# Study on Multi-Span Tension Coupling Relationship of Gravure Printed Electronic Equipment
## 2 Multi-Span Tension Model  
The multi-span web transmission system is composed of multiple single-span web transmission units connected in series, which is the most common form of web transmission system. In gravure printed electronic equipment, the printing tension subsystem is located between the unwinding tension subsystem and the rewinding tension subsystem. Its input is the output of the unwinding tension subsystem, and its output is the input of the rewinding tension subsystem. In the printing section, the tension system is coupled with the registration system, and the former is the foundation of the latter. The tension system of the printing section is mainly composed of several printing units sequentially arranged, and the adjacent printing units constitute a printing span, as shown in Figure 2. Therefore, the tension system of the printing section is essentially a typical multi-span web tension system, but a drying device is installed in each printing span. Figure 2: Schematic diagram of web transmission in the tension system of the printing section. In the gravure printed electronic equipment, due to the absence of additional registration adjustment mechanisms, the web length of each printing span remains unchanged. Therefore, based on the single-span web tension model in reference [21], combined with the relationship between the drying system and web tension, the multi-span tension system model for the printing section can be established:  
\[\left\{ \begin{array}{l}[L_{p1}+L_{Dp1}(\frac{E_{E}}{E_{D}}-1)]\frac{\text{d}T _{p1}(t)}{\text{d}t}=[AE_{E}-T_{p1}(t)]R_{p1}\omega_{p1}(t)-[AE_{E}-T_{u2}(t)]R _{p0}\omega_{p0}(t)\\ [L_{p2}+L_{Dp2}(\frac{E_{E}}{E_{D}}-1)]\frac{\text{d}T_{p2}(t)}{\text{d}t}=[ AE_{E}-T_{p2}(t)]R_{p2}\omega_{p2}(t)-[AE_{E}-T_{p1}(t)]R_{p1}\omega_{p1}(t)\\ \ldots\\ [L_{p\text{m}}+L_{Dpn}(\frac{E_{E}}{E_{D}}-1)]\frac{\text{d}T_{pn}(t)}{\text{d }t}=[AE_{E}-T_{pn}(t)]R_{p\text{m}}\omega_{pn}(t)-[AE_{E}-T_{p(n-1)}(t)]R_{p(n- 1)}\omega_{p(n-1)}(t)\\ \end{array} \right. \tag{1}\]  
The descriptions of the parameters in Equation (1) are shown in Table 1.  
Equation (1) represents a multi-span tension coupling system for gravure printing, where each printing span is represented by an equation, and adjacent printing spans are coupled together by the roller speed of the shared printing unit in the middle. In order to study the coupling characteristics of the multi-span tension system and facilitate the analysis and experimental verification, the printing tension system in Figure 2 is simplified by omitting the drying device, and the multi-span web transmission model shown in Figure 3 is obtained. Figure 3 shows the basic model of multi-span web transmission, which is composed of three spans, and a set of roller systems is shared between the two adjacent spans. According to Equation (1), the tension model of the multi-span system in Figure 3 can be obtained. \[\left\{ \begin{array}{l}L_{2}\frac{\text{d}T_{p}(t)}{\text{d}t}=[AE-T_{2}(t) ]R_{2}\omega_{2}(t)-[AE-T_{1}(t)]R_{1}\omega_{1}(t)\\ L_{3}\frac{\text{d}T_{3}(t)}{\text{d}t}=[AE-T_{3}(t)]R_{3}\omega_{3}(t)-[AE-T_ {2}(t)]R_{2}\omega_{2}(t)\\ L_{4}\frac{\text{d}T_{4}(t)}{\text{d}t}=[AE-T_{4}(t)]R_{4}\omega_{4}(t)-[AE-T_ {3}(t)]R_{3}\omega_{3}(t)\\ \end{array} \right. \tag{2}\]  
In the multi-span tension system, the input variables are roller speeds \(\omega_{1}\), \(\omega_{2}\), \(\omega_{3}\), \(\omega_{4}\), and web tension \(T_{1}\), and the output variables are \(T_{2}\), \(T_{3}\), and \(T_{4}\). From the mathematical model of the multi-span tension system, it can be seen that there is a coupling relationship between the adjacent spans, and there is also a coupling relationship between the input