Charactersitics of Salter’s cam for extracting energy from ocean waves

Lab.

y. Sc1ìeepstouwkutde

Technische Hogeschoot

Detli

Characteristics of Salter's cam for extracting

energy from ocean waves

ARTHUR E. MYNETT, DEMETRIO D. SERMAN and CHIANG C. ME!

Ralph M. Parsons Laboratorr, Department of Civil Enqineerinq, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received April 1978; revised June 1978)

The performance of Salter's cam as an energy extractor from ocean waves is studied numerically on the basis of linear theory of surface waves and floating bodies. Quantities of engineering interest such as efficiency, induced motion and reaction forces are presented and discussed both for fixed and for partly constrained cam shafts.

INTRODUCTION

Salter12 has demonstrated experimentally that a two-dimensional cam shaped as a tear drop mounted on a fixed shaft can be used for extracting wave energy. As the measured efficiency (power removed from the waves divided by incident wave power) has been shown to be as high as 90% for a wide frequency band, Salter's discovery has given new impetus to the study of wave energy; his device and its variations have in particular attracted much attention. Some related hydrodynamic theory has been advanced by Mei3, Evans4 and Newman5 which showed qualitative confirmation of Salter's original experiments

and revealed additional features of this and related

designs. In particular Evans et al.6 have recently treated extensively the circular cylinder for which numerical calculations are relatively convenient. Attempts to use computer methods to deal with Salter's special shape have

been made by Katory7 whose numerical results are

unfortunately marred by errors.*

In this paper a comprehensive numerical study is made for Salter's device. An effective tool based on the hybrid element method reported by Bai and Yeung8 is used. The objectives are to study the performance characteristics of

Salter's cam with fixed or moveable support so as to

provide a quantitative basis for judging its practical

potential. The laboratory case of regular waves and fixed shaft is dealt with first. Comparison with experiments by

Salter and by Carmichael and Davis are then made.

Effects of varying geometrical parameters such as shape, submergence and water depth will be investigated. The effects of non-rigid support will be studied at the end.

LINEARIZED DYNAMICAL EQUATIONS FOR

THE CONSTRAINED CAM WITH

THREE

DEGREES OF FREEDOM

We assume the cam to be a long rigid cylinder and the waves to be normally incident and simple harmonic in time. The problem is two-dimensional and only a unit length of the cam needs to be considered. We distinguish * _{For example, in Fig. 3 of ref 7 off-diagonal terms of the added mass}

matrix violated the symmetry property p, = ,which is an exact identity

in the linearized theory.

ARCHIEF

the three generalized displacements and forces by the

subscripts i= 1, 2 and 3 where i= I and 2 correspond

respectively to horizontal (x) and vertical (r) directions while i = 3 corresponds to rolling or moment about the = axis. The wave induced displacements are defined to be:

(t)= Re(A1e'')

where A, represents the incoming wave amplitude. By slightly generalizing the well-known theory for freely floating bodies in waves9"0 the equations for harnionic motion of the cam can be written as follows (the factor exp( - iat) being omitted):

- x2m1

_{= f +}

f + f + f

(I)

where the left-hand side represents the inertia forces of the cam. Summation of repeated indices is assumed. On the right-hand side, we may distinguish the exciting force/ due to wave diffraction on the stationary body, the added

hydrostatic buoyancy restoring force f

due to cam

displacement, the hydrodynamic reaction f due to cam motion and the reactionf due to the external constraints on the cam.

Specifically we may write*:

(2a)

(2

_{+ ia).frx (2b)}

If there are no fixed constraints applied directly on the cam,f can only come from the reaction from the energy extractor which may have its own apparent mass and damping matrices and

_{., or}

from the elastic restoring force with stiffness C. We may then write:

= (op + ic). - C;J)J

(2e)

where _{, ..} and C are prescribed. All three equations in (1) must be solved simultaneously. On the other hand, if the cam cannot heave or sway, but can roll about the fixed shaft, only f is expressed in the form of equation (2c). *

Whenever i does not appear as a subscript, it means

Salter's cam for enerqvfro;n ocean waves: A. E. Minert et al.

