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 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.

Direct Differentiation Methods for the Design Sensitivity of Multibody Dynamic Systems

1996 ◽  
Author(s):  
Radu Serban ◽  
Jeffrey S. Freeman
Keyword(s):  
Sensitivity Analysis ◽  
Dynamic Systems ◽  
Design Sensitivity ◽  
Differential Algebraic Equations ◽  
Mechanism Analysis ◽  
Algebraic Equations ◽  
First Order ◽  
Direct Differentiation ◽  
Multibody Dynamic ◽  
Standard Techniques

Abstract Methods for formulating the first-order design sensitivity of multibody systems by direct differentiation are presented. These types of systems, when formulated by Euler-Lagrange techniques, are representable using differential-algebraic equations (DAE). The sensitivity analysis methods presented also result in systems of DAE’s which can be solved using standard techniques. Problems with previous direct differentiation sensitivity analysis derivations are highlighted, since they do not result in valid systems of DAE’s. This is shown using the simple pendulum example, which can be analyzed in both ODE and DAE form. Finally, a slider-crank example is used to show application of the method to mechanism analysis.

scholarly journals New efficiency algorithm for solving fractional order differential-algebraic system

2016 ◽  
Vol 5 (3) ◽  
pp. 152
Author(s):  
Sameer Hasan ◽  
Eman Namah
Keyword(s):  
Exact Solution ◽  
Fractional Order ◽  
Lagrange Multiplier ◽  
Iteration Method ◽  
Algebraic System ◽  
Analytic Solution ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Differential Algebraic System ◽  
Fractional Differential

This work provided the evolution of the algorithm for analytic solution of system of fractional differential-algebraic equations (FDAEs).The algorithm referred to good effective method for combination the Laplace Iteration method with general Lagrange multiplier (LLIM). Through this method we have reached excellent results in comparison with exact solution as we illustrated in our examples.

scholarly journals L-Stable Method for Differential-Algebraic Equations of Multibody System Dynamics

2019 ◽  
Vol 2019 ◽  
pp. 1-11
Author(s):  
Bowen Li ◽  
Jieyu Ding ◽  
Yanan Li
Keyword(s):  
System Dynamics ◽  
Multibody System ◽  
Differential Algebraic Equations ◽  
Multibody System Dynamics ◽  
Dynamics Simulation ◽  
Time Interval ◽  
Algebraic Equations ◽  
Stable Method ◽  
Long Term Conditions ◽  
Equidistant Nodes

An L-stable method over time intervals for differential-algebraic equations (DAEs) of multibody system dynamics is presented in this paper. The solution format is established based on equidistant nodes and nonequidistant nodes such as Chebyshev nodes and Legendre nodes. Based on Ehle’s theorem and conjecture, the unknown matrix and vector in the L-stable solution formula are obtained by comparison with Pade approximation. Newton iteration method is used during the solution process. Taking the planar two-link manipulator system as an example, the results of L-stable method presented are compared for different number of nodes in the time interval and the step size in the simulation, and also compared with the classic Runge-Kutta method, A-stable method, Radau IA, Radau IIA, and Lobatto IIIC methods. The results show that the method has the advantages of good stability and high precision and is suitable for multibody system dynamics simulation under long-term conditions.

Multibody System Software Used for Research of Car Suspension System Dynamics

2014 ◽  
Vol 1036 ◽  
pp. 794-799 ◽  
Author(s):  
Łukasz Konieczny ◽  
Rafał Burdzik ◽  
Piotr Folęga
Keyword(s):  
System Dynamics ◽  
Modular Design ◽  
Differential Algebraic Equations ◽  
Suspension System ◽  
Motor Vehicles ◽  
Algebraic Equations ◽  
System Software ◽  
Lagrange Equations ◽  
Car Suspension ◽  
The Impact

The paper presents results of investigation of car suspension system dynamics. In this research the multibody (Multi Body System - MBS) system software MSC.Adams was used. ADAMS software (MSC.Software) is a commercial software to build a multibody structural models. Modular design allows for the usage of applications with different focuses, such as rail, aviation and motor vehicles. Models with a large number of freedom degrees of the components are built with mass concentrated on the assumption that the system is composed of a rigid (or deformable) bodies combined in a specific way (connection spherical, sliding, rotary), moving under the action of the forces and moments of different types (concentrated or distributed forces, the forces of contact). The complex multibody systems are automatically generated by the Lagrange equations of motion of the second kind in absolute coordinates. Integral procedures used to solve the differential-algebraic equations include multistep algorithms with variable row and a variable-and fixed-step and one-step algorithms. The Adams/Car module enables building and simulation-based examination of individual car subsystems such as, for instance, the suspension, steering or driving system as well as their combinations forming a complete car. The programme contains an extensive library of structural solutions applied in cars. The geometry and relationship data of individual components are stored in libraries, and software operation on a standard user level can be brought down to defining positions of constraints in space. The software is compatible with various CAD programmes, thus enabling import of elements created in other applications. The study was conducted for the vehicle model of Fiat Seicento. The examined system of the complete vehicle consists of 49 kinematic degrees of freedom. The article examined the impact of chosen parameters on vehicle vibration in an Adams Car Ride. Used in simulation Adams/Car /Ride module allows to test vehicle dynamics forcing the position of the plate of test stand . Virtual model of the vehicle is set on four servo-motors. They can control any excitation combination of individual actuators (dispalcement and amplitude, phase between extortion, etc.) and determine all kinds of vibration (vertical, lateral, angular). The study was conducted for selected parameters of the test rig.

Simulation of Constrained Mechanical Systems—Part II: Explicit Numerical Integration

2012 ◽  
Vol 79 (4) ◽  
Author(s):  
David J. Braun ◽  
Michael Goldfarb
Keyword(s):  
Mechanical Systems ◽  
Nonholonomic Constraints ◽  
Direct Consequence ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Constrained Motion ◽  
Constrained Mechanical Systems ◽  
Holonomic And Nonholonomic Constraints ◽  
Long Time ◽  
Constraint Correction

This paper presents an explicit to integrate differential algebraic equations (DAEs) method for simulations of constrained mechanical systems modeled with holonomic and nonholonomic constraints. The proposed DAE integrator is based on the equation of constrained motion developed in Part I of this work, which is discretized here using explicit ordinary differential equation schemes and applied to solve two nontrivial examples. The obtained results show that this integrator allows one to precisely solve constrained mechanical systems through long time periods. Unlike many other implicit DAE solvers which utilize iterative constraint correction, the presented DAE integrator is explicit, and it does not use any iteration. As a direct consequence, the present formulation is simple to implement, and is also well suited for real-time applications.

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