Set Of Constraint Equations Research Articles

Comparison of solution strategies for multibody dynamics equations

AbstractIn the last decades, several different formalisms have been proposed in the literature in order to simulate the dynamics of multibody systems. First, the choice of the coordinates leads to very different formalisms. The set of constraint equations appears very different in complexity or in size depending on whether either relative, natural or reference point Cartesian coordinates are adopted. Second, once such a choice has been made, different solution strategies can be followed. It seems that formalisms based on redundant coordinates are often used in commercial software, whereas those based on a minimum number of coordinates are usually preferred in real‐time computations.In this paper, with reference to models with redundant absolute coordinates, the problem of coordinate reduction will be discussed and a group of 11 different methods will be compared on the basis of their computational efficiency. In particular, methods based on constraint orthogonalization and on the use of pseudoinverse matrices have been selected, together with other methods based on least‐squares block solution. The methods have been implemented on three different test cases. The main purpose is to provide hints and guidelines on the choice and availability of solution strategies during simulation of moderate size multibody systems. Copyright © 2011 John Wiley & Sons, Ltd.

Cumulants and symmetries in a trivariate normal mixture

An alternative parameterization of a trivariate normal mixture for the stellar velocity distribution provides a set of constraint equations between global cumulants that are used to estimate characteristic constants of the mixture and population parameters. Under particular distribution symmetries these relationships become simpler and easy to evaluate, and they are used to test some meaningful mixtures. This method for the analysis of trivariate normal mixtures is applied to local star catalogs with known velocity space in order to find Gaussian components and underlying distribution symmetries. A large local sample (13 678 stars, 300 pc) obtained from HIPPARCOS catalog allows us to identify two mixed velocity ellipsoids with parameters corresponding to thin and thick disk populations, similar to those of samples selected from non-kinematical criteria. In addition to the usual assumptions of distribution symmetry plane and non-significant differential movement in the radial direction, our analysis also detects a slight but clear loss of axial symmetry associated with the vertex deviation of both population ellipsoids.

An Optimal Feedforward Design for Complete PMD Compensation Up to the Second Order

As an extension to a previous paper, this paper describes the optimization of a second-order, feedforward polarization-mode dispersion (PMD) compensation scheme by reducing its degrees of freedom (DOF) by two. The new design is optimal in the sense that the number of DOF used is the same as the minimal number of DOF required. Also derived is a set of constraint equations that govern the choice of various system parameters.

On noncommutative Calabi–Yau hypersurfaces

Using the algebraic geometry method of Berenstein et al. [hep-th/0005087], we reconsider the derivation of the noncommutative quintic algebra Anc(5) and derive new representations by choosing different sets of Calabi–Yau charges {Cia}. Next we extend these results to higher d complex dimension noncommutative Calabi–Yau hypersurface algebras Anc(d+2). We derive and solve the set of constraint equations carrying the noncommutative structure in terms of Calabi–Yau charges and discrete torsion. Finally, we construct the representations of Anc(d+2) preserving manifestly the Calabi–Yau condition ∑iCia=0 and give comments on the noncommutative subalgebras.

An Efficient Dynamic Formulation for Multibody Systems

The objective of this article is to present an efficient extension ofRosenthal's order-n algorithm to multibody systems containing closedloops. The equations of motion are created by using relative coordinatesand partial velocity theory. Closed topological loops are handled by cutjoint technique. The set of constraint equations of cut joints isadjoined to the system's equation of motion by using Lagrangemultipliers. This results in the equation of motion as adifferential-algebraic equation (DAE) rather than an ordinarydifferential equation. This DAE is then solved by applying the extendedRosenthal's order-n algorithm proposed in this article. While solvingDAE, violation of the kinematic constraint equations of cut joints iscorrected by coordinate projection method. Some numerical simulationsare carried out to demonstrate efficiency of the proposed method.

