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INTRODUCTION

PARAMETRIC FEA

MAKING MECHANISMS MOVE

INTRODUCTION

A finite-element model can be thought of as a system of solid springs. When a load is applied to the structure, all elements deform until all forces balance. For each element in the model, equations can be written relating displacements and forces at the nodes.

The element , for example, can be a 2D quadrilateral having four nodes. Each node has two degrees of freedom associated with it (displacements in X and Y directions), so that the element has a total of eight degrees of freedom. There must also be a nodal force for each nodal degree, so there are also eight nodal forces for the element.

These displacements and forces are identified by a coordinate numbering system for entry the computer program. For example, d§xi1 is the deflection in the X direction for element i at node 1, while d§yi1 is the deflection in the Y direction for the same node in the same element. Forces are identified in a similar manner, so that F§xi1 is the force in the X direction for element i at node 1.

An equation relating displacements and forces for the element takes the form of basic spring equation, F = kd. For four nodes

k§11 d§xi1 + k§12 d§yi1 + k§13 d§yi2 + k§14 d§yi2 + ... + k§18 d§yi4 = F§xi1

k§21 d§xi1 + k§22 d§yi1 + k§23 d§yi2 + k§14 d§yi2 + ... + k§28 d§yi4 = F§yi1

k§31 d§xi1 + k§32 d§yi1 + k§33 d§yi2 + k§34 d§yi2 + ... + k§38 d§yi4 = F§xi2

. . . k§81 d§xi1 + k§82 d§yi1 + k§83 d§yi2 + k§84 d§yi2 + ... + k§88 d§yi4 = F§yi4

The k factors are stiffness coefficients relating the nodal deflections and forces, and are calculated by the finite-element program from material properties such as Young's modulus and Poisson's ratio, and from the element geometry. Thus, in the example, coefficient k§13 relates deflection 3 and force 1.

The set of equations can be written in matrix form

If degrees of freedom and nodal forces are consecutively numbered (d§xi1 = d§1, d§yi1 = d§2, F§xi1 = F§1, F§yi1 = F§ 2, and so forth), the matrix can be renumbered to show how stiffness coefficients relate nodal forces and deflections.

When a structure is modeled, individual sets of matrix equations are automatically generated for each element. The elements in the model share common nodes so individual sets of matrix equations can be combined into a global set of matrix equations. This global set relates all the nodal degrees of freedom to the nodal forces, and the nodal degrees of freedom are solved simultaneously from the global matrix. When displacements for all nodes are known, the state of deformation of each element is known. And, when deformation of each element is know, the stress and strain within the element are also known.

For simple static analysis, the finite-element method is a two-step process. Nodal displacements are first simultaneously calculated from the element stiffness and the nodal forces, both internal and external. Next, stresses are calculated, generally at the each element's centroid. Because displacements are calculated for only a finite number of points in the structure, the finite-element method is a numerical approximation rather than an exact solution.

PARAMETRIC FEA

Parametric features are becoming more common in FEA packages. The key benefit of parametric features is that they let users see the effects of design changes quickly. With adequate planning, users can define an FE model entirely in terms of variables or parameters. Even mesh characteristics can be defined as parameters.

To illustrate, consider a simple design-analysis problem. A simply supported steel beam deflects excessively under load. To stiffen the design, reinforcing strips will be spot welded to 500 already-cut beams.

Theory says that the addition would provide adequate stiffness if the two components were perfectly joined. But spot welds provide a discontinuous interface. Therefore, a new requirement is to find the minimum number of spots welds to limit maximum deflection of the welded beam to no more than 110% of the deflection of a solid beam with the same cross section.

The parametric finite-element model must be flexible enough to simulate welded beams with a varying number of spot welds. However, with little additional effort we can create a more general model that also simulates welded beams with a range of dimensions. The payoff is that we are better prepared to evaluate reinforcement bars with different cross sections, should that become necessary. In addition, we want to define mesh characteristics as parameters. This helps with convergence or mesh-refinement studies. All the dimensions are treated as parameters. To make the finite-element model as general as possible, top and bottom sections are allowed a different number of elements across their widths. However, the two sections must have the same number of elements along their lengths.

