SES dynamics in the vertical plane

i

Introduction

Surface Effect Ships (SES) are known for offering a high quality iide in heavy sea states compared to conventional catamarans. However, in low and moderate sea states there are still comfort problems due to high frequency vertical accelerations induced by resonances in the pressurized air cushion. A high performance ride control system is required ou SES to achieve high quality passenger comfort and crew workability according to the ISO 2631 standard for human comfort and workability. In the development of a ride control system it is essential to use a siniplified and rational dynamic model.

Previous ride control systems have been based on the coupled equations of motion in heave and pitch as derived in Kaplan and Davis (197, 1978) and Kaplan et al.(1981). Their work was based on the assumption that the major part of the wave induced loads from the sea was imparted to the craft as dynamic uniforiiì ah pressure acting on the wetdeck, while a minor part of the wave induced loads froni the sea was imparted to the craft as dynamic water pressure acting on the side-hulls.

Sørcn.sen et al. (199.2) have further developed the work by Kaplan and Davis (197, 1978) and Kaplan et aI. (1981) by including the effect of spatial pressure variations in the air cushion caused by the two lowest acoustic niodes. It was found that the acoustic resonances in the air cushion excited by incident sea waves can result in significant pitch accelerations. Unpublished full scale measurements of a 35 m SES equipped with a rigid panel at the stern show the presence of acoustic resonance modes. Furthermore, full scale measurements of a 35 m SES equipped with flexible bag indicate that the presence of the flexible stern bag will affect the behaviour of the response caused by the acoustic resonance modes. In some operation conditions the response will be amplified, and the resonance frequencies seem to appear at lower frequencies for SES equipped with flexible stern hag system. This is related to the dynamics of the flexible bag system. The internal dynamics of the flexible bag influences the air cushion by modifying the cushion volume and by changing the air flow characteristics at the stern region due to dynamic leakage through the bag booster fan and due to leakage under the bag. This is still an active area of research. In Sørensen et aI. (1992) a distributed model was derived from a boundary value problem formulation where the air flow was represented by a velocity potential subject to appropriate boundary conditions on the surfaces enclosing the air cushion volume. It was assumed that the air cushion was sealed by a rigid panel at the stern. A solution was

found using the Helmholtz equation in the air cushion region.

In this paper the mathematical model presented by Sørensen et aI. (1992) is extended to consider an infinite number of acoustic resonance modes in the longitudinal direction. This mathematical model is then used to design a new ride control system which provides active damping of both the dynamic uniform pressure and the acoustic resonances in the air cushion. Special attention is given to the fan system and the ride control system placement to achieve robust stability and high performance.

SES Dynamics in the Vertical Planel

Asgeir J. Sørensen, Sverre Steen and Odd M. Faltinsen

The Norwegian Instilute of Technology2

'This work was sponsored by the Ulstein Group and the Royal Norwegian Council for Scientific and Industrial Research (NTNF).

Division of Marine Hydrodynamics, N 7034 Trondlielni, Norway

Schiffstechnik Bd. 40 - 1993 I Ship Technology Research Vol. 40 - 1993 71

TECHSCflE _UNJVERSITT

Laboratorium voor

Scheepshydromh

Archief

Mekeiweg 2, 2628 CD DelIt

Zg

x,

Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993

Fig. 1. Surface Effect Ship (SES) - coordinate fiame

2

Mathematical modelling

A moving coordinate frame is defined so that the origin is located in the mean water plane below the centre of gravity with the .r-. y- and z0-axes oriented positive forwards, to the port, and upwards respectively, see Fig. 1. The equations of motion are formulated in this moving frame. Translation along the :-axis is called heave and is denoted ï73(t). The rotation angle

around the yg-axis is called pitch and is denoted i75(t). Heave is defined positive upwards. and pitch is defined positive with the bow down.

The mathematical niodel is derived similarly to that of Kaplan and Davis (1974, 1978) and Kaplan et al. (1981), but with the significant extension that the effect of the spatially varying pressure in the air cushion is included. In this paper we are mainly concerned about the high frequency vertical vibrations. In this frequency range the hydrodynamic loads on the side-hulls are of minor importance. Nevertheless, since the mathematical model is supposed to be valid also for lower frequencies, hydrodynamic loads on the side-hulls are included. Furthermore, infinite water depth is assumed.

The dynamic response of the craft advancing forward in regular head sea waves is studied. The waves are assumed to have small slope and circular frequency . The circular frequency

of encounter w is

w=wo+kU

(1)

where k = 2ir/.\ is the wave number, . is the wave length and U is the craft speed. The incident surface wave elevation ç(x, t) for regular head sea is defined as

(Zy, t) = Ça Sifl (Wet + kxg) (2)

where is the wave amplitude. The waves are assumed to pass through the air cushion undisturbed.

The beam b and the equilibrium height h0 of the air cushion are assumed to be much less than the length L. Hence, a one-dimensional ideal and compressible cushion air flow in the longitudinal directiomi is assumed. The total pressure variationspc(Xg, t), inside the air cushion can be represented by

Pc (xg, t) = Pa + Pu (t) + p

(x,t)

(3)

where Pa jS the atmospheric pressure, pu(t) is the uniform excess pressure and Pjp(xg, t) is the

spatially varying pressure. When neglecting seal dynaniics. aerodynamics and viscous effects, the external forces are given by the water pressure acting on the side-hulls and by the dynamic air cushion pressure p(t)+p3(xg,t) acting on the wetdeck. We will assume that the dynamic cushion pressure is excited by incoming sea wave disturbances. In the absence of waves, the stationary excess pressure in the air cushion is equal to the equilibrium excess pressure Po For simplicity, at the equilibrium condition we will consider a rectangular cushion of length L, with height h0 and beam b. The air cushion area is then assumed to be given by  = Lb. This approximation of the air cushion area is believed to have minor influence on the results presented in this paper.

2.1

The global continuity equation

For analysing the dynamic air cushion pressure it is convenient to define a temporary coor-dinate frame with the origin located at the geometrical centre of the air cushion water plane area with the axes defined as before except for the longitudinal transformation of the origin to the mean centre of pressure. This means that x = x + x, y = yg and z = Zg. where x is the longitudinal position of the centre of pressure relative to the centre of gravity. In this

coordinate frame the cushion length is reaching from x = -L/2 at the stern to x = L/2 at the bow. The rate of change of the mass of air inside the cushion is equal to the net mass flux into the cushion. The linearized global continuity equation for mass flow into and out of the cushion follows from

V L/2

Po (Qj

(t) -

Q0 (t)) =

f

(x, t) dx + Pcoc(t) (4)

L _-/--L/2

where Q (t) is the volumetric air flow into the air cusluon from the lift fan system,

_Q0(

t) is the volumetric air flow out of the air cushion due to leakage, V(t) is the cushion volume. Pa jS

the air density at the atmospheric pressure Pa, p(x. t) is the density of the air at the pressure

Pc(X, t), and PCO is the density of the air at the equilibrium pressure Pa + ¡Jo. V0 = Lbh0 is the

equilibrium air cushion volume. The basic thermodynamic variations in the air cushion are assumed to be adiabatic. From the adiabatic pressure-density relation it follows that

p(x,t) =Po

L

rpa + J) (t) + P3p (s,t)1

Pa+Po J

(5)

where y is the ratio of specific heat for the air.

We define the nondimensional uniform pressure variations t) and the nondimensional spatial pressure variations

¡3(x,t)

as

ì Pu (t) - Po P

(s, t) Ip (X,t) =

Po Po

(6)

If we linearize equation (5) around the craft equilibrium operating point

_{t(t) = 1i3(x,t) = 0}

and differentiate with respect to time t, we find an expression for the time derivative of p( s. t)

as

PcO

Pc(X.t) = (/2u(t) + 3p(x,t))

7(1 + Pa/PO)

The rate of change of the air cushion volume can be written as

(t) =

A(3(t)

x5(t))

-The last terni in (8) represents the wave volume pumping; it can be written as

L/2 _{sin (kL/2)}

'0(t)=b

_f

((X,t)dX=Ac(ae

COSWet.

