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 (Ref1) 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 sinewave 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:
RESULTS 
Parameter 
Symbol 
Value 
Unit 
Natural frequency of spring 
f^{+} 


Hz 
Maximum allowed cyclic loading frequency 
f_{loading} 


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 sinewave 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 Ref1] .
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. [Ref1]. 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 [N_{t}] 
N_{a} 
N_{a}+1 
N_{a}+2 
N_{a}+2 
Free height [L_{f}] 
pN_{a}+d 
p(N_{a}+1) 
pN_{a}+3d 
pN_{a}+2d 
Solid height [L_{s}] 
d(N_{t}+1) 
dN_{t} 
d(N_{t}+1) 
dN_{t} 
Pitch [p] 
(L_{f}  d) / N_{a} 
L_{f} / (N_{a}+1) 
(L_{f} 3d) / N_{a} 
(L_{f} 2d) / N_{a} 
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 sinewave motion.
List of Parameters 
Symbol 
Definition 
D 
Spring mean diameter 
d 
Wire diameter 
ρ 
Spring material density 
k 
Spring rate 
N_{a} 
Number of active coils 
Reference:
 Courtesy of Associated Spring (1987)., Design Handbook
 Budynas.R , Nisbett.K. (2008) . Shigley's Mechanical Engineering Design
.8th edition. McGrawHill
 BS 1726 Part 1:1987, Guide For the Design of Helical Compression Springs