Investigation of motions of catamarans in regular waves-I

Ocean Engng, V o l . 23, No. 1, pp. 89-105, 1996 Copyright © 1995 Elsevier Science L t d Printed in Great Britain. A l l rights reserved

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INVESTIGATION OF MOTIONS OF CATAMARANS I N

REGULAR WAVES—I

C. C. Fang, H . S. Chan and A . Incecik*

Department of Naval Architecture and Ocean Engineering, University of Glasgow, Glasgow G12 8QQ, U.K.

(Received 25 April 1994; accepted 8 August 1994)

Abstract—In this paper an analytical technique based on the two-dimensional Green function method associated with a cross-flow approach for taking viscous effects into account to estimate the motion response of catamarans in the frequency domain is presented. In order to validate this method, the numerical results are compared with experimental values obtained for two different catamarans (ASR5061 [Wahab, R., Pritchett, C. and Ruth, L.C. 1971. On the behaviour of the ASR catamaran in waves. Marine Technology, 8, 334-360] and Marintek [Faltinsen, O., Hoff, J.R., Kvalsvold, J. and Zhao, R. 1992. Global loads on high speed ' catamarans. 5th Int. Symp. on Practical Design of Ships and Mobile Units, University of

Newcastle-upon-Tyne, 1.360-1.373]).

In the second part of the paper the tests carried out with a third catamaran configuration at the Hydrodynamics Laboratory of the University of Glasgow are presented to evaluate the non-linear effects. These test results cover different speeds and wave heights at a wide range of wave frequencies. The paper concludes that the two-dimensional method correlates very well with measurements of small amplitude motions. For large amplitude motion tests, the non-linear effects become significant when the model speed and wave ampUtudes increase. The peak values of heave and pitch motions measured around the resonance frequency are smaller than those obtained from the linear theory.

1. I N T R O D U C T I O N

Catamarans are the most accepted f o r m of high speed craft f o r passenger/vehicle transportation. Compared w i t h other high speed c r a f t , they possess good transport efficiency at moderately high speeds. The large deck area is one of the desirable characteristics of catamarans and gives a small rate and angle of roU and, consequently, good stability.

T o assess the seakeeping performance of a catamaran design either results of exper-imental measurements or those obtained mainly f r o m linear frequency domain methods are used [Incecik et al. (1991), L e e et al. (1973)]. A l t h o u g h a number of experimental and theoretical investigations o f catamaran motions have been conducted i n the past, there is a lack of understanding o f the non-linearity of large amplitude m o t i o n . Further-m o r e , the frequency-doFurther-main Further-m e t h o d is restricted to the prediction of sFurther-mall aFurther-mplitude motions because i t is assumed n o t only that the f r e e surface condition can be hnearised, b u t also that the ship displacements are small relative to t h e ship dimensions [Salvesen

et al. (1970)].

* Author to whom correspondence should be addressed. 89

Pergamon

To investigate the large amplitude motions of catamarans travelling i n waves, a series of experiments w i t h a catamaran model was carried out i n the Towing Tank of the Hydrodynamics Laboratory at the University of Glasgow. D u r i n g the experiments, the wave amplitudes and f o r w a r d speeds were varied.

2. T H E O R E T I C A L B A C K G R O U N D

A two-dimensional linearised method based on potential theory w i t h a cross-flow approach f o r taking viscous effects into account is used to predict the m o t i o n p e r f o r m -ance o f catamarans i n waves. A l t h o u g h the three-dimensional method is a more accurate technique than the two-dimensional technique f o r the calculation of the motions o f t w i n hulls, the computation time f o r the three-dimensional method is significantly higher than f o r the two-dimensional method.

Formulation of motion

Figure 1 shows the right-hand coordinate system o-xyz which moves i n the same direction and speed as the moving body. The ;c-axis is pointing upstream parallel to the longitudinal plane o f the body and the z-axis is pointing vertically upward through the centre of gravity of the body w i t h the origin i n the plane of the mean free surface. The body is assumed rigid and oscillates i n six degrees of f r e e d o m about its mean position w i t h complex amplitudes (/ = 1, 2, . . . 6 ) . H e r e ƒ = 1, 2, 3, 4, 5, 6 refer to surge, sway, heave, roU, pitch, and yaw modes of m o t i o n respectively.

