Torsional vibration

Last updated November 22, 2024

Torsional vibration is the angular vibration of an object - commonly a shaft - along its axis of rotation. Torsional vibration is often a concern in power transmission systems using rotating shafts or couplings, where it can cause failures if not controlled. A second effect of torsional vibrations applies to passenger cars. Torsional vibrations can lead to seat vibrations or noise at certain speeds. Both reduce the comfort.

In ideal power generation (or transmission) systems using rotating parts, the torques applied or reacted are "smooth" leading to constant speeds, and the rotating plane where the power is generated (input) and the plane it is taken out (output) are the same. In reality this is not the case. The torques generated may not be smooth (e.g., internal combustion engines) or the component being driven may not react to the torque smoothly (e.g., reciprocating compressors), and the power generating plane is normally at some distance to the power takeoff plane. Also, the components transmitting the torque can generate non-smooth or alternating torques (e.g., elastic drive belts, worn gears, misaligned shafts). Because no material can be infinitely stiff, these alternating torques applied at some distance on a shaft cause twisting vibration about the axis of rotation.

Sources of torsional vibration

Torsional vibration can be introduced into a drive train by the power source. But even a drive train with a very smooth rotational input can develop torsional vibrations through internal components. Common sources are:

Internal combustion engine: The torsional vibrations of the not continuous combustion and the crank shaft geometry itself cause torsional vibrations^[1]
Reciprocating compressor: The pistons experience discontinuous forces from the compression.^[2]
Universal joint: The geometry of this joint causes torsional vibrations if the shafts are not parallel.
Stick slip: During the engagement of a friction element, stick slip situations create torsional vibrations.
Lash: Drive train lash can cause torsional vibrations if the direction of rotation is changed or if the flow of power, i.e. driver vs. driven, is reversed.

Crankshaft torsional vibration

Torsional vibration is a concern in the crankshafts of internal combustion engines because it could break the crankshaft itself; shear-off the flywheel; or cause driven belts, gears and attached components to fail, especially when the frequency of the vibration matches the torsional resonant frequency of the crankshaft. Causes of the torsional vibration are attributed to several factors.

Alternating torques are generated by the slider-crank mechanism of the crankshaft, connecting rod, and piston.
- The cylinder pressure due to combustion is not constant through the combustion cycle.
- The slider-crank mechanism does not output a smooth torque even if the pressure is constant (e.g., at top dead centre there is no torque generated)
- The motion of the piston mass and connecting rod mass generate alternating torques often referred to as "inertia" torques
Engines with six or more cylinders in a straight line configuration can have very flexible crankshafts due to their long length.
2 Stroke Engines generally have smaller bearing overlap between the main and the pin bearings due to the larger stroke length, hence increasing the flexibility of the Crankshaft due to decreased stiffness.
There is inherently little damping in a crankshaft to reduce the vibration except for the shearing resistance of oil film in the main and conrod bearings.

If torsional vibration is not controlled in a crankshaft it can cause failure of the crankshaft or any accessories that are being driven by the crankshaft (typically at the front of the engine; the inertia of the flywheel normally reduces the motion at the rear of the engine). The couplings turn the vibration energy into heat. Therefore, and to ensure that the coupling is not damaged due to this (temperature could be very high, depending on the load), this is verified through torsional vibration calculation.^[3]

This potentially damaging vibration is often controlled by a torsional damper that is located at the front nose of the crankshaft (in automobiles it is often integrated into the front pulley). There are two main types of torsional dampers.

Viscous dampers consist of an inertia ring in a viscous fluid. The torsional vibration of the crankshaft forces the fluid through narrow passages that dissipates the vibration as heat. The viscous torsional damper is analogous to the hydraulic shock absorber in a car's suspension.
Tuned absorber type of "dampers" often referred to as a harmonic dampers or harmonic balancers (even though it technically does not damp or balance the crankshaft). This damper uses a spring element (often rubber in automobile engines) and an inertia ring that is typically tuned to the first torsional natural frequency of the crankshaft. This type of damper reduces the vibration at specific engine speeds when an excitation torque excites the first natural frequency of the crankshaft, but not at other speeds. This type of damper is analogous to the tuned mass dampers used in skyscrapers to reduce the building motion during an earthquake.

