On the Basic Dynamics of Extracting Power from Waves

ON THE BASIC DYNAMICS OF EXTRACTING POWER FROM WAVES P. C. Parks Department of Mathematics and Ballistics, Royal Military College of Science, Shriven ha m, Swindon, SN6 8LA, England

Abstract. There is much current interest in extracting power from waves in the sea. This paper examines some basic mechanisms f o r extracting power, first from waves on taut strings and then from waves in the sea. In the sea a fl oating "exponential wedge" containing a tuned mass-spring system is proposed. This has the mathematical attraction that exact solutions of the coupled wedge and wave system can be found. Single- and double-sided systems are considered: single-sided systems suitably tuned can extract all the power fr om an incoming harmonic wave train whereas double-sided systems can usually extract only half this power. It is hoped that the paper will stimulate further analysis and design of wave power devices. Keywords. machines.

Underseas systems;

optimal systems;

INTRODUCTION The energy crisis facing the countries of the world has led to many schemes for energy conservation and to a search for new sources of energy. One such source is to be found in waves on the surface of the sea. Sinusoidal waves of amplitude h on the surface of the sea give rise to an energy density ~gph2 per unit surface area of the sea, where g is the acceleration due to gravity and p the density of sea water. This energy is propagated in the direction of the waves at the group velocity which, in deep water, is half the velocity of the waves themselves. The power thus delivered is impressive. Consider, for example, waves of amplitude h = Im and period 8s. Their group velocity u is 6.25 m/s and so the power delivered which is ~gph2u per metre of wave front works out to be 30 kW per metre or 30 ~~ per kilometre. Evidence presented to Parliament (House of Commons, 1977) quoted an average power of 80 kW/m and a total availability of 120 GW of wave power within UK territorial waters. A major problem arises in extracting this power. An obstacle lying in the path of the waves will in general reflect and transmit waves and it is possible that no power at all will be absorbed. Clearly some dissipative process must take place in the obstacle itself and here some kind of damped mass-spring system would appear appropriate. In this paper, as a first step, we examine power extraction from waves in taut strings the mathematical models here are more easily handled - and we then progress to the more difficult problem of waves in the sea. We

1537

linear systems;

wave power

propose a mathematically attractive device for absorbing power from sea-waves in the form of an exponentially shaped wedge containing a tuned mass-spring system with damping. A number of different arrangements are examined and optimised. By a suitable optimised design it is possible to absorb all the power from an incoming harmonic sea wave, although this ideal is unlikely to be achieved in practice. Standard mathematics of mass-spring systems, wave propagation and frequency response is used throughout in what is a first examination of the problem from a control engineering point of V1ew. LIST OF SYMBOLS A a b C c d e g h I i K k t M

Amplitude of potential function for incoming wave Amplitude of potential function for transmitted wave Amp] . "ude of potential function for reflected wave on string, also amplitude of vertical motion of wedge Damping constant of viscous damper Wave velocity defined by c 2 = T/p for string and as c 2 = g/k for deep water sea waves Amplitude of potential function for transmi tted wave from wedge Exponential function Acceleration due to gravity Height of wave Power absorption ratio

r-T Spring stiffness Wave number = 2rr/A Width of wave absorbing wedge Mass in mass-spring system

P . C. Parks

1538

p 7 t u

u u

x

w

x y

z

o

r,

e )

w

P

w

w

o

Distance 1n the direct i on of outwar d !1orr::al Hater pre ssure Ten si on in string Time :;r oup velocity of wa ves cl/I:. f o r de ep '..Jate r sea .. aves Ve l ocity vector Component o f ~ in x- di re ction

Spring , sti ffness Y. 1·1as s , 11

Displac enent o f lunped nass 1n mass Spri!1g systen , r elative to '..Jedge ~orizontal distance Tr ansve rs e deflection of string V ert~ cal dista.'lce abo ve sea- sClrface iwt Vert1cal dlsplacement of wedge = b e

