Journal of Sound and Vibration (1973) 29(1), 113-128
THE FIFTH ANNUAL FAIREY LECTURE: ON THE LINEAR
REPRESENTATION OF FLUID FORCES AND MOMENTS IN
UNSTEADY FLOW
R. E. D. BISHOP, R. K. BtJRCnER AND W. G. PRICE Department of Mechanical Enghteerhlg, University College London, London WCIE 7JE, England
(Received 25 January 1973)
When a rigid body departs from a uniform motion in a fluid, it is usual to assume that the forces and moments exerted on the body at any instant by the fluid are determined by the motion at that instant. This is the assumption of "quasi-steady flow" and it is on this basis that the fluid forces and moments are represented. Although this approach often suffices for aircraft it is known to be quite inadequate for ships, with the result that com- plicated and diverse non-linear fluid representations have been devised.
It has recently been found, however [1, 2], that an unambiguous representation of the fluid forces and moments on a ship can be formulated in terms of Volterra series. This theory allows the assumption of quasi-steady flow to be relaxed and also has a simple approximate form. The latter does not destroy the linearity of the representation with which fluid forces and moments may be specified whilst the ship performs a non-steady motion. This fresh linear theory, which may well supplant the semi-empirical non-linear theories in current use, is explained and developed in simple terms. In particular, com- patible definitions of "fluid derivatives" and "oscillatory coefficients" are presented.
PREAMBLE BY R. E. D. BISHOP
First o f all I should like to thank the University o f Southampton for inviting me to give this, the fifth Fairey Lecture. It is a compliment that I much appreciate. When casting about for an appropriate subject m y thoughts naturally turned to aircraft and, thence, to a rather basic matter that has interested me for several years--in fact ever since my period as chairman o f the Oscillation Sub-Committee (as it then was) o f the Aeronautical Research Council. I refer to the way in which fluid forces and moments applied to a moving body are represented when the flow is unsteady.
Over the last few years, I have not merely remained very interested in this subject myself but have drawn the attention o f certain colleagues to it. Several o f its aspects have been looked at as a consequence and the occasion o f this lecture seems an opportune time to describe in simple terms the present state o f play and to develop one or two o f the more promising lines o f thought.
1. INTRODUCTION
A solid body moving in a fluid is subjected to a distributed force. This paper is concerned with the representation o f such a force for the purposes o f analysis. This is a subject that has received much attention in the past with particular emphasis having been placed on it by t To be presented at the Institute of Sound and Vibration Research, University of Southampton, on 16 July 1973. These annual lectures on topics in the general field of sound and vibration are made possible by a generous gift to the I.S.V.R. from the Fairey Technical Education Fund.
114 R.E.D. BISHOP ETAL.
aircraft and ships. As in most previous studies, but mainly for reasons o f simplicity, we shall here assume that the solid body is rigid.
When a rigid body, such as a ship, departs from steady motion ahead, the fluid in which it moves (i.e., the water) exerts a resultant force and a resultant moment about its centre o f mass. These have to be specified in terms o f the perturbation o f motion for the purposes o f analysis. It is commonly assumed that the force and moment are uniquely determined at any instant by the prevailing perturbation o f motion at that instant. "Slow-motion derivatives" are defined on this basis and the theory is known as one o f "quasi-steady flow".
Generally speaking this approach is quite satisfactory for aircraft. But it is not satisfactory for ships because (i) only "velocity derivatives" are capable o f direct measurement (the other sort--"acceleration derivatives"--requiring special apparatus whose results are by no means unambiguous); (ii) the predictions o f quasi-steady flow theory are accurate only for very gentle manoeuvres. This second defect has led to the adoption o f empirical representa- tions o f fluid forces and moments with little agreement as to which particular empiricisms should be taken as standard.
One outcome o f this unsatisfactory state o f affairs is that the m o d e m literature in ship dynamics has become inordinately complicated. It was, in fact, this complexity which provoked the investigation that is reported here, and so we shall refer specifically to ships. The arguments appear to be equally valid for any rigid body placed in a flowing fluid, though the problem o f a rigid aircraft is o f much less pressing importance than that o f a rigid ship. There is no doubt that the basic assumption o f a quasi-steady flow is vulnerable. It is well known (and, indeed, obvious) that the flowing water exhibits a form o f " m e m o r y effect". The adjustment o f circulation in the water around a hull is associated with vortex shedding. As long as the vortices remain in the immediate vicinity o f a hull they will influence the fluid force and moment exerted on the hull.