For dynamic equilibrium the total wave-induced forces must be equal to the mass

Catamaran motion in regular waves—I 91

inertia forces and the coupled Hnear equations of m o t i o n of the rigid body can be written as

k=i

{MJ, + A-zt) i \ + Bj, + C j , ^ = Fr+Fyj= 1,2,...6 (1)

where 1, and kk are m o t i o n acceleration and velocity respectively; Mj, is the mass matrix; Aj^is the added mass; Bjkis the damping; C,^ is the restoring coefficient; F j is the wave exciting force and F j is the excitation force due to viscous effects. The indices j and k indicate the direction of the fluid force and the mode of m o t i o n respectively.

I t is assumed that the catamaran has one longitudinal plane of symmetry. The symmetry of the huU w i t h respect to the longitudinal centre-plane of the t w i n - h u l l ship leads to decouphng of the vertical plane modes f r o m the horizontal plane modes. Thus the equations of m o t i o n can be divided i n t o surge-heave-pitch and sway-roll-yaw equations. The generahsed mass matrix [ M ] o f the ship whose centre of gravity is at (0,0, ZG) can be written as [ M ] = M 0 0 0 MZG 0 0 M 0 -MZG 0 0 0 0 M 0 0 0 0 -MZG 0 I44 0 ~h6 MZG 0 0 0 0 0 0 0 "•^64 0 •^66 (2)

where M is the mass of the ship, Ijj is the moment o f inertia about the origin i n the / t h mode of m o t i o n and I j , is the cross-product of inertia about the origin.

A two-dimensional pulsating source potential technique is developed to solve the unsteady velocity potential due to the incident, diffracted and radiated wave systems. W i t h the basic linear assumption, the d i f f r a c t i o n wave potential ^ij and the radiation wave potential i n the / t h mode of m o t i o n , must satisfy the f o l l o w i n g linearised boundary conditions:

Laplace's equation i n the fluid domain

V2 <^j = 0; (3) the linearised free-surface condition

(m + Udldxf ct)y + g ^ = 0

the kinematic body boundary condition

/ = l , 2 , . . . 7 a t z = 0 ; 3 ^ dn -mUj + Unij / = 1, 2 , . . . 6 on ; . (4) (5) and

Fig. 2. Segmentation of the ASR5061 catamaran.

the kinematic condition on the ocean floor

^ = 0 7 = l , 2 , . . . 7 a t z ^ - o o ; (7) where g is acceleration due to gravity; rij is the generalised direction cosine w i t h

{nx,n2,n^) = n and {n4,ns,ns) = r x n ; n is a unit n o r m a l vector outward f r o m the mean wetted body surface and r is a position vector of a point o n the mean wetted body surface; (mi,m2,m3) = - ( n • V ) W and {1714,1715,171^) = - ( n • V ) ( r x W ) ; and W is a steady velocity field. I f the body is slender, the steady perturbation potential due to f o r w a r d m o t i o n is neghgible i n the unsteady flow. Then my = 0 f o r ƒ = 1, 2, 3, 4; ms = «3 and mg = -«2> which are used i n the present study.

For the two-dimensional method, a high-frequency assumption is made that the frequency of oscillation w is much higher than the differential operator Udldx i n the f r e e surface boundary condition which reduces to

Catamaran motion in regular waves—

r^

O 2 - 1 1.5 H 0.5 H MARINTEK experiment 2D Potential K 2D Potential+Viscous 3 4

Fig. 6. Pitch motion response in 135° heading for the Marintek catamaran at F„ = 0.49.

This f r e e surface boundary condition ( 8 ) f o r the ship body osciUation at f o r w a r d speed requires that the wave length is approximately of the same order as the ship beam. This is a very critical assumption and it makes the theory somewhat questionable i n the low-frequency range since the f o r w a r d speed effects on the free surface are n o t included.

The incident wave potential o f unit amplitude è^Q satisfies the Laplace's equation, the linearised free surface condition and the sea b o t t o m condition can be represented by w i t h = _ J Qkz + ik(x cosp + y s i n p ) COo CO = |(Oo — Vk cosp ( 9 )

where COQ is a wave frequency; /: is a wave number; and (3 is an angle of incidence w i t h the X-axis (180° at head sea). I t is understood that the real part is to be taken i n all expressions involving e x p ( - i a ) r ) .

