**2. Shovel resistance forces and modeling**

**1. Introduction**

108 Lagrangian Mechanics

during its operation.

**Figure 1.** Nomenclature of a cable shovel.

Cable shovels are used as primary excavation equipment in large-scale surface mining operations. The overall efficiency of shovel-truck surface mining operations is largely dependent on shovel efficiency. Dipper payloads of the shovels have seen an increasing trend over the years, and current shovels have payloads in excess of 100 tons per scoop [1, 2]. The payloads, combined with dipper weight, rigging, and variable material diggability, result in varying mechanical energy inputs and stress loading of the shovel's front-end assembly across the working bench. Furthermore, the repeated shovel loading and unloading cycles induce fatigue stresses in shovel components. The induced stresses over time may exceed the yield strength of steel/material of the shovel leading to fatigue failure, teeth losses, and boom and handle cracks. High stresses and fatigue failure in shovel front-end assembly cause unplanned downtimes resulting in reduced efficiency and increased production costs. Dipper-related problem can be a significant contributor to the shovel downtime [3]. The current practice for the shovel front-end assembly

Electric rope shovel consists of the lower, upper, and the front-end assembly as illustrated in **Figure 1**. The lower assembly consists of the propel drive and crawler systems and provides a solid and stable base for the excavator. This helps excavator propel, reposition, and relocation

The shovel's upper assembly is a roller and left-pin system mounted on the lower mechanism. The upper assembly consists of multiple decks with housing for the hoist and swing machinery and electronic control cabinet on the lower deck and the operator's cab on the upper deck. Additionally, the upper assembly provides a platform for boom attachment and the counter weight for the dipper. The front end consists of the boom, crowd machinery, dipper handle,

repair is based on experience and history rather than science.

The dipper excavation processes can be categorized into penetration, cutting, and scooping processes [4, 5]. Penetration is the insertion of a tool into a medium, and cutting is the lateral movement of a tool, executed at a constant depth. The dipper teeth penetrate the formation, and the lip cuts the material. Excavation models are based on the formation resistive forces acting on the cutting tool. The resistive forces combine the cutting forces at the dipper teeth and lip and the excavation forces due to material movement along, ahead, and inside the dipper. Both the experimental and analytical models are based on these resistive forces. The model proposed by Hemami [6] is by far the most comprehensive model and consists of six component forces (f1 to f6), which must be overcome during excavation, as in **Figure 2**. All these forces, except f6, are dynamic forces. The six forces acting on the dipper, from the initial to the end point on trajectory, consist of the following:

f1: The force required to overcome the payload weight in and above the dipper

f2: The resultant resistive force due to material movement toward the dipper

f3: The friction force between the bucket walls and the excavated material as it slides into the dipper

f4: The resistance to cutting and/or penetrating that acts at the dipper tip and side walls

f5: The inertia force of the material inside and above the dipper

f6: The force required to move the empty dipper (modeled as part of f1)

**Figure 2.** Forces on a dipper during excavation [7].

The forces, f1 and f5, are the dynamic forces [7], where f1 changes both in magnitude and the point of application, and f5 depends on the bucket acceleration. The force (f6) was originally defined as a part of f1 and f5. The dipper payload force (f1) is the dominant force for the largecapacity dippers [6, 8, 9]. Awuah-Offei et al. [8] proposed a model based on the Balovnev [10] excavation model using the six forces. The force (f2) can be set to zero [6]. Forces, f3 and f4, are the cutting forces and can be combined as a single force and estimated using the empirical model [11] given by Eq. (1).

This empirical model is a result of extensive experimentation on frozen soils [11]. z is the coefficient that accounts for the blade impact on cutting force, which depends on w and d. **Table 1** is used to estimate z for d (between 25 cm and 50 cm). z increases as d decreases, and it also depends on the ratio Ts/Tw (Ts is the spacing between the teeth, and Tw is the tooth width). **Table 2** lists the multiplying factors for z based on Ts/Tw. Force f5 can also be set to zero if the dipper moves with a uniform velocity through the muck pile. Force f6 can be modeled as part of f1:



**Table 2.** Dependence of z on Ts/Tw [11].

$$\mathbf{P} \mathbf{=} 10 \mathbf{C}\_o \text{ d}^{1.35} \text{ (1+2.6w)} (1+0.0075 \mathbf{\hat{\beta}} \mathbf{\hat{\beta}}) \text{z} \tag{1}$$
