Identification

drag forces on the fin. The iigns of’L,, O f , and P, are assumed to always be positive, hence, the necessity for the ? signs.

The derivative of Yf with respect to v, taken at v, = 0 is the fin velocity-dependent force derivative,

mf:

where the negative sign in association with the abso- lute magnitude symbol

I I

is necessary because as shown in Section 4.2, Y, is always negative.

From Equation (50):

(53)

From Equation (51a) (Y,)f ⁼ 15, sin

pf

- (g),cos

- D,cos P, - -

(3,

sin

p f

and for = 0

(54)

The fin lift and drag can be expressed in terms of the lift and drag coefficients and the fin area, A,, as follows (see also Section 14.1):

L, ⁼ (CL),(P / 2)A,V2 (554

and from Equation (55a):

Substituting Equations (56) and (55b) into Equation (54) and Equations (54) and (53) into (52), one obtains finally for

&.

⁼ 0:

Previous Page

CONTROLLABILITY 235

4:;

Fig. 46 Fin arrangement showing forces and velocities (Abkowitz, 1964)

Values of the lift-curve slope, aC,/ap, needed for insertion in Equation (57), may be estimated from Equation (122b) of Section (14.5) or from Abbott and Doenhoff (1958). A much simpler expression of rea- sonable accuracy for very low aspect-ratio fins is the Jones’ formula (Abbott and Doenhoff, 1958)

a

^CL

__ =

(i)

a (per radian) ap

where a is the effective aspect ratio. This relationship is compared to (122b) Fig. 137 where it is seen that for a < 0.5 and for sweep angle A = 0 the two re- lationships are in excellent agreement. (Note that the values of a C L / a a are per degree whereas values needed for insertion in Equation (57) are per radian.) The drag coefficient (CJfat zero angle of attack may be estimated as essentially the skin-friction drag of the fixed fin; however, it is usually so small in relation to dC,/ap that it is frequently ignored. Thus:

and, with draft T introduced

(2

^- per radian)

= - l A l , g ) , j Because of the prominence of the ship’s draft, T, in the formulations of this and subsequent sections, it is often used in association with the ship’s length, L, as a nondimensionalizing parameter.

Also, from Equation (51b)

(”Jf

⁼ (YV)fx; ( 5 9 4 from which it may be seen that ( ~ V ’ ~ ) ~ i s positive (and hence stabilizing according to Table 1) if the fin is aft of the origin ( x ‘ ~ negative) and negative and destabi- lizing if the fin is forward of the origin.

The contribution of a fin to the derivatives

Y ,

and

N ,

is readily calculable from the expressions devel- oped in Section 4.2. Noting that xfin Fig. 46 is anal-

ogous to xa in Fig. 9, and neglecting the fin derivatives taken about the fin’s own midlength (which are neg- ligibly small), it follows from Equation (13) that:

VJf

^{= x;}

(Wf

(59b)

(”Jf

⁼ X’f2(Y’,)f (594 and

Equations (59a, b, and c) show that the contributions of a fixed fin to the velocity-dependent moment deriv- ative and to the rotary derivatives are all functions of the fin velocity-dependent force derivative ( Y J , . Sim- ilarly, the contributions of a fixed fin to the acceleration derivatives of a ship’s hull are functions of the accel- eration-dependent force derivative of the fin, (

Y ~ ) f .

This derivative corresponds to the “added mass” of a flat plate for accelerations perpendicular to the plate and may be approximated as follows:

(60)

- - 27rb’Aff

-

( a 2 + ^1lV2

where A,is the area of the fin, b is the geometric span, and a, is its geometric aspect ratio b2/Af. For a fin attached to a ship’s hull so that its effective aspect ratio is 2a,, (60) becomes:

(61)

- 47rb’A;

_ -

(a*

+

^I)’~

from which it may be seen that for the limiting case of zero aspect ratio the “added mass” of a fin with its root chord adjacent to a bounding surface is twice that of the same fin without the bounding surface.

The expressions for the contributions to the other acceleration derivatives of a fixed fin remote from the origin of the ship to which it is attached are exactly analogous to Equations (59a, b, and c)

(”& = df(Y’& ( 6 2 4

(“Jf

= ( x ’ f ) 2 ( y J f (624 The magnitudes of the fin acceleration derivatives determined from Equations (61) and (62) are generally small and of minor significance compared t o the fin velocity and rotary-dependent derivatives. The contri- bution that a fixed fin such as a rudder or deadwood makes to the latter derivatives is often decisive in determining whether a ship possesses control-fixed sta- bility or not. This is further discussed in Section 16.2.

