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Research Papers

Workpiece Positioning Analyses: The Exact Solutions and a Quadratic Variation Approximation Using the Method of Moments

[+] Author and Article Information
Jun Cao, Sun Jin, Zhongqin Lin

Body Manufacturing Technology Center, Shanghai Jiao Tong University, Shanghai 200030, China

Xinmin Lai1

Body Manufacturing Technology Center, Shanghai Jiao Tong University, Shanghai 200030, China

Wayne Cai

 General Motors Corporation R&D Center, Warren, MI 48090-9055

1

Corresponding author.

J. Manuf. Sci. Eng 130(6), 061013 (Nov 06, 2008) (10 pages) doi:10.1115/1.2976147 History: Received December 07, 2007; Revised June 15, 2008; Published November 06, 2008

This paper presents algorithms for workpiece positioning analysis under locating errors. Workpiece constraint equations are first constructed using the method of homogenous coordinate transformation. These constraint equations are solved numerically for exact workpiece positional deviations by means of deterministic analysis (using the Newton–Raphson method) and variation analysis (i.e., random analysis using a Monte Carlo simulation). To enhance numerical efficiency in variation analysis, we further propose a quadratic approximation solution using the method of moments instead of the Monte Carlo method. Several case studies are presented to exemplify the proposed algorithms, with comparisons to prior literature results on linear and quadratic analyses. The criterion for using the proposed quadratic variation analysis versus the linear method and Monte Carlo simulation is also presented. By using the proposed algorithms, the exact workpiece positioning errors or quadratic variation approximations can be calculated, with consideration of workpiece surface nonlinearity, interactions between locating errors, and the impact of noninfinitesimal locating errors. This paper represents algorithmic advancement in the field where exact solutions and approximations can all be obtained at users’ choice.

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Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 10

The standard deviation of workpiece displacement

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Figure 11

The expectation of workpiece displacement

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Figure 12

The standard deviation of workpiece displacement

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Figure 13

The expectation of workpiece displacement

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Figure 14

Euler angle definition and coordinate transformation

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Figure 1

Locating scheme for a 3D workpiece

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Figure 2

The locating scheme of a workpiece with two elliptical segments

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Figure 3

The locating scheme and dimensions (12)

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Figure 4

Comparisons of the linear (10), quadratic (12), and our linear method

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Figure 5

Flowchart of the variation analysis

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Figure 6

The locating scheme of a block

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Figure 7

The standard deviations of positional errors at origin

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Figure 8

The expectation of positional errors at origin

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Figure 9

Example 4: an ellipsoid workpiece

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