Equation (1) should then be solved lirst for -; afterwards the remaining equations for i = 1.2 give the horizontal and vertical reactions from the fixed shaft.

More compactly we define:

=

(Jxdxd.)/v.

K1=C1+C

M1=ni1+p1+p

(3)

so that equation (1) becomes:

[(Ko

2M) iaD1]1

_=f

(4)

For simplicity we assume that p, À

and C

are diagonal and are pure constants; the first assumption means that the reacting force components are uncoupled in different directions whereas the second assumption implies them to be independent of frequency. We define ). to include both useable and dissipated energy.

Because of the constraint the centre of rotation is

prescribed and need not coincide with the centre of mass, so that the details of m and C are somewhat different from those for a freely floating body. The following results can be easily obtained:

in O

ni(Y' Y°)1

= L

_{m(Y" Y°) ni(X'X°)}

o nl m(Xc X°)

133

_J

o i pqB(XA_xO) I(o)

{pg1+pgV(YV_Y0) I

inq( YC _{Y°)] I}

With reference to unit length of the cam the symbols are:

in, mass of cam;B,beam of cam at the water line;A,area of water plane ( =Bfor unit length); V, submerged volume of cam; X°, Y°, axis of rotation; X, yc centre of mass for the whole cam; XA, horizontal coordinate of centre of flota-tion (Jxds)/4: X', Y', centre of buoyancy

jvdxdv)/

,

1, moment of inertia of the water plane area about z-axis

=j(X_X0)2dx.

Among all these matrix coefficients, nit,, p, )

are prescribed, while

Ç

can be calculated from the geometry. The added mass (po) and damping (2e) tensor and the exciting force f must be calculated from the following formulae after the corresponding velocity potentials are

found:

14 Applied Ocean Research, 1979, Vol. 1, No. t

and

JL + ii./C = PjjIi1dS (8)

s

where S is the mean wetted boundary of the cam,(pd is the

potential for the diffraction problem with incident waves

from the left, and q, is the complex radiation potential for the jth generalized mode normalized for unit velocity amplitudes satisfying:

onS

(9)

with the generalized normal being:

{n} = n, (x - X°)n3,

(y - Y°)n}

(10)

All the hydrodynamic quantities F , p and have been

calculated using a hybrid element method8. Briefly, the idea of the method is to use finite element approximation near the body and analytical representations elsewhere; the latter satisfies the Laplace equation, the boundary conditions at infinity, on the free surface and on the sea

bottom. The matching conditions of two regions are

affected as two natural boundary conditions in a localized variational principle. The nodal coefficients of the finite elements and the expansion coefficients in the analytic representation are solved from a matrix equation with a symmetric and banded matrix. We used triangular instead

of the quadrilateral elements used8. A typical set of

computational data

is

as follows: number of nodal

unknowns =200, number of expansion coefficients=30, semi-bandwidth = 20, core storage = 500 K, CPU time on IBM 370-168=0.5 sec.

POWER EXTRACTED FROM THE WAVES

Because )., is diagonal the average power absorbed by the power extractor per period is

The incident wave power is:

2k/i

\

c8=(i

_sinh2kh)

is the group velocity. The ideal efficiency, corresponding to the upper limit of useable energy, may be defined as:

L =

= (i;.2IiI2)/ígcq

(14) o o o pqB u

_¡)gB(X"X°)

= d,idS (7) where

e X33 ?.33 _Eopt

where

=3a

and/.33=)«33(pa4/g/a)_ I

For a cam with one degree of freedom, say, roll with i = 3 only, Mci3 and Evans4 have shown independently that for a given cam and frequency the optimum ideal efficiency is:

opt=1

+IA/AJ2

(15)

where A and A are the radiated wave amplitudes

towards x

+ and x -

-

r

due to forced rolling

motion at unit amplitude in angular velocity. This

efficiency is achieved when the cam inertia is so adjusted as to resonate at the incident frequency, so that:

K33 2M33 =0

(16)

while the energy extraction rate must equal the rate of radiation damping:

A33 =A33 (17)

COMPARISON WITH EXPERIMENTS

In refs I and 2 not all necessary data were published to

allow a strict comparison with the present theory.

