A close-coiled helical spring has stiffness 900 N/m in compression with a maximum load of 45 N and maximum shear stress of 120 N/mm². The solid length of the spring is 45 mm. Find out wire diameter, mean coil radius and number of coils. Take G = 0.4 × 10^5 N/mm².

Question

A close-coiled helical spring has stiffness 900 N/m in compression with a maximum load of 45 N and maximum shear stress of 120 N/mm². The solid length of the spring is 45 mm. Find out wire diameter, mean coil radius and number of coils. Take G = 0.4 × [latex] 10^5 [/latex] N/mm².

Accepted Answer

We know deflection (δ ) and stiffness (k ) of the spring are

[latex] \delta=\frac{64 F R^3 n}{G d^4} \quad ext { and } \quad k=\frac{G d^4}{64 R^3 n} [/latex]

Putting k = 900 N/m or 0.9 N/mm and [latex] G=0.4 imes 10^5 N / mm ^2 [/latex], we get

[latex] 0.9=\frac{0.4 imes 10^5 imes d^4}{64 R^3 n} [/latex]

or [latex] d^4=\left\lgroup \frac{0.9 imes 64}{0.4 imes 10^5} ight group R^3 n [/latex] (1)

Torque ‘FR’ causes shear stress τ and is given by

[latex] au=\frac{16 F R}{\pi d^3} [/latex]

Putting [latex] au= au_{\max } [/latex] and substituting values, we get

[latex] au= au_{\max }=\frac{16 imes 45 imes R}{\pi d^3} [/latex]

or [latex] 120=\frac{16 imes 45 imes R}{\pi d^3} [/latex]

or R = 0.52d³ (2)

Now, solid length² of spring being 45 mm,

[latex] n=\frac{45}{d} [/latex] (3)

Combining Eqs. (1)–(3), we find

[latex] \begin{aligned} d^4 & =\left\lgroup\frac{0.9 imes 64}{0.4 imes 10^5} ight group imes\left(0.52 d^3 ight)^3 imes \frac{45}{d} \ & =(109.75)^{1 / 4} \ & =3.24 mm \end{aligned} [/latex]

The mean coil radius is

[latex] R=0.52 d^3=0.52 imes(3.24)^3=17.68 mm [/latex]

The number of turns is

[latex] N=\frac{45}{d}=\frac{45}{3.24}=13.88 \cong 14 [/latex]

Therefore, the wire diameter is 3.24 mm, mean coil radius is 17.68 mm and number of turns is 14.

[latex] ^2 [/latex] Solid length of the spring is its length when it is fully compressed and the coils of the spring are touching each other.

Question 2.14: A close-coiled helical spring has stiffness 900 N/m in compr...

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