Position of the origin of axes of a submerged body

Contents Surunary. 1. _{Nomenclature.} 1.1. ITTC Notation. Page iii 1 1

1.2.

Notation Concerning the PMM.

3

1.3.

Definitions. 3

2.

Introduction. 5

2.1.

Motion Requirements. 5

2.2.

The Planar Motion Mechanism. ₆

2.3.

Position of the Origin with Respect _{to the Body. 7'}

3.

Position of the Origin with Respect to the PMM. ₈

3.1.

Pure Heaving, Pure Swaying. ₈

3.2.

Pure Pitching Motion; Basic PMM. ₉

3.3.

Pure Pitching Motion; Adapted PMM. ₁₁

3.i.

Pure Rolling. ₁₇

1-p.

General Position of the Origin in the Body. _17'

1. _{Equations of Motion.}

18

5.

Application to PMM Experiments. ₁₈

5.1.

Equations of Disturbed Motion. ₁₈

5.2.

Hydrostatic and Gravity Force Increments. ₁₉

5.3.

Hydrodynamic Force Increments. ₂₀

5.14. _{Forces Exerted by the Swords.}

21

5.5.

PMM Equations of Motion. ₂₂

5.6.

Oscillatory Coefficients.

₂₃

6.

Conclusions. ₂₁₄

References. ₂₅

Tables 1 to

_5.

Oscillatory Coefficients. ₂₆ Appendix I. Setting the AEW PMM For Unequal Oscillation.

_32-33

Suinniary

The analysis of the trajectory of a submerged body is customarily referred to axes fixed in the body whose origin is _{at the centre of}

mass. Difficulties may then arise in hydrodynamic model _{tests, in} that (1) the centres of mass of body and model may not correspond geometrically and (2) it may not be possible to position the model so that the origin is in the correct relation to the test mechanism. Previous work has shown that the first difficulty can be partially overcome by positioning the origin at the point corresponding to the centre of mass of the full scale body rather than _{at the centre}

of mass of the model. In this note the remaining difficulties _are overcome, firstly by extending the theory to permit a more general position of the origin in relation to the body, _{and secondly by} examining the motion imparted by the test mechanism in _{order to} permit alternative positions of the origin in relation to the mechaniSm.

11]

Approved fo issue

POSITION OF THE ORIGIN OF AXES OF A SUBMERGED BODY By T B Booth 1. NOMENCLATURE 1. 1. _{ITTC Notation} MOMENT N YAW VELX R z FORCE Z HEAVE VELY. w

MOMENT M

PITCH VELY a

FIG. I. BODY AXES.

Body axes - a right handed orthogonal set Ax, Ay, Az. For a

subinar-me the axes are usually chosen to be in the following directions,

as shown in Figure 1.

Ax in the direction of the fore and aft axis. Ay normal to the plane of symmetry.

Az perpendicular to the other two.

FORCE X MOMENT K SURGE OR U FORCE Y

SWAY yELl V

x

amplitudes of

oscilla-tion 4o Oo io

Mass in kg in' = rn/p13

Co-ordinates of the centre of mass

Inertia components (with

respect to body axes)

Inertia cross products

(with respect to body axes) Buoyancy Co-ordinates of the centre of buoyancy 8 '1' rads 0 j) rads

Oscillatory coefficients are denoted by the eg Mq The non-dimensional form is denoted by '. eg Mq'

The divisors which give the non-dimensional form are given in

Section

5.6.

in

rn/s

V W rn/s v' = v/U etc

v w = v'/U etc

Q H rad/s P' = P1/U etc

q r p' = p1/U etc

Y Z N X' = X/pU2l2 etc

Y AXt = AX/pU2l2 etc

M N N.m K' = AK/pU213 etc

AN AK' = AK'/pU213 etc

xc yc zc m xc' = xc/l etc

Ix ly Iz kg.m2 Ix' = Ix/pl5 etc

Ir Ixz

Izx kg.m2 Ixy' = Ixy/p15 etc

B N B' = /p12U2

xb yb zb in xb' = xb/1 etc Dimensional Units Non-dimensional

Fluid density p kg/rn3 Characteristic length of the body 1 Reference velocity U Velocity components U small disturbances U

Angular velocity

compon-ents P small disturbances p Force components x small increments x Moment components K small increments

Angles of roll, pitch and yaw

(modified Euler angles) small increments

Small rotations about

Values of quantities in the undisturbed reference motion _{are denoted}

by . eg 0.

1.2. Notation Concerning the P4

Half-length between swords b

Displacement of swords 61 62

Amplitude of oscillation of swords al a2

Phase angles of oscillation of swords l 2

Amplitude of angular oscillation o Oo ipo

Frequency of oscillation w

Non-dimensional frequencies v = wl/U i w b/U

'-

/

- wl/g)

Direction of motion relative to direction

of tow x

Co-ordinates of P1 and P2 in body axes Co-ordinates of the origin with respect

to the sword attachments. Figure

_1.

Components of force, increments exerted by the swords on the model

Moment exerted via the roll gauge on

the model

Components of resultant force

increment

Components of resultant moment

increment

(xl, yl, zl)(x2, y2, z2)

(i,

_{n, çl)(2, n} , 2)

Fix Fly Flz F2x F2y F2z Gix

Fx F'y Fz Fx'= Fx/pl2U2 etc

Gx Gy Gz Gx' Gx/pl3U2 etc The 'in phase' and 'quadrature' components are denoted by suffices

IN and QUAD respectively.

1.3. Definitions See Figure 2

Planes of pitching and yawing - the planes Axz and Ayz respectively. Motion in the plane of pitching - any motion for which V = P = S = 0.

Each point of the body is moving in a plane, fixed in space, parallel

to the plane of pitching. Hence it is permissible to refer to the motion of the body as a whole as 'motion in the plane of pitching'.

Similarly for 'motion in the plane of yawing'.

Motion in pure heaving - motion in the plane of pitching for which

Q = 0 also. _{Since there is no rotation, all points of the body}

have the same velocity in both magnitude and direction. Hence it

is permissible to refer to the motion of the body as a whole as one of 'pure heaving'. Similarly for 'motion in pure swaying'.

Motion in pure pitching - a class of motions in the plane of pitching. The motion of a point of a body is one of pure pitching if there

FIG. 2(a) PURE HEAVING.