Carmichael and Davis have carried out experiments

similar to Salter's. Details of their experiments and the major part of their findings will be reported separately. All information quoted here corresponds to the cam inertia I33I33/pa4=3.57.* Energy was extracted from a tor-sion damper. They adjusted the damping rate at each frequency to achieve the best efficiency for the same inertia. By comparing the measured damping rate with the theoretical radiation damping rate as in Fig. 1, it is

clear that the condition (17) for optimum efficiency is not satisfied in the experiments for all <0.65. The theoretical optimum efficiency curve, calculated from equation (15) is

All quantities withA_{are dimensionless.}

Fiqure 1. Comparison between theory and experiments ht Carmichael and Dar is. The svnthols are:

Salter's cam for energy from ocean waves: A. E. Mynett et al.

also shown. To compare further with the experiments we used the experimental damping rates ().') and computed the efficiency and roll response; the agreement between theory and experiments is very good.

CALCULATION FOR SALTER'S LABORATORY MODEL

Assuming that the published drawings have been properly scaled, we estimated the following geometric properties of the cam:a =radius of the circular stern = 5cm; X° =0, Y°

= 5cm; X= 5.15cm, Y= 2.1cm; X' = 1.39 cm,

Y =4.56 cm; = 4.25 cm, Y =0; V° =displaced cam area= 103.5 cm2; B=8.5 cm, 14=204.7 cm3.

We remark that the centre of buoyancy depends solely on the geometry and was found to be somewhat different from that indicated in the drawing2. The mass was not given but can be inferred from hydrostatics to be:

1i = f) V0 XV/XC_{= 28.0 g/cm. ,îi =in/pa2= 4.14}

We assume that the shaft is sealed within the cam so that its volumecontributes to the buoyancy but its inertia is irrelevant to the dynamics of the cam.

Missing from refs I and 2 are the combined inertia /33

+/133 and the damping rate ).'3. We choose an

in-termediate value of /33±p'3=lI88 g/cm so that the

dïmensionsless value is (/33 +3) = I .9. Within the mea-sured range 0.4<&< 1.0, Salter's efficiency curve can be

fitted

perfectly on the curve with

= A33!pa

= 2.41 (see Fig. 2(a) hatched portion). But different choices

of /33 and /.33 can achieve similar agreement, however. Figure 2(a) also shows a family of efficiency curves for fixed inertia /33, each with a different value _,. held

constant over all frequencies. For comparison the op-timum efficiency curve is shown as a solid line. Clearly some values of .' give a broader bandwidth than others. The bandwidth depends however on the cam inertia and hence on the resonant frequency. We demonstrate this point by the broken curve in Fig. 2(a) which corresponds to an increase of inertia by a factor of 7. The efficiency curve is much more peaked around the resonance fre-quency a=0.3 and only for damping rates close to the

theoretical damping rate at &=0.3 is

it possible to approach t he opti mum efficiency.

For the interia /33 +j33 = 1.9 which may have been used by Salter, Fig. 2(b) shows the corresponding induced roll amplitudes and Fig. 2(c) the horizontal constraining forces on the shaft axis. Figure 2(d) shows the vertical constraining force f, as well as the exciting forcef; their difference is due largely to the added buoyancy when the

cam rolls away from its mean position. Hence the

difference is generally the greatest near resonance. The

order of magnitude of the forces can of course be

estimated by pgA,

(2a) which is consistent with the

calculations, and can be so large in real seas as to raise questions on the supporting system.

