Background
A marine propulsion shaft system rotates as a single mechanical entity but is mechanically a chain of inertias (the crankshaft, flywheel, intermediate shafts, propeller) connected by elastic shafting. Each combustion event in each cylinder produces a torque pulse that travels through this chain. The pulse is not steady; it varies through the firing cycle from peak combustion to scavenging, and it has strong harmonic content. If the frequency of one of these harmonics matches a torsional natural frequency of the shaft system, torsional vibration amplitudes can grow large enough to fracture shaft elements.
The discipline of torsional vibration analysis dates from the early twentieth century. Heinrich Holzer published his original tabular method in 1921 for solving torsional natural frequencies of multi-degree-of-freedom shaft systems. The method, refined and extended over the decades, remains the foundation of modern marine torsional analysis, now typically implemented in software with computer assistance.
Failure to perform adequate torsional analysis is a recurring cause of shaft failures in marine practice. Notable historical cases include cylinder cut-out operations on certain engine classes that produced previously unanticipated harmonics, and dual-fuel engine modes that shifted the firing harmonic content compared to single-fuel operation. Class societies (DNV, ABS, LR, BV, ClassNK, KR, RINA, CCS) require torsional analysis approval as part of new-build certification and after major modifications.
This article describes the analytical methods used, the practical results (mode shapes, critical speeds, barred speed ranges), the role of torsional dampers, and the operational implications.
The mass-elastic system
A propulsion shaft system is modelled as a chain of point masses (inertias) connected by torsionally elastic springs (shaft sections). For a typical slow-speed two-stroke engine plus shafting:
Engine mass elements
- Crankshaft throws (one per cylinder): each throw plus its share of the crankshaft modelled as a single inertia. The throw inertia includes the crank pin, web, and approximately half the journal mass.
- Crankshaft journal sections between throws: modelled as elastic springs. Stiffness depends on journal length, diameter, and the crank web geometry between adjacent throws.
- Flywheel at the engine output: a large inertia, typically 30 to 80 percent of the engine’s total rotating mass.
Shaft mass elements
- Intermediate shafting: divided into segments between bearings. Each segment is modelled as an inertia at its midpoint connected by springs to adjacent segments.
- Tail shaft: separate segment passing through the stern tube to the propeller.
Propeller
- The propeller inertia includes the propeller mass, the entrained water mass that moves with it, and any hydrodynamic added inertia. Entrained water adds typically 20-30 percent to the dry propeller inertia.
Auxiliaries
- Shaft generator (PTO/PTI): an additional inertia connected via a coupling, often modelled with the coupling’s elastic properties.
- Couplings: flexible couplings (rubber, viscous) appear as both elastic and damping elements.
The complete model typically has 8 to 20 inertia elements connected by elastic stiffnesses. Modern analysis includes damping (viscous, hysteretic, and structural).
Excitation
The engine excitation derives from cylinder firings. Each cylinder’s torque pulse repeats once per revolution (two-stroke) or every two revolutions (four-stroke). The pulse is decomposed into Fourier harmonics:
- Fundamental (1× firing frequency): peak amplitude
- Second harmonic (2× firing frequency): substantial amplitude
- Third harmonic (3× firing frequency): moderate amplitude
- Higher harmonics: progressively smaller
For a multi-cylinder engine, the cylinder firings are evenly distributed in crank angle (cylinder 1 fires, then cylinder 2 some degrees later, etc.). The total engine torque is the sum of individual cylinder contributions. Some harmonics constructively interfere across cylinders (these are the major harmonics of the engine), while others cancel.
For an N-cylinder two-stroke engine, the major harmonics are at orders k × N where k is a positive integer. For an 8-cylinder engine, major harmonics are at orders 8, 16, 24, etc. Other (minor) harmonics also exist but with smaller amplitude.
Operating at engine speed n (rev/s), the excitation frequencies are k × N × n Hz for the major harmonics.