FIG. 2(b) PURE PITCHING.

exists a line in the plane of pitching, fixed in the body, such that the velocity of that point is always in the direction of that line. In general not all points of a body can be in pure pitching motion

simultaneously. This description must therefore specify the point

concerned. Similarly for 'motion in pure yawing'.

Motion in pure rolling - motion for which there is a line parallel to the x axis, fixed in the body, such that the velocity of all points on that line is in the direction of that line. It can

readily be shown that this implies Q = R = 0.

Note that while the directions of the axes are fimdamental to the above definitions, none requires the position of the origin in the

body to be specified.

2. INTRODUCTION

2.1. Motion Requirements

The analysis of the motion of a submerged rigid body, such as a sub-marine, is usually based on equations of motion referred to axes

fixed in the body. These equations are in terms of the components of translation and rotation of the body along and about the body

axes. They consist of terms due to gravity, inertia, hydrostatic

forces and hydrodynamic forces. Of these the gravity, inertia and hydrostatic terms are readily calculated. The difficulty lies in

specifying the hydrodynamic forces. At present these are established by towing tank testing, principally by the use of the Planar Motion

Mechanism (commonly abbreviated to PMM).

In order to simplify the analysis of PMM experiments the model is rigged so that the motion of the origin of body axes A, consists of a steady reference motion, usually in the direction of the x

axis, on which is superposed a small sinusoidal disturbance consisting of only one component of velocity or angular velocity. The motions

are :

-Disturbance component w ; pure heaving.

Disturbance component v ; pure swaying.

Disturbance component q ; pure pitching such that A moves in the direction of the x axis.

Disturbance component r ; pure yawing such that A moves in the direction of the x axis. 5. Disturbance component p ; pure rolling.

Note that while it is customary for P experimenters to describe motion 3 as pure pitching, it is possible to have a pure pitching motion of A for which neither of the disturbance velocities u or w

is zero. The requirements of motion 3 are not satisfied unless

(1) the motion of A is pure pitching, (2) u = o, and (3) w = o. Similarly for motion 14

It is certainly possible to analyse motions in which the disturbance consists of more than one component, but the simpler, more direct

experiment is preferred. (It is sometimes argued that a disturbance velocity in u is tolerable since it merely introduces harmonics into the fluid force terms which are removed by the Fourier analysis.

This is not in general true. Equations (21) show that fundamental effects are introduced into the inertia terms of the surging, pitching and yawing equations, and they may be introduced into the

hydrodynamic terms also.) It will be assumed that the motions

required are those listed above. 2.2. The Planar Motion Mechanism

ROLL LINKAGE-.

SUPPORT SWORDS

ROLL GAUGE X V Z GAUGES FIG.3. PLANAR MOTION MECHANISM.

The Planar Motion Mechanism is mounted on the carriage of a towing tank for the purpose of imparting motions described in the preceding

paragraph. See Feferences [ 1 and [ 5

J .

Briefly, the model is

supported at points P1 and P2 say, by swords which can be oscillated

sinusoidally along their length. Figure 3. The model is restrained

from rolling about P1P2. Usually the swords are vertical one behind the other, P1P2 is in the direction of the tow, and the oscillations

are of equal amplitude. If the model is rigged so that its plane of pitching is parallel to the plane of the swords, the x axis parallel to P1P2, and the oscillations are in phase, then the motion of the

model is one of pure heaving. By setting a suitable phase differ-ence the motion of F, the mid point of P1P2, can be made one of pure pitching with P moving in the direction P1P2. Similarly by rigging the model on side the motions of pure swaying and pure yawing are

DIRECTION OF TOW

X Y Z GAUGES BACKBONE

imparted. _{Rolling is imparted by a special linkage which is used in}

lieu of the oscillatory motion of the swords to _{roll the model}

sinu-soidally about the line P1P2.

The forces exerted on the model by the swords are measured by sets

of gauges at Fl and P2. _{It is highly desirable that these should}

be orientated so _{.s to measure the components with respect to the body}

axes. _{Usually there is no difficulty in so doing. These components}

are further resolved into components 'in phase' and in 'quadrature' with the disturbance motion by Fourier analysis. _{The contributions} due to inertia, gravity and hydrostatic pressure can be calculated

and removed. _{Thus the PMM measures the 'in phase' and 'quadrature'} coirrponents of each of the force and moment components due to each of the motions.

2.3. Position of the Origin with Respect to the Body

So far the motions required have been discussed in relation to the body axes, as has the role of the P _{in imparting these motions,} but without considering how the axes themselves are to be chosen.

For a submarine, at least, there are preferred directions of _{the axes,} (see nomenclature) and it is likely that other bodies also have

their preferred directions, but the choice of origin is, as yet, open.

Such is the importance of the centre of mass of a body in dynanical

problems that it is the obvious choice for the origin. _{If possible,} the model is rigged such that this point is at F, the x axis lying along P1P2 and the plane of pitching or yawing coinciding with the plane of the struts. _{With this arrangement the PMM imparts the}

motions required in paragraph 2.1.

In practice this arrangement may not be possible, _{nor indeed is it}

necessarily desirable. Several objections may be cited.

It may not be possible to achieve the correct fore and aft position of the centre of mass vis a vis the full scale body

eg the weight of the propeller motor aft of the swords may

create this situation.

It may not be possible to position the model so that the origin

is at P eg structural difficulties.

For economy, the conversion of a submarine model from its inverted position (for oscillation in the plane of pitching) to its on-side position (for oscillation in the plane of yawing) is achieved by leaving the basic body of revolution _{attached to} the gauges and swords, and rotating only the appendages.

Unfortunately the centre of mass is not usually in the correct

position in both configurations.

1.. _{The centre of mass of the full scale body may not be in the}

position assumed at the time of the P experiments.

5 The choice of the centre of mass as origin. may introduce

unjustified complication into the expressions of hydrodynamic

revolution are conveniently specified with reference to an x axis which is also the axis of revolution, but the centre of mass is probably not on this axis.

