Nautica Digitale
Share this page
Tell a friend

Spring 2006

Article selected from our quarterly magazine dedicated to the largest and most luxurious boats with information, interviews, technical articles, images and yachting news



Yachting catalogue

Navigation tests

Used boats


Video Nautica

Article by
Angelo Sinisi



Firstly, what is a vibration?

A vibration is a relatively small amplitude of oscillation around a support point where its value is zero.

The vibrations of a hull, apart from creating disturbance on those parts and objects of the vessel that move with high frequency, cause noise that is equally negative for comfort on board. Vibration on board may be hull vibrations or may be produced by the propellers, the engines, the generators and all other plant with rotating or alternating masses, The greatest vibrations are caused by hull, propellers and engines. This article will deal with the first two.

A vessel's hull may be considered overall as an elastic body equipped with numerous natural frequencies which depend on its dimensions and the distribution of its masses and loads. Moreover the hull, like all elastic bodies, may undergo forced oscillations of a periodic nature if stressed by pulsating type actions.

Vessel-beam vibrations, like those of any beam, may therefore be free or natural and forced: they are free or natural when the beam is excited by a force and left free in its motion; they are forced when a force variable in time is applied to the beam.

In the first case the beam, when left free, begins to vibrate with what is called "first natural frequency". The amplitude of the vibration, which is the displacement value of the oscillation (Figure 1), is modest, compatibly with the intensity if the excitant cause, if the beam is made to vibrate with a frequency other than its natural one. Vice versa, if the excitant frequency coincides with the natural vibration frequency of the vessel-beam, this results in the condition of "resonance".

Figure 1

As a result of the resonance phenomenon, which takes place when one of the hull vibration frequencies coincides with or is very close to the frequency of any pulsating action discharged onto the resistant structures of the vessel, the amplitude (Figure 1) of the vibrations is heightened. Consequently the hull undergoes vibratory motions that are sometimes considerable and therefore troublesome. In fact if the propeller revs are very close to the hull's natural vibration frequency and the propeller itself is not perfectly balanced, oscillations are created which, even if they are not high in absolute value, can make the hull vibrate considerably.

The greatest oscillations are usually found in the longitudinal plane, and those that most concern the hull are:

Vertical flexional vibrations which occur in the longitudinal plane of symmetry; Horizontal or transversal vibrations which occur in a plane normal to the longitudinal plane of symmetry. The causes that excite vibratory motion of the hull and therefore bring about the condition of resonance are the propeller and shafting line, rolling-pitching movement and the alternative propulsion engines.

With regard to the above excitant causes, calculation of the frequencies, which are the number of vibration cycles (Figure 1) effected within the unit of time (which is the inverse of the vibration period) in Hertz (Hz) are:

The frequency transmitted by the propeller to the hull is given by


n = number of blades N = number of rpm of the propeller shaft

The frequency transmitted by the shafting line is given by

The frequency given by roll or pitch, whose half-cycle is equal to

The frequency given by the alternative propulsion engines is given by


NC = number of cylinders
NM = number of rpm of thermomotor shaft
K = number of strokes.

As illustrated above, for every direction of vibration a hull has several natural frequencies of resonance, and to every natural frequency there corresponds a mode or degree of hull vibration. The vibration modes are determined by the number of nodes (points of amplitude or zero displacement) (Figure 2) shown by the curve that represents the elastic line of the hull. For the free beam, such as, precisely, a floating vessel, the first mode of vibration has two nodes, the second has three nodes and so on.

Figure 2

In conclusion:

the 1st natural frequency corresponds to the 1st mode of vibration (2 nodes), the 2nd natural frequency corresponds to the 2nd mode of vibration (3 nodes), the 3rd natural frequency corresponds to the 3rd mode of vibration (4 nodes), and so on (Figure 2).

The earliest calculation systems for forecasting the natural vibrations of vessels were theoretical-experimental formulas. With time these formulas were refined and corrected in the light of surveys carried out on an ever growing number of vessels of different types. Generally, forecasting of the modes of vibration of the hull stops at the first mode, i.e. at hull vibrations with two nodes both vertically and transversally.

Below we shall examine some of the most widespread formulas - exclusively with regard to vertical vibrations - better known and more often suggested by the various literature.

Schlick's Formula


φ = coefficient depending on vessel type, on load conditions and displacement variation (98.000 - 129.000);
J = inertia moment of amidships section in inc² x feet²;
Δ = displacement in tons;
L = LOA in feet.

The indeterminate nature of coefficient , due to the too many variables on which it depends, leads to only very approximated knowledge of the vessel's natural vibration frequency.

Burril's Formula


J = inertia moment of the amidships section;
B = breadth of the waterline;
i = average draft;
L = vessel length;
β constant (= 200.000 if Lft and = 220.000 if Lpp);

= correction factor for shearing stresses;
= correction factor for water drawn by the vessel.

All values in metrical-decimal measurements.

Todd's Formula


&beta = coefficient that varies from 32.000 for small vessels to 62.000 for ships;
B = breadth in feet;
D = main deck height in feet;
Δ = displacement in tons;
L = length in feet;
i = draft in feet;
Δ1 = virtual displacement in tons.

