The vibratory damping of large high-speed catamarans

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Marine Structures 21 (2008) 1–22

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The vibratory damping of largehigh-speed catamarans

Giles Thomasa,�, Michael Davisb, Damien Hollowayb, Tim Robertsc

aAustralian Maritime College, P.O. Box 986, Launceston, Tasmania 7250, AustraliabSchool of Engineering, University of Tasmania, Private Bag 65, Hobart, Tasmania 7001, AustraliacRevolution Design Pty. Ltd., Bender Drive, Prince of Wales Bay, Hobart, Tasmania 7009, Australia

Received 14 March 2007; received in revised form 12 December 2007; accepted 13 December 2007

Abstract

Large high-speed catamarans may experience slamming, and subsequently whipping, when

operating in a large seaway. The rate at which the whipping behaviour decays after a slam is due to

the damping within the system. This paper reports on work to further understand the vibratory

damping of large high-speed catamarans. Full-scale measurements of slam events were conducted on

two large high-speed Incat catamarans during delivery voyages and regular service operations.

Exciter tests were also conducted on the vessels whilst stationary in calm water. An examination of

the components that contribute to the damping of the system was also undertaken. Estimates were

made of the relative magnitude of the various hydrodynamic components including: wave making

damping, viscous damping and acoustic damping. The total predicted damping, due to

hydrodynamic and material damping, was found to account for only a small proportion of the

measured total damping. The shortfall can only be presumed to be due to additional structural

damping which is estimated to be one order higher than the inherent material damping.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: High-speed catamarans; Damping; Whipping

1. Introduction

When a ship is operating in rough seas it may experience large relative motions betweenits hull and the water surface. An impact, known as a slam, may occur as the hull strikes

see front matter r 2007 Elsevier Ltd. All rights reserved.

.marstruc.2007.12.003

nding author. Tel.: +643 6335 4883; fax: +64 3 6335 4720.

dress: [email protected] (G. Thomas).

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dx.doi.org/10.1016/j.marstruc.2007.12.003

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ARTICLE IN PRESSG. Thomas et al. / Marine Structures 21 (2008) 1–222

the water surface upon re-entry of the keel after emergence. The shudder or vibration feltthroughout the hull after a slam is known as whipping. High-speed vessels tend toexperience increased encounter frequencies and greater motions in head or bow quarteringseas than slow-speed craft and are therefore more susceptible to slamming. Catamaransmay be subject to an additional form of slamming known as wet-deck slamming. Thisoccurs when the water surface impacts with the underside of the bridge structure betweenthe two hulls and has the potential to cause both local and global damage. In addition tolocal and global slamming damage, structural cracking due to fatigue, has been found tooccur on vessels, as shown in Fig. 1. Large aluminium high-speed vessels, due to theflexible nature of their hull girders, are especially susceptible to whipping vibrationsfollowing slam events. The corresponding vibratory stresses may be of equal order ofmagnitude to those induced by quasi-static wave bending moments, though they have ahigher fundamental frequency. Hence, whipping can make a very significant contributionto the total fatigue damage.The rate at which the whipping behaviour decays after a slam is due to the damping

within the system. As well as conducting a series of vibration measurements on a largetanker, Kumai [1] collated the first longitudinal mode (two-node vertical vibration)damping factors of seven additional steel monohulls determined by other investigators.For these vessels, the damping factors were found to be approximately inverselyproportional to their length. Betts et al. [2] conducted a survey of available full-scaleresults and found that values of hull damping varied widely. The damping was proposed toconsist of two components: hydrodynamic damping and structural damping. Thehydrodynamic damping was presumed to be small; however, no evidence was providedto support this assumption. In 1979 Bishop and Price [3] stated that with regard to hulldamping: ‘‘The simple truth is that knowledge is abysmal’’. In an analysis of the hydroelasticresponse of a high-speed monohull, Hermundstad et al. [4] concluded that the internal hulldamping had a negligible effect on the response of the first vibration mode. They suggestedthat the damping was dominated by hydrodynamic damping due to the forward speed.Sunnersjo and Janson [5] used finite volume calculations to study damping due to soundradiation. The results indicated that the hydrodynamic damping due to very low frequencypressure wave radiation contributes a significant part of the total modal damping. Since

Fig. 1. Structural cracking on a high-speed catamaran.

ARTICLE IN PRESSG. Thomas et al. / Marine Structures 21 (2008) 1–22 3

the survey of Betts et al. [2], there appears to have been little progress in increasingknowledge of damping, particularly for modern fast lightweight vessels. For example, themethod for estimating the structural response of a vessel to wave induced loads of Ramosand Guedes Soares [6] use the work of Kumai [1] in the absence more recent data.Therefore, further investigation into the levels of modal damping in ships and the relativemagnitudes of its components is overdue.

This paper reports on work conducted to ascertain the level of damping in the whippingbehaviour of large high-speed aluminium catamarans. In addition, the sources of thevarious damping mechanisms, and their contributions, were investigated. Extensive full-scale measurements of slam events were conducted on two large high-speed Incatcatamarans (86 and 96m in length) during delivery voyages and regular service operations.These measurements were utilised to investigate the damping of the whipping behaviour ofthe vessels. Exciter tests were also conducted on the vessels, whilst stationary in calmwater, in order to further examine their vibratory damping properties. An examination ofthe components that contribute to the damping of the system was also undertaken.Experiments were conducted to determine the inherent structural damping properties of6082 T6 grade aluminium. Estimates were made of the relative magnitude of the varioushydrodynamic components including: wave making damping, viscous damping andacoustic damping. The total calculated damping was then compared with the levels ofdamping found through the full-scale measurements and exciter tests.