:J: 7777

Danping ratio ~atio of lumped mass to mass o f st ri ng as defined by n = 2nM/pA ir, (l - 2 ~i /~)/(1 + 2~i~ - ~2) Wave length - i(l + 2~ i~ ) n /(1 + 2 ~i~ - ~2) , a l so ma ss ratio ~1/(!.1 + mass of we dge) Frequency ratio w/wo Line density of string or density o f sea water Density of wedge, but a less e r den s i ty (1 - w)p is als o used with Po = p o Velocity potential Frequency o f wave o r mass- spring system Undamped natural frequency of mass - spring system Laplace operator HAVES ON TAUT STRINGS

y

= Ae

i ( kx + wt)

A reflected wave is formed for x y

= be i( -

kx + wt)

and a transmi tted wave fo r x Y

= ae i (kx

and that K(a - z )e

iwt

o

We obtain a f ter some reduction a

2A , b =~ - e 2 - 9

=2

9

where

- in(l ~ 2~iq) 1 + 21:;1 ~ - ~

Here npA = 2nM , <

iwt

iwt (- Mz w2 + Ciwz ) e 0 o

0 as

>

Incident wave

Reflected wave

Waves in t aut st r ings

T(A - b - a)ike

We consider a mass - spring system with a damper s ituated at x = 0 on a taut st r ing with line density P and under tension T , (Fig 1). We suppose that a wave arrives f r om x = + = written 1n the complex form

777777

Transmitted wave

Fi g 1 .

Darnper , coo,tant C

~

w = ;;-,

o

0 as

C

w 2 o

M'

The abso r ption r atio 1S now

+ wt)

2

~

2

The wave velocity c is given by c 2 = T/p = w2 /k 2 , w being the f r equency and k the wave number whe re k = 2n/A , A being the wave - length.

This is maximised for any complex number 9 f o r 9 = - 2 when I = ~.

The energy in the first wave is arriving a t x = 0 at a rate ~pw2/A/2c and is being carr ied away by the reflected and t r ansmitted waves at r ates ~Dw2/b/2c and ;pw 2 /a/ 2 c respective l y .

If 9 = - 2 in this part icular case the n we must have I:; = n /4 1:; and 1:;2 = 1 - (I:;n/~) . Thus the mass - spri ng s y stem sho ul d be tuned so that w > w and the damp ing ratio I:; has to be

We ca.') define an abs orpti on r ati o I f or the device at x = o as 2 I =1 1*1 For the device shown in Fig '1 the bounda ry conditions at x o ar e t hat

chosen depending on the re l ation o f the ma ss r~ to t he mas s o f a gi ve n l e ngth of st r ing . For a solution to e xist this rat io a s give n by n must be such th at n < 1, when t wo solutions f o r ; and I:; a r e possib le . I f an alternative self- contained mass - spr ing system as s h own in Fig 2 i s us e d a s imi lar r es ult f or t he maxi mum absorpti on coe f ficient I = is obt a i ne d b ut wi th 9 r eplace d by ~ whe re

1~12

y(O , t)

ae

iwt

I

=1

- 12

= 91

- 12

91

o

1539

Dynamics of Extracting Power from Waves

i(l + 2I;il;)T) 1 + 21;if; - 1;2

Fig 2 .

Fig 4 .

Alternative absorber Fig 3.

Dashpot terminator Fig 5.

Dashpot alone

Two absorbers

This is maximised if 2k£ = 2rn r = 0, 1, 2 , . . = 94 whlch lS less than the best that

when I - 2 then

IfIJ

I;

'1/41;

<; .

as before

1 + 1;'11;, and so I; > 1 and it also follows that I; < ~ since '1 = 41;//1 - 41;2. So in this case the optimum tuning is for Wo to be less than w and for I; to be smaller than~. There is no restriction on '1 in this case. If a viscous damper lS used alone (Fig 3 ) then the absorption ratio I takes the same form once again with e replaced by _ Cw when I takes its maximum value of Tk

Cw

Tk= 2.

can be achieved by a single optimised absorber which is I = ~. However , the two absorbers together can in fact extract half the power if they are optimised together when e = - 1 for each , and '1 We see that the best that can be done with a mass-spring system in the middle of a taut string is to absorb half the incident power. A device at one end can absorb all the power if suitably tuned. In all these devices to achieve maximum absorption both the undamped natural frequency and the damping ratio must be adjusted correctly. There are some further interesting deductions that can be made in the calculations above, but we shall not pursue these further.