JL/2 kL/2

The volumetric flow into the air cushion is given by a linearization of the fan characteristic curve

about the craft equilibrium operating point (Fig. 2). It is assumed that q fans with constant turning rate are feeding the cushion, where fan i is located a.t the longitudinal position XFi.

The fan characteristic curve can be represented by

i=1 UP

Po [(t)

+ p(xF,t)]) (10)

PC

Po

Fig. 2. The fan characteristic curve

i= i

and A' are the stern and bow, respectively, equilibrium leakage areas at x = x4p and x = XFP. Dynamic leakage areas under the side-hulls and the seals due to craft motion are assumed to be negligible in this analysis. This type of leakage is a hard nonlinearity and can be analysed using describing functions (Gelb and Vander Velde, 1968). Computer simulations done by Sø'rense'n et al. (1992) indicate that the dynamic leakage terms due to craft motion can be neglected for small amplitudes of sea wave disturbances and associated small amplitudes of heave and pitch motions as long as the sealing ability of the stern and bow seal is good. The

volumetric mass flow out of the air cushion is then represented by

Qo (t) = Qp (t) + QFP (t) + QRcs (t) (15)

where

(14)

74 Schiffstechnik Bd. 40 - 1993 I Ship Technology Research Vol. 40 - 1993

Q0' _Out

where Q is the equilibrium air flow rate of fan i when p,(t) = Pu and (°Q/ip)I0 is the corresponding linear fan slope about the craft equilibrium operating point Q0 and Po. The total equilibrium air flow rate into the air cushion is then

Q0= Qui.

i= i

The volumetric flow out of the air cushion is proportional to the leakage area

AL(t)=AO+A(t)

(12)

-IL(t) represents the total leakage area and is expressed as the sum of an equilibrium leakage area A0 and a variable leakage area 4(t). The equilibrium leakage area will be divided into leakage areas under the bow and stern region, or more precisely under the rear and bow seals, in addition to the mean operating values or bias of the leakage areas of the louver systems.

4RCS(t) is the controlled leakage area of the ride control system. It can be written as

r

ARCS (t) =

(A[ + A5(x3, t))

(13)

i=i

where r is the number of louvers. ACS(x3. t) is defined as the comnianded variable leakage area of louver i, which is located at the longitudinal position x = Xt. In Section 3 we will derive an output feedback controller based on pressure measurements from pressure sensors located at the longitudinal position x x. ACS is defined as the mean operating value or

bias of the leakage area of louver i. This means that the total equilibrium leakage area is

Q4p(t) =

2(Pu (t) + p(x4p.t))

(16)

r

QRcs (t) = c ₌ ( 4RCS

+.ARCS

Ui i ix3, t))

i=1

Here c, is the orifice coefficient varying between 0.61 and 1.0 depending on the local shape of the edges of the leakage areas, see Sullivan et cl. (1992,). c, = 0.61 is used in the numerical simulations.

Using Taylor expansion around Po and A0. the air flow out of the cushion Q0(t) can be linearized and written as

Q0(t) =

A0 _3p(xFp,t)+

4jtF(xAp,t)

(

E

ASJip(xLj t) +

1 i=I i

+A0(t) + A0 +

i=1 - Í,P1xL, t)) Pa (18)

A5(x3t)) c/.

(19)

Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993 75

At stationary equilibrium the air flow into and out of the cushion is equal to

Qo = CriAO2P0/pa . (20)

Using the above relations, we find the uniform pressure eq. (55) (later in the text). 2.2

Boundary value problem for spatial pressure variations

The effect of spatial pressure variations caused by the two lowest resonance modes in the air cushion was investigated by Sørensen et al. (1992) in the frequency domain using Helmholtz equation. In this paper we extend the mathematical model to consider an infinite number of acoustic niodes. Furthermore the effect of a distributed ride control system is included. It is modelled as a boundary value problem. A one-dimensional ideal and compressible air flow in the x-direction is assumed justified by the high length/beam ratio, the high length/height ratio, and by the frequency range of practical interest. The motion of the craft is assumed to be small: so the body boundary conditions and the free surface condition can be linearized. The spatially varying air flow dynamics is formulated in terms of potential theory. where the velocity potential is denoted th3(x.z,t). We will assume that steady state conditions have been obtained. This means that the time dependence is harmonic and can be written as exp(iwt), where i is the complex unit. Hence, heave and pitch motion and nondimensional uniform and spatially varying pressure are written as

3(t) = 3aEt, _{(t) =} (t)

= uaCt,

(x,t) =

(s) et

(21)

where T)3a and 175a are the heave and pitch motion relatively to the coordinate frame located in the water plane below the centre of gravity, and and (x) are the nondimensional uniform and spatially varying pressure amplitudes respectively. These aniplitudes are generally complex and depend on . Like in the ship motion problem (Salve.sen et al. 1970) it is convenient to

split the velocity potential th5(x, z, t) in the air cushion into

c3p(x,z,t) =

(

o(X,Z)ua+3(X,Z)'O3a

+ _(x,z)

AS e't

(22)

i =

where o(x,z), 3(x,z). 5(x,z), 7(x,z), .bs(x,z),...,_çb7+r(x,z) are the spatially varying

pressure velocity potentials due to the effect of dynamic uniform ?ressure, heave and pitch

QFP(t) = 2 (p (t) + P3p(XFP,t)) (17)

L/2 L/2 >

Za

Xec k? Q, F?

AU

>

h.,

where mj(x) for n = 0, 3, 5, 7, 8, 7 + r are the one-dimensional spatially varying pressure velocity potentials due to the effect of dynamic uniform pressure, heave and pitch motion, incident waves and the r control actions respectively.

It is possible to derive an analytical solution of the boundary value problem. Because the boundary conditions on the wetdeck, such as the louver area and the outflow area of the lift fan system, do not cover the whole beam, we have to integrate the boundary conditions in y direction. The nondimensional spatial pressure variation 1i,(z.t), can be found as

X. t

u3(x,t) =

. (26)

Po (It

76 Schiffstechnik Bd. 40 1993 / Ship Technology Research Vol. 40 1993

i ."

=

-J

h0 Ø3(x.z.t)dz. (24)

Integrating the Helmholtz eq. (23) in z direction for n = 0. 3, 5, 7, 8,..., 7 + r gives

(LIJe

û2(x)

b/2

(3b,(x.z)

dy

û,(x,z)

=0

(25)

tÉJ,, (x) + , ., + --- f

OX- (100 J-1,/2 \ z

Qo X QZ

Fig. 3. The boundary value problem

motion. the incident waves and the r control actions respectively. The subscript i must not be confused with the complex unit i. The velocity potential

5(z,z,t) for the air flow is

assumed to satisfy the time independent Helmholtz equation in the cushion region and the boundary conditions enclosing the cushion volume, see Fig. 3. The Helmholtz equation is for

n=0,3,5,7,8...7+rgivenby

(e

2

82, (z

) (z, z)

,, (z, z) +

3x2 + az2 = o (23)

where c is the speed of sound in air. Implicitly (x,z) is a function of We.

In the following a simplified mathematical model will be proposed. The modal analysis will begin with introducing a one-dimensional approximation of the velocity potential for the air flow in the longitudinal direction. The air cushion velocity potential will be represented by a sum of an infinite number of acoustic modes. where each mode consists of a mode shape function and a frequency dependent modal amplitude function. The mode shape function will

be chosen so that the conditions at the seals are satisfied. The modal amplitude function will be determined according to the boundary value problem. After finding the modal amplitude functions to each of the modes shape functions, the frequency dependent spatially varying pressure and the resulting forces and moments on the craft can be determined.

