The Planar Motion Mechanism for Ship Model Testing, University College, London, UK, Admiralty Exeperiment Works, Volume 1

Technische Hogeschool

Delft

THE PLANAR MOTION MECHANISM

FOR SHIP MODEL TESTING.

Notes on a Course delivered at

A.E.W. Haslar, April 21st-23rd, 1969

UNIVERSITY COLLEGE, LONDON

ADMIRALTY EXPERIMENT WORKS.

PI969- i-2

FUNDAMENTALS OF SHIP DYNAMICS by RED, Bishop and A.G. Parkinson

The Equations of Motion

It is well known that the equations of motion of a rigid body may be expressed in the vectorial form as

4 F = mU , G = h

In words, the resultant force F acting on a rigid body is equal to the product of the mass. in of the body and the rate of change of the velocity U of its centre of mass. The resultant moment

about the centre of mass of all the external forces acting is

equal to the rate of change of the moment of the momentum, A, of the body relative to its mass centre.

Now there is no necessity to specify the various vectors

in equations (1) and (2) by reference to an inertial frame of reference. Thus even though U is an "absolute" velocity (i.e.

a quantity measured with respect to an inertial or a fixed frame

of reference) it may be specified in terms of body axes as

shown in fig. 1. Thus we may write

U = Ui Vj 4 Wk (3)

= hx h j _hzk (4)

where i, j, k are unit vectors in the directions Cx, Cy, Cz respectively, C being the centre of mass of the body.

If this particular set of body axes is employed, equations (I) and (2) assume the forms:

6i1 F = m-- + mO

x u

6t .4

al

₄

_-4 G =

+Oxh

6t

where, for example,

66 A :

= ti + vj + Wk and where _{represents the angular velocity of}

the body. We shall

not attempt to prove these results;

they are quoted as well established laws of vector dynamics.

Imagine that the body (which, for our present purposes, is a ship asshown in fig.1) starts from _{some reference configuration with}

Cxyz parallel to fixed axes OXYZ with the planes OXY and Cxy horizontal.

Suppose that the ship is then given

a yaw T about Cz

a pitch 0 about Cy

a roll 4) about Cx

in that.order. _{The weight of the ship mg}

will be parallel to OZ, but in terms of the unit vectors i, j, k _{it may be expressed as}

mg(-sine I + cos() sin(1) j + cos® cos)(1)

This vector is_the contribution of the weight to the force 1: of

equation (5).

The equations of motion that _{we seek are obtained by separating}

out the six scalar components of equations _{(5) and (6). They are}

X - mg sine = m(lj + QW - RV)

Y + mg cos0 _{+ RU - PW)}

and

L = II; - I

Q -I

R+

(IR -I P-I Q)Q - (IQ-I R-I 'OR

x xy xz z zx zy yz _YK

M =IQ-I

R-1

1;

+ (IP -I Q-I R)R - (IR -I P-I Q)P

(8)

X

yz _Yx xy )CZ z ' zx zy

N=I172-IP-I_+(IQ-I R-I P)P - (IP-I Q-I R)Q

1

zx zy Y yz Yx . x xy xz '

)

Here we have taken

= Pi + Qj + Rk (9)

The quantities X, Y, Z are components of applied force not

including weight; .i.e X, Y, Z may be taken as components of the

fluid force. Equally L, M, N are the components of the moment of

fluid forces about the centre of mass C.

Equations of Motion following a Disturbance

Equations (7) and (8) are of such a complicated mathematical

form that they are almost unusable, They are therefore often put into a "linearised" form, We now briefly remind the reader of

the nature of the approximations that are made,

Consider a rigid submarine executing a symmetric steady motion

For such motion, the equations reduce to

X - mg sin 7, -

0

o Z + mg cos

T

= 0

= o

R =

where bars over the symbols mean "steady value", We now contemplate

small departures from the steady reference motion - departures in

which:

(10)

fluid force components are :7 + AX, 31.- AY, Z + AZ fluid moment components are 1: + AL,M + AM, iv- + AN

i= (-17T. u)i OW w)k

2= pi + qj + rk

small displacements are made away from the position which

would have been occupied if the steady motion _{had continued}

undisturbed.

small angles, 4), e , 0 _{are made with the "steady motion"}

orientation.

Now the small displacements (e) may radically _{change the fluid forces}

and moments through the perturbations AX., AY, AZ, _{AL, AM, AN - a fact} that must be borne in mind _{- but they do not appear explicitly in}

the equations of motion (7) and (8).

Equations (7) and (8) govern-the perturbed-motion _{so that six}

more equations (comparable.withequations (10)) may be found when

the assumptions (a) - (f) are introduced. _{After elimination of the}

steady state values from equations (10) the equations for the perturbed

motion reduce to

AX + ms[in

e-

sin (0 0)] =

m[a + q(a +

+ mg cos _e) sine = naki + + u) 12,(c1- + w)]

AZ + mg cos[(6- +

e) coso

_{cos -CI}

_{m[* + pv gal + u)]}

AL=IP-I

4 -

+ (1r-I

z

p-I q)q-aq-I r-I p)r

x xy xz zx zy yz yx

AM =

Iy

-I f-I

yz _Yx

+ (Ixp -1 q-I Or - (I

xy xz _z

r -I p-I q)p

zx zY

N=IL

zf - _zx

P-I 4

+

r-I

p)pxp -1 q-I Oq

zy yz yx xy xz

In this form the equations are too complicated to be of much use.

In making further progress it is customary to reduce them to a linear

form, but before doing so we must discuss briefly the nature of the

Renesftatation of the Fluid Forces and Moments

For simplicity we shall coulience with a discussion of the fluid forces and moments experienced by a submerged submarine following

some disturbance from a straight-and-level reference motion parallel

to the undisturbed direction of Cx. By this means we can postpone consideration of free surface effects of the type (e) mentioned

above. Furthermore we can substitute

= o =

into equations (11),

To take one component of the fluid force by way of illustration, consider the increment LY in Y. In general the value of AY at

any instant t depends on the parametersdefining the instantaneous

motion together with the history:of the departure from the reference

motion, That is to say, AY is a function of the instantaneous

components of velocity, acceleration and where appropriate,

displacement of the ship, together with their previous values, Thus as the motion at any instant is

U = (U + u)i + vj + wk

= pi + qj + rk

the increment of force AY will be of the form

8Y = f(O,4 uvwpq,r

t; previous

values of O U and

other constant parameters) (13)

Now - to take just one of the independent variables - at least for

disturbances of the form _{v = voeut sin (wt + e), which are typical}

of the motions with which _{we are normally concerned, the value of}

at an instant t 7 _{can be represented-by the infinite Taylor series,}

v(t T) = v(t) -.T.0(t) +

2! 7 (t)

+ 00:

(14)

That is, the disturbed velocity _{at some instant t T}

is determined by the values of v V... at the time t.

Similar expressions _{can be formed for the values of all the}

parameters B, f at any time t T in the interval

to

t

T < t, tahere

to

is the time of onset of the disturbed motion. _{The previous history of the given type of disturbed motion}

can therefore be fixed by specifying the instantaneous _{values of}

and all their higher derivatives with _{respect to}

time. _{In these circumstances relation (13)}

can be rewritten in a

simpler form as

AY =

V"

...

t)

(15)

where the form of the function f( ) _{depends in part on the value}

of IT and other appropriate constant _parameters.