In two recent papers [1, 2] the restriction to quasi-steady flow has been relaxed. The work is based on the use o f Volterra series: i.e., on the use of"functionals". Now a Volterra series has a linear one-term approximation which is in the nature o f a convolution integral. It has been found, at least in the problems we have examined, that the linear approximation is surprisingly accurate. Once memory effects are allowed for, linear theory is by no means useless, even for ships.
The introduction o f an allowance for memory effects into linear theory strongly suggests that the concept o f a slow-motion derivative needs refinement. The purpose o f this paper, then, is to develop the linear theory in such a way as to include the memory effects (which certainly exist and, with ships at least, have large effects) and to arrive at a compatible definition o f derivatives. Further, the oscillatory coefficients, which have found a place in both the aeronautical and naval architecture literature, are similarly re-defined.
2. CONVOLUTION
The one-term approximation to a Volterra series can readily be introduced by analogy with the familiar Duhamel Integral. Suppose an excitation F ( t ) is applied to a linear time- invariant system.t The response q at some coordinate can be expressed as
co
q = I h(z) F ( t - "c) dz = q[F(t)], say. The function
h(z)
is the unit impulse response and it is such thath ( ' O = O, "r < 0 "j" A list of n~.ation is given in Appendix IL
FIFTH ANNUAL FAIREY LECTURE and the excitation
F(t -- t ) = 0, t > t.
This well-known result embodies an allowance for a form o f " m e m o r y " .
115
x U
v u
Figure I. Body axes fixed to a ship with the origin at the centre of mass C. The reference motion is O~ in the direction Cx and ~ represents the velocity of drift (or "sway").
F o r the sake o f definiteness, we now consider a ship which performs a motion o f drift (or "sway"), v(t), as it moves ahead with its reference velocity O as shown in Figure 1. The parasitic motion will be associated with fluid forces and m o m e n t s and we shall consider a t y p i c a l force component A Y. We know that A Y depends on v and we may express the
dependence in the f o r m A Y[r(t)]. By analogy with the Duhamel Integral we have oo
A Y[v(t)] = f hv(t)v(t - t ) d t .
The function h~,(T) is n o w the variation of A Y caused by a "unit impulse" o f v: i.e., by a sudden unit step to starboard. And, again,
h~(t) = 0, t < 0, v(t - - t ) = O , t > t.
It seems likely that the form of functions like hv(t) is o f cardinal importance in ship dynamics. It is therefore unfortunate that these quantities cannot be measured directly with much accuracy since the necessary sudden step o f displacement (or of rotation) cannot be imparted accurately, even to a model. On the other hand, it m a y well turn out that adequate theoretical techniques will be formulated so that we shall see the emergence o f the hydro- dynamic counterpart o f "indicial aerodynamics".
3. THE H A R M O N I C RESPONSE FUNCTION FOR A F L U I D FORCE OR MOMENT Reverting for a m o m e n t to the Duhamel Integral, we note that the response q(t) to excitation F(t) m a y be written in the alternative form
l ~o e-U~(t-~ d t ]
- c o
by the Fourier Integral Theorem [3]. The quantity co(co) is the appropriate receptance o f the linear system, being the complex response at q to a harmonic excitation F(t) o f u n i t amplitude
116 R . E . D . BISHOP E T A L .
and frequency 09. If now we put F ( t ) = 6(t), then q is by definition the unit impulse response so that
eo
,f
h(r) = ~ a(og) el'~*dr - - c O
The quantities h(T) and ~(o9) are a Fourier transform pair.
I f this result is re-interpreted in terms of the system of Figure I, we see that hv(~) is related to its Fourier transform H~(r by
1
h~(x) = ~ J n~(r eS'~'dog. ~ e o
And Hv(og) is the drift force A Y per unit amplitude of v(t) during sinusoidal drift motion of frequency co. The inverse relationship is
c0
H~(og) = f h~(lr)e-U~*dr
so that if we can measure H~(r.o) we can deduce ho(t) and vice versa. Whereas h~(T) would be very difficult to measure, its transform is quite straightforward in this respect. Harmonic response functions like Hv(r are what one really measurest with a planar motion mechanism (PMM) [4, 5, 6].
.J
:, ) v , A~.ody,
Figure 2. Representation of a planar motion test by rotating lines. The shorter lines represent amplitudes of drift force and the longer lines represent amplitudes of displacement and velocity during a parasitic drift motion.