2.2.1

Modal solution

The unbounded differential operator ¿92/Dx2 with boundary conditions on the finite interval s E {-L/2, L/2] appearing in eq. (25) lias a set of discrete eigenvalues (Keener 1988) called the discrete spectrum of the differential operator. The eigenvalue equation is given by

- c29 (r(x)) = wr (z)

,

j

1. 2,3,..., k

(27)

where r1(x) is the eigenfunction or the mode shape function of mode j, and w is the corre-sponding eigenfrequency. The eigenfunctions r1(x) are orthonormal, that is

rL/2

(r (s) ,r(x))

= J- L/2

r (x)r (s) dx =

(28)

where ö ist the Kronecker symbol. 4'(x;iw) has a unique representation

)n(e)= LP,(iwjr(x)

(29)

j=1

where P,(i) is the frequency dependent modal amplitude function for mode j due to action n. Due to linearity and orthogonality we can consider each mode separately and superpose the contributions from each of them.

2.2.2

Boundary conditions

We can set up the following boundary conditions on the surfaces enclosing the cushion air

volume:

1. On the rigid part of the wetdeck (z = ho):

¿We, Th=3

Ô: =

(Z+Xcp)e:

(30)

2. At the bow and rear seal systems (s = ±L/2) it is assumed that the seal systems are rigid:

-{

Poi7

where AF is the outlet area of fan i.

4. At all the controlled leakage areas (z = h0,x _{= xLi)}

n=0

0i

0, n=35,T.8,,7+r

Schiffstechnik Bd. 40 1993 / Ship Technology Research Vol. 40 - 1993 77

Oth (s, z; ioi) I _{Po-'-' .} r _-"OiIRCS

L L'n(XLi;iwe) (33) Ô: IRCS Po _i=1 0i RCS

n=0

n = 3,5,7

->i:::=

0,

119 ii

ACS where 112 =

Cr2Pü/pa.

₍₃₄₎ n(x,z;iwe)

0,

n = 0..3.5,7,8,,7+r

(31) = 3. At all the fan outlet areas (z = h0,z =

q (s, z; jwe) (xp; 2We) 01 (32) Pco2''e Ô: _¡=1 Fz P

5. At the bow leakage area (z = 0, x = = L/2): Assuming no motion induced leakage, we have - I 112/2, n = 0 (35)

On(x,z;iwe)

lPCOIi2i,(;j

0,

n = 3,5,7,8,,7+r

¿Iz 2 _Po

At the stern leakage area (z = 0,x =

= L/2):

Assuming no motioninduced

B0 (s) = Kc2 ( tAP

V:O \0

The mode shape functions will be chosen so that the boundary conditions on the seals are satisfied. The following mode shape functions, in genera.l infinitely many, will satisfy the boundary condition on the seals:

+

± >1

ACS cos ('Li + )

Tite frequency dependent modal amplitude functions will be determined according to eq. (25) and the remaining boundary conditions. By taking the inner product of eq. (25) with the mode shape function of mode j for j E {1, 2,3, ...} and using eq. (29), we find tile frequency dependent modal amplitude functions. However before doing so. it is convenient to divide the mode shape functions into even and odd modes around the origin of the coordinate frame.

Thus, i = 1,3,5,... are representing the odd modes, while i = 2,4,6.... are even modes.

Denoting s = We, eq. (29) is written as

P(s)r(x) =

A(s)r(x)+

B7(s)r(x)

(40)

j=1 j=,3,5,''' j=2,4,6,

where 4(.$) and B(.$) are the frequency dependent modal amplitude functions for the odd and even modes respectively due to action 'n. We then have

A0(s)

(41)

L _(A

- í-t'

_{+ -t cos} ('Li +

) - iPo

_{=i loi cos}

(xj + 4))

s2 + 2jw.s + (42)

- TPoIi

Iocos

(xp + 4))

2 + 2s + w]

A3 (s) = 0 (43) B3 (s) = 0 (44)

78 Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993

leakage, we llave n = 0 (36)

5n(x,z;ie)

PCOIt2j

(X4pi)

- 0,

n = 3.5.7,8,.-,7+r

ôz 2 _Po

7. At tile mean free surface (z = 0):

0,

n=0,3,5,8,.7+'r

n=7

(37)

J7r(

L L L

r (x) = cos + .rE

--i-.

--

,j=1,2,3....

(38)

From eq. (27) we find that the corresponding eigenfrequency j for mode j is

J7r

where A4 and K5 are A(T+1)j (s) = 4Lc A5 (s) ₌ 2 + 21ws +w B5 (s) = O A4 A1 (s) ₌ 2 + ₊

B7(s)=

_s2±2ws±w2

c0St (xLi + 4)

2 + 2ws + j

2.11:2 cos _{(xLi ± 4)} 2

+ 2ws + wj

- 4e2 kcos .)L4)O, h0L k2

_{- (fl}

= Jrb

+ A cos2 (XLI +

L

pL/2 (s) = p0b J (x; s) dx = O, 1L/2

= _pobJ

1i(x;s)xdx

L/2

3e2 k sin -o,

j=2,4,6,

h0L _k2 (L2fl L)

The relative damping ratio for all j values is found to be

j=

1,3,5,---=

1,2,3.--2p0bs

j-13

(L)2

[Adj (s) ILua (s) + A5 (s) 5a(s) + A7 (s) Ç (s) +

A7» (s) ¿ACS (x3i; s)] ,j = 1,3,5,.

0,

j=2,4,...

Due to symmetry around the origin of the temporary coordinate frame, there are no con-tributions from the spatially varying pressure on the heave motion. Since the vertical force is zero, F'(s) is the pitch moment about the xv-, y9- and - coordinate frame without further transformation.

2.3

Equations of motion and dynamic cushion pressure

The four coupled equations of motion and dynamic cushion pressure are: 1. Uniform pressure equation:

( ARCS

(4; s) + A/1spa (x; s) + ALspa

s))

2 &1 oi Pspa

q &Qi

PcOPO iIi1 jOiíLspa(4f; s) + (K1s + I13)/iva (s) - PCOACXPSS(s) + PcOA8h1aO (.$)

=

(s) + Po'2

A5 (x

s) ₍₅₅₎ (48)

i = 1,2,. -,r

(49)

i= 1,2,---,r

(50) (51) (52) .) j7r 01 L (XF +

cos_

-(53)

where A = A' +

The spatially varying pressure equation is given later (eq. 57).

The heave force and the pitch moment due to the spatially varying pressure are found to be

(54)

Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993 79

B(7) ( )=

ÔQ

where the time derivative of

V(t)

is given iii eq. (9) and

Pco1o4 Q0 ÛQ

'3Pc0Pc0P0

pO

2. Spatially varying pressure equation: Pco (Xg; s) =

-Po j=[,3

+

A

(s) ARCS (x s ) ì (x)

(

Oi

A01 (3)íLua (s) + A51 (s) (s) + A71 (s)Ç (s)

Po Po _=24, s

/

+ B71 (s) (s) + B(T±r)j (s) 4

(x;

s))

r1 (x)

where x = x +

Heave equation

((iii + A33) 2 _{+ B33s + C33) 73a (s) + (A35s2 + B35s° + C35) 15a (s) - Ap00 (s) = F (s)}

(58) where rn is the craft's mass.

Pitch equation

(uss+Asss2+Bsss+Css+2Pobs

()2ASl(s))

¡ì5(s)

+(A53s2 + B53s +C53) 13a (s) +

(ücp

+2Pob. j=1,3,

L)2

(s)) ua(s) (59)

-

2p0bs

=1,3

(L)2

(A71 (s) Ça (s) + -4(7+i)j (s)

ACS (d., s)) + F (s)

where 155 is the mass moment of inertia around the yg-axis.

