lournal of
SHIP RESEARCH
Estimation of Stability Derivatives and Indices
of Various Ship Forms, and Comparison With
Experimental Results
By W. R. Jacobs^
The analytical method of reference [1 ]^ for estimating stability derivatives, and hence stability on course, which combines Albring's empirical modifications of simplified flow theory with low aspect-ratio wing theory, is extended to take into consideration the effects on course stability of higher aspect-ratio fins as well. The method, which has been
applied in the earlier report to a family of eight hulls of 0.5 block coefficient, is tested
fui(fher by application to eight Series 60 forms differing in block coefficient as well as in beam, draft, and displacement—^with and without rudders; to an extreme vee
modifica-tion of a Series 60 model; and to three other forms—a Mariner Class model, a destroyer,
and a hopper dredge. Comparison with experimental results shows that the values of stability derivatives and indices determined by the analytical method are of the right orders of magnitude and indicate correct trends. Application to a variety of ship forms has demonstrated that the method can predict relative effects of changes in the geometry of a ship form, as well as the effects of changes in skeg and rudder a r e a .
Introduction
I n an earlier report [1], an analytical method was de-veloped for estimating the first-order stability derivatives (static and rotary lateral-force and yawing-moment rates) which would indicate the course stability and turning or steering qualities of ships. The method was applied to the case of a family of eight hulls of the same length and the same prismatic and block coefficient, but differing i n draft, beam, and displacement. The hulls were the 840 Series of the Taylor Standard Series type w i t h the after deadwood (faired-in skeg) removed. Ex-perimentally measured lateral forces and yawing mo-ments, f r o m Davidson Laboratory rotating-arm tests
1 Prepared for Bureau of Ships Fundamental Hydromechanics
Research Program (S-R009-01-01). Administered by David Taylor Model Basin under Contract Nonr 263(57), D L Project 2803/063.
2 Research Engineer, Fluid Dynamics Division, Davidson
Labora-tory, Stevens Institute of Technology, Hoboken, N . J.
2 Numbers in brackets designate References at end of paper.
Manuscript received at S N A M E Headquarters, December 3, 1965.
at different turning radii, were available for these hulls and for three of the hulls w i t h flat-plate skegs i n the place of the removed deadwood.*
Although the analytical method is based upon simple concepts combining simplified flow theory w i t h low aspect-ratio wing theory and using Albring's [3] em-pirical modifications for viscid flow, good correlation was attained between the stability derivatives calculated by this method and those determined f r o m experimental data. However, Albring's modification of the rotary-moment rate is a function of prismatic coefficient and, since all the hulls of the 840 Series have the same pris-matic (0.54), this modification was not tested f u l l y . I t was decided, therefore, to extend application of the prediction method to hulls of other prismatic, w i t h and w i t h -out skegs or deadwood aft, for which experimental data were available.
^ Results of several straight-course tests [2] confirmed previous experience at Davidson Laboratory that entirely reliable static force and moment rates for straight-course motion can be ob-tained from rotating-arm data at sufficiently large turning radii.
Fortunately, straight-course and rotating-arm model tests have been concluded on eight members of the Series 60 f a m i l y of ships [4], so that the effects on stability of varying block and prismatic coefhcients, beam, and draft—other f o r m characteristics remaining constant— can be ascertained f r o m the experimental measurements, for comparison w i t h theoretical predictions. These forms were not altered, as were the Standard Series types, b y removal of the after deadwood. Tests were made w i t h and without rudder and propeller, and one model was tested w i t h three rudders of differing chord length. I n addition, the analytical method was applied to the following four forms: A n extreme vee modification of a Series 60 ship [4], a Mariner class vessel [5], and the widely different destroyer [6] and hopper-dredge [7] forms. Consistent experimental techniques have been used i n tests of these forms conducted i n recent years at Davidson Laboratory.
W i t h the exception of the hopper dredge, all models had large areas of deadwood (faired-in skeg, including rudder) aft, w i t h maximum height at the stern f r o m
ex-tended keel line to load waterline. The aspect ratio of the skeg, equal to the square of the maximum skeg height divided by the skeg area and doubled to take into ac-count the free-surface effect, was, i n all these cases, less than unity. The hopper dredge, on the other hand, had a skeg of small area at the stern, masked f r o m the water surface by the broad bottom of the afterbody. The aspect ratio of this skeg, equal to the square of its maxi-mum height divided by its area, was greater than unity. For this form, and^for the Series 60 cases where the rud-der was removed or rudrud-der area was added, the effects of altering a body by adding or subtracting area having fin effect could not be treated by using low aspect-ratio wing theory.