236 PRINCIPLES OF NAVAL ARCHITECTURE

0 ) b.=o v = b ,

I

Fig.

-U

-

- - - -

-

.

^c--- _f--- ^{6.V.O '}

-

47 Relationship between V, and ( V J ,

SHIP Q

-

9.3 Prediction of the Control Derivatives. By treat- ing the fixed fin of the previous section as a control surface, values of the control derivatives Ys and N, can be computed in the same manner as the fixed fin contributions (YJf and (NJf. It is clear from Fig. 47 that neglecting interference effects from the hull and the propellers, the net effect of both a drift angle fl at the stern and a control-surface deflection angle 6 , is to introduce an angle of attack on the control surface so that in both cases the slope of the nondimensional Y-force versus angle of attack can be computed as a product of A', and aCL/aa, that is,

and ( 6 3 4

Y', = A f G

(=), ^ac, (2

per radian)

where aCL/aa may be estimated from Equation (122b).

From the preceding

y's = WP)f and since

(636) y = - y

P

Y , ⁼ - ( Y ) ^{v f} And from Equation (59a) for x; =

-a

N' = - ' y

2 6

9.4 Prediction of the Bare Hull Hydrodynamic Deriv- atives. It is shown in Chapter V that no adequate theory exists to predict the resistance of a ship in simple straight-line motion with zero angle of attack at constant speed; that to predict such resistance, one must resort to model data or to other empirical ap- proaches. The theory for predicting the hydrodynamic

derivatives of a ship's hull is even less well developed than that for predicting ship's resistance. What is pre- sented in this section is a procedure for predicting the hydrodynamic derivatives using a combination of the- oretical and empirical inputs.

For the purposes of this section, the ship's hull may be viewed as a very low aspect-ratio fin of very large area. The geometric aspect ratio of the hull is its draft- to-length ratio, its thickness-to-chord ratio is its beam- to-length ratio, its taper ratio is usually close to 1.0, its sweepback angle is usually slightly negative and its mean section shape corresponds to the shape of its water plane at half draft. At the low speeds (Fn <

0.25) where the influence of wavemaking may be neg- lected and to which the current approaches are strictly limited, it was shown by Tsakonas (1959) that the free water surface serves as a groundboard for the ship's hull, hence, the effective aspect ratio of the hull may be taken as 2T/ L. Because of its poor section shape as a lifting surface and because of its extremely low aspect ratio, it might be expected that a ship's hull would generate very small hydrodynamic forces and moments compared to its rudder; however, because of its very large profile area, a ship's hull does in fact generate forces and moments far larger than the con- trol forces generated by its rudder.

Viewed in this light, the velocity-dependent force derivative, (Yv)h, for the bare hull is identical in form to that of the fin ( Y J f given by (57):

where the subscript h refers to the bare hull and Ah is the profile area of the ship's hull. Since the effective aspect ratio of a ship's hull, 2T/L, is rarely greater than about

%,

the Jones formula shown in Fig. 137 is of ample accuracy. Thus,

($)h =

(;)

^{a = r T / L} per radian (65)

Substituting this expression in Equation (64a) and non- dimensionalizing on the basis of p , L, T, and V , we obtain:

where (CD)h may be obtained from the drag charac- teristics of the ship at zero angle of attack.

The velocity-dependent moment derivative of a ship's hull (NJh includes a term that is negligible for a fixed fin remote from the origin of the hull and is therefore not included in the expression for (N,!,given in (59a). This term, commonly called the Munk moment, Munk (1934), is derived in Lamb (1945) for an ellipsoid, deeply submerged in an ideal fluid. The nondimensional derivative of the Munk moment with respect to an

CONTROLLABILITY 237

:

¹⁰

-

C

2 0 8 0

E

⁰⁶

4"

% 0 4

*

^{0 2}

c

'

^{' 0}

C

"

'n c

0

01 02 0 3 0 4 0 5 0 6 0 7 0 8 0 9 10 Ratio of Minor/Mojor Axis

Fig. 4 8 Coefficients of accession to inertia for prolate spheroids (Davidson and Schiff, 1946)

angle of attack, p, for an ellipsoid may be expressed as follows:

( N ' J i = + ( k 2 - k l ) A (N'v)i = - ( k , - k,)A' or equivalently

where

the subscript i refers to the value of the derivative in an ideal fluid

k , ⁼ coefficient of accession to inertia in lateral, y- k , = Lamb coefficient of accession to inertia in lon- A' ⁼ nondimensional mass of ellipsoid

(66)

direction as given in Lamb (1945) gitudinal, x-direction

Fig. 48 shows values taken from Lamb, (1945) for k , , k , and k' (being the coefficient for accession to inertia in rotation). Since Fig. 48 indicates that k , is always larger than k , ,

(LV'~)~

is always negative according to Equation (66) and hence always destabilizing.

