A mathematical model for acceleration phase aerodynamically alleviated catamarans and minimizing the time needed to reach final speed

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D O I 10.1007/S00773-016-0368-Z CrossMark

O R I G I N A L A R T I C L E

A mathematical model for acceleration phase of aerodynamically

alleviated catamarans and minimizing the time needed to reach

final speed

M o r t a z a Azizi Yengejeh^ • H a n i i d Mehdigholi^ • M o h a m m a d Saeed Seif^

Received: 16 August 2015/Accepted: 9 January 2016/Published online: 16 February 2016 © J A S N A O E 2016

Abstract Racing catamarans use aerodynamic alleviadon concept which i n existing extreme ground effect signifi-candy enhances the performance. Beside design measures, controlling strategies may be employed as convenient solutions to improve the performance and address concerns regarding poor stability i n these crafts. B e i n g o f substantial importance f o r a racing catamaran to reach the final speed as soon as possible, this study attempts to find the optimal f o r m o f changing the drive angle (as control variable) to m i n i m i z e its acceleration time. I n this regard, a mathe-matical model is developed f o r f o r w a r d acceleration phase o f these catamarans based on empirical and theoretical methods. Then the formulation and solution algorithm f o r the time-optimal problem are described according to an indirect method. Results f o r a representative racing c r a f t have been presented i n uncontrolled and controlled con-ditions. Problem i n controlled condition has been solved w i t h o u t and w i t h a predefined constraint regarding stability margin. Optimal controlling o f the drive angle without stability constraint during the acceleration results i n 40 % reduction i n time required to reach the speed o f 110 k n and 14 % reduction i n resistance at this speed i n comparison to the uncontrolled case. A d d i t i o n o f the stability constraint changes optimal solution f o r drive angle and causes c r a f t t r i m angle f o l l o w a decreasing trend at higher speeds.

H Mohammad Saeed Seif seif@sharif.edu

' Center of Excellence in Dynamics and Hydrodynamics o f Marine Vehicles, Sharif University o f Technology, PO Box, Tehran 11365-9567, Iran

K e y w o r d s T i m e - o p t i m a l control • Racing catamarn • Aerodynamic alleviation • Extreme ground effect • M i n i m u m principle

1 I n t r o d u c t i o n

Aerodynamically alleviated catamaran concept has been used since more than a half century ago i n the f o r m o f tunnel hulls i n boat races. Superior performance o f these hybrid crafts i n comparison to conventional planing boats very soon led to introducing new separated racing classes [1]. These catamarans are indeed a subset o f the more general group k n o w n as aerodynamically alleviated marine vehicles ( A A M V ) . I n A A M V s aerodynamic and hydrody-namic l i f t s both are important i n cruise speed. I n catamaran configuration, hydrodynamic l i f t is generated by sponsons (Fig. 1) commonly i n the f o r m of asymmetric planing hulls. This asymmetry i n addition to g i v i n g stability i n turns [2] reduces wetted surface and removes the under h u l l spray drag [3, 4 ] . The structure between two sponsons is shaped c o m m o n l y i n the f o r m o f an a i r f o i l t o produce aerodynamic l i f t w i t h o u t any extra powered air support or skiiting. F r o m the aerodynamic viewpoint, this part acts as the ram w i n g or channel flow w i n g [ 1 ] . R a m - w i n g condi-tion happens when there is Httle or no gap between w i n g trailing edge and water surface, so a ( f u l l y or partially) stagnation flow occurs and nearly f u l l dynamic pressure w i l l exist under the w i n g . A t higher speeds w h e n w i n g sufficiently comes out f r o m water because o f presence o f side hulls, a restricted channel flow is f o r m e d under the w i n g . Aerodynamic configuration is very similar to the wings w i t h endplates.

As an eariy published works, Shipps [1] i n 1976 offered valuable discussions on different aspects o f ram w i n g

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Fig. 1 Aerodynamically alleviated racing catamaran

planing crafts. T w o years later, W a r d et al. [5] g i v i n g some technical data on the geometry, structure and propulsion system o f the racing catamaran K U D U I I , published results of some full-scale trial records on its smooth water per-formance. A t the same year, K a l l i o [6] performed com-parative trials between the K U D U I I and the conventional planing monohull, K A A M A . The measured data i n head sea state 2 showed that pitch m o t i o n f o r K U D U I I was about 40-70 % o f that f o r m o n o h u l l at speeds about 4 0 - 6 0 knots. Reif and Guenther [7] published the results o f w i n d tunnel and towing tank tests on model o f a tunnel racing boat. These results revealed the significant pordon o f aerodynamic drag i n total resistance o f the boat. Some years later, Reif [8] conducted another set o f model tests i n w i n d tunnel to study more accurately the aerodynamics o f a tunnel boat i n ground effect. I n another study i n these years, Nangia [9] reviewed some o f the design challenges of these crafts and made a f e w suggestions to address stability problems. One decade later i n 1997, Doctors [10] proposed a configuration called Ekranocat w i t h above mentioned principles and preformed a theoretical analysis to show its beneficial aspects. Some years later, M o r c h [11] by means o f experimental data and computational fluid dynamics (CFD) computations p r e f o r m e d an analysis on aerodynamic performance o f 7.5 m offshore racing cata-maran. M o r e recently, Russel [12] i n his book along w i t h reviewing the history o f tunnel boat design explains some design details o f tunnel boats w i t h a simple language. Recorded w o r l d speed records i n this reference f o r tunnel boats w i t h propeller propulsion systems and j e t engines are, respectively, about 180 and 275 knots. WiHiams [13] through his Ph.D. dissertation on aerodynamics o f high-speed multihuUed marine vehicles, using d i f f e r e n t analyt-ical and C F D analyses, tried to calculate the achievable superior peiformance f o r a planing catamaran as a result o f aerodynamic alleviation. Chancy and Matveev [14] used a mathematical model based on aerodynamic extreme-ground-effect theory [15] and hydrodynamic added-mass strip theory [16] to simulate the steady states and

vertical-plane motions o f a tunnel h u l l . Matveev [17] used the same theories f o r modeling the longitudinal motions o f a hydroplane boat i n w h i c h wing-like structure is extended behind the sponsons.