The first of these objections draws attention to a very important point, viz geometrical similarity does not imply similarity of

mass or its distribution. (It must be remembered that when gravity and inertia terms are removed from Pff measurements, as described in paragraph 2.2, it is the values appropriate to the model which must be used, not those appropriate to the full scale body.) On

the face of it, the origin of axes used for full scale analysis need not correspond geometrically to that used to analyse the model

experiments. However, the object of the model experiment is to

measure hydrodynamic forces. These are essentially dependent on the geometry of the body and are independent of mass distribution.

Consequently the data measured on the model is most easily applied to the full scale body if the axes correspond geometrically.

Moreover the expressions for hydrodynnric forces are neither short nor simple, and considerable complications can occur in conversion if the axes do not correspond. In this note it is assumed that

they do.

If the mass distribution of model and full scale body differ,

References [ 2 and

_E

5

3

show that the desirable location for the

origin of axes_is the centre of mass of the full scale body.

Reference [ 5

J

gives the data reduction equations on the assumption

that the mass distribution differs,but that the point in the model corresponding geometrically to the centre of mass of the full scale body can be positioned at F, and the model oscillated as required

in paragraph 2.1. This overcomes the first objection but not the

other four. In the following sections the following alternatives

are explored.

Firstly, accepting that it may be impossible to position the model so that the origin is at P, the possibility is explored of using or adapting the PNM to produce the motions required with respect

to points other than P. (No difficulty arises in rotating arm experiments, since there is no restriction on the position of the origin in relation to the arm.)

Secondly, the theory is ejctended to configurations in which the

origin is not related to either centre of mass, but is chosen at any convenient point of the body.

3. POSITION OF TEE ORIGIN WITH RESPECT TO THE PMM

In this section the positions of the origin to which the P4 can impart the required motions are established; firstly assuming the usual configuration of the mechanism; secondly assuming that it is adapted to our purpose.

3.1. Pure Heaving, Pure Swaying

The motions of pure heaving and pure swaying present no difficulty and require no adaption of the mechanism. When the model is

oscillated in this mode in the usual way, the motion of all points in it is identical. Hence the position of the origin is unrestricted.

3.2. Pure Pitching Motion; Basic PMM

First suppose that the PMM is rigged in the usual way so that the motion of P is one of pure pitching. Now examine the possibility that the motion of some other point of the model is also one of

pure pitching. Use can be made of the following lemma.

"If the motion of one point of a body, P say, is one of pure pitching, then

either:-those and only either:-those points in a plane through P normal to the direction of motion of P are also in pure pitching, and they are moving parallel to P, or

the body is rotating about a fixed axis normal to the plane of pitching, in which case all points of the body are in pure

pitching motion.

First note that each point is moving in a fixed plane parallel to the

plane of pitching. Since the body is rigid the motions of all points on a line normal to the plane of pitching are identical. Hence only

the motion of points moving in the same plane as P need be considered.

FIG. 4.

Let P be a point whose motion is one of pure pitching. Let its

velocity V be in a direction with a reference direction fixed in

space. Let Q be the angular velocity of the body. Let X be any other point of the body, distance c from P and making an angle a with

the direction of motion of P. Since the latter rotates with the body, a is constant.

If X also is in pure pitching motion its velocity vector also rotates

with the body. Hence its direction makes a constant angle with the direction of motion of P. The motion of X consists of the motion of

DIRECTION OF MOTION OF P DIRECTION OF MOTION OF X

REFERENCE DIRECTION

RIGHT ANGLES IC. PX

RIGHT ANGLES TO DIRECTION OF MOTION OF X

RIGHT ANGLES TO REFERENCE DIRECTION

V, VELOCITY OF P REFERENCE DIRECTION

10

FIG. 5(b)

P plus a velocity Qxc in a direction at right angles to PX. The sum

of these two ha-s a component at right angles to the direction of motion of X which is zero. Hence

V sin 8 = Q c cos (a + 8)

(1)

and the velocity of X is

V cos 8 + Q c sin (c + 8) (2)

Assum4ng that neither V nor Q is zero there are two solutions of

equation (1).

V/Q = c cos (a + 8)/sin 8 = C say

Since V/Q. is the distance of P from the instantaneous centre of

rotation and is constant, the motion is one of pure rotation about a fixed point - the point 0 in Figure 5a. The angular velocity need not be constant, but V and Q are not independent. This solution corresponds to a rotating arm experiment, and confirms that there is no restriction on the position of the origin in relation to the arm.

sin 8 = o, cos (a + 8) = o

Hence 8 = o or ie the motion of X is parallel to that of P

and a ± ir/2 le X lies on a line at right angles to the

velocity vector of F, as in Figure 5b.

The magnitudes of V and Q are independent, but the velocity vector of P rotates with the body, hence dA/dt = - Q.

Applying this result to the PMM it is seen that pure pitching motion is achieved if the swords oscillate in the usual way, the model being positioned such that its plane of pitching is parallel to the plane of the swords and the origin A lying in the plane through P normal to

P1P2. The motion of A will be parallel to that of F, hence if the model is orientated such that the x axis is parallel to P1P2, the

requirements that the reference motion shall be in the direction of the x axis and the disturbance velocity w shall be zero are both satisfied. Unfortunately if A is positioned away from P in this way equation (2) shows that there is a disturbance velocity component u in

addition to q unless c = o ie unless X is on a line through P normal

to the plane of pitching. Only then are the requirements of motion 3

of section 2.1 met.

FIG. 6. BASIC PMM.

3.3. Pure Pitching Motion; Adapted PMM

Now examine the possibility of adapting the motion of the P?4 to permit the origin to be positioned with less restriction and yet

impart the motion of pure pitching. The mechanism can readily be

adapted as

follows:-The amplitudes of oscillation of the swords need not be equal. The mechanism as a whole can be tilted so that the swords need

not be vertical.

Supposing these facilities to be available and the model to be rigged with its plane of pitching parallel to the plane of the swords,

we now establish which points, if any, of the model are in pure pitching motion as a result of small displacements of P1 and P2

CARRIAGE SPEED U

IA

P1

POINTS IN PURE PITCHING MOTION LIE ON A PLANE NORMAL TO P1 P2.

POINTS FOR WHICH U 0 ALSO, LIE ON

A LINE ThROUGH P NORMAL TO THE PLANE OF THE SWORDS.

being superposed on the reference motion. Only points in the plane of the swords need be considered since the motions of all points on

a line normal to this plane are identical.