The method of reference [1 ] was therefore extended by including the technique of reference [8] i n studying the effect on ship behavior of adding or subtracting fins. The l i f t on the f i n itself is calculated by using aerody-namic wing theory for wings of aspect ratio greater than unity. Then, b y assuming that the interference between fin and body is negligible, as i n the simplified theory used
-Nomenclature-A = profile area of wing or hull, sq
f t
M = aspect ratio of wing B = beam, f t
ft = local beam, f t
Cl = l i f t coefficient based on profile area
Cs = two-dimensional lateral added-mass coefficient (sec-tional-inertia coefficient) Cs = average Cs over hull
Do' = Rf + Rrj^ UHH = total-re-sistance coefficient of hull F = force,lb
F„' = FyJ^ UHH = measured lat-eral-force coefficient g = acceleration of gravity H = maximum draft, f t
h = local draft, f t
hf = maximum fin height, f t Jo = moment of inertia of hull,
Ib-ft-sec^
I 2 = added moment of inertia of
entrained water (see text), Ib-ft-sec^
kl, kijk' = Lamb's coefficients of acces-sion to inertia; longitudi-nal, lateral, and rotational l i f t , lb L L' I mo Lj'^ UHH = l i f t coefficient based o n a r e a i - J ï length, f t
A / g = mass of hull, slugs
TO/ A' .V' R Rf Rr
???o
ƒ
- iW-huU-mass coefficient kimo' = longitudinaladded-mass coefficient
lateral added-mass coeflficient (see text)
mo' -(- mi' = longitudinal vir-tual-mass coefficient mo' + m-i = lateral
virtual-mass coeflaoient
rotational added-mass coef-ficient (see text)
yawing moment, Ib-f t
iVƒ^ m m =
yawing-mo-ment coefficient
Jo + L^- I* H = virtual'" moment-ot-inertia coefli-cieiit
radius ot turning circle, f t frictional resistance, lb residual resistance, Ih I — = dimensionless angular JL t = U = X, y,z = velocitj' m I = dimensionless distance along path of center of gravity of hull
time, sec
velocity of center of gravity of hull, fps
coordinate axes fixed i n hull with origin at center of gravit.y .-Ss, Xb Y = 13 = 5 = A = P =
longitudmal distance from LOG, of center of gravity of lateral added mass, f t longitudinal distance from
LCG, of center of pressure at which lateral force Y acts, f t .
assumed longitudinal dis-tance from LCG, of center of pressure of tail surface or skeg, f t
^-coordinates of stern and bow, respectively
lateral hydrodynamic force, lb
Y
Subscripts f = H =
UHH = lateral hydro-dynamic-force coefficient yaw angle or drift angle rudder angle
displacement of hull, l b ; also increment
mass density of fluid, slugs/cu f t
stability indices
refers to high aspect-ratio fin (skeg or rudder)
refers to bare hull refers to ideal fluid
refers to derivative w i t h re-spect to ?•'
refers to derivative with re-spect to s
refers to derivative with re-spect to
Fig. 1 M o d e l orientation i n x-y plane
m reference [1 ], the changes i n static and rotary force and moment rates are computed.
Comparison w i t h experimentally derived stability derivatives and indices shows that the theoretically de-termined values are of the right orders of magnitude and indicate correct trends. The fact that the analytical method has the ability to predict relative effects of changes i n the geometry of a ship form, i n addition to the effects of changes i n rudder and skeg area, makes i t an acceptable working tool i n designing ships for greater course stability. I t is useful not only as an augment to experimentation but also i n planning an efhcient program of model testing.
The A n a l y t i c a l Method
Assumed Sfabiliiy Derivafives for Hulls Wifbouf Deadwood or Fins
I n the potential-flow theory the hydrodynamic-force and moment-rate coefficients, or stability derivatives, o f ' an elongated body of revolution without appendages are defined for the linearized region of small angles of attack and large radii of rotation as:
On straight course, r ' = l/R = 0: L,u' = Y,,' = 0
= m-i' - nil' = Nf,.' ( M u n k ideal moment) (1) I n turn, around = 0:
Yr„' = 0 NrJ = 0
The notation is that of The Society of Naval Architects and Marine Engineers (see Nomenclature and Fig. 1; subscript H refers to bare hull). The measured lateral-force coefficient is defined as
Fy' = F ' - {m,' + m i ' ) r '
and its deri^-ative w i t h respect to r' as My'
^ = Y / - (nio' + VI,') (2)
where nio' is the mass coefficient of the hull and vii' is the longitudinal added-mass coefficient. Lamb [9], considering the added-mass term as a hydrodynamic force, defines Yr„'' = -nn' = -fci7ïio'where is the co-efficient of longitudinal accession to inertia, Fig. 2. Equations (1) are equivalent to those derived by Breslin
° 0.4 0.6 0.8 1.0
RATIO OF MINOR AXIS/MAJOR AXIS
Fig. 2 Coefficients of accession to inertia f o r prolate spheroids (from H . Lamb's Hydrodynamics)
[10] for a long slender body w i t h tapered or pointed ends, f r o m three-dimensional singularity distributions.
I n Albring's [3] modification of potential-flow theory for a body of revolution moving i n a viscous and eddying fluid, the l i f t on the bare h u l l is no longer zero as i n po-tential theory, but the force developed as b y "obfique attack under an angle [/3] of a correspondingly shaped sohd without effect of curvature," which acts at a dis-tance Xp f r o m the center of gravity. I n determining this I f f t force on a surface ship, reference [1 ] follows Fedyaev-sky and Sobolev [11 ] i n identifying the bare hull of a ship (i.e., the ship Avithout deadwood, skeg, or any other area which has only fin effect) w i t h a low aspect-ratio wing. I n this analogy the span of the wing^is assumed to be double the d r a f t of the ship, to take into account the action of the free water surface. Tsakonas [2 ] shows that this "solid w a l l " method of accounting for free-surface effect is correct for moderate speeds when the influence of wave making can be neglected.