Torsional vibrations in electromechanical drive systems

Torsional vibrations of drive systems usually result in a fluctuation of the rotational speed of the rotor of the driving electric motor. Such oscillations of the angular speed superimposed on the average rotor rotational speed cause perturbations of the electromagnetic flux, leading to additional oscillations of the electric currents in the motor windings. Then, the generated electromagnetic torque is also influenced by additional time-varying electromechanical interactions, which lead to further torsional vibrations of the drive system. According to the above, mechanical vibrations/oscillations of the drive system become coupled with the electrical oscillations of the motor windings' currents. Such coupling is typically nonlinear and presents a high computational burden.

Due to the highly nonlinear and coupled nature of electromechanical oscillations, approximations are often used, enabling such oscillations to be characterized analytically. To simplify the characterization of the oscillations between mechanical and electric systems, it is common to assume the mechanical and electrical components are uncoupled. Then, by holding either the mechanical or electrical aspect in steady-state, the characteristic of the other can be calculated. A common method is to apply electromagnetic torques generated by the electric motors as assumed excitation functions of time or of the rotor-to-stator slip,^[4]^[5]^[6] which are usually based on numerous experimental measurements carried out for a given electric motor's dynamic behaviour. For this purpose, by means of measurement results, i.e., empirically, formulas have been developed that provide good approximations for the electromagnetic external excitations produced by the electric motor.^[7] Although the electric currents flowing in the electric motor windings are accurate, the mechanical drive system is typically reduced to one or seldom to at most a few rotating rigid bodies. ^[8] In many cases, such simplifications yield sufficiently useful results for engineering applications, but they can lead to inaccuracies since many qualitative dynamic properties of the mechanical systems, e.g., their mass distribution, torsional flexibility, and damping effects, are neglected. Thus, an influence of the oscillatory behaviour of drive systems on the electric machine rotor angular speed fluctuations, and in this way on the electric current oscillations in the rotor and stator windings, cannot be investigated with a satisfactory precision, excepting numerical methods, which can provide arbitrarily high accuracy.

Mechanical vibrations and deformations are phenomena associated with the operation of the majority of railway vehicle drivetrain structures. The knowledge about torsional vibrations in transmission systems of railway vehicles is of a great importance in the field of mechanical system dynamics.^[9] Torsional vibrations in railway vehicle drivetrains are generated by many coupled mechanisms, which are very complex and can be divided into two main parts:

The electromechanical interactions within the railway drivetrain system, including the electric motor, gears, and the driven parts of the disc and gear clutches.^[10]
Torsional vibrations of the flexible wheels^[11]^[12] and wheelsets caused by variations of the adhesion forces in the wheel-rail contact zone.^[13]

An interaction of the adhesion forces has nonlinear features which are related to the creep value and strongly depend on the wheel-rail zone conditions and the track geometry (especially when driving on a curve section of the track). In many modern mechanical systems, torsional structural deformability plays an important role. Often the study of railway vehicle dynamics using the rigid multibody methods without torsionally deformable elements are used ^[14] This approach does not enable analysis of the self-excited vibrations, which have an important influence on the wheel-rail longitudinal interaction.^[15] A dynamic modelling of the electrical drive systems coupled with elements of a driven machine ^[16]^[17] or vehicle is particularly important when the purpose of such modelling is to obtain an information about the transient phenomena of system operation, like run-up, run-down, and the loss of adhesion in the wheel-rail zone. The modelling of an electromechanical interaction between the electric driving motor and the machine also influence the self-excited torsional vibrations in the drive system.^[18]^[19]

Measuring torsional vibration on physical systems

The most common way to measure torsional vibration is the approach of using equidistant pulses over one shaft revolution. Dedicated shaft encoders as well as gear tooth pickup transducers (induction, hall-effect, variable reluctance, etc.) can generate these pulses. The resulting encoder pulse train is converted into either a digital rpm reading or a voltage proportional to the rpm.

The use of a dual-beam laser is another technique that is used to measure torsional vibrations. The operation of the dual-beam laser is based on the difference in reflection frequency of two perfectly aligned beams pointing at different points on a shaft. Despite its specific advantages, this method yields a limited frequency range, requires line-of-sight from the part to the laser, and represents multiple lasers in case several measurement points need to be measured in parallel.