In most practical. cases the conventional _{approach of expanding AY}

in terms of.a.7aylor series can be-adopteclo-Thus-for small disturbances the component ff _{is specified "to the first order" by}

AY=Y1111+Yo'd+T.31+

y

V +

Y.Nr + +

0..

Yww + + Y + E)

ci 4.+

+ gel y

_{+Yp+ Yo3}

Y-15 cPc _{D' P}

Yrr +

Yii + /ir +

+ Y(t)

0 0

where the Taylor series has been curtailed in the usual way to

exclude all non-linear terms and where Y(t) represents any time dependent forcing, In this equation the coefficients are defined by expressions such as

Y 13AY

v 3v

steady state

Here "steady state" means the condition in

which11

= Ui = 0 and the.partial differentials indicate, for example, that Yv is

associated with an increment AY generated by a perturbation v alone, with the remaining motion parameters held constant at their steady state values.

It is usually possible, however, to-eliminate many of the terms in equations (16) and in the corresponding expressions for the

other increments_ AX,,AL, ., The simplification arises from

the fact that Cxz is normally a plane of symmetry in a submarine (and also, for that matter, in a ship), In these circumstances it is possible.to distinguish between symmetric .and antisymmetric disturbances. The former are disturbances in the plane of

symmetry Cxz and are associated with perturbations 6, u, w, q and their time derivatives, whereas antisymmetric disturbances are

associated with the parameters 09 v9

p,

r and their time derivatives,

Consider, for example, a disturbed motion in which

Due to the symmetry of the vessel about. the plane Cxz a disturbance of this form will generate increments AX, AZ and ALK, but AY, AL

and AN will be zero, Thus, in particular,

Yw = 0 = L =N

w w steady state (18) 7 ( 1 7 )

disturbances 0, u, q, ü, 9

4, ...

In this way equation (16) can be abridged to

AY =Y v + Y,* + Y V +

'v V

This motion will produce non-zero valuesfor all the increments AX,

AY, ...,

A.

Considerations of symmetry, however, show that

Xv

= 0 = Z

V

=MV

regardless of the exact form of the variations of AX, AZ and AM with

v, Once more it is possible to establish similar results for

all the antisymmetric parameters cP, v, p, r, ..., so that the expression for AX analogous to equation (16) may be shortened to

AXXeO

x

x,

)(A

+ Xuu + +

""V7-1-X*17"-XVII"'° °

+ X(t) (21)

(20) + Yrr + Yti +

y.. r +

+ Y(t) (19)

with analogous expressions for AL and AN

Let us now turn our attention to the antisymmetric variables

O, v, p, r, ... and examine the effects-of.a disturbance for which

^

U + vi

9

Similar abridged results apply, of course, to AZ and AM,

In conclusion it should be noted that, if we had considered the

effect of disturbances from a general symmetric reference motion (for

which. U, W and 0 are all, non zero) the above atguments of symmetry

would have remained valid, The only effect of the non-zero values for W and 0 .wouLd have been to modify the form of the function.

f( ) in equation (15), which modification might also be reflected

in the values of the coefficients in relations (19) and (21),

Linear Equations of Motion

We can now proceed with the linearisation of equations (11)

by using linear expressions of the form (19) and (21) for the

increments AX, AY, AN and by discarding all products of small.

perturbations (such as iu and rq), Thus the linear equations for the motion following a small disturbance may be written in

the form. AX mg0 cos 75 = q ) AY 4 mgO cos 9 = m(4 4' r _{PW) L} (22) AE mg? sin 0 = m(w AL = Ix _{- IKZ} = I 4 (23) AN - Izi - I P XZ

It will be seen that as Cxz is a plane of symmetry, the equations

have been further simplified by noting that

T

=0 =I

=

=I

xy yx yz zy

In their linear form (22) and (23) the. equations of motion permit a further simplification. We have seen that the increments AX, AZ

and AM in the symmetric fluid loadings are-to,the first order

independent of the antisymmetric variables: 4), v, p, r, ... and that AY, AL and AN are independent of the symmetric parameters

e,

u5 w, q, _{In these circumstances-the six equations of} motion are uncoupled into two independent sets-of.three equations

as follows

Ax. mg _e cos .G = _citb AZ - mg 0 sin

T

= m(74 - 4U) AM = I 4 AY -4- mg4 cos 0 =

m(r

pk74-) AL - I - I f xz AN = I_{zf -} I P xz

In linear theory, therefore

it,is:possible'toconsider

separately two forms of disturbance-from-a-steady-symmetric,reference Motion,

namely symmetric and-antisymmetrit .disturbances .- Moreover, a general

motion is merely a superpositimof,these-two simpler forms of

disturbance. A symmetric disturbance produces a symmetric motion

and an antisymmetric departure' from a symmetric-refeTencl motion results in an antisymmetric disturbed motion,*

For a small symmetric motion we need onlyconsider non-zero

values of 0, u, w, q, AX, AZ and-AMandthe equations of motion

are the set (24), Conversely'for-small

antisymmetric

disturbances the--equations of motion (25) apply,and:they only concern the parameters

v, p, r, AY. AL and AN

(24)

(25)

----* It should be noted that a symmetric disturbed motion is possible

for large disturbances in non-linear theory. The existence of an independent antisymmetric motion and the possibility of reducing a general disturbance to separate symmetric and antisymmetric

components, however, both require the use of linear equations of motion andlinear approximations to the expressions for the force

It is worth while noting that the six equations (24) and (25) may be simplified further for surface .ships (and submarines with a horizontal, symmetric reference motion) since in these cases

14. = o =

-0"

That is for small symmetric departures of a symmetric ship from a

steady reference motion,

AX

mge =

AZ = m(* qU) (26)

AM = I CI

whereas the equations for antisymmetrfc-disturbed motion are

AY + mg O = + r17)

AL =

Ix - xzf.

AN = Izi - I f5 xz

Slow motion derivatives

We must now return to consideration of the linear expressions (19)

and (21) for the various fluid force and moment increments.

Consider, for example, the increment: AY represented by expression (19),

Conventionally an approximate form of this equation is found for

O.

The approximation rests upon the idea of "derivatives" or

strictly, "slow motion derivatives", Alternatively an exact

relation for equation (19) can be formed in terms of "oscillatory

coefficients", Both types of expression will now be discussed

briefly. It will be understood that AY is here used only by way

of example; similar arguments apply to AX, AZ, AL, AM and AN,

The concept of "slow motion" has been discussed in some detail

in an aerodynamic context by Duncan (1952),* Consider, by way of * References are listed alphabetically at the end of the paper,

(27) 11

example, a sway displacement .y _{of a ship, such that v = = y,}

The sway motion of the ship can be regarded _{as "slow", if it does}

not vary rapidly with time, _{We can: express this property by} postulating that, at any instant t, _{1,./v0 are all very}

much less than 11/2, where 2, is the length of the ship (or any

other convenient parameter with dimensions of length). _{Thus not} only are the parameters y, v, i _{.,. small, but their relative} rates of change are small. _{For example, v, the instantaneous rate} of change of y, and y _{are associated with a time scale which is} very much larger than

OU,

_{the time for the ship tc move through}

its own length with the forward speed --U. _{Mathematically the}

motion is slow, therefore, if

y 17-17 >> ₍₂₈₎

It should be noted that in this paragraph.y,-y, Y, _represent the absolute magnitudes of the variables and _{are all positive.}

Moreover, if the motion is.cscillatory, then _{y, v, ... here}

denote the amplitudes of the variables,

If _{therefore, .all the components of the disturbed motion}

are small in this sense, rhentheTaylor series (19) _{can be further}

curtailed by omitting theterms containing the higher order

derivat.ivesY.V,

. The component.