The dimensionless l i f t rate per unit lateral area of the hull is assumed as given by Jones' formula f o r a low aspect-ratio wing, derived f r o m the consideration of eUip-tic load distributions along the chord and the span of a thin foil. The Jones formula is
2IP
A (3)
The t o t a l bare-hull l i f t rate, nondimensionalized on the basis of area I X H,is then
TTH
I
L^„' (4)
On combining low aspect-ratio wing theory w i t h
SEPTEMBER 1 9 6 6
b / h
Fig. 3 Sectional inertia coefficient C, as functions of local beam-draft ratio hjh and section-area coefficient, f r o m Prohaska
Albring's empii'ically based formulas, the stability derivatives for a bare ship moving i n a viscous fluid are obtained as:
On straight course, r ' = = 0 The static force rate
= hu + -öo' = X + The static moment rate
(5)
(6)
I n t u r n , around /3 = 0 The t o t a l rotary force rate
ö r '
- m,' + Yr„.' = - («lo' + « i i ) - y i ^ / (7) ^nj = »io' + « u '
The rotary moment rate mo =
Do' = drag coefflcient at yaw angle |3 = 0, 'obtained by experiment or
esti-mated f r o m the Taylor Standard Series curves of resistance [12] AT^.' = M u n k ' s moment rate i n an ideal
fluid, equal to the difference be-tween the, lateral added-mass co-efficient mi! and the longitudinal added-mass coefhcient of en-trained water
Xf = distance f r o m L C G of the center of
pressure of lateral force, taken as the center of area of the h u l l pro-file (positive if f o r w a r d of the LCG)
= v i r t u a l longitudinal mass coefficient = where A is ship displacement, lb
A/" / = - JH. (8)
The various terms are determined as follows:
H/l = ratio of maximum d r a f t to length of
ship
?7ii' = hmo' = where h is Lamb's coefficient ,of longitudinal accession to inertia for an equivalent ellipsoid w i t h ratio of nnnor axis to major axis equal to 2H/1
Xo/l = half the prismatic coefficient Cp,
1,0 ( m ó + m 2 ) J ? / H = 2 C B / H + T T C B S E X P E R I M E N T A L DATA 8 o -O S E R I E S 6 0 n 8 4 2 , A M A R I N E R C L A S S • D E S T R O Y E R "EC E X T R E M E V E E M O D I F I C A T I O N O F S E R I E S 6 0
Fig. 4 Stability index in f o r hulls w i t h l o w aspect-ratio skegs (deadwood) to stern
vij = rotary added mass of entrained
water acting at distance x f r o m L C G
Tlie terms rrii, m/, and x are estimated as follows, ac-cording to the procedure advocated by M a r t i n [13]:
« 1 2 = m where V Xa m2 = h^TT \ Cshhlx in/ = y- 7112
ki, k' = Lamb's coefficients of accession to inertia, lateral and rotational, for an equivalent ellipsoid. Fig. 2
Xs, X!, = .r-coordinates of stern, bow h = local d r a f t at each section
Cs = two-dimensional lateral added-mass coefficient, determined at each section f r o m curves on two-dimensional forms of Lewis' sections, by Prohaska [14], Fig. 3
CJi\ J Xa
CM Xa
where x is positive forward of L C G
The various assumptions have been made primarily f r o m the pragmatic point of view. Tsakonas [2] and M a r t i n [13] found the Jones formula to be a good ap-proximation of the static l i f t rate of the Standard Series hulls. Albring [3] showed that a good estimate of rotary moment rate is obtained i f one assumes i n this case that the l i f t force acts at a distance xo f r o m L C G , equal to lC,/2.
The moment arm Xj, f o r the static moment rate is dis-cussed by M a r t i n [13 ], as follows. The disposition of the l i f t force over the bare hull can be surmised only on the basis of experimental data and some theoretical con-siderations. Measurements on slender bodies of revolu-tion indicate that the center of the l i f t force is between 0.2 and 0.3 of the length of the body a f t of the L C G . This force is associated w i t h vortex generation on the aft leeward side of the body. A similar effect might be expected on the bare h u l l of a ship. However, the bow of a ship is markedly different f r o m the nose of a body of revolution; the bow is slender and of uniform draft, and the keel is relatively sharp f o r about one tenth of the length a f t of the forward perpendicular. Because of the eddy generation along the keel forward of midships, an appreciable part of the l i f t force w i l l be located ahead of the L C G . A comparison of measured moment rates on the Standard Series hulls w i t h the M u n k ideal moment rate showed t h a t since the contribution of the fift force is very small, although the total fo^ce is not small, its center must be close to L C G .