The hydrostatic terms are found in the standard way by integration over the waterline area of the side-hulls. The hydrodynamic added-mass coefficients A1, the radiation damping coefficients B1, the heave excitation force F0(s) and the pitch moment Fa(5) are derived from hydrodynamic loads on the side-hulls. These may he calculated as presented by Faitinsen Zhao (1991e, b), Faitinsen et cii. (1991, 1992) and IYestegdra (1990,). However, since the main focus in this paper is on the high frequency range, we have used a simplified strip theory based on Saivesen et ai. (1970,) for calculating the hydrodynamic loads. Examples of the two-dimensional frequency dependent added-mass and wave radiation damping coefficients can be found in Faitinsen (1990). The high frequency limit of the two-dimensional added-mass coefficient is used, while the wave radiation damping coefficient depends on frequency.

2.4

Discussion of the mathematical model

It is seen from eqs. (.58, 59) that the heave and pitch motions are coupled to the dynamic air cushion pressure. This is expected since the major part of the SES buoyancy is due to the air cushion. The dynamic air cushion pressure is the sum of the dynamic uniform pressure and the spatially varying pressure which are coupled because of the asymnietric air flow caused by the inflow and outflow devices. The oniy way to avoid coupling is to place the fans at the same positions as the leakage areas so that the net air flow is zero all over the cushion length.

How many acoustic modes should be included in the mathematical model? Higher acoustic modes will be excited by sea waves containing very little energy; hence the dynamic response will be unimportant with respect to established criteria for passenger comfort and crew work-ability (ISO 2631). One should notice that the air cushion dimensions and the forward 3peed

(56)

B01 (S),Uua (s) (57)

strongly affect the vertical accelerations caused by the acoustic resonances. The acoustic reso-nance frequencies are inversely proportional to the air cushion length, as is seen from eq. (39). The wave excitation frequency w = w + kU increases with forward speed U. Thus waves of relatively low w0 may excite the craft in the frequency range of the acoustic resonances when U is high. This may result iii more wave excitation around the resonance frequencies since the maximum wave height tends to increase with wave period.

The one-dimensional approximation of the spatially varying pressure is valid only for fre-quencies below those corresponding to w, < 27rc/b,2lrc/ho such that the minimum acoustic wave length considered is much greater than the beam and height of the air cushion. Even if the solution is formally presented by an infinite number of acoustic modes, the number of modes that is reasonable to include is limited: As a rule of thumb, one should include tile first mode above the frequency range of interest. For frequencies above the specified maximum w,, it is necessary to take into account two- and three-dimensional effects which were neglected

here.

The relative damping ratio given by eq. (53) is important. The leakage and the fan inflow contribute to the damping. Tile longitudinal placement of the fan and the louver systems strongly affects the damping. In case of a single fan and a single louver system, it may seem natural to piace them in the middle of the air cushion. However, from eq. (53) we observe that the damping for the odd modes will be reduced significantly if XL and ip are O. Maximum damping of both odd and even acoustic resonance modes in case of a single lift fan system and a single louver system is obtained for Xp and XL equal to L/2 or L/2. Tile relative damping ratio of the first odd acoustic mode on a 35 m SES with tile ride control system turned off will increase from about 0.05 to 0.2 by placing the lift fan system at one of tile ends of the air cushion instead in the middle. This gives a significant improvement in ride quality even when the ride control system is turned off. In the same way the active damping due to the ride control system is maximized by placing the louver system at one of tile ends of tile air cushion. One should notice that .rg = x - x. This means that the mode shape function is written r(x9 + x) = r(r). Data for the 35 m SES are given in Appendix A. In the following we will use k acoustic modes, neglecting higher ones.

3

Controller design

The objective of the controller is to damp pressure fluctuations around the equilibrium pressure Po in the presence of waves. This can be formulated in terms of the desired value of the nondirnensional dynamic uniform pressure jt(t) = O and the nondimensional spatially varying pressure íi(x9, t) = O where time superscript d denotes tile desired value. The number of modes to be damped depends on the requirements related to established criteria for passenger comfort and crew workability.

The mathematical model of the craft dynamics is of high order as it contains a iligh number of acoustic modes. A practically impleinentable controller has to be of reduced order. When designing a controller based on a reduced order model, it may happen that the truncated or residual modes give a degradation of the performance, and even instability of the closed ioop system may occur. This is analogous to the so-called spillover effect in active damping of vibrations in mechanical structures (Balas 1978). The unwanted excitatioll of the residual modes has been termed control spihlover, while the unwanted contribution of the residual modes to the sensed outputs has been termed observation-spillover (Fig. 4). This problem was also discussed by Gevarter (197O) in connection with the control of flexible velmicles. Mode O in Fig. 4 is related to the uniform pressure, while the higher modes are related to the spatially varying pressure.

The controller must be robust with respect to modelling errors and parametric and

ConolIed Modes

Fig. 4. Observation and control spillover. where CkXL = cosk7r(xL + L/2)/L and Ckx. coskr(x, + L/2)/L

parametric uncertainties, iionlinearities in sensors and actuators and component failure. The use of collocated actuators and sensors pairs and output feedback controllers provides a design technique to circumvent these problems. The louver and sensor pairs may he distributed along the air cushion, preferentially in the longitudinal direction. The pressure sensor i is placed at the longitudinal position x and the louver system i is placed at the positions x.

In controller design it is convenient to transform the equations of motion and dynamic pressure as given by eq. (55-59) into the standard state space form, wlìere the equations are formulated in the time domain. The linear system is then assumed to be time-invariant. This implies that the coefficients are assumed to be constant. For notational simplicity time dependence is usually omitred in the equations but given in the declarations.

For simplicity and without loss of generality hydrodynamic and hydrostatic coupling terms and the coupling between the dynamic uniform pressure and the spatially varying pressure are assumed to be negligible in the controller design model. Furthermore the longitudinal position of the centre of pressure is assumed to coincide with the centre of gravity.

A linear time-invariant model with n degrees of freedom including heave, pitch, uniform pressure and k acoustic modes (i.e. n 5 + 2k) is then given by

* = Àx+Bu+Ev

y = Fx

(60)

where the n dimensional state vector x(t) is defined

= ['93, Th 173, 1], [, Pi, P2 .

. ., P, ji, P2,'

lk.jT (61)

82 Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993

Concro Sptllover S S S Observation Spiflover

A J k+1 C XL Mode k + I k+1 C x k C XL Mode k k

Cx

.

.

S

.

S S 2 C XL Mode 2 C XL Mode I X Mode O

Input Sensor Output

This nieans that the nondimensional spatially varying pressure 1i3(x, t) is represented by

(.r,t) ₌

>jj (t)r1 (x)

(62)

i=1

where the time derivative of p1(t) is the modal amplitude function for mode j. v(t) is the 3 + k dimensional disturbances vector. u(t) is the r dimensional control input vector

and r is the

number of louvers. The elements of u(t) are for i = 1, 2 r

u (t) =

\4Rcs(x9t)

(63)

where .45(x1,t) is defined in eq. (13). y(t) is the m dimensional measurement vector and ni is the number of pressure sensors. The symbolic expressions for the n x n system matrix A. n x r controller matrix B, n x (3 + k) disturbances matrix E and m X n measurement matrix F are found in Appendix B. One should notice that these matrices are assumed to have constant coefficients. The high frequency limit of the two-dimensional added-mass coefficient given by Faltinsen (1990) is used. Further. constant radiation damping coefficients were applied: In pitch it corresponds to the value at the pitch resonance frequency determined from structural mass forces acting on tile craft and liydrodvnamic forces ou the side-hulls. For heave we have chosen tile damping coefficient at tile resonance frequency that would exist without the excess air cushion pressure. These simplifications are motivated by fact that the effect of damping is most pronounced around tile correspondIng resonance frequency.

Since (A, B) can be shown to be controllable and (F, A) can be shown to be observable, there exists a minimal state space realization (A. B, F). where tile transfer matrix Hr(s) between input u(s) and the output ya(s) is

ye(s)

= F(sI - A)

Bu(s)

= H(s)u(s).

(64)

The transfer matrix Hd(s) between the disturbances input y(s) and the output y,(s) is found to be

yr(s)

= F(sIA)Bv(s)

= 11d (s) y(s). (65)

This means that y(s) = ye(s) + yr(s).

