(An) optimal design method of a four-bar linkage considering shape and velocity distribution of trajectory

Abstract

Four-bar linkage is one of the most versatile and useful mechanical devices. It is used as a transmission device like film advance mechanism or mechanical structure such as watts linkage. Because the shapes of the output motion are various though the input is a simple rotary motion, the four-bar linkage as a transmission device has high utilization. Nevertheless, research about the shape and velocity of trajectory synthesis, that figure out the four-bar linkage when the desired shape and velocity of trajectory is given, is very limited. This dissertation presents a new optimal design methodology for the shape and velocity of trajectory synthesis of four-bar linkage.

This dissertation is composed of two parts. The first part contains the development of a new design method for the shape and velocity of trajectory synthesis of four-bar linkage. The second part contains the practical applications of the proposed method. The method proposed in the first part consists of three steps. In the first step, the algorithm checks whether or not the shape of the generated trajectory of four-bar linkage belongs to the desired shape category. In this dissertation, we set a hypothesis that the trajectory of crank-rocker four-bar linkage is classified into four categories, and verified it. After the generated trajectory is passed the first step, the shape of trajectory is determined in the second step. The shape of the trajectory is obtained by minimizing the root mean square error (RMSE) between the slope (first-order derivative) of the generated trajectory and that of the target trajectory. At the final step, the size of the trajectory is then determined by minimizing the RMSE between the change in angle of slope (second-order derivative) of the generated trajectory and that of the desired trajectory. This proposed approach has three advantages: i) the desired trajectory can be set as a continuous and closed loop, ii) using the desired shape category, 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.

Because the design method of this dissertation is numerical method, the optimization algorithm to figure out the optimal link lengths of the four-bar linkage is required. In this research, the new hybridization optimization algorithm is also presented. The algorithm is called hybrid Taguchi-random coordinate search algorithm (HTRCA). The HTRCA combines two algorithms of the Taguchi method (TM) and the random coordinate search algorithm (RCA). The RCA adjusts one variable simultaneously with two directions and various step sizes in random order to escape a local minimum. Since the RCA modifies one variable at a time according to the random order, the result of the RCA is sensitive to an initial condition of variables. The TM was adopted to generate nearly optimal initial conditions. Using the TM, the approximate optimal value can be found in even multi-modal functions. Also, the TM is used with different setting values to escape a local minimum point by changing two or more variables in one step. By combining the two methods, a global optimal value can be found efficiently. Seven test functions were optimized and the robust and efficient performance was verified by comparison with other hybrid optimization algorithms. Finally, conventional path synthesis (not the proposed path synthesis) of four-bar linkage is done by the developed optimization algorithm, and the result is compared to previous works on the same problem with evolutionary algorithms.

Three case studies were conducted to verify the advantages of the new design methodology with HTRCA based on a new index called the goodness of traceability.

The practical application of the proposed method is presented in the second part of the dissertation. That is rotational transmission mechanism for the automatic tool changer of the tapping machine. The mechanism is based on the ducal four-bar linkages. One four-bar linkage start the contact with output plate, the other four-bar linkage loses contact. To doing so, the output plate can rotate continuously. The proposed mechanism is totally different from the mechanism using single four-bar linkage the motion of which is intermittent. In case of the rotational transmission mechanism using single four-bar linkage, the shape of the trajectory is the only thing to be considered. However, in case of dual four-bar linkage, both the shape and the velocity of the contact and non-contact paths of the trajectory are considered. In this part, the design methodology, for the new rotational transmission mechanism using the method proposed in the first part, is proposed. The design is mainly based on kinematic and singularity analysis and optimization. Dynamic simulation and prototype results are given for validation. Finally, the mechanism is applied to the tapping machine. From this case study, I conclude that the proposed methodology can be adopted to many engineering problems to figure out the four-bar linkage when the desired trajectory is given.