Torsional Spring Calculator — Torque, Energy & Natural Frequency
Enter a torsional spring's stiffness constant and deflection angle to get the restoring torque and stored elastic energy. Add the load's moment of inertia to find the natural oscillation frequency.
N·m/rad
degrees
kg·m²
Torque the spring exerts to return to zero angle: T = k × θ
- 1
Angle in radians
45° × π ÷ 180 = 0.7854 - 2
Restoring torque T = k × θ
10 × 0.7854 = 7.8540
How does this calculator work?
A torsional spring stores rotational energy. Restoring torque T = k × θ; stored energy U = ½kθ². With a known load inertia I, the natural frequency is f = (1/2π)√(k/I). Enter spring constant and deflection angle to get torque and energy; add inertia for the oscillation frequency.
Formula
How this is calculated
A torsional spring applies a restoring torque proportional to the angular displacement, following the rotational analogue of Hooke's law: T = k × θ, where k is the torsional spring constant (N·m/rad) and θ is the deflection angle in radians. The elastic potential energy stored in the spring is U = ½·k·θ² — the same quadratic form as for a linear spring. At the design load angle, this energy can be released to do work (as in a clockwork mechanism or a snap-action switch).
When the spring is attached to a body with moment of inertia I (kg·m²), the system oscillates at a natural angular frequency ω = √(k/I) rad/s, giving a natural frequency f = ω/(2π) Hz and period T = 1/f. This is the rotational equivalent of a mass-spring system. The formula assumes the spring mass is negligible compared to the load, linear behaviour (k constant), and no damping — real systems have friction and structural damping that reduce amplitude over time.
The torsional spring constant k depends on the material and geometry: for a coil spring wound from wire, k = Gd⁴/(64DR·N_a) where G is shear modulus, d is wire diameter, D_R is coil mean diameter, and N_a is active coils. This calculator works with k directly, whatever its source.
Frequently asked questions
The SI unit is N·m/rad (Newton-metres per radian). Some datasheets quote N·m/degree — multiply by 180/π ≈ 57.3 to convert to N·m/rad. Other common units are lb-in/rad or ozf-in/degree for imperial springs.
A linear spring resists linear displacement (F = k·x, J), while a torsional spring resists angular displacement (T = k·θ, N·m). The maths is identical in form — just replace force with torque, displacement with angle, and linear mass with moment of inertia.
Beyond the yield point, the spring permanently deforms and k is no longer constant. This calculator assumes linear elastic behaviour throughout. In practice, springs are rated to a maximum torque or deflection angle — stay within that rating to avoid plastic deformation.
Also known as
TG we-Calculate Editorial Team. (2026). Torsional Spring Calculator — Torque, Energy & Natural Frequency [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/torsional-spring-calculator
TG we-Calculate Editorial Team. "Torsional Spring Calculator — Torque, Energy & Natural Frequency." TG we-Calculate. 2026. https://we-calculate.com/calculator/torsional-spring-calculator.
TG we-Calculate Editorial Team, "Torsional Spring Calculator — Torque, Energy & Natural Frequency," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/torsional-spring-calculator
@misc{wecalculate_torsional_spring_calculator, title = {Torsional Spring Calculator — Torque, Energy & Natural Frequency}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/torsional-spring-calculator}}, year = {2026}, note = {TG we-Calculate} }
Did this calculator help you?