Holzer’s method
Holzer’s method computes torsional natural frequencies and mode shapes by iterating through the shaft system from one end to the other. Starting with an assumed angular displacement at one end and zero displacement at a reference, the method calculates the resulting force balance through the system. At a natural frequency, the force balance closes (residual torque = 0); at non-natural frequencies, residual torque ≠ 0. Iteration converges on the natural frequencies.
Holzer’s algorithm
- Choose a trial frequency ω
- Set angular displacement at element 1: θ_1 = 1 (arbitrary unit)
- Set angular momentum at element 1: M_1 = -ω^2 × J_1 × θ_1 (with J_1 the inertia of element 1)
- For each subsequent element i:
- Calculate displacement: θ_i = θ_(i-1) - M_(i-1) / k_(i-1)
- Calculate inertia torque: M_i = M_(i-1) - ω^2 × J_i × θ_i
- The boundary condition is M_N = 0 (free end at element N)
- Iterate ω until M_N = 0; that ω is a natural frequency
Modes (1st, 2nd, 3rd, etc.) appear as successive natural frequencies, each with its own mode shape (displacement pattern across elements).
Computer implementation
Holzer’s method is now implemented in dedicated torsional analysis software (Frahm, ARGO, MTS). The software accepts the mass-elastic model and returns natural frequencies, mode shapes, and (with damping included) response amplitudes at each excitation frequency.
Mode shapes
A natural frequency has an associated mode shape: the relative angular displacement pattern across the inertias when oscillating at that frequency.
First mode
The first mode (lowest frequency) typically has the propeller oscillating against the engine crankshaft, with the flywheel as a near-node. First mode frequency is typically 1 to 3 Hz on a typical slow-speed shaft system.
Second mode
The second mode often has the engine forward end oscillating against the engine aft end (with the flywheel and propeller relatively quiet). Frequency is typically 5 to 10 Hz.
Higher modes
Higher modes have multiple internal nodes within the engine and shafting. Frequencies climb to 20+ Hz.
Mode shape determines critical speeds
A mode is excited by a harmonic that matches its natural frequency. Critical engine speed:
n_critical = f_natural / k_harmonic
where f_natural is in Hz, k_harmonic is the harmonic order (k × N for major harmonics), and n_critical is in Hz (multiply by 60 for rpm).
For a first-mode natural frequency of 2.0 Hz and an 8-cylinder engine major harmonic at order 8: critical speed = 2.0 / 8 × 60 = 15 rpm. This is below operating range; the first mode is not a normal-operation problem on this engine.
For higher modes and minor harmonics, critical speeds may fall within or near the operating range.
Barred speed ranges
When a torsional resonance falls within the engine’s normal operating range, the affected speed range is barred: the engine is not allowed to operate continuously at those speeds.
Definition of the bar
Barred speed ranges are typically expressed as percentages of MCR speed, with a width sufficient to cover the resonance peak plus margin. A typical bar might be: barred from 65 to 72 rpm.
Operational impact
Operators avoid sustained operation in barred ranges. The engine can pass through the range during acceleration or deceleration; sustained operation is limited to a few minutes.
Origin of bars
Bars arise when:
- A natural frequency falls between operating extremes, and
- The mode is excited by a major harmonic at typical operating speeds, and
- The amplitude exceeds the allowable torsional stress at any shaft element
Some engines have no bars within their operating range; others have one or two narrow bars.
Torsional dampers
When unmodified analysis predicts unacceptable torsional stresses, dampers are added to suppress amplitudes:
Viscous damper
A flywheel-like component with a smaller inertia free to rotate within an oil-filled enclosure attached to the main shaft. Relative motion between the inner inertia and the shaft generates viscous shear in the oil, dissipating torsional energy. The viscous damper suppresses resonance peaks across a range of frequencies.