The Munk moment arises from the fact that in an ideal (nonviscous) fluid, an elongated three-dimensional body a t an angle of attack experiences a pure couple tending to increase the angle of attack. This is shown in Fig. 49. This couple is composed of equal and op- posite forces acting over the bow half and over the stern half of the body so that in an ideal fluid, there is no resultant lateral force acting on the body, only a destabilizing moment. The magnitude of this moment according to Fig. 49 is

(N), = 21YB so that

where 1 is taken as always positive and YB is negative

if it is directed to port a t the bow and positive if it is directed to starboard.

The relationship between this expression and that given in Equation (66) for the Munk moment derivative is as follows:

ayB 21 - av

(67)

The preceding development is based on ideal, poten- tial-flow considerations. In a viscous fluid, a deeply submerged ellipsoid a t an angle of attack generates vortices on the after, or downstream, side of the body which can be represented as reducing the pressures over the stern of the body as shown in Fig. 49 with little or no influence on the pressures over the bow.

In spite of this representation, the usual simplifying assumption in hydrodynamics is that potential-flow ef - fects and viscous-flow effects are independent of one another. Hence, it is assumed that the lateral force acting on the ellipsoid owing to vorticity is independent of and does not react with the force distribution in an ideal fluid. The total moment, N , acting on the sub- merged ellipsoid a t an angle of attack in a real fluid is taken to be the sum of two independent parts, one the ideal Munk moment and the other the moment of the lateral force arising from real fluid effects. Thus, following Fig. 49:

(68) where xp is the distance from the origin (taken at @) to the point of application of the real-fluid lateral force,

YL (negative if Y, is aft of 0).

According to Fig. 49, the real-fluid lateral force YL will always act in the same direction as YB. Further-

- -

( p / 2 )

LZ

TV

N = 2YB1

+

^xpYL

NET FORCE ON BOW IN LATERAL FORCE ON STERN

F R ~ M R ~ A L ?<$'/8OTH IDEAL AND REALFLUID

DISTRIBUTION IN A REAL FLUID

AN NET FORCE O N ST'ERN

NET

^{FORCE ON STERN IN} IDEAL FLUID IN REAL FLUID Ys( t l ( EOUPL AND OPPOSITE TO Ye)

Fig. 4 9 Forces acting on a submerged ellipsoid at an angle of attack in an ideal and a real fluid

238 PRINCIPLES OF NAVAL ARCHITECTURE

1.0

-

0.9 -

n

b-

a

^0.8

-

W P

b. WATERLINE BEAM OF EACH SECTION T * DRAFT OF EACH SECTION

0.5

-

0 I 2 3 4 5

4 T/b

Fig. 50 Sectional inertia coefficients, C, as functions of local b e a d d r a f t ratio and section area coefficient (Prohasko, 1947)

more, experiments with slender bodies of revolution reported by Johnson (1951) have shown that x, is al- ways negative and lies between 0.2 and 0.3 of the length of the body a f t of the midlength. If these signs are associated with Equation (68), it will be seen that N, the upsetting moment in a real fluid, is smaller in magnitude than the upsetting moment in an ideal fluid which is represented by the first term of (68).

For shiplike bodies at a free surface, the same basic expression shown in (68) is used but the particulars differ significantly. If the derivative is taken with re- spect to v and the result is nondimensionalized, the following is obtained:

+

Substituting from Equation (67) for the first term on the right of (69), we obtain for the ellipsoid:

N’, = - ( k 2 - k , ) A ’

+

² _L

(ayL)

^- _avl ⁽⁷⁰⁾

For the surface ship, Jacobs (1964) wrote an analogous expression:

X

( N l v ) h = - (A’, - k , A ’ )