EFFECTS OF CHANGING GEOMETRY

For the same water depth, we generate three groups of geometries. The submerged part of the cam is formed by a circular stern of radiusa,the bow is formed by a straight tangent inclined at the angle O with the horizontal. The

Applied Ocean Research, 1979, VoI. I, No. I 15

q, «o e 2 1.0 08 06 0.4 0.2 o

s -o

/

I

_f -'q-, /

/

o o _D

.*

D 2 Io 8

6,,

o 'q, 4 2 .0° 0.4 0.6 0.8 Theory Experiment A n

E

Salter's cam for energy from ocean waves: A. E. Mynelt et al.

LO 0.8-0.6 0.4 - 0.2-o 3.0 2.0 Iî I .0 X, fi

F'

-'

I

\

X33

N4

\\ '2.41

\.. .00

\

1.10 ¿

16 Applied Ocean Research. 1979, Vol. 1, No. I

." 0.20 I I

!\\ "

lîI

"..,,?

\

:0 \ 2À ....

\'\Il

1"

X33o.z 0 0.4 08 1.2 1.6

Fiqure 2. Calculated results for Salter's geonetrvfir rar ring frequencies and extraction ¡ Except otherwise indicated

each curte is for a fixed ¿' and¡33 = 133/pa4 =1.9.(a) EffIciency: the curte with narrow peak (...) is/or 133 = 13, ).' = 0.4; (b)

Roll ainplii,ide: (e) Horizontal total constraininq f oce ,f '. The horizontal exitittq force is 015(3 shown; (d) Vertical iota!

cons! rai;li,tq torce f',. The rertical excitinq force is a/SO sItot,, ( - -). All are Ilornulli:e(l bu pga for unit n'aie anl/)/iiti(le.

three groups correspond to three different depths of

submergence with the centre of the circle being (0, c:/2 and a) beneath the mean water surface. Within each group four dtfferent Osare taken: 0=45,600,750 and 90'.* Thus both the submergence and the keel slope O affect the degree of asymmetry. The water depth is kept constant: li

= 4a.

For each geometry all the important hydrodynamic quantities for both scattering and radiation are computed and are presented in Figs. 3(a) to 3(1).

The hull of conventional ships increases in width

monotonically from keel to beam, and all known calcu-lations show that the reflection and transmission coef-ficients for a fixed body, R and T, are monotonic in . On the other hand, when a body is completely submerged, the

R and T curves are known to oscillate in

due to

interference of waves which travel in both directions above the body. Even for a submerged vertical plate of zero thickness, R and T curves are also not monotonic''.

* _{Salters geometry is close to}

I) = 75 and submergenceawithh/u>> 1

whereas the geometryofCarmichael and Davisisbest approximated by

0=60 . submergencea/2,and h/a=4.

3.0 2.0 6.0 2.0 2.41 N. \. .0 5.00 'S._...

"

"_S o

As the submergence of a Salter cam increases, the corner beach on the shadow side also increases in extent. It ts possible that the wave motion in the corner contributes to the non-monotonie variation in R and T curves (Fig. 3a). We stress that in the present mathematical formulation the velocity is finite everywhere and there is no infinite value at the beach (the origin). Also. the segment of the x axis intersected by the cam is assumed not to be part of the water surface. Clearly in reality non-linear effects could be important in this case.

As the degree of asymmetry is reduced either by

increasing O or decreasing submergence, we notice that, interestingly, the optimal efficiency (Fig. 3b) remains almost the same. Especially for fixed submergence, the curves for different slope O can sometimes hardly be distinguished. However, this fact does not imply that the cam shape is immaterial in practice. Because .. dimin-ishes to zero as the wetted contour becomes more and more circular, (Fig.

3c), for finite power output the

amplitude of the induced roll (ch) must increase without bound, which not only invalidates the linearity assum-ptions but may also be undesirat for converter design.

b X33 20 ¡! !.!l. IO ,'! j.,,., \. \,. #7

\

". \.

\

'I,.

\\

f 003 d _{X33 0.2j''} I,.

