Rotational Stiffness Calculator (k = M / θ, N·m/rad, torsional frequency)

Understanding the rotational stiffness calculator — concept and principle

This calculator finds rotational (torsional) stiffness, the moment needed per unit of rotation, from an applied moment and the measured twist angle. It also gives the stiffness per degree, the flexibility, the energy stored and, with an attached moment of inertia, the torsional natural frequency. Engineers use it for shafts, couplings, joints and column bases.

Stiffness is moment per radian; with a known inertia it also sets how fast the shaft would ring.

How it is calculated

k = M ÷ θ (θ in radians)

Stiffness per degree = k × π ÷ 180; flexibility = 1 ÷ k

Energy stored = ½ k θ²

Torsional natural frequency f = (1/2π) √(k ÷ I)

M in N·m, θ in degrees or radians, I in kg·m².

The principle

Rotational stiffness is the turning version of a spring constant: twice the moment gives twice the twist in a linear-elastic part. Like a spring, a twisted shaft stores energy ½kθ². If a mass with moment of inertia I is attached, the combination oscillates like a torsion pendulum, with a natural frequency set by k and I, just as a mass on a spring has √(k/m).

Worked example

Default inputs: M = 50 N·m, θ = 2°, I = 0.05 kg·m².

θ = 2 × π ÷ 180 = 0.034907 rad.
k = 50 ÷ 0.034907 = 1,432 N·m/rad = 1.432 kN·m/rad; per degree 25 N·m/°.
Flexibility = 1 ÷ 1,432 = 0.0006981 rad per N·m.
Energy = ½ × 1,432 × 0.034907² = 0.8727 J.
f = (1/2π) √(1,432 ÷ 0.05) = 26.94 Hz.

Each point is the calculator re-run with only “Rotation or twist angle θ” changed; the gold point is the default example. Change the inputs above and this chart follows your numbers.

Rotation or twist angle θ	Rotational stiffness k (N·m/rad)
0.5	5,730
1	2,865
1.5	1,910
2	1,432
2.5	1,146
3	954.9
4	716.2

How to read the result

Higher stiffness means less twist under load and a higher natural frequency. In structures, the rotational stiffness of a connection decides whether it behaves as pinned, semi-rigid or fixed. In drive trains, keep the torsional natural frequency away from the operating speeds to avoid resonance.

Common mistakes

Using degrees in k = M ÷ θ when radians are expected.
Measuring twist beyond the linear range.
Including the attached inertia in the wrong units (g·cm² instead of kg·m²).

Keep in mind: Classification of structural connections follows design codes. Real designs need applicable standards and a qualified engineer.

Frequently asked questions

What units is rotational stiffness in?

N·m per radian.

How is it related to a spring constant?

It is the rotational analogue: moment instead of force, angle instead of displacement.

How do I find the natural frequency?

f = (1/2π)√(k/I), with I the attached moment of inertia.

Why does stiffness matter for joints?

It determines how much moment a connection can transfer for a given rotation.

Sources: OpenStax, University Physics Volume 1 (2016), chapter 15 key equations (torsional pendulum)

Checked 9 Nov 2026. Definition of rotational stiffness κ = τ/θ (rotational analogue of Hooke’s law) and torsional pendulum period T = 2π√(I/κ): OpenStax, University Physics Volume 1 (2016), Chapter 15 Key Equations.

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