Simulation of Mating Between Nonanalytic Surfaces Using a Mathematical Programing Formulation

[+] Author and Article Information
Robert Scott Pierce

Department of Physics and Engineering,  Sweet Briar College, Sweet Briar, VA 24595spierce@sbc.edu

David Rosen

 George W. Woodruff School of Mechanical Engineering, 801 Ferst Drive, Atlanta, GA 30332-0405david.rosen@me.gatech.edu

J. Comput. Inf. Sci. Eng 7(4), 314-321 (Apr 17, 2007) (8 pages) doi:10.1115/1.2795297 History: Received January 31, 2006; Revised April 17, 2007

In this paper, we describe a new method for simulating mechanical assembly between components that are composed of surfaces that do not have perfect geometric form. Mating between these imperfect form surfaces is formulated as a constrained optimization problem of the form “minimize the distance from perfect fit, subject to noninterference between components.” We explore the characteristics of this mating problem and investigate the applicability of several potential solution algorithms. The problem can be solved by converting the constrained optimization formulation into an unconstrained problem using a penalty-function approach. We describe the characteristics of this unconstrained formulation and test the use of two different solution methods: a randomized search technique and a gradient-based method. We test the algorithm by simulating mating between component models that exhibit form errors typically generated in end-milling processes.

Copyright © 2007 by American Society of Mechanical Engineers
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Figure 1

A high-speed stapling mechanism composed of multiple prismatic joints

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Figure 2

Mating planar faces. (a) “Against” condition for faces with perfect form. (b) Faces with imperfect form. How should “perfect fit” be defined for these faces?

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Figure 3

Clearance∕interference measurement. The magnitude of the errors has been increased by a factor of 10 for the purposes of this figure. (a) The mating surfaces and grid points. (b) Measurement of the closest-point distance.

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Figure 4

Topography of the feasible regions of the solution space for the four-surface prismatic joint example near the “best” mating solution. For the purposes of these diagrams, infeasible regions are shown to lie in the “Objective=0” plane. (a) Roll versus Yaw. (b) Yaw versus y.

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Figure 5

A single grid point/closest-point pair

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Figure 6

Pseudocode description of the calculation of the mating objective function

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Figure 7

Convergence of the BFGS and Hooke–Jeeves algorithms for the two validation problems. (a) The four-surface prismatic joint problem. (b) The perfect-fit prismatic joint problem.

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Figure 8

Convergence of the hybrid BFGS∕HJ algorithm near the optimum answer for the perfect-fit prismatic joint problem

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Figure 9

Mating results for a series of slider∕groove components. Each pair exhibits a different combination of cutter-deflection and spindle-tilt errors that are characteristic of end-milling processes. The error magnitudes have been scaled by a factor of 10 in order to make the errors visible.



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