,:/

-

'\ '".

\.

_r/

0.4 I.? t6 o 0.4 0.8 1.2 16 Eopt uf 0.8 1.2 .6 40 "C 2

This point has been made by Evans in his study of a

circular cylinder. The results here suggest that one should decide the cain profile by imposing restrictions on the roll amplitude; requirements on efficiency alone can be met by a wide range of shapes. In the same view, for a semi-circle

(O = it/2, s = 0), the constant optimum efficiency ( = 0.5) is a

mathematical limit having no practical significance.

Salter's cam fbr energy orn ocean waves: A. E. Myneit et al.

0.2

j

..;...-1774

_-.'i

,,e' /

j

1774 a/2-d../

/ 9="4,s0

z

,

fr-

'72 O //I

/

II

ç

_1/ ç o.'. '-. ...-.

p-I

_1T/20

I, _-<--'

It may also be noted from Fig. 3 that at

=0.5 the optimum efficienty r.

= I and at the same time RI =

and T=0. In order to check this numerical concurrence (without explaining it better, however), we investigate theoretically the condition that = I. From equation (IS) we must have A =0. Invoking a theoretical identity of Newman'':

Applied Ocean Research, 1979, Vol. I, No. I 17

s

1 33 and >33 0 7.0 5.0 3.0 .0 0.4 0.8 .2 .6

-ir /2,o \

/

--. "s

O0/2

X C 15 Q . 4.0 -30 -20 -I.0 -o 0.4 1.2 1.6 o 0.4 2 6 4.0 4.0 I--' / s / s 3-I î I 2.0 2.0 - _{ir/4 0/2} /2, Q .- r,4 -o 0.4 1.2 1.6 0.4 0.8 1.2 1.6

Figure 3. Hydrodvnamic properties/or atiimily of Salter cams/br various keel anqles(0) and submergence (s). (a) Reflectance IR! Wut trans/HiSS )1 ITI coefficients: (b) optimum effIciency: (C) added mass = p 3/pa4 wut radiatiie dampimi cocffìde;its

i 33/paJg/a (d) inagnaude of horizontal excitinq forc f

=11/pga (e) ,naqnuud of ICI tic al e ¡tlncj foice ft

=f/pga; (f) magnitude of exciting ,noment about 0,f =f/pga2.

0.4 1.2 1.6 d i' e 8 4, s O ir/ 2, 0 1774 _0/2

-1l'/2 0/2 O 0.8 Rl 0.6 and ITI 0.4 0.2 1.0 0.8 0.6 E0pt 0.4

Sa!ter.s cam for enerqt' from ocean waves: A. E. Mvnett ei al.

A+RA*_TA*=0

(18)

we conclude that Rj= 1,

since AO. Energy

con-servation of the pure scattering problem immediately

gives T=0. The general statement that energy for the

moving camwater system must also be conserved is

expressed by:

=I - IR +A - T-l-- A +/112 (19)

which leads to the condition that IR + A /AI =0, which iii turn implies the following relation between the phases:

IÒRò31=ir where R=IRIeR, A =IAIe1ôi

(20)

All these inferences are in agreement with the numerical results at & =0.5.

Lastly we also studied the effect of water depth by performing computations for Salter's geometry and h/a =4, 6, 12, and 24, keeping the same frequency range. The last ratio corresponds effectively to infinite water depth for the frequency range of Salter's experiments. The main conclusion is that water depth affects the hydrodynamical quantities and _'°r only in minor ways. This result, while theoretically not exciting, may be useful in design.

EFFECT OF NON-RIGID SUPPORT

Almost all supports to be built in reality will not be

perfectly rigid and it is therefore of practical interest to examine the effect of movable support. Although it is in principle possible to extract energy from all three degrees of freedom by suitable optimization34, this must involve additional difficulty in design and maintenance. More likely, the support will move in a passive, non-optimal manner. Since it is difficult to extract energy when the cam is totally free, motion of the support will normally be accompanied by a loss of efficiency. How much efficiency is lost depends of course critically on the type of support. A large variety of devices such as floating platforms,

mooring lines, steel frames can be imagined and the

number of the corresponding parameters can be large. To get some ideas, we have selected a model which has only a few new parameters and can be conveniently tested in the laboratory.

