An Analytic Expression for the Inverse Involute

Alberto López Rosado,Federico Prieto Muñoz,Roberto Alvarez Fernández

doi:10.1155/2019/3586012

Alberto López Rosado, Federico Prieto Muñoz + Show 1 more

Open Access

https://doi.org/10.1155/2019/3586012

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Abstract

This article introduces new types of rational approximations of the inverse involute function, widely used in gear engineering, allowing the processing of this function with a very low error. This approximated function is appropriate for engineering applications, with a much reduced number of operations than previous formulae in the existing literature, and a very efficient computation. The proposed expressions avoid the use of iterative methods. The theoretical foundations of the approximation theory of rational functions, the Chebyshev and Jacobi polynomials that allow these approximations to be obtained, are presented in this work, and an adaptation of the Remez algorithm is also provided, which gets a null error at the origin. This way, approximations in ranges or degrees different from those presented here can be obtained. A rational approximation of the direct involute function is computed, which avoids the computation of the tangent function. Finally, the direct polar equation of the circle involute curve is approximated with some application examples.

Highlights

Which is valid for x ≤ 0.5 with a maximum error for u 45° and Emax 0.0012 rad 4.12529′. is error has been calculated when comparing it with equation (3)
Is article introduces new types of rational approximations of the inverse involute function, widely used in gear engineering, allowing the processing of this function with a very low error. is approximated function is appropriate for engineering applications, with a much reduced number of operations than previous formulae in the existing literature, and a very efficient computation. e proposed expressions avoid the use of iterative methods. e theoretical foundations of the approximation theory of rational functions, the Chebyshev and Jacobi polynomials that allow these approximations to be obtained, are presented in this work, and an adaptation of the Remez algorithm is provided, which gets a null error at the origin. is way, approximations in ranges or degrees different from those presented here can be obtained
New approximations to the inverse involute function have been presented. ey are very accurate for a low degree polynomial, requiring a much reduced number of operations

Summary

A E B r u x

Its error grows with x, and it takes values for u 45° and Emax 0.000017 rad 3.506′′ with two iterations (computation until u2). E previous expression in these 9 terms has a very low error, which takes values for u 45° and Emax 1.58E − 9 rad 3.2589E − 4′′ (see [10]). To justify the resulting final expressions and compute some others, the concepts of expansion in a series of the Chebyshev polynomials, economization, and approximation of functions are briefly reviewed, and the series of Jacobi polynomials allowing for the selection of areas with a lower error in a flexible way are described.

Theoretical Basis of the Approximations Used in This Article

Conclusions