whirling of a propeller shaft of a mega-yacht, leading to serious complaints and high costs

	cases Techno Fysica BV is a Dutch-based engineering company, highly specialised in solving all kinds of problems related to the dynamic loading of machinery and installations. For this, several measuring and analysis techniques are used. In a series of articles, we will give examples of what can go wrong in the engineering and operating of vessels, and what errors lie at the root of the damages observed.
	whirling of propeller shaft of a mega-yacht, leading to serious complaints and high costs
	In a Dutch-build, twin propeller, aluminium mega-yacht, serious vibration problems were experienced during the sea trials of the ship. The ship suffered from severe vibrations of the aft ship, which were excited, as was determined by means of vibration analysis, by 1^st order excitations of the propeller shafts. In the engineering phase, the lateral vibrations of the propeller shaft were calculated by means of whirling calculations carried out by the supplier of the shaft installation. During trials, a considerable discrepancy was found to exist between predicted natural frequencies and actual natural frequencies. Even worse, it appeared that the first node bending mode of the shaft was excited at or near nominal operating speed of the shaft, and was excited by first order forces. The effect of this is that small unbalance forces are amplified by the resonance, which leads to significant vibrations, felt throughout the ship. In order to solve the problem, the ship was docked, and the propellers and couplings were balanced. In addition, the shafts were checked for straightness. Although this lead to some improvement, the vibrations were still excessive. In order to solve the problem, serious modifications were made to the aft ship, in order to increase the overall stiffness of the ship. Again, this did not solve the problem, and even appeared to make the problem worse. Since this unexpected problem led to very large additional costs for the yard, despite the fact that elaborate calculations had been made in advance, the question was posed whether or not vibration prediction by means of calculations was useful and reliable, or that these tools are only useful as a rough estimate. As Techno Fysica is convinced of the usefulness of calculations in an engineering phase, or as a means of solving problems in a later phase, the offer was made to make a study of the problem, and to show that a problem such as this can be successfully addressed by means of measurements in combination with advanced calculation techniques. This offer was made free of costs, and simply served to restore the confidence in the usefulness of advanced calculations by all parties involved. The study included the calculation of the whirling frequencies of the shaft on the basis of design parameters, from scratch and with our own added experience towards the calculation of additional parameters such as added water and elastic support. In addition, a finite element model was made of a large part of the aft ship, including the strut and bearing support. This was done with regards to the suspected behaviour of this part and the large influence of the relatively low stiffness of this part of the ship. This stiffness and the effect on the problems of the experienced vibrations was studied in combination with the whirling behaviour of the shaft.
		first mode of vibration, showing relative displacement of the aft section of the strut
	Because Techno Fysica had not performed the design calculations of the whirling frequencies, we do not know whether or not a stiffness value for the strut has been incorporated in the original design calculations. Normal practice is to use values for horizontal and vertical stiffness (estimated, or deducted by means of finite element calculations) and combine these stiffness values with the stiffness of the elastic support of the shaft in the point of support in the aft bearing. For steel ships, the additional stiffness of the strut is usually very high and is therefore often neglected in the whirling calculations. Furthermore, some calculation methods neglect the elastic support in the bearings completely, and enter the bearing as a single point of support.
	In order to tackle the problem, the following approach was used: 1. Calculation of the whirling frequency including only the effect of the rubber elements of the support in the bearing. This has been done for the propeller in water, including coefficients for entrained water and “dry” in order to simulate excitation tests. This provides the results which can be expected in an engineering phase. 2. Prepare a finite element model of the aft ship, in order to calculate the stiffness of the strut, and to compare this calculated stiffness with the measured stiffness. This is done for the original situation, and for the modified condition. 3. Tune, where necessary, the calculation model in order to match the calculated stiffness to the measured stiffness. It is assumed that the measurements of the stiffness of the aft ship are reliable. 4. Perform the whirling calculations including all parameters such as rubber stiffness, and stiffness of the aft ship. Again, this has been done “dry” and “in water”, and compare the results with design values. 5. Perform a forced response calculation for the original situation and tune force and an assumed value for overall damping in order to match actual vibration levels. 6. Use the values for force and damping in the calculation of the modified system

A finite element model of the propulsion shaft, up to the output shaft of the gearbox, was constructed. The model is based on dimensions as given in relevant drawings.

For the stiffness of the support in the rubber bearing, the values for static and dynamic stiffness were supplied by the manufacturer of the bearings, and are as follows:

Bearing position	Static stiffness	Dynamic stiffness
Middle	1.4E7 N/m	5.6E4 N/m
Aft	5.6E7 N/m	22.4E7 N/m

For the propeller, the amount of entrained water has been calculated on the basis of formulas from Schwanecke. Basis for the calculation of the entrained water are the dimensions of the propeller.

With these values, the following natural frequencies are calculated:

Situation	1^st node vertical	1^st node Hor.	2^nd node hor.	2nd node ver.
In air	15.2	16.0	33.2	33.9
In water	14.2	14.3	30.7	30.6

In the original calculation, made by the manufacturer of the installation, the calculated natural frequency for the 1^st and 2^nd node shaft frequencies are 16.9 Hz and 36.2 Hz respectively. This means a difference with Techno Fysica’s calculation of 6-10% when the influence of additional water is neglected and 14-15% if the effect of additional water is taken into account.