+

² ( Y ’ v ) h (71) L

where

A‘ = mass of ship, A, nondimensionalized by (p/2)L2T A’, ⁼ A,/(p/2)L2T

C, = two-dimensional lateral added-mass coeffi- cient (sectional-inertia coefficient) deter- mined for each section strip of width dx along x-axis. The C, may be determined from several sources, e.g., Fig. 16 of Pro- haska (1947) reproduced in Fig. 50.

k , A ’ ⁼ the so-called “added” mass of ship in the longitudinal x-direction; k , A’ = - ( X i ) h A’, = the so-called “added” mass of ship in the

transverse y-direction; A‘, = - ( Y e ) h h = local draft at each station

k , , k , = as defined for Equation (66) and as given in Fig. 48. For surface ships, the abscissa of Fig. 48 is defined as 2 T / L , where T corresponds to minor axis and L to major axis

x, = as defined in Equation (68) ( Y v ) h = as given in Equation (64a)

The results for C, given by Prohaska (1947) are for very-high-frequency oscillations in heave. However, it is assumed by Martin (1961) and confirmed by Porter (1966) that if the scale of the abscissa of Fig. 16 of Prohaska (1947) is treated as a scale of 4Tlb instead of blT, the Cs’ values of that figure are applicable to near-zero frequency oscillations in sway. As indicated in Section 8.5, interest in this chapter centers on near- zero frequency oscillations.

The first term on the right of Equation (71) differs from that of (70) only in the expression for A’,. The quantity under the integral sign in the expression for A12 represents the summation of the added masses of two-dimensional strips taken over the length of the ship. This first term can be estimated for any given ship form on the basis of the data given in Figs. 48 and 50.

Prediction of the second term on the right of Equa- tion (71) depends on knowledge of x, for shiplike bod- ies, since ( Y J h can be readily calculated from (64a).

The major difference between ellipsoids and ships in- sofar as x, is concerned is that ship bows are more slender, have a constant local draft forward and, for bows that are not bulbous in shape, have section shapes forward that are relatively sharp at the keel for a short distance aft of the bow. Therefore, these ship hulls a t an angle of attack generate vortices on the downstream side of both the bow and stern, whereas the ellipsoid hull was assumed to generate vortices only downstream of the stern. This change tends to move forward the point of application of the

No.

1 2 3 4 5 6 7 8

9 10 11 12 13

7

Model description Ta lor’s

sti

Series

Series 60 Normal form

Series 60, Ext. V Model (Eda, 1965) Mariner

Destroyer DD692 Hopper Dredge Hopper Dredge

Table 5-Hull Form and Appendage Configurations Studied by Tsakonas, Martin and Jacobs Dav. Lab.

model no.

843 845 847 842 846 848 844 3,0,0 & 3,1,1 841 6,0,0 & 6,1,1 5.0.0 & 5.1.1

i,o,o; 2,ly;

2,1,2 & 2,1,3 7,0,0 & 7,1,1 8.0.0 & 8.1.1 1;o;o & 1;1;1 9,0,0 & 9,lJ 4,0,0 & 4,1,1

Heavy Light

L/B 4.36 4.36 6.90 6.90 6.90 8.68 10.89 10.89 6.0 7.0 7.0 7.0 7.0 7.5 8.0 7.5 7.0

6.84 9.45 6.00 5.78

B/T 2.92 4.62 1.85 2.92 4.62 2.92 1.85 2.92 3.12 2.07 3.28 2.68 2.68 2.68 2.50 2.34 2.50 3.10 2.90 2.41 3.46

L/T 12.74 20.13 12.74 20.13 31.9 25.42 20.13 31.9 18.75 14.50 23.00 18.75 18.75 18.75 18.75 18.75 18.75 21.19 27.40 14.51 20.00

x’m -0.019 1-0.020 -0.019

i

f0.015

I

f0.005 -0.025 +0.015 +0.015 f O . O 1 l +0.023 +0.022

0 +0.012

Propeller None

None((

witk

None With 2 props.

None None

With ( s e e y l ] ) With 2 ru ders With

With

s

^{k e g ( s )}

None (see Fig. 81) None

None

None and to sta 17, 18 and 20 None and to sta 20 None and to sta 20 None

None

Normal Single Screw :e Fig. 83) Stern (E

n 0 Z

As built (see [41]) As built

~~- .

Like Fig. 51(c) Like Fig. 51(c)

240 PRINCIPLES OF NAVAL ARCHITECTURE

lateral force Y L , shown in Fig. 49 for the ellipsoid.

