A new design methodology for four-bar linkage mechanisms based on derivations of coupler curve

• A four-bar mechanism is optimized based on derivation of a coupler curve.
• Two-step optimization is performed to minimize the RMS error of the derivatives.
• Three case studies are provided to verify the effectiveness of the proposed method.
• The method can take account of the output velocity with a constant velocity input.

This paper presents a new design methodology for crank-rocker four-bar linkages based on differential objective functions. Firstly, to avoid obtaining a mechanism with an unintended trajectory shape, we classified the trajectory of the linkage into four types of shapes. Two-step optimization for a specific shape type is proposed. In the first step, the shape of the trajectory is obtained by minimizing the root mean square error (RMSE) between the slope (first-order derivative) of the coupler point and that of the target trajectory. The size of the trajectory is then determined by minimizing the RMSE between the change in angle of slope (second-order derivative) of the coupler point and that of the desired trajectory. This new approach has three advantages: i) the desired trajectory can be set as a continuous and closed loop, ii) the optimal solution can be obtained without the possibility of generating a mechanism with an unintended coupler curve, and iii) the method can take account of the velocity of the output for each section with a constant input velocity. Three case studies were conducted to verify the advantages of the new approach based on a new index called the goodness of traceability.