A concentric spring consists of two helical compression springs having the same free length. The composite spring is subjected to a maximum force of 2000 N. The wire diameter and mean coil diameter of the inner spring are 8 and 64 mm respectively. Also, the wire diameter and mean coil diameter

Question

A concentric spring consists of two helical compression springs having the same free length. The composite spring is subjected to a maximum force of 2000 N. The wire diameter and mean coil diameter of the inner spring are 8 and 64 mm respectively. Also, the wire diameter and mean coil diameter of the outer spring are 10 and 80 mm respectively. The number of active coils in the inner and outer springs are 12 and 8 respectively.
Assume same material for two springs and the modulus of rigidity of spring material is 81370
N/mm². Calculate:
(i) the force transmitted by each spring;
(ii) the maximum deflection of the spring; and
(iii) the maximum torsional shear stress induced in each spring.

Accepted Answer

[latex] ext { Given } P=2000 N \quad G=81370 N / mm ^{2} [/latex].

Step I Force transmitted by each spring
Suppose suffix i and o refer to inner and outer spring respectively

[latex] D_{i}=64 mm \quad d_{i}=8 mm \quad N_{i}=12 coils [/latex].

[latex] D_{o}=80 mm \quad d_{o}=10 mm \quad N_{o}=8 coils [/latex].

[latex] ext { Since both springs have same deflection, } \delta_{i}=\delta_{o} [/latex]

From Eq. (10.8),

[latex] \delta=\frac{8 P D^{3} N}{G d^{4}} [/latex] (10.8)

[latex] \frac{8 P_{i} D_{i}^{3} N_{i}}{G d_{i}^{4}}=\frac{8 P_{o} D_{o}^{3} N_{o}}{G d_{o}^{4}} [/latex].

or [latex] \frac{P_{i} D_{i}^{3} N_{i}}{d_{i}^{4}}=\frac{P_{o} D_{o}^{3} N_{o}}{d_{o}^{4}}[/latex].

[latex] \frac{P_{i}(64)^{3}(12)}{(8)^{4}}=\frac{P_{o}(80)^{3}(8)}{(10)^{4}} [/latex].

[latex] \frac{P_{o}}{P_{i}}=1.875 [/latex] (a).

Also, [latex] P_{o}+P_{i}=2000 N [/latex] (b).

Solving equations (a) and (b) simultaneously,

[latex] P_{o}=1304.35 N \quad P_{i}=695.65 N [/latex] (i).

Step II Maximum deflection of the spring
From Eq. (10.8),

[latex] \delta_{i}=\delta_{o}=\frac{8 P_{o} D_{o}^{3} N_{o}}{G d_{0}^{4}}=\frac{8(1304.35)(80)^{3}(8)}{(81370)(10)^{4}} [/latex] (ii).

Step III Maximum shear stress

[latex] C=\frac{D_{o}}{d_{o}}=\frac{80}{10}=8 \quad C=\frac{D_{i}}{d_{i}}=\frac{64}{8}=8 [/latex].

From Eq. (10.7),

[latex] K=\frac{4 C-1}{4 C-4}+\frac{0.615}{C} [/latex] (10.7).

[latex] K=\frac{4 C-1}{4 C-4}+\frac{0.615}{C}=\frac{4(8)-1}{4(8)-4}+\frac{0.615}{8}=1.184 [/latex].

Outer spring
From Eq. (10.13),

[latex] au=K\left(\frac{8 P C}{\pi d^{2}} ight) [/latex] (10.13).

[latex] au_{o}=K\left(\frac{8 P_{o} C}{\pi d_{o}^{2}} ight)=(1.184)\left[\frac{8(1304.35)(8)}{\pi(10)^{2}} ight] [/latex].

[latex] =314.61 N / mm ^{2} [/latex] (iii).

Inner spring

[latex] au_{i}=K\left(\frac{8 P_{i} C}{\pi d_{i}^{2}} ight)=(1.184)\left[\frac{8(695.65)(8)}{\pi(8)^{2}} ight] [/latex].

[latex] =262.18 N / mm ^{2} [/latex] (iii).

Question 10.18: A concentric spring consists of two helical compression spri...

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