ANGLE OF TILT. REFERENCE POSITION OF P1 P2. AX NORMAL TO THE PLANE OF THE _'7 SWORDS. _x 8a84 6 62 P2 A VELOCITY OF X OR A. CARRIAGE SPEED U FIG. 7. ADAPTED PMM.

Let the mechanism be tilted through an angle 8, and let the dis-placements of P1 and P2 in the directions of the swords be 61 and

62. Let P1P2 = 2b, and let X be any point fixed in the model in the

plane of the swords, the co-ordinates of X relative to P1 and P2

being (ci,

ci),

(2,

2) as shown in Figure 7.

The rotation of P1P2 relative to the reference position is 0 where

tan 0 = (61 - 52)/2b (3)

Provided 61 and 62 are small compared with_b, tan 8 = 8 and P1P2 in its displaced position makes an angle 0 = 0 + with the direction of

the tow. The angular velocity of A1A2 is q = 0 = (51 - 62)/2b.

()

The motion of X is conveniently calculated as the reference motion, plus the motion of P1 due to the displacements 61, plus the motion

of X relative to P1. Denoting by u and w the disturbance velocity components in the direction of, and at right angles to, the reference

motion:

u = 51 sin 8 + q 1 w = 61 cos B - q 1 Observing that

cos 8 = (i - 2)/2b sin 8 = (ç2 - 1)/2b

₍₅₎

= (C2 (Si - i (S2)/2b

w = (-

2 61 + i (S2)/2b

(6)

The velocity vector of X makes an angle A with the direction of the tow, given by

tan A = w/(U + u) _(i)

Provided the displacements are not too violent*, u is small

compared with U and may be neglected, so that to the first order

tanX=vC/U.

(8)

Sinusoidal Motion

Now suppose (Si and (S2 to be sinusoidal at frequency w, but not

necessarily in phase or of the same amplitude. Let

(Si = al cos (wt + ci)

_{(S2a2cos(Wt+C2)}

in which the phase angles ci and c2 will be chosen so that 8 = o when t = o.

From equation (3)

tan 8 = 0 = (al cos (wt + ci) - a2 cos (wt + c2))/2b

cos ci - a2 cos c2)cos wt + (-al sin ci + a2 sin c2)sin wt)/2b

(io)

0 therefore oscillates sinusoidally at frequency w. For 0 to be zero

when t = o

al cos ci = a2 cos c2 ki say

so that 0 0 sin wt, where 0

80

= (-

ai sin ci + a2 sin c2)/2b From equations (9)

= - ai w sin (wt + ci) = - a2 u sin (wt + c2)

and on substituting in equations

₍₆₎

and (8)

tan A = A = (2 al w sin (wt + ci) - i a2 w sin (wt -I- _c2fl/2bTJ

(*This must be so for the motions with which we are concerned. (Si and (S2 are small compared with b, hence 0 is small, hence if the motion is pure pitching A is small, hence (Si and (S2 are small compared with ii.)

= ((2 al cos ci - i a2 cos c2) sin wt

+ (2 al sin ci - 1 a2 sin c2) cos wt)/(2bU/w) (13)

If the motion of X is to be pure pitching, the velocity vector of X

must rotate with the model. Hence A = - 0 + B here B is a constant.

Comparison of equations (10) and (13) shows that B must be zero, and

by putting t = o and 1T/2w

(2 al cos ci - 1 a2 cos c2)/(2bU/w)= -00 = -(- al sin ci + a2 sin c2)/2b

2 al sin ci = i a2 sin c2 = k2 say (iIi)

Hence using equation (ii)

For the amplitude_Oo to be positive ki must be positive, since, in equation (15) b, U, w and i - 2(= 2b cos 8) are all positive. Hence

from (ii) ci and c2 lie in the range - ii/2 (ci or c2)

Equations

(16)

determine the phase angles for pure pitching motion of

X. If, as is probable, X is between P1 and P2 so that i is positive and 2 negative, then ci is negative and c2 positive, implying that the motion of the forward strut (z2) leads the pitch angle, while the motion of the after strut (zi) lags. See Figure 8.

PITCH DATUM ON FRAME

FIG.8. PHASE RELATIONS FOR PURE PITCHING.

POTS ZERO ON FRAME

FIDUCIAL MARK ON POTa SPINDLE. OUTPUT OF SINE P0Th IS ZERO

WHEN THIS MARK IS IN LINE WITH POTENTIOMETER ZERO.

DIRECTION OF ROTATION

(2 -

i)ki = - 2bU Oo/w = (k2/2 - k2/i)U/w (15)

ie p ki 1 2 = - k2 b where p = wb/U

The ratio of amplitudes of strut motions is determined by equations (ii) or (15), or can be expressed as

al/a2 = ((b2 + 2 i2)/(b2 + 312 22))

(ii)

The value of Oo is determined by equation (15). _{Note that when}

8 _{o equation (15) reduces to 0o = 31} kl/b

Note that the values of al/a2, ci and c2 are independent of 1 and

- a result to be expected from the lemma of section 3.2.

Since A = - e the mean value of A is zero. Hence the requirement that w should be zero is satisfied if the model is orientated _such that in undisturbed reference motion the x axis is in the direction

of the tow.

The requirement that u should be zero is not in general satisfied. To the first order u = u _{and, from equation (6), is zero if and}

only if

- 2 1 + 1 2 = 0

(Note that tan A is then equal to wC/U without approximation).

On inserting the sinusoidal values of i and 2

2 al sin (ot + ci) - 1 a2 sin (wt + c2) = 0

ie (C2 al cos ci - i a2 cos c2) sin wt + (2 al sin ci - _{i a2 sin c2) cos wt} and ±'or this to be true at all times

(2 al cos ci - i a2 cos c2) = (C2 al sin ci - i a2 sin c2) = 0 from which

2 al sin (ci - c2) = 1 a2 sin (ci - c2) = 0 ₍₁₈₎

Assuming al and a2 are not both zero (in which case there is no

disturbance motion), these equations have four solutions.

sin(cl-c2)=o a. i and 2 same sign, in which case ci = c2 and ai/a2 = 1/2.

b. 1 and C2 opposite signs, in which case ci = ir + c2

and ai/a2 = - çi/2

a2 = 2 = o in which case the disturbance motion consists of rotary oscillation about P2, and X lies in the

reference direction from P2.

ai = 1 = o in which case the disturbance motion consists o± rotary oscillation about P1, and X lies in the

reference direction from P1.

