A semi-elliptic leaf spring used for automobile suspension consists of three extra full-length leaves and 15 graduated-length leaves, including the master leaf. The centre-to-centre distance between two eyes of the spring is 1 m. The maximum force that can act on the spring is 75 kN. For each leaf

Question

A semi-elliptic leaf spring used for automobile suspension consists of three extra full-length leaves and 15 graduated-length leaves, including the master leaf. The centre-to-centre distance between two eyes of the spring is 1 m. The maximum force that can act on the spring is 75 kN.
For each leaf, the ratio of width to thickness is 9:1.
The modulus of elasticity of the leaf material is 207 000 N/mm². The leaves are pre-stressed in such a way that when the force is maximum, the stresses induced in all leaves are same and equal to 450 N/mm². Determine
(i) the width and thickness of the leaves;
(ii) the initial nip; and
(iii) the initial pre-load required to close the gap C between extra full-length leaves and graduated-length leaves.

Accepted Answer

[latex] ext { Given } \quad 2 P=75 kN \quad 2 L=1 m \quad b=9 t \quad n_{f}=3 [/latex].

[latex] n_{g}=15 \quad E=207000 N / mm ^{2} \quad \sigma_{b}=450 N / mm ^{2} [/latex].

Step I Width and thickness of the leaves
2P = 75 kN or P = 37 500 N
2L = 1 m or L = 500 mm.

[latex] b=9 t \quad n=n_{f}+n_{g}=3+15=18 [/latex].

From Eq. (10.41),

[latex] \sigma_{b}=\frac{6 P L}{n b t^{2}} [/latex] (10.41).

[latex] \sigma_{b}=\frac{6 P L}{n b t^{2}} \quad ext { or } \quad(450)=\frac{6(37500)(500)}{(15+3)(9 t) t^{2}} [/latex].

∴ t = 11.56 or 12 mm
b = 9 t = 9 (12) = 108 mm (i)
Step II Initial nip
From Eq. (10.39),

[latex] C=\frac{2 P L^{3}}{E n b t^{3}} [/latex] (10.39).

[latex] C=\frac{2 P L^{3}}{E n b t^{3}}=\frac{2(37500)(500)^{3}}{(207000)(18)(108)(12)^{3}} [/latex].

= 13.48 mm (ii).

Step III Initial pre-load
From Eq. (10.40),

[latex] P_{i}=\frac{2 n_{g} n_{f} P}{n\left(3 n_{f}+2 n_{g} ight)} [/latex] (10.40).

[latex] P_{i}=\frac{2 n_{g} n_{f} P}{n\left(3 n_{f}+2 n_{g} ight)}=\frac{2(15)(3)(37500)}{18(3 imes 3+2 imes 15)} [/latex]

= 4807.69 N (iii).

Question 10.24: A semi-elliptic leaf spring used for automobile suspension c...

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