In general Hv(og) is complex because A Yis not in phase with v(t). Perhaps the obvious way to show the form of Hv(o)) is to present it as a locus in the Argand Diagram. This, however, is not easy to do with much conviction as we are unsure how typical such a plot would be and because published results do not exist for very high frequencies co (and are also open to question at very low). We prefer, instead, to follow a practice that has grown up in the naval architecture literature, since these defects can then be allowed for.
Consider the way in which H~(og) is measured with a ship model. The model is towed along a towing tank at the constant reference speed and, at the same time, it is oscillated in drift by superimposing the lateral displacement
y ( t ) = Yo sin tot.
The resulting fluid force A Y ( t ) is measured--or, rather, its "in phase" and "quadrature" components are. Let the amplitudes o f these components be A Yi, and A Y~uad so that the
t The writers have come to the belief that this is the only logical interpretation of oscillatory testing, either in a wind tunnel or (with a P M M ) in a towing tank. In short, a P M M is a "Fourier transform machine".
rotating vector representation is that shown in Figure 2. If the amplitude of drift velocity is Vo, we have
A Y[v(t)] = f ha(z) Vo cos co(t - r) dr
--co
= Vo cos cot f h~(r)cos (or dr + Vo sin o)t f h,(r)sin o~r dr
-co
--co
But
Therefore
= A
Yl.
sin tot + AY.u~a
cos cot.H~(og) = f h~('c) e -''~' dr
--co
f hl:(r)coso~'~dr- i ; ht~(r)sin('~
A Yquad
- - , /JoAY,.
VoThe quadrature and in-phase components for a particular ship model are shown plotted in Figure 3. The results were obtained by van Leeuwen [7]. Now although they are admittedly
_--- (say) -JC ~ ( .~ -39' ~" _ 4 0 I < ~ * ~ ( o ) - 5 C f L I 0 n u I ~ _~- ~,~ x....~-.-x-..ff/.._ x
f
9 ~
(soy)
~, -~ x x ~'~ ~-" (b) <31~ -io i x v 0 5 I 0 15 ~ ( r o d / s )Figure 3. The results [7] of a planar motion test in which a ship model executed a sinusoidal drift motion as it was towed along a tank. Curve (a) shows the variation with impressed frequency of force in quadrature with drift displacement (and in phase with drift velocity); curve (b) shows the force in phase with drift displace- ment. Curve (b) shows the experimental points (o) and also points calculated from curve (a) under the assumption of linearity (x).
118 R . E. D . B I S H O P E T AL.
incomplete in the sense that the frequency range is not as large as one would wish, these curves a p p e a r to exhibit the salient properties that are o f interest.
T h e curve o f
A Y,~,JVo
falls as co is increased, but it is uncertain whether it will fall to zero or not. W e shall h e n c e f o r t h assume thatA Y,~,aa/Vo
has a n o n - z e r o asymptote and, indeed, p r o d u c e evidence to the effect that this is so.T u r n i n g to Figure 3(b), we perceive that
A Y~,[(-vofo)
has a l m o s t certainly g o t a n o n - z e r o a s y m p t o t e as co - + co. This is foreshadowed by potential flow theory and, as we shall see, is associated with the concept o f " a d d e d mass".4. T H E F O R M OF T H E U N I T I M P U L S E RESPONSE F U N C T I O N
F r o m measured functions A Yin, A Y~uad we can deduce the forms o f H~(fO) and HR(fO), respectively. We therefore have a means o f finding h~(r).
Let
lim [A
Y~n/(-VofO)]
= lim [H~(fO)/fO] = Y~(,z), ~-->cO [;J --~ zOmerely e m p l o y i n g this as an abbreviation for the time being. Then, H~(fO) = 09 Y~(co) + H*'(fo), where H~*r(fO) is such t h a t
lim H*l(fo) = O. But this implies t h a t
r
f
h~(r)
sin for d r = - f o Y~(co) - H*I(fo) - - c of r o m which we see t h a t h,,(r) must have the f o r m d
hv(~) = - r'~(co),~(r) ~ + h*(~),
where h*(r) is a function possessing a Fourier t r a n s f o r m such t h a t co
f h*(x)
sin fOT d r = - H * I ( w ) . This is because + h*(r) sinfOrdr = -fOY~(co) + h * ( r ) s i n f O r d r --cO = - c o Y~(co) -- H*'(fo). We have already agreed to allow a non-zero a s y m p t o t e f o rA Yqu~a/Vo.
