Q$1.(10$ marks$)$ A circular steel pressure$-$diaphragm rosette shown in the figure below is used for design of a pressure transducer. The deflection $(y_c)$ of the diaphragm center is given by:

Where $P=$ pressure,

$y_c=(\frac{3}{16})(\frac{P{a}^4}{E{t}^3})(1-v^2)$

$a=$ radius of the diaphragm $=7\mathrm{.}5cm,t=$ thickness of the diaphragm $=1\mathrm{.}28mm,E=$ the modulus of elasticity $=200{10}^{9}Pa,v=$ Poisson's ratio $={?}^{-}0\mathrm{.}26.$

The pressure transducer uses rosette strain gauges and bridge circuit to convert pressure to voltage.

The output voltage of the bridge is given by:

$E_o=0\mathrm{.}820\frac{P{a}^2(1-v^2)}{E{t}^2}{E}_S$

The natural frequency is given by:

$f_r=\frac{10\mathrm{.}21}{a^2}\sqrt[\phantom{2}]{\frac{E{t}^2}{12{}_d(1-v^2)}}Hz$

Where, $E_o$ is the output voltage $($volt$)$

$E_s$ bridge supply voltage $=5$ volts

${}_d$ diaphragm density is $6{10}^{3}K\frac{g}{m^3}$

a$)$ Explain how the gauges should be arranged on the bridge $(2$ marks$)$

b$)$ Find the sensitivity of the pressure transducer in $(m\frac{V}{P}a)?(2$ marks$)$

c$)$ Find the natural frequency and operating frequencies range of the transducer $(2$ marks$)$

d$)$ What would be the maximum allowable pressure difference that could be applied if the maximum deflection at the center is restricted to $y_{c_{\text{m}\text{a}\text{x}}}\frac{t}{3}?(2$ marks$)$

e$)$ What is the output voltage if the maximum pressure found above is measured? $(2$marks $)$