Torsional vibration software

There are many software packages that are capable of solving the torsional vibration system of equations. Torsional vibration specific codes are more versatile for design and system validation purposes and can produce simulation data that can readily compared to published industry standards. These codes make it easy to add system branches, mass-elastic data, steady-state loads, transient disturbances and many other items only a rotordynamicist would need. Torsional vibration specific codes:

AxSTREAM RotorDynamics, ( SoftInWay) - Commercial FEA-based program for performing the full scope of torsional analyses on the complete range of rotating equipment. Can be used to perform steady-state and transient, modal, harmonic and reciprocating machines analysis, and generates stability plot and Campbell diagrams quickly.
ARMD TORSION (Rotor Bearing Technology & Software, Inc.) - Commercial FEA-based software for performing damped and undamped torsional natural frequencies, mode shapes, steady-state and time-transient response of mechanical drive trains with inputs of various types of external excitation, synchronous motor start-up torque, compressor torques, and electrical system disturbances.

Bond Graphs can be used to analyse torsional vibrations in generator sets, such as those used aboard ships.^[20]

Bibliography

Nestorides, E.J., BICERA: A Handbook on Torsional Vibration, University Press, 1958, ISBN 0-521-04326-3
Parikyan, T. (2011). "Multi-cycle torsional vibration simulation with AVL EXCITE Designer". ASME 2011 Internal Combustion Engine Division Fall Technical Conference. Vol. ASME Paper ICEF2011-60091. pp. 1009–1018. doi:10.1115/ICEF2011-60091. ISBN 978-0-7918-4442-7.

Related Research Articles

A synchronous electric motor is an AC electric motor in which, at steady state, the rotation of the shaft is synchronized with the frequency of the supply current; the rotation period is exactly equal to an integer number of AC cycles. Synchronous motors use electromagnets as the stator of the motor which create a magnetic field that rotates in time with the oscillations of the current. The rotor with permanent magnets or electromagnets turns in step with the stator field at the same rate and as a result, provides the second synchronized rotating magnet field. Doubly fed synchronous motors use independently-excited multiphase AC electromagnets for both rotor and stator.

A brushless DC electric motor (BLDC), also known as an electronically commutated motor, is a synchronous motor using a direct current (DC) electric power supply. It uses an electronic controller to switch DC currents to the motor windings producing magnetic fields that effectively rotate in space and which the permanent magnet rotor follows. The controller adjusts the phase and amplitude of the current pulses that control the speed and torque of the motor. It is an improvement on the mechanical commutator (brushes) used in many conventional electric motors.

<span class="mw-page-title-main">Synchro</span> Variable transformers used in control systems

A synchro is, in effect, a transformer whose primary-to-secondary coupling may be varied by physically changing the relative orientation of the two windings. Synchros are often used for measuring the angle of a rotating machine such as an antenna platform or transmitting rotation. In its general physical construction, it is much like an electric motor. The primary winding of the transformer, fixed to the rotor, is excited by an alternating current, which by electromagnetic induction causes voltages to appear between the Y-connected secondary windings fixed at 120 degrees to each other on the stator. The voltages are measured and used to determine the angle of the rotor relative to the stator.

A drive shaft, driveshaft, driving shaft, tailshaft, propeller shaft, or Cardan shaft is a component for transmitting mechanical power, torque, and rotation, usually used to connect other components of a drivetrain that cannot be connected directly because of distance or the need to allow for relative movement between them.

Hybrid Synergy Drive (HSD), also known as Toyota Hybrid System II, is the brand name of Toyota Motor Corporation for the hybrid car drive train technology used in vehicles with the Toyota and Lexus marques. First introduced on the Prius, the technology is an option on several other Toyota and Lexus vehicles and has been adapted for the electric drive system of the hydrogen-powered Mirai, and for a plug-in hybrid version of the Prius. Previously, Toyota also licensed its HSD technology to Nissan for use in its Nissan Altima Hybrid. Its parts supplier Aisin offers similar hybrid transmissions to other car companies.

A linear actuator is an actuator that creates linear motion, in contrast to the circular motion of a conventional electric motor. Linear actuators are used in machine tools and industrial machinery, in computer peripherals such as disk drives and printers, in valves and dampers, and in many other places where linear motion is required. Hydraulic or pneumatic cylinders inherently produce linear motion. Many other mechanisms are used to generate linear motion from a rotating motor.