2. Full-scale trials

To investigate whipping behaviour, full-scale measurements of slam events wereconducted, during delivery voyages and regular service operations, on two large high-speedIncat catamarans, Hulls 042 and 050.

2.1. Vessel details

Incat Hulls 042 and 050 are large high-speed aluminium catamaran ferries, as pictured inFig. 2. The principal parameters of the vessels are shown in Table 1.

2.2. Measurements

Hull 042 was monitored during a delivery trip from Sydney, Australia to Portland, UKand during regular services across the English Channel. The monitoring of Hull 050 took

Fig. 2. Incat Hulls 042 and 050.

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Table 1

Principal parameters of Hulls 042 and 050

Hull 042 Hull 050

Length overall (m) 86.6m 96.0m

Length waterline (m) 76.4m 86.0m

Beam overall (m) 26.0m 26.0m

Draft (m) 3.5m 3.7m

Hull beam (m) 4.33m 4.5m

Deadweight (ton) 415 tonnes 800 tonnes

Speed, fully loaded condition (knots) 40 knots 40 knots

Number Location

1 Port steel chevron brace at frame 2on CL

2 Stb. Steel chevron brace at frame 2on CL

3 Top aft transverse box at frame 14on CL

4 Bottom aft transverse box at frame14 on CL

5 Inner x-brace at frame 14

6 Bat wing on x-brace at frame 17

7 Longitudinal girder, 4600mm offCL, at frame 24

8 Port portal top at frame 24


10 Transverse girder at frame 35 on CL

11 Stb. hull keel at frame 24.5



14 Port steel post at frame 54

15 Stb. steel post at frame 54

16 Cross bridge web at frame 24 on CL

Fig. 3. Hull 042 strain gauge locations.

G. Thomas et al. / Marine Structures 21 (2008) 1–224

place during regular services across Cook Strait between the North and South Islands ofNew Zealand. A series of 16 strain gauges was fitted to each vessel at the locations shownin Figs. 3 and 4. The motions of the vessels were also monitored using accelerometers andrate gyroscopes; whilst an on-board radar-based wave meter, supplied by TSK, was fittedto the bow of each vessel to give readings of instantaneous absolute wave height. Furtherdetails on the measurement systems may be found in Thomas et al. [7,8].

2.3. Whipping analysis

To investigate the structural whipping response, spectral analysis was conducted on thestrain gauge data traces of slam events. The 20Hz raw data were highpass filtered at0.04Hz to remove low frequency drift and windowed using a Hanning window to reduce

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Number Location

1 Top rider at frame 69 on CL




5 Stb. fwd diagonal at frame 62

6 Stb. aft diagonal at frame 59


8 Port portal top at frame 24

9 Port lower steel post at frame 63

10 Stb. lower steel post at frame 63

11 Stb. portal x-brace at frame 41

12 Stb. portal x-brace at frame 23

13 X-brace vehicle deck




Fig. 4. Hull 050 strain gauge locations.

σj

σj+1loge

1π� =

�j+1

�j

Fig. 5. Definition of decay coefficient or damping factor.

G. Thomas et al. / Marine Structures 21 (2008) 1–22 5

spectral leakage. The power spectra for the strain gauge records for a 15 s windowsurrounding the slam event were then determined. These spectra were used to determinethe whipping response frequencies of the vessel. The decay coefficient was estimatedfrom the decaying oscillation by determining the ratio between pairs of successiveamplitudes. The decay coefficient, sometimes known as damping factor, is given in termsof successive stress peak values as defined in Fig. 5. Comprehensive analysis of otheraspects of the slamming events including a definition of a slam event, frequency ofslamming occurrence and magnitude of impulsive slam peak forces may also be found inThomas et al. [7,8].

Fig. 6 shows examples of raw strain gauge data traces of a slam event measured on Hull042. The traces clearly illustrate the initial dynamic impact loading due to the slam event

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80

100

120

140 Steel Chevron Brace

-20

0

20

40

60

10

20

30Inner X-Brace

-30

-20

-10

0

1520253035

Keel Plate at Frame 24.5

-15-10-505

10

20

25

30

35

Str

es

s (

MP

a)

Str

es

s (

MP

a)

Str

es

s (

MP

a)

Str

es

s (

MP

a)

Keel Plate at Frame 41.5

-10

-5

0

5

10

15

160 165 170 175 180 185

Time (sec)

Time (sec)Time (sec)

Time (sec)

160 165 170 175 180 185

160 165 170 175 180 185

160 165 170 175 180 185

Fig. 6. Slam event—Hull 042 raw strain gauge data traces.


(at time t ¼ 168 s) and the subsequent whipping of the structure, which decays away byapproximately t ¼ 178 s for the gauges on the steel chevron brace and the inner crossbrace,and by t ¼ 175 for the two keel gauges.For each vessel the principal frequencies of the whipping responses were averaged for

individual strain gauges and are shown in Figs. 7 and 8. The range bars for each pointindicate the range of the data prior to averaging. The results for four gauges were notincluded in Fig. 7 since they were located in close proximity to other gauges and exhibitedidentical frequencies. For Hull 042, information was available on the vessel displacementduring the delivery voyage and thus the whipping frequencies were found for two differingloading conditions. The results for Hull 042 clearly show two distinct frequencies for all ofthe gauges, except the steel chevron braces and the cross braces which only featured asingle whipping frequency. The two frequencies are at approximately 1.5 and 2.6Hz. Theshift in frequencies for the two displacements is small, though it may be clearly seen. Twowhipping response frequencies are also clearly visible in the Hull 050 results, except for thesteel posts and cross braces. The frequencies are at approximately 1.3 and 2.8Hz. Modalanalysis conducted utilising finite element analysis, including the fluid-structure interac-tion, showed that the higher frequency corresponded to the first longitudinal mode, whilstthe lower frequency was consistent with the lateral torsion mode (pitch connectingmoment) [9].The average decay coefficients for Hull 042 are shown in Fig. 9 for the signals from four

strain gauges: inner cross brace at frame 14 (calculated for lateral torsion mode), transversegirder at frame 35 (calculated for first longitudinal mode) and frames 24.5 and 41.5 at thekeel (calculated for first longitudinal mode), where the average values from a number of

