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Dimensional synthesis of rack-and-pinion steering mechanism using a novel synthesis equation

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Abstract

This article presents a method for the dimensional synthesis of the rack-and-pinion steering mechanism by optimization technique based on a novel synthesis equation. The proposed kinematic model allows obtaining a polynomial synthesis equation to formulate the objective function as a sum of squares. Then, the computation of the objective function derivatives is straightforward compared to existing formulations. Finally, the application of the proposed method is shown through a numerical example implemented in Matlab®.

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Abbreviations

δ :

Steering angle

Y ijk :

Bilateration matrix

s ij :

Square distance between points i and j.

f :

Synthesis equation

z :

Design variables vector

g :

Objective function

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Acknowledgements

This study was financed in part by the Coordenacão de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001 and CAPES- PRINT/UFSC AUXPE 2835/2018 and CNPq under project PQ 312117/2017-5.

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Authors and Affiliations

  1. Federal University of Santa Catarina (UFSC), Centro Tecnológico, Campus Universitário Reitor João David Ferreira Lima, Trindade, Florianópolis, SC, CEP: 88.040-900, Brazil

    Neider Nadid Romero Nuñez, Rodrigo S. Vieira & Daniel Martins

  2. Faculty of Engineering, University of Antioquia (UDEA), calle 67#53-180, El Carmen de Viboral, Colombia

    Anderson Romero Florez

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  1. Neider Nadid Romero Nuñez
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  3. Rodrigo S. Vieira
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Correspondence to Neider Nadid Romero Nuñez.

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Appendix 1: Transformation matrix

Appendix 1: Transformation matrix

This appendix presents all the coefficients of the fourth-degree polynomial expressed in Eq. (13).