Assumed Stability Derivatives for Hulls with Large Areas of Deadwood Aft
For hulls w i t h large areas of deadwood or low aspect-ratio skegs aft, extending to the water surface at the stern, and including rudders parallel to the centerline (rudder angle 5 = 0), simplified theory assumes t h a t there is no interference between the bodies and these surfaces, so that the effects of the skeg area are simply additive. This is admittedly crude, b u t an appraisal of this assumption given i n reference [8], based on experi-mental results for torpedoes w i t h and without fins and ships w i t h and without skegs of various sizes, indicates that noninterference is an acceptable working assumption and yields results of the right order of magnitude. The static l i f t rate per u n i t skeg area is assumed given by equation (3). Since the length of the deadwood, or skeg, plus rudder is small i n comparison w i t h the length of the
Bare hull (no deadwood) I E q s . ( 5 ) - ( 8 )
H u l l with large dead- E q s . (13) - (16) wood and rudder '
H u l l w i t h small skeg E q s . ( 5 ) - ( 8 ) plus E q s . ( 1 8 ) - ( 2 I )
Hul 1 w i t h large dead-wood, rudder removed
Fig. 5 Configurations
E q s . ( ] 3 ) - ( 1 6 ) plus E q s . ( l 8 ) - ( 2 1 )
hull, the distance between the stern and the center of pressure on the skeg is a negligible part of the nroment ai-m about the L C G . I t will therefore be assumed that the additional l i f t acts as x^, the distance of the ship stern f r o m the LCG.
The increments i-fesulting f r o m deadwood, and so on, to be added to the bare-hull stability derivatives given by equations (5)-(8), are (see reference [1])
AL,' = AYs' = ^ AN/ = AYr'' = ANr'' = -I l ) I Xs\ TTH I J I XsV TTH I I (9) (10) (11) (12) where Xs is negative.
As shown i n reference [1], these formulas give es-sentially the same results as those obtained i n the proce-dure suggested by M a r t i n [13]. M a r t i n modified the linearized equations of motion i n the horizontal plane, given i n reference [15], by including terms involving two-dimensional lateral added mass at the stern to ac-count f o r the sudden change of section of the hull and skeg at Xs. When the section at the stern is that of a flat plate, the added lateral mass is numerically the same as the Jones formula f o r a low aspect-ratio wing of span equal to twice the sliip's draft.
The total values of the static and rotary force and moment rates i n the case of huUs w i t h large skeg area a f t extending to the load waterline at the stern are:
On straight course, r' = l/R = 0 Y 2wH + Do' Nf,' = mi' - m,' + TTH I (13) (14) I n turn, around fi = 0 dF,' N/ = Xp + Xs\ tH I J I Xo^ + x j \ tH Z2 ) I (15) (16)
Changes in Sfabilify Derivatives Due to Adding or Subtracting Fins of Aspect Ratio Equal to or Greater Than Unify
I n the case of hulls like the hopper dredge, w i t h very little deadwood or skeg area a f t (and that masked f r o m the water surface by the hull bottom), a different treat-ment is required. The bare-h\ül stability derivatives are obtained f r o m equations (5)-(8) as before, b u t the effect on the derivatives of adding area having fin effect and an aspect ratio which cannot be considered low is deter-mined as i n reference [8] by using aerodynamic wing theory applicable to Avings of higher aspect ratio, equal to or greater than unity. The latter theory is also em-ployed i n studying the effects of adding or subtracting rudder area in the hull cases w i t h large deadwood aft.
The dimensionless l i f t rate per unit of fin area Af in such case is
0(3 2 (17)
Because the skeg or rudder is below the hull bottom and does not extend to the water surface, i t is assumed that there are no free-surface effects. Thus the fin aspect ratio Mf is the ratio of the square of the fin span (maximum height /) to the fin area Af.^ The increment
* I n this case the effective aspect ratio is equal to the geometric. I n predicting drag of keeled yachts at Davidson Laboratory, i t has been found that use of the geometric aspect ratio of the keel gives much more satisfactory results than when the keel span is doubled. Possibly, the increment i n l i f t on the skeg, due to its junction with the huh, is balanced by the decrement in l i f t on the hull, due to the presence of the skeg; if not exactly balanced, at least to a degree which would preclude use of a factor as large as 2.
or decrement to the static lateral-force rate, nondimen-sionalized on the basis of area l-H, w i l l be
where
( A F / ) , < 0 when subtracting fin area.
Again the assumptions are made that interference be-tween f i n and body is negligible and that the center of pressure of the f i n is at the stern at a distance a f t of the L C G . The other stability-derivative changes are then ( A A T / ) , = Y ( A F / ) , ( A F . / ) , = - Y ( A F / ) , ( A A ^ / ) , = - f ( A F / ) , (19) (20) (21)
Although equations (19)-(21) are derived for f i n area at the stern, they are, through substitution of the correct moment arm i n place of x^, apphcable also to added or subtracted f i n area at the bow.
I n the case of a ship w i t h small skeg of relatively high aspect ratio at the after end of the underwater hull, the stability derivatives F^', A^^', ( m / - F , / ) , and A^,/are obtained f r o m equations (5)7-(8), modified by equations (18)(21) w i t h (AF^O/ positive. When r u d -ders are removed f r o m hulls w i t h large deadwood area aft, the stability derivatives are defined by equations (13)-(16), modified by equations (18)-(21) w i t h ( A F ^ ' ) / negative.
