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This page contains the equations used in the Compression Spring Designer calculator.
The spring constant k is function of the spring geometry and the spring material's shear modulus G,

where G is found from the material's elastic modulus E and Poisson ratio n,

and D is the mean diameter of the spring (measured from the centers of the wire cross-sections),

Spring Geometry
The distance between adjacent spring coils (defined as the coil pitch) is found by dividing the spring free length by the number of active coils,

The rise angle of the spring coils (the angle between the coils and the base of the spring) is found from the arctangent of the coil-pitch divided by the spring circumference,

The solid height of the spring is found by summing the widths of all the spring coils. The total number of spring coils is equal to the active coils in the spring interior plus the 2 coils at the spring ends (more),

The length of wire needed to make the spring is found from,

Spring Force and Stress
The maximum force the spring can take occurs when the spring is deformed all the way to its solid height,

The maximum shear stress in the spring associated with the maximum force is given by,

where W is the Wahl correction factor (accounting for spring curvature stress) and C is the spring index (essentially an aspect ratio of the spring cross-section),

Spring Resonance
Finally, the lowest resonant frequency (in Hz) of the spring is found from the simple equation,

where k is the spring constant from above and M is the spring mass (see derivation). The spring mass M can be found by weighing the spring, or by finding the spring volume and multiplying by its material density,

We can express the spring's lowest resonance in terms of basic spring geometry if we substitute for k and M in the equation for fres (and then eliminate Lwire). Doing so gives,

For springs with small rise angles and several active coils we can make the approximation,

If we also allow the approximation,

we can then simplify the resonant frequency formula to a form that can be found in several reference books,