scholarly journals Automated Linearization of Nonlinear Coupled Differential and Algebraic Equations

1991 ◽  
Author(s):  
Martin J. Vanderploeg ◽  
Jeff D. Trom
Keyword(s):  
Finite Difference ◽  
Dynamic Systems ◽  
Large Scale ◽  
Algebraic Equations ◽  
New Approach ◽  
Large Scale Systems ◽  
Equation Formulation ◽  
Equilibrium Configurations ◽  
Multi Body ◽  
Body Dynamic

Abstract This paper presents a new approach for linearization of large multi-body dynamic systems. The approach uses an analytical differentiation of terms evaluated in a numerical equation formulation. This technique is more efficient than finite difference and eliminates the need to determine finite difference pertubation values. Because the method is based on a relative coordinate formalism, linearizations can be obtained for equilibrium configurations with non-zero Cartesian accelerations. Examples illustrate the accuracy and efficiency of the algorithm, and its ability to compute linearizations for large-scale systems that were previously impossible.

scholarly journals Automated Linearization of Nonlinear Coupled Differential and Algebraic Equations

1994 ◽  
Vol 116 (2) ◽  
pp. 429-436 ◽  
Author(s):  
J. D. Trom ◽  
M. J. Vanderploeg
Keyword(s):  
Finite Difference ◽  
Dynamic Systems ◽  
Large Scale ◽  
Algebraic Equations ◽  
New Approach ◽  
Large Scale Systems ◽  
Equation Formulation ◽  
Relative Coordinate ◽  
Equilibrium Configurations ◽  
Multibody Dynamic

This paper presents a new approach for linearization of large multibody dynamic systems. The approach uses an analytical differentiation of terms evaluated in a numerical equation formulation. This technique is more efficient than finite difference and eliminates the need to determine finite difference pertubation values. Because the method is based on a relative coordinate formalism, linearizations can be obtained for equilibrium configurations with non-zero Cartesian accelerations. Examples illustrate the accuracy and efficiency of the algorithm, and its ability to compute linearizations for large-scale systems that were previously impossible.

Floating Time Algorithm for Time Optimal Control of Multi-Body Dynamic Systems

2005 ◽  
Vol 219 (3) ◽  
pp. 225-236 ◽  
Author(s):  
G Nakhaie Jazar ◽  
A Naghshineh-Pour
Keyword(s):  
Optimal Control ◽  
Dynamic System ◽  
Dynamic Systems ◽  
Time Optimal Control ◽  
Coordinate Space ◽  
Algebraic Equations ◽  
Initial State ◽  
Time Optimal ◽  
Multi Body ◽  
Body Dynamic

Moving a dynamic system in minimum time from a given initial state to a desired final state on a prescribed path is one of the oldest and most enduring technological dreams of the scientific and industrial communities. In this research, the problem of bounded-input time optimal control for applied multi-body dynamic systems subject to a full nonlinear dynamical model is solved. To solve the problem, an innovative method, called the ‘floating-time’ method is introduced and utilized. Compared to traditional methods, the floating-time method is an applied method not based on variational calculus. It can be applied to the full nonlinear model of the dynamical system and can handle static and dynamic constraints defined by differential or algebraic equations. The problem of time optimal control is as follows. Find the control law of bounded inputs that drive a given multi-body dynamic system (such as the gripper of a manipulator) along a pre-specified trajectory (in either configuration space or generalized coordinate space) from a given initial position to a given final position, minimizing the time of the motion as a performance index. Using variable time increments, the equations of motion of the system will be reduced to a set of algebraic equations. Searching for a set of time increments (floating-times) that make the equations to exert the maximum available effort produces the minimum possible floating-times, and minimizes the total time of motion. The applicability of the method will be shown by using three examples: a point mass sliding on a rough surface, a 2R robotic manipulator, and the well-known Brachistochrone.

scholarly journals Chebyshev Wavelet Finite Difference Method: A New Approach for Solving Initial and Boundary Value Problems of Fractional Order

