**5. Numerical modeling and simulation of the dipper-formation interaction**

The dynamic model is a system of ordinary differential equations (ODEs), which results from an iterative process and includes a number of ODE subprocesses. The ODEs are numerically solved in MATLAB using the embedded Runge-Kutta algorithm. The simulation model consists of MATLAB programs (.m files) and SIMULINK design-based models and submodels. The simulation model consists of the main model and sub-models. These sub-models define the dipper's trajectory, the crowd-arm extension and rotation, and the resistive forces (cutting forces, material, and dipper's weight) on the dipper. The following sub-models and main model are created:

**1.** *Test bench geometry and trajectory*: **Figure 4** shows the test bench geometry created for the digging process simulation. The excavated material characteristics can be selected for various digging conditions. A simulation step size is selected to make the dipper move with a constant linear velocity following field experimental results [17]. The failure surface is modeled as a quadratic function given by Eq. (27), and bench face is modeled as a straight-line function L(x). During the simulation process, the coordinates of the dipper tip (O4(x,y)) and the dipper depth into the working bench (d) are continuously computed at every time step using Eqs. (28) and (29), respectively:

$$\mathbf{y} = 0.9927\mathbf{x}^2 - 22.557\mathbf{x} + 117.68\tag{27}$$

$$O\_4\text{x}, \mathbf{y} = \left[ -0.4837t^2 + 2.4351t + 12.053, \qquad 0.9927\,\mathrm{^{\circ}x} - 22.557\,\mathrm{^{\circ}x} + 117.68 \right] \tag{28}$$

$$d = O\_4(\mathbf{x}) - L(\mathbf{x})\tag{29}$$


**Figure 4.** Representative bench geometry.

4. Determine the force (F*<sup>i</sup>*

5. Compute the joint torque (N*<sup>i</sup>*

of the link.

118 Lagrangian Mechanics

**interaction**

main model are created:

) acting on every link at the centroid of the link using and mass

) for every link.

**5. Numerical modeling and simulation of the dipper-formation**

acceleration of the centroid are computed for every link.

at every time step using Eqs. (28) and (29), respectively:

2 2

4 , 0.4837 2.4351 12.053, 0.9927 \* 22.557 \* 117.68 *Oxy t t* =- + + é ù *x x* - + ë û (28)

The force and torque are computed at the centroid of each link. Therefore, the velocity and the

The dynamic model is a system of ordinary differential equations (ODEs), which results from an iterative process and includes a number of ODE subprocesses. The ODEs are numerically solved in MATLAB using the embedded Runge-Kutta algorithm. The simulation model consists of MATLAB programs (.m files) and SIMULINK design-based models and submodels. The simulation model consists of the main model and sub-models. These sub-models define the dipper's trajectory, the crowd-arm extension and rotation, and the resistive forces (cutting forces, material, and dipper's weight) on the dipper. The following sub-models and

**1.** *Test bench geometry and trajectory*: **Figure 4** shows the test bench geometry created for the digging process simulation. The excavated material characteristics can be selected for various digging conditions. A simulation step size is selected to make the dipper move with a constant linear velocity following field experimental results [17]. The failure surface is modeled as a quadratic function given by Eq. (27), and bench face is modeled as a straight-line function L(x). During the simulation process, the coordinates of the dipper tip (O4(x,y)) and the dipper depth into the working bench (d) are continuously computed

<sup>2</sup> *y* = -+ 0.9927x 22.557x 1 17.68 (27)

*d O x Lx* = - <sup>4</sup> () () (29)

At each simulation step, the (x, y) trajectory coordinates and the area excavated (*A*c) are numerically computed using Eq. (30) [8] and built-in routines in MATLAB R2012a. This area is used to calculate the force (*f*1) due to the payload weight using Eq. (31) [8]. An optimization algorithm [8] is used to define the geometry of the payload based on the material distribution from Hemami [6]. The centroid of each material geometry, a polygon inside the dipper, is computed using a special algorithm [18]. This centroid is a dynamic point, which is used as the point of application for the dynamic force f1. This force (*f*1) is computed continuously at every instant of the excavation process:

$$A\_c = \frac{1}{2} \left(\boldsymbol{\chi}\_l - \boldsymbol{\chi}\_o\right)^2 \tan \alpha \mathbf{z} - \iint\_{\boldsymbol{\chi}\_o} \mathbf{f}\left(\mathbf{x}\right) d\mathbf{x} \tag{30}$$

$$f\_1 = A\_c a \rho \mathbf{g} \tag{31}$$

**4.** *Material resistive force f6*: The force, due to the weight of the dipper, is calculated during the digging cycle along the trajectory. The computation of centroids of payload geometry and dipper suggests that these two centroids can be considered concentric. Therefore, both f1 and f6 are combined into single force acting at the dynamic left of the payload geometry.


During this numerical simulation process, four of the six resistive forces (f1, f3, f4, f6) are computed as separate subsystems, while the other two resistive forces (f2 and f5) are set to zero. The resistive force f2 is set to zero by selecting an appropriate trajectory of the dipper [6]. The excavation trajectory is selected in such a way that the dipper stays clear off the material and does not compress the material. This assumption is reasonable in the sense that it involves proper bench geometric design and operator skill. An improper bench geometric design would lead to undue stresses on the shovel, which must be avoided during the excavation process. The force f5 represents the dipper and payload inertia. This force can be set to zero if the dipper moves through the material with a constant velocity and hence with zero acceleration. For this research, it is assumed that the dipper moves through the bench with a constant velocity and hence a zero acceleration. This assumption is consistent with the field observations [17] for hoist rope extension.