Motor drive means a system that includes a motor. An adjustable speed motor drive means a system that includes a motor that has multiple operating speeds. A variable speed motor drive is a system that includes a motor and is continuously variable in speed. If the motor is generating electrical energy rather than using it – this could be called a generator drive but is often still referred to as a motor drive.

Engine balance refers to how the inertial forces produced by moving parts in an internal combustion engine or steam engine are neutralised with counterweights and balance shafts, to prevent unpleasant and potentially damaging vibration. The strongest inertial forces occur at crankshaft speed and balance is mandatory, while forces at twice crankshaft speed can become significant in some cases.

A jackshaft is an intermediate shaft used to transfer power from a powered shaft such as the output shaft of an engine or motor to driven shafts such as the drive axles of a locomotive. As applied to railroad locomotives in the 19th and 20th centuries, jackshafts were typically in line with the drive axles of locomotives and connected to them by side rods. In general, each drive axle on a locomotive is free to move about one inch (2.5 cm) vertically relative to the frame, with the locomotive weight carried on springs. This means that if the engine, motor or transmission is rigidly attached to the locomotive frame, it cannot be rigidly connected to the axle. This problem can be solved by mounting the jackshaft on unsprung bearings and using side-rods or chain drives.

<span class="mw-page-title-main">Swashplate</span> Mechanism to convert between reciprocating and rotary motion

A swashplate, also known as slant disk, is a mechanical engineering device used to translate the motion of a rotating shaft into reciprocating motion, or vice versa. The working principle is similar to crankshaft, Scotch yoke, or wobble, nutator, and Z-crank drives in engine designs. It was originally invented to replace a crankshaft, and is one of the most popular concepts used in crankless engines. It was invented by Anthony Michell in 1917.

An induction generator or asynchronous generator is a type of alternating current (AC) electrical generator that uses the principles of induction motors to produce electric power. Induction generators operate by mechanically turning their rotors faster than synchronous speed. A regular AC induction motor usually can be used as a generator, without any internal modifications. Because they can recover energy with relatively simple controls, induction generators are useful in applications such as mini hydro power plants, wind turbines, or in reducing high-pressure gas streams to lower pressure.

In electrical engineering, electric machine is a general term for machines using electromagnetic forces, such as electric motors, electric generators, and others. They are electromechanical energy converters: an electric motor converts electricity to mechanical power while an electric generator converts mechanical power to electricity. The moving parts in a machine can be rotating or linear. While transformers are occasionally called "static electric machines", since they do not have moving parts, generally they are not considered "machines", but as electrical devices "closely related" to the electrical machines.

A harmonic damper is a device fitted to the free end of the crankshaft of an internal combustion engine to counter torsional and resonance vibrations from the crankshaft. This device must be an interference fit to the crankshaft in order to operate in an effective manner. An interference fit ensures the device moves in perfect step with the crankshaft. It is essential on engines with long crankshafts and V8 engines with cross plane cranks, or V6 and straight-three engines with uneven firing order. Harmonics and torsional vibrations can greatly reduce crankshaft life, or cause instantaneous failure if the crankshaft runs at or through an amplified resonance. Dampers are designed with a specific weight (mass) and diameter, which are dependent on the damping material/method used, to reduce mechanical Q factor, or damp, crankshaft resonances.

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<span class="mw-page-title-main">Geislinger coupling</span>

The Geislinger coupling is an all-metal coupling for rotating shafts. It is elastic in torsion, allowing it to absorb torsional vibration.

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Electromagnetically induced acoustic noise, electromagnetically excited acoustic noise, or more commonly known as coil whine, is audible sound directly produced by materials vibrating under the excitation of electromagnetic forces. Some examples of this noise include the mains hum, hum of transformers, the whine of some rotating electric machines, or the buzz of fluorescent lamps. The hissing of high voltage transmission lines is due to corona discharge, not magnetism.

This glossary of automotive terms is a list of definitions of terms and concepts related to automobiles, including their parts, operation, and manufacture, as well as automotive engineering, auto repair, and the automotive industry in general. For more specific terminology regarding the design and classification of various automobile styles, see Glossary of automotive design; for terms related to transportation by road, see Glossary of road transport terms; for competitive auto racing, see Glossary of motorsport terms.

