# Mathematical Study of a Product-Gripping Mechanism for Industrial Transportation
## 2 Materials and Methods  
In this study, a crane tong lifting system consisting of binary elements clamped together by means of coupling was investigated (Figure 1). This crane tong lifting system is found in the literature in various forms and is used for clamping and transporting various types of materials. The working steps used in the study of the gripping and lifting mechanism are as follows:  
* Identify the mathematical calculation relationships required to determine the value of various components of the mechanism. Only their final forms are presented in this paper. * Realization of the mechanism operation by means of a simulation program. For this purpose, the use of the Linkage program (free program) was chosen. * A comparison was made between the values obtained using the two work methods: mathematical and simulation. The following programs were used to study the motion of this mechanism:  
* Mathcad v.15 (PTC Mathcad) [52] was used to perform mathematical calculations using the computational relations corresponding to the coordinates of the analyzed points. * Linkage v. 3.11.3 (developed by [PERSON]) [53] was used to simulate the movement of the studied mechanism and to verify the correctness of the results obtained from the mathematical calculation. Figure 2 shows the schematic of the clamping and lifting mechanism. It should be noted that this representation (Figure 2) does not correspond to reality (the mode of operation and use is that of Figure 1). The mechanism is rotated 90\({}^{\circ}\) in order to occupy less space in this article. The geometrical components of the mechanism under study are as follows:  
* Coupling (Figure 2): Representation of the crane tong system. The fixed position of coupling A with coordinates (0, d), where the coordinate is 0 on the OX axis, and for the OY axis, it is d (distance from the origin of the OY axis to coupling A). Coupling B--mechanism articulation. Coupling D--fixed and positioned at the origin of the coordinate system. Coupling C--rigid, to which the CE element is connected. Coupling E--the end of the CE element, representing the zone of contact between the lifting mechanism and the moved product. Binary elements (Figure 3):  
Element AB, for which its dimensions are a. Element BC, for which its dimensions are b. Element CE, for which its dimensions are c.  
Element BD, for which its dimensions are e.  
Angle \(\theta\) is the angle generated by elements BC and CE (Figure 3). The value of the angle is given by the device and can be modified according to the type and nature of the material used. The CE element is rigidly attached to the BC element. Because the analyzed mechanism is composed of two symmetric structural groups, it was chosen to analyze the mechanism presented in Figure 3. As in the case of Figure 2, the mechanism shown in Figure 3 is flipped (rotated 90\({}^{\circ}\)) to occupy less space within the article. The geometrical analysis of the mechanism under study is based on the following premises:  
The mechanism is analyzed in relation to a coordinate system with an origin corresponding to point D.  
Points A and D are on the same axis OY. The distance between points A and D varies. The movement of point A allows the closing or opening of the gripping claws (EC element) by decreasing or increasing the distance between point E and the vertical OY axis, thus allowing the operation of gripping or detaching the load. The calculation relations obtained for determining the movement executed by the various components of the mechanism are given below:  
Figure 3: Representation of the geometrical characteristics of the mechanism under analysis. As a result of the movement of point A, several elements of the mechanism change their position. Also, the angles that these form in relation to the OX axis will vary in value.