When a spring is going to be used in an environment where there is a cyclic loading, the spring shall be designed to have enough margins between cyclic loading frequency and spring natural frequency. If not designed so, the resonance situation occurs. This will amplify the stress levels which are calculated for static case. To avoid resonance, the natural frequency of the spring shall be minimum 13 times (Ref-1) larger than cyclic forcing frequency.

The critical frequency calculator of coil springs calculates natural frequency of a spring whose one end is against flat plate and other end is moved by a cyclic sine-wave motion. Input parameters of wire diameter, spring diameter, number of active coils, spring rate and material density are required for the calculations.

The formulas and parameters used in the calculator are given in the " List of Equations " section of this page.

Mechanical springs - Compression spring terminology

Note: This calculator was developed by mainly using Shigley's Mechanical Engineering Design book. For further information on subject , this reference source can be used.


Parameter Symbol Value Unit
Wire diameter d
Number of active coils Na ---
Spring rate k
Spring material density ρ
Min. design margin (spring natural frequency / cyclic loading frequency) nf ---

Parameter Symbol Value Unit
Natural frequency of spring f+ --- Hz
Maximum allowed cyclic loading frequency floading ---
Mass of the active coils  m ---
Spring index C* --- ---
Spring outer diameter OD ---
Spring mean diameter D ---
Spring inner diameter ID ---

Note 1 : +  One end is against flat plate and the other end is moved by a cyclic sine-wave motion.

Note 2 : * Preferred index range is 4 to 12. Springs with high indexes tangle and may require individual packaging, especially if the ends are not squared. Springs with indexes lower than 4 are difficult to form [From Ref-1] .


Spring Rate: The parameter which shows relation between applied force and deflection. In other words, reaction force per unit deflection or spring resistance to length change.

Number of active coils: The coils of a spring that stores and releases energy. Number of active coils cannot be directly measured . It can be calculated by subtracting number of inactive coils from total number of coils.

Spring index: The ratio of spring mean diameter to coil diameter. As a general rule, the ratio shall be between 4 and 12. Spring sizes out of this interval increases cost and manufacturing process is harder. [Ref-1]. According to BS1726:Part 1:1987 , advised index range is between 3.5 to 16. A low index value indicates a very tightly wound spring with a relatively large wire or bar being coiled sharply around a relatively small coil diameter. This results very high axial stiffness. A high index value means an open wound spring which will be very flexible along its axis (low spring rate)


Parameter Open or plain (Not ground) Open or plain (Ground) Squared or closed (Not ground) Squared or closed (Ground
Total coils [Nt] Na Na+1 Na+2 Na+2
Free height [Lf] pNa+d p(Na+1) pNa+3d pNa+2d
Solid height [Ls] d(Nt+1) dNt d(Nt+1) dNt
Pitch [p] (Lf - d) / Na Lf / (Na+1) (Lf -3d) / Na (Lf -2d) / Na
Guidelines for Dimensional Characteristics of Compression Springs
Source : From Design Handbook [Ref 1] page 32

List of Equations:

Parameter Symbol Formula
Spring index C Spring Index
Spring outer diameter OD Spring outer diamete
Spring inner diameter  ID Spring inner diameter 
Mass of the active coils m Mass of the active coils of the spring
Natural frequency of the spring f+ Natural frequency of the spring

Note 1 : + One end is against flat plate and the other end is moved by a cyclic sine-wave motion.

List of Parameters
Symbol Definition
D Spring mean diameter
d Wire diameter
ρ Spring material density
k Spring rate
Na Number of active coils

  • Courtesy of Associated Spring (1987)., Design Handbook
  • Budynas.R , Nisbett.K. (2008) . Shigley's Mechanical Engineering Design .8th edition.  McGraw-Hill
  • BS 1726 Part 1:1987, Guide For the Design of Helical Compression Springs