**5. NUMERICAL TESTING AND SIMULATION RESULTS**

**5.1. KINEMATIC SIMULATIONS RESULTS AND DISCUSSIONS**

The kinematic model of the dragline front-end assembly was built using the concept of generalized coordinates and generalized motion variables (generalized speeds). It was verified and validated in Section 4 using real world data. Section 5 focuses on the numerical implementation of the digging phase and the loaded bucket swinging to dump material on spoil piles. This model contains all the relevant data and information about the linear and angular measurements of displacements, velocities, and accelerations of the machine house and the hoist, dump, and drag ropes. The operational cycle of a dragline usually starts when the dragline machinery, with its empty bucket, begins swinging back from the spoil piles to the digging area. In the kinematic simulations, the time is set to zero at the beginning of the digging phase.

As can be seen in Figure 5.1 (a), during the digging phase the hoist rope extends and the drag rope retracts to move the bucket towards the bank and excavate the overburden. The linear displacements of the hoist and drag ropes are given in equation (4.1). Thus, the linear velocities of the hoist and drag ropes are fixed and are of magnitude 2.54 m/s and β1.32 m/s, respectively. The linear and angular displacements of the drag and hoist ropes govern the motion of the dragline bucket and constrain it from colliding with the machine house or the boom. Figure 5.1 (b) shows an agreement among the linear and angular displacements of both ropes. The hoist rope trajectory increases with the digging time as a result of its extension to permit more mobility to the bucket. Releasing the hoist clutch and engaging the drag motor cause the drag rope to start dragging the bucket. This rope gets shorter as the bucket is partially submerged in the bank under its weight and the weight of the filled materials in the loading process. This occurrence causes

an additional angular displacement to occur in the drag rope and it improves the diggability of the bucket. It can also be seen from Figure 5.1 (b) that the variation of the angular displacement of the hoist rope is less than that for the drag rope and this behavior is also governed by the profile of their linear displacements. The initial angular displacement of the hoist rope is zero and that for the drag rope is -30 Β°, which is the slope of the digging face. All angular displacements are consistent with the machine operating limits and the model is accurate for simulating the real kinematic operations of a dragline.

Figure 5.1. Displacements of hoist and drag ropes: (a) linear and (b) angular

The trajectory of the rigging system is characterized by a polynomial function of
third order and it is depicted in Figure 5.2. This trajectory was established based on the
numerical integration of the nonlinear constraints differential algebraic equation (4.4) at
the acceleration level. The trajectory q_{5} is also a representative of the bucket trajectory
according to the assumption that the bucket is rigidly attached to the rigging system. The
variation of the trajectory function of the dump rope ππ_{5} is relevant to its behavior in a real
digging event. It begins with positive orientation and changes rapidly to position the bucket

properly against the digging phase. At the end of the digging, it has a steady slow change to eliminate spilling of the filled materials. The field observations have also shown that the orientations of the dump rope slightly change during the hoisting of a loaded bucket. In addition, the dump rope and boom are parallel to each other during the swinging of the loaded bucket onto the spoil piles. This feature plays a key role in stabilizing the bucket carry angle and provides a momentary dynamic balancing until the dumping happens.

Figure 5.2. Angular displacements of dump rope

The behavoir of the angular velocity of the hoist and drag ropes is shown in Figure 5.3 (a) and it captures the real operation. The hoist rope is not under direct tension from the hoist motor and its velocity changes at a slow rate. On the contrary, the drag rope rotates at higher velocity to peferctly position the bucket during the digging operation. The angular velocity of the dump rope is plotted in Figure 5.3 (b) and it is much higher than that for the hoist and drag ropes. This is because the dump rope has more mobility to rotate than other ropes.

Figure 5.4 shows the variations in the angular accelerations of the hoist, drag, and dump ropes. These accelerations provide valuable information about the dynamics of the dragline machinery during the digging phase. It can be seen that all ropes rapidly change the acceleration within the first 4 seconds of the digging time and then deceleration occurs during the bucket loading process. The acceleration profiles for each rope approximately reaches steady-state after 7 seconds when the bucket is almost filled and is ready to be lifted off the bank.

