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TECHNICAL BRIEFS

# Analytical Behavior Law for a Constant Pitch Conical Compression Spring

[+] Author and Article Information
Emmanuel Rodriguez

Laboratoire de Génie Mécanique de Toulouse, INSA, 135 avenue de Rangueil, F-31077 Toulouse, Cédex 4 Franceemmanuel.rodriguez@insa-toulouse.fr

Manuel Paredes, Marc Sartor

Laboratoire de Génie Mécanique de Toulouse, INSA, 135 avenue de Rangueil, F-31077 Toulouse, Cédex 4 France

J. Mech. Des 128(6), 1352-1356 (Dec 29, 2005) (5 pages) doi:10.1115/1.2338580 History: Received July 01, 2005; Revised December 29, 2005

## Abstract

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.

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## Figures

Figure 1

Parameters of the studied conical spring

Figure 2

Deflection curve and successive coil arrangements according to compression phases (only active coils are displayed)

Figure 3

Distribution of active coils, at any step of the nonlinear phase

Figure 4

Experimental data and analytical model of two constant pitch conical springs, No. 1 is telescoping, No. 2 is nontelescoping

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