Letlim [A
YquaJvo]
= lim HR(fO) = Y,(co), O - o c O rwhere again the c o n s t a n t Y~(co) is to be interpreted merely as an abbreviation at this stage. T h a t is to say
HR(fO) = Y,(co) + H*R(fO), where
FIFTH ANNUAL FAIREY LECTURE 119 This in its turn implies that
eO
f h,.(r)cos
cot dr = Yv(oo) + H*R(~o) - - t oso that h~(~) must have the form
h~(r) = y~(oo),~(r) + h**(r),
where h~**(z) is such that
c o
f h**('r) cos o r dr = H*R(r - c o
The two expressions for h~(~) are not incompatible. There is no real need to distinguish between h*(r) and
h**(r).
Moreover the terms containing Y;.(~o) and Y,(~) may be added together since the former plays no part in determining HR(r and the latter has no effect on H~(a 0. To sum up, then, the curves of Figure 3 imply thatd
11#) = - y~(~o) 6 ( , ) ~ + y~(oo) ~(r) + h*~(r),
whence
A Y[v(t)] =
- y~(oo) 6(r)-~r + Y~(oo) 6(z) + h*(r) vCt - r) dr
= Y~.(oo)~(t) + Y~(oo)v(t) + f h*(x)v(t -
r)dr.- - r
This general result calls for certain observations.
First we must remember that y~(oo), y~.(oo) are still to be thought of as mere abbreviations. They are not, in particular, to be interpreted as slow-motion derivatives. (It is tempting to identify them with Y~ and Yv respectively and to regard the integral
e o
f h*(r) v(t - r) dr
as a "correction" for time history effects.)
Secondly we may note that the term containing the Dirac function and operator may be integrated by parts to give
f Y 6 ( o o ) f ( r ) ~ v ( t - z ) d r =
]Y~(oo)6(r)v(t-r)
-
Y~(oo)r(t-r)~6(r)dr.
l O O --0O - - o o
That is, we may write
ho(r) = y~(oo) ~(r) + yo(~o) 6(,) + h~*(r).
Thirdly it is worth noting that, just as h*(r) can theoretically be deduced from either of the curves in Figure 3, so one can start from one curve and
deduce
the other (apart, that is, from the constant asymptotic values). This has been done [2] for the particular set ofexperi, mental results employed in Figure 3 and the computations gave remarkably accurate confirmation of the linearity upon which the whole of this theory is based.120 R . E . D . BISHOP E T A L .
5. INDICIAL PARASITIC VELOCITY
The force A Y m a y be described in an alternative, though related, way. Still retaining drift motion by way o f illustration, suppose that the ship acquires a sudden constant velocity to starboard instead o f a sudden step displacement. In other words, having considered Case I o f the following table we now contemplate Case II:
Motion to starboard superimposed on steady forward velocity
A
Case "Indicial (step) Impulsive (delta) 9
I Displacement Drift velocity v
II Drift velocity v Drift acceleration b The alternatives are those shown in Figure 4.
v ( t Cosel
Cose tl
0 I
Figure 4. The two types of "unit motions". Case I is a sudden unit step to starboard taken during the (otherwise steady) reference motion. Case II is the sudden acquisition of a unit drift velocity to starboard.
It is evident that we must now employ the form
co
A Y[t)(t)] = 1
h~(z)(;(t - z)dz,
where
h~(z)
is the variation o f A Y caused by a unit impulse o f b, such that h~(~) = 0, z < 0,~(t - 0 = 0, z > t.
A unit impulse o f b, being the imposition o f a unit indicial drift velocity, would be impossible to impose accurately so that, like
h~(z), h~(O
would be impossible to measure directly. Once again it is necessary to have recourse to oscillatory testing and, accordingly, we now re- interpret the results shown in Figure 3.Let us return to our analogy with the Duhamel Integral. We can conveniently borrow another result from the linear theory o f transient vibration. Figure 4 indicates that the quantities
h*(t)
andh~(t)
are analogous to an impulsive receptance c~,~(t) and an indicial receptance c~,~s(t), respectively. Now we have freely employed the relationship that each o f these types o f receptance has with its corresponding harmonic receptance ~,(r this being based on the Fourier transform theory. But it is well known (e.g., see reference [8]) that suchFIFTH ANNUAL FAIREY LECTURE 121 receptances are also related to each other. In the present context the a p p r o p r i a t e relationship is
d
h*(t)
=
h*(O)
6(t) + ~th*(t)
if we leave out the p a t h o l o g i c a l c o n t r i b u t i o n s to
h,,(t)
a n dhi(t)
t h a t we have isolated. O u r a n a l o g y implies t h a t there exists a functionh~(t)
such t h a tA Y =
y~(oo)t3(t)+Y~(oo)v(t)+
h*(O)6(t)+-d-~-Jv(t-z)d~
fdh
= Y~(~o)i;(t) +
[Yv(o~) +h*O)]v(t) + j dr v(t --
z)dz.- o o
I f the definite integral is integrated b y parts this m a y be written oo
a r[~Ct)] = r~(oo)i;Ct) + [y~(oo) + h'~(O)]v(t) + j h*(O~Ct - ~)d~, w h e r e again
h * ( z ) = O if z < O , b ( t - x ) = O if x > t .