I. i = 2 = a in which case S = o and X lies in P1P2, extended if necessary.

x

1(a)

SAME SIGN; ZI,Z2 IN PHASE; øh/02 =

I/2.

P2

I(b)I,

2 OPPOSITE SIGN; ZI, Z2

Ir OUT OF PHASE;aI/a2"fl/'2.

P2

P2

P2

2. 02= 2O

3. aI=IzO

4 _S P1 X P2

4. 6"O,

Iz2=O

DIRECTION OF TOW

FIG.9. PMM CONFIGURATIONS GIVING LLO.

Of these solutions only the fourth is compatible with the requirements

for pure pitching motion. If any of the first three solutions are combined with equations (ii) and (1)4) the result is al = a2 = o.

Thus tilting the mechanism precludes a pure pitching motion which meets

the requirements of Section 2. 1 also. On the other hand, the ability to set unequal amplitudes of oscillation of the swords permits the origin to be positioned anywhere in the plane through P1P2

perpend-icular to the plane of pitching. Figure 10. Although this falls short of complete freedom, it is nevertheless a most useful

relaxation.

I I

Ii VIlIlI/Kill

/CARRIAGE

SPEED

PURE PITCHING MOTION WITH IL 0 CAN BE IMPARTED ONLY TO POINTS IN THE PLANE THROUGH P1 P2 NORMAL TO THE PLANE OF THE SWORDS. THE TILT ANGLE MUST BE ZERO.

FIG. 10. ADAPTED PMM.

3.)4. Pure Rolling

The P4 cannot readily be adapted to roll the model about an axis

other than P1P2.

)4. GENERAL POSITION OF THE ORIGIN IN THE BODY

In order to refer the analysis to a general position of the origin it is necessary, firstly, to restate the equations of motion accord-ingly and, secondly, to apply them to the analysis of a PNM

experiment. The revised equations extend those derived in References [ 3 and [ 5 J in two respects.

1. _{References [} 3 and [ 5 derive equations for a model whose centre of mass differs from that of the full scale body, the origin being positioned at the point corresponding to the

centre of mass of the full scale body. In so doing it is

tacitly assumed that the centre of buoyancy lies on the z

axis. The revised equations permit general positions of the

2. _{References [ 3}

7

_{and [ 5}

7

assume symmetry of shape and mass

distribution in the Axz plane. The revised equations permit

general shape and mass distribution. 4.1. Equations of Motion

The equations of motion with respect to a general position of the origin are given in [ 2 as follows

m(1J - VP + WQ - xc(Q2 + B2) + yc(PQ. - R) + zc(PR + Q)) = X m(1T - WP + UP - yc(R2 + p2) + zc(QR - P) + xc(QP + R)) = Y - UQ + VP - zc(P2 + Q2) + xc(RP - ) + yc(RQ + F)) = Z

Ix + (Iz - Iy)QR - ( + PQ)Izx + (B2 - Q2)Iyz + (PR - )Ixy

+ m(yc(W UQ + VP) - zc(V - + UP)) = K

ly + (Ix - Iz)RP - ( + QR)Ixy + (P2 - R2)Izx + (QP - )Iyz

+ m(zc(fj - VP + WQ) - xc(1 - UQ + VP))

=M

Iz + (ly - Ix)PQ - (. + RP)Iyz + (Q2 - F2)ixy +(RQ

+m(xc(T-WP+UR)-yc(fJ-VP+WQ))

=N

(19)

in which the X, Y, Z, K, M, N are the force and moment components acting on the body including fluid forces, gravity forces and, in restrained model experiments, the forces exerted by the swords on the

model. Of these the hydrostatic and gravity terms are readily and exactly specified, viz

To X - (mg - B) sin e

To Y (mg - B) sin cos 8 To Z (mg - B) cos cos 8

To K (mg yc - B yb) cos cos 8 - (mg zc - B zb) sin cos 8

To M (mg xc - B xb) cos cos 9 _{- (mg zc - B zb) sin 8}

To N (mg xc - B xb) sin cos 8 + (mg yc - B yb) sin 8 ₍₂₀₎

The hydrodynamic terms are less well defined and may be lengthy.

The expressions used the Naval Ship Research and Development Center USA are given in [ 2

J

5. APPLICATION TO PMM EXPERIMENTS

5.1. Equations of Disturbed Motion

Theprinciples on which the theory of the P?1 is based are discussed

the procedure and the main equations and results will be presented.

The body is not assumed to be symmetrical.

In application to a PMM experiment equations (19) are linearised by assuming that the motion consists of a steady reference motion U in the direction of the x axis from which force and moment increments AX, AY, AZ, AK, AM, AN produce small disturbances whose products can

be neglected.

AX=m(ü - yc F + zc

₄₎

+ Ur -

zc + xc F) - Uq - xc

4 + yc

AK= Ix - Ixy

4 -

Ixz F + m yc(r - Uq) - m zc( + Ur)

AM=Iy

4 -

Iyz F - lyx + m zc - m xc( - Uq)

ANIz F - Izx

- Izy

₄

+ m xc(r + tJr) - m yc ü (21)

The force and moment increments are composed of increments in the gravity forces, hydrostatic forces, hydrodynamic forces and forces

exerted by the swords. Of these the gravity and hydrostatic force increments depend on the instantaneous orientation of the model but not on its motion, which the hydrodynamic force increments depend on the motion relative to the fluid but are independent of the

orienta-t ion.

5.2. Hydrostatic and Gravity Force Increments From equations (20) these are

AX = - (mg - B) cos 8 A8

AY = (mg - B)(cos cos 8 A - sin sin 8 A8) AZ = (mg - B)(- sin cos 8 A - cos sin 8 A8

AK = (mg ye - B yb)(- sin cos 8 A - cos sin 8 A8)

- (mg xc - B xb)(cos cos 8 t - sin sin 8 A8) AM = (mg xc - B xb)(- sin cos 8 A - cos sin 8 A8)

-(mgzc-Bzb)cosOAO

AN = (mg xc - B xb)(cos cos 8 A - sin sin 8 A8)

+ (mg yc - B yb) cos 8 A8 (22)

where A and A8 are the increments in c and 8.