References

↑ Den Hartog, J. P. (1985). Mechanical Vibrations. Nineola, N.Y.: Dover Publications. p. 174. ISBN 0-486-64785-4.
↑ Feese, Hill. "Prevention of Torsional Vibration Problems in Reciprocating Machinery" (PDF). Engineering Dynamics Incorporated. Archived from the original (PDF) on 19 October 2013. Retrieved 17 October 2013.
↑ VULKAN Couplings System Competence - Torsional Vibration Calculation, archived from the original on 2021-12-22, retrieved 2021-06-06
↑ B. F. Evans, A. J. Smalley, H. R. Simmons, Startup of synchronous motor drive trains: the application of transient torsional analysis of cumulative fatigue assessment, ASME Paper, 85-DET-122, 1985.
↑ A. Laschet A., Simulation von Antriebssystemen, Springer-Verlag, Berlin, Heidelberg, London, New-York, Paris, Tokio, 1988.
↑ P. Schwibinger, R. Nordmann, Improvement of a reduced torsional model by means of parameter identification, Transactions of the ASME, Journal of Vibration, Acoustics, Stress and Reliability in Design, 111, 1989, pp. 17-26.
↑ A. Laschet A., Simulation von Antriebssystemen, Springer-Verlag, Berlin, Heidelberg, London, New-York, Paris, Tokio, 1988.
↑ L. Harnefors, Analysis of subsynchronous torsional interaction with power electronic converters, IEEE Transactions on power systems, Vol. 22, No. 1, 2007, pp. 305-313.
↑ R. Bogacz, T. Szolc, H. Irretier, An application of torsional wave analysis to turbogenerator rotor shaft response, J.Vibr. Acou. -Trans. of the Asme, Vol. 114-2 (1992) 149-153.
↑ O. Ahmedov, V. Zeman, M. Byrtus, Modelling of vibration and modal properties of electric locomotive drive, Eng. Mech., Vol. 19: 2/3 (2012) 165–176.
↑ S. Noga, R. Bogacz, T. Markowski, Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation, J.Sound Vib., Vol. 333:24, (2014) 6706-6722.
↑ R. Bogacz, R. Konowrocki, On new effects of wheel-rail interaction, Arch. Appl. Mech, Vol.82 (2012)1313-1323.
↑ 5. V. Zeman, Z. Hlavac, Dynamic wheelset drive load of the railway vehicle caused by shortcircuit motor moment, App. & Comp. Mech., Vol.3, No.2 (2009)423–434.
↑ B.S. Branislav, Simulation of torsion moment at the wheel set of the railway vehicle with the traction electromotor for wavy direct current, Mech. Trans. Com., Issue 3 (2008) 6-9
↑ J. Liu, H. Zhao, W. Zhai, Mechanism of self-excited torsional vibration of locomotive driving system, Front. Mech. Eng.China, Vol.5:4 (2010,) 465-469.
↑ Szolc T., Konowrocki R., Michajłow M., Pręgowska A., An investigation of the dynamic electromechanical coupling effects in machine drive systems driven by asynchronous motors, Mechanical Systems and Signal Processing, ISSN 0888-3270, Vol.49, pp.118-134, 2014
↑ Konowrocki R., Szolc T., Pochanke A., Pręgowska A., An influence of the stepping motor control and friction models on precise positioning of the complex mechanical system, Mechanical Systems and Signal Processing, ISSN 0888-3270, doi : 10.1016/j.ymssp.2015.09.030, Vol.70-71, pp.397-413, 2016
↑ Konowrocki R., Szolc T., An analysis of the self-excited torsional vibrations of the electromechanical drive system, Vibrations in Physical Systems, ISSN 0860-6897, Vol.27, pp.187-194, 2016
↑ Konowrocki R., Analysis of electromechanical interaction in an electric drive system used in the high speed trains, ART Conference 2016, ADVANCED RAIL TECHNOLOGIES - 5th International Conference, 2016-11-09/11-11, Warsaw (PL), pp.1-2, 2016
↑ Heeringa, T (2018-10-03). "Torsional Vibration Analysis by Bondgraph Modelling. A practical approach". Proceedings of the International Naval Engineering Conference and Exhibition (INEC). Vol. 14. Glasgow, UK. doi: 10.24868/issn.2515-818X.2018.034 .{{cite book}}: CS1 maint: location missing publisher (link)

External links

Torsional Vibration Application Case for Vehicle Frontend Accessory Drive

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[1] Den Hartog, J. P. (1985). Mechanical Vibrations. Nineola, N.Y.: Dover Publications. p. 174. ISBN 0-486-64785-4.

