Helical Coil Springs

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Helical coil springs are used widely in applications such as washing machines, compressors, etc. It is important to know the spring stiffness/spring rate when designing a spring-damper-mass system. Natural frequency of the system is affected by the spring stiffness and mass of the system. Spring stiffness can be calculated theoretically if the data is not available from the manufacturer. 

A spring deforms reversibly under force. Meaning it goes back to the original position after the applied force is removed. It is said that the spring is deformed in elastic region. If the force applied is too high, the spring experiences plastic deformation i.e. the spring does not go back to the original position after the applied force is removed. When designing a particular spring, the applied force has to be within elastic region. The force, F applied is proportional to the displacement, d.

F = k x d

where k is the spring stiffness. 

Consider a spring is manufactured from a rod of circular cross section of diameter D. The shear modulus of the rod is G. The rod is formed into a coil of N turns of radius r.  It is assumed that the coil radius is much larger than the radius of the rod, and that the normal to the plane of one coil nearly coincides with the axis of the spring. 

By using principles of mechanics of materials and examining free body diagram on the spring, a helical coil spring can be modeled as a linear spring of stiffness:

k = GD^4/64Nr^3  

A simple test can be conducted to confirm the stiffness value of the spring by fixing it on one end and putting a weight (force) on the other end. The stiffness is the ratio of the force divided by the corresponding displacement.