Mechanics of Actuated Disc Cutting

Abstract

This paper investigates the mechanics of an actuated disc cutter with the objective of determining the average forces acting on the disc as a function of the parameters characterizing its motion. The specific problem considered is that of a disc cutter revolving off-centrically at constant angular velocity around a secondary axis rigidly attached to a cartridge, which is moving at constant velocity and undercutting rock at a constant depth. This model represents an idealization of a technology that has been implemented in a number of hard rock mechanical excavators with the goal of reducing the average thrust force to be provided by the excavation equipment. By assuming perfect conformance of the rock with the actuated disc as well as a prescribed motion of the disc (perfectly rigid machine), the evolution of the contact surface between the disc and the rock during one actuation of the disc can be computed. Coupled with simple cutter/rock interaction models that embody either a ductile or a brittle mode of fragmentation, these kinematical considerations lead to an estimate of the average force on the cartridge and of the partitioning of the energy imparted by the disc to the rock between the actuation mechanism of the disc and the translation of the cartridge on which the actuated disc is attached.

Appendices

Appendix 1: Calculation of Angular Position of Left end of Contact Segment

The angle $\varphi _{L}(\theta )$, which identifies the left extremity of the contact arc, and thus, the contact angle $\psi (\theta )=\frac{1}{2}(\varphi _{L}(\theta )-\varphi _{R}(\theta ))$ can be determined from geometric and trigonometric considerations. Point L is the intersection of the disc perimeter and the cutting front at the end of the previous actuation.

By reference to Fig. 19, it can be observed that $\varphi _{\tilde{R}}(\tilde{\theta })$ is the angular position of point L in terms of the angle $\tilde{\theta }$ identifying the position of the disc centre at the previous iteration. In other words, $\tilde{\theta }$ is the position of the disc centre at the previous actuation cycle when the right extremity of the contact arc was at the current point L.

Fig. 19

Contact length of disc cutter with rock

Fig. 20

Schematics of determining the boundaries of ADC/rock interaction

The angle $\varphi _{L}(\theta )$ can be determined by solving a system of three equations involving the two auxiliary unknown $\tilde{\theta }$, $\varphi \tilde{_{R}}$ that are themselves functions of $\theta$:

$$\begin{aligned}&\epsilon \left( \cos \theta -\cos \tilde{\theta }\right) +\cos \varphi _{L}-\cos \varphi _{\tilde{R}}=0 \end{aligned}$$

(33)

$$\begin{aligned}&\epsilon \left( \sin \theta -\sin \tilde{\theta }\right) +\sin \varphi _{L}-\sin \varphi _{\tilde{R}}+\epsilon \upsilon (\theta -\tilde{\theta })=0 \end{aligned}$$

(34)

$$\begin{aligned}&\tan \varphi _{\tilde{R}}=\frac{\sin \tilde{\theta }}{\cos \tilde{\theta }+\upsilon } \end{aligned}$$

(35)

Equations (33) and (34) are obtained by projecting the vectors $\overrightarrow{\tilde{S}S}$, $\overrightarrow{\tilde{S}\tilde{O}}$, $\overrightarrow{\tilde{O}L}$, $\overrightarrow{SO}$, $\overrightarrow{OL}$ on the x- and the y-axis, respectively, while enforcing that

$$\begin{aligned} \overrightarrow{\tilde{S}S}+\overrightarrow{SO}+\overrightarrow{OL}=\overrightarrow{\tilde{S}\tilde{O}}+\overrightarrow{\tilde{O}L} \end{aligned}$$

(36)

and noting that $|\tilde{S}S|=\upsilon (\theta -\tilde{\theta })$. Equation (35) simply expresses that point L is on the cutting front at the previous actuation.

Expressing $\cos \varphi _{\tilde{R}}$ and $\sin \varphi _{\tilde{R}}$ in terms of $\tan \varphi _{\tilde{R}}$ given by (35) yields

$$\begin{aligned} \cos \varphi _{\tilde{R}}=\frac{\cos \tilde{\theta }+\upsilon }{\sqrt{1+2\upsilon \cos \tilde{\theta }+\upsilon ^{2}}},\quad \sin \varphi _{\tilde{R}}=\frac{\sin \tilde{\theta }}{\sqrt{1+2\upsilon \cos \tilde{\theta }+\upsilon ^{2}}} \end{aligned}$$

(37)