$$\begin{aligned} r_{1} = & 2(c_{o} + c_{i} - 2) \\ r_{2} = & 4(s_{o} - s_{i} ) \\ r_{3} = & 6( - c_{o} + c_{i} + 2)w \\ r_{4} = & 8( - s_{o} + s_{i} )w \\ r_{5} = & 6(c_{o} + c_{i} - 2)w^{2} \\ r_{6} = & 4(s_{o} - s_{i} )w^{2} \\ r_{7} = & - 2(c_{o} + c_{i} - 2)w^{3} \\ r_{8} = & (3c_{o}^{2} + 10c_{i} c_{o} - 8c_{o} + 3c_{i}^{2} - 8c_{i} ) \\ r_{9} = & 8(c_{o} + c_{i} )(s_{o} - s_{i} ) \\ r_{10} = & 4(s_{o} + s_{i} )^{2} \\ r_{11} = & - 2(3c_{o}^{2} + 10c_{i} c_{o} - 8c_{o} + 3c_{i}^{2} - 8c_{i} )w \\ r_{12} = & - 8(c_{o} + c_{i} )(s_{o} - s_{i} )w \\ r_{13} = & (3c_{o}^{2} + 10c_{i} c_{o} - 8c_{o} + 3c_{i}^{2} - 8c_{i} )w^{2} \\ r_{14} = & 4(c_{o} + c_{i} )(2c_{i} c_{o} - c_{o} - c_{i} ) \\ r_{15} = & 8(c_{o} + c_{i} )(c_{i} s_{o} - c_{o} s_{i} ) \\ r_{16} = & - 4(c_{o} + c_{i} )(2c_{i} c_{o} - c_{o} - c_{i} )w \\ r_{17} = & 2(s_{o} - s_{i} ) \\ r_{18} = & - 4(c_{o} + c_{i} - 2) \\ r_{19} = & - 6(s_{o} - s_{i} )w \\ r_{20} = & 8(c_{o} + c_{i} - 2)w \\ r_{21} = & 6(s_{o} - s_{i} )w^{2} \\ r_{22} = & - 4(c_{o} + c_{i} - 2)w^{2} \\ r_{23} = & - 2(s_{o} - s_{i} )w^{3} \\ r_{24} = & 2(3c_{o} s_{o} + 5c_{i} s_{o} - 4s_{o} - 5c_{o} s_{i} - 3c_{i} s_{i} + 4s_{i} ) \\ r_{25} = & 8(s_{o}^{2} - 2s_{i} s_{o} + s_{i}^{2} - c_{o}^{2} - 2c_{i} c_{o} + 2c_{o} - c_{i}^{2} + 2c_{i} ) \\ r_{26} = & - 8(c_{o} - c_{i} )(s_{o} + s_{i} ) \\ r_{27} = & - 4(3c_{o} s_{o} + 5c_{i} s_{o} - 4s_{o} - 5c_{o} s_{i} - 3c_{i} s_{i} + 4s_{i} )w \\ r_{28} = & - 8(s_{o}^{2} - 2s_{i} s_{o} + s_{i}^{2} - c_{o}^{2} - 2c_{i} c_{o} + 2c_{o} - c_{i}^{2} + 2c_{i} )w \\ r_{29} = & 2(3c_{o} s_{o} + 5c_{i} s_{o} - 4s_{o} - 5c_{o} s_{i} - 3c_{i} s_{i} + 4s_{i} )w^{2} \\ r_{30} = & 8(2c_{i} c_{o} s_{o} - c_{o} s_{o} + c_{i}^{2} s_{o} - c_{i} s_{o} - c_{o}^{2} s_{i} - 2c_{i} c_{o} s_{i} + c_{o} s_{i} + c_{i} s_{i} ) \\ r_{31} = & 8(c_{i} s_{o}^{2} - 3c_{o} s_{i} s_{o} - 3c_{i} s_{i} s_{o} + c_{o} s_{i}^{2} - 2c_{i} c_{o}^{2} + c_{o}^{2} - 2c_{i}^{2} c_{o} + 2c_{i} c_{o} + c_{i}^{2} ) \\ r_{32} = & - 8(2c_{i} c_{o} s_{o} - c_{o} s_{o} + c_{i}^{2} s_{o} - c_{i} s_{o} - c_{o}^{2} s_{i} - 2c_{i} c_{o} s_{i} + c_{o} s_{i} + c_{i} s_{i} )w \\ r_{33} = & (s_{o} - 3s_{i} )(3s_{o} - s_{i} ) \\ r_{34} = & - 8(c_{o} + c_{i} - 2)(s_{o} - s_{i} ) \\ r_{35} = & 4(c_{o} - c_{i} )^{2} \\ r_{36} = & - 2(s_{o} - 3s_{i} )(3s_{o} - s_{i} )w \\ r_{37} = & 8(c_{o} + c_{i} - 2)(s_{o} - s_{i} )w \\ r_{38} = & (s_{o} - 3s_{i} )(3s_{o} - s_{i} )w^{2} \\ r_{39} = & 4(2c_{i} s_{o}^{2} - s_{o}^{2} - 4c_{o} s_{i} s_{o} - 4c_{i} s_{i} s_{o} + 2s_{i} s_{o} + 2c_{o} s_{i}^{2} - s_{i}^{2} ) \\ r_{40} = & - 8(2s_{i} s_{o}^{2} - 2s_{i}^{2} s_{o} + 3c_{i} c_{o} s_{o} - 2c_{o} s_{o} + c_{i}^{2} s_{o} - 2c_{i} s_{o} - c_{o}^{2} s_{i} - 3c_{i} c_{o} s_{i} + 2c_{o} s_{i} + 2c_{i} s_{i} ) \\ r_{41} = & - 4(2c_{i} s_{o}^{2} - s_{o}^{2} - 4c_{o} s_{i} s_{o} - 4c_{i} s_{i} s_{o} + 2s_{i} s_{o} + 2c_{o} s_{i}^{2} - s_{i}^{2} )w \\ r_{42} = & - 8s_{i} s_{o} (s_{o} - s_{i} ) \\ r_{43} = & - 8(s_{o} - s_{i} )(c_{i} s_{o} - s_{o} - c_{o} s_{i} + s_{i} ) \\ r_{44} = & 8s_{i} s_{o} (s_{o} - s_{i} )w \\ \end{aligned}$$

where \(s_{i} = \sin \delta_{i}\), \(s_{o} = \sin \delta_{o}\), \(c_{i} = \cos \delta_{i}\), \(c_{o} = \cos \delta_{o}\).

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Nuñez, N.N.R., Florez, A.R., Vieira, R.S. et al. Dimensional synthesis of rack-and-pinion steering mechanism using a novel synthesis equation. J Braz. Soc. Mech. Sci. Eng. 45, 411 (2023). https://doi.org/10.1007/s40430-023-04317-4

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