Consider the design shown in the insert of Fig. 4. The shaft of the cam is mounted rigidly on vertical rods which can move in vertical sleeves on a carriage. The carriage can move only horizontally with negligible friction. For simplicity we ignore the inertia and hydrodynamic re-action of the vertical rods and sleeves, assuming that they are sufficiently streamlined. In the laboratory this support system can be kept above water. The vertical motion is resisted by a damper with coefficient À,. We shall study two values for the mass of the carriage ,n:m=O and 'nc =

in, ignoring the added mass and damping of the

carriage.

All necessary hydrodynamic quantities are presented in Figs. 4(a) to 4(c) for Salter's shape. We now show some sample numerical results in order to assess the perfor-mance and efficiency. We base our computation on the values 133 +'33 =(3.27)pa4 and Ä' =1.10 pa4/g/a which would give l00 efficiency if the cam has only one degree of freedom at & =0.57. Several values of),, will be

18 Applied Ocean Research, 1979, Vol. I, No. 1

taken =0, 1, 10, and 100. The normalized mass of

the support will be taken to be ii=m/pa2 =0 and

,îi.

=4.14.

Figure 4(d) shows the normalized amplitudes of the induced motions for a typical case of2',2/).'33 =10. lt is

noted that z

becomes large as &-0. This singular

behaviour can be explained theoretically as follows. For heave and roll amplitudes z, and z the transfer matrix

(coefficient of z1 on the left of equation (4)) is dominated by

the finite quasi-static restoring force due to buoyancy, as & f0; the inertia and damping forces vanish in the limit. For the present example it can be calculated that:

o o o

= 0 1.7

1.445

0

1.445

1.355

The computed limiting values of the exciting forces are:

Îj

,Î/pga

1.8, _.Î

_T/pga2

1.4

which can be checked by a quasi-static theory. lt is then found that:

&,z,/A,=20.8,

3=z3a/A1=23.2

which are finite but too large for the drawing. On the other hand, owing to the lack of restoring force (buoyancy or mooring) the sway mode is resisted only by the body inertia, the added hydrodynarnic inertia and radiative damping. i.e.

f1=[aMi1,a

(21)

Note that p are finite constants. Hence in the long

wave limit the inertia term above is of the orderO(a2). We

now argue that D11=.1 is at best proportional topali2,

hence the bracket in equation (2) is of O(a2). The reasoning is as follows. From equation (8) ).xpaImq. Since the cam size is comparable to h by assumption, it follows from mass conservation that for kh-0, xh, hence:

¿i xf)cr1m2 (22)

The exciting force

f

on the left-hand side of equation (21) can be estimated via Flaskind relation:

/1+Çsinh2k/i

,,pgkA1

C

a2 cosh2kh

which gives, for small k/i:

I'

2q/m a

lt is further known that the radiative damping is:

I, _i_±' cinh2 L-Ii

O(2 (25)

cc1

j

pG

K33/a3-4.0 3.0

IîI

2.0 LO o 2.0 8.0 4.0 0.0 -4.0

\

'--.5-. 0.5 'S SS 33

'

'I2

Figure 4. Salter's cam with three degrees of freedom h

= 4a. (a) Exciting forces; (b) apparent mass coeffìcients: = p33/pa4, = /111/pa2, fL = i13/pa3; i = 3: (c) damping

coelficiciits: . = I.

/)(I/ J

LI. .jj =

i.1j)a2 Jj/a,

= ¿3/pa\/g/a: i = 3: (d) induced disp1acenents. Nunthers refer to the carriage mass 0: m = 0.4: th = 4.14. Cam data:

inertia 133 +/L'33 = 3.27: cam mass th =4.14; rate of energy

extraction from roll /.33= 1.1: water depth h/a =4: (e)

ef)ìciencv: th=O (--'-); th=4.14 (----)

1.0 .5 2.0

Salter's cam for energy from ocean waves: A. E. Mynett et al.