T h e first o f these conditions follows f r o m that previously i m p o s e d o n h~(z) while the second m u s t be true if 1)(t - ~) = 0 f o r z > t.
F o r oscillatory m o t i o n
v(t)
= VoCOScot
this result b e c o m e sA Y[~(t)] =
Y~(oo)(-Vowsintot) +
[ y~(~o) +h*(O)](VoCOStot)
- Vo r sin cot. H*R(to) -- Vo to cos tot. H*I(r = A Y~, sincot + A YquadCOS tot.
T h a t is
Y[I1
= y ~ ( c o ) - H * R ( c o ) , (-Vo~O) A Y~u~d ro(oo) + h*(0) - to H~ (oJ).*' VoT h e s e are the quantities t h a t are conventionally plotted in P M M tests and it is n o w o f interest to e x a m i n e their b e h a v i o u r at the limits co = 0 and to --- ,z.
Since
lim H*R(to) ---- lim ~o h*(z) cos toz d r = 0 J
- c o
because, b y hypothesis, h ~ ( 0 possesses'a F o u r i e r integral, it follows that lim (
A Y,._____~ = y~(oo)
~ \-OoCO/ as one would expect. Also,
lim = y ~ ( o o ) - h~(z)d~. o-~ \-Vo to/
122 40 2(3 C ~ - 2 0 ~" - 4 0 - 6 0 - 8 0 30 2O
"~ .,~
R . E. D . BISHOP E T AL. i I i i (a) I T I I i I i i/
(b) I I l l 0 0 5 I 15 2 2 5 t (s)Figure 5. Unit response functions (minus the de|ta function contributions) calculated from the results shown in Figure 3. Curve (a) is h*(t) and curve (b) is h~(t).
Turning to A Yauaa, we first note that
co
* $ I f
h~(O) - ~oH,~ (o9) = h*(O) + h*(r) a~sin o.~r d r
--co co
= h * ( O ) - f h*(r) d -co cl-~r c~ ~ dr"
W h e n the integration is performed by parts it is found that
co ,
fdh:
h*(0) - o~H'1(o9) = h~(0) + j dr cos~ordr - c o co = h*(0) + f [h*(r) - / 1 " ( 0 ) 5(x)l cos wr dr - t o co = f h~*(r) cos ~or dr. --coSince this quantity tends to zero as o9 tends to infinity, it follows that
lim'(A Yquaal = y~(oo)
o-,~\ Vo /
again as one would expect. At the other limit, we have
lira [
[A r"~
= lim [ Y~(r + h*(O) - coH*'(co)] ~,~o\ Vo ] ~,~oFIFTH ANNUAL FAIREY LECTURE 123 I f we denote this value by the abbreviation Y~(0), then we have
Y~(oo) = L(0) - / , * ( 0 ) and from Figures 3(a) and 5(a) it would appear that, here,
- Y ~ ( ~ ) - 5 + 2 3 = 2 8 .
This value by no means approximates to zero so we have justified the retention o f y,(oo) in the text.
It will be seen that, in this expression, Y~(0) can be measured accurately (by an oblique towing test). On the other hand, y,,(oo) is the quantity about which we voiced some initial uncertainty. This issue now seems to be resolved. Figures 5(a) and 5(b) show the results o f
an attempt to compute functions h*(O and h*(z) from the data of Figures 3(a) and 3(b).
6. MOMENTS OF THE h*(r) CURVE
Looking ahead a little, we may note that terms containing Dirac functions in the expressions for h~(z) call for little comment. By contrast, the remaining component, h*(z), is unfamiliar, cannot be ignored and is used in a convolution integral:
oo
f h*(O r ( t - r) dr.