ARTICLE IN PRESS

0

1

2

2.5

3

Top ri

der f

r. 69

Top ri

der f

r. 67

Top ri

der f

r. 66

Top ri

der f

r. 65

Stb

. Fw. D

iag

Stb

. Aft

Diag

Pt.

Fw. D

iag

Pt A

ft. D

iag

Por

t Pos

t

Stb

. Pos

t

X-b

race

fr. 4

1

X-b

race

fr. 2

3

Ste

el V

dk.

Kee

l fr.

49.5

Kee

l fr.

40.5

Kee

l fr.

24.5

Fre

qu

en

cy (

Hz)

3.5

1.5

0.5

Strain Gauge Location

Fig. 8. Hull 050 whipping frequencies.

0

1

2

3

Ste

el B

race

Ste

el B

race

Aft

Trans

Box

Inne

r X-b

race

X-b

race

Gird

er fr

. 24

Por

tal T

op

Gird

er fr

.32

Gird

er fr

. 35

keel fr

. 24.

5

keel fr

. 35.

5

keel fr

. 41.

5

Fre

qu

en

cy (

Hz)

Displacement = 1150 tonnes

Displacement = 970 tonnes

3.5

2.5

1.5

0.5

Strain Gauge Location

Fig. 7. Hull 042 whipping frequencies.


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0.2

0.3

0.4

0.5

Inner Cross Brace at Frame 14

-0.3

-0.2

-0.1

0

0.1

2nd 3rd 4th 5th 6th 7th 8th 9th

Avera

ge D

ecay C

oeff

icie

nt

Structural Response Cycle Number

Longitudinal Girder at Frame 35

Keel Plate at Frame 24.5Keel Plate at Frame 41.5

0.2

0.3

0.4

0.5

-0.3

-0.2

-0.1

0

0.1

2nd 3rd 4th 5th 6th 7th 8th

Avera

ge D

ecay C

oeff

icie

nt


0.2

0.3

0.4

0.5

-0.3

-0.2

-0.1

0

0.1


Avera

ge D

ecay C

oeff

icie

nt


0.2

0.3

0.4

0.5

-0.3

-0.2

-0.1

0

0.1


Avera

ge D

ecay C

oeff

icie

nt


Fig. 9. Hull 042 full-scale slam decay coefficients.


slam events are shown with range bars. The decay of the signal for the inner cross bracestrain gauge at frame 14 appears to be fairly constant across the number of detectablecycles at approximately 0.1. This gauge only exhibited the lower frequency and therefore itappears that this frequency has lower damping levels than the higher frequency mode. Thedecay coefficient for the transverse girder strain gauge at frame 35 tends to start at a higherlevel of approximately 0.25, and then this slowly decreases until the seventh cycle, when thecoefficient turns negative before growing rapidly to diminish the whipping response. Thegauge at frame 24.5 at the keel appears to have three distinct parts: the large decaycoefficient (between 0.3 and 0.4) for the first cycle, a negative coefficient for the secondcycle and then a relatively steady lower coefficient (between 0.05 and 0.2) for thesubsequent cycles until the signal becomes indistinct. It should be noted that a negativedecay coefficient simply means that the signal amplitude of the subsequent cycle of thepoint of measurement increases. The decay at the gauge at the keel at frame 41.5 is similarto that at the keel at frame 24.5 although the value at the second cycle does not appear tobe so low. For the first longitudinal mode, the first cycle number tended to have asignificantly higher decay coefficient than subsequent cycles. This non-linear effect may bedue to the initial transfer of the slam energy throughout the vessel’s structure occurring inthis first cycle. When a slam occurs, the centrebow has entered the water and the increase insurrounding fluid may also influence the first one or two vibratory cycles.The average decay coefficients are shown in Fig. 10 for the signals from four strain

gauges for the first longitudinal mode of Hull 050: frame 67 in the centre bow, frame 41 atthe keel, frame 25 at the keel (all calculated for first longitudinal mode) and frame 23 onthe starboard portal crossbrace (calculated for lateral torsion mode). The decay of the

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0.4

0.6

0.8Top Rider at Frame 67

-0.4

-0.2

0.0

0.2

2nd 4th 6th 8th 10th 12th 14th

Avera

ge D

ecay C

oeff

icie

nt


Keel Plate at Frame 25

Keel Plate at Frame 14 Cross Brace at Frame 23

0.4

0.6

0.8

-0.4

-0.2

0.0

0.2

2nd 4th3th 6th5th 8th7th 9th

2nd 4th3th 6th5th 8th7th 9th2nd 4th3th 6th5th 8th7th 9th 10th

Avera

ge D

ecay C

oeff

icie

nt


0.4

0.6

0.8

-0.4

-0.2

0.0

0.2

Avera

ge D

ecay C

oeff

icie

nt


0.4

0.6

0.8

-0.4

-0.2

0.0

0.2A

vera

ge D

ecay C

oeff

icie

nt


Fig. 10. Hull 050 full-scale slam decay coefficients.


