Search

In this article, using the hybrid method defined by Nakajo and Takahashi [17], we first obtain a strong convergence theorem for two noncommutative nonlinear mappings in a Banach space. Next, using the shrinking projection method defined by Takahashi, Takeuchi and Kubota [25], we prove another strong convergence theorem for the mappings in a Banach space. Using these results, we get well-known and new strong convergence theorems by the hybrid method and the shrinking projection method in a Hilbert space and a Banach space.

In this study, we developed prototypes of boring tools with anisotropic dynamic stiffness, and their chatter stability was investigated analytically and experimentally. In Part 1, a novel design of boring tools with an anisotropic structure was proposed to improve the nominal stiffness in boring operation, thus resulting in higher chatter stability in simulation. In this part (Part 2), anisotropic boring tools with a holder length-to-diameter ratio (L/D) of 4 and 10 designed in Part 1 were prototyped, and their frequency response functions were evaluated. Then, the chatter stabilities were evaluated through turning experiments. With respect to a L/D4 boring tool with anisotropic structure, the nominal dynamic stiffness was significantly improved within the range of machining conditions that satisfies the appropriate combination of cutting force ratio and chip flow direction, compared to a conventional tool with isotropic structure. We confirmed that the proposed boring tool with an anisotropic structure increases the critical radial depth of cut by approximately 17 times compared with conventional tools. Even with an L/D10 anisotropic boring tool, a similar effect was observed wherein the dynamic stiffness of the tool was improved. In contrast, the effect of improving the dynamic stiffness was insufficient; thus, chatter free cutting could not be realized. Analytical investigations verified the importance of further improvement of dynamic characteristics. To realize stable boring with L/D10 boring tools, it is necessary to further reduce the system compliance while also improving the similarity of the frequency response function.

This paper presents a novel design method of the anisotropic structure to attain infinite dynamic stiffness to avoid chatter vibration in boring operations. Because a long and slender tool is used for boring operations, the stiffness of the tool holder is likely to decrease, resulting in low chatter stability. Although it is difficult to improve the stiffness of the boring holder itself, the nominal dynamic stiffness for the cutting process can be improved by designing an appropriate anisotropy in the dynamic stiffness of the boring tool. In this study, we formulate a theoretical relationship between the mechanical structural dynamics and chatter stability in boring operation and present the basic concept of tool design with anisotropic structure. In the actual tool design, ideal anisotropy may not be realized because of the influence of design error. Therefore, an analytical study was conducted to clarify the influence of the design error on the vibration suppression effect. Analytical investigations verified that the similarity of the frequency response functions in the modal coordinate system and the design of the compliance ratio according to the machining conditions are important. Furthermore, we designed a boring tool with an anisotropic structure which can achieve the proposed anisotropic dynamics. The frequency response function was evaluated utilizing FEM analysis. The estimated anisotropic dynamics of the proposed structure could significantly improve the nominal dynamics for boring operations.