Provided that these increments are small, they may be regarded as the result of small rotations , 0, about the body axes, to which

they are related as

follows:-= _{+ sin tan 8.0 + cos tan 8.} = cos ct.0 - sin .ij

Hence the following relations are obtained for the four test

con-figurations.

Modelupright

8=O

=L8=O

Modelinverted

_=7r,8=O

Model on-side (starboard) = r/2, 0 = 0 = _{8 =} -1. _{Model on-side (port) =}

_-ff12,

_{8 = 0 t4 =}

4 A8 =

Hence the following expressions for the increments of the hydrostatic

and gravity forces.

X=+(mg-b)O

_-U

-i-I

=+(mg-b)p

_+S

_-P

AY+(mg-B)

_+U

_-I

= 0

SorP

AZ=

0

UorI

=±(mg-B)4

_-s

+P

K=±(mgzc-Bzb)

-Ti

+1

=T(mgyc-Byb)

-s

+

M=+(mgzc-Bzb)e

-U

+1

=+(mgxc-Bxb)$+(mgzc-Bzb)p

-s

-i-p

AN=±(mgxc-Bxb)+(mgyc-Byb)e

+U

-I

=(mgyc-Byb)

-s

+P

(24)

where U, I, S, F, denote the upright, inverted, onside (starboard)

and onside (port) configurations. 5.3. Hydrodynauiic Force Increments

In References [ and

1

5 the increments in fluid force, both

hydro-static and hydrodynamic are included in the oscillatory coefficients. By stating the hydrostatic increments explicitly (as will be done here) the oscillatory coefficients need express only the

hydro-dynmc force increments. Since these do not depend on the orienta-tion of the body, coefficients such as O and Kc are eliminated.

All the hydrodynamic force increments can be expressed in terms of oscillatory coefficients and increments in velocity and acceleration eg the pitching moment due pitching is expressed as q +

_{M4 4.}

5.1. _{Forces Exerted by the Swords}

BODY AXES FOR MODEL INVERTED

FIG. II.

Denote the force component increments exerted on the model at P1 by

Fix Fly Fiz and those at P2 by F2x F2y F2z. Denote the moment exerted at P1 via the roll gauge by Gix. Figure 11. Denote the

co-ordinates of P1 and P2 by (xi, yi, zi) (x2, y2, z2). Then the

resultant can be expressed generally by force components Fx Fy Fz acting at A together with moment components Gx Gy Gz where

Fx = Fix + F2x Fy = Fly + F2y

Fz = Flz + F2z

Gx = Gix + Flz yl - Fly zi + F2z y2 - F2y z2 Gy = Fix zi - Fiz xl + F2x z2 - F2z x2

Gz = Fly xl - Fix yl + F2y x2 - F2x y2

For the arrangement shom in Figure 1, and with the x axis in the direction of the tow the co-ordinates of P1 are:

Test Configuration xi zl

upright l

inverted - i -

-onside (starboard) - i

-onside (port) - i - ri

with similar expressions for x2 y2 z2.

5.5. P Equations of Motion

By substitution for AY, AZ, AK, AM, AN in equations (20) the following equations are obtained for the motions imparted by PMM

tests. These equations assume that the disturbances are small

and sinusoidal.

Pure heaving - model upright or inverted

Yww+ii+Fy =

AY = 0

ww+ri+Fz =

AZ = m

Kww+Ki+Gx =

AK

= myc

Mww+Mr+Gy =

AM

= -mxci

Nww+N+Gz =

AN = 0

Pure pitching - model upright or inverted

Pure swaying - model on side

Yvv+Yrr+Fy =

AY = m

Zvv++Fz =

AZ = 0

Kvv+Krr+Gx =

AK

= -rnzcr

Mv v + Mr T + Gy = AM = 0

Nvv+N+Gz =

AN

= mxci

Pure rolling - model upright or inverted

Ypp+Y+Fy±(mg-B)

AY-mzc

Zpp+Zf+Fz

AZmyc

Kp p + K + Gx + (mg zc - B zb)$ = AK = Ix

Mp p + + Gy = AM = - lyx

Npp+M+Gz+(mgxc-Bxb)4 = AN- Izx

where U and I denote model upright and inverted.

U or I U or I U or I U or I U or I (25)

+U -I

U or I

-U+I

U or I

+U- I

(21) S or P S or P S or P S or P S or P q + + Fy = = 0 U or I Zq q + + Fz = AZ = - m Uq - m xc U or I Kq q + + Gx = AK = - m yc Uq - Ixy U or I

Mqq+M+Gy+(mgzc-Bzb)e AMmxcUq+Iy

-U +1

Nqq+N+Gz+(mgyc-Byb)eAN-Izy

+U -I

(26)

Pure rolling - model on side

Ypp++Fy

=AY=-mzc

Zpp+Z+Fz+(nig-B)

=

Z=myc

Kpp+K+Gx±(mgyc-Byb) =K=Ix

p++Gy±(mgxc-Bxb)=M=-Iyx

Npp+N+Gz

=

N=-Izx

where S and P denote model onside, starboard and port.

5.6. Oscillatory Coefficients

Finally since the motion is the result of sinusoidal displacements

eg $ = o sin wt, the force increments Fx Fy Fz can be expressed in the form F ₌ FIN sin wt ₊ FQUAD cos wt with similar expressions

for Gx, Gy, Gz. On equating the in phase and quadrature terms the

relations listed in Tables 1-5 are obtained. These are given in non-dimensional form, the divisors relating the oscillatory

coefficients to the dimensional form being listed also. Those

coefficients which have gravity and hydrostatic terms are given

in dimensional form also.

The only extra information required in comparison with [ 5

_II

is

the buoyancy of the model and the centre of buoyancy. This is

readily determined, being a function of the geometry only. The

measurement of model mass and centre of mass still presents a problem to which the most satisfactory solution is to ballast the model to neutral buoyancy and neutral stability in pitch and roll.

In this condition the mass equals the buoyancy and the centres of mass and buoyancy coincide, so both are knom accurately. Moreover

the hydrostatic and gravity terms (those in the same bracket as B) cancel, thus simplifying the analysis.

