The spring actuator shown in Fig. 3.42 operates by using the air pressure from a pneumatic controller \((p)\) as input and providing an output displacement to a valve \((x)\) proportional to the input air pressure. The diaphragm, made of a fabric-base rubber, has an area \(A\) and deflects under the input air pressure against a spring of stiffness \(k\). Find the response of the valve under a harmonically fluctuating input air pressure \(p(t)=p_{0} \sin \omega t\) for the following data: \(p_{0}=65 \mathrm{kPa}, \omega=8 \mathrm{rad} / \mathrm{s}, A=0.065 \mathrm{~m}^{2}, k=75 \times 10^{3} \mathrm{~N} / \mathrm{m}\), mass of spring \(=8 \mathrm{~kg}\), and mass of valve and \(\operatorname{rod}=12 \mathrm{~kg}\).

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Input (air under pressure,p) Diaphragm (area, A) Valve rod Spring (stiffness, k) Valve FIGURE 3.42 A spring actuator. Output air (pressure controlled by motion of valve)