The Sfabilify Indices
The criteria for inherent dynamic stability of a free bo'dy moving on a straight course i n the horizontal plane are the damping exponents ai and 0-2 i n the solution
13 = /3ie-» + i3,e"% r' = )'i'e"* + 7-2'e™ • of the homogeneous linearized equations of motion [15]
(?"/ - Yr''y - 771//3, - F//3 = 0
(22) n/r/ - Nr'Y - N / f i = 0
Here s = Ui/l. The damping exponents are given by
- ( 7 i / F / - 7 « / A ^ / ) ± { (T I / F / - my'Nr''}' + 4n/m/[N-/Y,' + ( 7 » / - F , / ) A f / ] } ' A
0"1,2
where
•nil,' = ' « 0 ' + 7 ^ 2 ' virtual lateral mass coefficient (23)
Table 1 Pertinent Characteristics of 840 Series (Taylor Standard Series)
Hulls Model (to. 81)2 81)6 81)8 Length I , f t 6 - 0 5 . 0 6 . 0 Beam B, f t 0 . 8 / 0 0 . 8 7 0 0 . 6 9 t D r a f t H , f t , o . z g B 0 . 1 8 8 0 . 2 3 6 Di s p l a c e m e n t A , lb 1(8.1)0 3 0 . 5 0 3 0 . 5 0 P r i s m a t i c c o e f f i c i e n t (= '"o/l) O . f l f 0 . 5 4 0 . 5 4 B l o c k c o e f f i c i e n t Cg 0 . 5 0 0 . 5 0 0 . 5 0 L C G / i , from bow 0 . 5 2 0 0.1)81 0.1)81 B/H 2 . 9 2 l ( . 6 2 2 . 9 2 i/B 6 . 5 0 6 .90 8 . 6 8 l/H 2 0 . 13 3 1 - 9 0 25-1)2 Lamb's C o e f f i c i e n t s o f A c c e s s i o n to I n e r t la f o r E a u i v a l e n t E l 1 DSO ids M a j o r a x i s / m i n o r a x i s , 1/2H 1 0 . 0 6 1 5 . 9 5 12-71 kl ( l o n g i t u d i n a l ) . 0 2 0 . 0 1 2 -017 kg ( l a t e r a l ) . 9 6 0 •978 -967 k' ( r o t a t i o n a l ) . 8 8 5 •935 .902 O t h e r P h y s i c a l C h a r a c t e r i s t i c s m ' , mass c o e f f i c i e n t .11(5 • I i t 5 -115 m^', l o n g i t u d i n a l added-mass c o e f f i c i e n t .003 .002 .002 m ' , l a t e r a l added-mass c o e f f f i c i e n t . 1 2 9 •oak .103 m ' , r o t a t i o n a l added-mass c o e f f i c i e n t . 1 1 9 . 0 8 0 .096 n ^ , v i r t u a l m o m e n t - o f - i n e r t l a c o e f f i c i e n t .0165 .011)1 . 0 1 3 2 x / j J , C G o f l a t e r a l added mass from LCG 110 . 0 7 0 .070 X p / / , c e n t e r o f a r e a o f p r o f i l e from LCG 031 . 0 5 2 . 0 5 2 ( e s t i m a t e d d r a g c o e f f i c i e n t a t P = 0 ) D\k . 0 1 8 • Oil)
n/ = lo + h = virtual moment-of-inertia coefficient
/ o
2 7^0
^ (assuming the radius of gyration is equal to Z/4)
= moment of inertia of the ship
= k' Y' J ^ ' CJiVdx (reference [13]) = moment of inertia of the entrained
mass of water
The derivatives, F / , N / , (m/ - F , / ) , and A",/, are defined f o r the various configurations as shown i n Fig. 5.
The index a^, obtained by using the minus sign i n equa-tion (23), is always negative. Therefore the stability of the motion depends on the sign of ai or its real part; the more negative ai, the sooner an i n i t i a l disturbance w i l l damp out, and hence the greater the stability. I f ai or its real part is positive, the motion is unstable and the h u l l cannot be kept on straight course w i t h o u t applying a corrective rudder.
The stability criterion ci is also an index of the turning qualities of a huh i n turns that are not too tight, i.e., when nonlinearities can be neglected. A more dy-namically stable h u l l w i l l t u r n i n a larger radius than w i l l a less stable hull w i t h equal rudder force. Con-versely, the more stable hull w i l l require greater rudder
SEPTEMBER 1966
Fig. 6 Body plan o f model parent (840 Series) DECK
1
L . W . U . /////7777-7 É -A.P. 1 9 - 1 8 1 7 16 ? 5 1 4 DECK1 . 6 1 . 7 P R O F I L E A R E A , S Q . F T .
1.6
Fig. 8 Comparison o f calculated and experimental stability derivatives and indices f o r 842 h u l l w i t h various skegs
force than the less stable huU to t u r n i n a given radius. On the other hand, an unstable ship, defined by positive
0-1, may t u r n i n a direction opposite to that called for by the applied rudder, i n which case i t w i l l need a large force to bring i t around.
