Formula and Method for Rotational Stiffness
Rotational stiffness — also called torsional stiffness or angular stiffness — measures how strongly an elastic system resists being twisted. It is defined as the applied torque divided by the resulting angular displacement: k = T / θ, where T is torque (N·m) and θ is the angle of twist in radians. This is the direct rotational counterpart of a linear spring's stiffness, F = kx: torque plays the role of force, and angular displacement plays the role of linear displacement. This calculator also derives the elastic energy stored in the twisted system and, if you supply the rotating body's moment of inertia, the natural frequency at which it would oscillate.
How the calculation works
Enter the applied torque and its unit, then the resulting angular twist and its unit. The calculator converts both to SI units (newton-meters and radians) and divides to get stiffness, k = T / θ, in N·m/rad. It also reports the same stiffness per degree (multiplying by π/180 ≈ 0.017453) and the strain energy stored while twisting the system from rest, U = ½kθ². If you enter a moment of inertia I (in kg·m²) for the body attached to that rotational spring, the tool computes the natural angular frequency of torsional oscillation, ω = √(k / I), along with the corresponding frequency in hertz and the oscillation period, T = 2π/ω.
Common mistakes
- Mixing angle units: stiffness in N·m/rad and N·m/degree differ by a factor of π/180 ≈ 0.01745 — always confirm which unit a quoted stiffness value uses before comparing numbers.
- Using degrees inside the raw formula: k = T/θ only gives N·m/rad directly when θ is in radians; plugging in degrees without converting understates the true stiffness by roughly a factor of 57.3.
- Confusing torsional stiffness with a shaft's polar stiffness: k = T/θ is the general, measured definition; a shaft's geometric stiffness k = GJ/L (shear modulus × polar moment of inertia ÷ length) is a separate way to predict the same quantity from material and cross-section.
Real-world applications
- Mechanical design uses torsional stiffness to size drive shafts, couplings, and torsion-bar springs so they twist within an acceptable range under load.
- Structural and connection engineering uses rotational stiffness to classify beam-column or bolted joints as rigid, semi-rigid, or pinned.
- Robotics and automotive suspension design use rotational stiffness together with a component's moment of inertia to predict torsional vibration frequencies and avoid resonance.
- Instrumentation such as torsion balances and torsional pendulums relies on a known, stable rotational stiffness to convert a measured twist angle into a torque reading.