That is, x, is less negative and may even be positive for some ship hulls, whereas it is always large and negative for ellipsoids.

On the basis of experiments with a group of eight Taylor Standard Series models of varying B/T ratio and L/B ratio (see Table 5), Martin (1961) showed that the distance x, lay between 2 0.1L from the center of gravity of the model which was 0.02L forward of the midlength. Jacobs (1964) suggests that x, be mea- sured from G to the center of area of the hull profile.

Thus, according to Jacobs, x, is likely to be positive for ships without bulbous bows. Accepting the Jacobs’

suggestion, it is apparent that whereas for ellipsoids the moment derivative in a real fluid is less in mag- nitude than the ideal moment derivative, for many ships the real moment derivative may be greater than the ideal moment derivative.

Fundamentally, since x, is a function of external body geometry, it is more accurate to relate it to some geometric position on the body, such as its midlength rather than to G, which is a function of the location of the internal weights of the body. This was one of the reasons why the equations of motion (10) were so written that the origin could be taken a t

0

rather than at the center of gravity.

It will be noted that the second term of Equation (71) is equivalent to the expression for ( NJ , given in (59a) with the important distinction that x, is very small compared to x,. Strictly, (594 should include a term representing the Munk moment of the fin. How- ever, because of the assumed short chord length of the fin relative to the distance x,, the Munk moment for the fin is negligible. For the ship’s hull, on the other hand, where x, is very small, the Munk moment term is of dominant importance.

It follows from these remarks that the Munk mo- ment is important for fins when moments are taken about an axis located in the fin. An important practical case is the prediction of rudder torques about the rud- der stock shown partly in (124).

The rotary-force derivative for the bare hull, ( Y J h , like the velocity-dependent moment derivative (N,)h just discussed, also includes two terms, one arising from ideal fluid considerations and the other from real- fluid effects. For a ship in a real fluid, the expression is as follows:

(‘,Ih

= - klA’

+ ³

( Y v ) h (72)

(p/2) L 2 W L

W , ) h =

with symbols as defined for Equation (71).

The first term corresponds to an outward (centrif- ugal) force exerted by the fluid on a body in rotary motion. This force is due to the uniform rate of change of direction of the longitudinal momentum of the fluid which has been imparted to the fluid by the body mov- ing in a circular path. The second term arises from viscous-flow effects and is identical in form to the con-

tribution of a fixed fin to the rotary-force derivative shown in Equation 59b. Since both k , (see Fig. 48) and x,/L are very small for slender, shiplike bodies, the rotary-force derivative for the bare hull is always small. This is in accord with Section 4.2. If x, is positive, as suggested by Jacobs (1964) ( Y J h will be negative since ( Y J h is always negative.

As noted in (ll), the derivative

Y ,

is only one of the terms in the coefficient of r’; the other is A‘. If these two terms are grouped together as they are in (ll), and combined with (72), the following expression used by Jacobs (1964) is obtained:

(A’ - Y , ) h ⁼ (1

+

^{k,)A‘ -} X ² L ( Y v ) h (73) The first term on the right of Equation (73) is the actual mass plus the “added mass” in the x-direction.

Jacobs uses the symbol A’, for the first term and refers to the second term as ( Y , ) h . Since ( Y r ) h , by definition, should encompass all of its hydrodynamic parts, it should be defined as in Equation (72).

Jacobs (1964) following Martin (1961) expresses the rotary-moment derivative as the sum of a potential and a viscous term although Lamb’s potential-flow analysis indicates that N , should be zero in an ideal fluid (Lamb, 1945). According to Jacobs:

where A: is the so-called rotational added-mass coef- ficient acting at a distance

X

from G

A12 = (k‘/k,)A‘, = A 2 / ( p / 2 ) L 2 T

stern

[

^C,h2xdx

1

^Csh2dx

ow

x ⁼ distance from 0 to section strip of width, dx, positive if forward of 0, negative if a f t

x,l L ⁼ taken as half the value of the prismatic coef- ficient of ship and is assumed to be always positive

All other symbols are as defined for Equa- tion (71)

While X / L may be either positive or negative, its value is always small. On the other hand, xo is much larger than

X

and since ( Y J h is always negative, (N‘r)h will always be large and negative. This is in accord with Section 4.2.

The two cross-coupled acceleration derivatives, Y+

and N* have very small values for the bare hull and in practice are often assumed to be zero. The linear acceleration derivative

Y ,

is equivalent to the term