Magnet system design with cost optimization

The problem of optimizing the design of magnet systems to simultaneously satisfy a set of constraint equations for field, geometry, protection, and stability while minimizing the total system cost is considered. A mathematical description of the problem is given, along with details concerning possible constraint algorithms and cost functions. Rules for the optimal selection of magnet technology (potted, ventilated, and cable-in-conduit coils) are presented. The resulting sets of equations can be solved by different numerical methods; the final choice is based on considerations of functional form as well as computational speed and accuracy. Depending on the choice of constraint equations, the resulting problem may or may not be smooth, and may or may not admit multiple minima. The choice of minimization algorithm can be tailored to the situation at hand to yield optimal performance. We present results pertaining to an alternating-field solenoid system being considered by the MuCool collaboration. The system design requires consideration of field profile, magnet stability, magnet structure, and protection, as well as total cost. A key application of the optimization code is the analysis of cost trade-off with respect to the imposed constraints, and examples are given in the alternating-solenoid case.

Redesigning the Dynamics of Structural Systems

The critical link between two powerful modeling techniques, component mode synthesis and large admissible perturbation theory, Is established. This new redesign process allows structural systems that are composed of several substructures to be redesigned. The process is developed in three steps. First, the analytical first-order perturbation relationships between the structural system eigenvalues and the eigenvalues of each independent substructure are developed. Second, a transformation matrix is developed for each perturbation relationship, then combined to form the ultimate structural transformation matrix. Third, a minimization problem is defined that uses the ultimate transformation matrix as a set of constraint equations. The result of this minimization is a set of changes to the substructure eigenvalues that approximately affect the objective changes to the structural system eigenvalues. The success of the redesign process is demonstrated by an isolated platform example. The changes in the substructure eigenvalues are predicted via the redesign process, then substituted back into the full nonlinear equations for the structural system, and the results are discussed. This work forms the basis for future redesign developments when other static and dynamic objectives are to be achieved for structural systems.

On the kinematic of a 3-DOF Stewart platform

The kinematic behavior of the Stewart platform has been discussed in the literature by many researchers. In the present article, a special form of the Stewart platform, namely, the 3-degrees of freedom platform with triangular shaped base and upper platforms, is considered by thoroughly deriving a new set of constraint equations which makes it possible to discover several new properties of the platform. Both the inverse and direct kinematic problems are discussed and a new approach for solution of the direct kinematic problem is described which gives all the real (physical) solutions. ©1999 John Wiley & Sons, Inc.

Modeling of multiple mobile manipulators handling a common deformable object

The dynamic equations of motion for a system of multiple mobile manipulators carrying a common deformable object are developed. The derived centralized model is based on Kane's approach to dynamics. The imposed kinematic nonholonomic constraints are included and incorporated into the dynamics. It is pointed out, however, that these kinematic constraints are only necessary, and they only become sufficient to impose nonholonomic motion when accompanied by the dynamic counterparts. Sufficient conditions for avoiding tipping over by the mechanisms are also provided. The whole set of constraint equations is analyzed and useful properties for the set of admissible solutions for the ground reaction forces are obtained. The deformable object under manipulation is modeled and the simplest approximating grid structure is indicated. © 1998 John Wiley & Sons, Inc.

EVALUATING GEOMETRIC SENSITIVITIES AT SINGULAR AND NON-SINGULAR POINTS OF VARIATIONAL CAD MODELS FOR DESIGN OPTIMIZATION

Much has been published on the topic of design optimization of mechanical components. Most of the research has concentrated on the finite element or boundary element part of the problem. Little effort has been applied to integrating design optimization with CAD. With the introduction of parametric and variational CAD, it is more desirable than ever to merge these technologies, i.e., to perform the analysis directly on the CAD geometry and to use the CAD parameters/dimensions as design variables. In this paper, one part of the problem is examined, the calculation of geometric sensitivities on variational CAD geometry. It is shown that for a well-conditioned set of constraint equations, the geometric sensitivities are easily obtained by a straight-forward application of the implicit function theorem. When the constraint equations become singular, the situation is more complex. The nature of singularities is explored, and a method, based on rational transformations that are common in algebraic curve tracing, is suggested to resolve singular points. It is shown that the geometric sensitivity is a natural by-product of the transformation. The paper concludes with some symbolic algebra software, a Mathematica package, that was found to be useful in the investigation.