Substituting the above expressions (37) in (33) and (34) and writing that $\cos ^{2}\varphi _{L}+\sin ^{2}\varphi _{L}=1$ lead to an implicit equation in $\tilde{\theta }$, which can be numerically solved to give $\tilde{\theta }=\tilde{\theta }(\theta ;\epsilon ,\upsilon )$. Finally, $\varphi _{L}(\theta ;\epsilon ,\upsilon )$ is deduced from (33) and (34) by expressing $\tan \varphi _{L}$ as

$$\begin{aligned} \tan \varphi _{L}=\frac{\sin \varphi _{\tilde{R}}-\epsilon \left( \sin \theta -\sin \tilde{\theta }\right) -\epsilon \upsilon (\theta -\tilde{\theta })}{\cos \varphi _{\tilde{R}}-\epsilon \left( \cos \theta -\cos \theta ^{'}\right) } \end{aligned}$$

(38)

where the right-hand side of (38) is now a function of the form $f(\theta ;\epsilon ,\upsilon )$ and can be evaluated numerically for every selected values of $\theta$, $\epsilon$ and $\upsilon$.

Appendix 2: Determination of Transition Angles $\theta _{\mathrm{I}}$, $\theta _{\mathrm{II}}$, and $\theta _{\mathrm{III}}$

The angular boundaries between the start $\theta _{\mathrm{I}}$ and end $\theta _{\mathrm{III}}$ of cutter/rock interaction can be determined based on the kinematics of the cutter. As described in the main text, phase I commences as soon as the edge of the disc makes contact with the cutting trace from previous actuation cycle. This is when the locus of disc centre intersects with its path as illustrated in Fig. 5a. The interaction is, hence, initiated with a discrete point of contact at L, and the boundary point $\theta _{\mathrm{I}}$ can then be determined based on the coordinates of the intersection point and by solving equation below for $\theta _{\mathrm{I}}\in [-\pi ,0]$:

$$\begin{aligned} \sin \theta _{\mathrm{I}}+\upsilon \left( \theta _{\mathrm{I}}-\pi \right) =0 \end{aligned}$$

(39)

Phase II commences at the moment at which points R and $R'$ catch up, which is when $x_{R}=x_{R'},\,y_{R}=y_{R'}$ (Fig. 20a). With OR and $\tilde{O}R$ being of equal lengths (radius of the disc) in the isosceles triangle $O\tilde{O}R$, it can be written that $\varphi =2\pi -\tilde{\varphi }$. Replacing this expression in $x_{R}=x_{R'}$ yields $\epsilon \left( \cos \theta -\cos \tilde{\theta }\right) +\cos \varphi -\cos \tilde{\varphi }=0$ and simplifying this equation leads to $\cos \theta =\cos \tilde{\theta }$ and $\theta =2\pi -\tilde{\theta }$. Similarly, substituting $\varphi =2\pi -\tilde{\varphi }$ and $\theta =2\pi -\tilde{\theta }$ in the equation for $y_{R}=y_{R'}$ and simplifying it yields

$$\begin{aligned} \frac{1}{\epsilon }\sin \tilde{\varphi }+\sin \tilde{\theta }+\upsilon \left( \tilde{\theta }-\pi \right) =0 \end{aligned}$$

(40)

Using Eq. 37, $\sin \tilde{\varphi }$ can be rewritten as $\epsilon \sin \tilde{\theta }/v_{O}$, where $v_{O}=\epsilon \sqrt{1+2\upsilon \cos \tilde{\theta }+\upsilon ^{2}}$. Substituting this expression for $\sin \tilde{\varphi }$ and replacing $\tilde{\theta }=2\pi -\theta$ in (40) yields the equation to be solved for $\theta _{\mathrm{II}}\in \left[ -\pi ,0\right]$:

$$\begin{aligned} \sin \theta _{\mathrm{II}}+\upsilon \left( \theta _{\mathrm{II}}-\pi \right) +\frac{\sin \theta _{\mathrm{II}}}{v_{O}}=0 \end{aligned}$$

(41)

Similarly, as shown in Fig. 20b, the interaction between the disc and the rock ceases when points $L\equiv R$ . In isosceles triangle $RO\tilde{O}$, $OR=O\tilde{R}$ so it can be concluded that $\varphi =2\pi -\tilde{\varphi }$. Replacing this in the equation for $x_{R}=x_{L}$ gives $\cos \theta =\cos \tilde{\theta }$. The acceptable solution is $\theta =4\pi -\tilde{\theta }$, considering Fig. 20b. Replacing $\theta$ and $\varphi$ in the equation for $y_{R}=y_{L}$ and substituting $\sin \tilde{\varphi }$ by $\epsilon \sin \tilde{\theta }/v_{O}$ and $\tilde{\theta }$ by $4\pi -\theta _{\mathrm{III}}$ yields the equation to be solved for $\theta _{\mathrm{III}}$:

$$\begin{aligned} \sin \theta _{\mathrm{III}}+\upsilon \left( \theta _{\mathrm{III}}-2\pi \right) +\frac{\sin \theta _{\mathrm{III}}}{v_{O}}=0 \end{aligned}$$

(42)