2013 ◽  
Vol 2013 ◽  
pp. 1-15 ◽  
Author(s):  
A. Kazemi Nasab ◽  
A. Kılıçman ◽  
Z. Pashazadeh Atabakan ◽  
S. Abbasbandy
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Finite Difference Methods ◽  
Algebraic Equations ◽  
Chebyshev Wavelets ◽  
New Approach ◽  
Nonlinear Algebraic Equations ◽  
Chebyshev Wavelet ◽  
Fractional Differential

A new method based on a hybrid of Chebyshev wavelets and finite difference methods is introduced for solving linear and nonlinear fractional differential equations. The useful properties of the Chebyshev wavelets and finite difference method are utilized to reduce the computation of the problem to a set of linear or nonlinear algebraic equations. This method can be considered as a nonuniform finite difference method. Some examples are given to verify and illustrate the efficiency and simplicity of the proposed method.

scholarly journals A PARALLEL MULTIRATE ALGORITHM FOR THE NUMERICAL INTEGRATION OF SYSTEM OF NONLINEAR DIFFERENTIAL EQUATIONS

2014 ◽  
pp. 34-41
Author(s):  
Petro Stakhiv ◽  
Serhiy Rendzinyak
Keyword(s):  
Differential Equations ◽  
Numerical Integration ◽  
Dynamic Behavior ◽  
Large Scale ◽  
Nonlinear Differential Equations ◽  
New Approach ◽  
Large Scale Systems ◽  
Computing Process ◽  
Parallelization Efficiency

The new approach to calculate dynamic behavior of large-scale systems, separated on subsystems is presented. Parallelization efficiency of computing process is described.

Concept Of The Decomposition And Aggregation Method With Example: Part 1

1988 ◽  
pp. 27-40
Author(s):  
Dr. Zainol Anuar Mohd. Sharif ◽  
Ng Boon Choong
Keyword(s):  
Stability Analysis ◽  
Dynamic Systems ◽  
Basic Concept ◽  
Large Scale ◽  
Direct Method ◽  
Aggregation Method ◽  
Control Engineering ◽  
Large Scale Systems ◽  
Engineering Economics ◽  
The Stability

This paper describes the basic concept of the decomposition and aggregation method. It shows the feasibility of the method and its advantages when applied, particularly to large scale systems. This method is extensively used in solving problems related to control engineering, economics, optimization and stability. This paper also illustrates specifically the application of the method of decomposition and aggregation in the analysis of dynamic systems. It is divided into two important parts, namely; the decomposition part which involves breaking up a large system into subsystems and the aggregation part which is obtained through a reformulation of the Liapunov's second method (direct method). The relation between the decomposition and the aggregation methods is also shown. The procedure for checking the stability based on this concept is also outlined.For further illustration, an example of a dynamic system has been included. It shows how the system is decomposed and aggregated to suit the requirement for stability analysis.

Variational Analysis of a Two Link Slider-Crank Mechanism Using Polynomial Chaos Theory

2017 ◽  
Author(s):  
Paul S. Ryan ◽  
Sarah C. Baxter ◽  
Philip A. Voglewede
Keyword(s):  
Chaos Theory ◽  
Closed Loop ◽  
Polynomial Chaos ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Advantages And Disadvantages ◽  
Multi Body ◽  
Polynomial Chaos Theory ◽  
Body Dynamic ◽  
Crank Mechanism

Variation occurs in many closed loop multi-body dynamic (MBD) systems in the geometry, mass, or forces. Understanding how MBD systems respond to variation is imperative for the design of a robust system. However, simulation of how variation propagates into the solution is complicated as most MBD systems cannot be simplified into to a system of ordinary differential equations (ODE). This paper investigates polynomial chaos theory (PCT) as a means of quantifying the effects of uncertainty in a closed loop MBD system governed by differential algebraic equations (DAE). To demonstrate how PCT could be used, the motion of a two link slider-crank mechanism is simulated with variation in the link lengths. To validate and show the advantages and disadvantages of PCT in closed loop MBD systems, the PCT approach is compared to Monte Carlo simulations.

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