Presentation and Discussion of Results
Lateral-force and yawing-moment coefficients and sta-b i l i t y derivatives determmed f r o m experimental measure-ments are compared graphically w i t h those computed by the linear theory of the present report.
840 Series Hulls
Particulars and charts of reference [1] for three Taylor Standard Series models, w i t h deadwood removed and w i t h flat-plate skegs added i n lieu of deadwood are given
in Table 1 and Figs. 6-10. These models had been tested at Davidson Laboratory i n 1959 and 1951. The tests were made at Froude numbers of 0.16 and 0.23; hence the assumption that wave-making effects can be neglected is tenable.
Fig. 6 shows the body plan of the parent h u l l and Fig. 7 the stern profile w i t h skeg installed. Fig. 8 is a summary chart comparing the calculated Y / , N / , Y / , N / , and o-i w i t h values obtained i n 1959 f r o m measurements on the 842 h u l l [2], without skeg and w i t h three skegs of d i f -ferent sizes. The calculated and experimentally meas-ured static rates, Y/ and N / , are identical. The calcu-lated and experimental magnitudes of the rotary deriva-tives and stability index differ slightly, but the stability predictions err on the conservative side.
SEPTEMBER 1966
1 . 0 - 1 . 0 0 . 2 0.1 G.H -OH -o.iH 0 . 1 -0 . 9 1 . 0 | . I P R O F I L E A R E A . S Q , F T . 1 - 2
Fig. 9 Comparison of calculated and experimental stability derivatives and indices for 846 h u l l without and w i t h skeg
I t is seen t h a t the analytical method predicts the trends i n stability derivatives w i t h increase i n profile area. This conclusion is confirmed by Figs. 9 and 10 for the 846 and 848 models, although these models were tested i n 1951 by experimental techniques not quite consistent w i t h those of more recent years. As skeg area is increased and extended farther aft, Y/ and Y / , be-come more positive, N^'' more negative, and A''^' less jDositive. A l l trends are i n the direction of greater course stability, as indicated by the progressively more negative value of (71.
Series 60 Hulls
The new results for the eight Series 60 models [4], w i t h and without rudder and propeller a r è presented. The
forces and moments are for zero-rudder angle and a Froude number of approximately 0.20.
Table 2 notes the hull partictilars and the various co-efficients of added mass and center of pressure computed by the present analytical method. Table 3 gives the stability derivatives estimated f r o m theory, and also those estimated f r o m a "least squares" fit of the experi-mental data. I n the latter procedure the force and mo-ment coefficients are assumed to be of the following poly-nomial .form.• F ' = F/ + N'
mo'r'
2 , . /Co + Cil3 + dr' - f c-S
+ cSr'^
+ - fc^r'i
(24)MnHpl M . l L e n g t h I , f t ( L B P ) 5 0 Beam B, f t 0.667 D r a f t H, f t 0-267 D i sp 1 a c e m e n t A , l b 33-27 P r i s m a t i c c o e f f i c i e n t , P o 0.6I?( B l o c k c o e f f i c i e n t C„ 0 0.6 L C G / i from bow 0 - 5 1 5 B/H 2-50 l/B 7-5 i / H 1 8 - 7 5 Rudder s p a n , f t 0-200 Rudder c h o r d , f t 0.105
Table 2 Pertinent Characteristics o f Series 6 0 Hulls
I . I 2, 1, 2 2, 1, 3 3, 1 , 1 h,\,\ 5, 1 , 1 6, 1 , 1 7, 1 , 1 8, 1 , 1 0 - 8 3 3 0 - 6 2 5 O - y i f t ^ 0 2 1 7 5 03't5 0 . 2 6 7 -' t l- 5 6 3 1- 1 9 2 9 - 1 0 1(6.07 h\-dh 1(7-50 0 - 6 1 6 0 - 6 I A - 0 - 7 1 3 0 - 8 0 7 ^ 0-7 0 - 8 0 - 5 0 5 0-1)75 ! . 6 8 ^ 3 - 1 2 2 - 3 4 3 - 2 8 2 - 0 7 2 - 6 8 2 - 6 8 •'•0 • ^ 6 - 0 8 - 0 7 - 0 7 - 0 7 - 0 7 - 0 e - 2 3 - 0 0 i i ( - 5 0 1 8 - 7 5 ^ ^ 0 - 1 6 4 0 - 2 5 8 0 . 2 0 0 0 - 2 0 0 0 - 1 0 5 0 - 1 0 5 0 - 1 6 7 . 0 - 0 8 0 0 - 1 0 5 L a m b ' s C o e f f i c i e n t s o f A c c e s s i o n to I n e r t i a f o r E q u i v a l e n t E l l i p s o i d s M i n o r a x i s / m a j o r a x i s , 2H/.e 0 - 1 0 6 7 0 - 0 8 7 0 o . o 6 g o 0 - 1 0 6 7 — ~ ( l o n g i t u d i n a l ) ^ „ . Q ^ ^ ^ 0 - 0 1 9 0 - 0 3 3 0 - 0 2 2 kg ( l a t e r a l ) . 0 - 9 5 7 0 - 9 6 8 0.91(0 0 - 9 5 7 ^ k' ( r o t a t i o n a l ) 0 - 8 7 5 — 0 - 9 0 3 0 - 8 2 0 0 - 8 7 5 ^ O t h e r P h y s i c a l C h a r a c t e r i s t i c s m^, mass c o e f f i c i e n t 0 - 1 6 0 O- I 7 ! 1~ 0 . 2 0 0 0 . 1 5 0 O. I 7 I 0 - 1 7 ) 0 . 2 0 0 0 - 2 2 9 m^', l o n g i tud I n a l a d d e d