SY is now adequately specified

to the first order" as

AY =Y0

4.Yyri-

Yr4Ypi, Y,45 +Yr+ Y,.'r

Y(t) (29)

0

where now we must regard Y _{as "slow motion derivatives".} The fundamental assumption is made that motion is so slow that only

orientation (0), velocity (y, p, r) and acceleration 0°,,15, i) _{of the} antisymmetric terms need be retained in _{a Taylor series expansion of}

the function (15) - aside, that is from a-possibie time 'dependent forcing term Y(t),

13

Proper justification of this approximation _{is by no means easy,}

but the approach is now well established _{and-a-further discussion} of it will not be attempted here. The approximation is important because many problems facing the analyst in _{ship dynamics relate to}

disturbed motions which are slow -as, for example, in the

transition from a stable to _{a divergent,mon-oscillatory, unstable}

reference motion.

In essence this is the basis of _{the most -common technique for} specifying the components AX, AY, AZ, AL, AM, N. As we shall discover

later, however, it has some _{drawbacks.which'can, in fact, be avoided}

by the adoption of an alternative _{approach using 'oscillatory coefficients.}

Oscillatory coeffitients

For simplicity consider 'atest.in which a planar motion mechanism imposes a pure sinusoidal sway motion _{y = yo sin wt}

on a model that is towed.at a test speed

_If

_{along a tank.} If y = yo sin wt then

V = yow cos wt

2 . (30)

= -yow sin wt

In these circumstances the _{general linear expression (19) for} AY reduces to

6X=YITIT4**Y"*Ye+

_1). 014 2 ₂ 2 v w Y. ...)(yow cos wt) y, w .00)(-yow sin wt) (31) as the parameters

_{q), p,}

_{r, 15,}

it etc0 are all held _{zero during the} test. _{This expression can be rewritten in the form}

AY =

coefficients". The value of Y _{is obtained from that component}

of

a

_{which is in quadrature with-the}

sway displacement, while Y, _{is found from the "in-phase component".}

Similar arguments _{may be advanced for the dependence of}

a

on

and its derivatives and on r and its derivatives, _{so that in} all

Ly = (I)

+ y p

Yvv + Y.17 + rr + Y. + Y(t) Yo

(33)

Notice that there is no oscillatory coefficient Y15 since it is

no longer possible to distinguish between _{Y and 'Y..}

(both being

in phase with roll). _{Observe too that all the oscillatory}

coeff-icients are frequency-dependent,

This property is suggested by the

2 4

explicit appearance of factors like w , w _{60.1 in theexpressions} for Yv, *Y" ,

To retutn to the oscillatory coefficients _{Y and Y} t

will be seen that if they are measured as-a pair and if the frequency is made very small, then they approximate closely to

theslowmotionderivat.ivesYvandYi.yrespectively.

For example,

2

Yv = lim [Y j _{Y + w}

...]

Yv (34)

Ci.) W

We have seen therefore that _{there are at least two special}

types of disturbed motion for which it is _{possible approximately}

to express the incremental fluid forces _{and moments (such as AY)}

as linear combinations of the instantaneous _{orientations,}

velocities and accelerations (such as

O, v, p, r, V, P

_{and i),} These special classes are (i) _{very slow motions for which we may}

15

use slow motion derivatives as in equation (29), and (ii) oscillatory

motions for which we may use oscillatory coefficients as in expression (33).

Fortunately one is able in this way to deal with some types of

ship motions which are of interest to the analyst. In many problems the slow motion derivatives are adequate, but for some purposes

the oscillatory coefficients may be required. This is the case when

the ship motion is sinusoidal and of significant frequency, either

as a consequence of imposed motions of.control surfaces (rudders, hydroplanes or stabilizers) or alternatively as a result of operating

at the boundary of an oscillatory instability.

Thedeerp_ideapie:fluid forces'and.momentson'diSplaCeteritt'from.

the reference tOtion

Expressions of the form (13) for the increments in the fluid

forces and moments are valid for fully-immersed vessels, such as a

submerged submarine provided, of course, that the fluid is of

approximately constant density and that the submarine is well away

from the surface, the sea bed or other boundaries. The forces and moments experienced by a surface ship, however, during a disturbed

motion, may also depend on linear displacements of the centre of

mass C in addition to the orientation, velocity and acceleration

parameters considered earlier.

We must now, therefore, make an allowance for such displacement

effects and for this purpose it is convenient to introduce the

additional set of axes Ax*y*z* shown in fig. 2In this figure

Cxyz are the body axes attached to the ship. OXYZ are stationary axes with OXY _{horizontal and parallel to the directions of Cxyz}

4- In so far as the axes Cxyz move and rotate with the vessel, it is not possible to express the displacement of the vessel by reference to

=

wat - TY

I

eat = z

-TT

r

edt

during the steady reference motion. The axes Ax*y*z* coincide with

Cxyz _{during the reference motion, that-is they move with a velocity}

if along Ax* and they continue to move with this steady velocity

without rotation during the disturbed-motion. _{Thus at any instant} the position of the vessel is determined by the vector

OC = OA AC

such that OA represents the displacement due.to.the steady state

motion U along Ax* and AC -represents the displacement du t to

any small disturbance. Thus, if- i, 3,

I

are unit vectors along OX, OY, OZ respectively, then

AC = x*I y*J +

z*k

(35)

where x*, y*, z* are the coordinates ofC relative to A.

The displacement of the ship at any instant during the disturbed

motion can therefore be specified by-the coordinates, x*, y*, z*

which are related (to the first order) to the perturbation components

by the following expressions,

= u

= v + Ttp

(36)

i* = w

1Tt)

If these expressions are integrated then the actual displacement

can be determined in the form

x* = .1) udt = x

0

t t_

(39) 17

where x, y, z are parameters with dimensions of length, but they

do not represent readily identifiable displacements.

We must now consider the possible dependence of AX, AY, AZ,

AL, AM and AN (i.e. the force-and moment increments along the body axes) on the displacement parameters x*, y*, z* in addition to the variables in the functions of the type (15), For a ship on the open sea it is easy to deduce that the fluid-forces and moments are not changed by translational displacements x* and y*

which only represent a horizontal displacement of the ship to a

different location.+

It remains to examine the effects of the z*' displacements on these increments which may require further modification for ships

in shallow water (see, for-example, Fujino (1968)), That is we must consider by analogy with equation (15) functions of the form

AY = f(z*,8,qh, u, v, w..,, ₍₃₈₎

the normal Taylor series expansion technique is used once more and we confine our attention to the linear terms then in the example under discussion the expression for AY becomes

_A = Y ,z* + terms in equation (16) z*

If Cxz is a plane of symmetry in the vessel, the simplifications

outlined earlier still apply, _{That is in a linear theory one can still} treat symmetric and antisymmetric motions separately and moreover

AY, AL and AN are independent of z* alone, so that Yz*, Lz*

and

Nz* are all zero. Thus we must merely include components

X*

z z*, Zz*z*, Mz*z* in the expressions for .4X9.4Z, AM..respectiyely. + Examples of circumstances in which these horizontal displacements

and also yaw angle 4) are significant have been given by Fujin° (1968) and Paulling and Wood (1962),

Equations (27) for antisymmetric motion are therefore unchanged

by displacement effects, but those (26) for symmetric motion must be

satisfied to include the extra terms obtained above. Examine first the nature of equations (26), if the z* terms are ignored. If

we express AX, AZ and AM in terms of slow motion derivatives then

in matrix form equations (26) become

q,

= fx

z

el

Q {X(t) z(t)

m(01

(44)

Notice that the column matrix q is not to be confused with the angular

-,A

velocity of pitching, q. The parameters x and z appearing in the column vector for a are defined in equations (37). The matrix equation

(40) comprises three simultaneous equations in the independent variables x, z, 8.