L i f t to drag ratio increases dramatically f o r a w i n g especially w i t h endplates i n extreme ground effect by decreasing the ground clearance. Rozhdestvensky [15] uses the'term "extreme ground e f f e c t " when this clearance lies below 10 % o f the cord. I n F i g . 2, Carter's experimental result [18] f o r m a x i m u m l i f t to drag ratio (which occurs at attack angle of 2 . 5 ° - 3 ° ) has been plotted against height to cord ratio w i t h and without endplates. W i n g aspect ratio is 1 and the height has been measured at traihng edge. A s can be observed at the most close proximity to the surface, i.e. hi c = 0.015, LID w i t h endplates is three times o f that at hi c — \. Endplates effect is noticeable f o r hic < 0.2 and at hi c — 0.015, L/D w i t h endplates is about 65 % more than that without them. However, such a gain i n l i f t is accompanied by g r o w t h i n aerodynamic pitching moment w h i c h makes the boat more susceptible to longitudinal instability. Indeed, at the h i g h speeds i n waterborne condition, there is a fine and f r a g i l e balance between the aerodynamic, hydrody-namic and propulsive forces. As a matter o f fact, the small aspect ratio wings used i n racing boats, on their o w n , are unstable i n the v i c i n i t y o f the surface [19]. Such critical nature o f the stability and safety concerns are the main reasons f o r the l i m i t e d fields o f application f o r these crafts.

Stability analysis o f A A M V s has been the subject o f some studies i n recent years. CoUu [20] provided a math-ematical f r a m e w o r k f o r the dynamics o f generic A A M V s and proposed a static stabihty criterion f o r them i n a sim-ilar f o r m to that o f I r o d o v [ 2 1 ] . Irodov static stability cri-terion is f o r W I G s and states; the aerodynamic center i n height should be located upstream o f the aerodynamic center i n pitch. The criterion proposed by C o f l u is inde-pendent f r o m aerodynamic and hydrodynamic centers i n pitch and so seems to be unrealistic. Irodov stability

S 12 s

11

=• With End] = Without E jlatc >

i'

'—-=• With End] = Without E ndplat \ ' 1%

11

II

il

! "

: •-a !

!i

1 1 J 1 i 1 0.01 0.1 1 h/c

Fig. 2 M a x i m u m l i f t to drag ratio versus the height to cord ratio at trailing edge o f a w i n g w i t h aspect ratio of 1 [18]

460 J Mar Sci Teclinol (2016) 2 1 : 4 5 8 ^ 7 0

criterion may be used directly f o r A A M V s similarly as that in Matveev and Kornev [19] by using aero-hydrodynamic centers i n height and pitch. I t means derivatives o f sum o f aerodynamic and hydrodynamic l i f t force and pitching moment should be used i n calculation o f centers i n height and pitch.

Kornev et al. [22] developed a dynamical model f o r racing boats based on combination o f theories developed for planing boats and W I G crafts. They preformed a linear stability analysis and studied the influence o f different geometric and mass parameters on dynamic stability bor-ders. I n another related study, Gu et al. [23], along w i t h p e r f o r m i n g a C F D analysis on aerodynamic aspects and using a simple mathematical model, discussed the initial development o f pitch stabilizing systems f o r a 21-foot-long planing catamaran.

As another design challenge, A A M V s may o f f e r poor performances i n inidal acceleration mode because these vehicles are designed to operate at very high speeds where there is a very little draft and aerodynamic l i f t force con-tribution is sufficiendy increased. H i g h value o f the hydrodynamic drag i n acceleration phase, w h i c h manifests itself as the resistance hump, may stop speed-up phase and prevent i t f r o m reaching final cruising speed. Such a problem exists more seriously f o r W I G crafts w h i c h are designed to operate i n airborne conditions and should successfully pass the take-off phase and even f o r conven-tional planing crafts during transition f r o m displacement into planing mode. Besides design measures, f o r example, i n the power o f propulsion system to cope this challenge, craft attitude control mechanisms may be employed i n acceleration mode. Trimmable drives i n propulsion system, control surfaces such as t r i m tabs and approaches such as throttling the engine power are used c o m m o n l y i n planing vessels f o r this puipose. T r i m m i n g the drive angle may be the most c o m m o n and feasible solution. I n the case o f using such controlhng mechanisms f o r a racing boat, e f f i -cient and o p t i m u m control policies should be applied to cause the craft not only easily but also rapidly pass acceleration phase. I n this regard, the present article tries o f f e r a solution to this question as to how the thrust angle should be changed i n a trimmable drive system, i n order to craft pass the acceleration phase and reach a final speed i n the m i n i m u m possible time. This is a time-optimal control problem w i t h the drive angle as its control variable. Opti-mal control theory has been used previously f o r the m o t i o n control o f marine vessels i n the applications such as sea-keeping and maneuvering. Examples o f application o f optimal control f o r high-speed crafts can be f o u n d i n [24¬ 31]. I n the present study, authors use the theory introduced in [32] to solve above time-optimal control problem f o r an aerodynamically alleviated racing catamaran without and w i t h additional constraint regarding stability margin.

