Optimum dimensional synthesis of planar mechanisms with geometric constraints
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ORIGINAL PAPERS
Optimum dimensional synthesis of planar mechanisms with geometric constraints V. Garcı´a-Marina R. Ansola
. I. Ferna´ndez de Bustos
. G. Urkullu
.
Received: 18 January 2020 / Accepted: 19 September 2020 Ó Springer Nature B.V. 2020
Abstract The deformed energy method has shown to be a good option for dimensional synthesis of mechanisms. In this paper the introduction of some new features to such approach is proposed. First, constraints fixing dimensions of certain links are introduced in the error function of the synthesis problem. Second, requirements on distances between determinate nodes are included in the error function for the analysis of the deformed position problem. Both the overall synthesis error function and the inner analysis error function are optimized using a Sequential Quadratic Problem (SQP) approach. This also reduces the probability of branch or circuit defects. In the case of the inner function analytical derivatives are used, while in the synthesis optimization approximate V. Garcı´a-Marina (&) Department of Mechanical Engineering, Faculty of Engineering of Vitoria-Gasteiz, University of the Basque Country (UPV/EHU), Nieves Cano 12, 01006 Vitoria-Gasteiz, Spain e-mail: [email protected] I. Ferna´ndez de Bustos G. Urkullu R. Ansola Department of Mechanical Engineering, Faculty of Engineering of Bilbao, University of the Basque Country (UPV/EHU), Alamedade Urquijo s/n, 48013 Bilbao, Spain e-mail: [email protected] G. Urkullu e-mail: [email protected] R. Ansola e-mail: [email protected]
derivatives have been introduced. Furthermore, constraints are analyzed under two formulations, the Euclidean distance and an alternative approach that uses the previous raised to the power of two. The latter approach is often used in kinematics, and simplifies the computation of derivatives. Some examples are provided to show the convergence order of the error function and the fulfilment of the constraints in both formulations studied under different topological situations or achieved energy levels. Keywords Optimum dimensional synthesis Nodal coordinates Deformation energy Geometric constraints Mathematics Subject Classification 65F30 1504 6504
65F05
1 Introduction 1.1 Background The design of mechanisms is a complex field in mechanical engineering, and many methods for general mechanism design have been proposed in the literature. Some among them are found in [1] for kinematic synthesis, an Assur Group based method for type kinematic synthesis and Genetic Algorithms for dimensional kinematic synthesis as in [2], an Exact
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Meccanica
Gradient method in [3] for dimensional kinematic synthesis, Neural Networks for dimensional kinematic synthesis [4], Geometric Constraint Programming for kinematic synthesis as in [5] and [6] among others, or Kinematic Mapping for kinematic synthesis in [7], Circular Proximity Function for optimal synthesis in [8], or synthesis of constrained chains in [9]. Given that
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