On spin-(3/2) systems in Ricci flat space–times

The Dirac formulation of massless spin-(3/2) fields is discussed. The existence and uniqueness for the solutions of the spin-(3/2) field equations in Dirac form is proven. It is shown that the system of equations can be split into a symmetric hyperbolic system of evolution equations and a set of constraint equations. The constraints are shown to propagate on a curved manifold if and only if it is an Einstein space. The gauge freedom present in the spin-(3/2) system is discussed and it is shown that the complete system ‘‘solutions modulo gauge’’ has a well posed Cauchy problem if and only if the Einstein equations hold.

The Joint Displacement Method for Multiloop Kinematic Analysis

We present a kinematic formulation, referred to as the joint displacement method. In the kinematic analysis, the closed-loop kinematic chain of interest must be transformed into several open-loop chains by imagining that some joints are relaxed (disconnected), and by replacing them as physical entities with a set of constraint equations. This formulation requires only a minimal number of generalized coordinates, hence it provides a highly efficient approach for real-time kinematic analysis of spatial mechanisms, heretofore unseen in the literature. For example, the Jacobian matrix for the RSCR mechanism is 3×3 and the CPU time required to perform Gauss elimination is 182 times shorter than that in the traditional link-coordinate method which requires a 17×17 Jacobian matrix. This method is applicable to both single loop and multiloop spatial mechanisms with revolute, cylindrical, translational, spherical and universal joints. A general-purpose computer program that implements the joint displacement method has been developed and tested on a variety of mechanisms.

Dynamic stress analysis of rotating turbo-machinery bladed-disk systems

The steady state dynamic stress and deformation analysis of high pressure stage turbomachinery bladed-disks is carried out taking into account all the geometric complexities involved. The contributions due to initial stress and membrane behaviour are included. Only one substructure is used for finite element modelling and analysis, taking advantage of rotational periodicity. A triangular shell element with six degrees of freedom per node is employed for this purpose. The blade and disk attachment is established by a set of constraint equations obtained by the Love-Kirchhoff hypothesis. The final set of equations are solved by an out of core submatrices elimination method. The influence of different levels of approximations and various geometric parameters on the stresses and deformations is discussed.

Covariant gauge theory of strings

String theories are reformulated as gauge theories based on the reparametrization invariance. The differential geometry needed for such an approach is developed, leading to a set of constraint equations and an invariant action.

Correlation-coefficient optimisation in integrated-circuit design

This paper reviews the accomplishments to date in the use of correlation coefficients between circuit elements as design variables in the computer-aided tolerance design of integrated circuits. Two optimisation schemes have been developed which make use of the interior-point and exterior-point penalty function methods. Each method requires a different set of constraint equations to provide a positive definite correlation-coefficient matrix. A comparison of the two approaches is given, using a practical circuit tolerance design example.

Time-domain design of frequency-sampling digital filters for pulse shaping using linear programming techniques

The use of time-domain specifications is demonstrated for designing frequency-sampling digital filters that generate data pulses for transmission over an ideal band-limited channel. Desired characteristics of the transmitted pulse are used to formulate the set of constraint equations and objective function used in linear programming to obtain an optimum set of filter coefficients, i.e., frequency samples {| H(k) |} - The constraints are the amplitudes assigned to the set of regularily spaced samples taken from the pulse. The objective function is either to minimize the maximum absolute error between desired and generated pulse samples over the specified pulse duration or to provide zero crossings in the transmitted pulse with near-zero slope in order to protect against intersymbol interference (ISI) due to timing error (jitter).

A Computer Algorithm for the Synthesis of Memoryless Logic Circuits

A method has been investigated for the synthesis of memoryless logical networks using a restricted repertoire of functional modules. The method is based on the reduced general solution to a generalized system of Boolean equations (BE) as applied to the decomposition of Boolean functions. The aim of the synthesis is to obtain the most constrained circuit having at most two levels of gating. The constraints take the form of single input variables or constant logic levels applied to the inputs of the first level gate. This is achieved by assembling a set of constraint equations which are then a part of the generalized system of BE. The method is then tested on some synthesis examples of single and multiple output functions in terms of the NAND and (WOS) modules.