We will consider the case where the sensors and actuators are ideal, that is linear and instantaneous with no noise. It is assumed that the control input matrix B can be related to

the measurement matrix F so that

F=BTP

(66)

where P is a n X n diagonal positive definite matrix providing correct scaling of the BT matrix to the F matrix. This is the case when there is perfect collocation between the sensors and the

louvers. i.e. 4 = .c for all i and r = in. P is defined in Appendix C.

Let the the controller be defined as the linear time-invariant operator H between the input

y(t) = y(Ì) + Yv(t) and tile output u(t). The transfer matrix of H is denoted Hjs). The

controller given by He(s) stabilizes the process Hr(s) if the controlled system consisting of stabilizable and detectable realizations of He(s) and He(s) in standard feedback configuration, see Fig. 5, is asymptotical stable when y(s) = O.

Control law

A proportional output feedback controller between the input y(s) and the output up(s) is proposed according to

ut(s)

= Hy(s)

H,.(s)

= Gp

(67)

up

p

u

Fig. 5. Feedback system

where G = diag [gpi] > O is a constant diagonal feedback gain matrix of dimension r x r. This control law provides enhanced damping of the pressure variations around the resonance frequencies. The diagonal feedback gain matrix G caii be determined for instance by pole placement techniques. Connecting the transfer matrices Hr(s), Hd(.$) and He(s) together, we obtain the feedback system illustrated in Fig. 5. We can now state the main result regarding controller design.

Theorem: Consider the transfer matrix Hr(s) with ya(s) as the output and u(s) as the input. Then the proportional output feedback controller He(s) witlì u(s) as output and y(s) as input stabilizes Hr(s).

Proof: The stability proof is based on Lyapunov's direct method applied on a linear time-invariant system. The closed-loop system is bounded input bounded output (BIBO) stable if the equilibrium point x(t) = O of the autonomous closed-loop system is asymptotical stable when v(t) = O. Applying the proportional output feedback control law, the linear time-invariant autonomous closed-loop system becomes

*=(ABGF)x=(A--PFTGF)x=A.x

(68)

where the n x n closed-loop system matrix A is defined in Appendix C. Define the Lyapunov function candidate

'T

V(x)=x Px>0

(69)

where the n x n diagonal positive definite matrix P is given in Appendix C. V(x) is positive definite. The time derivatives of V(x) is negative semidefinite, that is

V(x)= xT (AP+PA)x = _XTQX <0

(70)

where the n x n symmetric positive semidefinite matrix Q is given in Appendix C. From eqs. (68.70) it is seen that

x=xo=[3,q5,O,0,O,p1,p2...pk,0,0...,0]T.

(71)

However, from ecl. (68) follows

7Ì3=1]5L,P1=P2=..=Pk=O

(72)

only if

T73=q5=J)1=9=PkO.

(73)

Hence, by the invariant set theorem (Vidyasagar 1993,) the equilibrium point of the closed-loop system x(t) = O is asymptotically stable and the result follows. D

Hence, stability of the closed-.loop system using cohlocated sensor and actuator pairs is main-tained regardless of the number of modes, and regardless of the inaccuracy in the knowledge of the parameters. Thus the spillover problem is completely avoided and the parameters do

not have to be known in advance to obtain stability. Notice that there are no restrictions to the placement of the collocated sensor and actuator pairs with respect to stability. However,

84 Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993

q

H

yv V

optimizing the perfomance, the longitudinal placement of the sensor actuator pairs is crucial as pointed out in Section 2. To show stability in the presence of actuator and sensor nonlinearities and actuator and sensor dynamics, it is convenient to use the theory of passivity (Sørensen and Egeland 1993 and Sorensen 1993).

4

Simulation examples

In this section numerical simulation results of the vertical motions and pressure variations of a 35 m. 55 m and loo m long SES advancing forward in head sea waves at 50 knots are presented. The fully coupled model presented in Section 2.3 is used in the simulations. The effects of actuator and sensor placement on the dynamic response are illustrated. The transfer functions of heave and pitch motion and acceleration, nondimensional uniform and spatially varying pressure relative to the incident wave amplitude are shown for the 35 ru SES. The effects of collocation and noncollocation of the sensor and actuator pairs for the 35 in SES will be investigated by studying the diagonal entries of the transfer matrix ff(i) between the control input u(s) and the sensor output Yu()- We will assume that the three craft are equipped with one single fan system and two louvers and two pressure sensors. At the end of this section a study of the importance of spatially varying pressure for increasing craft length is shown. This is done by studying the operability limits in different sea states for craft of lengths 35 m. 55 ni and 100 m. Main dimensions and data of the SES craft are given in Appendix A. The number of acoustic modes considered in the simulation model is four. i.e. k = 4 and

n = 13.

In Fig. 6 the transfer functions of the heave motion and heave acceleration are presented.

For We O the craft follows the incoming waves. At sorne frequencies the transfer functions

show almost no response. The wave volume pumping is zero at these frequencies because the integrated volume change due to the incident waves is zero. The cancellation effects and the low-pass qualities of the dynamic system lead to no significant peaks of the heave motion amplitude around the uniform amid spatially varying resonance frequencies. However, the heave acceleration amplitude is strongly influenced by the resonances of the dynamic excess pressure. The shape of the heave motion and acceleration transfer functions for a SES are quite different from those of a conventiona.l catamaran. This is due to the strong coupling to the dynamic excess pressure acting on the wetdeck of a SES.

In Fig. 7 the transfer functions of the pitch motion and pitch acceleration are presented. The low frequency peak around 0.20 Hz seen in the pitch motion amplitude is at the pitch resonance frequency which is determined from structural mass and hydrodynamic forces on the side-hulls, like on a conventional catamaran. The shape of the pitch motion transfer function for a SES is similar to that of a conventional catamaran. Due to the low-pass qualities of the pitch motion, the high frequency spatially varying pressure force acting on the wetdeck is seen only in the pitch acceleration.

Fig. 8 shows the transfer function of the nondimensional dynamic uniform pressure. When - 0. the nondimensional uniform pressure tends to zero, i.e. the uniform pressure in the cushion is equal to the equilibrium excess pressure. The high value around 2 Hz is due to the resonance of the dynamic uniform pressure. This resonance may cause excessive vertical acceleration if the craft is advancing in sea states containing energy at this frequency. The same resonance is observed in full scale measurements in moderate sea states; however, the dynamic uniform pressure response is overestimated by the present linear theory. This tendency was also observed by McHermry et al. (1991). Measurements of response in calm water with specified dynamic leakage area variations through the louvers indicate that one of the reasons for the overprediction is the excitation force due to the wave volume pumping given in eq. (9) in the nondimensional uniform pressure eq. (55). Sullivan et al. (1992) report that the effect of dynamic fan response, which is neglected here, will in some specific cases significantly reduce

-. 0.8 E o. E 0.6 02

Fig. 6. Numerically calculated transfer functions of the heave a) ¡notion and b) ac-celeration; 5L1 = 12 m, 5L -12 m. ip = 5 m, pQ = 500 mmWc, U = 50 knots 4.5 2 0.5 86 0 0.5 2 10° 1O-iO-z 10-° 1 0-E O -50--LOO -150

Heave Heave AT1

1.5 2 2.5 3 3.5 Frequency of Encounler (Hz! 10-2 4.5 -35 5

Fig. 10. Numerically calculated Bode matrix H(iw0); 5L1 = = 12 m, 5L2

mmWc

SoC. Lot - Ii/LI

0.5

1O 100 ìO

Boce pLot - Phe(yL/i>

10° 10°

102 Hz

102 Hz

plot of the first diagonal entry of the 2 x 2 transfer

= = -1.2 iii, xp = 5 in, U = 50 knots, Po =500

Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993

0.1 - P!tcfr PEch Acc 0.45 i t A 0.4 - 0.08 E n 0.35 0.08 I L r r. 0.3 0.25 0.2 0.04

--

t.