S or P

-S +P

-S +P

-s

+P

S or P

Pure yawing - model on side

Yr r + YF F + Fy = AY = mUr + m xc F S or P

Zr r + ZF F Fz = AZ = 0 S or P

Kr r + K' F + Gx = AK

= -

Ixz

F -

m zc Ur S or P Mr r + MF F + Gy + (mg zc - B zb) AM

= -

Iyz F

+S -P

Nr r + NF F + Gz -s- (mg yc - B yb)p = AN = Iz F + in xc hr

-s

+

6. CONCLUSIONS

The expressions obtained in section 5 permit the analysis _{of PMM} experiments with respect to any position of the origin in _{the body,} provided only that the origin can be given the motions _{required by}

the experiment. _{The results of section 3 show that, given the}

facility to set unequal oscillations of the swords, the five motions required can be imparted to any point on the line joining the

points of attachment of the swords, or its extension. Thus the

origin need not be positioned mid way between the points of attachment

as is the present custom. _{An even wider range of positions of the}

origin is possible if not all five motions are required.

The greater freedom in positioning the origin both with respect

to the body and to the mechanism is likely to overcome the objections cited in section 2.1 for all but the most extreme body shapes.

The problem of correctly positioning the origin in relation _{to the} swords does not arise in rotating arm experiments.

References

Reference 1. Gertler M, Syinposiuni on Towing Tank Facilities,

Zagreb,

_1959.

Reference

2.

Gertler, M and Hagan G H, Standard Equations of

Motion for Submarine Simulation. June

_1967.

NSRDC Report

2510.

Reference 3. Bishop R E D, and Parkinson A G. On the Choice of Origin for Body Axes Attached to a Rigid Vehicle.

1969 Journal of Mechanical Engineering Sciences 11 ,55 1.

Reference 1. Bishop H E D, and Parkinson A G. On the Planar Motion Mechanism Used in Ship Model Testing.

1970.

Phil Trans Roy Soc (A) Vol

266.

Reference

5.

Booth T B and Bishop R E ID. The Planar Motion

Table 1

PURE SWAYING - MODEL ON SIDE (PoRT OR STARBOARD)

yyosinwt

V

yocoswt

= - yo w2 sin wt

V1 = yo'u eQs wt = - yo'v2 sin wt

Divisor Coefficient

p12 _{Yv' = - FYQUAD} =

-p1 ¶ Ky' =

_{- GxQU'}

/Yo'V

pl3U Mv' = - GYQU'/Yo'v pi3U Nv' = -pl3

Y' =

m' + FYIN'/yo'v2 p13 Zr' FzINh/yotv2 = - m'zx' + GxIN'/yo'V2 p1 Mr' _GyIN'/yo'V2 pl Nir' = mlxct _{+ GzIN'/yo'v2}

Table 2

PURE HEAVING - MODEL UPRIGHT OP INVERTED

z = zo sin wt w = zo w cos uit = - zo w2 sin t w' = zo'' cos wt = - zo'2 sin u.t

Divisor Coefficient p12 = -

QUJ'OV

pl2i Zw' = _{- FzQU'/zo'\} pl3U Kw' =

_{- GxQU'/zo'v}

pl3U Mw' = -pl3U _{Mw' = - GzQU'/zo'\.} pl3

_{YT' =}

pl3

Z' =

_{in' + FzIN'/zo'\2} pl Ki' = m'yc' + GxIN'/zo'v2

p1 M'i' = - m'xc' + GYIN'/zo'v2

PURE ROLLING - MODEL uPRIGHT, INVERTED OR ON SIDE 4> = 4>o sin wt Table 3 p = 4>0w cos wt. p' = 4>0 \) cos wt = - 4>0 >2 sin wt = - 4>0

.2

sin wt Divisor Coefficient Test Configuration

(ii upright, I inverted,

on side (starboard) S, (port)P)

pi3U ' =

-Qu'4>°

U, I, S or P pi3U Zp' = - FzQU'/4>o U, I, S or P pl'U Kp' = - GxQU'/4>O v U, I, S or P plL>U Mp' = - GYQU'/4>o v U, I, S or P

p1U

Np' = - GzQU'/4>o V U, I, S or P = - m zc + IN"4>° w2

±

(mg - B)/w2 + U - I pl pl ,

y'

= = -+ IN"4>° 2

±

(m'/(w')2 - m'zc' + IN'4>° v2

-B'/v2)

+ u

- i

S or P pl4

Z' =

m'yc' + FzIN'/4>o V2 U or I = m ye + FzIN/4>o w2 + (mg - B)/w2

-+ p

3p1

Z' =

m'yc' + FzIN'/4>o v2 (m'/(w')2 - B'/v2) - S + P

Table 3 (continued)

Divisor

Coefficient

Test Configuration

(U upright, I inverted,

onside (starboard) S, (port) P)

= Ix + GxIN/4o w2 + (mg zc - B zb)/w2 -u + i pl5 = Ix' GxIN'/co v2 (in'zc'/(w')2 - B'zb'/2) -u + i = Ix + GxIN/o w2

+ (

yc - B yb)/w2 -s + p pl5

K'

= Ix' GxIN'/o v2 (m'yc'/(w')2 - B'yb'/v2) -s + p pl5

M'

=

-lyx' + GyIN'/o v2 U or I

=

-lyx + GyIN/o w2 + (mg xc - B xb)/w2 - S + P p15

M'

=

-lyx' + GyIN'/o v2 (m'xc'/(w')2 - B'xc'/v2) - S + p - Izx + GzIN/o w2 + (mg xc - B xb)/w2 + U - I pl5 - Izx' + GzIN'/o 2 + (m'xc'/(j')2 - B'xc'/2) + U - I pl5

=

-Izx' + GzIN'/4o v2 S or P

Divisor Coefficient Test Configuration (U upright, I inverted) pi3U = - FYQUJ\JJ/Oo v U or I p13U Zq' = --- FzQU'/8o v U or I

plU

Kq' = - m'yc'

- GxQU'/0o

V U or I

plU

Mq' m'xc'

- GYQUh/eo v

U or I

plU

Nq' = - GZQJ'/O0 V U or I pl

Y4' =

IN/0O v2 U or I pl = - m'xc' + FzIN'/Go v2 U or I pl5 1(4'