The extra terms

Xz*z* Zz*z* Mz* are not readily expressed B = -M. u -X u -M, w -X w 1 - M, Y q -X q (42) -Z u -Zw -(miT + Z_q) -M u -Mw -Mq C = 0 0 -mg (43) i'Aft 0 0 0 0 0 0 Ati + 134 + Cq = Q (40) ,A where A = m - X. u -Z. U

-W m - X. w -X. q Z. q (41)

19

in terms of x, z, 6 due to the nature of the transformations (37). Indeed some writers merely equate

Xz*z* = X z, _{= Zzz' Mz*z* =M}

zz (45)

and use equation (40) with A, and B unchanged, but with _C

Ano. \

modified to

Such a simplification is not, however, really satisfactory (see Abkowitz (1957)),

A better approach is to express w and in terms of i*

and using equations (36), The matrix equation (40) can then be rewritten for the perturbation vector

q=

fx z* 61 = fx* z* (47)

as x* = x to the first order, Following this transformation the new matrix equation of the form (40) contains matrices A, 13 C of

which only matrix _A (defined in equation (41)) is unchanged. The

remaining matrices B and C, however, must be rewritten as

B

-x

-x-Xw -1 -q w (48) -Zu -2w -MU Zq w -Mu -mw -m_q

m:d

w C ₀ -Xz* -mg

-

x

e xwu (49) 0 -Zz* -z zw 0 -Mz*

-M6 -MU

0 0 0 -Xz -Zz -Mz -mg 0 0 (46)

A somewhat similar approach, which does not involve the use of

matrices, has been proposed by Abkowitz (1957) who publishes _results

indicating the importance of the z* terms.

Substitution of the matrices (41), (47), (48) and (49) into

equation (40) produces valid and useful equations of motion for a

surface ship in pitch and heave, but their form is-not suitable for

analysing the results of experiments performed with'a'planar motion

mechanism. The forms of excitation which produce-uncoupled motions in

pitch and heave (for velocities and accelerations) in-terms of body

axes are not independent of the displacement- z*. Consequently, for example, independent measurement of Z and -Z is not easy,

w z*

To obtain simple meaningful equations of symmetric-motion for use

with a surface planar motion mechanism it is necessary to recast the

whole theory in terms of the axes Ax*y*z*, Moreover the forms of

motion corresponding. to pure heave and pitch are also changed. Such modifications, however, are beyond the scope of the present

intro-duction. We just conclude here by noting that the equations (27) for

antisymmetric motion are unchanged in form by surface effects,

whereas equations (26) for symmetric motion are not readily applicable

References

Abkowitz, M.A. 1957, Proc. Symp. on "The Behaviour of Ships

in a Seaway", Wageningen, pp.178-189.

Duncan, W,J. 1952 "Control and Stability of Aircraft",

Cambridge University Press.

Fujino, M. _{1968. Int.Shipbuilding Progress, vol. 159 pp.279-301,} "Experimental Studies on Ship Manoeuvrability in

Restricted Waters, Part I".

Paulling, J.R. and Wood, L.W. 1962. Univ. of Cal. Inst. of Engng Research. "The Dynamic Problem of Two Ships Operating on Parallel Courses in Close Proximity:'

Y

/

Z

FIG. 2.

October 1968

On the Planar Motion

Mechanism used in

Ship Model Testing

by

R.E.D. Bishop and

R.E.D. Bishop* and A.G.

Parkinson**

Summary

In the linear theory of small departures from steady reference

motions of submarines and ships it is standard practice to employ

the idea of hydrodynamic "derivatives". These derivatives permit the magnitudes of

fluia forces and moments to

be specified. In recent

years it has become common to measure the derivatives by means of a "planar motion mechanise which is essentially a device for

oscilla-ting a ship (or submarine) model whilst it is being towed in a

testing tank.

The derivatives referred to in the maritime literature have invariably

been "slow motion" derivatives. The theory of the planar motion

technique is recast in terms Of "oscillatory" derivatives - or, better, "oscillatory"coefficients - since they are more appropriate for use

where the mechanism is concerned. The idea behind these quantities

is borrowed from aeronautical practice, but it requires some

adapta-tion because (a) ship models work at the water surface and (b) ships

and submarines arc subject to significant buoyancy forces. There can be

little doubt that the planar motion mechanism is a powerful tool and a

re-appraisal is perhaps

timely

since the first mechanism of this sort to be installed in the U.K. is to be commissioned this year (1968).

* Kennedy Professor of Mechanical Engineering, University College London. ** Lecturer in Mechanical Engineering, University College London.

Introduction

In recent years it has become common to test models of ships

and submarines using a "planar motion mechanism. This is a

de-vice that imparts a pure sinusoidal motion in one degree of

free-dom - in yaw for instance - to a model that is towed along a

test-ing tanks While this motion is executed, measurements are made

of the forces acting on the model, whence the fluid forces may be

deduced. The technique is of fairly recent origin, having been

pioneered in the U.S.A. by Gertler (1959; and Goodman

(l960)

but there is already no doubt as to its value in the measurement of

hydrodynamic forces and in particular, of the hydrodynamic

°deri-vatives" with which this paper is largely concerned, It is at the same time more versatile and more ecOnomical in use than

alterna-tive techniques.

The value of the planar motion mechanism rests on its assumed

ability to impose sinusoidal motions that are pure and inexorable.

In reality, of course, this is not strictly possible and it is

conceivable that for extreme accuracy p an alternative approach

might be needed. In theory at least, it may be better from the

point of view of accuracy to measure the impressed forces and

the motions so that the analysis has to be based on the coupled

equations of motion (rather than On the equations taken one at a

time), But_the case for such a sophisticated approach has by no

means been made where

Ehip

models are concerned and the planar motion mechanism is probably the most promising practical

In this paper the theoretical background of the planar motion mechanism

is presented in a new way. The Concept of the "oscillatory derivative"

is adapted for this purpose (although for reasons that will be explained

the name "oscillatory cOefficient" is preferable and will be adopted

here). For while oscillatory derivatives (or coefficients) are familiar

in aeronautical practice, they appear not to have found any place

what-soever in the maritime literature. This fresh approach is thought to have

intrinsic merit and it also suggests a line of speculation that may be

of some significance. Thus if oscillatory coefficients could be found

for a sufficient range of frequency they could be used in conjunction with

Fourier integral techniques for the study of transient behaviour of ships

and submarines, even though in most cases the response of ships and

submarines is much too slow to warrant the use of such a technique.

