Cylindrical compression spring behavior has been described in the literature using an efficient analytical model. Conical compression spring behavior has a linear phase but can also have a nonlinear phase. The rate of the linear phase can easily be calculated but no analytical model exists to describe the nonlinear phase precisely. This nonlinear phase can only be determined by a discretizing algorithm. The present paper presents analytical continuous expressions of length as a function of load and load as a function of length for a constant pitch conical compression spring in the nonlinear phase. Whal’s basic cylindrical compression assumptions are adopted for these new models (Wahl, A. M., 1963, Mechanical Springs, Mc Graw-Hill, New York). The method leading to the analytical expression involves separating free and solid/ground coils, and integrating elementary deflections along the whole spring. The inverse process to obtain the spring load from its length is assimilated to solve a fourth order polynomial. Two analytical models are obtained. One to determine the length versus load curve and the other for the load versus length curve. Validation of the new conical spring models in comparison with experimental data is performed. The behavior law of a conical compression spring can now be analytically determined. This kind of formula is useful for designers who seek to avoid using tedious algorithms. Analytical models can mainly be useful in developing interactive assistance tools for conical spring design, especially where optimization methods are used.

1.
Wahl
,
A. M.
, 1963,
Mechanical Springs
,
McGraw-Hill
, New York.
2.
Ding
,
X.
, and
Selig
,
J. M.
, 2004, “
On the Compliance of Coiled Springs
,”
Int. J. Mech. Sci.
0020-7403,
46
, pp.
703
727
.
3.
Chassie
,
G. G.
,
Becker
,
L. E.
, and
Cleghorn
,
W. L.
, 1997, “
On the Buckling of Helical Springs Under Combined Compression and Torsion
,”
Int. J. Mech. Sci.
0020-7403,
39
(
6
), pp.
697
704
.
4.
Becker
,
L. E.
,
Chassie
,
G. G.
, and
Cleghorn
,
W. L.
, 2002, “
On the Natural Frequencies of Helical Compression Springs
,”
Int. J. Mech. Sci.
0020-7403,
44
, pp.
825
841
.
5.
Jiang
,
W. G.
, and
Henshall
,
J. L.
, 2000, “
A Novel Finite Element Model for Helical Springs
,”
Finite Elem. Anal. Design
0168-874X,
35
, pp.
363
377
.
6.
Todinov
,
M. T.
, 1999, “
Maximum Principal Tensile Stress and Fatigue Crack Origin for Compression Springs
,”
Int. J. Mech. Sci.
0020-7403,
41
, pp.
357
370
.
7.
IST, 1980–2005
, “
Essential Spring Design Training Course
,” Handbook, The Institute of Spring Technology, Sheffield, United Kingdom.
8.
Wolansky
,
E. B.
, 1995, “
Fundamental Frequency
,”
Springs
,
34
, pp.
61
66
.
9.
Yildirim
,
V.
, 2002, “
Expressions for Predicting Fundamental Natural Frequencies of Non-Cylindrical Helical Springs
,”
J. Sound Vib.
0022-460X,
252
(
3
), pp.
479
491
.
10.
Yildirim
,
V.
, 2004, “
A Parametric Study on Natural Frequencies of Unidirectional Composite Conical Spring
,”
Commun. Numer. Methods Eng.
1069-8299,
20
, pp.
207
227
.
11.
Wolansky
,
E. B.
, 1996, “
Conical Spring Buckling Deflection
,”
Springs
,
35
, pp.
63
68
.
12.
Wolansky
,
E. B.
, 2001, “
,”
Springs
,
40
, pp.
95
98
.
13.
Rodriguez
,
E.
, and
Paredes
,
M.
, 2005, “
Ends Effect on Conical Spring Behavior
,”
Springs
,
44
, pp.
32
36
.
14.
Spring Manufacturers Institute, Inc., 2001 Midwest Road, Suite 106, Oak Brook, IL 60523-1335, www.smihq.orgwww.smihq.org
15.
Universal Technical Systems, Inc., 202 West State Street, Suite 700, Rockford, IL 61101, www.uts.comwww.uts.com
16.
The Institute of Spring Technology, Henry Street, Sheffield. S3 7EQ. UK., www.ist.org.uk/index.htmlwww.ist.org.uk/index.html
17.
Paredes
,
M.
,
Sartor
,
M.
, and
Masclet
,
C.
, 2002, “
Obtaining an Optimal Compression Spring Design Directly From a User Specification
,”
Proc. Inst. Mech. Eng., Part B
0954-4054,
216
, pp.
419
428
.
18.
Paredes
,
M.
,
Sartor
,
M.
, and
Masclet
,
C.
, 2001, “
An Optimization Process for Extension Spring Design
,”
Comput. Methods Appl. Mech. Eng.
0045-7825,
191
, pp.
783
797
.
19.
Wu
,
M. H.
, and
Hsu
,
W. Y.
, 1998, “
Modelling the Static and Dynamic Behavior of a Conical Spring by Considering the Coil Close and Damping Effects
,”
J. Sound Vib.
0022-460X,
214
(
1
), pp.
17
28
.