The first step i n dealing w i t h the problem is to define its plant or dynamic model on which the control process w i l l be applied. I n the present problem, this plant is the model of vertical plane m o t i o n o f the racing catamaran during f o r w a r d acceleration i n respect o f the time. I n order to develop this 3-degree-of-freedom ( D O F ) model, hydrody-namic, aerodynamic and propulsion system l i f t and drag forces and pitching moment should be calculated to be used in the equations o f motion. These equations then are written i n a state-space f o r m and comprise the dynamic model of the problem. For this puipose, empirical and theoretical methods are employed f o r calculating the hydrodynamic and aerodynamic forces and moments. For the propulsion system, we use a convenient model w h i c h includes the operational effect of the propeller, drive and engine. B y developing such dynamic model we w i l l be able to model the craft m o t i o n during acceleration time f o r any defined variations i n drive angle as input. I n the present article after introducing this dynamic model its results f o r a racing boat i n an uncontrolled condition are presented. Then solution algorithm o f the time-optimal control prob-lem is described according to an " i n d i r e c t " method and applied f o r obtaining the optimal drive angle signal f o r this boat. The paper closes w i t h some conclusions and recommendations.

2 D y n a m i c m o d e l

As mentioned, dynamic model w i l l be based on empirical and theoretical methods f o r calculating the forces and moments. Most o f these methods are based o n the assumption o f constant f o r w a r d velocity o f vehicle or steady-state condition. Thus, although the model is time dependent, we neglect the unsteady effects o f the craft f o r w a r d acceleration on forces and moments. Another part of the unsteady effects is related to the aero-hydrodynamic forces and moments associated w i t h craft unsteady surge, heave and pitch motions. These terms, conventionally, are considered by added mass and damping coefficients i n equations o f motions. These coefficients have been used i n linear stability and seakeeping analysis o f planing crafts b y many authors. See f o r example [16, 33, 37]. Katayama and Ikeda [34] used these coefficients to carry out the time-domain simulation o f a personal watercraft i n f o r w a r d accelerated m o t i o n f r o m rest. They calculated hydrody-namic coefficients f r o m their measured data or theoretical methods based on a potential theory and f o u n d that the added mass forces contribution are small enough to be ignored. The calculation o f these coefficients f o r A A M V s w h i c h should be performed by taking into account both hydrodynamic and aerodynamic flows is beyond the a i m and scope o f this w o r k . I n the present study, equations o f

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motion are solved in a quasi-steady manner by neglecting tliese unsteady effects. B y quasi-steady, we mean craft speed changes f r o m time step to time step as a result o f f o r w a r d acceleration, but steady-state attitude at corre-sponding speed is used at each time step.

2.1 Hydrodynamic force and moment model

Using xyz coordinate system w i t h its origin fixed on c a l m water level ( F i g . 3), x^it) and , r 5 ( / ) are defined as time-dependent vertical position o f C G and t r i m angle o f h u l l , respectively. Then FIH and F^H, the hydrodynamic forces in X and z direction and F j / / , hydrodynamic pitch moment about center o f gravity (CG), w i l l be a f u n c t i o n o f c r a f t f o r w a r d speed ii(t), Xsit) and xsQ).

FiH =fiH{u,X3,Xi) / = 1,3,5 ( 1 )

For prismatic planing h u l l , Savitsky [35] gives empirical formulas w h i c h can be used to calculate l i f t force F^H and the longitudinal center o f pressure Lp (see F i g . 3). Savitsky method p r i m a r i l y has been introduced f o r monohulls but i t can be used f o r catamaran configuration by some m o d i f i -cations. L i u and Wang [36] introduced interference coef-ficient Ai i n Savitsky's equations f o r l i f t and center o f pressure f o r this purpose. For t w o l i m i t cases (i.e. f u l l interaction and no interaction between demihulls), A,-equals 1 and \ / 2 , respectively. A t zero spacing (i.e. A,- = 1) two asymmetric demihulls f o r m one symmetric h u l l and u n m o d i f i e d equations are acquired. The interference coef-ficient value between t w o above l i m i t cases depends highly upon the h u l l separation distance, speed and wetted length of the vessel. Savistky equations f o r Cto (the l i f t coefficient f o r zero deadrise) and Lp after introducing the interference coefficient A,- w i l l be modified as f o l l o w s :

CiO = (I8O.V5/71)(0.012;. "'YA,- + 0.0055A,-A2-Y(c2r)) (2) -1 •

Lp = bXl 0.75 5.21C,2r/(A,;0 -F2.39 ₍₃₎

where x^ is i n radian, b the effective beam (summation o f two demihulls' beam), X mean wetted length-beam ratio, /• separation ratio (the ratio o f b to overall beam o f the catamaran B) and C,, ul{gBf'^ speed coefficient (beam Froude number). I t should be noted that I and C„ are defined here based on overall beam. N o w , Fj^ can be calculated as f o l l o w s :

F^H = O.SCtppii^B'-. (4)

where Cip is l i f t coefficient f o r deadrise angle o f [i i n degrees and is calculated as f o l l o w s :

CLU = Q o - 0.0065^ C; .0.6 _{10 '} (5) where 1 is expressed w i t h respect to keel and chine wetted lengths Lf: and Lc according to E q . 6. L^ and Lc f o r pris-matic hull can be approximated by Eqs. 7 - 9 w i t h respect to

X3 and Xs [ 3 7 ] : X = 0.5{LK+LC)/B LK = LCG + VCG/tanA-5 -l-xs/sinxs LC=LK- 0.50 t a n / J / [ X 5 ( 1 + 2 n , a x / V , ) ] Z m a x / V , = - 2 X lO"^/?^ - 7 X IQ-^ - 10'^p • 0.575 (6) (7) (8) 5 (9) In Eq. 7 LCG and VCG are the longitudinal and vertical position o f C G , respectively, f r o m transom stern and keel Hne as shown i n F i g . 3. I n Eq. 8 ZmaJV, is a parameter

dependent on deadrise angle according to E q . 9 w h i c h has

Fig. 3 The used coordinate system, hydrodynamic forces on hull and some other geometric definitions

A

FfiCosx5=Fi,f{+ Fysinxs

Ffl sin,T5 = ( F 3 H+Fr s i n x j )tan X5

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been obtained by curve f i t t i n g on results o f a similarity solution presented i n [ 3 7 ] . Equations 6-9 are used f o r racing catamarans i n this study by neglecting interference and asymmetry effects on wetted area pattern. B y esti-mating the viscous drag, Fy, by an empirical f o r m u l a such as I T T C - 5 7 [37], FIH and F^H can be obtained according to Fig. 3 as f o l l o w s , hy is the moment arm o f Fy about C G .

