Hooke's Law

Spring

Springs are fundamental mechanical components which form the basis of many mechanical systems. A spring can be defined to be an elastic member which exerts a resisting force when its shape is changed. Most springs are assumed linear and obey the Hooke's Law,

where F is the resisting force, Δ is the displacement, and the k is the spring constant.

Basic Spring Types

Springs are of several types, the most plentiful of which are shown as follows,

Compression Spring

Figures : Spring Constant Dependencies [1]

We can expand the spring constant k as a function of the material properties of the spring. Doing so and solving for the spring displacement gives,

where G is the material shear modulus, n_a is the number of active coils, and D and d are defined in the drawing. The number of active coils is equal to the total number of coils n_t minus the number of end coils n* that do not help carry the load,

The value for n* depends on the ends of the spring. See the following illustration for different n* values:

Figure : An illustration of compression spring with different n* values [1]

Geometrical Factors

The spring index, C, can be used to express the deflection,

The useful range for C is about 4 to 12, with an optimum value of approximately 9. The wire diameter, d, should conform to a standard size if at all possible.

Shear Stress in the Spring

The maximum shear stress τ_max in a helical spring occurs on the inner face of the spring coils and is equal to,

where W is the Wahl Correction Factor which accounts for shear stress resulting from spring curvature,

To be continued

References :

1. http://www.efunda.com/DesignStandards/springs/spring_design.cfm
2.