Knowledge / Ship systems

Torsional vibration and the barred speed range

Understand excitation orders, shaft modes, alternating stress and why time spent crossing a resonance matters.

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A barred speed range is a vessel-specific operating restriction associated with excessive torsional response over a band of shaft speeds. The shaft can twist back and forth around its mean rotation even while the tachometer looks steady. Understanding the restriction requires excitation frequency, natural modes, stress amplitude and passage duration together; an engine speed alone is not a universal danger threshold.

Mean torque and alternating torque are different

Mean torque transmits useful power: P = Q_mean ω, with power in watts, torque in newton metres and angular speed in radians per second. The fluctuating component changes elastic twist and may produce substantial alternating shear stress. A small average load does not prove that this alternating component is small, particularly near resonance.

Torsion rotates shaft sections relative to one another. Lateral whirling moves the shaft centreline sideways, while axial vibration moves components along the axis. They can interact, but a low lateral bearing-vibration reading does not directly establish low torsional stress. MIT’s dynamics teaching example explains fluctuating electric-motor torque as a source of fluctuating shaft twist.

Excitation order turns shaft speed into frequency

An order h means h excitation cycles per revolution of the selected reference shaft. For shaft speed N in r/min, f_exc = hN/60 in hertz. Engine gas forces, reciprocating inertia, propeller loading and connected electrical machinery can contribute different orders. The strongest order depends on machinery arrangement, firing sequence and operating condition; it should not be guessed from cylinder count alone.

On a speed-frequency diagram, order lines rise with speed while a simple constant-property natural frequency appears horizontal. Their intersections identify candidate resonances. An intersection locates a frequency match, not the stress amplitude: excitation magnitude, phase, damping and mode shape determine how strongly the shaft responds. Geared installations also require a consistent reference speed.

A mode describes where the twist occurs

A basic model uses rotary inertias connected by torsional springs. For a uniform circular shaft, stiffness K = GJ_p/L, where G is shear modulus, J_p the polar second moment of area and L length. J_p has units of m⁴; it is different from mass moment of inertia I, measured in kg·m². Confusing the two produces dimensionally incorrect dynamics.

One mode can have large angular movement at an end but high alternating torque in a different shaft segment. A torsional node is a location of zero modal angular displacement relative to mean rotation, not necessarily zero stress. Increasing diameter raises stiffness strongly, but shifting one resonance can expose another excitation order rather than solve every operating case.

Worked example: a two-inertia resonance

Consider two free rotary inertias I₁ = 40 000 kg·m² and I₂ = 10 000 kg·m² joined by a massless shaft of stiffness K = 8.0 × 10⁶ N·m/rad. Ignore damping and external restraint. This idealisation has a zero-frequency rigid-rotation mode and one elastic mode, with ω_n = √[K(1/I₁ + 1/I₂)].

Substitution gives ω_n = √1 000 = 31.6228 rad/s and f_n = ω_n/(2π) = 5.0329 Hz. A fifth-order excitation crosses that frequency at N = 60f_n/5 = 60.3951 r/min. These calculations locate one hypothetical resonance. They do not establish a barred band around it, because no forcing amplitude, damping, stress limit or instrument allowance has been supplied.

The elastic mode also satisfies I₁θ₁ + I₂θ₂ = 0, so θ₂ = −4θ₁: the smaller inertia swings through four times the angular amplitude in the opposite phase. This illustrates why a measurement at one position cannot be interpreted without the mode shape. A real engine crankshaft and propeller system needs more than two inertias.

Translate torque into the relevant stress

For a smooth, solid circular shaft under an illustrative alternating torque amplitude Q_a = 120 kN·m and diameter d = 0.320 m, nominal surface shear-stress amplitude is τ_a = 16Q_a/(πd³) = 18.65 MPa. Torque amplitude means half the peak-to-peak variation for a centred sinusoid. Mixing amplitude and peak-to-peak conventions would introduce a factor-of-two error.

This nominal value excludes fillet, keyway and other local stress concentration, as well as mean-stress and material influences on fatigue. A hollow shaft requires its own polar section modulus. Comparing 18.65 MPa with an arbitrary material strength would not establish acceptance. Fatigue assessment needs the applicable stress definition, detail geometry and number of cycles.

Crossing time changes fatigue exposure

If an illustrative vibration frequency stayed near 5 Hz for a 20 s passage, it would impose about 100 cycles; a 60 s passage would impose about 300. The actual count is the integral of frequency over time, and damage also depends strongly on the varying amplitude. Three times as many cycles is not a general threefold increase in whole-life damage.

DNV’s 2018 technical note connects long passage times with shaft-fatigue concerns and discusses acceleration capability. Its numerical rule examples are historical and are not universal operating limits. The engine-maker feeder paper also relates acceptable passage to stress and lifetime crossing frequency.

Measurement must match the calculation

Strain measurements can estimate shaft torque and torsional stress; angular encoders measure rotational fluctuation that must be interpreted through the model. Sensor location, bandwidth, order tracking, calibration and speed history affect the result. A short record at steady speed and a rapid run through resonance answer different questions.

The operating case must also be stated: engine load, propeller condition, direction of running and any permitted degraded configuration. DNV’s vibration-service description distinguishes natural-frequency calculations, forced response and transient shaft dynamics. Agreement in natural frequency alone does not validate predicted peak stress or the response during a passage.

A changed propulsion plant needs a changed assessment

A damper dissipates oscillatory energy; a tuning wheel changes inertia; altered shaft stiffness shifts modal frequencies. These mechanisms are different, and each affects more than a single plotted peak. Damper condition and coupling properties can also change with service, so original calculations depend on the maintained configuration.

Engine derating, propeller replacement or a new power take-off can alter excitation, inertia or acceleration capability. A previous restriction should not be silently removed or copied to the changed plant. The operative band and passage procedure must come from its approved documentation and machinery instructions; a general educational resonance calculation cannot set bridge or engine-room controls.

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