Solids[1]
- 1. SOLIDS Application of Solid Models In mechanical engineering, a solid model is used for the following applications: 1. Graphics: generating drawings, surface and solid models 2. Design: Mass property calculation, interference analysis, finite element modeling, kinematics and mechanism analysis, animation, etc. 3. Manufacturing: Tool path generation and verification, process planning, dimension inspection, tolerance and surface finish. 4. Component Assembly: Application to robotics and flexible manufacturing: Assembly planning, vision algorithm, kinematics and dynamics driven by solid models. Solid Model Representation There are three different forms in which a solid model can be represented in CAD: • Wireframe Model • Surface Model • Solid Model Wireframe Models: Joining points and curves creates wireframe models. These models can be ambiguous and unable to provide mass property calculations, hidden surface removal, or generation of shaded images. Wireframe models are mainly used for a quick verification of design ideas. Surface Models: Surface models are created using points, lines, and planes. A surface model is unable to identify points that do not lie on the surface, and therefore, the moment of inertia, volume, or sections of the model cannot be obtained. A surface model can be shaded for better visibility. Surface models are used for modeling surfaces of engineering components. Solid models : Solid models are the most preferred form of CAD models. and represent unambiguous image of a component. A solid model can be used to analyze the moment of inertia, mass, volume, sections of the model, etc. Solid models are mathematical models of objects in the real world that satisfy specific properties, listed below. 1. Bounded: The boundary must limit and contain the interior of the solid. 2. Homogeneously Three-Dimensional: No dangling edges or faces be present so that the boundary is always in contact with the interior of the solid. 3. Finite: The solid must be finite in size.
- 2. Solid Model Creation Scheme A solid model can be generated by the following schemes. 1. Constructive Solid Geometry (CSG) 2. Boundary Representation (B-Rep) 3. Sweeping A brief description of these schemes follows. Constructive Solid GeometryScheme This scheme is based on the principle that two primitives can be combined to produce a new solid model. This method is also known as ‘Building Block’ method. The scheme uses the Union, Intersection, and Subtraction techniques to create three- dimensional models, which are based on the Boolean operation. The steps involved in generating a solid model are: 1. Select the primitives from a library 2. Go through the scaling, dimension modification, and any other transformations. 3. Combine the primitives to create the desired solid model. Since CSG method uses solid primitives, internal details of the object are automatically contained in the model. The model can be sectioned to reveal internal details and can be used for calculating mass, volume, moment of inertia, etc. New solid models can be created from the primitives or other solid models by the following operations: • Union (U): Two solids are joined and the common volume of one of the primitives is neglected in the resulting solid. • Subtraction or Difference (-): One solid is subtracted from the other and the resultant solid retains only the uncut portion of the solid. • Intersection (П): When two solids are combined, the resultant solid represents the common volume of the two solids. The most common primitive solids found in a CAD program are: Block, Cylinder, Cone, Sphere, Wedge, and Torus. Boundary Representation(B-Rep)Scheme This scheme is based on the concept that a physical object is bounded by a set of faces. A solid model is created by combining faces and contains vertices, edges, loops, and bodies. Only the boundary surfaces of the model are stored and the volumetric properties are calculated by the Gauss Divergence theorem, which relates volume integral to surface integrals. This scheme can model a variety of solids depending on the primitive surfaces (planar, curved, or sculptured). There are two types of solid models in this scheme:
- 3. 1. Polyhedral solids 2. Curved solids 1. Polyhedral Solids: Polyhedral models consist of straight edges, e.g., a non- cylindrical surface: box, wedge, combination of two or more non-cylindrical bodies, etc. Polyhedral solids can have blind or through holes, and two or three-dimensional faces, with no dangling edges. A valid polyhedral abides by the Euler’s equation: F – E + V – L = 2 (B-G) Where, F = Face E = Edge V = Vertices L = Inner Loop B = Bodies G = Through holes A simple polyhedral has no holes; each face is bounded by a single set of connected edges (bounded by one loop of edges). Euler’s equation for a simple polyhedral can be reduced to: F - E + V = 2 Example: For the box shown, F = 6, E = 12, and V = 8 Examples of other types of polyhedral are shown below. Polyhedral with two loops Polyhedral with a blind hole 2. Curved Solids: A curved solid is similar to a polyhedral object but it has curved faces and edges. Spheres and cylinders are examples of curved solids. Sphere with F = 1, V = 1, E = 0 Cylinder: F = 3, E = 3, V= 2
- 4. Primitives: In B-rep, a model is made up of the following primitives: • Vertex: A point in space • Edge: A finite, no-intersecting space curve bounded by two vertices that are not necessarily distinct. • Face: A finite connected, non-self-intersecting, region of a closed oriented surface, bounded by one or more loops. • Loop: An ordered alternating sequence of vertices and edges. A loop defines a non-self-intersecting closed space curve, which may be a boundary of a face. • Body: Entity that has faces, edges and vertices. A minimum body is a point. B-rep scheme is closely related to the traditional drafting method. SweepRepresentation Sweeping can create a solid model. The method is useful for creating 2 ½ – dimension models. The generated models are axisymmetric and have uniform thickness (i.e., extruded models). There are two types of sweeps: linear and rotational. In linear sweep, a closed 2-D sketch is extruded through the desired length, creating a homogeneous and axisymmetric model, as shown in the figure. Sweep direction Linear sweep – Creating a box by sweeping a rectangle In rotational sweep, a closed sketch is rotated around an axis. The generated model is always axisymmetric. In addition to the two sweeps described above, a model can also be created by a non- linear sweep. In this type of sweep, a closed sketch is sweeped along a non-linear path.
- 5. Cell Decomposition Most commercial software use the sweeping scheme. ProE and Solidworks are good examples of software that utilize sweep technique to generate a 3-D model. In both of these programs, a 2-D sketch is created and extruded to generate a 3-D base-model. The base model is then used to add or modify features. Most engineering components can be created by this technique. Relatively new software, Ironcad utilizes the CSG technique to create 3-D models. There are pre-build models in the library (catalog) of the software that can be retrieved and modified as needed. The availability of the primitives (basic solid models) accelerates the process of model generation. All the three software mentioned above are parametric modelers. Another popular software, AutoCAD is capable of generating 3-D models; however, this software is basically a 2-D modeler and lacks the parametric feature. AutoCAD is capable of creating a solid model with CSG, B-rep, and the sweep methods, but limited to only very simple models.