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| Scenario | Inputs | k (N·m/rad) | Notes |
|---|---|---|---|
| General spring | T = 120 N·m, θ = 5 deg | 1375.099 | Useful for joints and couplings. |
| Torsional shaft | G = 79 GPa, d = 30 mm, L = 1.2 m | 1453.869 | Solid round polar moment is used. |
| Cantilever end moment | E = 200 GPa, b = 50 mm, h = 120 mm, L = 1.0 m | 120000.000 | Moment–rotation stiffness at free end. |
Rotational stiffness (k) links torque to angular rotation. Higher k means less twist for the same moment, which improves alignment, vibration behavior, and control response in mechanisms and structures. In dynamics, k also influences natural frequency and steady-state deflection strongly.
If you already know the applied torque T and measured rotation θ, the simplest stiffness is k = T/θ. This fits flexible couplings, hinge joints, bushings, and test rigs where torque and angle are recorded directly.
For a uniform shaft, stiffness depends on shear modulus G, polar moment J, and length L. Shorter shafts, stiffer materials, and larger diameters raise k. For steel, G is about 79 GPa; for aluminum, near 26 GPa. Because J scales with diameter to the fourth power, increasing diameter 10% raises stiffness about 46%.
When a beam is loaded by an end moment, the free-end rotation is θ = ML/(EI), so k = M/θ = EI/L. This is useful for brackets, cantilever arms, and fixture plates where rotational compliance drives tip angle. For many steels, E is near 200 GPa, while polymers are far lower.
The calculator supports direct J or I input, round sections from diameter, hollow sections from inner and outer diameters, and rectangular approximations. Rectangular torsion is nonuniform, so the J estimate is approximate but practical for early sizing.
The core formulas use radians. The tool converts to torque-per-degree for easier interpretation on drawings. It also reports common imperial outputs, helping teams compare catalog spring rates and legacy inch-pound specs without hand conversions.
Validate inputs: L must be positive, and inner diameter must be smaller than outer diameter. Compare results with expected ranges. Machine couplings may fall near 10^2–10^5 N·m/rad, while thick steel shafts and beams can be far higher.
It measures how much torque is needed to produce a given rotation. A larger stiffness means less angular deflection for the same moment, which helps maintain alignment and reduce vibration.
Use T/θ when you have measured torque and rotation. Use GJ/L for a shaft in torsion. Use EI/L when a cantilever is rotated by an end moment.
For round shafts, the polar moment J depends on diameter to the fourth power. Small diameter increases can dramatically raise stiffness, while small reductions can greatly increase twist.
The per-degree value is smaller because one degree is only 0.01745 radians. Convert using k(N·m/deg) = k(N·m/rad) × (π/180).
Yes. Many catalogs list torque per degree or torque per radian. Enter torque and angle as a pair, or convert the catalog rate to an equivalent torque and rotation point for k = T/θ.
No. Rectangular torsion has nonuniform shear, so the torsional constant is approximate. It is suitable for quick sizing, but final designs should confirm stiffness with validated formulas or FEA.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.