However, according to the measurements, the problem existed due to excitation of the first order, which amounts to approximately 11.3 Hz at nominal speed. Even in the worst case of the above mentioned calculation results, a safe margin between maximum excitation and the closest natural frequency should be present, and the problem should not have existed.

However, the original calculation omitted the stiffness of the strut, which proved to be of vital importance.

Based on the geometry of the aft ship, a finite element model was built. With this model, the stiffness of the support of the aft bearing has been calculated by applying a unit force at the location of the point of support of the shaft in the aft bearing and determining the corresponding displacement. This was done for both the original condition as well, in a later stage, for the modified construction

The results between the measurements and the theoretical stiffness were compared and this yielded the following results:

Situation	Calculated stiffness	Measured stiffness
Original	3.92E6 N/m	4.04E6 N/m
Modified	7.6E6 N/m	10.1E6 N/m

The calculated stiffness of the model of the original condition closely matched that of the measured stiffness. The stiffness of the modified installation is wrong by almost 25%. This is probably caused by the fact that only a part of the ship has been modelled. Since it is assumed that the measurements have provided the correct stiffness, the modified model has been altered in order to match it to the measured value.

This has been done by increasing the thickness of the used plates. The final stiffness of the modified model is 10.4E6 N/m, which closely represents the measured stiffness.

In the model, the propulsion shaft is modelled correctly with regards to mass and inertia properties, and with respect to the support of the stern tube and the rubber bearing.

It would also have been possible to just use the value of the calculated stiffness in the restricted shaft model, but in this way it is more easy to visualize what actually happens.

With the combined system, the natural frequencies have been calculated. Again, this has been done with the properties of the propeller in air and in water.

The following frequencies and mode shapes are calculated:

In air:

Node	Frequency	Mode shape
I	10.6	Shaft and strut parallel and in phase, horizontal direction
II	14.8	Vertical, shaft only
III	17.2	Shaft and strut in opposite direction, horizontally
IV	23.1	Vertical, shaft and strut in opposite direction

And in water:

Node	Frequency	Mode shape
I	9.7	Shaft and strut parallel and in phase, horizontal direction
II	13.7	Vertical, shaft and strut in phase
III	16.3	Shaft and strut in opposite direction, horizontally
IV	22.5	Vertical, shaft and strut in opposite direction

From the calculations can be deducted that the problem is caused by the fact that the combination of the shaft and the low stiffness of the strut shows resonance at the low frequency of, in the calculation model, 10.6 Hz in air, and 9.7 Hz in water. This is very close to the maximum excitation at operating speed, and will be excited in the case of existing first order excitation forces.

The finite element model was adapted on the basis of the modifications as given in the mentioned drawing. With this modified model, the following frequencies were calculated

In air:

Node	Frequency	Mode shape
I	14.7	Shaft and strut parallel and in phase, horizontal direction
II	16.2	Vertical, shaft only
III	19.6	Shaft and strut in opposite direction, horizontally
IV	30.7	Vertical, shaft and strut in opposite direction

And in water:

Node	Frequency	Mode shape
I	12.9	Shaft and strut parallel and in phase, horizontal direction
II	14.9	Vertical, shaft and strut in phase
III	20.6	Shaft and strut in opposite direction, horizontally
IV	29.9	Vertical, shaft and strut in opposite direction

As can be seen, there is a considerable difference before and after modification. The most critical first node shifts approximately 33%.

The measured natural frequency (in air) after modification , measured in the dock, varied between 12 Hz (SB) and 14 Hz (PS). Therefore, there is a quite reasonable agreement between measurements and calculations, especially on PS installation.

However, the calculated margin between nominal excitation and closest calculated natural frequency is still too small (13%) to ensure absence of vibrations. Based on the results of the calculations this can be explained by the fact that, before the modifications, the installation operated over-critical, on the flank of the first node resonance. After modification, and despite a 33% shift, the installation operates under-critical, but still in a flank of the same resonance. Therefore, the stiffness will have to be increased further in order to increase the margin between excitation and resonance.

The purpose of this exercise was to restore the confidence in the usefulness of applying advanced calculation methods in a design stage. This report shows that the problem can be well reproduced in a computer simulation. Ergo, the problem could have been prevented by performing in an-depth analysis prior to construction.

The analysis as originally performed was not extensive enough, and had not captured all problems. This is mainly due to the fact that the material used, aluminium, has lower stiffness than is usually expected in a steel ship, which is something an experienced engineer had realized.

However, this fact was ignored and based on a standard approach, no problems were expected.

For more info, please contact techno Fysica BV in the Netherlands at:

Techno Fysica BV

p.o. box 351

2990 AJ Barendrecht

The Netherlands

Tel. +31 180 620211

Fax. +31 180 620705

info@technofysica.nl

http://www.technofysica.nl/

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