Torsion Spring Calculator

Torsion springs store and release rotational energy by twisting their coils. The mechanical behavior of a torsion spring is governed by several core mathematical relationships. First, the spring index ($C$) compares the mean spring diameter ($D$) to the wire diameter ($d$) using the equation $C = D / d$. The total active turns ($N$) incorporate full coils plus contributions from the attached arm lengths. Torque ($ au$) is generated by applying a force at a specific distance or arm length ($L_{arm}$), expressed as $ au = F \times L_{arm}$. To evaluate structural integrity, the Wahl stress factor accounts for direct shear and curvature effects, multiplying standard bending stress to find the true inner fiber stress. Finally, angular displacement ($\theta$) and spring rate ($k$) relate the applied torque to the Young's modulus ($E$) of the material and the physical geometry of the spring coils.

Consider a custom steel torsion spring with a wire diameter ($d$) of 2 mm and a mean spring diameter ($D$) of 16 mm, meaning the spring index is 8. Let the material be music wire with a high Young's modulus of 200,000 MPa. If the spring possesses 5 active turns ($N$) and experiences a torque ($ au$) of 1.5 N·m, we can determine its angular displacement and spring rate. Using the angular displacement formula: $\theta = (64 \times 1.5 \times 0.016 \times 5) / (200,000 \times 10^6 \times (0.002)^4)$, the resulting angular deflection is approximately 1.52 radians (about 87 degrees). The resulting spring rate ($k = \tau / \theta$) evaluates to roughly 0.987 N·m/rad.

When designing or analyzing torsion springs, always verify your spring index ($C$). An ideal index falls between 4 and 12; values below 4 make manufacturing difficult and introduce high residual bending stresses, while values above 12 lead to floppy springs that waste space. Additionally, always account for friction and the exact angle of deflection, as binding against a shaft during rotation will drastically alter the operational torque.