Helical extension springs store energy and provide resistance to tensile loads that are applied through appropriate spring ends. Typical spring ends include different types of hooks or loops. Both ends of a helical extension spring are attached to other components. When the two components move apart and their distance is increased, the helical extension spring exerts a tensile force between the two components and tries to decrease their distance. There are various applications for helical extension springs that include automobiles, toys, hand tools, agriculture machines, textile machines, and medical devices. The common configuration of helical extension springs uses straight cylindrical shape that has constant coil diameter and pitch. Unlike regular helical extension springs, variable-diameter helical extension springs do not employ constant coil diameter. Their variable coil diameter enables them to produce desired force deformation relationships and reduce stress concentration. The distinctive features of variable-diameter helical extension springs also raise their synthesis challenges. To surmount these challenges, a method is introduced in this paper to model and design variable-diameter helical extension springs. The configuration of a synthesized spring is described by a composite parametric curve. The entire spring is defined by its control parameters. Synthesizing the spring is systematized as optimizing its control parameters. Examples on modeling, analyzing and designing springs are presented in the paper to demonstrate the procedure and verify the effectiveness of the introduced synthesis method.

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