The subject matter falls conveniently into three parts and

these will be separated for clarity of presentation. They are

I - "Slow motion derivatives" and "oscillatory coefficients"

II - Theory of the planar motion mechanism in terms of

oscillatory coefficients and applied to submarine

models

Notation

centre of mass of model P forces applied to model

Fn Froude number

H moments applied to model

metacentric height of submarine

Ty'

Iz moments of inertia of model about pitch and yaw axes respectively j, k unit vectors in directions of body axes Cx, Cy, Cz; Cxz

is a plane of symmetry, and Cxy is parallel to the undis-turbed water surface

"

L, M, N moments of fluid force about C parallel to i, j,/k

length of model"between perpendiculars"

Zo see figs,

3,

5

mass of model

r perturbations of components of angular velocity in

^ : ^

directions i, j, k; i.e. angular velocities of roll, pitch and yaw

Re Reynolds number

axial torque applied to model time

velocity of centre of mass

reference velocity in direction i

u, V, W perturbations of components of velocity of C in

directions i, j, k; i.e. velocities of surge, drift (or sway) and heave

x, z coordinates of mass centre of model in Appendices 2 and 3

phase difference (see equation (41))

C,1-1 angles of rudder and hydroplane deflection

dimensionless

(or

reduced") frequency = w9.117

[N.B. a second dimensionless frequency is used in

figures 8 and 9, viz w' = wi/(/g)] small angles of pitch, roll and yaw

angular velocity of ship or submarine

frequency of oscillation (rad/s)

Subscripts and dressings

steady state value amplitude (except Zo)

1, 2 forward and after

in in phase with displacement or orientation

out in quadrature with displacement or orientation

A increment

oscillatory coefficient dimensionless

6

Part I ..'"Slow Motion

Derivatives" and "Oscillator Coefficients" Delmrture

fro= stead Motion

in a strai ht line

A number of writers have discussed problems of directional

sta-bility and control of

ships

and submarines in calm water [see, for example, Abkowitz

(1964)],

The problems that they seek to elucidate arise from the fact that - to take a surface ship as an example

-mall departures from a steady reference motion

= + Oj + Ok (1)

are associated with

small

variations of the hydrodynamic forces and moments.

As is

particularly well known to aeronautical engineers

(who are faced with comparable problems) this means that the

stabi-lity Of the reference motion is open to question. To take one

com-ponent of the fluid

force by way of illustration consider the

com-ponent

Yj. A small departure from the motion (1) will produce a

variation AY

in

Y.

In general the value of AY at any instant t depends on the parameters defining the instantaneous motion together with the

history of the departure from the reference motion. That is to says AY is a function of the instantaneous components of

velo-city s acceleration and where appropriate, displacement of the

ship, together With their previous values. Thus if the motion at

any instant iS

n

U = ( + u)i + vj + wk

* In a linear theory the independence of the symmetric and anti-symmetric motions i8 readily justified.

and

if,

in the interests

of

brevity, the symmetric variables u, w, q are omitted from the discussion of the antisymmetric quantity* AY, then

the increment of

force

AY

will be

of the form Al = f(4), v,Per91.19 previous

values of (I), _°.., I., TT and

other constant parameters) (3)

where the symbols have the meanings given in the list of notation.

Fortunately it is possible to simplify the form of the function (3)

for most of the small disturbed motions with which one is normally

concerned. Consider, for example, the problem of specifying v at

any instant t T during the disturbed motion (i.e. t < t - to) where

to denotes the time of onset of the disturbed motion). It

is shown in the Appendix that for disturbances of the form

v =

vet

sin (olt + c), which are typical of the motions with which we are normally concerned, the value of v at the instant t T can

be represented by the infinite Taylor series,

2 3

v(t - T) = v(t) - Tr(t) + - *17(t) +

2, 3!

That is, the disturbed velocity at some instant t T is determined

by the values of v, at the time t.

Similar expressions can be formed for the values of all the

parameters cp, v, P at any time t - T in the interval

to < t T < t. The previous history of the given type of disturbed motion can therefore be fixed by specifying the instantaneous values

8

time. In these circumstances relation (3) can be rewritten in a simpler

form as

AY = f(b, v, p, r,

b,

is, i:, t) (5)

where the form of the function f( depends in part on the value of

and other appropriate constant parameters.

Expression (5) for AY is similar to the one normally used in

for-mulating problems of ship dynamics, except that instead of neglecting

the history of the motion we have allowed for it, at least for

expon-entially growing or decreasing oscillatory motion, by including the

higher order derivatives of v, p and r, We have therefore a relation

for AY as a function of all the variables 4), v

V, 00.,"V.,

In most practical cases the conventional approach of expanding AY

in terms of a Taylor series can be adopted. Thus for small disturbances

the component AY is specified "to the first order" by

AY = Y,4)

+ Y p + Y+

+ PP Yv + Y,v + Yv + V . On V V Y r + Y. + Y-r + Y(t)

where the Taylor series has been curtailed in the usual way to exclude

all non-linear terms and where Y(t) represents any time-dependent forcing. In this equation the coefficients are defined by

ex-pressions such as LY Y. steady state ' v = a Y steady state

(6)

(7)

Here "steady state" means the condition in which U = Ui, = 0

and the partial differentials indicate, for example, that Yv is

associated with an increment AY generated by a perturbation v

alone, with the remaining motion parameters held constant at their

steady-state zero values.

Conventionally an approximate form of equation (6) is found

for AY, This rests upon the idea of "derivatives", which may

be of two types - namely "slow motion derivatives" and "oscillatory

coefficients", Both types will now be discussed briefly. It will

be understood that AY is here used only by way of example;

similar arguments apply to AX, AZ, AL, AM and AN.

Slow motion derivatives

The concept of "slow motion" has been discussed in some detail

in an aerodynamic context by Duncan (1952). Consider, by way of

example, a sway displacement y of a ship, such that

v =

y,

= y, The sway motion of the ship can be regarded as "slow", if it does not vary rapidly with time We can express

this property by postulating that, at any instant t, v/y,

v/i'r are all very much less than U/Z, where k is the length

of the ship (or any other convenient parameter with dimensions

of length). Thus not only are the parameters y, v, .. small,

but their relative rates of change are small. For example, v, the

instantaneous rate of change of y, and y are associated with a

time scale which is very much larger than 9W/U, the time for the

ship to move through its own length with the forward speed U.

Y >> v9/ET >> 1.7.Q.20 >> (8)

It should be noted that in this paragraph y, v, represent the

absolute magnitudes of the variables and are all positive.

More-over, if the motion is oscillatory, then y, v, here denote

the amplitudes of the variables.

If, therefore, all

the

components of the disturbed motion are small in this sense, then the Taylor series

(6)

can be further cur-tailed by omitting the terms containing the higher order derivatives

Y-v, ..._v The component AY is now adequately specified "to the

first order" as

0 p Y.f) Yrr Y(t) (9)

where now we must regard Y4), Yv, as "slow motion

deriva-tives". The fundamental assumption

is

made that the motion is so slow that only positions (0)) velocity (v, p, r) and acceleration

bt 1%) _{terms need be retained in a Taylor series expansion of}

the function

(5) -

aside that

is,

from a possible time-dependent forcing term Y(t).

