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**Chapter 9 **

**SPRINGS**

Spring is a device, in which the material is arranged in such a way that it can undergo a considerable change, without getting permanently distorted. A spring is used to absorb energy due to resilience which may be restored as and when required. The quality of a spring is judged form the energy it can absorb. The spring which is capable of absorbing the greatest amount of energy for the given stress is the best one.

**Stiffness of a Spring**

- The load required to produce a unit deflection in spring is called spring stiffness.

**Types of Springs **There are two types depending upon the type of resilience:

- Bending spring (leaf spring)
- Torsion spring (helical spring)

**Leaf Springs **

Central deflection d is given by,

Where,

l = Span of spring

t = thickness of plates

b = Width of plates

n = Number of plates

W = Load acting on the spring

E = Young's modulus

**Closed coiled helical springs subjected to axial loading **

Deflection in the spring due to load W,

- Stiffness of the spring,

Where, W = axial load n = no. of turns of spring d = diameter of the rod of the spring R = mean radius of the coil G = modulus of rigidity for the spring material.

- Energy stored,

**Closely coiled helical spring subjected to axial twist**

- Total angle of bend = φ

where l = 2 pRn = 2pR ' n '

Thus, the change in curvature or angle of bend per unit length, is constant, throughout the spring.

- Energy stored =

Where, R = mean radius of the spring coil

n = no. of turns or coils

M = Moment or axial twist applied on the spring

R = decreased mean radius due to twist

n = increased no of turns due to twist

I = moment of inertia of spring rod section

E = Young's modulus

**Open coiled helical spring subjected to axial load**

- In this case load W will cause both twisting and bending of coils.

- Deflection of the spring as a result as a result of axial load,

Where,

d = Diameter of spring wire

R = Mean radius of spring coil

P = Pitch of the spring coil

n = no. of turns of coils

G = modulus of rigidity for spring material

W = axial load on the spring

a = angle of helix

E = Young's modulus

- If we put µ = o

= deflection of closed coiled spring.

**SPRINGS IN SERIES AND PARALLEL**

(i) Springs in series

- Total extension = å (individual extensions)

d = d_{1} + d_{2}

- load applied (W) will be same on both the springs.
- The equivalent stiffness is given by

where K_{1} & K_{2} are individual stiffness of the springs.

**(ii) Springs in parallel**

Both the springs have same extension.

δ = δ_{1} = δ_{2}

Load applied (W) is shared by the springs.

W = W_{1} + W_{2}

The equivalent stiffness is given by,

K_{eq} = K_{1 }+ K_{2}

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