F\H = -[Fv cos A-5 - I - {F-iH + Fy siuAs) tanA-5] (10) FsH = {Fm + Fv sin A's)/cos A-5 x {Lp - LCG) - Fyhy

( 1 1 )

Even though these formulas are v a l i d f o r extensive range o f beam Froude number, they can be used only when the bow is out o f water. The bow i m m e r s i o n may occur at l o w Froude numbers as a result o f l o w t r i m angle. W e assume that h u l l length is enough to prevent bow immersion.

2.2 Aerodynamic force and moment model

A l l o f the vessels experience aerodynamic force because o f the air flow on their unwetted parts. For the tunnel hull boats there is an additive portion due to the air flow w i t h i n the tunnel under the h u l l . These t w o sources o f aerody-namic force are treated i n d i f f e r e n t manners and are cal-culated separately i n the present study.

Force and moments due to the out o f water parts o f sponsons and i n general, upper structure, are estimated by convendonal empirical aerodynamic coefficients according to the craft's instantaneous attitude. On the other hand, force and moments due to the flow w i t h i n the tunnel are calculated by I D mathematical model o f w i n g w i t h end-plates i n the extreme ground e f f e c t introduced by Rozhdestvensky. For a w i n g w i t h endplates i n a close p r o x i m i t y o f surface when there is a small clearances under the tips o f the endplates, the flow nature is dominantly one-dimensional [ 1 5 ] . Simple nonlinear one-one-dimensional models has been used previously f o r example by Gafling-ton et al. [38] f o r steady flow under a w i n g w i t h endplates or by T u c k [39] f o r unsteady flow under 2 D w i n g section i n the extreme ground effect. Gallington et al. [38] considered the effect o f leakage o f flow under the tips o f the endplates

but chordwise variations o f flow velocity (and so pressure) under the w i n g have not been taken into account. Rozhdestvensky [15], later, extended this model, by taking into account the chordwise variations and inserting unsteady effects. He also, by i n v o l v i n g the v e l o c i t y c o m -ponent due to the laterally leaking flow, represented an analogous 2 D model f o r the problem. These I D or 2 D models have been used frequendy by matveev et al. (see for example [14, 17, 40]) f o r different types o f A A M V s . 2D model is suggested i n cases w i t h significant side leak-age. The authors o f the present study utilized b o t h I D and 2D steady models f o r some cases including typical range o f operation o f tunnel boats and i n spite o f significantly more computational e f f o r t o f 2 D model, observed accept-able coincidence f o r l i f t and drag coefficients ( w i t h error approximately 10 % i n average) between results.

Similar to Matveev and Kornev [19] i n c a l m water we consider a flat tunnel ceiling and an undisturbed water surface. Effects o f deformable water were recognized to be small f o r wings w i t h no augmented airflow b y Matveev [40].

The coordinate axis and geometry is depicted in F i g . 4. The governing one-dimensional continuity equation f o r steady inviscid and incompressible airflow w i l l be as f o l l o w s :

; f [ / . . ( x ) « . ( A ) ] - H ^ s i g n ( t / , ^ - . ? ( A ) )

<i-^' (12) X .^Uj - 11^{x) = 0, Us{0) = -Us

where ii^ is local spanwise-averaged velocity and the span. Sign f u n c t i o n has been added to ensure correct flow direction i n gap under endplates. Span-wise pressure dis-tribution according to B e r n o u l l i equation w i l l be as f o l l o w i n g :

Ps{x) = ^^P„x[U^-i<M (13)

Subsequently, aerodynamic l i f t , drag and p i t c h moment due to the air flow i n the tunnel can be calculated b y integration o f pressure distribution along x and adding viscous and leading edge suction effects according to [ 1 5 ] .

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Fig. 5 Wetted lengths on flat plate

Total aerodynamic drag, l i f t and pitching moment about CG denoted, respecdvely, by FIA, F^^ and F j ^ w i l l be determined by adding upper structure and tunnel flow portions. Aerodynamic model i n general can be represented by f o l l o w i n g functions:

FiA =fiAill,X3,X5) / • = 1,3,5 (14)

It has to be noted when trailing edge is below the undis-turbed water level, as a result o f airflow blockage i n tunnel.

Us = 0 and stagnation pressure w i l l be exerted under the

dry part o f the w i n g . For wetted part, wetted length-beam rado can be calculated f r o m Eq. 15 [ 3 5 ] . Ave and Xwi are defined i n F i g . 5. Hydrodynamic force on flat plate (i.e. /? = 0) can be calculated according to the same relations w h i c h were used i n Sect. 2 . 1 :

, r i . 6 1 , , 1 - 0 . 3 4 , ( 0 < i „ i < l ) / , 5 ^ U „ a + 0 . 3 ( l < A i n < 4 ) ^ ^

2.3 F o r c e and moment model for propulsion system

Considered model f o r calculating the force and moment exerted by the propulsion system on h u l l is same as that used by the present authors i n [ 3 2 ] . The model gives pro-peller thrust by taking into account the propro-peller-engine operational interaction i n a quasi-steady manner and also most o f the concerned peiformance factors. A c c o r d i n g to this model, propeller r p m and thrust change during the c r a f t f o r w a r d acceleration. A t n m m a b l e sterndrive is used as drive system and hydrodynamic drag and l i f t on its immersed part are considered.

This model finafly can be represented by f o l l o w i n g functions ĥ/>:

NpFip =fip{ii,X3,X5,(Pp) r = 1,3,5, (16)

where Np is the number o f engine-drive sets used i n propulsion system. Fip and Fi,p are horizontal and vertical forces due to the propulsion system on h u l l and F^p is their moments about C G o f vessel, (pp which w i f l be considered as control variable i n next sections is the propeller shaft angle relative to keel line, positive when propeller goes d o w n .

