Helical conical compression spring design

[jstluise]

I am designing a constant pitch, helical conical compression spring (round wire cross section) that will be compressed into a final desired shape. So my issue at hand is figuring out the pre-compressed geometry that will give me my desired final geometry once compressed.

The compression will be accomplished by longitudinal tension lines (fixed at each point on the spring for constant pitch), so the spring should not be torqued at all when compressed. As opposed to keeping the two end orientations fixed when compressing.

I have a spring I've been playing with, and my observations are that when the spring is compressed (with no torque), the number of turns decreases and the diameter of each turn increased.

I've been unsuccessful so far in finding equations to help me figure this out...

Final Shape, Known:
Dmin - small diameter, bottom of spring
Dmax - large diameter, top of spring
N - number of turns
h - height
L - total arc length, function of other variables, remains constant


Uncompressed Shape:
Dmin0 - Unknown
Dmax0 - Unknown
N0 - Unknown
h0 - Known, based on the compression I want (~50%)
L0 = L


Can anyone shed some light on this? I'm not sure if material properties come into play...I know they will for the compression force required, but I don't know about the geometry.

Thanks!

 

[Seth.T]

The formula for conical springs is based on ordinary compression springs ie for each turn on the conical spring
here are some formula :-the rate for each individual turn or
fraction of a turn = k=Gd^4/(8*D^3*Na)

total rate k for a complete spring
of different diameter turns = k= 1/((1/k1)+(1/k2)etc)To calculate stress in the wire the mean diameter of the
largest active coil is used at the required load ie:-

S= 8*P*D*Kw/(3.142*d^3)

The solid height of a conical spring of a uniform taper but not telescoping, with squared and ground ends can estimated
from :-

La=Na*(d^2-u^2)^0.5 + 2d

where G= modulus of rigidity of material
d= wire dia
D= mean coil diameter
Na = number of active turns
k = spring stiffness
S = stress
P= spring load
La= solid height of spring
u = o.d of large coil - o.d. of small coil/(2*Na)
Kw1= (4*C-1/(4*C+1))+ 0.615/C
C = mean dia of largest coil/ wire diameter

 

[jstluise]

Thanks for the response, but that is not quite what I am after. Basically I am looking for an equation for angular deflection due to axial loading (ie change in height) with both ends free to rotate.

I'll focus my search on finding it for a regular (non-conical) compression spring, then I can separate my conical spring into several compression springs in series. My taper angle is pretty low, so that should give me a good approximation.