Rotational Stiffness Calculator

Background.

Twist a shaft, a coupling or a bolted beam-to-column joint and it resists by turning back against you. How hard it resists for each unit of twist is its rotational stiffness — the turning-motion counterpart of a spring constant. A stiff drive shaft barely winds up under engine torque; a flexible coupling is designed to wind up noticeably so that shock loads are cushioned before they reach the gears and bearings.

The number is simply torque divided by the angle it produces. Apply 120 newton-metres and measure 0.02 radians of rotation, a little over one degree, and the stiffness is 6,000 N·m per radian. That one figure feeds straight into the questions engineers care about. In a drivetrain or a gauge it sets the torsional natural frequency, √(k/J) in radians per second for a rotor of inertia J, and so decides whether the system will resonate at a running speed. In a structure it decides how much a connection lets a beam end rotate, which shifts bending moments between members.

Two things are easy to get wrong. The angle must be in radians: one radian is about 57.3°, so a deflection read off in degrees and typed in as if it were radians makes the stiffness come out 57.3 times too small. And the ratio describes the component only while its response stays linear. Rubber couplings, slipping joints and anything approaching yield get softer as the load rises, so a stiffness worked out from a single load and angle is a secant value for that load. It may not hold at another.

For a solid round shaft, the stiffness can also be predicted from its geometry and material as G × J_p ÷ L, but that prediction is not made here; this page starts from a measured or specified torque and angle.

What is rotational stiffness calculator?

Rotational stiffness, also called torsional stiffness or a torsional spring constant, is the torque required per unit of angular deflection: k_θ = T ÷ θ. In SI units it is measured in newton-metres per radian. It describes how strongly a component resists being twisted, in the same way that a linear spring constant in newtons per metre describes resistance to being stretched.

How to use this calculator.

1. 01Enter the applied torque in newton-metres. If you have force and lever arm, multiply them; 400 N on a 0.3 m arm is 120 N·m. To convert from pound-feet, multiply by 1.3558.
2. 02Enter the rotation that torque produced, in radians. To convert from degrees, multiply by π ÷ 180, so 1.146° is 0.02 rad.
3. 03Use the twist of the part you care about only. If your measurement includes slack in the clamps, keys or bolts, the result will describe the whole assembly, not the part.
4. 04Read the stiffness in N·m/rad. For a per-degree figure, multiply it by π ÷ 180.

The formula.

k_θ = T ÷ θ, where T is the applied torque in newton-metres and θ the resulting angular deflection in radians. Because the radian is dimensionless, the unit N·m/rad is sometimes written simply as N·m; keeping the rad makes clear that the value is per radian rather than per degree. The relation is the rotational form of Hooke's law, T = k_θ θ, and holds while the material and joints respond elastically and in proportion. Doubling the torque for the same twist doubles the stiffness; doubling the twist for the same torque halves it. Two torsional springs in series combine as 1/k = 1/k₁ + 1/k₂, and in parallel their stiffnesses add. The energy stored in the twisted part is ½ k_θ θ².

A worked example.

Frequently asked questions.

References& sources.

1. [1]Inman, Engineering Vibration, 4th ed. (PRINT), section 2.2, torsional spring stiffness and torque-displacement relation.
2. [2]NPTEL, Mechanical Vibrations, torsional vibration model and torsional spring relation.
3. [3]BIPM, The International System of Units (SI Brochure), 9th ed., version 3.01, coherent derived units and quantity equations.