- o o
It will be helpful therefore to examine this integral more closely.
Consider a Taylor's series representation of the drift velocity v(t) in which we " r u n the clock backwards". Thus
T 2 r 3
v ( t - ~) = v(0 - ~,~(t) + ~ ( t ) - ~ ~(t) + . . . . Evidently this permits us to write
; [
1
A Y[v(t)l = Y~(~) i~(t) + Yv(~) v(t) + h*~(O v(t) - ~iJ(t) + ~iJ(t) - . . . d~ so that the drift force can be represented in the series form
[
;
] [
!
]
A Y[v(t)] = Yo(~) + h*~(O d~ L'(t) + Y~(~) - zh*(t) dr i~(t) +
- o o
This result shows that we can replace the convolution integral containing h*(r) by a series whose coefficients are determined by successive moments o f area o f the h*(r) curve. We have thus an alternative way o f allowing for time history effects, as was foreshadowed in reference
[6]. This aspect is examined in further detail in Appendix I.
7. FLUID DERIVATIVES
The concept o f a force or moment derivative (a "slow-motion derivative") is due to Bryan [9]. For the problem we have taken, he assumes that at any instant the fluid force depends on the prevailing parasitic velocity v(t) and acceleration ~(t). Thus
A Y = r v v ( t ) + r ~ ( t ) where Yv and Y~ are constants o f proportionality.
124 R . E . D . BISHOP E T A L .
Justification for calling Yv and Y~ "derivatives" can be had from the common practice [10, 11, 12] of treating (in this case) A Y as a function o f the two quantities v, ft. If A Y(v, O) is now expanded in a Taylor series about the steady reference motion, for which v = 0 = 0, it is seen that
aA Y , OA Y
ro = ~ o~o~ r~ = ~ ~ o ~ "
Bryan's constants of proportionality are then partial derivatives. The weakness o f this particular argument is that v and b cannot be truly independent o f each other as is required by the Taylor series.
The sheer brilliance o f Bryan's hypothesis needs no emphasis since it continues to this day to serve in aeronautics, the field for which he formulated it. The basic assumptions were made on the strength o f physical intuition, being (a) that A Y depends linearly on o and O, and (b) that time history effects can be ignored. The standing of the second of these supposi- tions is examined in the light of the functional analysis in reference [1] and we merely note here that if h~*(0---A~(z) (so that the fluid forces are "instantaneous") Bryan's theory is completely justified. This is because our analysis gives
A Y[v(t)] = Y~(oo)b(t) + [ Yv(oo) + A i r ( t ) so that
Y~ = Y~(~o), Yv = y~(oo) + ,4.
Suppose that we do not make the second of Bryan's assumptions, but instead allow that history effects may not be entirely negligible. Instead of being a Dirac function, h*(t) has now to have some less specialized form. (It cannot be zero.) The series specification of the fluid force A Y now shows that Bryan's formulation gives a first approximation to the truth if we take co y ~ = y~(oo)- f zh~*(z)dr, - - o 0 cO
Yo = ro(~) + f h*(O dr.
- - c OThis is, of course, under the assumption that successive terms in the series are less and less significant so that curtailment after the second term is admissible.
It therefore appears that, by suitably interpreting the constants of proportionality in Bryan's theory, we can extend its applicability to systems in which time history effects are non-negligible. The theory then becomes an approximation. We may note that if Bryan's assumption of quasi-steady flow is made, so that the fluid force is "instantaneous", then that approximation becomes exact.
8. OSCILLATORY COEFFICIENTS
It has already been indicated that oscillatory testing of aircraft and ship models is by no means uncommon. Also we have suggested that the purpose of such testing should be the determination of Fourier transforms like H~(og). This, however, is not the standpoint adopted hitherto and we come now to examine what is normally done, and to do so in the light of the functional analysis.
F I F T H A N N U A L F A I R E Y L E C T U R E 125 If the motion v(t) --- VoCOSo9t is imposed on a model, then the force deviation A Yis given by
A Y [ t ' ( t ) ] = Y~(~o)+ h~*(z)d~ _ ~ 2 = z 2 h . ( z ) d T + 2 ! -oo
).co.o.+
+ Y~(~) - _ ~h~*(~) d~ + o9~ ~ -~ ~h~*(~) d~ -
) ),
- o9 4 z s h*(z) d~ + . -Vo o9) sin ogt.
