Q$3.$ A vertical spring$-$loaded valve is used for a compressed air receiver. The valve opens at a pressure of $1\frac{N}{m}{m}^2$ gauge and must be fully open with a lift of $4$ mm at a pressure of $1\mathrm{.}2\frac{N}{m}{m}^2$ gauge. The diameter of the port is $25$ mm $.$ Assume the shear yield strength in steel as $480$ MPa $,$ shear ultimate strength is $643\mathrm{.}2$ MPa and shear modulus as $80k\frac{N}{m}{m}^2.$ Assume the helical compression spring has squared and ground ends, minimum factor of safety, $n=1\mathrm{.}5$ and spring index, $C=6.$ Determine for shot$-$peened wires

a$)$ initial compression

$[2$ Marks$]$

b$)$ coil diameter $d$ using Zimmerli data, Goodman theory and Langer theory. Select standard size of Washburn & Moen wire

$[7$ Marks$]$

c$)$ helical diameter $D$

$[1$ Marks$]$

d$)$ number of actual turns, $N_a$ and total turns, $,N_t$

$[2$ Marks$]$

e$)$ free length

$[2$ Marks$]$