## 2 Models of the Drilling Torque  
### Previous Models  
Typically, the drilling torque model, \(T\) is practically expressed in an exponential form as [32]:  
\[T=C\cdot f^{\alpha}\cdot d^{\beta} \tag{1}\]  
where \(C\), \(\alpha\), \(\beta\) are constants for particular workpiece material and cutting conditions. The torque is dependent on the drilling diameter, \(d\) and the feeding rate, \(f\) but independent of the drilling depth. The form of the drilling torque model as shown in Equation (1) can be found in most engineering textbooks and handbooks (See [33; 34; 35]. It can be useful to predict the drilling torque in normal drilling, where the depth-to-diameter ratio is smaller than 5 [10]. In such kinds of drilling, the chip evacuation force is small and thus can be neglected.  
In the torque model for deep hole drilling of [PERSON] et al. [8], the drilling torque can be considered as representative of the chip-evacuation performance. In this model, the total drilling torque was considered as a combination of two parts: the cutting torque and the chip-evacuation torque. The cutting torque was considered to be constant and independent of the drilling depth. The chip-evacuation torque increases with the drilling depth as a result of the growing force required to move the cumulative amount of chips that fill the flutes. The chip-evacuation torque, \(M\) was expressed in an exponential form as:  
\[M=\frac{R\mu_{w}S_{w}F_{c}(0)}{B}\left(e^{(kBD/A_{0})z}-1\right) \tag{2}\]  
where \(R\) and \(D\) are the radius and diameter of the drill bit, respectively; \(\mu_{w}\) is the friction coefficient between the chips and hole wall; \(S_{W}\) is the interface length between the chip and the hole wall; \(F_{c}(0)\) is the initial chip-evacuation force when the depth, \(z\) is equal to zero; \(B\) and \(A_{0}\) are the parameters dependent on the workpiece material and other characteristics of the cross section of the chip and the tool flutes as well as the friction force among the chips, the flutes, and the hole wall. The constant coefficient \(k\) is a workpiece material property, reflecting the ratio between the lateral and axial pressures from the chip onto the hole wall and the tool flutes.  
The [PERSON]'s model of the chip-evacuation torque was applied in the study of [PERSON] et al. [18] for drilling simulation of bone material in orthopedic surgery. It can be seen that the [PERSON]'s model includes various parameters, which are difficult to measure or need abundant experimental works to be calibrated. Since the geometrical parameters of a specific drill and the value of each parameter of the model vary with cutting conditions, it is difficult to measure or identify each parameter individually. A recent investigation by [PERSON] et al. [14] tried to reduce the number of the parameters of the chip-evacuation torque model. According to the work of [PERSON] et al., the chip-evacuation torque, \(T_{ch}(z)\) was computed using the following practical model:  
\[T_{ch}(z)=K_{tch}\left(e^{K_{ch}z}-1\right)\frac{D}{2} \tag{3}\]  
where \(K_{tch}\) and \(K_{ch}\) are coefficients, determined from calibration tests by fitting experimental data, \(D\) and \(z\) are the tool diameter and the drilled depth, respectively.  
This simplified model makes it easier to be practically applied since it consists of only two coefficients, \(K_{tch}\) and \(K_{ch}\), which can be experimentally calibrated for particular cutting conditions. However, the two-coefficient model still causes difficulties in comparing several processes. Moreover, all previous models either neglected the fluctuated component in chip-evacuation torque or applied a low pass filter with small threshold frequency to cut off this component, leading to an omission of a substantial part of the total torque. The new model proposed below aims to address these issues.