signal at frame 67 has two parts: the large decay coefficient (between 0.3 and 0.4) for thefirst two cycles and then the steady lower coefficient (o0.15) for the subsequent cycles untilthe signal becomes indistinct. Since this strain gauge was situated in the centre bow, andwas very close to the slam impact region on the hull, the initial large decay value was mostlikely to be due to transient energy transfer in the beam-like structure towards the aft of thevessel in order to set up the modal vibration. This effect is borne out further in the straingauge results from further aft in the vessel where the initial decay factor is negative (seesecond cycles in Fig. 10(c) and (d)), meaning an increase in oscillation strength, as theenergy is transferred aft. After this initial effect, the decay coefficients for the gauges atframes 23, 25 and 41 are consistently low at less than 0.15.

3. Exciter tests

Vibration exciter tests were conducted on the two vessels to determine the frequency,damping and modal shape of the primary longitudinal mode. Unfortunately, Hull 042 wasnot available for testing and hence Hull 045, an identical sister ship, was utilised instead.These tests were carried out in controlled conditions with the vessels being stationary incalm water. The vessels’ anchors (Hull 050 mass ¼ 1.8 ton, Hull 045 mass ¼ 1.4 ton) wereused to excite the longitudinal mode of vibration by dropping the anchor and theninstantaneously restraining it with the electric winch. Four accelerometers, distributedalong the length of the vessel on the centreline, measured the structural response.Unfortunately, due to practical constraints, it was not possible to conduct exciter tests forthe torsional modes. Hull 045 was in the lightship condition whilst the displacement ofHull 050 was close to that recorded from the full-scale trials.


3.1. Data analysis

The 100Hz raw data were highpass filtered at 0.6Hz to remove low frequency drift. Thepower spectra for the accelerometer records were then determined and used to identify themodal response of the vessel. The decay coefficients were estimated from the decayingoscillation by determining the amplitude reduction between successive peaks. The modalshape of the dominant structural response was found by comparing the response at thefour accelerometers distributed along the centreline with regard to magnitude and phase.

3.2. Results

An example of the exciter test raw data is shown in Fig. 11. The oscillatory nature of thestructural response may be clearly seen, along with the decay of the signal. The responselevels on Hull 045 were lower than for Hull 050 in the water which suggests that the anchorexcitation method was not as successful at exciting the longitudinal mode of Hull 045. Thismay have been due to a combination of the lighter anchor and less powerful winch fittedon Hull 045.The average first longitudinal natural frequencies from the tests were 3.01Hz for Hull

045 and 2.89Hz for Hull 050. For Hull 045, the decay coefficient ranged between 0.007 and0.14, as shown in Fig. 12, with an average value of 0.069. The Hull 050 results shown inFig. 13 indicate that the decay factor was generally in the range of 0.01–0.06 with anaverage decay coefficient for all cycles of 0.035. These values of decay coefficient fall withinthe range determined through the analysis of the full-scale slam events. It is noted thatdecay coefficient for the exciter does not exhibit the increased value for the first cycle; thismay be due to the lack of water surrounding the centrebow for the exciter tests comparedto the slam events.

-1

0

1

2

0 8

Accele

rati

on

(m

/s2)

Time (sec)

1.5

0.5

-0.5

-1.5

1 2 3 4 5 6 7

Fig. 11. Hull 050 exciter test raw data.

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0

0.05

0.1

0.15

0.2

0.25

0.3

2nd 3rd 5th 6th 7th 8th

Av

era

ge

De

ca

y C

oe

ffic

ien

t

-0.2

-0.15

-0.1

-0.05


4th

Fig. 12. Hull 045 exciter test decay coefficients.

0

0.05

0.15

0.25

0.3

2nd 4th 6th 8th 10th 14th 16th 18th

Avera

ge D

ecay C

oeff

icie

nt

0.2

0.1

-0.05

-0.1

-0.15

-0.2

12th


Fig. 13. Hull 050 exciter test decay coefficients.


The range of results appear to be smaller for Hull 050 than for Hull 045 which may bedue to the heavier anchor used for the excitation producing a larger and clearer structuralresponse. This is also borne out by the number of cycles that the data were able to be


analysed for: 19 cycles for Hull 050 and eight cycles for Hull 045. The measured mode wasidentified as the first longitudinal mode of vibration and the decay coefficient wasdetermined for this mode. The damping level recorded for Hull 050 is significantly smallerthan that measured for Hull 045, although it falls within the measured damping range ofHull 045.

4. Comparison of full-scale and exciter test results

The results from the full-scale trials measurements and exciter tests were compared andthe comparison is shown in Fig. 14. It is clear that the damping coefficients found throughthe full-scale measurements were higher than those assessed by the exciter tests. There aretwo possible reasons proposed for this difference. Firstly following a slam event, in the full-scale trials, the vessel may have become more immersed in the water which may have asupplementary damping effect. Additionally, when the vessels underwent exciter tests therewas no cargo on board, i.e. cars, trucks or passengers, the absence of which may havereduced the damping effect. The full-scale damping results and those derived from theexciter tests are also summarised in Table 2, with data on the mean decay coefficient andstandard deviation of the decay coefficient being included. The mean and standarddeviation were also determined when disregarding the first two cycles of the vibratoryresponse. The large dispersion of the data is reflected in the significant standard deviationvalues for both the full-scale and exciter test data.The values of decay coefficient determined through the full-scale trials measurements

and the exciter tests were also compared with those collated by Betts et al. [2]. Thecomparison shows that the decay coefficient values for the two large high-speedcatamarans lie at the lower end of the range of values. It should be noted that all theother vessels were of steel construction.

