ComputerAided_Design_Engineering_amp_Manufactur.pdf
ComputerAided_Design_Engineering_amp_Manufactur.pdf
ComputerAided_Design_Engineering_amp_Manufactur.pdf
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The system may be written in matrix notation as:<br />
where<br />
n 1<br />
x x is the vector of displacements of iteration n<br />
J is the Jacobian matrix<br />
r is the vector of residuals.<br />
When formulating the equations system, all equations are satisfied, but if a change in the value of one<br />
or more variables is made, the residuals are no longer zero. The original values of the variables are used<br />
as initial guesses, and a new set of values of the variables is generated and used to solve the above equation.<br />
The process is iterative until the residuals tend to zero within a tolerance value.<br />
�<br />
x n<br />
� �<br />
fij �<br />
�fi��xj 9.6 Other Researches and Developments<br />
In previous sections, we focused on the introduction of the connectivity design module of ProMod, in<br />
short, it is an assembly modeling technique that uses spatial relationships to constrain and maintain<br />
assembly relationships between components. In the following sections, we will show other related developments<br />
derived from ProMod, i.e., stability analysis of assembly, feasible approach directions and precedence<br />
constraints, and, finally, kinematic and world modeling.<br />
Stability Analysis of Assembly<br />
Stability analysis is a mathematical model that simulates the stability of assembly placement during each<br />
assembly stage. Research has been carried out in production planning, robot path planning, grasp<br />
planning, and fixture planning, but the problem of subassembly stability planing has been continually<br />
ignored. It has been implicitly assumed that parts and subassemblies will retain their configuration all<br />
through the manufacturing process. For instance, when a one-hand robot picks up a part and places it<br />
in some locations, it is assumed that the configuration of the part will not change until the robot moves<br />
it. In cases where the parts “looked unstable” during design, either 1. fixturing was taken as the default<br />
without even considering changes in the part or assembly design that would make the system stable and<br />
eliminate unnecessary fixturing, or 2. changes were made to the design to make the subassembly stable,<br />
without giving a thought to what minimal changes in the design would bring in stability.<br />
An assembly process is basically a sequence of steps in which the parts are to be placed on a subassembly<br />
to achieve the final goal. However, it is necessary to satisfy certain constraints such as geometric and<br />
manufacturing resource constraints to realize the assembly. In addition, it is also important to consider<br />
stability constraints. At every instant of the manufacturing process, the components of the subassembly<br />
must be stable. Analyzing for stability is also helpful in determining the limiting values of the assembly<br />
forces and torques. If excessive forces or torques are applied while mating a certain component with a<br />
partial assembly, then that can result in instability of the subassembly.<br />
There are quantitative and qualitative considerations in assembly stability analysis methods. Numerical<br />
simulation technique and equivalent parameter identification are considered in quantitative analysis. In<br />
assembly stability simulation, the system is excited with the assembly torques and forces. The response<br />
of the system varies with its structural properties, such as inertia, mass, d<strong>amp</strong>ing, and stiffness of the<br />
joints. In addition, the response is also dependent on geometric characteristics, such as the location of<br />
the mass centers of the bodies. The equivalent parameter identification is to develop a system to evaluate<br />
the behavior of the mechanism at the point of interest with equivalent in mass and inertia or equivalent<br />
in stiffness and d<strong>amp</strong>ing. With this in view, it is convenient to reduce the complex dynamics of the<br />
multiDOF assembly to that for a single body, namely, a scalar quantity for mass. The inertial matrix with<br />
body is known as the virtual mass (Vishnu, 1992).<br />
© 2001 by CRC Press LLC<br />
J � �x � r