5. NUMERICAL TESTING AND SIMULATION RESULTS
5.2. DYNAMICS SIMULATION RESULTS AND DISCUSSION
The mathematical model of the dragline machinery has been verified and tested using the correct angular displacements as described in Section 4.2. The results of the kinematic simulations generated in Section 5.1 are used as inputs for the dynamics simulation. The dynamic model contains three actuators, which are used to generate the required torques and force and operate the machine in a cycle time of 60 seconds. These actuators may provide simultaneously a swinging torque 𝜏𝜏1, a hoisting torque 𝜏𝜏2, and a dragging force 𝜎𝜎 or a hoisting and a dragging force depending on the underlying operational task. Thus, a dragline machinery can be regarded as a robotic excavator with three DOF. The solution of the dragline force and torques is a problem of inverse dynamics. The latter requires providing accurate values of the trajectory functions that must meet the machine limits in the course of the solution.
Generally, a loading cycle of a dragline starts when the operator has already positioned an empty bucket and engages the drag motor to begin excavating the materials. In the digging phase, the bucket penetrates the bank with a prescribed dragging velocity 𝑞𝑞̇7 of approximately -1.32 m/s. The digging extends for 15 seconds and terminates when the bucket is fully loaded with the materials. Previous studies have shown that the maximum loads are more likely to develop, in a dragline machinery front-end assembly, at the end of digging cycle (Nikiforuk and Ochitwa, 1964; Nikiforuk and Zoerb, 1966; Nichols et al., 1981). Thus, the simulation of the digging cycle is of particular interest and would result in valuable information about the dynamic loading of a dragline machinery. The developed dynamic model, in equation (3.87) in Section 3, is a robust model that helps to accurately predict the unknown dynamic loading.
The solution process of the dynamic model was given in Figures (4.8) and (4.9). It starts with the definition of the geometry of the dragline front-end assembly, masses and inertia of components, and prescribed trajectory inputs. The solution of the DAE, in equation (3.87), during digging eliminates the needs for the swinging torque, as the machine house is fixed during this phase. The dynamic solution algorithm includes the kinematic algorithm with its Baumgarte’s Stabilization Technique (BTS) and the dynamic equations of motion in equation (3.87). Mathematical functions are developed and contain all relevant information about the dynamic and kinematic analyses. The resistance force to cutting was included in the analysis and it follows the model given in equation (3.70). However, the developed friction model of the drag rope, in equation (3.80), was not included in the dynamic simulation based on the assumption of limited contact with the ground. Its parameters need further research and that is out of the scope of this dissertation.
The bucket payload model was developed using an input function given in equation (3.72). The maximum load (in kg), that the drag rope pulls at the end of the digging phase is depicted in Figure (5.8). It includes the bucket tare mass and the overburden (waste materials) mass, which is a maximum of 80,000 kg per the machine allowable payload (Nikiforuk and Zoerb, 1966).
Figure 5.8. Bucket mass variation during digging phase
During the digging operation, the bucket tip interacts with the ground and a resistance force to cutting the material develops and increases with time. The cutting resistance model is included and was developed according to Poderni (2003). It was assumed that the bucket has already made an angle, called “the carry angle ca” with the horizontal and its magnitude is 35°. This value was chosen on the basis that this angle provides a good estimate to the cutting force. It also reduces the bucket tipping-over during the digging and minimizes material spillage during a completely loaded bucket swinging onto the spoil piles. The resulting resistance force to cutting is calculated in accordance
with equation (3.69) and is plotted in Figure 5.9. Two components of the resistant cutting force affect the diggability of materials and the horizontal component is always maximum. The variation of the horizontal cutting resistance force is around 400 KN, whereas the maximum value of the horizontal component, at 4 seconds, is 100 kN. This model captures the complex digging scenarios when the bucket starts penetrating the ground. It can be seen that the vertical cutting force reduces with time and this is an indication of the reduction resistivity of ground to digging. The orientation of the bucket and its carry angle, as well as the fragmentation of the rock and its resistance to cutting and the bucket capacity influence this behavior.
Figure 5.9. Cutting resistance force during digging phase
During the empty-bucket lowering phase, the hoist rope carries most of the load, which includes its own weight, bucket weight, and a partial tension from the drag rope. On the other hand, the drag rope has less load acting on it. The operator releases the hoist clutch quickly and that results in increasing the hoist rope length. The drag motor is also
engaged to position the bucket properly. Both ropes change their lengths and their masses. This change also occurs during the digging phase, in which, the hoist rope becomes longer and the drag rope becomes shorter as in Figure 5.1 (a). This implies that hoist rope mass is linearly increasing with time, whereas the drag rope mass is linearly decreasing with time. In this dissertation, the dynamic simulation takes into account these simple variations to correct the pitfalls of using constant rope mass during the analysis. The variable masses of the hoist and drag ropes are shown in Figure 5.10 for given inputs of 𝑞𝑞8 and 𝑞𝑞7.
Figure 5.10. Variation of rope mass during digging: (a) hoist rope, and (b) drag rope
The dynamic model is developed to cover an operational period of 40 seconds, which encompasses the digging phase and the loaded bucket swinging phase. However, it can be extended to consider the full cycle with appropriate changes to the input functions. The solution of the dynamic model during digging only includes the drag force and hoist force at the beginning of excavation. That corresponds to the solution of the second and third equations in the complete dynamic model captured by equation (3.87). Figure 5.11 (a) shows the variation of the drag force with time and it follows a polynomial function.
Figure 5.11. Rope loads during digging and full-bucket swinging back motions
It can be seen that the drag rope has some tension at the beginning of digging to allow the orientation of the bucket tip towards the operator. The tension in the drag rope increases rapidly with time as the bucket is being filled with the materials. Also, the resistance to cutting increases, as shown in Figure 5.9, during an operational time span between 15 and 20 seconds. A maximum dragging force, which approximately measures 1.375 × 106 𝑁𝑁, occurs when the bucket has already moved for a period equivalent to three-quarters of the digging time (27 seconds).
The hoist force is also shown in Figure 5.11 (b) for the digging phase and the loaded bucket swinging onto the spoil piles. At the beginning of the digging phase, the hoist rope has some tension due to the empty- bucket weight. This force decreases with time to allow the bucket to move freely under the effect of the drag force. At 5 seconds of digging, the hoist force changes direction to hosit the bucket and to prevent it from being tipped over. The tension in the hoist rope increases with time until the bucket is filled at time 30 seconds into the digging operation. At the end of the digging phase, the bucket is lifted off the bank, which requires significant hoisting torque to lift a loaded bucket of 80,0000 kg. It can be seen that the maximum hoisting torque, during time interval [15-30] seconds is 690.39
KN.m. However, during the swinging of the loaded bucket onto the spoil pile, at interval of time [30-45] seconds, the maximum hoisting torque is 917.87 KN.m at time 40 seconds.
Nikiforuk and Zoerb (1966) reported the maximum allowabe drag and hoist forces on the dragline Marion 7800. The maximum drag force is 300,000 lb, which is equvailent to 136,077.7 Kg and 1,334 KN and the maximum hoist force is 277,000 lb, which is equvailent to 125,645.1 Kg and 1,232.5 KN. These values (in Figure 5.11) are within the machine limits. Thus, the dynamic model of the dragline front-end assembly is capable of generating accurate results. These values validate the results for the finite element analysis on wire ropes. The maxiumum drag force 1,375 KN was used as a buoundry condition in analyzing the maximum stresses in several wire rope constructions. The results of the stress analysis are provided in the Section 5.3 and for the geometries given in Figures 4.19-4.21 (see Section 4.4.1).