Proper justification of this approximation is by no means

easy, but the approach is now well established and a further

discuss-ion of it will nOt be attempted here. The approximation is important

because many problems facing the analyst in ship dynamics relate to

disturbed-mctions_which are slow - as, for example, in the transition

from a stable to a divergent, non-oscillatory, unstable reference

motion.

equations of motion they have usually to be measured, since theoretical

methods have not proved sufficiently reliable for experiments to be

dispensed with. The quantity Yv, for example, can be found from

towing tests in a long tank using a yawed model [see Abkowitz

(1964)].

Other experimental approaches may be relevant for other

derivatives but, for the sake of explanation, we shall refer only to

the sway derivatives,

ThederivativeYi..ris usually large and is therefore of importance.

It is, however, almost impossible to measure it directly in a towing

tank without special apparatus, since such measurement would require

the model to suffer a sway acceleration with no sway velocity. (A

whirling arm gives centripetal acceleration without centripetal velocity,

but of course with rotary motion as well).

This difficulty can be avoided by the use of a "planar motion

mechanism" . The technique which appears to be full of promise

-was first described by Certler

(1959),

who referred particularly to its use with submarine models.* In one form of test the mechanism is

adjusted, to impart a sinusoidal sway displacement y to a model that is towed at the test speed U down a tank. If y = yo sin wt then

v = yow cos wt

(10) =

-y0U)sin wt

In effect it is suggested that, during such a test, variation of the

* The technique does not appear to be as useful for accurate measurements with aircraft models because it is difficult to prevent important

12

fluid force acting athwartships arises only from the terms

Yvv and

in equation

(9)

so that

py = y (y w cos wt)

V0

w2 sin wt) (11) v o

Thus the amplitude of the measured component of AY that is in phase

with the imposed displacement gives -w2y

oYv . and the amplitude of

the observed component of AY that is in quadrature with y gives

WYOYv'

When such test b are performed it is found that the derivatives so measured are frequency-dependent. Since the object is to

dis-cover the values of the slow motion derivatives, we may suppose that

interest should be focussed on the values to which

Yv and Y. in equation (11) tend as w is made smaller and smaller.

Leaving aside the case of a sinusoidal instability of

non-negligible frequency for a moment, there are two possible motions

that can be executed exactly at the boundary between stable and

unstable motion. These are (a) slow sinusoidal motion and (b) slow

non-oscillatory motion, these being associated respectively with the

onset of an oscillatory- instability and a divergent, non-oscillatory instability. In the former case inequality (8) stipulates that the

frequency w should be mall (in fact w << 117Q); the second

poss-ibility requires that w is exactly zero and that inequality (8)

is stilJ cnt'isfied. While this approach appears to be quite simple, it is perhaps likely to raise questions. Indeed it is not clear

theoreticallythatthevaluesofYandY.Ineasured with a

plarer motion mechanism for w 4- 0 are identical to the

condition w 0 leads to a motion in which v = 0 = whereas slow

non-,oscillatory motiOn corresponds to non-zero but small v and V'.

As noted earlier, Yv (though not TO can be measured directly

by a non-oscillatory test in which a yawed model is towed in a long

tank so that, in theory at least, this point can be clarified by

comparing the results of both types of test.

Again, just as the definition of Yv clearly relates to a

hypo-thetical measurement that is made when = 0, so the definition of

refers to a condition when v = 0 (i.e. the Lartial

deri-vatives

(7)

are evaluated for the condition 4, = 0 = =

Thus the planar motion method of obtaining Y fundamentally

from the towing test determination of I, since Y. is not always

measured directly in a manner suggested by its definition. Some

reassurance can be found, however, in that it is possible in principle

to derive

Yv by measuring AY when = 0 and likewise to determine Y. from AY when v = 0 LEee Ahkowitz (19f1)1)]. Tor:

v = 0; AY = -Y1.1.(yow2 sin wt) = Ye, when t = it/2w and

= 0; AY = Yv(yow cos wt) = Yvv, when t =

In actual fact there appears to be no real doubt as to the

correctness of the oscillatory technique as a practical means of

determining slow motion derivativet for non-oscillatory motion. But

the fact that the newcomer to the technique must be forgiven if he

queries its theoretical background is unfortunate since the

114

technique is undoubtedly of vital importance and may even supplant

direct measurement in the fullness of time. Moreover very little

reassurance can be found in published data.

The planar motion mechanism can serve at least two other

purposes. First it provides a means of measuring derivatives such

as Yr and N' so removing the need for a rotating arm facility -

_r:

in theory at least, (The rotating arm mechanism imposes a

non-oscillatory rate of yaw, r, whereas the planar motion device

pro-duces a sinusoidal yaw so that, as before, the distinction between

oscillatory and non-oscillatory slow motion derivatives must be

remembered). Secondly the planar motion mechanism can be used to

measure oscillatory derivatives for frequencies which are too high

to permit one to Use the sloW motion approximation. Frequency.7

dependent oscillatory derivatives would be needed, for example,

to estimate the onset of a general oscillatory instability. An

alternative approach will therefore be described now in terms of

"oscillatory coefficients" - quantities which are potentially

more useful, since they give rise to "slow motion derivatives" as

a special case.

Oscillatory coefficients

For simplicity consider a towing test in which the planar

motion mechanism imposes a pure sinusoidal sway motion y = yo sin wt

on a model. In these circumstances the general linear expression

AY = Yvv + +

.=(Y.st.-.2Y4-...)(you)Cosos0.4-(Y.--(Av+...)(-yow2 sin wt) (13)

This expression can be rewritten in the form

-The quantitieS Yv sr Y. may be referred to as "oscillatory

coefficients'-'. Quantities of this type are used in aero-,

nautics. The value of

Yv is obtained from that component of AY which is in quadrature with the sway displacement, while is

found from the "in-phase component". It should be noted that _Yv and Y. are just coefficients multiplying

7

and r respectively and are not true derivatives in the sense of the definitions

(7)

even though they are sometimes referred to in the aeronautical literature

as "oscillatory derivatives". Nevertheless if AY were determined

for a chosen frequency and for various values of yo, then the

quadrature and in-phase components could be plotted against the

amplitude of

v

and

respectively

and Yv and Y. estimated

from the gradients of the appropriate graphs at yo = 0 (i.e.

v = 0 =

Similar arguments may be advanced for the dependence of AY on cp

and its derivatives and on r and its derivatives, so that in all

16

Notice that there is no oscillatory coefficient _{since it is no}

longer possible to distinguish between and Y.

-(both being in

(15

phase with roll). Observe too that all the oscillatory coefficients

are frequency-dependent. _{This property is suggested by the explicit}

appearance of factors like

w2w4

in the expressions for

Y , Y..

V v

To return to the oscillatory coefficients _{Yv and Yi,r, it will}

be seen that in general no question _{arises now of determining one}

with-out the other. _{They are, so to speak, placed on the same footing. If}

they are measured as a pair and if the frequency is made _{very small,}

then they approximate closely to the slow motion derivatives Yv and

Y. _{respectively (at least in their oscillatory form).}

For example,

limYv = lim [Yv - w2Y- * w ...] _{= Yv}

w-*0

We have seen therefore that there are at least two special types

of disturbed motion for which it is possible approximately to express

the incremental fluid forces and moments (such as LY) as linear

com-binations of the instantaneous displacements, velocities and

accelera-tions (such as qb, v, p, r, and

r).