-0.15 0.02

-

0.1 t i 0.05 O 2 3 4 5 6 7 10 Frequency of Encounter [Hz]

Fig. 7. Numerically calculated transfer functions of the pitch a) motion and b) ac-celeration; 5L1 = 12 m, 5L2 = -12 m, Sp = 5

m, Po = 500 mmWc, U = 50 kiiots.

6 8 1 12 14 16 18 20

Frequency of Encounter [Hz]

Fig. 8. Numerically calculated transfer function of the notidirnensional dynamic uni-form pressure; 5L1 = 12 m, 5L2 = -12 m, 5F = 5 m, Po 500 mmWc, U = 50 knots

2 6 8 10 12 14 16 18 20

Frequency at Encounter (Hz]

Fig. 9. Numerically calculated

trans-fer function of the nondimensional spatially varying pressure at the stern and centre of pressure; 5L1 = 12 m, 5L2 = -12 m, x = 5 = 500 mmWc, U = 50 knots -200 25 20 15 10 30 E o. E a a a

the uniform pressure response. Nonlinear leakage under seals and side-hulls wiLl also tend to reduce the response, especially at large amplitude heave and pitch motions. Even if the response level nlay be overpredicted, the resonance frequency is well predicted.

In Fig. 9 the transfer function of the nondimensional spatial pressure variations at the stern and at the centre of pressure are presented. The transfer function at the bow is similar to that at the stern. At the centre of pressure the transfer function shows little response around 6 Hz and 18 Hz while the response around 12 Hz is almost the same as at the bow and stern. The response around 6 Hz and 18 Hz is due to the First and second odd acoustic resonance modes, while the response around 12 Hz is related to the first even acoustic resonance mode. For small encounter frequency the nondiniensional spatially varying pressure tends to zero. Generally, the aiiiplitude of the spatial pressure variations is lower than that of the uniform dynamic pressure. Because of poor wave measurements during the fuR scale rneasuremeiits, it is difficult to compare measured and numerically predicted response levels. However, full scale

measurements indicate that the predicted response is of the correct order ofmagnitude. Fig. 10 shows the Bode plot of the first diagonal entry in Hp(ie) between the pressure sensor Yui() and the louver ui(s) when the two sensor and actuator pairs are fully collocated. A Bode plot is a graphical representation of the transfer matrix, consisting of separate plots of the magnitude and the phase as functions of frequency. Sensor I (2) and louver I (2) are placed at the bow (stern) respectively. When - O. the dynamic pressure tends to a static value proportional to k1/I2. This indicates that the equilibrium pressure o will decrease when the

equilibrium leakage area increases and vice versa .Ar ound 0.3 Hz there is almost no response. This is related to the action of structural mass and hydrodyiìaniic forces on the side-hulls. The high response around 2 Hz is due to the resonance of the uniform pressure. The high values around G, 12, 18 and 24 Hz are due to acoustic resonance modes. From the phase plot we observe that the phase is varying between +900 and _900 the whole frequency range. This is expected from Section 3 using collocated sensor and actuator pairs. The second diagonal

entry in H, ( ¿w. ) behaves in a siniilar nianner and is hot presented here.

Fig. 11 shows Bode plots of the two diagonal entries of Hp(iWe) for the case where the two pressure sensors are placed at the bow. one of the louvers is placed at the centre of pressure and one at the stern. From the magnitude plot of the first diagonal entry we observe that there are almost no peak values around 6 Hz and 18 Hz. Since louver i is placed at tue centre of pressure, it will have no influence on the first, third and the higher order odd acoustic resonance modes. From the phase plot of the first entry we observe that the phase is decreasing below 90° around 7 Hz. At the second acoustic resonance frequency around 12 Hz the pressure signal at the bow is 1800 out of phase compared to that at the centre of pressure. From the second diagonal entry of fIp(i'e) referring to the sensor at the bow and louver 2 at the stern, we observe that the pressure signal at the how is 180° out of phase compared to that at the stern. This is expected if non-collocated sensor and louver pairs are used which introduce loss of phase and may lead to instability.

Fig. 12 shows the transfer function of the nondimensional spatially varying pressure at the stern when the ride control system is off. and when the two sensor and actuator pairs are collocated and non-collocated, with tite feedback gains given as Ypi = gp2 = 1. In the latter case louver I at the bow is reading the sensor signal 2 at the stern. wiule louver 2 at the stern is reading sensor signal 1 at the bow. The ride control system contributes to reduced response around the resonace frequencies when collocated sensor and actuator pairs are used. However, iii the case of non-coLlocated sensor and actuator pairs the response around the first odd resonance frequency increases significantly. This is expected since, in this case, the ride control system reduces the relative damping ratio (see eq. 53).

Fig. 13 shows the transfer function of the vertical acceleration at the stern when the ride control system is off, and when the two sensor and actator pairs are collocated and

E 1.5

11

0.5 10°

400

Fig. 11. Numerically calculated Bode plot of the diagonal entries of the 2 x 2 transfer matrix H(iti4); a,1 = a,2 = 12 m, vLl = O In, .rL9 = -12 ru, XF =5 m, U =50 knots, m =500 mmWc

O

e e

Frequency al Encounter tHzl

Fig. 12. Numerically calculated

trans-fer function of the nondimensional spatially varying pressure at the stern with, a) RCS off; b) RCS on: gpi = gp2 = 1, L1 = a,1 = 12 ru,

XLO = = -12m; C) RCS on: gpi = 9p2 =1.

= x, = 12 ru, XLO x, = -12 m, where xp=5 m, Po = 500 mmWc, U = 50 knots Vertical Acceleration at AP 0.5 0.4 C' - 0.3 a) > a) 0.2 C o-1 O

Cade plat - lUI/uil

ROSolI

- CoIIoled

12 14

Fig. 14. Operability limits for a 35 m SES based on the ISO 2631 "Reduced Comfort Boundary", with and without the effect of spatially varying pressure. Head sea, X? = 5

flï, PO = 500 mmWc, U = 50 knots 200 100 -300 o 1.5 10-' H:

500e olot - haseIy2/u2l

102

t:'

Fig. 13. Numerically calculated transfer function of vertical acceleration at the stern with a) RCS off; b) RCS on: gpi = gp2 = 1, XLI = = 12 m, XL2 = a,2 = -12 m; c) RCS on: gi. = gp2 = 1, = .S, = 12 m, .rL2 = 1,1 = -12 m. where X? 5 m, ₌ 500 rnmWc, U =50 knots Vertical Acceleration at AP 2.5 3 3.5 4 4.5 5

Peak Penod Tp (sec]

Fig. 15. Operability limits for a 55 m SES based on the ISO 2631 "Reduced Comfort Boundary", with and without the effect of spatially varying pressure. Head sea, Xp = 7.9 'II. Po = 800 mmWc, U = 50knots

10-3r

10.2 10-' 10° 102 102

10-'

Hz

Sode Olot - Pliaselyl/ull 200 100-N E -LOO-O -200-- -300--400 ° L0' LO 102 10' 10' LO-'

88 Schiffstechnik Bd. 40 - 1993 I Ship Technology Research VoI. 4.0 - 1993

10-' 10-' 10° 10° 10' $ 8 10 12 14 Frequery al Errcourrer (Hz] 4 3.5 1:5 2 2.5

Peak Period Tp [Sec]

O

E -LOO

C

collocated. with the feedback gains given as gpi = gp2 = 1. The same tendency as mentioned above is also seen in the acceleration amplitude. Notice that the dynamic uniform pressure is not directly influenced by using collocated or non-collocated actuator and sensor pairs.

Figs. 14-17 show calculated operational limits for SES of lengths 35, 55 and loo ni. The criterium used for the operational limit is the International Standard (ISO) 2631/l&3 (1985). It considers the effect of vertical acceleration on human performance and comfort. It contains criteria for two frequency bands. The high frequency band, which is of most importance here. is based on three limit levels:

Exposure Limit, giving the limit which can not be exceeded without reduced safety Fatigue-decreased Boundary, related to maintaining the working efficiency of the crew Reduced Comfort Boundary, related to evaluation of passenger comfort.

