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

\text { Given } P=2000 N \quad G=81370 N / mm ^{2} .

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

D_{i}=64 mm \quad d_{i}=8 mm \quad N_{i}=12 coils .

D_{o}=80 mm \quad d_{o}=10 mm \quad N_{o}=8 coils .

From Eq. (10.8),

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

\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}} .

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

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

\frac{P_{o}}{P_{i}}=1.875            (a).

Also,          P_{o}+P_{i}=2000 N             (b).

Solving equations (a) and (b) simultaneously,

P_{o}=1304.35 N \quad P_{i}=695.65 N               (i).

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

\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}}               (ii).

Step III Maximum shear stress

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

From Eq. (10.7),

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

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 .

Outer spring
From Eq. (10.13),

\tau=K\left(\frac{8 P C}{\pi d^{2}}\right)               (10.13).

\tau_{o}=K\left(\frac{8 P_{o} C}{\pi d_{o}^{2}}\right)=(1.184)\left[\frac{8(1304.35)(8)}{\pi(10)^{2}}\right] .

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

Inner spring

\tau_{i}=K\left(\frac{8 P_{i} C}{\pi d_{i}^{2}}\right)=(1.184)\left[\frac{8(695.65)(8)}{\pi(8)^{2}}\right] .

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