3.0-ii 2.0 0.5 l'o 0.8 0.6 E 0.4 0.2 5.0 4.0 2.0- 1.0-1.5 l'o 05 1.0 1.5 C I-2.0 \ t \\\ \ \ 'S

\

-.

''S

\\

T:;

'aIl-\

'k-\( -

\

O 54

\

)\ 4

I2i

-d

(26) since 1, and are finite. Heuristically, in a steady

poten-tial flow a body can move steadily at an arbitrary

velocity without experiencing a force, therefore it is not surprising that as the frequency 0 the displacement tx. From Fig. 4(e) it is important to note that the efficiency drops rapidly with increasing freedom in heave and sway.

( _{When the vertical motion is nearly eliminated, ).2=100}

X ,., the maximum efficiency is still somewhat less than

that of the lixed shaft case, this is clearly due to the

freedom of sway. Also the effect of the finite mass of the

(28) support seems to increase the efficiency slightly.

.50

A ¡plied Ocean Research. 1979, Vol. 1, No. I 19

0.0 0,5 5 2.0

o 0.50 1.00 1.50

which implies finally from equation (21) that: So that for small kh:

=0('ï"') for

or kh. I (29)

o 050 IDO

2

(.11)=poIm2(kliL3

kli1

Upon equating equations (22) and (26) we get:

A/1I=0(â312)

It follows from equation (24) that:

Salter's cam for energy front ocean waves: A. E. Mynett et al.

CONCLUSIONS

From these calculated results the following observations can be made:

For fixed shaft (one degree of freedom) calculations conlirm the experiments of Carmichael and Davis as well as Salter's in that Salter's cam can give very high efficiency over a broad frequency range. For fixed shaft the submergence and asymmetry in profile can be used advantageously to control the roll amplitude in design. The reacting force on the shaft is, however, always very large.

For a Salter cam with only partial constraint in

sway, heave and roll, the added degrees of freedom can reduce both the efficiency and the reacting force. Thus, safety and efficiency are two opposing factors in practical design.

Can there be wave energy devices which will require less support and still possess high efficiency? The floating mat suggested independently by Glen [-lagen of the USA and Christopher Cockerell of Great Britain'3 appears very attractive. Research is underway both in Great Britain and by ourselves.

ACKNO WLE DG EM ENTS

We thank Mr Dick K. P. Yue for his important assistance

in developing the computer program for the hybrid

element method and Professor Carmichael and Mr Brye Davis for providing us the experimental data. Financial support has been received from the National Oceanic and

20 Applied Ocean Research, 1979, Vol. 1, No. I

Atmospheric Administration, M.I.T. Sea Grant Program

and the Fluid Dynamics Program, Office of Naval

Research.

REFERENCES

I Salter, S. H., Wave power, Nature 1974, 249, 720

2 Salter. S. H., Jeffrey, D.C. and Taylor, J. R. M. The architecture of

nodding duck wave power generators, Natal Arc/nt. January

1976, pp 21-24

3 Mci. C. C., Power extraction from water waves, J S/tip Res..

1976,30,63

4 Evans, D. V. A theory of wave power absorption by oscillating bodies. J. Fluid Mec/i. 1976, 77, I

5 Newman, J. N. The interaction of stationary vessels with regular

waves, Proc. I Ot /i Svinp Natal Hvdrodrna,nics, London 1976 6 Evans, D. V.. Jeffrey. D.C.. Salter, S. H. and Taylor, J. R. M. The

submerged cylinder wave energy device: theory and experiment, App!. Ocean Res, to be published

7 Katory, M. On the motion analysis of large asymmetric bodies

among sea waves: an application to a wave power generator. Naco! Arc/nt. September 1976, pp 158-159

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