Figure 7.4 shows a pneumatic piston, which can exert a force in one direction and serve as a translational actuator. The displacement of the piston is \(x\), the total mass of the piston and the load is \(m\), and the damping coefficient is \(b\). Use subscript \(i\) to denote the chamber index and \(i=1,2\). As shown in Figure 7.4, \(p_{i}\) is the absolute pressure in the chamber, \(q_{\mathrm{m} i}\) is the mass flow rate at the port, and \(A_{i}\) is the effective area of the piston. The two chambers have the same initial volume of \(V_{0}\). Assuming ideal gas and isothermal process, \(R\) is the specific gas constant and \(T\) is the absolute gas temperature. Derive the dynamic equations of the pressure change in the pneumatic chambers and the equation of motion of the mass block.

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P A2 P2 m 4ml Im2 FIGURE 7.4 Problem 4. b