These special classes are (i) very slow motions (both oscillatory and non-oscillatory) for which

we may use slow motion derivatives in equation (9), and (ii) oscillatory

motions for which we may use oscillatory coefficients as in expression

(15).

Fortunately one is able in this way to deal with some types of ship (16)

motions which are of interest to the analyst In many problems the

slow motion deriwtilies are adequate, but for some purposes the

oscillatory coefficients may be required. This is the case when the

ship motion is si=oidal and of significant frequency, either as

a consecuerce of imposed motions of control. surfaces (rudders, IydroPlahes or stabilizers) or alternatively as a result of

18

.L3.1...z.Thear_zoLthe4olleChanism in terms of Oscillatory,

Coefficients and Applied to

Submarine Models

Osc_LIatay Motions in tut e

'heave or

itch

It is a familiar feature of linear systems in general that "a

sinus-oidal cause will produce a sinussinus-oidal effect having the same frequency".

This effect may only emerge as a steady state after the effects of initial

conditions have died out but emerge it will eventually if the system is

stable. Suppose, then, that the planar motion mechanism imparts a

sinus-oidal heaving motion to a model that is towed at some constant speed 7.

while it is submerged (fig. 1). The steady reference motion about which the

heaving motion-takes place is

= Oj Ok

The imposed disturbance is such that

u = 0 = q

z = z sin wt

The heaving motions modify the steady fluid forces and moments by the

addition of force components AXi, AZk and a moment AMj. The quantities

AX, AZ, AM will vary sinusoidally with the frequency w after a steady

state of motion has been reached but there is no ground for supposing either

that they will be in phase with z or that they will be in phase with each

other.* Consider just one of these quantities, LM for instance; it (18)

* This is a consequence of the so-called "Wagner Effect" that has been widely studied in aeronautics but has received almost no attention in the maritime literature.

can be expressed in the form

AM = A cos wt + B sin wt

where A and B. can be measured. If

A

M =

, M =

w z w z z

This may be written in the form

-AM = M w + Mzz

since = _{zow cos} wt = w. The constants M and

Mz are tyPical "oscillatory coefficients".

If we were studying, not sinusoidal but slow unidirectional

disturbances from the steady motion we should assert that AM has no

direct dependence on z - only on its time derivatives. For reasons

that will become obvious, then, we should prefer to write

M. = 2 z w whence AM = Mww + M.* (19) since Z' = -w2z0 sin wt = *.

In just the same way the two symmetric components of the fluid

force can be written in the form

AX = _{Xww + X.}

(20) AZ

20

in which the oscillatory coefficients can be measured by test. In all three cases, the first term (that is proportional to w)

repre-sents the quadrature component while the other reprerepre-sents the

in-phase component,

Using the same reference motion (17) we could impart a

sinus-oidal surge motion

x =

xo sin wt

U

xow cos wt

*0004 0003

instead of the heave z. Alternatively a sinusoidal pitching motion

0=o

sin wt eow cos wt

99999999 O0.00,

could be imposed, although in that particular case the towing speed

would only be equal to the reference speed to the first order of

approximation Yet again the hydroplane angles and

n2 could be varied sinusoidally so that

n1 = (n1 sin wt 11= (n1)ow cos wt

if .1,014100000

or n2 = (n2)0 sin wt = _{(n2)0w COS}

51,11.909,000oeee.o.

AX = Xu + X.A + Xviw + X.* + Xqq + u u w X.&q - -+ _{(X)1n1 +(x.)1}fl 1

+ (x

)2 n2 + n n 71

2 2

AZ=Z la +Z1+Zwy+Z.W+Z q +

Z.61 U w q ci + (Z)1n1

_{+ (z.)1}

1 i'l + (Z)2n2

+ (Z.)

11 n

n 2

2 AM =-"11.1.1 ww -"1.*

+ m q

.(-111021 11

nil

(-1;1)2n2

(V22

>

(21)

For all of these imposed motions, expressions of the type (19), (20)

may be written down and the multipliers

x,

x.

(m ) (M.)

u u

n 2

n 2

found from measurements of the in-phase and quadrature components.

Suppose that the sinusoidal disturbances u, w, q,

ni,

n2 are imposed simultaneously with the same frequency. For a submerged

submarine model we should obtain the expressions

Notice that, in writing these expressions we have

invoked the usual assumption that the symmetric forces and

moments are independent of the antisymmetric variables v, p, r,

and

used M80 for the in-phase moment rather than M.(1 because,

when slow motion derivatives are employed a term M00 = -mghe is

22

a height h above the centre of mass and

(c) omitted a term M comparable With M. in the "slow motion

(1. (11.

equations" because one can only discriminate between the in-phase

andquadraturesothatM.is indistinguishable from

M06. (1(5'

Expressions like those of equations (21) are to be used in the

equations of motion governing symmetric disturbances, namely

AX mg() =nid

AZ = m(* q7) (22)

AM = I

But it must be remembered that if equations (21) are relevant, all the

disturbances are sinusoidal and of the same frequency w.

It will be appreciated that vertical symmetric oscillations of

a submerged submarine have been chosen for the purposes of

explana-tion, Similar arguments apply to antisymmetric disturbances which,

for a submerged submarine model, may be imposed by mounting the model

on its side and oscillating it in the vertical plane. It is also

possible to impart a sinusoidal rolling motion to the model by suitably

adapting the mechanism; but again this requires no fundamental

modi-fication of the underlying theory.

Some properties of the measured oscillatory coefficients

Py means of dimensional analysis it can be shown that, for

example, hi , , f(n, Re, ---) j (23)

lim w.4.0 lim w40

M =M

w w = M.

That is to say a "non-dimensionalised" oscillatory coefficient depends

(for a given shape of model) on

Fn, the Froude number

Re, the Reynolds number

, the dimensionless "reduced frequency"

In general, then, the constants that we have called "oscillatory

coefficients" are frequency-dependent. In this respect they differ

in a significant manner from ordinary slow-motion derivatives.

Un-fortunately, no data have yet been published for a submarine showing

this variation of a typical oscillatory coefficient with frequency.

It is to be expected, however, that as the frequency is made very

small the oscillatory coefficient becomes equal to the more familiar

slow motion derivative. Thus we should expect that

(24)

as has already been mentioned.

Suppose that it is necessary to find the slow motion derivatives

M0, M, Y._{rt q} for use in the third of equations (22). It would appear

that

only

N

wed M, can be found, beinic inseparable from M

1

0.

By analogy with equation (13) the oscillatory coefficient Mo is

24

* In order to prevent interference of the flow round the fin by the supporting struts, the model is usually held upside down.

= Me w4 (25)

If therefore the pitch 0 = 00 sin wt _{is imposed and a curve is} plotted of the in-phase component

Me against w2 one may seek to

findbothMandM,using the technique indicated by the sketch

0

in fig. 2. In as much as it

i8

usual

to

take Me = -mgh in prac-tice, one should also be able to applya useful check on this part

of the result.

The techniAte of measurement

It i8 usual in the Study of ship and submarine dynamics to

sep-arate the

analysis

of surging motions u from that of the remaining symmetric motions w and q. We shall therefore focus our attention

on the latter. _{We shall examine briefly (a) the method by which the}

desired sinusoidal motions are imparted to a model and (b) the way in

which the oscillatOry-coefficionts may be calculated from the measured

data.

