# CRITICAL FREQUENCY OF COIL SPRINGS

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.

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.

##### Calculator:

 INPUT PARAMETERS Parameter Symbol Value Unit Wire diameter d mm m inch ft Spring outer diameter - [OD] Spring mean diameter - [D] Spring inner diameter - [ID] Number of active coils Na --- Spring rate k N/mm lbf/inch lbf/ft Spring material density ρ g/cm^3 kg/m^3 lb/in^3 Min. design margin (spring natural frequency / cyclic loading frequency) nf ---

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

##### Definitions:

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)

##### Supplements:

 TYPE OF SPRING ENDS 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 outer diameter OD Spring inner diameter ID Mass of the active coils m Natural frequency of the spring f+

###### 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

##### Reference:
• 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