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Research Papers

On the Identification of Machine Settings for Gear Surface Topography Corrections (DETC2011-47727)

[+] Author and Article Information
Marco Gabiccini1

Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione,  University of Pisa, Largo Lucio Lazzarino 2, 56122 Pisa, Italym.gabiccini@ing.unipi.it

Alessio Artoni

Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione,  University of Pisa, Largo Lucio Lazzarino 2, 56122 Pisa, Italya.artoni@ing.unipi.it

Massimo Guiggiani

Dipartimento di Ingegneria Meccanica, Nucleare e della Produzione,  University of Pisa, Largo Lucio Lazzarino 2, 56122 Pisa, Italyguiggiani@ing.unipi.it

1

Corresponding author.

J. Mech. Des 134(4), 041004 (Mar 19, 2012) (8 pages) doi:10.1115/1.4006002 History: Received June 16, 2011; Revised September 15, 2011; Published March 07, 2012; Online March 19, 2012

In this paper, we set out to investigate the performances of some algorithms proposed in the gear literature for identifying the machine-tool settings required to obtain predesigned gear tooth surface topographies, or needed to compensate for flank form deviations of real teeth. For ease of comparison, the problem is formulated as a nonlinear least squares problem, and the most widely employed algorithms are derived as special cases. The algorithms included in the analysis are (i) one-step methods, (ii) iterative methods, and (iii) iterative methods with step control. The performance index is devised in their ability of returning practical solutions in the presence of (i) strong model nonlinearities, (ii) ill-conditioning of the sensitivity matrix, and (iii) demanding topographic shapes. Instrumental here is an original classification of topographic modifications as either “simple” or “complex,” based on the singular value decomposition (SVD) analysis of the sensitivity matrix. Some selected numerical examples demonstrate that iterative techniques with step control are the most convenient in terms of reliability and robustness of the obtained solutions. The generation process considered here is face-milling of hypoid gears, although the methodology is general enough to cope with any gear cutting/grinding method.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 1

Initial residual h0(x)=[h10(x)…hm(0)(x)]T. pi(0), basic points on Γ(x 0 ); pi*, target points on the target surface Γ(x *).

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Figure 2

Residual h (x ) = [h1 (x )…hm (x )]T . pi(0), basic points on Γ(x 0 ); pi*, target points on the target surface Γ(x *); Γ(x ), surface with generic parameters x .

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Figure 3

Eigen-topographies u i , with (i = 1,…,20)

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Figure 4

Effects of model nonlinearities on the solution. Solid, nonlinear cost function f(x); dashed, quadratic model q0 (s); xGN solution obtained by linear regression; x6  = x* global solution obtained by an iterative method with step control [16].

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Figure 5

Lengthwise crowning defined by variation of Rp , S0 , and q0 through Stadtfeld’s formulas [28]

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Figure 6

Complex ease-off. Modification proportional to the 12th eigen-topography u 12 , i.e., h (x 0 ) = 0.25u 12

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Figure 7

Complex ease-off. Predesigned and obtained topographies after six steps of the Levenberg–Marquardt algorithm [16] (iterative method with step control).

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