## 4 Guiding Forces Estimated from Empirical Formulas  
The standard in [26] with the commentary in [39] allowed the estimation of guiding forces. The forces from the roller guides were straightforward to calculate provided that the stiffness of the rolling guides and the operating range of the guideway were known. Forces from the sliding plates involved many assumptions and more complex calculations. The value of forces from the sliding plates is given by Formula (1):  
\[F=\alpha_{n}\left(P_{b}\frac{400m_{e}v^{2}e}{L^{2}}\right) \tag{1}\]  
where \(\alpha_{n}\) is the conveyance impact factor (corresponding to the dynamic amplification factor); in the absence of data, it is assumed to be equal to 2.0; \(m_{e}\) is the proportion of the conveyance mass that effectively acts as a slipper plate; \(L\) and \(v\) are the guide span (button-to-button) and the winding velocity, respectively; \(P_{b}\) is the sliding plate's load coefficient read from the nomogram in [26], which takes into account the stiffness of the steelwork at the mid-span of the guide and at the connection to the button; \(e\) is a measure of the misalignment of the steelwork.  
Many simplifications are assumed in this formula. One of them is the approximation of the extraction conveyance to a cuboid with the mass of the conveyance including the excavated material (Figure 6).  
Figure 6: Simplified diagram of the skip.  
Depending on the directions, the mass portion of the conveyance to the force acting on the guide is calculated. The mass contribution when acting in the face direction (F\({}_{\rm C}\)) is  
\[m_{e}=\frac{m_{s}\cdot I_{z}}{\left(I_{z}+m_{s}{h_{y}}^{2}\right)} \tag{2}\]  
In the side direction (F\({}_{\rm B}\)), it is  
\[m_{e}=\frac{m_{s}\cdot I_{x}I_{y}}{\left(I_{x}I_{2}+m_{s}I_{x}{h_{y}}^{2}+m_{s} I_{y}{h_{x}}^{2}\right)} \tag{3}\]  
where \(m_{s}\) is the mass of the conveyance including the excavated material, and \(I_{x}\), \(I_{y}\), and \(I_{z}\) are the mass moments of inertia about the \(x\), \(y\), and \(z\) axes (Figure 6).  
The physical interpretation of the magnitude of the nonlinearity coefficient \"e\" is shown in Figure 7.  
Such a rail deformation is actually rarely observed; the primary occurring deformations are discontinuous shifts on adjacent guides. This is also very difficult to measure. In practice, it is assessed on the basis of the maintenance category of the shaft (maintenance category). Its lowest value is 5 mm for shafts with a high level of maintenance. Because of the requirements of Polish regulations [30], all the shafts in operation in Poland are included in this category.  
For the shaft under consideration, the mass of the skip including the excavated raw material was \(m_{s}\) = 34,716 kg. The mass moments of inertia, assuming a uniform mass distribution with respect to the axis, were \(I_{x}\) = 701,214 kg\(\cdot\)m\({}^{2}\), \(I_{z}\) = 669,476 kg\(\cdot\)m\({}^{2}\), and \(I_{y}\) = 39,907 kg\(\cdot\)m\({}^{2}\). Assuming nominal thickness of the steelwork elements and the articulated connections of the steelwork elements (as recommended in [26]), the stiffness (understood as the value of the force causing unit displacement) in the middle of the guide was \(k_{g}\) = 5.56 \(\times\) 10\({}^{6}\) N/m, and at the point of connection with the button, it was \(k_{b}\) = 37.60 \(\times\) 10\({}^{6}\) N/m. The distance between the buttons was \(L\) = 4.5 m, and the coefficient was \(P_{b}\) = 0.1. For the speed of \(v\) = 16 m/s, the face guiding force from the sliding plate was \(F_{c}\) = 44.09 kN.