Since ~(t) = -ogvosinogt, this result may be written in the form A Y[v(t)] = ~'~b + ~'vv where
+
I
~ = y ~ ( o o ) - Th*~(z) dz + o9 2 ~. z 3h*o(z) dz - . . . ,
c o
It is important to notice that in this general result the multipliers f~, Yv approach the preferred (i.e., revised) definitions for the coefficients Y~., Yv o f derivative theory as o9 is made very small. These revised quantities are in fact what are found in oscillatory testing [6]; but as we shall now show, this is a case o f the right result obtained for the wrong reason.
Suppose that, instead o f countenancing time history effects, we adopt Bryan's assumptions from the outset so that h*(z) = A6(t). In that case we find that
~ = y~(oo), Yv = yv(oo) + A, and the theory collapses to Bryan's original form.
Instead of going as far as assuming instantaneous fluid forces, let us admit a quantity h*(z) that is not a Dirac function. By analogy with the (admittedly somewhat spurious) analysis o f Taylor's series we note that A Y depends on v, b,/3, V , . . . and so adopt the symbolism
A Y [ v ( t ) ] = Y v v + Y ; b + YviJ+ . . . .
Notice that Y~ and Y~ are the revised quantities and not those o f Bryan's more rudimentary analysis. It follows that
? = y , _ o g z yv+o9+ y ~ _ . . . , ? ~ = y~--'og~ y.~.+ o9, Y.~-.-...,
where, it must be remembered, Y~, Y~, etc., are not strictly "derivatives". In the past [4, 5, 6] these equations have been employed in such a manner as to imply that
lim ~ = Y~, lim ~ = Y~,
o--~0 o-=)O
where Y~ and Y~ are the slow=motion derivatives of quasi=steady flow (i.e., under the mistaken assumption that h * ( z ) = AcS(t)). In this context, the quantities Y~, Y~ have been called
126 R. E. D. BISHOP E T A L .
"oscillatory derivatives" or (better) "oscillatory coefficients", but we now see that the t e r m - - preferably the latter--is best kept for the general result in which no special assumptions are made.
9. CONCLUSIONS
When a ship or aircraft departs f r o m a uniform reference motion the surrounding fluid exerts time dependent forces and m o m e n t s on the vehicle. These forces and m o m e n t s depend in part on the past history o f the parasitic motion and, with ships in particular, this is o f cardinal importance. In previous papers it has been shown how the time history effects can be allowed for by the use of functional analysis. In this p a p e r we have seen how c o n t e m p o r a r y techniques (which are o f long standing) fit in with the functional approach.
It has been shown in particular that, suitably reinterpreted, con;centional slow-motion derivatives can be adapted to non-steady flow by way o f a first approximation. It has also been shown that, explicit assumptions to the contrary notwithstanding, this reinterpretation is made in oscillatory testing techniques. Indeed without the more general interpretation given here, oscillatory coefficients are strictly meaningless.
REFERENCES
1. R. E. D. BISHOP, R. K. BURCHER and W. G. PRICE 1973 Proceedings ofthe Royal Society A332, 23-35. The uses of functional analysis in ship dynamics.
2. R. E. D. BISHOP, R. K. BURCHER and W. G. PRICE 1973 Proceedings of the Royal Society A332, 37--49. Application of functional analysis to oscillatory ship model testing.
3. T. APOSrOL 1957 Mathematical Analysis. Cambridge, Massachusetts: Addison-Wesley Press. 4. A. GOODMAN 1960 Proceedings of the Third Sympositun of Naval Hydrodynamics, Scherenhtgen 379--449. Experimental techniques and methods of analysis used in submerged body research. 5. M. GERXLER 1959 Symposium of Towhtg Tank Facilities, Zagreb, Yngoslavia, paper 6.
6. R . E . D . BISHOP and A. G. PARKINSON 1970 Philosophical Transactions of the Royal Society A226, 35-61. On the planar motion mechanism used in ship model testing.
7. G. VAN LEEUWEN 1964 Technological University, Delft, Report No. 23. The lateral damping and added mass of a horizontally oscillating ship model.
8. R. E. D. BIsHoP, A. G. PARKINSON and J. W. PENDERED 1969 Journal o f Sound and Vibration 9, 313-337. Linear analysis of transient vibration.
9. G. H. BRYAN 1911 Stability hi Aviation. London: MacMillan & Co. Ltd. 10. I. E. GARRICK 1959 Manual on Aero-elasticity AGARD 2, 1-26.
