# Estimation of the Frequency Response Function of the Rotational Degree of Freedom
## 1 Introduction  
To optimize the machining process and respond to abnormal conditions, it is essential to analyze clutter stability using the dynamic characteristics of the equipment and cutting tools. A chatter stability lobe diagram is utilized for chatter stability analysis, and an essential parameter is the dynamic characteristic of the machining system. The dynamic characteristics of machining are significantly affected by the equipment, material, and process characteristics, and the dynamic characteristics change as the cutting tool changes. The types of cutting tools utilized in processing equipment are very diverse, and it is very inefficient to change the cutting tools to analyze the dynamic characteristics of various cutting tools, perform vibration tests of processing equipment, and obtain dynamic characteristics. Acceptance coupling substructure analysis (RCSA) decomposes a complex structure into relatively simple substructures and obtains the frequency response function (FRF) of the substructures using experiments or FEM. Subsequently, by synthesizing the FRF of the substructure, the FRF of the entire system is obtained, and the dynamic characteristics are predicted. Regarding a machining center, the FRF of the entire machining equipment system is obtained by synthesizing the FRFs of the cutting tool and machining equipment, and the dynamic characteristics are predicted. Through the RCSA of various cutting tools, the FRF and dynamic characteristics of the equipment can be predicted. It is more efficient in terms of time and cost to change solely the cutting tool and perform a vibration test rather than change the cutting tool repeatedly and perform a vibration test on the entire machining equipment [1; 2; 3; 4; 5; 6]. To perform RCSA, the FRF for translational and rotational degrees of freedom of the substructure is required. In previous studies, errors occurred when RCSA was performedusing only the translational FRF. When RCSA is performed using FRF in the rotational direction, the accuracy is improved [2; 3; 4; 6]. However, it is difficult to obtain the rotational displacement and moment during the actual vibration test, and there is a disadvantage, in that the cost of additional experimental equipment is incurred. Rotational FRF has been known as a problem since 1972, as can be seen in the paper of [PERSON] and [PERSON], and many studies to estimate accurate rotational FRF are still in progress [5; 7]. Although FEM can be achieved without major problems, finding rotational FRF through practical experiments is rarely achieved. Experimentally, the block, mass additive, finite difference, estimation, angular transducer, and laser techniques have been used [5; 6; 7; 8; 9; 10]. In this study, the problem was analyzed from an experimental point of view. Experimentally, the finite-difference method is used, which is low in cost and relatively simple to obtain accurate results, does not require additional equipment, and performs post-processing using translational FRF. When estimating the rotational FRF using the finite difference method, it is difficult to select the order of the finite difference method and the distance of the response point. In previous studies, only the order of the finite difference method was studied. In addition, there is only the suggestion of the distance of the measurement point without considering the size of the structure and the size of the sensor [6]. However, in this study, a suitable ratio of the measuring point distance to the size of the structure and the maximum measuring distance considering the sensor size during actual testing are presented. The FRF of rotational degrees of freedom was estimated and analyzed using the FRF of translational degrees of freedom according to the response distance [6; 7; 8; 9]. To estimate and verify the rotational FRF, a vibration analysis was performed using Abaqus, a commercial program.