2.4 Equations of motions and the solution method in time domain

As mentioned, h u l l heave and pitch velocity and acceleration effects on fluid forces are neglected here. I n the other words, added mass and damping coefficients w i l l not exist i n equations. Only added mass i n advance direction o f craft due to fhe acceleration i n this direction, i.e. w i l l be used i n equations o f motion. As mentioned before, h u l l m o t i o n w i l l be considered accelerated only i n f o r w a r d direction so f o r vertical forces and pitch moments only e q u i l i b r i u m condi-tion w i l l be considered at each time step. Such assumpcondi-tion is true i n c a l m water because o f small rate o f change i n craft attitude i n these directions relative to change i n x-coordinate. The governing equations i n respect to introduced f o r c e models are as Eqs. 17-19. These equations should be solved simultaneously f o r 3 unknowns, ii, x^ and ^ 5 . (pp is k n o w n or w i l l be determined f r o m o p t i m a l control law.

{in + an)ii = a ' s . A s ) +fiAiii,X3,X5)

+ flp{ll,X3,X5,(pp)

= fi{ii,X3,xs,(pp) (17)

0 =f3H{u,X3,X5) +f3A(ll,X3,X5) +f3p{ll,X3,X5,(pp) - mg

= f3{u,X3,X5,(pp)

(18) 0 =f5Hiu,X3,X5) +f5Aill,X3,X5) + fip{u,X3,Xf,, (pp)

= fs{ll,X3,X5,(pp) (19)

I n order to solve these equations i n each time step we use simply a f o r w a r d finite difference f o r m o f Eq. 17 as

i>n + an)'-^^^^=AH{'r,xrA^')

+frAiir,xr,xr)

+f,p{u'+\x'+\x'+\(pp) (20) I n this equation superscripts indicate time step number and

«' is k n o w n f r o m i n i t i a l condition or previous t i m e step. Replacing x'3'^ and xs^' (unknowns at present time step) i n Eqs. 18 and 19 beside Eq. 20, we reach a system o f three nonlinear (algebraic) equations w h i c h can be solved by a numerical routine. Such procedure w i l l be continued at next t i m e step.

2.5 Modeling the acceleration phase

of an aerodynamically alleviated c a t a m a r a n without control

I n this section results o f introduced mathematical model are presented f o r a racing boat w i t h specifications very similar to K U D U I I [ 5 ] . The main characteristics o f the c r a f t are presented i n Table 1. Sponsons have been

464 J Mar Sci Technol (2016) 21:458-470

Table 1 Main characteristics of K U D U I I craft [5]

Sum o f demihulls' beam 2 X 0.96 m

Overall length 10.36 m

LCG 2.29 m

V C G 0.85 m

Mass 5361.6 kg

Deadrise angle 13°

Cord o f wing section 7.77 m

Span o f wing section 2.27 m

ho (Fig. 4) 0.32 m

a (Fig. 4) 5°

considered as single-deadrise prismatic Irulls w i t l i neglec-ted inteiference effect. Before modeling the acceleration phase some comparisons between ward et al.'s results [5] and present model f o r attitude of the vessel i n steady-state condition have been presented i n Table 2. Measurements have been performed during fuU-scale trials i n three con-stant speeds. The c r a f t attitude has been calculated by present model o f hydrodynamic and aerodynamic forces using the k n o w n propulsion forces at these speeds. W h i l e general agreement between calculated and measured values is encouraging, significant discrepancy i n some points can be observed w h i c h is not so unexpected due to the sim-plifications i n model. It has to be noted that part o f the discrepancies are attributed to measurement errors as a result o f difficulties i n record o f t r i m angle and keel wetted length as mentioned by ward et al. [ 5 ] . However, there is a very little published data on performance o f these cata-marans and p e r f o r m i n g more accurate comparisons needs further experimental data.

I n this part we use the introduced dynamic model to sim-ulate the acceleration phase o f this racing boat. Solution algorithm i n time domain is according to the Sect. 2.4 and time step is 0.05 s w h i c h is sufficiently small to prevent f r o m time-step-dependent results. A t this stage, no control is per-f o r m e d on drive angle and it is per-fixed on - 9 . 5 ° . The same angle has been used f o r steady-state measurements by ward et al. [5]. A more p o w e r f u l propulsion system i n comparison to that f o r K U D U I I is considered i n order that the craft be able to reach speeds more than 100 k n . A couple o f sterndrive- engine systems, each w i t h m a x i m u m power of 1100 hp and the gearbox reduction ratio o f 1.25, are used i n propulsion system. The engine f u l l throttle power versus its i p m and open water performance curves o f propeller including thrust coefficient KT, torque coefficient Kg and efficiency ;/ versus advance ratio J are shown i n F i g . 6.

I n F i g . 7a craft speed variation has been plotted during i n i t i a l 30 s o f acceleration phase. A f t e r 22.6 s speed reaches 110 k n and finally steady-state condition has been achieved.

