Torsional Vibration: The Twisting Resonance Hidden in Every Rotating Shaft

A crankshaft, a propeller shaft, a turbine rotor — each looks like a solid piece of steel that simply turns. But every rotating shaft is also a spring. Apply a sudden twist at one end and the far end lags, winds up, and then snaps back, oscillating in torsion long after the disturbance is gone. Most of the time this twisting is invisible. Then an engine reaches a particular speed, the twisting locks into resonance, and a shaft that was perfectly balanced begins to fail.

This article explains torsional vibration: how a disk on a shaft behaves as a rotational oscillator, how to find its natural frequency, and why that frequency is one of the first numbers to pin down when designing rotating machinery.

Why this calculation matters

Torsional vibration is dangerous precisely because it is hard to see. A shaft can be twisting back and forth thousands of times a second while its axis stays still and its bearings feel nothing unusual. There is no obvious wobble to warn you. The damage shows up instead as fatigue cracking, failed couplings, gear-tooth wear, or a snapped crankshaft.

The reason it matters is resonance. Every shaft-and-disk system has one or more natural frequencies of torsional oscillation. If a periodic torque — the firing pulses of an engine, the blade-passing excitation of a propeller, a gear-mesh harmonic — happens to match a natural frequency, the twisting amplitude grows dramatically. The rotational speed at which that match occurs is a critical speed. Engineers compute torsional natural frequencies early so they can keep operating speeds away from critical speeds, or add a damper if a clash is unavoidable. Skip this step and the failure can be sudden and complete.

The core method

Picture the simplest case: a single disk fixed to one end of a shaft whose other end is clamped. The shaft acts as a torsional spring, and the disk acts as a rotational mass. Twist the disk and the shaft pulls it back; this is exactly the rotational version of a mass on a spring.

The natural angular frequency of that oscillator is:

omega_n = sqrt(k_t / I)

where k_t is the torsional stiffness of the shaft and I is the polar mass moment of inertia of the disk. The torsional stiffness — how much torque is needed per radian of twist — comes from the shaft's geometry and material:

k_t = G * J / L

Here G is the shear modulus of the shaft material, L is the shaft length, and J is the polar second moment of area of the cross-section. For a solid circular shaft of diameter d:

J = pi * d^4 / 32

The fourth-power dependence on diameter is striking: a small increase in shaft diameter stiffens it sharply and raises every natural frequency. The natural frequency in hertz follows from omega_n:

f_n = omega_n / (2 * pi)

The structure of omega_n = sqrt(k_t/I) carries the design intuition. A stiffer shaft raises the natural frequency; a heavier or larger-radius disk lowers it. To move a natural frequency away from a troublesome excitation, you change one of those two things.

A worked example

Take a steel shaft carrying a single disk. The shaft has shear modulus G = 79 GPa, diameter d = 30 mm, and length L = 0.5 m. The disk has a polar mass moment of inertia I = 0.05 kg.m^2.

Step 1 — polar second moment of area. For the solid circular cross-section:

J = pi * d^4 / 32
J = pi * (0.030)^4 / 32
J = pi * 8.1e-7 / 32
J = 7.95e-8 m^4

Step 2 — torsional stiffness. Combine material and geometry:

k_t = G * J / L
k_t = 79e9 * 7.95e-8 / 0.5
k_t = 12,560 N.m/rad

So the shaft resists with about 12,560 newton-metres of restoring torque for every radian it is twisted.

Step 3 — natural angular frequency. Use the oscillator relation:

omega_n = sqrt(k_t / I)
omega_n = sqrt(12,560 / 0.05)
omega_n = sqrt(251,200)
omega_n = 501 rad/s

Step 4 — natural frequency in hertz. Convert to cycles per second:

f_n = omega_n / (2 * pi)
f_n = 501 / 6.283
f_n = 79.8 Hz

The system has a torsional natural frequency of about 79.8 Hz. In practical terms, any excitation near 79.8 cycles per second — which at this frequency corresponds to a rotation rate of roughly 4,800 rpm if the shaft were excited once per revolution — would drive this disk into torsional resonance. That is the number a designer must keep clear of the operating range.

Common mistakes

Confusing torsional vibration with lateral whirl. Lateral or bending vibration moves the shaft axis sideways and is what most balancing work targets. Torsional vibration twists the shaft about its own axis with no sideways motion at all. The two have different natural frequencies and different fixes — solving one does not address the other.

Using the wrong second moment. Torsional stiffness uses the polar second moment of area J = pi*d^4/32. Bending uses the area moment I = pi*d^4/64. They differ by a factor of two for a circular section, and swapping them throws every stiffness and frequency off.

Mixing up the two "I" quantities. k_t uses J, a geometric property of the cross-section in metres to the fourth. omega_n uses I, the disk's mass moment of inertia in kg.m^2. They share a symbol family but are entirely different physical quantities — keep them straight.

Forgetting the strong diameter dependence. Because J scales with d^4, a modest change in shaft diameter shifts stiffness and natural frequency far more than intuition suggests. Small geometry tweaks can move you into or out of a critical speed.

Ignoring damping and multiple discs. A real driveline often has several inertias and stiffnesses and behaves as a multi-degree-of-freedom system with several natural frequencies. The single-disk formula is a clean first estimate, not the final answer for a full engine train.

Try the interactive NovaSolver calculator

The single-disk formula is an excellent starting point, but real shafts carry several discs, and the mode shapes matter as much as the frequencies. The Torsional Vibration Analysis — Multi-DOF Shaft System tool on NovaSolver computes natural frequencies, mode shapes, and Campbell diagrams for 2- to 4-disc torsional systems in real time. You set each disc inertia, each shaft stiffness, and a damping value with sliders, and it returns the first and second natural frequencies, omega_1, the mode-1 node location, and a critical-speed view for engines, turbines, and gear trains.

Related calculators

• Torsion of a Shaft — work out the static side: shear stress and angle of twist from torque, geometry, and material.
• SDOF Vibration Response — the single-degree-of-freedom oscillator that the one-disk torsional system maps directly onto.
• Eigenvalue Vibration Analysis — the general approach to natural frequencies and mode shapes for multi-degree-of-freedom systems.

You can browse the rest in the vibration tools hub.

Closing note

Torsional vibration is a quiet failure mode, and that is exactly what makes the calculation worth doing. A shaft can twist itself into a fatigue crack while looking perfectly steady from the outside. The remedy is straightforward in principle: model the shaft as a torsional spring and the disc as a rotational mass, find omega_n = sqrt(k_t/I), convert to a frequency, and compare it against every periodic torque the machine will see. Keep the natural frequencies clear of the excitation frequencies — or damp them — and a rotating system that would otherwise shake itself apart runs smoothly for its full design life.