Geometrical and Topological Issues in NMT-Based Modeling

The Non-Manifold Topology (NMT) Radial Edge data structure, along with the supporting set of Euler operators, provides a versatile environment for modeling non-manifold domains. The operators provide the basic tools to construct and manipulate model topology. However, an implementation of the base functionality in a geometric modeling environment raises some geometry-related issues that need to be addressed to ensure the topological validity of the underlying model. This paper focuses on those issues and emphasizes the use of geometry in the implementation of topological operators. Enhancements to the topology manipulation operations are also discussed. Specifically, this paper describes (i) a geometry-based algorithm for face insertion within the Radial Edge data structure, (ii) a manifestation of the face-insertion algorithm to resolve topological ambiguities that arise in the design of topological glue operators, and (iii) enhancements to the topology deletion operators to meet application-specific requirements.

Minimization of the Vibration Energy of Thin-Plate Structures

An optimization method is proposed to reduce the vibration of thin-plate structures. The method is based on a finite-element shell analysis, a modal analysis, and a structural optimization method. In the finite-element analysis, a triangular shell element with 18 degrees of freedom is used. In the optimization, the overall vibration energy of the structure is adopted as the objective function, and it is minimized at the given exciting frequency by varying the thickness of the elements. The technique of modal analysis is used to derive the sensitivity of the vibration energy with respect to the design variables. The sensitivity is represented by the sensitivities of both eigenvalues and eigenvectors. The optimum value is computed by the gradient projection method and a unidimensional search procedure under the constraint condition of constant weight. A computer code, based on the proposed method, is developed and is applied to design problems using a beam and a plate as test cases. It is confirmed that the vibration energy is reduced at the given exciting frequency. For the beam excited by a frequency slightly less than the fundamental natural frequency, the optimized shape is close to the beam of uniform strength. For the plate, the optimum shape is obtained such that the changes in thickness have the effect of adding a stiffener or a mass.

Three Dimensional Robot Path Planning With Workspace Considerations

This paper considers the problem of robot path planning by optimization methods. It focuses on the use of recursive quadratic programming (RQP) for the optimization process and presents a formulation of the three dimensional path planning problem developed for compatibility with the RQP selling. An approach 10 distance-to-contact and interference calculations appropriate for RQP is described as well as a strategy for gradient computations which are critical to applying any efficient nonlinear programming method. Symbolic computation has been used for general six degree-of-freedom transformations of the robot links and to provide analytical derivative expressions. Example problems in path planning are presented for a simple 3-D robot. One example includes adjustments in geometry and introduces the concept of integrating 3-D path planning with geometric design.

Tilt-Wing Rotorcraft Dynamic Analysis Using Multibody Formulation

A model is presented that can be used to simulate the highly nonlinear transient dynamics associated with advanced rotorcraft conversion processes. Multibody equations of motion of the fuselage, the tilting wing, and the rotor assembly are derived using a minimal set of coordinates. An enhanced aerodynamics model is employed to account for unsteadiness and nonlinearity in the near-wake aerodynamics, with a dynamic uniform inflow to compute the far-wake aerodynamics, and a flight control system is employed to compute the blade pitch settings that are necessary to achieve a desired flight path. The model is subjected to a demanding flight path simulation to illustrate that it can perform vertical take-off, hover, tilt-wing conversion, and high-speed forward flight maneuvers effectively.

Object Extent Determination for Algebraic Solid Models

This paper presents methods for determination of spatial extent of algebraic solid models. Algebraic solid models (ASM) are a variation of implicit solid models defined by implicit polynomial functions with rational coefficients. Spatial extent information, which can be used to enhance the performance of visualization and property evaluation, includes silhouettes, outlines and profiles. Silhouettes are curves on the surface of the solid which separate portions of the surface which face towards or away from a given viewpoint. The projection of the silhouette onto the viewing plane gives the outline of the solid, and the bivariate implicit function which defines the area enclosed by the outline is called the profile. A method for outline determination is demonstrated using concepts from algebraic geometry including polar surfaces and variable elimination via the Gröbner basis method and/or resultants. Examples of outline generation are presented and a sample profile function is constructed.

Hertz Contact Force Model With Permanent Indentation in Impact Analysis of Solids

A continuous analysis method for the direct-central impact of two solid particles is presented. Based on the assumption that local plasticity effects are the sole factor accounting for the dissipation of energy in impact, a Hertzian contact force model with permanent indentation is constructed. Utilizing energy and momentum considerations, the unknown parameters in the model are analytically evaluated in terms of a given coefficient of restitution and velocities before impact. The equations of motion of the two solids may then be integrated forward in time knowing the variation of the contact force during the contact period. For Illustration, an impact of two soft metallic particles is studied.

A Model for Mechanism Data Storage

A data model for kinematic structure of mechanisms and its coding principle are proposed, based on the topological graph and contract graph. In the model every basic chain is mapped by a code of 5 decimal digits and a mechanism is mapped by a set of code of basic chains. The model occupies minimal memory, and contains a complete set of useful primary parameters of structure, and significantly reduce computer time for isomorphism identification.

A Methodology for Compliant Mechanisms Design: Part I — Introduction and Large-Deflection Analysis

A broader research proposal seeks to systematically combine large-deflection mechanics of flexible elements with important kinematic considerations, in yielding compliant mechanisms which perform useful tasks. Specifically, the proposed design methodology will address the following needs: development of the necessary nomenclature, classification and definitions, and identification of the kinematic properties; categorization of mechanism synthesis types, both structurally as well as by function; development of efficient computational techniques for design; consideration of materials; and application and validation. Contained herein, in particular, is an introduction to the state-of-the-art in compliant mechanisms, and the development of an accurate chain calculation algorithm for use in the analysis of a large-deflection, cantilevered elastica. Shooting methods, which permit specification of additional boundary conditions on the elastica, as well as compliant mechanism examples are presented in a companion paper.

Boundary Surface Recovery From Skeleton Elements: Part I — Skeleton Curves

Skeleton curves and surfaces have many applications in computer aided design and analysis. Construction of skeletons is an active area of research. We consider the inverse problem that of recovering boundary surfaces from given skeleton elements. The skeleton of any 3D object will, in general, consist of curves and surfaces. Therefore, any boundary reconstruction algorithm must systematically process the surfaces generated by the skeletal curves and the skeletal surfaces. In this paper (Part I) we present algorithms for reconstructing boundary surfaces corresponding to skeletal curves. Implemented examples are also included. In a companion paper (Part II) we consider skeletal elements that are surfaces.

Geometric Design and Fabrication of Developable Bezier Surfaces

In this paper we study Computer Aided Geometric Design (CAGD) and Manufacturing (CAM) of developable surfaces. We develop direct representations of developable surfaces in terms of point as well as plane geometries. The point representation uses a Bezier curve, the tangents of which span the surface. The plane representation uses control planes instead of control points and determines a surface which is a Bezier interpolation of the control planes. In this case, a de Casteljau type construction method is presented for geometric design of developable Bezier surfaces. In design of piecewise surface patches, a computational geometric algorithm similar to Farin-Boehm construction used in design of piecewise parametric curves is developed for designing developable surfaces with C2 continuity. In the area of manufacturing or fabrication of developable surfaces, we present simple methods for both development of a surface into a plane and bending of a flat plane into a desired developable surface. The approach presented uses plane and line geometries and eliminates the need for solving differential equations of Riccatti type used in previous methods. The results are illustrated using an example generated by a CAD/CAM system implemented based on the theory presented.