Design a steel helical compression spring to satisfy the following requirements:
Spring stiffness \((k) \geq 8000 \mathrm{~N} / \mathrm{mm}\)
Fundamental natural frequency of vibration \(\left(f_{1}\right) \geq 0.4\mathrm{~Hz}\)

Spring index \((D /

d) \geq 6\)

Number of active turns \((N) \geq 10\).

The stiffness and fundamental natural frequency of the spring are given by [1.43]:

\[k=\frac{G d^{4}}{8 D^{3} N} \quad \text { and } \quad f_{1}=\frac{1}{2} \sqrt{\frac{k g}{W}}\]

where \(G=\) shear modulus, \(d=\) wire diameter, \(D=\) coil diameter, \(W=\) weight of the spring, and \(g=\) acceleration due to gravity.