Here the Reduced Comfort Boundary is used. The lower frequency band (0.1-0.63 Hz) in the standard is related to discomfort such as motion sickness. In both frequency bands, limits depend on exposure time. Here four hours were used. The limiting significant wave height H, as a function of peak period T is specified in the figures. Pierson-Moskowitch spectra were used for all sea states. For low peak periods the maximum possible significant wave height (given in the figures as Max Hs) is limited by the maximum wave steepness:

H, O.1gT/(27r). (74)

Fig. 14 shows the limiting significant wave height for a 3.5 mn SES. The first acousticresonance is near T=1.7 s, while the uniform pressure resonance is at T=3.5 s. Neglecting the spatially varying pressure will lead to too high operational limits for small peak periods. However Fig. 14 shows that the uniform dynamic pressure, dominating the response for T values froni 2.0 to 4.0 s. will be most critical for the operation of the craft. As already mentioned. the uniform

dynamic pressure might be overestimated; so the actual operation limits might be less severe thami shown here.

Figs. 15 and 16 show the limiting significant wave height for 55 and 100 m long SES

respec-tively. They display the saine basic features as Fig. 14, but the importance of including the spatially varying pressure seems to increase with increasing SES length. For the 55 (100) m

SES, the first acoustic resonance is at T = 2.2 (3.1) s, and the uniform pressure resonance at T =4.3 (5.3) s.

Fig. 17 shows how the ride control system affects the operability limits on a 35 rn SES. The ride control system configuration is the same as used in Fig. 12. The ride control system is most effective (linuting wave height almost doubled) in the frequency range dominated by uniform pressure variations, but its effect is significant also at higher frequency. The strong adverse effect of the non-collocated louver and sensor pairs in the frequency range dominated by spatial pressure variations is also shown in Fig. 17. This clearly stresses the importance of using cohlocated sensor and actuator pairs.

5 _Conclusions

The pressure variations in the air cushion of a SES can be divided into a dynamic uniform and a spatially varying pressure term. Depending on the craft dimensions and the sea state, the pressure behaviour is dominated by the dynamic uniform pressure in the lower frequency range and by the spatially varying pressure in the higher frequency range. Resona.nces of the dynamic uniform pressure and the spatially varying pressure _{may cause excessive vertical accelerations} if the craft is advancing in sea states containing significant energy at the resonance frequencies. To achieve high passenger and crew comfort, it is necessary to reduce these accelerations using a distributed ride control system. Such a system is developed based on a niathematical model where both the dynamic uniform pressure and spatially varying pressure are included. Spillover

0.8 0.6 0.4 C) o 2

loom SES

Vertica[ Acce]erat]on at AP 2.5 3 3.5 4 4.5 Peak Period Tp [sec]

WI acoustic - wio acoustic Max Hs

55

Fig. 16. Operability limits for a 100m SES based on the ISO 2631 'Reduced Comfort Boundary", with and without the effect of spatially varying pressure. Head sea, Xp =

14.3 m, Po = 940 mrnWc. U = 50 knots 0.5 42 0.4 C) U) o

35m SES

Vert]ca] Acce]eraon at AP CSon

Max C4, -e- RCS xonaxle

Peak Period Tp [sec]

Fig. 17. Operability limits for a 35 in SES based on the ISO 2631 'Reduced Comfort Boundary", with and without the effect of spatially varying pressure. Head sea, XF = 5 m, = 500 mmWc. U = 50 knots.

a) RCS off: b) collocated R.CS on: = = 1, £L1 = X,1 = 12m, XLC = = -12 in; c)non-cohlocated RCS on: gpi = gp2 = 1.

XL1 s2 = 12 m, XLC = X,1

12 m

effects, i.e. excitation of modes neglected in the mathematical model, is avoided when using cohiocated sensor and actuator pairs, regardless of the parameter values and the number of modes considered. An analysis of operability limits according to ISO 2631 criteria for SES of lengths 35. 55 and 100 iii shows that iieglectiiìg the spatial pressure variations will lead to overprediction of the operational capability. It shows also the importance of including spatial pressure variations especially for greater SES length.

References

BALAS, M. (1978), Feedback control of flexible systems, IEEE Transaction of Automatic Control, Vol. AC-23 (4), 613-679

GEVARTER, W.B. (1970), Basic relations for control of flexible vehicles, AIAA Journal Vol. 8 (4) FALTINSEN, O.M. (1990), Sea loads on ships and offshore structures, Cambridge University Press FALTINSEN, O.M., HELMERS, J.B., MINSAAS, N.J. and ZHAO, R. (1991), Speed loss and operability ofcatamarans and SES in a seaway, First mt. Conf. on Fast Sea Transportation FAST'91, Trondheim,

N or way

FALTINSEN, 0M. and ZHAO, R. (1991 a), Numerical predictions of ship motions at high forward

speed, Phil. Trans. of the Royal Society, Series A

FALTINSEN, 0M. and ZHAO, R. (1991 b), Flow predictions around high-speed ships in waves. Math. Approaches in Hydrod., SIAM

FALTINSEN, 0M., HOFF, JR., KVÀLSVOLD, J. and ZHAO, R. (1992), Global loads ou high-speed catamarans, P RADS '92, Newcastle, England

GELB, A. and VANDER VELDE, WE. (1968), Multiple-input describing functions and nonlinear

system design, McGraw-Hill Book Company

KAPLAN, P. and DAVIS, S. (1974), A simplified representation of the vertical plane dynamics of SES craft, AIAA Paper No. 74-314, AIAA/SNAME Advanced Marine Vehicles Conference, San Diego, Cal. KAPLAN, P. and DAVIS, 5. (1978), System analysis techniques for designing ride control system for SES craft in waves, 5th Ship Control System Symp., Annapolis, MD.

KAPLAN, P., BENTSON, J. and DAVIS, S. (1981), Dynamics and Hydrodynamics of surface effect

ships, SNAME Tr. 89

90 Schiffstechnik Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993

4.5 5

KEENER, J.P. (1988), PrincipIes of Applied Mathematics, Addision-Westley Pubi. Comp., Readwood City, Cal.

MCHENRY, G.. KAPLAN, P., KOR.BIJN, F. and NESTEGARD. A. (1991), Hydrodynarnic analysis of surface effect ships: Experience with a quasi-linear model, FAST'91, Trondheim, Norway

NESTEGARD, A. (1990), Motions of surface effect ships, A.S. Ventas Res. Rep. 90-2011

SALVESEN. N.. TUCK, EQ. and FALTINSEN, O.M. (1970), Ship motions and sea loads, Tr.SNAME 78, 345-356

SORENSEN. A.J. (1993), Modelling and control of SES in the vertical plane. Dr-Ing. thesis, Dep. of Eng. Cybernetics, The Norwegian Inst. of Technology

SORENSEN, A.J. and EGELAND. 0. (1993), Ride control of SES, submitted to Automatica, J. of

hit. Fed. of Automatic Control

SORENSEN, A.J., STEEN, S. and FALTINSEN, 0M. (1992), Cobblestone effect on SES, Intersociety High Performance Marine Vehicle Conf. - HPMV'92, American Soc. of Naval Eng., Washington D.C. STRANG, G. (1988), Linear algebra and its applications, Third Edit. Harcourt Brace Jovanovich Pub!., Orlando, Florida

SULLIVAN, P.J., GOSSELIN, F. and HINCHEY, M.J. (1992), Dynamic response of an air cushion lift fan, HPMV'92, American Socc of Naval Eng., Washington D.C.