Figure 5.4. Angular accelerations: (a) hoist and drag ropes and (b) dump rope

(a) (b)

Figure 5.3. Angular velocities: (a) hoist and drag ropes and (b) dump rope

The acceleration curves for the drag and dump ropes have decreasing profiles at
different rates. This behavior indicates that both ropes operate consistently during the
digging phase. It can be explained that the dump rope accelerates to quicly respond to the
motion of the drag rope. As the bucket slides on the ground, the acceleration of the dump
rope gradually decreases to 1 deg/s2. It was seen in Section 4.1.1.1 that the initial
conditions of the trajectory functions ππ_{4}, ππ_{5 }and ππ_{6} have significant impact on the kinematic
solutions. This impact also affects the results of the dynamic simulations. Wrong initial
values are more likely to give incorrect solutions, especially when dealing with a stiff
mathematical DAE model. The invariants of the kinematic model are a key to performing
full and acceptable kinematic and dynamic simulation experiments. They provide
consistency in the behavior of the model during the integration and reduce the possibility
of increasing the drift error.

The mathematical model also provides information about the velocity and acceleration variations of ropes with respect to their resulting trajectories, as shown in Figure 5.5. The experiment shows that the hoist rope velocity is minimal when its trajectory is β 40 Β°, as given in Figure. 5.5 (a). The hoist rope, at this angle, is close to the boom and its angular speed is already decreased to avoid collision with it. The angular velocity of the dump rope is much faster than that of the hoist rope allowing for a quick adjustment to the rigging system, as depicted in Figure 5.5 (b). Figure 5.5 (c) shows the variation of the angular velocity of drag rope versus its trajectory and it is similar to that of the hoist rope. When the bucket is filled, the drag rope angle is maximum and the velocity is minimal to prevent the bucket from hitting the machine.

(a) (d)

(b) (e)

(c) (f)

Figure 5.5. Angular velocity versus trajectory: (a) hoist, (b) dump, and (c) drag ropes angular acceleration versus trajectory: (d) hoist, (e) dump, and (f) drag ropes

The behavior of the dynamic model is also investigated during the digging phase for the acceleration and deceleration of the ropes. Figure 5.5 (d, e, f) show the resulting angular acceleration of each rope versus its trajectories at time interval [0, 10] seconds. The angular acceleration profile is steep at the beginning of the excavation process and it gradually decreases through the excavation process. The acceleration is minimal at the end of digging and this behavior is explainable from an operational viewpoint. The rapid change in the accelerations, at the beginning of the digging phase, may be attributable to the stiffness of the constraint, velocity and acceleration equations. The initial conditions of the constraint and velocity functions are also the cause of such behavior. The acceleration of a structural member in the front-end assembly is due to its interaction with other members. Equations (3.42) and (3.43) provide an explanation to this interaction.

Figure 5.6. Angular displacement versus linear displacement for hoist and drag ropes

The variations in both the hoist and drag rope lengths also affect their trajectories during the digging. They also change the trajectory of the dump rope, whose length is fixed

for all experiments. Figure 5.6 (a) shows that by increasing the hoist rope length ππ_{8}, its
trajectory increases and it has a maximum magnitude of -22.5Β° at 100 m. The drag rope
trajectory also increases with time and it has a maximum magnitude of -72.5Β° at 62 m of
its length, as given in Figure 5.6 (b). At the end of digging, the operator switches the control
from digging to full-bucket swinging motion and the drag rope extends while the hoist rope
retracts. Thus, the behavior of the trajectory functions depends on the underlying task that
the dragline machinery performs and on its technological limitations.

Figure 5.7. Filled bucket trajectory in a 3 D space during swinging-back phase Machine

House

Bucket Drag

The front-end assembly of a dragline machinery, with its three ropes (hoist, dump,
and drag), governs the bucket motion in a (x-z) plane of digging as shown in Figure (3.2)
(see Section 3). The ropes also determine the trajectory of the bucket in a 3D space during
a loaded bucket swinging operation. The resulting trajectories of the kinematic model are
also used to simulate the bucket motion after completing the digging phase. In a loaded
bucket swing motion, the bucket moves in the x-z plane and changes its coordinates in a
global reference frame, as depicted in Figure 5.7. The trajectories ππ_{1}, ππ_{4} and ππ_{6}
simultaneously change with time to return the filled bucket to the dumping area. It can be
seen that the machine house makes a rotational displacement of -80Β°, whereas the
trajectories of the drag and hoist ropes make 22.5Β° and 25Β°, respectively.