These potentials are obtained by means of a two-dimensional source distribution o f the f o r m

(10)

2'ui^{p-)= u(q)G(p;q)dsiq)

CO

Catamaran motion in regular waves—I 95 o MARINTEK experiment 2D Potential 2D Potential+Viscous 1 2 3 4

coVlVg

Fig. 7. Heave motion response in 135° heading for the Marintek catamaran at F„ = 0.49.

source o f u n k n o w n strength <j{q) at the source point q. The solution f o r G(p;q) is given by Wehausen et al. (1960). Here CQ is the immersed contour of the body. Using the body boundary conditions, the u n k n o w n source strength (j{q) can be determined. Thus TTa(p)+ (T(q)—--—dy(^) = 27T dn dn (11)

wwvwiiiir////.

.wwwva

Fig. 9. Segmentation of the V-1 catamaran model.

Once the source densities a ( ^ ) are k n o w n , the velocity potential ^{p) can be solved by E q u a t i o n (10). The hydrodynamic force and moment can be obtained by substituting the k n o w n velocity potential into the linearised Bernoulh equation and integrating the resultant f o r m u l a t i o n over the mean wetted body contour. Then, the sectional hydrodynamic coefficients and wave exciting force can be obtained by a strip approxi-mation. These quantities are integrated along the length of the ship to obtain the total hydrodynamic coefficients.

W i t h one longitudinal plane o f symmetry, the hydrostatic coefficients Cj, are given by

C33 = pgA„;c44 = pgVGMr;c55 = p g G M ^ ;

C35 = C53 = - p g A ^ , (12) where A,^ and Ay are the area and the first moment of the waterplane area at z - 0

respectively; V is the volume displacement of the catamaran; GMT is the transverse metacentric height and GM^ is the longitudinal metacentric height above the origin.

The hydrodynamic forces due t o viscous effects can be separated into viscous damping forces, viscous restoring forces and viscous excitation forces i n the f o r m

Fj= É {bjAk + èikik)-Fj (13)

where bj, and Cj, are viscous damping and restoring coefficients respectively. Details on the cross-flow approach f o r taking viscous effects into account can be f o u n d i n Chan (1993). The hydrodynamic coefficients i n the equations o f m o t i o n may be considered as linear dependence of fluid forces due to non-lift potential flow and cross-flow effects such that: Bj, = V + ^Jk, Q/t = + Cj,.

Correlation studies

The validation of the mathematical analyses was carried out by comparing results obtained f r o m the two-dimensional method w i t h those obtained f r o m experiments w i t h two catamaran models. The values of the drag coefficient C^, used i n the cross-flow

Catamaran motion in regular waves—I 97

Draft 6.72cm (V-1 Catmaran)

0 1 2 3 4 5

Model Speed(ni/s)

Fig. 10. Resistance measurements of the V-1 catamaran.

O I I 1 ( I O 1 2 3 4

(oVïVg

Fig. 13. Heave motion response in 180° heading for the V-1 catamaran at F„ = 0.0.

calculations were taken as 0.4 f o r the ASR5061 catamaran and 0.01 f o r the models of M a r i n t e k and V - 1 catamarans. The principal dimensions of the ASR5061, M a r i n t e k and V - 1 catamarans are given i n Table 1. Figures 2, 3 and 9 illustrate the segmentation of three catamaran models f o r the computation o f the two-dimensional method.

Motion response of the ASR5061 catamaran

The most comprehensive sea load measurements were carried out i n the Naval Ship Research and Development Centre by Wahab et al. (1971) f o r the ASR5061 catamaran model advancing obhquely i n deep water waves. The demi-hull was asymmetrical f o r w a r d and symmetrical aft. I n this correlation study, the experimental values o f the A S R m o d e l w i t h a h u l l separation/beam = 1.58, wave direction p = 180° and Froude number F„ ( = U/VgL) = 0.31 were compared w i t h the results obtained f r o m the t w o -dimensional method.