Table 2 Comparisons between the results of the of Ref. [5] and present model f o r K U D U I I craft

Speed (knot) Ref. T r i m angle do

61 Present 4.1 2.5 0.14 Ref. [5] 3.7 3.41 0.1 69 Present 3.9 2.2 0.17 i _{Ref. [5] 4.6 2.13} 0.15 78 Present 4.2 1.7 0.19 Ref. [5] 5.1 1.37 0.2 \ ^ • lOKQ e O O 0 O Eff. ( c ' ' O

•

{ 1 r 0.5 1 1.5 2

J

(b) 900 - 1 •• 1 1 800 -• i I - - - - ¬ 700 T ; ~ 600 -! i ' ^ ~ ~ [ 500 -i - Z f i ^ f \ I I 400 T ^1 r 300 ---^„0^\ 1 1-200 - T ' 1 100

\ 1 1

0-J i 1 i 2000 3000 4000 5000 6000 R P M

Fig. 6 Open-water performance curves o f the used propeller and f u l l thi-ottle power generated by engine at each rpm

I n Fig. 7b the height o f the trailing edge o f the w i n g and craft C G f r o m undisturbed water level have been plotted against the time. Negative values f o r trailing edge height u n t i l f = 3.6 s mean i t is below the undisturbed water level and the stagnation occurs f o r air flow i n the tunnel. W e c a l l this situation as wetted-wing condition. I n order that traibng edge can emerge f r o m water, a 0.5-m rise o f C G is needed w h i c h takes place after speed o f 40 k n as a result o f craft planing. As mentioned before hydrodynamic l i f t is imposed on wetted part o f the w i n g .

Variations o f the w i n g wetted length, besides hull keel and chine wetted lengths, have been shown i n F i g . 7c. B y

J Mar Sci Teclinol (2016) 2 1 : 4 5 8 ^ 7 0 465 (b) 10 15 20 time(s) 25 30 ( C ) i ' ' K e e l Wetted Length ' ' K e e l Wetted Length

I A i = " Chine Wetted Length _{• • • W i n e Wetted Lenath}

1 S 1 i > 1 ^ A l : • i\ ' ^ i L l • ' ^ \ '. '"^; ^>». — — '. '"^; ^>». — — \ J j ; ^ (e) 0.; 10 15 20 time (s) 25 30 © 1

-<

0.2 0.1 0 10 15 20 time (s) 25 3D 1,2 1 0.8 0.6 ' 0.4 0.2 0 - 0.2 - 0.4 - 0.6 0

= " Trailing Edge Heiglit — — C G Height 15 20 time (s)

id)

<^ 'ail a < g •C 10 15 20 time (s) 25 (f) 15 20 time (s) 25 \ \ \ \

4

\

. 1 .

\ / \

\---Total Resistance — — Acceleration \ \ \ '• \ \ S 30 - 7 • 6 • 5 - 4 • 3 - 2 1 0 30

Fig. 7 Variations o f some performance characteristics o f the racing boat during forward acceleration time without the control of drive angle

Starting the channel flow under the tunnel, as a result o f different pressure distribution pattern under the w i n g rel-ative to the stagnation condition, the slope o f decreasing trend i n keel and chine wetted lengths changes signifi-cantly. D r y chine occurs after / — 6.75 s or u — 75 k n . Keel wetted length retains the decreasing trend u n t i l reaching final steady-state value o f 0.62 m . It has to be noticed that although the Savitsky's empirical relations are k n o w n to be used f o r wetted chine planing, they are used here also f o r dry chine condition by taking into account the change i n wetted beam o f planing surfaces. A c c o r d i n g to Payne analysis [41] such assumption is not so unrealistic f o r prismatic huUs w i t h deadrise angles between 10° and 15° i n t r i m angles between 4 ° and 10°.

Boat t r i m angle variations during acceleration are shown in F i g . 7d. T r i m angle experiences a hump and h o l l o w i n

wetted-wing condition. I t experiences second m a x i m u m and m i n i m u m , respectively, where w i n g t r a i l i n g edge clears the water and dry-chine condition occurs. Then i t foUows an increasing trend until reaching steady-state value.

The ratio o f aerodynamic l i f t to craft weigh during acceleration has been shown i n F i g . 7e. A t the final steady-state condition (ii = 110.5 k n ) aerodynamic l i f t contribu-tion reaches about 43 % o f weight. The remaining part o f weight is supported by sum o f the hydrodynamic l i f t and the vertical component o f the thrust.

Resistance and horizontal component o f the thrust dur-ing acceleration time have been plotted i n F i g . 7 f . I n this figure f o r w a r d acceleration also has been shown w i t h secondary vertical axis at the right-hand side. A t the beginning of the m o t i o n , the difference between thrust and

466 J Mar Sci Technol (2016) 21:458-470

resistance is significantly high and reladvely high values o f acceleration w i t h a m a x i m u m o f 6.5 m/s^ can be observed. The local m a x i m u m i n resistance at initial seconds is the convendonal resistance hump i n planing crafts which appears markedly small beside final resistance that resulted f r o m the high attainable speeds.

3 T i m e - o p t i m a l c o n t r o l p r o b l e m

As mentioned i n the introduction, the aim o f this study is to solve the optimal control problem o f m i n i m i z i n g the dme needed to reach a final speed f o r a racing catamaran. I n time-optimal control problems, the m a i n objective is find-ing control signals as input f o r a plant i n such a way that t i m e needed to go f r o m an initial to a final state be m i n i -m u -m . Also at the sa-me d-me, so-me constraints -must be satisfied. Constraints may be on control or state variables. I n the present problem, physical constraint on control variable, i.e. drive angle, is (pp^min < Vpi'^) — Vp.mm- Plant of the problem, which was expressed eventually by Eqs. 17-19, can be represented i n a more compact f o r m as:

[u{t) 0 0]^ =f{X{t),(pp) = \/{m + ax^WM.X3,Xi,q)p)

h{u,X-i,Xi,<Pp) fs{u,Xn,,X5, (pp)^

= \fl {ll,X3, Xs, (pp) ƒ 3 ( M , A-3, A-5, (pp)

fs{ll,X3,Xs,(Pp)]^ (21)

where superscript T stands f o r transpose, X ( 0 = [u(t)

A-3(f) xsit)f is the state vector, and X(t) and (ppit) are unknowns that should be determined i n time interval to to t f . The above equations construct a differential-algebraic system o f equations.

3.1 Solution algorithm for time-optimal control problem

T o solve the abovementioned problem different numerical approaches may be used i n the optimal control theory. W e use i n present study an indirect method w h i c h is based on M i n i m u m Principle. A l t h o u g h f r o m mathematical view-point this principle offers the necessary condition and on its o w n does not guarantee the globality o f the optimal solu-tion, it has been used since decades, i n solving many dif-ferent problems i n engineering applications.

