Dynamic Analysis of Closed-Loop Forging System

Abstract

The previous chapter mainly considers the dynamic analysis of the open-loop forging system. This chapter will develop an approach to estimate the dynamic behavior of the closed-loop forging system. The model of the closed-loop forging system is first derived and a solving method is then developed in order to find the velocity expression of the closed-loop forging system. Using this velocity expression, the dynamics of the closed-loop forging system is further estimated and the conditions of stability, vibration, and creep, as well as the relationships between the controller parameters and the constraints are also derived. These derived dynamic characteristics, conditions and relationships for different workpieces are further integrated and used to design the controller.

Appendices

Appendix A

$$\begin{aligned} & w_{10}^{2} = \frac{1}{2}\left( {w_{1}^{2} + w_{2}^{2} + \sqrt {(w_{1}^{2} - w_{2}^{2} )^{2} + 4m_{4} n_{6} } } \right),w_{20}^{2} = \frac{1}{2}\left( {w_{1}^{2} + w_{2}^{2} - \sqrt {(w_{1}^{2} - w_{2}^{2} )^{2} + 4m_{4} n_{6} } } \right), \\ \mu_{2} = \frac{{w_{1}^{2} - w_{2}^{2} \sqrt {(w_{1}^{2} - w_{2}^{2} )^{2} + 4m_{4} n_{6} } }}{{2m_{4} }}, \\ & \mu_{3} = \frac{{w_{1}^{2} - w_{2}^{2} - \sqrt {(w_{1}^{2} - w_{2}^{2} )^{2} + 4m_{4} n_{6} } }}{{2m_{4} }},M_{1} = 1 + \mu_{2}^{2} ,M_{2} = 1 + \mu_{3}^{2}, \\ = H \frac{{\varepsilon (m_{1} + \mu_{2} n_{1} + \mu_{2}^{2} n_{5} )}}{{2M_{1} }} + \frac{{\varepsilon^{2} (m_{2} + \mu_{2} n_{2} )}}{{2M_{1} }}(\frac{{\mu_{2} n_{4} }}{{M_{1} w_{10}^{2} }} + \frac{{\mu_{3} n_{4} }}{{M_{2} w_{20}^{2} }}), \\ & Z = \frac{{\varepsilon (m_{3} + \mu_{2} n_{3} )}}{{8M_{1} }}, A_{0} = \varepsilon n_{4} (\frac{{\mu_{2} }}{{M_{1} w_{10}^{2} }} + \frac{{\mu_{3} }}{{M_{2} W_{20}^{2} }}),\\ E = w_{10} - \frac{{M_{1} A_{{{\kern 1pt} 11}}^{2} + \varepsilon^{2} (m_{1} + \mu_{2} n_{1} + \mu_{2} \mu_{3} n_{5} )B_{0}^{(1)} w_{10} }}{{2w_{10} M_{1} }}, A_{12} = \frac{{\varepsilon (m_{3} + \mu_{2} n_{3} )}}{{8M_{1} }} \\ & B_{0}^{(1)} = - \frac{{w_{10} (m_{1} + \mu_{3} n_{1} + \mu_{2} \mu_{3} n_{5} )}}{{M_{2} (w_{20}^{2} - w_{10}^{2} )}} \\ & B_{0}^{(2)} = - \frac{{w_{10} (m_{3} + \mu_{3} n_{3} )}}{{4M_{2} (w_{20}^{2} - w_{10}^{2} )}}, A_{11} \frac{{\varepsilon (m_{1} + \mu_{2} n_{1} + \mu_{2}^{2} n_{5} )}}{{2M_{1} }}, B_{0} = \varepsilon (B_{0}^{(1)} + B_{0}^{(2)} a^{2} ), \\ C_{0} = \frac{{\varepsilon (m_{2} + \mu_{2} n_{2} )}}{{6M_{1} w_{10} }} - \frac{{\varepsilon w_{10} (m_{2} + \mu_{3} n_{2} )}}{{2M_{2} (w_{20}^{2} - 4w_{10}^{2} )}}, \\ & F = - \frac{{3M_{1} A_{11} A_{12} }}{{w_{10} M_{1} }} - \frac{{\varepsilon C_{0} (m_{2} + \mu_{2} n_{2} )}}{{4M_{1} }} - \frac{{\varepsilon^{2} (m_{1} + \mu_{2} n_{1} + \mu_{2} \mu_{3} n_{5} )B_{0}^{(2)} }}{{2M_{1} }} - \varepsilon A_{12} B_{0}^{(1)} , \\ & G = - \frac{{3A_{12}^{2} + 2A_{12} D_{0} w_{10} + 2\varepsilon A_{12} B_{0}^{(2)} w_{10} }}{{2w_{10} }}, D_{0} = \frac{{\varepsilon (m_{3} + \mu_{2} n_{3} )}}{{32M_{1} w_{10} }} - \frac{{\varepsilon w_{10} (m_{2} + \mu_{3} n_{3} )}}{{4M_{2} (w_{20}^{2} - 9w_{10}^{2} )}}, \\ C_{1} \,{\text{is}}\,{\text{constant}} \\ \end{aligned}$$

Appendix B

The quantization [E(k), EC(k)] of [e(k), ec(k)] is divided into seven fuzzy rules according to e(k) and ec(k), as shown in Table 9.4. The adaptive increments $\Delta k_{p}$, $\Delta k_{i}$, and $\Delta k_{d}$ are calculated as follows

Table 9.4 Fuzzy rule

$$\Delta k_{p} (k) = 0.1T_{kp} [E(k),EC(k)],\quad \Delta k_{i} (k) = T_{ki} [E(k),EC(k)],\Delta k_{d} (k) = T_{kd} [E(k),EC(k)] \times 10^{ - 6}$$

Here, $T_{kp} [E(k),EC(k)],T_{ki} [E(k),EC(k)]$, and $T_{kd} [E(k),EC(k)]$ are obtained using look-up Tables 9.5, 9.6, and 9.7 respectively.

Table 9.5 $T_{kp} [E(k),EC(k)]$

Table 9.6 $T_{ki} [E(k),EC(k)]$

Table 9.7 $T_{kd} [E(k),EC(k)]$