## 2 Experimental and Methodology
### Background
The physical principle of conservation of energy states that the energy input to the hydraulic cylinder must be equal to the energy output, as shown in Figure 3 and expressed in Equation (3). The input energy for the hydraulic cylinder is interpreted as the hydraulic energy of the fluid's movement and pressure characteristics. The intensity of the hydraulic energy depends on the hydraulic valve opening. In addition, the output energy is measured as the kinetic and potential energy produced by the hydraulic cylinder. The presented mathematical model is developed to overcome the challenge of including external disturbances of the forming process and frictional characteristics in the analysis of the hydraulic press. It is important to emphasize that the mathematical model does not include factors such as heat losses, internal fluid friction losses, pressure losses, mechanical losses, and other minor effects on the overall system dynamics. This deliberate simplification is intended to maintain the simplicity of the model. \[E_{cyl,in}=E_{cyl,out}\ \rightarrow\ \ E_{cyl,out}=E_{cyl,kinetic}+E_{cyl,potential} \tag{3}\]  
Consider the time interval \(t_{T}=[t_{i},t_{i+1}]\) from the initial \(i\)-th event occurring at time \(t_{i}\) to the \((i+1)\)-th event occurring at time \(t_{i+1}\). To further explain this phenomenon, the hydraulic energy depends on the opening of the hydraulic valve and can be determined as a multiplication of the hydraulic fluid volumetric flow rate in the hydraulic cylinder chambers \(Q_{A}\)= \(\{Q_{A,1},\ \ldots,\ Q_{A,n}\}\), \(Q_{B}\)= \(\{Q_{B,1},\ \ldots,\ Q_{B,n}\}\) and the pressure in the hydraulic cylinder chambers \(p_{A}\)= \(\{p_{A,1},\ \ldots,\ p_{A,n}\}\), \(p_{B}\)= \(\{p_{B,1},\ \ldots,\ p_{B,n}\}\) in the interval \(t_{T}\) as follows:  
\[E_{flow}=t_{T}\int_{t_{i}}^{t_{i+1}}(Q_{A}\cdot p_{A})\ dt+t_{T}\int_{t_{i}}^{t_ {i+1}}(Q_{B}\cdot p_{B})\ dt \tag{4}\]  
According to the principle of [PERSON]'s law, in a hydraulic system, a fluid pressure \(p_{i}\) exerted on a piston with a surface area \(A_{i}\) generates the force \(F_{i}\). During various conditions in the sheet metal bending cycle, the pressure in the hydraulic cylinder chambers \(p_{A}\) and \(p_{B}\) changes dynamically and exerts on the piston surface \(A_{1}\) and \(A_{2}\) of the hydraulic cylinder, producing the forces \(F_{A}\)= \(\{F_{A,1},\ \ldots,\ F_{A,n}\}\) and \(F_{B}\)= \(\{F_{B,1},\ \ldots,\ F_{B,n}\}\). During the movement of the hydraulic cylinder, the frictional force \(F_{\mu}\) acts on the energy losses and is the effect of the frictional characteristics of the sealing and guiding technology of the hydraulic cylinder as well as the frictional characteristics in the guides of the hydraulic press. In addition, the dynamic external force \(F_{E}\)= \(\{F_{E,1},\ \ldots,\ F_{E,n}\}\) acts in the opposite direction to the movement of the hydraulic cylinder and is the cause of the forming process. In the context of a hydraulic cylinder, the generated hydraulic cylinder force \(F_{C}\) is the  
Figure 3: Energy conversion in a hydraulic cylinder. resultant of the forces \(F_{A}\), \(F_{B}\), and \(F_{\mu}\) acting in opposite directions and can be expressed as follows:  
\[ \begin{array}{ll}F_{C}(t)&=\big{|}F_{A}(t)-F_{B}(t)-F_{\mu}(t)\big{|}\\ &=\big{|}p_{A}(t)\cdot A_{1}-p_{B}(t)\cdot A_{2}-F_{E}(t)-F_{\mu}(t)\big{|}\end{array}  \tag{5}\]  
To determine the output energy of a hydraulic cylinder, both the kinetic and the potential energy must be considered. The potential energy is evident by [PERSON]'s second law, which considers the union of the hydraulic cylinder force \(F_{C}\) and hydraulic cylinder displacement \(X_{C}\) in the time interval \(t_{T}\), since the hydraulic cylinder displacement \(X_{C}\) is not constant during the sheet metal bending cycle. The kinetic energy is associated with the motion of the object and is proportional to the mass of the hydraulic cylinder and the square of the hydraulic cylinder velocity \(v_{C}\).