A concentric spring is used as a valve spring in a heavy duty diesel engine. It consists of two helical compression springs having the same free length and same solid length. The composite spring is subjected to a maximum force of 6000 N and the corresponding deflection is 50 mm. The maximum

Question

A concentric spring is used as a valve spring in a heavy duty diesel engine. It consists of two helical compression springs having the same free length and same solid length. The composite spring is subjected to a maximum force of 6000 N and the corresponding deflection is 50 mm. The maximum torsional shear stress induced in each spring is 800 N/mm². The spring index of each spring is 6. Assume same material for two springs and the modulus of rigidity of spring material is 81370 N/mm². The diametral clearance between the coils is equal to the difference between their wire diameters. Calculate:
(i) the axial force transmitted by each spring;
(ii) wire and mean coil diameters of each spring; and
(iii) number of active coils in each spring.

Accepted Answer

[latex] ext { Given } \quad P=6000 N \quad \delta=50 mm \quad au=800 N / mm ^{2} [/latex].

C = 6 G = 81370 N/mm²
Step I Axial force transmitted by each spring
The diametral clearance between the coils is equal to the difference between their wire diameters. From Eq. (10.26),

[latex] \frac{d_{1}}{d_{2}}=\frac{C}{(C-2)} [/latex] (10.26).

[latex] \frac{d_{1}}{d_{2}}=\frac{C}{(C-2)}=\frac{6}{(6-2)}=1.5 [/latex].

From Eq. (10.23),

[latex] \frac{P_{1}}{P_{2}}=\frac{d_{1}^{2}}{d_{2}^{2}} [/latex] (10.23).

[latex] \frac{P_{1}}{P_{2}}=\frac{d_{1}^{2}}{d_{2}^{2}}=\left(\frac{d_{1}}{d_{2}} ight)^{2}=(1.5)^{2}=2.25 [/latex] (a).

[latex] ext { Also, } \quad P_{1}+P_{2}=P=6000 N [/latex] (b).

Solving equations (a) and (b) simultaneously,

[latex] P_{1}=4153.85 N ext { and } P_{2}=1846.15 N [/latex] (i)

Step II Wire and mean coil diameters
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(6)-1}{4(6)-4}+\frac{0.615}{6}=1.2525 [/latex].

Outer spring
From Eq. (10.13),

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

[latex] au=K\left(\frac{8 P_{1} C}{\pi d_{1}^{2}} ight) ext { or } 800=(1.2525)\left[\frac{8(4153.85)(6)}{\pi d_{1}^{2}} ight] [/latex].

Outer spring
From Eq. (10.13),

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

[latex] au=K\left(\frac{8 P_{1} C}{\pi d_{1}^{2}} ight) ext { or } 800=(1.2525)\left[\frac{8(4153.85)(6)}{\pi d_{1}^{2}} ight] [/latex].

[latex] d_{1}=9.97 ext { or } 10 mm [/latex].

[latex] D_{1}=C d_{1}=6(10)=60 mm [/latex] (ii).

Inner spring
From Eq. (10.13),

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

[latex] au=K\left(\frac{8 P_{2} C}{\pi d_{2}^{2}} ight) \quad ext { or } \quad 800=(1.2525)\left[\frac{8(1846.15)(6)}{\pi d_{2}^{2}} ight] [/latex].

[latex] d_{2}=6.65 ext { or } 7 mm [/latex].

[latex] D_{2}=C d_{2}=6(7)=42 mm [/latex] (ii).

Step III Number of active coils
From Eq. (10.8),

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

[latex] \delta=\frac{8 P_{1} D_{1}^{3} N_{1}}{G d_{1}^{4}} \quad ext { or } \quad 50=\frac{8(4153.85)(60)^{3} N_{1}}{(81370)(10)^{4}} [/latex].

[latex] N_{1}=5.67 ext { or } 6 ext { coils } [/latex] (iii).

It is assumed that the springs have square and ground ends. Therefore,

[latex] \left(N_{t} ight)_{1}=N_{1}+2=6+2=8 ext { coils } [/latex].

Since the springs have the same solid length,

[latex] \left(N_{t} ight)_{1} d_{1}=\left(N_{t} ight)_{2} d_{2} \quad ext { or } \quad 8(10)=\left(N_{t} ight)_{2}(7) [/latex].

[latex] \left(N_{t} ight)_{2}=11.43 ext { or } 12 ext { coils } [/latex]

[latex] N_{2}=12-2=10 ext { coils } [/latex] (iii).

Question 10.17: A concentric spring is used as a valve spring in a heavy dut...

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