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Impact Factor (2025): 6.9
DOI Prefix: 10.47001/IRJIET
Impact Factor (2025): 6.9
DOI Prefix: 10.47001/IRJIET
Vol 10 No 5 (2026): Volume 10, Issue 5, May 2026 | Pages: 304-315
International Research Journal of Innovations in Engineering and Technology
OPEN ACCESS | Research Article | Published Date: 15-05-2026
The nonlinear load-deflection response of telescopic conical compression springs is due to the gradual stacking of the coils and the change of the diameter along the spring axis. The classical theory for cylindrical springs is well established, but analytical treatments for spiral-defined conical springs are fragmented and often based on incremental coil-by-coil approaches that may lead to more inaccuracies in the prediction of their load-deflection response than the more comprehensive approaches proposed in this paper. A unified closed-form theoretical framework is presented in this paper for the load-deflection and stiffness behaviour of telescopic conical springs, whose coil centrelines follow Archimedean, logarithmic and parabolic spiral laws. The elastic deflection is determined exactly by a continuous turn-index formulation. A load-radius boundary mapping models the progressive stacking regime explicitly. Closed-form expressions are obtained for the linear and stacking regimes, the transition and collapse loads and the evolution of instantaneous stiffness. We formally prove the mathematical continuity, monotonicity, and limit correctness. By means of comparative analysis we reveal a consistent stiffness hierarchy between spiral types and the geometric mechanisms governing nonlinear response. The framework avoids incremental summation methods and offers design-ready analytical expressions, which are appropriate for optimization and parametric synthesis of nonlinear conical springs.
Conical compression spring, spiral geometry, nonlinear stiffness, telescopic spring, closed-form model, stacking regime, analytical mechanics.
Ali Khayoun Al-Janabi, & Essam L. Esmail. (2026). Unified Closed-Form Load–Deflection Modeling of Spiral-Defined Telescopic Conical Compression Springs. International Research Journal of Innovations in Engineering and Technology - IRJIET, 10(5), 304-315. Article DOI https://doi.org/10.47001/IRJIET/2026.105041
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