Stability of Linear Mechanical Systems With Holonomic Constraints

1993 ◽  
Vol 46 (11S) ◽  
pp. S160-S164 ◽  
Author(s):  
Peter C. Mu¨ller
Keyword(s):  
Singular Systems ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
Descriptor Systems ◽  
Algebraic Equations ◽  
Holonomic Constraints ◽  
Lyapunov Matrix Equation ◽  
Lyapunov Matrix ◽  
Complete Set ◽  
And Control

Singular systems (descriptor systems, differential-algebraic equations) are a recent topic of research in numerical mathematics, mechanics and control theory as well. But compared with common methods available for investigating regular systems many problems still have to be solved making also available a complete set of tools to analyze, to design and to simulate singular systems. In this contribution the aspect of stability is considered. Some new results for linear singular systems are presented based on a generalized Lyapunov matrix equation. Particularly, for mechanical systems with holonomic constraints the well-known stability theorem of Thomson and Tait is generalized.

scholarly journals Estimation and Global Stability Analysis of PDAEs with Singular Time Derivative Matrix and Wetland Conservation Application

2016 ◽  
Vol 2016 ◽  
pp. 1-10 ◽  
Author(s):  
Yushan Jiang ◽  
Qingling Zhang
Keyword(s):  
Global Stability ◽  
Singular Systems ◽  
Time Derivative ◽  
Estimation Method ◽  
Differential Algebraic Equations ◽  
Wetland Conservation ◽  
Descriptor System ◽  
Algebraic Equations ◽  
Energy Estimation ◽  
Singular Time

Some partial differential algebraic equations (PDAEs) system with singular time derivative matrices is analyzed. First, by PDE spectrum theory, this system is formulated as infinite-dimensional singular systems. Second, the state space and its properties of the system are built according to descriptor system theory. Third, the admissible property of the PDAEs is given via LMIs. Finally, the developed energy estimation method is proposed to investigate the global stability of PDAEs. The proposed approach is evaluated by an application in numerical simulations on some wetland conservation system with social behavior.

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.

Towards an Energy-Based Linear Analysis of Nonholonomic Systems

1999 ◽  
Author(s):  
Marwan Bikdash ◽  
Richard A. Layton
Keyword(s):  
Steady State ◽  
Singular Systems ◽  
Nonholonomic Constraints ◽  
Nonholonomic Systems ◽  
Linear Analysis ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Indirect Approach ◽  
Physical Systems ◽  
Future Work

Abstract Guidelines toward an energy-based, linear analysis of discrete physical systems are presented, based on previous work in systematic modeling using Lagrangian differential-algebraic equations (DAEs). Recent work in this area is extended by accommodating nonholonomic constraints and explicit inputs. An equilibrium postulate is proposed and equilibrium is characterized for static and steady-state conditions. Lagrangian DAEs are linearized using a local, indirect approach. Alternate descriptor formulations leading to different linear singular systems are compared and one formulation is determined to be a good foundation for future work in linear analysis using Lagrangian DAEs.

Estimation of Clutch Parameters for Online Diagnostics and Control Applications

2005 ◽  
Author(s):  
S. Joshi ◽  
J. Liu ◽  
S. Ananthakrishnan
Keyword(s):  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Torque Estimation ◽  
Clutch Engagement ◽  
Online Implementation ◽  
Engagement Angle ◽  
Shaft Angle ◽  
Clutch Control ◽  
And Control ◽  
Clutch Torque

The primary goal of this research was to identify the role of non-linear parameter estimation in clutch torque estimation for automotive applications. The benefit of this approach in estimating parameters of a system defined by a set of differential algebraic equations (DAEs) that represents, say, clutch torque profile is investigated. In addition, online implementation, albeit at a slow rate compared to control loop rates is demonstrated. This method of analytical DAE based estimation has significant advantages over purely empirical methods in that it directly identifies the relationships between system problem variables, such as, engine speed, cross shaft angle to variable of interest, say, clutch torque, unlike, indirect approaches [1]. In addition, in contrast with time series evolution of discrete system models based on ARMAX models, this approach allows a designer to know the relationship between system parameters such as friction coefficient, clutch engagement angle, etc and the estimation process leading to better design of clutch control algorithms. However, unlike, the direct digital control methods, DAE based approaches are computationally more intensive resulting in a need for additional onboard processing. One of the goals of this initial research is to study this to identify practical analytical and numerical approaches that will lead to onboard implementation of these algorithms for truck applications, specifically in automated mechanical transmissions (AMTs) [2, 3].

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.

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.

Force Equilibrium Approach for Linearization of Constrained Mechanical System Dynamics

2003 ◽  
Vol 125 (1) ◽  
pp. 143-149 ◽  
Author(s):  
Ju Seok Kang ◽  
Sangwoo Bae ◽  
Jang Moo Lee ◽  
Tae Oh Tak
Keyword(s):  
Mechanical Systems ◽  
Nonlinear Dynamic Analysis ◽  
Differential Algebraic Equations ◽  
Dynamic Equations ◽  
Small Displacement ◽  
Algebraic Equations ◽  
Small Motion ◽  
Constrained Mechanical Systems ◽  
External Forces ◽  
Force Equilibrium

The purpose of this study is to derive a linearized form of dynamic equations for constrained mechanical systems. The governing equations for constrained mechanical systems are generally expressed in terms of Differential-Algebraic Equations (DAEs). Conventional methods of linearization are based on the perturbation of the nonlinear DAE, where small amounts of perturbations are taken to guarantee linear characteristics of the equations. On the other hand, the proposed linearized dynamic equations are derived directly from a force equilibrium condition, not from the DAEs, with small motion assumption. This approach is straightforward and simple compared to conventional perturbation methods, and can be applicable to any constrained mechanical systems that undergo small displacement under external forces. The modeling procedure and formulation of linearized dynamic equations are demonstrated by the example of a vehicle suspension system, a typical constrained multibody system. The solution is validated by comparison with conventional nonlinear dynamic analysis and modal test results.

A Computer-Aided Environment for Analysis and Design of Mechanical Systems

1991 ◽  
Author(s):  
Hamid M. Lankarani ◽  
Behnam Bahr ◽  
Saeid Motavalli
Keyword(s):  
Equations Of Motion ◽  
Mechanical Systems ◽  
Differential Algebraic Equations ◽  
General Purpose ◽  
Design System ◽  
Integrated Analysis ◽  
Algebraic Equations ◽  
Efficient Design ◽  
Analysis And Design ◽  
Wide Range

Abstract This paper presents the description of an ideal tool for analysis and design of complex multibody mechanical systems. It is in the form of a general-purpose computer program, which can be used for simulation of many different systems. The generality of this computer-integrated environment allows a wide range of applications with significant engineering importance. No matter how complicated the mechanical system under consideration is, a numerical multibody model of the system is constructed. The governing mixed differential/algebraic equations of motion are automatically formulated and numerically generated. State-of-the-art numerical techniques and computational methods are employed and developed which produce in the response of the system at discrete time junctures. Postprocessing of the results in the form of graphical images or real-time animations provides an enormous aid in visualizing motion of the system. The analysis package may be merged with an efficient design optimization algorithm. The developed integrated analysis/design system is a valuable tool for researchers, design engineers, and analysts of mechanical systems. This computer-integrated tool provides an important bridge between the classical decision making process by an engineer and the emerging technology of computers.

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