0

0.05

0.15

0.25

0 50 100 150 200

De

ca

y C

oe

ffic

ien

t

Betts et al. (1977)Exciter TestsFull Scale Trials

0.1

0.2

Vessel Length (m)

250

Fig. 14. Decay coefficient: comparison of full-scale and exciter test results with Betts et al. (1977) data.

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Table 2

Summary of decay coefficients from full-scale measurements and exciter tests

Vessel Strain gauge and vibratory mode Decay coefficient

All cycles Disregarding first

two cycles

Mean S.D. Mean S.D.

Hull 042 (full-scale) Inner cross brace at fr. 14 (lateral torsion mode) 0.0724 0.0896 0.0696 0.0991

Longitudinal girder at fr. 35 (first longitudinal

mode)

0.1224 0.1098 0.1036 0.1094

Keel plate at fr. 24.5 (first longitudinal mode) 0.1186 0.1696 0.1104 0.1244

Keel plate at fr. 41.5 (first longitudinal mode) 0.1295 0.0953 0.1076 0.0900

Top rider at fr. 67 (first longitudinal mode) 0.0954 0.1629 0.0904 0.1343

Hull 050 (full-scale) Keel plate at fr. 25 (first longitudinal mode) 0.1405 0.2171 0.0680 0.1274

Keel plate at fr. 14 (first longitudinal mode) 0.0511 0.1031 0.0706 0.0816

Cross brace at fr. 23 (first longitudinal mode) 0.0357 0.1234 0.0587 0.0946

Hull 045 (exciter

test)

First longitudinal mode 0.0685 0.0904 0.0530 0.0865

Hull 050 (exciter

test)

First longitudinal mode 0.0350 0.0531 0.0322 0.0458


5. Damping mechanisms

As noted in Section 1, there is scarce knowledge available on the damping of thewhipping behaviour of ships. The reported analysis of the full-scale slam measurementsand exciter test results has enabled estimates to be made for the total damping for largehigh-speed catamarans. It has been proposed previously [2] that the damping consists oftwo components: hydrodynamic damping and structural damping. Methods for estimatingthe various components of hydrodynamic damping, including wave making, viscous,appendage and acoustic damping, are now introduced. These methods were utilised toestimate the hydrodynamic damping of a large high-speed catamaran, Incat Hull 050, forthe first longitudinal and lateral torsional modes, and the results are presented. Anexperimental investigation into the material damping of the aluminium utilised in theconstruction of large high-speed catamarans was also conducted. This work has enabledconclusions to be drawn on the relative importance of the various components of damping,in particular the hydrodynamic and structural damping.

5.1. Energy dissipation in the system

For a linear, second order, damped system [10], the total force resisting motion may beexpressed as

F ¼ �kx� b _x, (1)

where k is the stiffness and b the damping. If the motion is assumed to be simple harmonic,then Eq. (1) becomes

F ¼ �kX sin ot� boX cos ot. (2)


The energy dissipated in a complete cycle will be

DW ¼ pboX 2. (3)

The energy dissipated by each identified damping mechanism, in a complete cycle, DW,may therefore be found. In order to achieve this, the vessel is split into transverse sections(nominally corresponding to the structural frames) and the damping contribution for eachsection found, bsX

2s , where bs is the sectional damping and Xs is the sectional amplitude of

motion or mode shape displacement. Using the mode shapes, for the first longitudinal andlateral torsion modes, found through finite element analysis, the total dampingcontribution, bX2, may be found by summing the contributions of each section alongthe vessel as follows:

bX 2 ¼XL

0

bsX2s , (4)

and the energy dissipated in a complete cycle found from Eq. (3).The fraction of the total energy of the vibrating system which is dissipated in each cycle

may then be calculated: where the fraction of the total energy of the vibrating system whichis dissipated in each cycle, DW, is divided by the total energy in the system, W. The totalenergy in the system W may be expressed either as the maximum potential energy ð1

2kX 2Þ or

the maximum kinetic energy ð12mv2 ¼ 1

2mo2X 2Þ since they will be equal for low damping

levels. It thus follows that

DW

W¼

pbo12mo2

X 2 ¼ 22po

� �b

2m

� �. (5)

Again, a ‘‘generalised’’ mass is used where mX 2 ¼PL

0 msX2s analogous to Eq. (4). The

total energy in the system may be found from finite element modal analysis which gives thetotal strain energy in the vessel’s structure, for each modal frequency, normalised againstthe point of maximum deflection in the structure. The damping loss may be defined asfollows:

DW

W¼ 2d ¼ 4pz, (6)

where d is the decay factor or logarithmic decrement and z is the ratio of damping tocritical damping. The loss coefficient, Z, is defined as the ratio of the energy dissipated perradian and the total strain energy:

Z ¼DW

2pW¼

dp. (7)

For clarification purposes, a summary of definitions of damping measures is given in Table3.

5.2. Structural damping

Vibrational energy can be dissipated within a volume element of material as it iscyclically deformed [11,12]. There is a range of mechanisms associated with internalreconstructions of the micro- and/or macro-structure, ranging from crystal lattice tomolecular scale effects which cause material damping.