11. W.J. DUNCAN 1959 The Principles of the Control and Stability of Aircraft. Cambridge University Press.
12. B. ETKIN 1959 Dynamics of Flight. New York: John Wiley & Sons.
APPENDIX I
OSCILLATORY MOTIONS OF LOW FREQUENCY
As we have seen, particular significance is attached to harmonic motions at low frequencies. N o w we have already seen that
/I Yl, HJ(to) ~ sin toe
= l / l # ) d~
(-toVo) to J~ to
~' 9 sinto~ = Y , ; ( o o ) - J ~ h v ( z ) to dr.
FIFTtl ANNUAL FAIREY LECTURE 127 M o r e o v e r
A Y, auaa HR(tO) f hv(z) cos ogr d r
U0 -:0
c~
= Yv(~) + f h~*(r)cos~ordr.
A t zero frequency, these results reduce to
lim [A Y,,[(-COVo)] = y;(oo) -
0,)-40
co
f rh*(r) dr,
- 0 o
c o
[A YquaJVo] =
Y~C oo)
+ fh*(z)
dr.- o o
W e n o w e x a m i n e the question o f h o w these results are modified if the frequency is small but finite. In d o i n g so we consider the F o u r i e r t r a n s f o r m
H * ( w ) = H*R(og)
+
iH*t(oo) o f h*(z).It will be seen t h a t
1
A Y~. y~(oo) + - - H*l(eo).
(-,Ovo) ,o T h e F o u r i e r t r a n s f o r m t e r m m a y n o w be expanded in a M a c l a u r i n series d to 2 d 2 H*'(og) = H * ' ( 0 ) + ~ [H*'(0)] + ~-! ~--~m 2 [H*'(0)] + a n d , since d r co
dco' [H*'(oJ)] = - f h*(r) dr -~o ~m-;m~ (sin c~ d r '
all the'even derivatives at co = 0 are zero (along with H~*~(0)). T h a t is to say
where A Y,. d " 012 d 3 (-COVo) - y~(oo) + ~ [n*'(0)] + ~ . ~5~3 [ n*t(0)] + " - d co [H*'(O)] TI1*(z) dr, d ~ r f r 3
hv*(T)
d r , de)3 [H*I(O)] = -~ etc.T u r n i n g next to the q u a d r a t u r e c o m p o n e n t , we have A Yqua~ = y.(oo) + H*•(w). V0 N o w d to 2 d 2 H*R(to) = H'R(0) + t.~--~ [H'R(0)] + -~-! ~-i~ 2 [H'R(0)] + . . .
128 R . E . D . BISHOP E T A L .
and all the derivatives o f odd order vanish so that
where
A
Yquad
G)2 d2- - = Y~(r + H~*R(O) + ~ ~ [H*R(O)I + . . .
/)0 d 2 co
do92 [H'r(0)] = _ f t2h*(t) dr,
- c o d 4 o0 dto* [H'R(0)] = f t'h*(t) dz, --co etc. APPENDIX II NOTATIONFor general linear mechanical system: F(t) h(t) q(t) ~,,(oJ) &,(t) ~z,l,(t) applied force unit impulse response response
harmonic receptance function impulsive receptance function indicial receptance function For rigid body in flowing fluid:
Hv(co) Fourier transform ofhv(t) H~(to) Fourier transformofh~(t) H*(co) Fourier transformofh*(t) H*(co) Fourier transformofh~(t)
h~(t) variation of ,4 Yfollowing unit step displacement to starboard h~(t) variation of A Yfollowing unit step variation of v
h*(t) function h~(t) without delta function constituents (the same as h**(t)) h~(t) function h~(t) without delta function constituents
U steady forward reference velocity v drift ("sway") velocity
Vo amplitude of harmonic drift (sway) velocity
Y fluid force to starboard with perturbations expressible as .4 Y(t), A Y[v(t)] or zt Y[~(t)] Yv slow-motion velocity derivative
Y~ slow-motion acceleration derivative 17v oscillatory velocity coefficient 174 oscillatory acceleration coefficient y~(oo) value of d
Yquao/Vo
at infinite frequency y~(r value of A Yin/(--VoCO) at infinite frequencyy(t) displacement to starboard
yo amplitude of harmonic displacement to starboard A Y~, harmonic component of A Yin phase with Yo
zt Yqu~a harmonic component of A Yin quadrature with Yo (and in phase with vo) r time variable
o~ circular frequency