If the string te:rminates at x = 0

with a viscous damper (Fig 4) then the reflected wave has an amplitude b given by

\{AVES IN THE SEA Here we shall assume standard wave theory (Coulson, 1949) introducing a velocity potential ~ so that u = - grad ~ where ~ satisfies Laplace's ~quation V2~ = O. At the surface of the sea (z = 0) we have a 2

A(l -

boundary condition

~)

1 + Cw Tk

and clearly the reflected wave is suppressed

= 1,

when ~

= 1.

If two

absorbers of the type shown in Fig 1 are attached at x = 0 and x = - £ as in Fig 5 where each is optimised so that e = - 2 then the combined absorption coefficient turns out to be given by 2 2 I

=1

+ g

object immersed in the sea

b

entirely, so that I

a 4> W

- 14 _ :_2ik£1

-

14 _ :_2ik£1

.£.1 az

= 0 and on

~ in the

direction of the outward normal equals (-1) x the velocity of the object in that direction. The height h of a wave is given by h

=~ ~

evaluated at z

= O.

We shall

also assume the deep water dispersion relationship that the wave speed c is glven by c 2 = g/k or w2 = gk where k = 2n/A is the wave number. A wave coming from x = + 00 towards the origin may be written in terms of ~ as

P. C. Parks

1540

$ = Aekz + i(kx + wt)

and a reflected $

= ae

~ave

may be

$ = Aekz + i(kx + wt) + aekz + i(- kx + wt) ~ritten

as

kz + i (- kx + wt)

z is negative belo~ the surface of the sea. No~ it is difficult in general to match these solutions at x ~ + 00 ~ith local conditions in the vicinity of an obstacle at x = O. Ho~ever, noting the exponential form in z, ~e devise an exponential ~edge sho~ in Fig 6 as a tentative ~ave absorption device. We suppose that the ~edge bobs up and do~n ~ith a harmonic displacement in the vertical direction given by

~here

z +'

3 • .-<

o

= be

iwt

p

o

. iwt ( A + a ) £plwe

£

"z

0

Z

0

Jz=-oo

rl

al C

£

u 0

o

+'+'

x

0

El

p

~I at x=o

£ke kz dz and the second term

If the density of the

2rr/k

Reflected ~ave: velocity potential kz+i (- kx+wt) $ = ae

Fig 6.

Exponential

Incoming ~ave: velocity potential kz+i (kx+wt) $ = Ae

~edge

_ £ke kz biwe iwt

Po = P, as it

.

mass-sprlng system

.

~lth

a mass M =

~

~

k

(0 < ~ < l),(Fig 7). Thus the ~edge of lesser mass together ~ith its interior mass ~ill just float in a calm sea.

In the sea

Transmitted

II ( _ Aik + aik)e kz + iwt

Reflected

Fig 7.

Wedge

~ith

~ave

internal absorber

Fig 8.

ax

~ave

-

= ux

assuming the slope of the ~edge is steep compared ~ith the horizontal, ie £k « 1. The motion of the ~edge may no~ be related to the ~ave motion by ~riting

assuming

~edge

~edge

must be for the ~edge to float (by Archimedes' principle), then since w2 = gk the equation of motion for z ~ill be excited at the o resonant frequency of the bobbing motion. This is an unexpected property of the exponential ~edge. We shall no~ ho~ever assume that the ~edge is of lesser density (1 - ~)p and therefore of massp(l - ~)£ /k , but that it carries in its interior a damped

The velocity in the x-direction at the point z lS then

x

x

comes from the loss of buoyancy as the rises from the sea.

'r-l'r!

u

~idth,

is, for a unit

On the right-hand side the first term comes from integrating the resolved pressure term

N

(l!

t; - gpz.

The second term gives rise to the usual buoyancy force ~hich ~e assume is equal to the ~eight of the floating ~edge. If the density of the ~edge is p then its mass per £ 0 Po unit ~idth is k and the equation of motion

k

(l!

