# Mathematical Study of a Product-Gripping Mechanism for Industrial Transportation
## 1 Introduction  
Different material handling systems are used to carry out the transport process in industrial activities, such as individual, bulk, palletized, or containerized items. These systems have been adapted considering the dimensional characteristics of the material/product (i.e., the size, shape, density, and surface condition of the product being transported), as well as the operator's requirements and local resources [1, 2, 3, 4, 5, 6, 7]. The systems for handling the transportation of products involve carrying out three main operations: loading, moving, and unloading the product [8]. The equipment used to carry out these operations, in addition to the source generating the movement and the load-bearing elements, also has systems for clamping the material so that it can be handled as easily as possible. A number of different devices are used in order to safely grip and handle the product, such as textile lifting slings; lifting-and-lashing rings; lifting rings; lifting clamps; lifting beams; chain grabs; overhead crane grabs; general hoists and electric hoists; electric and manual winches; hydraulic lifting platforms; lifting magnets; general clamps and clamps for curbs and pipelines; vacuum grabs; forklift grabs; gripping systems for forkkifts; gripping systems for slewing cranes; and gripping systems for hydraulic cranes [9, 10, 11]. These product clamping systems are based on a series of mechanisms (mechanical components) and hydraulic or electrical components, which makes it possible to carry out simple or complex movements, i.e., clamping or unclamping [12; 13; 14; 15; 16; 17; 18]. Referring only to the mechanical systems used for explanations from a mathematical point of view and to optimize the movements performed by these elements, different methods of analysis have been used, starting from mathematical, vectorial, or computerized analysis (using a series of programs to simulate the movement of the elements and to identify the stresses occurring in the active elements of the mechanism) [13; 15; 16; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46]. In this article, we will study, from a mathematical point of view, the movement of a simple mechanism for gripping materials, which is a mechanism used mainly in transportation installations. Crane tongs (Figure 1) are used to lift different types of loads. These types of mechanisms are characterized by the fact that the clamping force of the product is minimal (which is why it cannot generate deformations on the product's surface), which is realized by means of gravitational forces, the clamping force directly proportional to the weight of the supported product [38; 47]. From a constructional point of view, these devices are designed to hold the product via clamping or supporting operations [48; 49]. Due to the simplicity of their construction, the mechanical crane tong lifting systems found in transport installations have not been studied, which is why this article focuses on such mechanisms. The specialized literature is rich in studies presenting theoretical analyses and simulations aimed at identifying the deformations that occur in different clamping systems encountered in different fields [21; 25; 26; 27; 28; 29; 30; 35; 41; 42; 43; 51]. Standard analyses of a mechanism involve decomposing it into small structural groups (of at least two elements) and identifying the mathematical relationships in them [24; 24; 25; 26; 27; 28; 29; 30; 36; 39; 44; 45; 46]. In this article, we did not use this method. The whole mechanism, which contains three elements, was taken into consideration, and its motion was studied. The aim of the article is to present a mathematical analysis of the behavior of the mechanical elements that make up such gripping and transport systems used in various industrial fields (handling of railway rails, blocks of material, sheet metal rolls, pipes, battened products, boxes, etc.) and beyond.