[2] Feese, Hill. "Prevention of Torsional Vibration Problems in Reciprocating Machinery" (PDF). Engineering Dynamics Incorporated. Archived from the original (PDF) on 19 October 2013. Retrieved 17 October 2013.

[3] VULKAN Couplings System Competence - Torsional Vibration Calculation, archived from the original on 2021-12-22, retrieved 2021-06-06

[4] B. F. Evans, A. J. Smalley, H. R. Simmons, Startup of synchronous motor drive trains: the application of transient torsional analysis of cumulative fatigue assessment, ASME Paper, 85-DET-122, 1985.

[5] A. Laschet A., Simulation von Antriebssystemen, Springer-Verlag, Berlin, Heidelberg, London, New-York, Paris, Tokio, 1988.

[6] P. Schwibinger, R. Nordmann, Improvement of a reduced torsional model by means of parameter identification, Transactions of the ASME, Journal of Vibration, Acoustics, Stress and Reliability in Design, 111, 1989, pp. 17-26.

[7] A. Laschet A., Simulation von Antriebssystemen, Springer-Verlag, Berlin, Heidelberg, London, New-York, Paris, Tokio, 1988.

[8] L. Harnefors, Analysis of subsynchronous torsional interaction with power electronic converters, IEEE Transactions on power systems, Vol. 22, No. 1, 2007, pp. 305-313.

[9] R. Bogacz, T. Szolc, H. Irretier, An application of torsional wave analysis to turbogenerator rotor shaft response, J.Vibr. Acou. -Trans. of the Asme, Vol. 114-2 (1992) 149-153.

[10] O. Ahmedov, V. Zeman, M. Byrtus, Modelling of vibration and modal properties of electric locomotive drive, Eng. Mech., Vol. 19: 2/3 (2012) 165–176.

[11] S. Noga, R. Bogacz, T. Markowski, Vibration analysis of a wheel composed of a ring and a wheel-plate modelled as a three-parameter elastic foundation, J.Sound Vib., Vol. 333:24, (2014) 6706-6722.

[12] R. Bogacz, R. Konowrocki, On new effects of wheel-rail interaction, Arch. Appl. Mech, Vol.82 (2012)1313-1323.

[13] 5. V. Zeman, Z. Hlavac, Dynamic wheelset drive load of the railway vehicle caused by shortcircuit motor moment, App. & Comp. Mech., Vol.3, No.2 (2009)423–434.

[14] B.S. Branislav, Simulation of torsion moment at the wheel set of the railway vehicle with the traction electromotor for wavy direct current, Mech. Trans. Com., Issue 3 (2008) 6-9

[15] J. Liu, H. Zhao, W. Zhai, Mechanism of self-excited torsional vibration of locomotive driving system, Front. Mech. Eng.China, Vol.5:4 (2010,) 465-469.

[16] Szolc T., Konowrocki R., Michajłow M., Pręgowska A., An investigation of the dynamic electromechanical coupling effects in machine drive systems driven by asynchronous motors, Mechanical Systems and Signal Processing, ISSN 0888-3270, Vol.49, pp.118-134, 2014

[17] Konowrocki R., Szolc T., Pochanke A., Pręgowska A., An influence of the stepping motor control and friction models on precise positioning of the complex mechanical system, Mechanical Systems and Signal Processing, ISSN 0888-3270, doi : 10.1016/j.ymssp.2015.09.030, Vol.70-71, pp.397-413, 2016

[18] Konowrocki R., Szolc T., An analysis of the self-excited torsional vibrations of the electromechanical drive system, Vibrations in Physical Systems, ISSN 0860-6897, Vol.27, pp.187-194, 2016

[19] Konowrocki R., Analysis of electromechanical interaction in an electric drive system used in the high speed trains, ART Conference 2016, ADVANCED RAIL TECHNOLOGIES - 5th International Conference, 2016-11-09/11-11, Warsaw (PL), pp.1-2, 2016

[20] Heeringa, T (2018-10-03). "Torsional Vibration Analysis by Bondgraph Modelling. A practical approach". Proceedings of the International Naval Engineering Conference and Exhibition (INEC). Vol. 14. Glasgow, UK. doi: 10.24868/issn.2515-818X.2018.034 .{{cite book}}: CS1 maint: location missing publisher (link)

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