## 3 Suggesting Rolling Force on a Single Disc Cutter for the Design of Cutterhead Drive
### Suggestion of Relationships between Normal and Rolling Force Acting on a Single Disc Cutter  
Taking the developed relationships between thrust force and torque into account, the relationships between normal and rolling force were acquired using equations developed in the NTNU model [3]. Normal force was calculated by dividing thrust force by number of disc cutters and rolling force was calculated based on torque, as shown in Equations (1)-(3), considering relative distance between disc cutters, diameter of cutter head, and number of disc cutters. \[F_{n}=T_{thrust}/N \tag{1}\]  
Figure 8: Relationships between thrust force and torque at different RPM levels of cutterhead, (**a**) At 1.5 RPM; (**b**) At 4.5 RPM; (**c**) At 6.0 RPM; (**d**) At 7.5 RPM. Figure 9: Regression lines of the relationships between thrust force and torque according to the UCS (Uniaxial compressive strength) of rock mass. here, \(F_{n}\) is the normal force on disc cutter (kN), \(T_{thrust}\) is the thrust force (kN), and N is the number of disc cutters. \[T=r_{mc}\times D_{TBM}\times N\times(F_{n}\times k_{c})/2 \tag{2}\]  
\[T=r_{mc}\times D_{TBM}\times N\times F_{r}/2 \tag{3}\]  
here, \(T\) is the torque (kN.m), D\({}_{TBM}\) is diameter of cuttherhead (m), N is the number of disc cutters, \(F_{r}\) is the rolling force on the disc cutter (kN). As mentioned before, the torque measured during these tests was calculated based on the oil pressure from the operating hydraulic pumps, because torque was not found to be acting on the disc cutters. Therefore, the calculated normal and rolling forces were taken as the average values acting on all the disc cutters in cutterhead. The relationships between thrust force and torque mainly for the conditions of 4.5 RPM and above were integrated into a single relationship because of the almost absolute similarity between them and the red line in Figure 10 shows the relationships between normal and rolling forces obtained from them according to four UCS values. In order to visualize a comparison between various cases and methods, the relationships obtained at 1.5 RPM and the two limit lines obtained by the NTNU model are presented in Figure 10. [PERSON] (2000) [3], at NTNU, suggested that the limit lines ranged between 0.12 to 0.08. Figure 10: _Cont._  
In this study, the regression lines obtained in case of 4.5 RPM and above have shown different trends as compared to those for 1.5 RPM and limit lines, but this difference tends to become smaller with the increase in UCS of rock mass. However, when the TBM machine is planned to be operated specially at slower rotational speed of the cutterhead when facing the fractured zone of rock mass, then this machine should be designed following the relationships at 1.5 RPM considering the development of higher torque values. The trends of \(k_{c}\) values, defined as the ratio of rolling force to normal force (slopes of regression lines in Figure 10), obtained according to the UCS of artificial rock mass are shown in Figure 11. The \(k_{c}\) value obtained in 27 MPa (Test# 1) of artificial rock mass was about 0.10, which helps in deducing that the developed rolling force is about 10% of the normal force. The \(k_{c}\) value obtained in 99 MPa (Test# 4) of artificial rock mass was 0.055 deducing the developed rolling force to be about 5.5% of the normal force, which is almost half as compared to that developed in the 27 MPa case. In this study, for the cutterhead rotational speed exceeding 4.5 RPM and above, the \(k_{c}\) values are suggested to be lying between 0.10 and 0.04 according to the UCS ranging between 27 to 200 MPa, as shown in Figure 11.  
\[k_{c}=-0.03\text{ln}(\text{UCS})+0.2 \tag{4}\]  
Figure 8:0 Relationships between normal and rolling force on single disc cutter derived from regression analysis at different UCS levels of artificial rock mass, (**a**) In 20 MPa UCS; (**b**) In 50 MPa UCS; (**c**) In 70 MPa UCS; (**d**) In 100 MPa UCS. here, \(k_{c}\) is the ratio of rolling force to normal force.