-mass c o e f f i c i e n t ' O-OO3 0 - 0 0 4 r~ 0 - 0 0 4 O-OO3 0 - 0 0 4 0 - 0 0 6 0 - 0 0 4 0 - 0 0 5
m^, l a t e r a l a d d e d - m a s s ^ c o e f f i c i e n t O. I 7 I O. I 7 O ^ 0 - 1 6 9 O. I 7 2 O- I 3 8 0 - 2 2 0 O. I 8 O 0 . 1 9 4 m ^ , r o t a t i o n a l a d d e d - m a s s c o e f f i c i e n t 0 - 1 5 3 0 - 1 5 2 ^ O. 1 5 I 0 - 1 5 4 0 - 1 2 7 0 . 1 9 2 0 - 1 6 5 O. I 7 5 n ^ , v i r t u a l m o m e n t o f -i n e r t -i a c o e f f -i c -i e n t 0 . 0 2 1 3 0 . 0 2 1 9 »~ 0 - 0 2 3 7 0 . 0 2 0 6 0 - 0 2 0 2 0 . 0 2 3 9 0 . 0 2 3 7 0 . 0 2 7 1 x / / , CG o f l a t e r a l
added mass from LCG 0 . 0 4 8 0 - 0 4 9 ^ 0 - 0 4 8 0 - 0 4 9 0 - 0 4 8 0 - 0 4 9 0 - 0 3 9 0 - 0 0 5 X /I,center o f a r e a o f
^ p r o f i l e from LCG 0 - 0 2 8 — 0 . 0 3 3 0 . 0 2 8 0 . 0 2 6 - 0 . 0 1 6
D ^ ( e s t i m a t e d d r a g c o e f
-° f i c i e n t a t 0=0) 0 - 0 1 5 • '• * - 0 - 0 1 7 0 . 0 1 4 0 - 0 1 7 0 . 0 1 5 O. O I 9 , 0 . 0 2 1
Each model is designated by a sequence of three digits. The first signifies change i n h u l l ; the second signifies presence, 1, or absence, 0, of propeller; the t h i r d signifies presence, 1, or absence, 0, of design rudder. The digit 2 or 3 i n t h i r d place refers to rudder w i t h larger or smaller chord, respectively, than the original rudder.
For the models labeled (—, 1, 1), w i t h rotating pro-peller and design rudder, equations (13)-(16) are used for the theoretical derivatives w i t h Do = 0, since the propeller revolutions are adjusted to obtain zero-drag condition. The tests of Models 6, 1, 1 and 7, 1, 1 i n both clockwise and counterclockwise turns showed practically no asymmetry i n the data w i t h change i n angular velocity f r o m negative to positive. Therefore l i f t owing to pro-peller operation can be assumed negligible. For models labeled ( —, 0, 0 ) , without propeller or rudder, equations (13)-(16) are modified by equations (18)-(21). I n these
cases Do is the experimentally measured drag coefficient at zero-yaw angle. Models 2, 1, 2 and 2, 1, 3 w i t h rotat-ing propeller and larger or smaller rudder chord, respec-tively, than the original 2, 1, 1 are treated by subtracting the l i f t due to the original rudder and adding that due to the replacement.
Figs. 11-13 show typical planforms and F i g . 14 a plan of the design rudder i n location. F i g . 15 is a summary chart for all eight models, w i t h and without rudder and propeller, showing analytically calculated and experi-mentally derived stability index o-i versus ship-mass co-efficient nio = 2CbB/1. Fig. 16 shows o-i and the stability
derivatives versus rudder area for model 2 Avith rudders of varying chord, again comparing theoretical values w i t h those obtained f r o m a least squares fit of the experi-mental data. Reference [16] presents graphs of lateral-force and yawing-moment coefficients versus yaw angle
1.4
P R O F I L E A R E A , S Q . F T
Fig. 10 Comparison of calculated and experimental stability derivatives and indices f o r 848 hull without and w i t h skeg
and angular velocity r' for individual models, Figures B-7 to B-40, showing the experimental data and values computed on the basis of the linear theory; i.e., first-order variation w i t h fi and r'.
The correlation between theoretical and experimental derivative estimates is good for the huhs w i t h rudder and propeller, slightly less good for the huhs w i t h o u t rudder and propeller. The discrepancies are for the most part w i t h i n the experimental error.