The oscillatory coefficients may be measured conveniently in a

towing tank by the use of a planar motion mechanism which imparts a

known vertical displacement at each of two points P and Q, of a model.

These are best chosen on the centre line of the model

and

at equal distances-AL0 - say, fore and aft of-the centre of mass C (fig.3).*

The displacements, which will be denoted by z1 and z2, are usually

made to-vary_sinusoidally by slider-crank mechanisms or by Scotch

place to diScuss them here). It must be remembered that the displacements

z1 and z2 are vertical and so not necessarily parallel to the moving axis Cz.

The forces that have to be applied to the model to maintain

the sinusoidal motion are measured at the two points P and Q,

where the planar motion mechanism is attached. To be more exact, it is the components of these forces in the Ct direction which are measured; they will be denoted by F1 and F2. The planar

motion mechanism iS mounted on the carriage of the towing tank so

that the model may be given a velocity U along the centre line of

the tank together with a harmonic vertical motion relative to the

carriage. We shall assume that the towing speed may be treated as

the reference speed, any fluctuation of the speed in the direction

Cx being negligible.

Heave coefficients

of

a submerged submarine model

-V

When the coefficients ZZ., M , M. are measured, the model

w w w w

is made to undergo a pure heaving motion whilst it is towed at

constant speed along the tank. The planar motion mechani= is adjusted

so that

z1 and z2 have the same amplitude and are in phase, being given by z = zo sin wt so that

w =

zd1)cos

c4

1

1 * = -zo w2 sin wt

/I

The displacements

z1 and z2 are now parallel to Cz, which (26)

26

remains vertical.

The

steady

state

_{amplitudes and phases of the forces}

1 and _F2 are recorded; suppose they are given by

Fi = Fl, + (Fi)in sin wt +

(Fl)out cos wt

F2 = F2'4

(F2 )inwt

(F2 )outcos wt

where, it will be noted allowance has been made for Constant as well as

fluctu-ating components. The oscillatory coefficients:, for heave may now

be found from these measurements, but their derivation requires the use

of equations ,,of

motion-which,are

not only simplified -relationships but

have

been simplified in,more ways than_one-in

_{the literature.}

Remembering that-we are concerned with sinusoidal'heaving motions

of a submerged submarine consider the heave equation of motion. It is the second of equations (22) with the expression for .AZ that is

given in equations (21).

That is

zla

la.

z w + (m

_v.

Z.)* - (Z

w

+ M(7)q -

.

ci.

= z z(t) + _{(zn)1n1 + (z1.)11 + (Zn)2n2 + (Z.)}

n 2 2 (28)

Notice that we have here included-two extra terms - Z, and Z(t). The

first of these .Z is a constant for any given reference speed U;

it represents the dependence on the speed U of the normal force at zero

angle of attack.- This dependence exists because the plane Cxy is not

a plane of symmetryvif only because of the presence of the fin, and it

would normally be-counteracted by

adjustment of the zero-settings of the

hydroplanesThe other forces. Z(t) is the imposed sinusoidal force

which causes the harmonic-displacements and which requires us to use (27)

oscillatory coefficients (as opposed to slow-motion derivatives)

in the equation.

The planar motion mechanism imparts a sinusoidal motion of

pure heaving while the hydroplanes are held fixed so that

equation (28) becomes

Fout = (F1)outout

+(F)

According to equation (26) we have

-Zwzow cos wt (m - Z.)z_w o w2 sin wt = F* + F.in sin wt + Fout cos wt

( 31 ) so that F. in 2 z

Thus we can find Z,, Z_w and Z. from readings obtained for a

given speed. (32)

+(m-Z.)*=Z A-F*+F.

_in i-Zvw out s _n Lot F os wt ( 29 ) where F* = F1* F2* Fin = (F1)

in

(F2)in ( 30 )

28

Thus

The pitch equation (22)

is

-Muu

-Mww-y -MO-Mq+I

0

= m, + m(t) +

(m)in

+

(m)1I',1

+ (M

)2n2

191

n 2n 2 +

(M22

n

(33)

For the motion (26)t1ierefore,

Z W GOS Wt M.Z W2 sin wt =M* + G* + G. sin wt + Gout cos wt

WO w in

Pitch-coefficients.of a submer ed submarine model

The model may be given a pure pitching motion while it is moving

along the tank

at

constant speed.

The model moves along a path like that shown

in fig, 4,

The angle made by the axis Cx of the model with the horizontal is

-1 z1

8 = sin

0

(36)

(37)

where

(34)

G* =

-Zo(F1*

F2*)

Gin = -Zo

Pl)in

(F2)in

(35)

The vertical downwara Velocity

of C

is

(

z1

z2)

so

that the resultant

velocity makes an

angle

X = trim

d( Zi + Z2)

-[

-1 dt 2

with the horizontal,

If the motion is

to be one of pure pitching, it is necessary that w = 0 = so that the velocity of C must always

be tangential to the Path of C. That is 0 = -X, so that for small

angles

this relationship. It is easy to show that this requirement is met if

z1 and z2 have the form

z1o

cos (wt + z2 = zo cos ( wt - 2

where the phase lead of zl with respect to z2 is

w2, -E 2 tan

u

(38) z (zi z2,) -2 (39)

01

U

To produce a pure pitching motion, then, z1 and z2 must satisfy

(no)

30

That is to

say, a

_{pure pitching motion is obtained}

if

the amplitudes

of

z1 and 22 are the

same and

the phase difference between them depends upon the

frequency of

_{oscillation and the towing speed in this}

way; z1

must lead _z2- by the angle*

E.

If the displacements

z1 and. are of the form given in equation

(40),they

_{are no-longer parallel to Cz as the direction of}

the

latter

varies.

The angle of pitch is given by cos-(wt - cos (wt + sin 0 =

o

according to equation (37). This may be simplified to

sin 0 =

--a

sin wt sin

S.--Zo 2

so that, for small

angles

of

pitch, we may write

2.

= e = e

_0wsin

wt

* An alternative

expression for

e is commonly used in the literature;

it is

-,

c =

cos1

2)/(l

)/(l + '1,42)1

where 1.1 = . where

It follows

that O = eo sin wt o o o = sin 2 = = eow cos wt (42) (43) ()4)

If the steady state force amplitudes F1 and F are measured

and have the form

(27)

it is posSible to calculate the pitch coeff-icients. Notice that having arranged _z1 and z2 to give a pure

pitching motion (by adjustment of s, their difference of phase) we

can conveniently refer the phase of the applied forces F1 and F2

Whence

Fin Z. = ---7

q

0ow

The pitch equation (33) is now

_ 2 .

-M O.

sin wt -MOwcos wt -10w sin wt

e q Y

-= M* + G* + G. sin wt + Gout cos wt in

.0"

to 09 the angle of pitch - rather than to the dummy variable with

reference to which z and

z2

are set.

1

The heave equation (28) now becomes

-(Z + mU)(0w cos wt) + Z.(0 w2 sin wt)

o q o + F. sin wt + F' cos wt in out

(45)

so that Z* = -F* F '7

out

₍₄₆₎

`111.0

Z=

---q 0w