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Table 3

Definitions of damping measures

Symbol Name Definitions

d Logarithmic decrement ln x1

x2

� �12DWW

Z Decay coefficient, damping factor, loss factor/coefficient 1p ln

x1

x2

� �12p

DWW

z Damping ratio 12p ln

x1

x2

� �14p

DWW

� Specific damping capacity 2 ln x1

x2

� �DWW


Xie et al. [13] stated that ‘‘there exists little information about damping in commercialaluminium alloys’’. They conducted tests that showed the loss factor for three commercialaluminium alloys lay in the range of 0.2� 10�3–1� 10�3. The loss factor for aluminiumhas also been given as ranging between 5� 10�5 and 7� 10�3 by White and Walker [14]and approximately 1� 10�4 by Beranek and Ver [15].

The total structural damping for a structure will be significantly greater than thematerial damping property of the material of construction [15,16]. Nashif et al. [11] statethat built-up structures usually have high initial structural damping, with loss factors ashigh as 0.05. Depending on the joints used to create the structure the built-up structuremay increase the material damping by a factor of 10. White and Walker [14] quote valuesfor structural damping of thin aluminium structures measured experimentally of 0.004 fora model structure and 0.04 for an aeroplane elevator panel. These values are significantlygreater than the inherent material damping of aluminium by factors of between 5 and 800.Ungar [17] also gives typical loss factors for materials as 10�4–10�3 and for an aluminiumaircraft structure as 10�2 due to the effect of joints. These values give increases in dampingdue to the structure of between 10 and 100 times the loss factor of aluminium. Values forloss factors of 0.016 and 0.018 are given by Clarkson and Ford [18] and of 0.017 by Mead[16] for the damping of aluminium aircraft panels. Again, these values are significantlygreater than the inherent material damping of aluminium by factors of between 2 and 360.It should be noted that most of the data for aircraft structures are for relatively highfrequencies that relate to jet noise.

There is considerable uncertainty concerning the mechanism that dominates thedamping of built-up structures at low frequencies [17]; however, various factors areproposed for the increase in damping for a complex structure. Firstly, the dampingproperties of a metal tend to increase with increasing stress [12], which might havesignificant implications for welded aluminium vessels where significant welding residualstresses [2] may be present (other stress concentrations will also be present due to localdesign and manufactural influences). Secondly, energy dissipation is likely to occur at thestructural joints [15], but how the energy is dissipated, and what parameters this dampingobeys, appears to be unknown. For higher frequencies, above 30Hz, Beranek and Ver [15]provide a method for calculating the loss factor for a structure accounting for varyingabsorption coefficient. The absorption coefficient is dependent upon the type of fastening,i.e. rivets, bolts or welds. The type of weld may also affect the damping properties of awelded built-up structure. Betts et al. [2] reported on tests conducted into the damping of awelded beam where it was found that the damping was significantly influenced by the type


of weld with the stiffest weld connection having the least damping. This may be significantfor aluminium-welded vessels since much of the welding is not continuous along the wholelength of a join but instead extended spot welding is employed. Grice and Pinnington [19]conducted a series of experiments on plate stiffened beams in order to investigate thevibration analysis of built-up structures. They investigated the typical setup in a shipmachinery foundation where flexible plates (hull and deck), which do not carry much load,are separated by stiff beams (frames), which carry significant loads. They surmised that thespeed of the long vibratory waves in the beams is high and the beams form the primarypath for vibration transmission. The flexural waves travelling along the stiff beams radiateshort wavelength flexural waves into the flexible plates which remove energy from thebeams. The relative proportion of the total power carried by the two waves depends on theinherent damping of the long waves (e.g. the material loss factor) and the level of couplingbetween the two waves at the joints where the stiff and flexible components meet.Therefore, built-up structures may increase damping if more energy can be transmitted tothe short waves in the more flexible plate. Energy dissipation will also occur due to thenon-structural contents of the vessel, i.e. the outfit and on-board cargo. The outfit willinclude cables, pipes, stowed equipment, carpets, ceiling and wall fittings, etc. whoseattachments will not be perfectly rigid, will allow some relative movements and so willallow energy to be dissipated.Unfortunately, it is difficult to examine and quantify all of these effects on the overall

damping of a structure. Out of the water exciter tests were conducted on Incat Hull 050 tomeasure the total structural damping, by comparing the results with the inherent dampingproperties of aluminium as measured on simple beams. However, the presence of the drydock vessel supports meant that the measured damping was greater than that measuredwhen the vessel was in water. This led to the verdict that the results could not be regardedas conclusive.There is therefore no experimental evidence that quantifies the increase in low frequency

damping from the inherent material damping to the structural damping for a structuresuch as a large aluminium catamaran. The only practical solution is to experimentallydetermine the total damping, calculate the hydrodynamic damping and subtract this fromthe total damping. This will provide an estimate of the total structural damping of thesystem. The total structural damping may then be compared with the inherent damping ofthe material to give an indication of the increase in structural damping due to the built-upstructure.

5.2.1. Measurement of material damping

In order to ascertain the damping properties of the commercial grade aluminium (6082T6 grade aluminium) used in the construction of large high-speed catamarans, tests wereconducted on extruded beams typical of those used by Incat. Two beams were tested:firstly, an I-Beam section of length 6505mm with web 150� 6mm and flange 100� 10mm,and secondly, a square section beam of length 5913mm with an outside dimension of47mm and wall thickness 3mm. Both sections were simply supported at the ends, with theI-Beam being tested for both the weak and strong axes. An accelerometer was fixed to themid point of the beam and the data logged at a sampling rate of 100Hz by a notebook PC.The beams were excited by a small impact and then the response allowed to decay. Theexperimental test was designed to have a frequency similar to that of the ship modes ofinterest.