Catamaran motion in regular waves—I 99 bi) 180-1 90' OH -90 H -180-f 1.25 coVL/g

Fig. 14. Fitch motion response in 180° heading for the V-1 catamaran at F„ = 0.0.

Figures 4 and 5 show the comparison o f experimental data and the present theoretical results w i t h and w i t h o u t viscous effects f o r the pitch and heave motions o f the ASR5061 catamaran. The comparison shows that the two-dimensional potential theory gives good agreement both i n heave and pitch motions except around the resonance regions. A f t e r incorporating the effects of viscous damping, the theoretical predictions correlate very well w i t h the experimental data.

Motion response of the Marintek catamaran

Tests w i t h the M a r i n t e k catamaran model were carried out i n the Ocean Environment Laboratory of M a r i n t e k [Faltinsen et al. (1992)]5 The demi-huh of this catamaran was r o u n d bottomed, symmetrical w i t h respect to the longitudinal vertical plane and w i t h a transom stern. A free r u n n i n g model was used and measurements were carried out at F„ = 0.49. The heave and pitch transfer functions i n 135° obhque waves f o r the

0 1 2 3 4 1

o)VÏ7g

Fig. 15. Heave motion response in 180° heading for the V-1 catamaran at F„ = 0.226.

catamaran model were compared w i t h the numerical values obtained f r o m the t w o -dimensional method as shown i n Figs 6 and 7. The heave results indicate a good comparison between the numerically predicted and experimentally measured heave m o t i o n responses. Some discrepancies were observed i n the prediction o f pitch motions at the smaller wave frequencies. For the autopilot system, experimental errors may have occurred because it was not possible to keep the heading constant during these tests as discussed by the authors [Faltinsen et al. (1992)].

3. E X P E R I M E N T A L S E T - U P

The V - 1 model [Incecik et al. (1991)] is a high speed catamaran h u l l f o r m . The demi-hull is o f the planing type, featuring a V-type section and cut-off transom stern as shown i n Fig. 8. I t was constructed of glass-reinforced plastic fibres. The turbulence simulation comprised studs of 3 m m diameter and 3 m m height at a spacing of 25 m m .

Catamaran motion in regular waves— 101 OJ a s -180 1.25 0.75 H 0.25 H Experimental Data ^0 = 1cm O ^0 = 3cm ^0 = 4.5cm

Fig. 16. Pitch motion response in 180° heading for the V-1 catamaran at F„ = 0.226.

The studs were situated f r o m 1.9 to 2.0 m f o r w a r d of stern. N o other underwater appendages were attached t o the V - 1 model during the tests.

I n order to evaluate the non-linear motions o f the V - 1 catamaran i n regular waves, tests were carried out at three d i f f e r e n t model speeds of 0, 1 and 3 m/s corresponding to the Froude numbers of 0.0, 0.226 and 0.677, three d i f f e r e n t wave amplitudes of 1, 3 and 4.5 cm and 14 d i f f e r e n t wave frequencies i n the head sea condition.

A measurement system [Fang (1994) and Stevens and Crago (1966)] was designed and tested to investigate the non-linear effects during large amplitude motions of a catamaran i n head sea condition. The m o d e l was towed by a vertical post which allowed freedom i n pitch and heave motions w i t h restraints i n r o l l and yaw motions. The towing point was positioned at the centre of gravity o f the model.

The total resistance was measured by a force gauge transducer instaUed o n the middle of the towing post f r a m e ; the accuracy of the measurement was w i t h i n ± 0.012 k g . T h e

heave and pitch osciUations of the model were measured w i t h a pair of small hght-emitting diodes ( L E D , selspot system) positioned on the deck of the model; the accuracy o f the measurement was w i t h i n ± 0.01 cm f o r heave and ± 0.02° f o r pitch m o t i o n . A gravity-type accelerometer was used to measure the vertical bow acceleration at FP. One resistance-type wave probe whose measurement accuracy was w i t h i n ± 0.02 c m was situated o n the carriage and was parallel to the bow Hne (122.5 c m f o r w a r d of C G ) . The phase difference between the wave excitation and the model m o t i o n responses was measured. I n order to measure the incident wave amplitude, three resistance-type wave probes were located at B/2, B/3 and B/4 f r o m the tank side waU, where B is the tank w i d t h , and approximately 5.5 m i n f r o n t of the wavemaker.

