## 5 A Mathematical Model of a Platform Motion  
This section contains the mathematical model of the platform motion during its actuation. The aim of this model is to analyse the waveform of the main acting moments in the mechanism. The mathematical model is described by an equation of motion derived by means of [PERSON]'s equations of the second kind, which the general form is known as the following:  
\[\frac{d}{dt}\frac{\partial E_{k}}{\partial\dot{q}_{j}}-\frac{\partial E_{k}}{ \partial q_{j}}+\frac{\partial E_{D}}{\partial\dot{q}_{j}}+\frac{\partial E_{ p}}{\partial q_{j}}=Q_{j};j=1,2,\ldots,n, \tag{48}\]  
where \(E_{k}\) is the kinetic energy of a mechanical system, \(E_{D}\) is the Rayleigh dissipative function, \(E_{P}\) is the potential energy of a mechanical system, \(Q_{j}\) is the vector of external loads and \(q_{j}\) are generalised coordinates of the mechanical system. We apply this method for our solved mechanism based on the scheme in Figure 15. We can consider the simplified mechanical model of the platform, which consists of rigid bodies. As is identified in the previous chapter, the drive shaft of the gearbox is the most loaded component. Figure 15 depicted the entire platform mechanism as well as the detail of the driving system. Figure 14: Distribution of the stress in the analysed shaft for the maximal considered load. Figure 13: (a) Definition of the boundary conditions for calculation; (b) the shaft mesh model. We consider that the simplified mechanical system has one degree of freedom, i.e., the rotation of the drive-shaft marked as \(\varphi\). Moreover, we can consider, that other bodies of the platform perform translation movement in the \(x\)-axis direction. We need to substitute into [PERSON]'s equation of the second kind their first time derivation, which represent velocities as the following:  
* The rotational velocity of the drive-shaft is \(\dot{\varphi}\);
* The translational velocity of other components is \(\dot{x}\). Other rotating components of the platform have a negligible effect on the dynamic properties of the platform in terms of the investigated problem, and therefore their moments of inertia are much smaller in comparison with the moment of inertia of the drive-shaft. Based on the described assumptions, the mechanical system of the platform is described by means of one equation of motion. For its derivation, we need to identify the individual energies of the mechanical system (Equation (48)), which are as follows:  
* The kinetic energy \(E_{k}\): \[E_{k}=\frac{1}{2}\cdot I_{S}\cdot\dot{\varphi}^{2}+\frac{1}{2}\cdot m_{2}\cdot \dot{x}^{2},\] (49) where \(I_{S}\) is the moment of inertia of the drive-shaft and \(m_{2}\) is total weight of the platform. As we consider that the motion of the mechanical system of the platform will be described only by the one coordinate, namely the \(x\) coordinate, we must express the angle \(\varphi\) by a constraint equation. The platform is driven by the gearing of the drive-shaft without slipping. Thus, the translational velocity of the platform \(\dot{x}\) equals the peripheral velocity \(\dot{\varphi}\) of the drive-shaft on a pitch circle with the radius \(R_{G}\). The same relation is valid between translation motion of the platform \(x\) and angular deviation \(\varphi\) of the drive-shaft. Then, the constraint equation is:  
\[x=R_{G}\cdot\varphi. \tag{50}\]  
The final form of the kinetic energy is:  
\[E_{k}=\frac{1}{2}\cdot\left(I_{S}+m_{2}\cdot R_{G}^{2}\right)\cdot\dot{\varphi }^{2}. \tag{51}\]  
* The potential energy \(E_{P}\): \[E_{P}=m_{2}\cdot g\cdot h,\] (52)  
Figure 15: Indication of parameters for derivation of the equation of motion of the platform motion. where \(g\) is the gravitational acceleration and \(h\) is the total height which the platform must overcome during operation (Figure 1). Again, we must express the potential energy by means of the \(\varphi\) coordinate.