ARTICLE IN PRESS

Table 4

Beam section decay coefficient values

Beam section Decay coefficient

I-Beam (strong axis) 1.15� 10�3

I-Beam (weak axis) 1.55� 10�3

Square section 2.18� 10�3

Average 1.71� 10�3


The loss factor for each test was determined and averaged for each beam and the resultsare given in Table 4. When the values for loss factor were averaged for all the beams avalue of decay coefficient of 1.718� 10�3 was obtained. The values for the loss factorappeared to vary slightly for the varying beam section; however, the values for materialdamping fall within the range provided by the literature.

5.2.2. Other structural damping contributions

Another factor that has been identified as possible contributor to the overall damping isthe effect of the superstructure rubber mounts. On Incat catamarans, the superstructure isconnected to the main hull structure by way of a series of rubber mounts. The rubbermounts are used to reduce the level of vibration felt by the passengers from sources such asthe engines and water jets which is in the range of 25–35Hz. The rubber mounts may alsohave an effect on the damping of lower frequency vibratory modes of the vessel. Themanufacturer’s specification for the rubber mounts used, HDA2P, quotes a loss factor of0.05.

The influence of the mounts may be investigated by estimating the strain energy in themounts during the modal deflections and then attributing this percentage of the total strainenergy to estimate the contribution of the rubber mounts. This exercise was carried out forIncat Hull 050 where it was found that for the first longitudinal mode the rubber mountsare subjected to 2% of the total strain energy. The damping contribution of the rubbermounts may therefore be estimated as a loss factor of 0.05� 0.02 ¼ 1� 10�3. This suggeststhat the rubber mounts do not have a significant influence on the total structural dampingof the vessel.

5.3. Hydrodynamic damping

The hydrodynamic damping is defined as the component of force in phase with thevessel’s velocity exerted by the body on the water for a unit amplitude velocity of the body.The hydrodynamic damping arises from a number of mechanisms. The primarycomponent is due to the waves created was the oscillating vessel, called wave makingdamping; this has been calculated using a steady periodic Green function panel method[20]. Energy may also be dissipated by friction; although these viscous effects are likely tobe small [21] they are also calculated. Since the damping due to sound radiation washighlighted as a possible hydrodynamic damping source by Sunnersjo and Janson [5], atechnique was also investigated for estimating this form of damping. The presentedtechniques were then utilised to estimate the hydrodynamic damping of a large high-speedcatamaran, Incat Hull 050.


5.3.1. Wave making damping

The wave making damping arises because the oscillating vessel generates waves whichradiate outwards and dissipate energy. It may be calculated utilising a steady periodicGreen function panel method as described in Holloway and Davis [20]. The formulationfor estimating damping was extended to account for forward speed [22] as follows:

bu ¼ b 1�U2

o2

Z00oZo

� ��U

da

dxþ

Z0oZo

2aþU

o2

db

dx

� �� , (8)

where a and b are the added mass and damping at zero speed, respectively, U the ship’svelocity, o the angular frequency and Z is the local vertical displacement of a point on thehull.

5.3.2. Viscous damping

As the vessel oscillates, water flows past the hull and exerts frictional forces on the hullsurface which may provide a damping mechanism. Since the oscillating flow for whippingwill be of fairly high frequency and small amplitude, it is proposed that the flow regimedoes not fit the typical ship motion solutions [21]. This was confirmed by analysing theexciter test results from Incat Hull 050 where it was found that the vertical amplitude ofdisplacement during a typical first longitudinal modal vibration at the bow wasapproximately 2.5mm and the maximum velocity was approximately 0.05m/s.The oscillating flow therefore has a very low Reynolds number (�216.8) and will be

laminar in nature. The oscillating flow is therefore similar to a flat plate with sinusoidaloscillations parallel to itself, which is sometimes termed Stokes’s second problem [23]. Ifonly the steady periodic solution is considered, after the starting transients have died out,there are no initial conditions to satisfy.As described in detail by Thomas [24], the work done by the fluid on the plate per cycle

per unit area, DW, may then be found as

DW ¼ pU2

ffiffiffiffiffiffiffiffirm2o

;

r(9)

with m being the dynamic viscosity and r the fluid density. If we assume for a cross sectionwith vertical velocity amplitude oXs that UEoXs sin a, where a is the angle between atangent to the cross section and the horizontal, then the work at a cross section per unitlength of hull will be

DW ¼

Zsection

DW dl ¼ X 2spo

2

Zsection

sin2odl, (10)

and for the whole ship

DW ¼

Z L

0

DW s dx ¼ DxX

DW s, (11)

summed for all cross sections, in which Dx is the section spacing.

5.3.3. Acoustic damping

Sound radiated by a vibrating structure transports energy from the structure and thuscontributes damping to the system [14]. Sunnersjo and Janson [5] conducted a study intothe hydrodynamic inertia and damping of ship hull vibrations. They used a finite volume

ARTICLE IN PRESSG. Thomas et al. / Marine Structures 21 (2008) 1–22 19

method to investigate the damping due to sound radiation and found that thehydrodynamic damping due to this mechanism may contribute significantly to the totalmodal damping.

The present study used an approach proposed by Holloway et al. [25]. Propagation ofsmall amplitude acoustic waves through water at constant frequency is governed by theHelmholz equation, a condition of compatibility of velocity on the surface of the radiatingsource (ship hull), zero pressure on the free surface (due to the great disparity between thedensities and speeds of sound in air and water), and a far-field condition of outgoingwaves. The power radiated by the flexing body, in this case a ship, may be determined as

P ¼

Z~V � n̂pdS ¼

X 1

4iorf0

~V � n̂A, (12)

where n̂ is the normal to the element i, dS an element of the boundary surface, p the surfacepressure (�irof0), f the velocity potential, ~V the velocity, o the angular frequency, r thefluid density and A is the element area. From this the loss factor is easily obtained in termsof energy radiated per cycle as 2pP/o.

