# Differential-algebraic Equations (differential-algebraic + equation)

Distribution by Scientific Domains

## Selected Abstracts

### Strategies for the numerical integration of DAE systems in multibody dynamics

COMPUTER APPLICATIONS IN ENGINEERING EDUCATION, Issue 2 2004
E. Pennestrì
Abstract The number of multibody dynamics courses offered in the university is increasing. Often the instructor has the necessity to go through the steps of an algorithm by working out a simple example. This gives the student a better understand of the basic theory. This paper provides a tutorial on the numerical integration of differential-algebraic equations (DAE) arising from the dynamic modeling of multibody mechanical systems. In particular, some algorithms based on the orthogonalization of the Jacobian matrix are herein discussed. All the computational steps involved are explained in detail and by working out a simple example. It is also reported a brief description and an application of the multibody code NumDyn3D which uses the Singular Value Decomposition (SVD) approach. © 2004 Wiley Periodicals, Inc. Comput Appl Eng Educ 12: 106,116, 2004; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20005 [source]

### Rigid body dynamics in terms of quaternions: Hamiltonian formulation and conserving numerical integration

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2009
Peter Betsch
Abstract In the present paper unit quaternions are used to describe the rotational motion of a rigid body. The unit-length constraint is enforced explicitly by means of an algebraic constraint. Correspondingly, the equations of motion assume the form of differential-algebraic equations (DAEs). A new route to the derivation of the mass matrix associated with the quaternion formulation is presented. In contrast to previous works, the newly proposed approach yields a non-singular mass matrix. Consequently, the passage to the Hamiltonian framework is made possible without the need to introduce undetermined inertia terms. The Hamiltonian form of the DAEs along with the notion of a discrete derivative make possible the design of a new quaternion-based energy,momentum scheme. Two numerical examples demonstrate the performance of the newly developed method. In this connection, comparison is made with a quaternion-based variational integrator, a director-based energy,momentum scheme, and a momentum conserving scheme relying on the discretization of the classical Euler's equations. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### Numerical method to solve chemical differential-algebraic equations

INTERNATIONAL JOURNAL OF QUANTUM CHEMISTRY, Issue 5 2002
Ercan Çelik
Abstract In this article, the solution of a chemical differential-algebraic equation model of general type F(y, y,, x) = 0 has been done using MAPLE computer algebra systems. The MAPLE program is given in the Appendix. First we calculate the Power series of the given equations system, then we transform it into Padé series form, which gives an arbitrary order for solving chemical differential-algebraic equation numerically. © 2002 Wiley Periodicals, Inc. Int J Quantum Chem, 2002 [source]

### dsoa: The implementation of a dynamic system optimization algorithm

OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 3 2010
Brian C. Fabien
Abstract This paper describes the ANSI C/C++ computer program dsoa, which implements an algorithm for the approximate solution of dynamics system optimization problems. The algorithm is a direct method that can be applied to the optimization of dynamic systems described by index-1 differential-algebraic equations (DAEs). The types of problems considered include optimal control problems and parameter identification problems. The numerical techniques are employed to transform the dynamic system optimization problem into a parameter optimization problem by: (i) parameterizing the control input as piecewise constant on a fixed mesh, and (ii) approximating the DAEs using a linearly implicit Runge-Kutta method. The resultant nonlinear programming (NLP) problem is solved via a sequential quadratic programming technique. The program dsoa is evaluated using 83 nontrivial optimal control problems that have appeared in the literature. Here we compare the performance of the algorithm using two different NLP problem solvers, and two techniques for computing the derivatives of the functions that define the problem. Copyright © 2009 John Wiley & Sons, Ltd. [source]

### Direct optimization of dynamic systems described by differential-algebraic equations

OPTIMAL CONTROL APPLICATIONS AND METHODS, Issue 6 2008
Brian C. Fabien
Abstract This paper presents a method for the optimization of dynamic systems described by index-1 differential-algebraic equations (DAE). The class of problems addressed include optimal control problems and parameter identification problems. Here, the controls are parameterized using piecewise constant inputs on a grid in the time interval of interest. In addition, the DAE are approximated using a Rosenbrock,Wanner (ROW) method. In this way the infinite-dimensional optimal control problem is transformed into a finite-dimensional nonlinear programming problem (NLP). The NLP is solved using a sequential quadratic programming (QP) technique that minimizes the L, exact penalty function, using only strictly convex QP subproblems. This paper shows that the ROW method discretization of the DAE leads to (i) a relatively small NLP problem and (ii) an efficient technique for evaluating the function, constraints and gradients associated with the NLP problem. This paper also investigates a state mesh refinement technique that ensures a sufficiently accurate representation of the optimal state trajectory. Two nontrivial examples are used to illustrate the effectiveness of the proposed method. Copyright © 2008 John Wiley & Sons, Ltd. [source]

### Dynamics and control of underactuated mechanical systems: analysis and simple experimental verification

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2009
Wojciech Blajer
Underactuated mechanical systems are systems with fewer control inputs than the degrees of freedom, m < n, the relevant technical examples being e.g. cranes, aircrafts and flexible manipulators. The determination of an input control strategy that forces an underactuated system to complete a set of m specified motion tasks (servo-constraints) is a demanding problem. The solution is conditioned to differential flatness of the problem, denoted that all 2n state variables and m control inputs can algebraically be expressed, at least theoretically, in terms of the desired m outputs and their time derivatives up to a certain order. A more practical formulation, motivated hereafter, is to pose the problem as a set of differential-algebraic equations, and then obtain the solution numerically. The theoretical considerations are illustrated by a simple two-degree-of-freedom underactuated system composed of two rotating discs connected by a flexible rod (torsional spring), in which the pre-specified motion of the first disc is actuated by the torque applied to the second disc, n = 2 and m = 1. The determined control strategy is then verified experimentally on a laboratory stand representing the two-disc system. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Numerical integration of differential-algebraic equations with mixed holonomic and control constraints

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2008
Mahmud Quasem
The present work aims at the incorporation of control (or servo) constraints into finite,dimensional mechanical systems subject to holonomic constraints. In particular, we focus on underactuated systems, defined as systems in which the number of degrees of freedom exceeds the number of inputs. The corresponding equations of motion can be written in the form of differential,algebraic equations (DAEs) with a mixed set of holonomic and control constraints. Apart from closed,loop multibody systems, the present formulation accommodates the so,called rotationless formulation of multibody dynamics. To this end, we apply a specific projection method to the DAEs in terms of redundant coordinates. A similar projection approach has been previously developed in the framework of generalized coordinates by Blajer & Ko,odziejczyk [1]. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]

### Energy consistent time integration of planar multibody systems

PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2006
Stefan Uhlar
The planar motion of rigid bodies and multibody systems can be easily described by coordinates belonging to a linear vector space. This is due to the fact that in the planar case finite rotations commute. Accordingly, using this type of generalized coordinates can be considered as canonical description of planar multibody systems. However, the extension to the three-dimensional case is not straightforward. In contrast to that, employing the elements of the direction cosine matrix as redundant coordinates makes possible a straightforward treatment of both planar and three-dimensional multibody systems. This alternative approach leads in general to differential-algebraic equations (DAEs) governing the dynamics of rigid body systems. The main purpose of the present paper is to present a comparison of the two alternative descriptions. In both cases energy-consistent time integration schemes are applied. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source]