VIDYASAGAR, M. (1993), Nonlinear system analysis, Second Edit. Prentice Hall, Englewood Cliffs, New Jersey

Appendix A SES Main Dimensions

Appendix B - Symbolic model matrices

The 3 + k-dimensional disturbance vector v(t) is defined as

v(t)= [Fi, F, t'o,

'k

]T

(75)

where F(t) and E(t) are the hydrodynamic excitation forces on the side-hulls in heave and pitch

respectively, see Salvesen e al. (1970). The time derivative of '(t), where i = 0, 1, 2...k, are the

excitation of the dynamic uniform pressure and the spatially varying pressure due to the wave volume pumping found in Section 2. Hence, the n x (3 + k) disturbance matrix is defined as

Schiffstechnik Bd. 40 - 1993 i' Ship Technology Research Vol. 40 - 1993 91

Length overall Loa [rn] Air cushion length L [ni] Air cushion beam b [m]

35 28 8 55 44 12 100 80 22

Mean side-hull beam bs [ni] Cushion height ho [m] 0.5 2.0 0.8 3.1 1.5 5.7

Vessel total mass m [tons] 140 543 2108

Max. velocity U [knots] 50 50 50

Mean cushion pressure po [mrnWc] 500 823 958

Mean fan flow rate [rn3/s] 150 526 2346

Linear fan slope [m2/s] 140 276 676

First acoustic resonance [Hz] 5.89 3.75 2.06

Corresponding sea wave excitation period [s] 1.75 2.23 3.08

Uniform pressure resonance [Hz] 1.74 1.10 0.76

Corresponding sea wave excitation period [s] 3.38 4.34 5.37

E=

o O F.. 0 0

00

00

O O K O O o 0 0 O . O o P0'k><k 0 0 O O o (76) O o Okx3

The n x r contro! input matrix is given by

FT

where = L x and c1 = 2poAT2c2/(poVo). The ni x n measurement matrix is given by

o o o o o o o o

...

o o o

...

o i I. i °kxr

cj cos (x31 + 4) c1 cos _{(x,2 + 4)} c1 cos _{(sm + 4)}

31r

I

2

c1 cos - (z1 + -) c1 cos - (X39 + -) c1 cos - (xgm + ₎

cicos(x31+4) cicos(x2+4)

CiCOS(Xm+4)

Define tile n-dimensional state vector x(t) as

x =

T

X1 = [173,775,7/3,175,/Au] X2

The n x n system matrix is then given by

s15x5 = T T = [Pi P2 pk] X = [Pi1P2,.. pk] 0 0 1 0 0 o 0 0 1 0 C.13 B33 Ap0

m+A33 m+A33 ru-f-A33

o _155+A55 B,5 o

0 O poA_{K1 K1}

000

0

(79)

(81)

92 Schiffstechnik Bd. 40 1993 / Ship Technology Research Vol. 40 1993

For simplicity hydrodynamic coupling terms like A35, A53, £35 and B53 and hydrostatic coupling terms C35 and C53 are neglected.

0 0 0 0 0

...

O

00000

.0

where 5'25xk = O O O O O d1 O d O d5

_o

o o o o 0

...

(82) d 2pob (83) 155 + A55 \jirj S3k5 is defined as 0 0 0 gi O

00000

S3k5 = O O 1) g3 O (84)

B=

o o o o poK3 o o

...

o o poK2 o o o o poK3 (77) K1 K1 Okxr -. K1

c1 cos (XL1 + 4) C1 COS - + 4) C1 COS (XLr + 4)

c1 cos _{- (LL1 + -)}2 c1 cos 2:

- (xL2 + -s-) C1 COS _{XLr +}

c1 cos _{(LL1 + )} c1 cos (xL2 + 4) Ci COS y LLr + 4)kT f

s15x5 °sxk 525xk

A= °/cx5 Okxk 'kxk (80)

83kx5 S4kk S5kxk

w here

4poLc2 poh0 (jir) S4/c/c and S5kk are defined a.s

S4krk = diag [-jj (86)

S5kk = diag[-2ç11J (81)

Appendix C - Proof of closed-loop stability The n x n closed-loop system matrix A is defined as

(815X5 - C155) OSxk (S25Xk - C2sk)

= _{OkxS Okxk} 1/cx k (88)

(83kx5C3kx5) S4/cX k (S5kXk - C4kXk)

5'5x5, 825x., 33kx5, S4L-xk and are defined in Appendix A.

If the sensor and actuator pairs are fully collocated (Lj = .zj), then C4kk is symmetric and positive

semidefinite. This follows since all the underdeterminants of order 2 are zero according to

p7r

cos _{LLi +} L'\ vn

L\

kír

-7) cos y + _-) cos

Hence, by induction all principal submatrices (Sfrang, f988) of C4kXk with order > 2 have zero deter-min ants.

The n x n diagonal positive defiiìite P matrix is choosen to be

P = diag[P],i= 1,2,3....,n, j = 1,2,3,...,k,

(95) = 3 C33 , P2 = C55 , pco(rn+ A33), (96) m+A33 155±A55 (85)

pir(

L\

qir(

L'\ /cim-( L

ros

--

XLi +

.7) CO5y XLi + -i-) cos XLi +

L'\

cL+

0.

2j

(94)

Schiffstechruk Bd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993 93

C25Xk =

_0

0

00

00

00

h1 h2 C3kxs = where h3 _0 O O O

000...

000

000

000

h4 h5 .. O O O t1 O O O t2 O O O t3 O O O t/c C4kxk = Umn = gpjcos-7-O O 0 0 (J h/c iL11 U21 U31 U/cj O O _{PU jYp} where h1 = pj r where lj = r YpiCOS ¿=1

U19 _U13 U1/c

U29 U33 2k U32 U33 _113k U/c2 U/c3 ... mír7

L\

L2)L

jim-I gpi cos

jîr(

L Li+ 2 L £3j ± + L (90) (91) (92) (93) Upq Up/c Ukq Ut-k i=1

-0

0 0 0 0

0000

0

C15X5 = 0000

0 ₍₈₉₎

0000

0

The 3 x 3-dimensional matrix Q'3x3 is (the subscripts a.re in accordance with the dimensions of the full

Q matrix)

03x2 ,/Q13X3

Okx2 Okx3

lt. cati he shown by inspection that

= diag[q], i = °3x Okxk \/Q1SQ23Xk Q3kxk -Q2'XkQ13Q23xk (99) (107)

94 SchiffstechnikBd. 40 - 1993 / Ship Technology Research Vol. 40 - 1993

2B33 2B55 2 -q55 =

PS 1\3

+PcO'2

/

j=1 r (100) q33 = P3 rn + A33 q44 = P4 155 A55 The 3 x k-dimensional matrix Q23xk is

o o o o

Q23Xk = o o o

..

o q5(5+k+1) 15(5+k+2) q5(5+k+3) q5(5+2)

(101)

qs(5+k+j) =

P5h -

PS+kjtj (102)

The k x k-dimensional matrix Q3kxk is

q(5k+1)(5+k+1) q5+k+i)(s+k+2) (5+k+l)(5+2A) -q( S+k+2)(5+k+1) ?(5+k+2){5+k+2) q(5+k2)(52k) Q3kxk = (103) q(+2k 5+k-f-1) q(5+2k h 5+k+2) q(52k)(5+2k) q(5+k-1-j)(5+k4-j) = + (104) l(5+k+j ±k+i) = P5+k+j ji + P5+k+iUij ,i,

j

= 1,2,. . .,k (105)

P4=---,

P5=

_{P;+j=w1P5+k+j,}

P+k+j=-Po'2 C1 - °2x2 02x3 O2xk O2xk -03x Q13x3 _{O3xk Q23xk} °x2 Okx3 °kxk °kxk Okx2 Q2'3 Okxk Q3kxk (97) (98) The n x n. symmetric matrix Q is found to be

Q=(AP+PA1) where

Q=

Q133 > 0; Q3t-xk > 0; Q3tx - > 0. (108)

Since LLi =

zj,

C4kxk is positive sernidefinite and S5kk is positive definite, the matrix Q3kxk is

symmetric positive definite. Hence the n x n symmetric matrix Q is positive semidefinite according to

Strang (1988) since

Q =

R'R

(106)