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There are very few theoretical studies about a constant-force spring. Considering the strain energy necessary for flattening a coiled spring, Votta proposed an analytical theory of such a spring. This theory, however, could not be applicable for all stages of displacements, especially for small displacements, because the uncoiled length of the spring can not be flattened.In this report, we perform a new theoretical analysis of the constant-force spring based on the theory of Elastica. Then, the extension mechanism and the deformed shape of the spring, which could not be clarified in the formulas given by Votta, are analyzed. As a result, it is found that the force rises rapidly and approaches a constant value as the spring is extended infinitely. Furthermore, the experimental verification of this analysis is carried out using a commercially available constant-force spring. The theoretical values are in good agreement with the experimental ones. Consequently, the new theory is proved to be of practical use.

Analysis on Characteristics of Constant-Force Spring Based on Large Deformation Theory