A U the analog signals passed through multi-channel amphfiers and filters before entering into the A M U X - 6 4 system which is an analog-to-digital converter. The digital signals were then recorded by a Macintosh-Ilci microcomputer and displayed graphically to ensure that the acquisition and measuring systems w o r k e d properly.

Catamaran motion in regular waves—1 103

4. R E S U L T S A N D D I S C U S S I O N

The total resistance, C G rise and t r i m change of the V - 1 catamaran m o d e l were measured over a speed range up to F„ = 1.129 i n the stiU water condition. These results are presented i n Figs 10-12.

The heave and pitch m o t i o n results of the V - 1 catamaran based upon the theoretical method just described were computed and compared w i t h experimental data. N o n -dimensional amplitudes and phase angles are plotted as a f u n c t i o n of non--dimensional wave frequency i n Figs 13 and 14 f o r zero f o r w a r d speed. Figs 15 and 16 f o r F„ = 0.226, Figs 17 and 18 f o r F„ = 0.677.

I n general, the magnitude of the m o t i o n amphtude at the resonant encountering frequencies is overestimated by the two-dimensional potential method and increases w i t h the f o r w a r d speed. A f t e r incorporating the empirical method based on the steady

cross-flow assumption, the theoretical predictions have been improved as shown i n the comparison between theory and experiment.

For the zero speed case, the theoretical results correlate very w e l l w i t h measurements f o r heave and pitch motions. A t the^resonance p o i n t , the results show that the m o t i o n responses slightly decrease when the wave amphtude increases.

Figures 15 and 16 are the experimental m o t i o n response values and phase angles of the V - 1 catamaran model at F„ = 0.226. Compared w i t h the experimental data, the numerical predictions shghtiy overestimate the heave motions at the resonance region. The prediction of pitch motions does show some disagreement w i t h measurements i n the low-frequency region. W h e n the wave amplitudes increase, the rnotion responses at the resonance point decrease.

The non-linear effects at high speeds can be examined i n Figs 17 and 18 f o r

Fn = 0.611. The numerical results f o r heave are i n good agreement w i t h experimental

measurements. For the pitch motions, the resonance point of the experimental measure-ments is slightly shifted towards the low wave frequency region. I n long waves, the pitch m o t i o n responses measured f r o m experiments do not approach 1.0. This means that pitch motions of the V - 1 catamaran at low wave frequencies are not equal to the wave slope. Furthermore, i t can be confirmed by these test data that the non-linear effects are more significant when the f o r w a r d speed and wave amplitudes increase around the resonance frequency.

5. C O N C L U S I O N S

I n this study, the two-dimensional potential theory associated w i t h a cross-flow approach f o r taking viscous effects into account f o r estimating the smah amphtude motions i n the frequency domain has been vahdated by experimental results o f three catamarans. Some conclusions may be drawn as fohows:

1. The two-dimensional method calculations can provide reasonable predictions of motion responses of catamarans at low f o r w a r d speed. There are some discrepancies observed i n the prediction of pitch motions i n the l o w wave frequency region.

2. Through the experimental investigation of the V - 1 catamaran model w i t h variable f o r w a r d speeds and wave amplitudes, i t can be confirmed that the non-linear effects become significant when the model speed and wave amphtudes increase. The peak values o f heave and pitch motions measured around the resonance are smaller than those obtained f r o m the small amplitude m o t i o n predictions.

Because of the linearised boundary value assumption, the frequency domain method wiU be restricted to the prediction o f smah amplitude motions. T o investigate the non-linear effects of large amplitude motions of catamarans i n waves through numerical predictions, f u r t h e r studies backed-up by experiments are being carried out and wiU be reported as part I I i n the near f u t u r e .

Acknowledgement--i:h& first author acknowledges support received from Professor D. Faulkner, Head of

Department, in obtaining financial support through the Overseas Research Student Awards scheme and a Postgraduate Scholarship from the University of Glasgow.

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Catamaran motion in regular waves— 105

Int. Symp. on Practical Design of Ships and Mobile Units, University of Newcastle-upon-Tyne, U.K.

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Fang, C.C. 1994. Experimental investigation of large amplitude motions of a catamaran in waves. Report

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