Costantini's Formula


C1 = coefficient, in function of vessel type, of the total fineness coefficient and the ratio;
C = 0.91 for narrow vessels, C = 0.94 for large vessels;
K = coefficient in function of L/B;
J = inertia moment of the amidships section in inc²ft²;
B = width in feet;
i = draft in feet;
Δ = displacement in tons;
Δ1 = virtual displacement in tons;
L = length in feet;

Bunyan's Formula


K = 34.000 for cargo ships with transversal structure and K = 48.000 for tankers with longitudinal structure;
n = 1.23 for tankers and n = 1.165 for cargo ships;
Cb = total fineness coefficient;
i = loaded vessel draft in feet;
i1 = maximum draft in feet for the condition under examination;
L = vessel length in feet;
D = overall height in feet;

Over and above the more or less reliable formulas worthy of consideration there are other calculation systems for a more accurate forecast of the hull's natural vibration frequency.

One is the "integration" method which with a progression of approximations determines the exact frequency of vertical natural vibration regarding the first mode of vibration (2 nodes). The calculation method is based on the hypothesis that with natural vibration at 2 nodes displacements are proportional to accelerations, i.e. the amplitude of vibration is proportional to acceleration.

Another system is the matrix calculation method which can be used with the aid of a computer, considerably simplifying the procedure.

As already explained, one of the causes that heightens vibrations in the vessel or one of its component parts - producing dangerous stress on materials and creating unbearable living conditions for passengers and crew - can be ascribed to the propellers.

When vibrations are produced by the propeller they may often be eliminated or at least reduced to tolerable limits by intervening on their frequency and amplitude by varying the number of blades and sometimes by modifying the design (total pitch fraction, surface distribution, variation of pitch along the blade, inclination and form of generating force, blade outline, choice of blade profiles etc.).


Figure 3
Distribution of wake flow
in the airscrew disc of a single screw vessel

Figure 4
Thrust curve in function of the rotation angle
of a 3, 4 and 5 blade propeller

A propeller with a limited number of blades does not produce any vibration in twisting moment transmitted and thrust produced as long as it is sufficiently immersed in a uniform current flow. Any vibrations produced by the propeller can be attributed to a non-uniform wake current. Measuring the axial velocity of the current in the airscrew disc, curves are obtained like those shown in Figure 3 for a single screw vessel with normal fineness of keel. The form and the position of the velocity curves depend on stern shape, the position of the propeller with regard to the hull and the form and position of the appendages. Figure 3 shows that the propeller blade on each revolution enters zones where water speed is considerably different in a radial and circumferential direction.

Due to the non-uniformity of the wake, the propeller produces three different kinds of impulses which are examined below in order of importance:

Thrust fluctuation - Fluctuations of thrust act without producing disturbance in the vessel through the shafting line and the thrust bearing, since their frequencies are less than the natural frequency of the shafting line in the direction of thrust, and dampening is not considerable.

Twisting moment fluctuations - Twisting moment fluctuations are highly influenced both by the relatively high dynamic inertia moment of the propeller, which allows only small angular variations, and by the torsion elasticity of the shafting line. With a torsigraph a rapid diminution of the torsional angular variations produced by the propeller along the shafting line may be noted. Twisting moment fluctuations, if they get as far as the engine, are transmitted to the vessel by the engine bedding.

Flexion fluctuations - As a result of continual variation of load on the rotating blades, the centre of thrust application shifts from the centre of the propeller, thus giving rise to flexion stresses on the propeller shaft, though they are small in comparison with those caused by the weight of the propeller immersed in seawater. These stresses are transmitted to the vessel in a plane perpendicular to the shafting line through the stern tube bearings. And since these stresses have the same frequency as the thrust, the effects are added together.

With a diagram of thrust values during rotation of propellers with a different number of blades we obtain curves like those shown in figure 4. Since the high frequency and small amplitude vibrations are dampened, rapidly shifting away from the centre of vibration, and since the vibrations in question generally have a frequency greater than those of hull structure resonance, one can understand how a five blade propeller produces considerably less vibrations than a propeller with fewer blades.

Knowledge of the maximum oscillations of the twisting moment absorbed and the

thrust supplied by the propeller is important for calculation of the blade's toughness. The frequencies of these oscillations do not depend on the number of blades: they are multiples of the number of revs, and may produce vibrations in the blades and therefore added stresses on the material.

To avoid vibrations produced by the propeller, the following main aspects must be kept in mind:

One must try to obtain a wake flow as uniform as possible in the propeller zone, refining the stern waterlines and adopting a large screw aperture and a good distance from the propeller and rudder posts.

The frequencies of the propeller and engine vibrations should not coincide with or be in the reduction ratio of the reduction gear.

Natural frequency and the vibrations of structural parts of the vessel which fall in the field of propeller frequency must be eliminated by reinforcing these structural parts.

Over and above the vibrations of the hull there may be noises transmitted from the propeller to the hull through the water, amplified by the hull plating which starts vibrating. These noises, whose frequency is given by the product of revs by number of blades, are caused by local formations of cavitation and great differences in wake speed during the passage of the blade in correspondence to the propeller post or propeller bracket.

In conclusion, the more blades a propeller has - since the frequencies will be greater and the vibratory amplitudes less - the more certainty there is of avoiding hull vibration.