Tuned damper
A flywheel-like component connected by a relatively soft elastic element (rubber, spring) to the main shaft. The damper is tuned to a specific resonance frequency: at that frequency, the damper inertia oscillates 180 degrees out of phase with the shaft, cancelling the resonance. Tuned dampers are highly effective at the design frequency but less so at others.
Combined dampers
Some installations combine viscous and tuned damping for broadband suppression with a sharp peak removal at a specific frequency.
Damper placement
Dampers are typically located at the engine’s forward end (opposite the flywheel) where torsional displacements are largest in the first mode. For higher-mode suppression, damper placement at intermediate locations is sometimes used.
Class society approval
Class societies require torsional analysis approval at design and after major modifications:
New-build approval
The shipyard or engine builder submits a complete torsional analysis report to the class society, covering:
- Mass-elastic model details
- Excitation assumptions
- Calculated natural frequencies and mode shapes
- Critical speeds and predicted stresses
- Barred speed ranges
- Damper design (if used)
The class society reviews and either approves or requires modifications.
Sea trial verification
After delivery, sea trials include torsiograph measurements to verify that actual torsional behaviour matches predictions. Discrepancies may require additional dampers, modified barred ranges, or operating restrictions.
Modification approval
Major modifications (engine derating, propeller change, addition of shaft generator, cylinder cut-out enabling) require renewed torsional analysis and class approval before commissioning.
Cylinder cut-out implications
Cylinder cut-out, sometimes used for super-slow-steaming, changes the firing pattern and therefore the harmonic content of the excitation. With cut-out:
- The major harmonic order decreases (from k × N to k × (N-1) or k × (N-2) etc.)
- The minor harmonics gain relative amplitude
- New torsional excitations may emerge that were not present in full-cylinder operation
Cut-out modes must be approved by the class society and may have their own barred speed ranges. Some engines support only certain cylinder cut-out patterns to keep torsional behaviour acceptable.
Dual-fuel torsional considerations
Dual-fuel engines have potentially different torsional behaviour in gas vs liquid mode:
- Different combustion characteristics produce different torque pulse shapes
- Pilot ignition vs full diesel injection give different harmonic profiles
- Some dual-fuel engines require separate torsional approval for each mode
In practice, the differences are usually small; the basic harmonic structure is preserved across modes. Major differences would require separate analyses.
Ice impact loads
Ships operating in ice (icebreakers, polar-class container ships) experience occasional impulsive torsional loading from propeller-ice contact. These transient impacts excite torsional modes briefly and produce stress peaks well above normal operation. Ice-class ships have specific torsional analysis requirements and often use stronger shafts or additional dampers.
Operational monitoring
Some ships monitor torsional vibration in service via:
- Torsiograph: a torsion-measurement device installed on the shaft, recording amplitude and frequency
- Strain gauges: bonded to the shaft, with telemetry to onboard recording
- Vibration sensors at engine bearings: indirect indicators of torsional behaviour
Continuous monitoring is rare on commercial ships but increasingly common on specialised vessels (icebreakers, naval ships).
Related Calculators
- Torsional Natural Frequency Calculator
- Mass-Elastic System Calculator
- Critical Speed Calculator
- Barred Speed Range Calculator
- Torsional Damper Sizing Calculator
- Crankshaft Torque Calculator
See also
- Crosshead Diesel Engine Architecture Overview
- Two-Stroke Marine Diesel Engine Fundamentals
- Engine Derating for Slow Steaming
- Engine Power and BMEP Relationships
References
- Holzer, H. (1921). Die Berechnung der Drehschwingungen. Springer.
- Wilson, W. K. (1956). Practical Solution of Torsional Vibration Problems. Chapman & Hall.
- DNV. (2023). Rules for Classification of Ships, Pt.4 Ch.4: Rotating Machinery — Torsional Vibration.
- Lloyd’s Register. (2023). Rules and Regulations for the Classification of Ships, Part 5 Chapter 8: Shaft Vibration.
- IACS. (2020). Unified Requirements M68: Torsional Vibration Calculations.