>

the pressure is given by p = P

.

p

H

for an incident and reflected ~ave. There is no transmitted ~ave. The equation of motion of the ~edge in the vertical direction may no~ be ~ritten do~, bearing in mind that

Double-exponential

~edge

1541

Dynamics of Extracting Power from Waves Let w be the upward displacement of the interior mass M relative to the wedge. The equation of motion for this mass ~ill be

= - K~ - cW o and the equation of motion for the no~ be M(w

f2.!.

as before but

CONCLUSIONS

~edge ~ill

(A + a) iwt + K~ + cW p1.iwe 2 - p1.gz o

0

(A + a) iwt p1.iwe 2

b iwt e ,

or putting Zo

remembering that gk

.l!£.!. ("~ k

= ~ 0 e iwt

~

= w2

+

"z

) 0

and

~e obtain

2lJw2~

____ 0

A+ a

s =~ k1.

I = ~. This is in fact the greatest possible value of I in this case.

+ ~ )

(1 - i1 )~

k

which is a maxlmum when

+ w 2)kiw

kiw

.

on puttlng

K

o

M = Wo

2

and

C

M = 2~wo'

We have a second equation for a and b coming from the expression for u this is x A - a

A preliminary investigation into absorption of po~er from harmonic ~aves in strings and ~aves in the sea has been undertaken. It is possible t o absorb only half the po~er using damped mass-spring systems in the middle of a taut string. This is true also for t~o­ sided symmetrical shapes bobbing in the sea, but an unsymmetrical design such as an exponential ~edge ~ith one plane vertical face ~ill not transmit ~aves on from the vertical face. The exponential face ~ith a .. proflle glven by x = 1.e kz( - 00 < z < 0 ) can absorb completely the po~er delivered in a harmonic ~ave train of ~avelength 2'TT/k given an internal damped mass-spring system ~ith appropriate resonant frequency and damping. The resulting amplitudes of the ~edge and the resonant internal mass can be calculated in terms of the ~ave height. Based on these considerations a 1/100 scale model has been built for ~ave tank trials, Fig 9.

1.bw.

We should no~ like to absorb all the incoming power, in ~hich case there ~ill be no reflected This will imply that ~ave so that a = O. supporting strings b

A

kiw(- w2 + 2~iww

o

This is possible only if Wo

+ w 2) 0

=w

and ~

= Tk

.

We can examine the resulting amplitudes of b and w in terms of the wave h, wave-length A and wedge beam dimension 1.. The wave amplitude h

= Aw g

from standard wave theory and A

= k2'TT

Hence Ibl Iwl

A _~_£L 1.w - 1.w - 1.gk 1 h Ib I = 2; . 2~

Since 1.k «

1 or 2'TT1. « A

h

x

A

2TI1.

and

1 it follows that the

bobbing motion b will be much greater than the wave amplitude h and the motion of the mass M relative to the wedge will be of the order of h depending on the proportion lJ that M bears to the mass of the whole device. The vertical side at x = 0 is an advantage since the vertical bobbing motion does not generate transmitted waves on the x < 0 surface of the sea. If we consider a double exponential wedge (Fig 8) then a modification of the theory above gives Fig 9. I

(1

+

~~1.)

2 when wo

=w

Photograph of 1/100 scale model wedge: height of wedge 35 cm

1542

P. C. Parks

Many further problems remaln: in particular, the problem of converting the work done by the viscous damper into hydraulic or electrical energy and mathematical modelling of this process. Here there is also scope for non-li near oscillation theory and also for adaptive control concepts in this new field of endeavour. ACKNOWLEDGEMENT This paper was prompted by discussions ~t the Oxford Study Groups in Applied !4athemahcs, March 1977 , in which the more difficult • ft hydrodynamic problem of a rotatlng Salter duck" was discussed. (Salter , 1974, Evans , 1976, Longuet-Higgins, 1977).

REFERENCES Coulson , C A (1949) Waves, Olive r and Boyd, Edinburgh Evans, D V (1976) A theory for wave-power absorption by oscillating bodies , J Fluid 14ech 77 pp 1-25 House of Common~(1977) The development of alternative sources of energy for the United Kingdom, Vols 1-3, HMSO, London Longuet-Higgins, M S (1977) The mean f orces exerted by waves on floating or submerged bodies with applications to sand bars and wave power machines. Proc R Soc A, 362, pp 463-480 Salter, S H (1974) Wave power. Nature,~, pp 720-724