem data. They are (1) the d'Alembert variational formulation, (2) a broadly applicable manifold theoretic extension of Maggi's equations that is a system of first-order ordinary differential equations (ODE), (3) Lagrange multiplier-based index 3 differential-algebraic equations (index 3 DAE), and (4) Lagrange multiplier-based index 0 differential-algebraic equations (index 0 DAE). The ODE formulation is shown to be well posed, as a direct consequence of the theory of ODE. The variational formulation is shown to be equivalent to the ODE formulation, hence also well posed. Finally, the index 3 DAE and index 0 DAE formulations are shown to be equivalent to the variational and ODE formulations, hence also well posed. These results fill a void in the literature and provide a theoretical foundation for nonholonomic mechanical system dynamics that is comparable to the theory of ODE.

scholarly journals On the integrable deformations of a system related to the motion of two vortices in an ideal incompressible fluid

2019 ◽  
Vol 29 ◽  
pp. 01015 ◽  
Author(s):  
Cristian Lăzureanu ◽  
Ciprian Hedrea ◽  
Camelia Petrişor
Keyword(s):  
Incompressible Fluid ◽  
Mechanical System ◽  
Integrable Systems ◽  
Degrees Of Freedom ◽  
Mechanical Systems ◽  
Ideal Incompressible Fluid ◽  
First Integrals ◽  
Integrable Deformation ◽  
Integrable Deformations ◽  
The Given

Altering the first integrals of an integrable system integrable deformations of the given system are obtained. These integrable deformations are also integrable systems, and they generalize the initial system. In this paper we give a method to construct integrable deformations of maximally superintegrable Hamiltonian mechanical systems with two degrees of freedom. An integrable deformation of a maximally superintegrable Hamiltonian mechanical system preserves the number of first integrals, but is not a Hamiltonian mechanical system, generally. We construct integrable deformations of the maximally superintegrable Hamiltonian mechanical system that describes the motion of two vortices in an ideal incompressible fluid, and we show that some of these integrable deformations are Hamiltonian mechanical systems too.

A Geometric Derivation of the Equations of Motion for Mechanical Systems With Scleronomic Constraints

2015 ◽  
Author(s):  
Sotirios Natsiavas ◽  
Elias Paraskevopoulos
Keyword(s):  
Numerical Methods ◽  
Lagrange Multipliers ◽  
Equations Of Motion ◽  
Mechanical Systems ◽  
Main Idea ◽  
Constraint Equation ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Motion Constraints ◽  
Stiffness Properties

A new set of equations of motion is presented for a class of mechanical systems subjected to equality motion constraints. Specifically, the systems examined satisfy a set of holonomic and/or nonholonomic scleronomic constraints. The main idea is to consider the equations describing the action of the constraints as an integral part of the overall process leading to the equations of motion. The constraints are incorporated one by one, in a process analogous to that used for setting up the equations of motion. This proves to be equivalent to assigning appropriate inertia, damping and stiffness properties to each constraint equation and leads to a system of second order ordinary differential equations for both the coordinates and the Lagrange multipliers associated to the motion constraints automatically. This brings considerable advantages, avoiding problems related to systems of differential-algebraic equations or penalty formulations. Apart from its theoretical value, this set of equations is well-suited for developing new robust and accurate numerical methods.

Sign in / Sign up

Export Citation Format

Share Document