# Energy Consumption Model for Drilling Processes Based on Cutting Force
## 2 Energy Consumption in Drilling Processes
### The Cutting Power  
In this paper, cutting force was adopted to calculate cutting power as shown in Figure 2. The cutting forces in drilling can be modelled as [24; 25]:  
\[\mathrm{d}F_{t}(z)\ =\ K_{tc}\mathrm{d}A+K_{te}\Delta b \tag{3a}\] \[\mathrm{d}F_{f}(z)\ =\ K_{fc}\mathrm{d}A+K_{fe}\Delta b\] (3b) \[\mathrm{d}F_{r}(z)\ =\ K_{rc}\mathrm{d}A+K_{re}\Delta b \tag{3c}\]  
where \(\ K_{tc}\), \(\ K_{fc}\), and \(\ K_{rc}\) (N/mm\({}^{2}\)) and \(K_{te}\), \(K_{fe}\), and \(K_{re}\) (N/mm\({}^{2}\)) are the specific cutting and edge force coefficients; \(\mathrm{d}A\) (mm\({}^{2}\)) is an area of a chip differential lip removed, and it can be calculated by \(\mathrm{d}A(z)=\ \Delta b\)-\(h\); \(h\) (mm) is the chip thickness, and it can be calculated by \(h=\ \hat{\sharp}sin\kappa_{t}\); \(\Delta b\) is the chip width, and it can be calculated by \(\Delta b=\ \frac{dz}{cos\kappa_{t}}\); \(\kappa_{t}\) is the taper angle; and \(z\) (mm) is the axial position.  
The components of the elemental cutting force components (\(\mathrm{d}F_{t}\), \(\mathrm{d}F_{f}\), \(\mathrm{d}F_{r}\)) can be calculated in the \(x\), \(y\), \(z\) directions (see Figure 2):  
\[\left[\begin{array}{c}\mathrm{d}F_{x}(z)\\ \mathrm{d}F_{y}(z)\\ \mathrm{d}F_{z}(z)\end{array}\right]=\left[\begin{array}{ccc}-cos\theta& sin\gamma_{d}&-sini\\ -sin\theta&-cos\gamma_{d}cos\kappa_{t}&sini\text{-}sin\gamma_{d}cos\kappa_{t}\\ 0&cos\gamma_{d}\text{-}sin\kappa_{t}&cosi\text{-}cos\kappa_{t}+sini\text{-} sin\gamma_{d}sin\kappa_{t}\\ \end{array}\right]\left[\begin{array}{c}\mathrm{d}F_{t}\\ \mathrm{d}F_{f}\\ \mathrm{d}F_{r}\end{array}\right] \tag{4}\]  
Figure 1: The power profile in a drilling process.  
where \(\theta\) is the angle between the cutting velocity and the \(x\)-axis; \(\gamma_{d}\) is the angle among the two velocity components (a component perpendicular to the cutting edge and a component of the cutting speed in the \(y\)-axis); \(i\) is the angle between the cutting velocity and normal to the cutting edge.  
Energy consumption during the drilling process is caused by the relative motion of the tool and workpiece material. Rotary drive and feed drive are the main modes of this relative movement. Modelling two different modes of motion can be used to calculate the cutting energy consumption of the drill during machining.  
In the rotary motion, since d\(F_{f}\) and d\(F_{r}\) are perpendicular to the cutting speed, they do not contribute to the generation of rotational motion power, and the differential powers consumed at each tiny finite element cutting edge in tangential, feed, and radial directions can be established as:  
\[\mathrm{d}P_{rotation,t}\ =\ \mathrm{d}F_{t}\cdot V(z) \tag{5a}\] \[\mathrm{d}P_{rotation,f}\ =\ 0\] (5b) \[\mathrm{d}P_{rotation,r}\ =\ 0 \tag{5c}\]  
where \(V(z)\  
Substituting Equation (4) into Equation (8), Equation (8) can be expressed as:  
\[\mathrm{d}P_{feat}=\ \mathrm{d}F_{z}(z)\cdot f/60000 \tag{9}\]  
Finally, the instantaneous cutting power consumption can be calculated by:  
\[\begin{array}{ll}P_{cutting}=\ P_{feat}&+P_{rotation}=\int\limits_{\mathrm{d }P_{feat}+\int\limits_{\mathrm{d}P_{rotation}}}\\ &=\ f/60000\cdot\int\limits^{\mathrm{o}}\mathrm{d}F_{z}(z)+\int\limits^{ \mathrm{o}}V(z)\cdot\mathrm{d}F_{t}(z)\end{array} \tag{10}\]