5.3.4. Other components of hydrodynamic damping

Two other forms of hydrodynamic damping may be present. Eddy making damping isdue to the eddies which are shed when relatively sharp corners move through the water; itis proposed that this mechanism of damping may be insignificant for the small amplitudehigh frequency motion present. Appendages, such as the ride control surfaces, may alsocontribute a damping force. However, the areas of these surfaces are small and the effect isalso likely to be minor.

5.3.5. Total hydrodynamic damping—calculation and results

The components of hydrodynamic damping were estimated for a large high-speedcatamaran, using Incat Hull 050 as an example vessel. The method outlined in Section 5.1was used to estimate the total hydrodynamic damping given the section damping of thevarious components.

The hydrodynamic damping results for Hull 050 are shown in Table 5. These resultsshow that the wave making damping contribution was very low for both modes. This leadsto the conclusion that, for vessels such as the one examined, the waves radiated by thevibratory whipping are very small and consequently contain little energy. From the wavemaking damping results, it was clear that the frequency of whipping vibration is too high

Table 5

Hydrodynamic damping decay coefficient values

Decay coefficient

First longitudinal mode Lateral torsion mode

Wave making damping 0.23� 10�3 2.31� 10�3

Viscous damping 78.6� 10�6 38.2� 10�6

Acoustic damping 24.0� 10�6 24.0� 10�9

Total hydrodynamic damping 0.23� 10�3 2.31� 10�3


to make the wave making damping a large component of the total damping. The frictionaldamping component is insignificant due to the small oscillation displacements andvelocities.The acoustic damping is also insignificant. At low frequencies (i.e. when the acoustic

wavelength is significantly longer than the typical ship dimensions, such as is the case here),the radiated acoustic power in theory is proportional to the sixth power of frequency;hence, the energy loss per cycle is proportional to the fifth power of frequency. However, ifthe frequency increase is a result of increased structural stiffness (with constant mass) thenthe modal energy increases with the second power of frequency, thus the loss factor will beproportional to the third power of frequency. Acoustic damping therefore may besignificant at higher frequencies.

6. Comparison of measured and predicted damping

The predicted damping values, for both hydrodynamic and structural damping, werecompared with the measured damping values. The measured total damping used forcomparison purposes was that measured during the exciter tests on Hull 050. The excitertest results were chosen, rather than the full-scale trials results, because they were madeunder more controlled conditions with a predominance of a single frequency of vibration.The Hull 050 results were used rather than the Hull 045 results since the Hull 050 excitertest results appeared to be a much clearer and consistent set of measurements due to thelarger anchor. The comparison is shown in Fig. 15. This plot shows that there is a largediscrepancy between the total measured damping value and the combined material andhydrodynamic damping found through measurement and calculation. The major factorproposed for this discrepancy is that the total structural damping for a large structure maybe considerably greater than the inherent damping of the component material. Thisstructural damping component, due to the built-up structure rather than the inherentmaterial damping, has been estimated as being 16 times the inherent material damping.This structural damping component, excluding the inherent material damping, is alsoshown in Fig. 15.Since the vibratory damping influences the rate at which whipping will decay following a

slam event, knowledge of the damping magnitude is important in order to make validestimates of the fatigue of a high-speed catamaran. The results gained through this workprovide such data for designers. In addition, the finding that the total damping isdominated by the vessel’s structural damping suggests that efforts to increase damping toreduce whipping cycles should focus on increasing the structural damping.

7. Conclusions

This paper has reported on work which has improved the understanding of vibratorydamping of large high-speed catamarans. Full-scale measurements of slam events wereconducted on two large high-speed Incat catamarans (86 and 96m in length) duringdelivery voyages and regular service operations. These measurements were used toinvestigate the damping of the whipping behaviour of the vessels. Exciter tests were alsoconducted on the vessels, whilst stationary in calm water, in order to further examine theirvibratory damping properties. An examination of the components that contribute to thedamping of the system was also undertaken.

ARTICLE IN PRESS

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

Lo

ss F

acto

r

Measured - Exciter TestMeasured - Beam Damping ExperimentCalculated - Hydrodynamic DampingEstimated - Structural Damping

Total Damping Material Damping Hydrodynamic

Damping

Structural

Damping

Fig. 15. Hull 050—comparison of measured, calculated and estimated damping.


Estimates were made of the relative magnitude of the various hydrodynamiccomponents including: wave making damping, viscous damping and acoustic damping.The total calculated damping was then compared with the levels of damping foundthrough the full-scale measurements and exciter tests. The total predicted damping, due tohydrodynamic and material damping, was found to account for only a small proportion ofthe measured total damping. The shortfall can only be presumed to be due to additionalstructural damping which is estimated to be one order higher than the inherent materialdamping. This ratio is similar to that found in aeronautical and similar structuralapplications [14,16–19]. This additional structural damping is likely to be composed ofmany items of damping which are not accounted for in a simple structural model,including: fuel and water in tanks, all fittings, soft materials and furnishings, bonded joints,fireproofing and pipe work.

Acknowledgements

This work has been partly supported by an Australian Research Council SPIRT grant.The authors would like to acknowledge the help of their colleagues at the University ofTasmania, Incat Tasmania and Revolution Design Pty. Ltd. In particular, the contributionof Nigel Watson in acquiring the Hull 042 full-scale trials data is acknowledged.

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The vibratory damping of large high-speed catamarans

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