I n indirect methods the problem is converted to a boundary value problem w i t h state and costate equations as its governing equations. State equations are indeed the plant equations and costate equations are added to find costate or adjoint variables.

Hamiltonian function f o r this problem is defined as f o l l o w s :

H{X, X,(pp) = l+ J2 ¥i(u,X3,X5,<Pp), '=1,3,5

(22)

where I = [li A3 Xsf is the vector o f costate variables. According to M i n i m u m Principle, as a necessary con-dition f o r (p*p, it should be obtained i n such a way that above H a m i l t o n i a n (when considered as a f u n c t i o n o f cpp) b c m i n i m i z e d f o r physically admissible values o f (pp so at each time we have

^ A ; / ; ( « * , A - ; , A - ; , r p ; ) < ^ ; . ; ƒ • ( « * ,A - ; , A - ; , rp^) (23)

'•=1,3,5 (=1,3,5

Star sign * denotes the optimal values o f variables.The state equations are obtained according to the dynamic model equations on the optimal trajectory as f o l l o w s :

f,{it*,x*,xi,(p*p)-ir =0 (24)

Mu*,xl,x*s,cp'p)=0 (25)

fsiit*,x*„x*s,cp;)=0 (26)

The costate equations w h i c h are obtained f r o m H a m i l t o -nian derivative w i t h respect to state variables on optimal trajectory, w i l l be i n the f o l l o w i n g f o r m : J2 KMit*,xl,x*s,(p*p) + X\=0 '•=1,3,5 i*fi,,{i,*,x;,x*s,cp;) = Q 1=1,3,5 ^ X%,{u*,xl,x*s,tp*p)=0 '•=1,3,5 W i t h Initial condition at t

It* Xj X*sY=^ [llo Xo3 •

fo as i T (Pp\, = <PPO (27) (28) (29) (30)

A t t = t f , craft should reach final speed and H a m i l t o n i a n must vanish as final condition:

1 + '•=1,3,5 X*Mit*,xl,x*,(p*p) 1 + r « * = 0, II \, = u„ i + xiMtt*,x;,x*s,cp*p) (31)

I n Eq. 27 fi„[ii*, xl,x*s, (p*p) is the partial derivative o f f u n c t i o n / i w i t h respect to u at the point (ii*,xl,x*s, (p*p). /ivs and fixs i n Eqs. 28 and 29 are also derivatives w i t h respect to X3 and A5 at this point. Above boundary value problem is solved here by a single shooting method. A t first time step, solution algorithm starts w i t h i n i t i a l con-dition and an i n i t i a l estimation f o r A j . Then A3 and A j are determined f r o m the solution o f Eqs. 28 and 29 w h i c h are linear algebraic when we use initial values. H a v i n g adjoint variables we can f o r m Hamiltonian as Eq. 22 and solve numerically, relevant constrained minimization problem to find (p*p at this time step. N o w we can solve state equations

J Mar Sci Teclinol (2016) 2 1 : 4 5 8 ^ 7 0 467

F i g . 8 The numerical solution algorithm of the time-optimal problem 'Costate Equations: =1,3,5 =1,3,5 '=1,3,5 A ^ O p t i m i z a t i o n P r o b l e m Minimize: / / ( X , ) . , ^ p ) = l + =1 ,3,5 Dynamic M o d e l (Pp State Equations: X 3 , X 5 , ^ ? ^ ) - Z / * = 0 y ^ ( i / * , x 3 , x 5 , ^ p ; ) = o

to update ( P j X j j X j . Finally, E q . 27 w i t h updated values is used to f i n d ?*•^ f o r next dme step. This procedure should be repeated at each time step u n t i l reaching the final condition, i.e. u* = Hma,x- A t this point i f the other f i n a l condidon o f E q . 31 was not satisfied by a defined error this numerical procedure should be repeated entirely, w i t h a new i n i t i a l guess (shooting) f o r X\. The numerical solution algorithm has been illustrated i n F i g . 8. A t the end o f the solution procedure, the final time t f , optimal

control signals (p*p{t) and optimum trajectory i n time domain, i.e. u*{t),xl{t) a n d x 5 ( ? ) are determined.

3.2 Solution of time-optimal control problem for K U D U I I without state constraint

I n Sect. 2.5 results o f the mathemadcal model f o r accel-eration phase i n a fixed tpp were presented. I n this secdon we apply the optimal control algorithm to find the (pp{t)

468 J Mar Sci Technol (2016) 21:458-470 ^\ \ /' I 1 ! i T -30 25 20 15 10 5 10 12 14 time (s)

F i g . 9 The optimal solution f o r drive angle and variations of the craft total resistance and trim angle during the control

w i t h w h i c h the same craft reaches the final speed of 110 k n i n the m i n i m u m possible time. Physical constraint on (pp is — 10° < <PP < 5 ° and at this stage no constraints on state variables are considered.

A s can be observed, f o r example, f r o m F i g . 7d, the state variables such as t r i m angle are not differentiable at points where the w i n g trailing edge clears the water and chines become dry. Thus to be able o f using above mentioned optimal control algorithm, according to the optimality principle, problem can be broken into three sub-problems i n w h i c h state variables are continuously differentiable. Indeed, the same procedure as F i g . 8 should be utilized f o r each sub-problem. However tbe boundary condidons f o r these sub-problems must be matched correcdy. I t means the i n i d a l condition of second sub-problem should be set same as the final state o f first sub-problem and so on f o r third sub-problem.