The model consists of two layers of quadrilateral shell elements. The bottom layer of nodes sits in the plane y = -(t1)/2 where t1 is the thickness of the reinforcement bar.

The sketch of corner nodes shows them at the bottom section of the model along with parameters that define two corner nodes. Coordinates (X, Y, Z) for node 1 are (0, -(t1)/2, (b1)/2). These coordinates become arguments to the first command which defines the first node of the model. The last node, nend, on the same edge is at coordinates (len, -(t1)/2, (b1)/2).

The value of nend is defined in terms of the number of welds and number of elements needed between welds. Coordinates for node nend become arguments to the second command. A Fill command simply fills in the nodes between nodes 1 and nend.

If we increase the number of welds or elements between welds, we increase the value of nend. In other words, we have automated part of the mesh generation. By continuing this process, we automate every aspect of the finite-element mesh. This gives us precise control over dimensions, nodes, and elements in the model.

Loads are also parameterized. For the example, the model is subjected to a unit distributed load. One parameter denotes the load. Another (beam length × width) defines the load area on the upper section. Two other parameters calculate the pressure required for the upper surface. In this way, changing the model size does not affect the total load on the beam.

Even boundary conditions can be applied with parameters. For instance, we use parameters to select nodes at the ends of the upper layer. Thus, we are assured that the simply supported boundary conditions are correctly applied for each run of the study, regardless of the changes that might be made to either geometry or mesh.

In the example, parametric modeling also specifies the spot-weld locations. We want to spot welds to be placed at the beam's symmetry plane and equally spaced along its span. Linear constraint equations, automatically generated by the program, model the stiffening effect of spot welds.

Assigning parameters that describe linked nodes lets the model simulate any number of spot welds. Model dimensions and mesh density are changed with equal ease. Parametric models also help with convergence or mesh-refinement studies which estimate the discretization error, an accuracy indicator.

A convergence study lets us select the FE mesh that provides the quickest acceptable solution. This is important when the model is used many times in parametric studies. In such case, it's advantageous to generate several solutions with a finite, but acceptable error in each.

MAKING MECHANISMS MOVE

General-purpose motion-analysis packages run on PCs. They are aimed at designers who want to try out a few ideas quickly. Other packages, designed for more demanding tasks, are available on Unix workstations. These might not only analyze loads on pivots and pins, but also optimize designs by determining the best length for moment arms, and so forth. Packages designed for use on PCs typically are mechanical dynamics simulators. The user defines objects with specified masses, then watches as the program shows what happens under given sets of conditions. Geometries can either be devised in the program itself or imported via .dxf files.

Models have physical properties that include mass, static and kinetic friction, elasticity, moment of inertia, and electrostatic charge. Available constraints in PC-based programs include pulleys, joints, dampers, ropes, inflexible rods, springs, actuators, and motors to join masses. Users constrain masses with pin, slot, keyed slot, and rigid joints, and specify forces acting on the model.

Motion simulators calculate equations of motion through numerical integration. Users define the time steps in the simulation to determine accuracy. Large time steps are acceptable for slowly moving objects, but may throw off accuracy. These simulators provide several different numerical methods that trade off accuracy versus speed of calculation.

PC-based motion simulators work strictly with 2D rigid bodies at this point. For situations where deflections play a part in the simulation, and where mechanisms move in 3D, workstation simulators are required. The two packages that are most well known are Adams from Mechanical Dynamics Inc. and Dads from Computer Aided Design Software Inc. Among the factors that distinguish these packages from PC-based versions is the use of high-level integration techniques such as implicit integration, useful for solving problems involving extremely stiff springs, high frequencies, or both.

A related kind of software called kinematic synthesis programs develop mechanism designs when given the parameters of the mechanism - number of links, number of joints and joint types, connectivity of links and joints, and specification of which links are grounded. Basic linkage types are defined within programs as kinematic chains. Component dimensions are not specified. The synthesis program then determines mechanism structures required to perform specific tasks.

Kinematic synthesis programs are used principally to design four-bar linkages, although some also synthesize six-bar linkages. At present, only one synthesis package is available for desktop use. It is strictly 2D, runs on Macintosh machines, and is primarily a teaching tool.