=

-Ixy' + GxET'/Oo \2 U or I = ly + Gy11/Oo w2 (mg zc - B zb)/2 - Ti + I

M4' =

ly'

+ GYIN'/Oo v2 (m'zc'/(w')2 - B'zb'/V2) - U + I

N4' =

-Izy + GzIN/Oo w2 ± (mg yc - B yb)/w2

+ U

- I

pl5

N4'

-Izy' + Gz1111/Oo \)2 ± (m'yc'/(w')2 - B'yb'/V2)

+ U

- I

Table

)4

PURE PITCHING - MODEL UPRIGHT OR INVERTED

8 = Oo

Sifl

wt = 00 w cos wt - Go t2 sin wt q' = Go \) cos wt = - Go v2 ,jri wt

Divisor

Coefficient

Test Configuration

(on side (starboard) S, (port) P)

pl3U = m' - FYQUA'/1pO V S or P pi3U Zr' = - FzQUAD'/lpo v S or P pl'U Kr' = - m'zc' - GxQUAD'/lpo v S or P

plU

Mr' = - GYQuD'/410 v S or P

plU

Nr' = m'xc' - GzQUh/1Po v S or P 1 Y1' = m'xc' + IN/'0 v2 S or P pl Zt = FzIN'/tpo V2 S or P pl5 = - Ixz' + GxIN'/410 'v2 S or P = - Iyz + Gy1/ipo w2 ± (mg zc - B zb)/2 + S - P

15

pl M' = - Iyz' + GYIN'/ipo v2

±

(mtzct/(wt)2 - B'zb'/v2 + S - P N1 = Iz + GzIN/410 w2 + (ing yc - B yb)/2 - S + P pl5

N' =

Iz' + GzIN'/410 u2 (m'yc'/(w')2 - B'yb'/V2) - S + P Table 5

PURE YAWING - MODEL ON SIDE

= J)O sin wt

r

= 1)o (Si) COS wt = - 410 W2 sin wt

r' = 1)o v cos

wt = -v2 sin wt

Appendix I SETTING THE AEW PMM FOR UNEQUAL OSCILLATION

Setting the phase.

Knowing the point X, whose motion is to be pure pitching, 1 and 2 are known (Figure 7).

Knowing the carriage speed, oscillation frequency and sword

spacing, p = b/U is known.

Hence compute ci = tan1(- p 1/b)

c2 = tan1(- p 2/b)

ci and c2 are to be in the range - ir/2 (ci or c2) ir/2.

(Note that if X lies between the points of attachment of the

swords i will be positive and 2 negative, so that ci will be negative, c2 positive.)

Set the phase difference between the swords, indicated by the after scale on the phase changer such that the forward sword

leads by c2 - ci. Secure.

Using the pitch datum, set the reading on the forward scale of the phase changer to 2Xc2 (since this scale is in degree units the actual phase being set is c2). Now set the sine/

cosine potentiometer to give zero output from the sine winding, either by setting the fiducial mark to zero or by nulling the

output. Secure. Setting the amplitude.

Compute the amplitude ratio ai/a2 = cos c2/cos ci. Select al (or a2), compute a2 (or al) and set sword amplitude.

Compute the oscillation amplitude Oo = p (al/b) cos ci. Measuring the forces and moments.

Denote the components of force exerted by the swords by Fix F2x Fly etc and the moment exerted via the roll gauge by Gix. Denote the

distance of the origin from the plane of the swords by r; positive

into the paper in Figure 7. The following expressions apply to any tilt angle and any position of the origin, provided the x axis is in the direction of the tow and the plane of pitching (or yawing) is

parallel to the plane of the swords. For zero tilt angle i = ç2.

Model upright or inverted; U, upright; I, inverted

Gx = (Gix + (Fly i + F2y 2) ± (Fiz + F2z)r

-+U +-T

Gy = (Fiz l + F2z 1) -i- (Fix l + F2x 2)

+U

-T

Model on side; s, on side (starboard); P, on side (port)

Gx = Glx + (Flz

i

+ F2z

_{2) ± (Fly + F2y)n}

Gy = (Flz

l

+ F2z

₂₎

(Fix + F2x)n

Gz = - (Fly

1

+ F2y

2) + (Flx

i

+ F2x

2)

Any configuration

Fx

Flx + F2x

Fy = Fly + F2y

Fz = Fiz + F2z

Volume:

1 =

35.315 ft3

1 cm3 0.0610214 in3

Mass:

Appendix M

SI CONVERSION FACTORS TO ASSIST COMPARISON WITH DATA IN IMPERIAL UNITS

Length:

1 km =

_0.53961

UK nautical mile

=

0.53996

International nautical mile

1 in =

3.2808

ft

1 turn =

0.039370 in

Area: Pressure, stress:

1 in2 = 10.7614 ft2 1 MN/rn2 0.06147149 tonf/in2 1 n2 =

0.0015500

in2 = 1145.014 _lbf/in2 = 10 bar Viscosity (dynamic): 1 Ns/m2 =

0.020885

lbf s/ft2 = cP 1 m3/tonne =

35.881

_ft3/ton

Prefixes (in order of magnitude) (standard specific volume for

sea water is 0.97514 m3/tonne tera T

iol2 = 35 ft3/ton) giga G io mega M = 106 Force: kilo k = io 1 kN =

0. 10036

tonf hecto h = 102 1 N = 0.221481 lbf deca da = 10 Torque: deci d = 10 1 kNm =

0.32927

tonf ft centi C = 10-2 1 Nm =

0.73756

lbf ft .

milli

in = 10-3 micro i = nano n = 10 1 tonne

= 0.981421

ton Viscosity (kinematic):

1 kg

= 2.20146

lb 1 m2/s = 10.7614 ft2/s = 106 cSt Density: 1 kg/in3 = 0.0621428 lb/ft3 Power: Specific Volume: 1 kW = 1.31410 hp 1 W =

0.73756

ft lbf/s

Distribution

Copy Numbers 1-4 Director General Ships

(3 copies DPT; 1 copy DWD)

5 Director Research Ships

6 Superintendent AEW 7 Staff Officer (C) BNS 8-114 US NAVSHIPSTECHREP 15-19 CDRS(L) 20 Director ARL 21-36 DRIC