I n Fig. 9 the solution f o r (p), it) has been shown. I n order to facilitate tracing the dependencies on control variable, variations o f the craft total resistance and t r i m angle also are shown i n this figure. The resistance value should be read f r o m the axis at the right-hand side. The solution intervals f o r sub-problems have been indicated by vertical Unes. According to the obtained optimal solution, to m i n i m i z e acceleration time, drive angle should be adjusted at upper l i m i t o f - F 5 ° since r = 1.6 s (where t r i m angle experiences first local m i n i m u m ) then be lowered as i n F i g . 9 since reaching final value of about - 3 ° . A t the boundaries o f the separated intervals f o r sub-problems slight jumps (about 0.15° and 0.25°) can be observed as a resuk o f discontinuity i n derivatives. B y optimally controlling the drive angle, the boat reaches the final speed o f 110 k n i n 14 s. A t this speed t r i m angle and resistance are, respectively, 4 . 8 ° and 25 k N .

3.3 Solution of time-optimal control problem with additional constraint regarding stability

As mentioned i n the introduction, preserving the stability is one o f the main concerns i n design o f aerodynamically

time (s)

F i g . 10 The optimal solution f o r drive angle and variations o f the craft total resistance and trim angle during the control, w i t h additional constraint regarding stability

alleviated racing catamarans. Instability can be avoided by appropriate control strategies on craft t r i m angle. B y suf-ficiently l o w e r i n g the bow, craft skipper can prevent f r o m undesired pitch instability. Kornev et al. [22] using a hnear stabihty analysis f o r a racing catamaran presented stability border curves indicating the critical t r i m angles at w h i c h instability starts f o r each craft speed. B y such a manner, stable regions can be determined w i t h respect to speed and t r i m angle. I n other words, stability condition can be rep-resented by an inequality relation between u and xs- Such an inequality can be considered as a state constraint i n optimal control problem to avoid unstable planing. Con-cerning this, i n present section, the above t i m e - o p t i m a l problem is solved by additional constraint regarding sta-b i l i t y margin on state variasta-bles u and X5. Calculating the stability margins is out o f scope o f the present study so we assume i n this article a representative stability margin similar i n trend to those obtained by Kornev et al. [ 2 2 ] . There are d i f f e r e n t approaches such as the penalty f u n c t i o n method f o r solving the optimal control p r o b l e m w i t h state constraints. I n this method, inequality constraint is con-verted to equality constraint and the problem is solved by augmenting the H a m i l t o n i a n w i t h convenient additional term including this constraint.

I n F i g . 10 the obtained optimal solution f o r drive angle after implementing the additional constraint regarding stability has been shown along w i t h the craft t r i m angle and total resistance. Referring to F i g . 9 we can see that addi-tional constraint on state variables is active just after ? = 7.7 s where unconstrained t r i m angle curve violates the stability threshold. As can be observed f r o m F i g . 9, after this time, <ip*p stops the decreasing trend and starts to increase u n t i l reaching near the upper physical l i m i t o f -t-5°. This increasing trend o f thrust angle generates neg-ative pitch moment w h i c h tends to push d o w n the b o w . Optimal solution f o r drive angle is that w h i c h enforces t r i m angle to f o l l o w the predefined stabihty border curve. B y such trajectory f o r craft attitude, i t never reaches final

J Mar Sci Teclinol (2016) 21:458-470 ₄₆₉

Speed o f 110 k n because o f higher total resistance. So a slightly lower value equal to 108 kn has been considered as final condition. The craft reaches this final speed at 16.8 s and at this time, resistance value is 29.3 k N .

4 C o n c l u s i o n s a n d r e c o m m e n d a t i o n s

Aerodynamically assisted catamarans are kinds o f A A M V s w h i c h fill the gap between conventional planing and W I G crafts. The review performed i n this article on the pub-lished studies on these crafts indicates that there is a sig-nificant lack o f dedicated investigations and experimental data i n comparison to conventional planing and W I G crafts. A m o n g design challenges f o r these marine vehicles, their instability and poor performance during acceleration phase were emphasized i n this article. Beside primary design measures, controlling strategies were proposed as convenient solutions to counter these concerns. I n this regard, an optimal control problem f o r acceleration phase o f these crafts was intended i n which we seek f o r the optimal drive angle signal to m i n i m i z e the craft accelera-d o n time.

Results were presented i n uncontrolled and controlled conditions. Uncontrolled case was preformed w i t h fixed drive angle and simulation was continued until reaching steady-state condition. About 43 % o f weight o f the craft is supported by aerodynamic l i f t at final steady-state speed o f about 110 kn. The analysis prefoimed on relevant variables during the acceleration revealed a different behavior before and after when trailing edge o f the w i n g section clears the water.

The problem i n controlled condition was solved without and w i t h a predefined constraint regarding stability margin. Optimally controlling the drive angle without stabihty constraint during the acceleration resulted i n 40 % reduc-tion i n time required to reach the speed o f 110 k n and 14 % reduction i n resistance at this speed i n comparison to uncontrolled case w i t h fixed drive angle o f —9.5°. The addition o f the constraint regarding stability changes opti-m a l solution f o r drive angle i n such opti-manner that causes craft t r i m angle f o l l o w the stability border curve while the unconstraint problem violates the stability condition. This results i n higher resistances and slighdy lower final speed. A p p l y i n g this additional constraint increases the time required to reach final speed o f 108 k n about 33 % and resistance at this speed about 21 %.

As development o f work i n the future, more exact models f o r calculating hydrodynamic and aerodynamic force and moment may be used. Supplying more experi-mental data is necessary f o r validation purposes i n these models.

A more accurate dynamic model may be used by including aero-hydrodynamic added mass and damping

coefficients f o r solving the time-optimal problem i n rough water and more realistic sea condition i n further develop-ments. Optimal problem can be solved f o r other marine vehicles such as W I G crafts before take-off phase and w i t h other controlling mechanisms. This study can be consid-ered as i n i t i a l steps f o r designing and f u n c t i o n i n g the automatic close loop controllers f o r marine vehicles, w i t h the 'aim o f keeping the best performance at the same time preventing instability risks.

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