## 5 A Mathematical Model of a Platform Motion
When we take into account the inclination angle \(\alpha\), we obtain the equation:  
\[h=x\cdot\sin\alpha \tag{53}\]  
and, considering the Equation (50), we obtain the final form of the potential energy as the following:  
Figure 15.0  
\(\bullet\) The Rayleigh dissipative function \(E_{D}\):  
Figure 15.1  
where \(b\) is the coefficient, which includes energy losses in the mechanical system. In this case, it represents mainly friction losses in the mechanism which we can also call passive resistances. Now, we derivate all energies in accordance with the Equation (41) and we obtain the final form of the equation of motion:  
Figure 15.2  
where \(M_{h}\) is the drive torque. Solving of the derived equation of motion of the platform mechanism is performed using the Matlab 2017a program (MathWorks Inc., Natick, MA, USA). The driving moment is the input to the solution by means of an input file, which contains characteristics of the driving electromotor. For calculation we consider the initial conditions as the following:  
* The platform starts from rest,
* For time \(t=0\) s, the initial angular velocity \(\dot{\varphi}=0\) rad \(\cdot\) s\({}^{-1}\), the initial angular deviation \(\varphi=0\) rad,
* Time interval for calculation was set for 0.5 s.  
Figures 16-18 show waveforms of the driving torque of the electromotor, the torque of passive resistances and the torque of slope resistance for weight \(m_{2}\) of 225 kg, 300 kg and 375 kg, respectively. Figure 15.3 Waveforms of investigated torques for the weight \(m_{2}=225\) kg. These figures show that waveforms of the investigated torques correspond to values introduced in Figure 15.4 Individual torques are marked as the following: \(M_{h}\) is the drive torque of the electromotor, \(M_{PR}\) is the torque of the passive resistance, \(M_{\rm S}\) is the torque of slope resistance and \(M_{\rm axi}\) is the total resistance torque, i.e., \(M_{\rm axi}=M_{\rm S}+M_{PR}\). We can observe that the start of the platform motion is accompanied with increasing drive torque of the electromotor. Then, the drive torque gradually decreases, while the equilibrium of the total resistance torque and the drive torque is reached. Further waveforms depict the steady state of the platform motion. It is obvious from the figures, that achieving equilibrium of the drive torque and the torque of total resistance depends on the weight. This weight causes directly proportional inertial effects of the mechanism. The change of the weight from 225 kg to 375 kg increases the achieving the steady state approx. twice. Figure 19 shows acceleration of the platform for its starting. We can see that acceleration of the platform reaches high values. This phenomenon is caused due to the fact that we do not know the exact characteristics of the control system used for controlling the electromotor. Finally, it should be mentioned, that the wavy waveforms of the drive torques in Figures 15-17 and acceleration in Figure 18 are caused due to set errors in the integration method used for calculation of the mathematical model as well as due to dynamic effects of the platform motion. Figure 15.5 Waveforms of investigated torques for the weight \(m_{2}=300\) kg. Figure 15.6 Waveforms of investigated torques for the weight \(m_{2}=375\) kg.