## 6 Analysis of the Platform Deflection  
As the STN EN 81-40 standard allows the maximal deflection (\(w_{max}\)) of the platform to be \(w_{max}\) = 20 mm [22], it is necessary to analyse this deflection. In the device operation, such a situation can happen that the total load of the platform will be transmitted to columns either by one tube (upper or bottom) or the load will be distributed to both tubes. For both exceptional cases, dependencies of the load were made (Figures 15 and 16). These dependencies influence the deflection of the end platform point when the distance between columns is changed. The authors introduce an example of the calculation in the case of transmitting of loads by means of the upper guiding tube. The load, which is caused by the platform weight and the transported goods, represents only one component of force \(A_{yll}\) (the reaction force due to the own weight load of the platform and the goods), because the axis force \(F_{oll}\) is considered in the pinion gearing (this is the drive force of the pinion and it acts in the platform motion direction), i.e., it influences the strain of the bottom guiding tube. Figure 20 shows the considered loading forces. The acting force \(A_{yll}\) is depicted in the cutting-plane \(A\)-\(A\). For the relation for calculation of deformation, we can use a simplified model of a beam with two supports representing columns. These supports remove three degrees of freedom. A distribution of the resulting bending moment and the lateral force regarding to the upper guiding tube is shown in Figure 21.  
Figure 21: A simplified model of a beam and the distribution of the bending moment and the lateral force. Figure 20: Waveform of acceleration of the platform mechanism for time interval of 0.0–0.5 s.  
Forces marked as \(F_{H\!I}\) and \(A_{x\!I}\) cause deformation in two different planes. Therefore, we analyse deflections individually (we use the superposition method) and subsequently, we determine the resulting value of the deflection by means of the Pythagorean theorem as follows:  
\[\sum_{i=1}^{n}F_{iy}=0\Rightarrow R_{1}+R_{3}-F_{k}^{II}=0\,, \tag{60}\]  
\[\sum_{i=1}^{n}F_{iz}=0\Rightarrow R_{2}=0\,, \tag{61}\]  
\[\sum_{i=1}^{n}M_{iR_{1}}=0\Rightarrow R_{3}\cdot l-F_{k}^{II}\cdot\frac{l}{2}=0\,. \tag{62}\]  
We get Equations (63)-(65) for the calculation of reaction force from Equations (60)-(62), respectively:  
\[R_{3}=\frac{F_{k}^{II}}{2}, \tag{63}\]  
\[R_{2}=0, \tag{64}\]  
\[R_{1}=F_{k}^{II}-R_{3}=F_{k}^{II}-\frac{F_{k}^{II}}{2}=\frac{F_{k}^{II}}{2}. \tag{65}\]  
Calculation of deformation and lateral forces is performed applying the [PERSON]'s theorem, which comes from Figure 22.  
Then, we proceed as follows:  
* The first section: Figure 21:0 (66) The bending moment \(M_{O(z_{1})}\) is: Figure 21:1 (67) and the lateral force \(T_{(z_{1})}\) is calculated as: Figure 21:2 (68)  
Figure 21:3 Application of the [PERSON]’s theorem: (**a**) a first cut of the upper guiding tube, (**b**) a second cut of the upper guiding tube. * The second section: Figure 21:4 (69) The bending moment \(M_{\text{O}(z_{2})}\) is: Figure 21:5 (70) and the lateral force \(T_{(z_{2})}\) is calculated as: Figure 21:6 (71) The entire calculation is performed by means of the differential equation of the elastic curve as follows: Figure 21:7 (72) After integration and substituting corresponding parameters, the differential equation of the elastic curve is: Figure 21:8 (73) where Figure 21:9 and Figure 20:0 are sections of the analysed track and Figure 20:1 is the force acting on the track in the analysed sections. The complete calculation is very extensive; therefore, it cannot be completely described. Comparing results shown in Figures 23 and 24, we can see that both load cases are very similar even with the same values of the deflection. Therefore, we assume that the load transmission by this tube does not have the significant effect on the limit value of the platform deflection.