Extreme Vee ModiFiation
The theoretical and experimental results are compared for an extreme vee modification [4] of Series 60 Model 1, Fig. 17, which was developed at the University of M i c h i -gan. Table 4 tabulates particulars of this h u l l (model 9), w i t h and w i t h o u t propeller and design rudder, the
calcu-lated added-mass and center-of-pressure coefficients, and t h e ' stability index ai as computed f r o m theoretical derivatives and f r o m experimentally measured rates. Figs. 18 to 21 are graphs of the lateral-force and yawing-moment coefficients for models 9, 1, 1 and 9, 0, 0, similar to those for the Series 60 hulls.
While the calculated static and rotary lateral-force derivatives apply as weU to the experimental data, the discrepancies between calculated and experimentally measured moment derivatives are larger than for the normal Series 60 forms. However, the contrary effects of lower static instability and lower rotary stability, pre-dicted by theory, appear to cancel each other i n the calculations of the stability index. The experimental and theoretical estunates of a, are close.
MODEL 2 SERIES 60, C,=0.6 B/H = 2.6a i / H = l8.75
Fig. 11 Body plan of Series 60 model 2
Fig. 12 Body plan o f Series 60 model 5
MODEL 5 SERIES 60, Cj = 0.6 B/H = 3.28, i/H=23.0
Table 3 Stability Derivatives for Series 60 Hulls i ) M o d e l s w i t h R u d d e r a n d P r o p e 1 l e r * 1,1,1 2 , 1 , 1 2 , 1 , 2 2, 1 , 3 3, l - , l 4 , 1 , 1 5, 1 , 1 6 , 1 , 1 7, 1 , 1 8, 1, 1 E s t i m a t e d f r o m T h e o r v 0 . 3 3 5 0 . 3 3 5 0 . 3 4 7 0 . 3 2 9 0 . 3 3 5 0 - 3 3 5 0-273 0 . 4 3 4 0 . 3 3 5 0- 3 3 5
%
0 . 0 8 8 0 . 0 8 6 0 . 0 8 0 0 . 0 8 9 0 . 0 8 5 0 . 0 8 9 0-07I 0 . 1 1 4 0 . 0 9 7 0 - 0 9 5 0 . 0 8 3 0 . 0 9 5 0 . 0 8 9 0 . 0 9 8 0 .1 2 4 0-083 0- 1 1 0 0-077 0 . 1 2 5 0 - 1 4 0 K' - 0 . 0 6 6 - 0 . 0 6 6 - 0 . 0 6 9 - 0 . 0 6 5 - 0 . 0 6 6 - 0 . 0 6 6 - 0 . 0 5 4 - 0- 0 8 1 - 0 . 0 6 8 - 0 - 0 7 7 - 0. 5 9 - 0 . 5 5 - 0. 6 4 - 0 . 4 9 - O . I f l - 0 - 6 1 - 0 - 3 3 - 0 . 7 6 - 0- 3 5 - 0 . 3 3 E s t i m a t e d f r o m " L e a s t S a u a r e s " F i t o f E x o e r i m s n t a l D a t a ^3 0 . 2 5 5 0 . 3 0 5 0 . 3 I I 0 . 2 9 3 0 . 3 0 8 0-283 0. 2 6 0 0-387 0.335 0 . 3 2 3 - 0 . 1 1 0 0 . 0 9 5 0.081 0 J O O 0 . 0 8 9 0-091 0 . 0 7 5 0 .1 3 2 O-O96 0-086 0 . 0 4 0 0.081 0 . 0 7 5 0 . 0 8 9 0. 111 0. 0 6 2 0-077 0.077 0 . 1 2 7 0 . 1 2 5 - 0 . 0 8 0 - 0 . 0 7 0 - 0 . 0 7 6 - 0 . 0 7 3 - 0 . 0 7 5 - 0- 0 6 6 - 0 - 0 5 7 - 0. 0 8 1 - 0 - 0 6 8 - 0 . 0 7 0 - 0 . 5 7 - 0 . 5 2 - 0 . 6 2 - 0 . 4 5 - 0 . ^ 2 - 0 . 5 6 - 0- 4 5 - 0. 6 0 - 0 . 3 4 - 0 . 3 4 P r o p e l l e r r e v o l u t i o n s a d j u s t e d t o o b t a i n z e r o d r a g Mode ft w i t h o u t R u d d e r o r P r o p e l l e r 1 , 0 , 0 2 , 0 , 0 3 , 0 , 0 c o n d i t i o n 4 , 0 , 0 5J 0 , 0 6 , 0 , 0 7, 0 , 0 8 , 0 , 0 E s t I m a t e d f r o m T h e o r v y , = L ' . D ; t > 0 . 3 0 3 0 . 3 0 3 0 . 3 0 5 0- 3 0 2 O-2I47 0,-395 0 . 3 0 6 0 . 3 0 9 0 . 1 1 2 0 . 1 1 0 0 . 1 0 9 0 . 1 1 3 0.092 0 . 1 4 0 0.121 0. 1 2 1 m' - Y ' , X r ' 0 . 1 0 8 0 . 1 1 9 0. 1 4 8 0 . 1 0 8 0.131 0 .1 0 3 0 . 1 4 9 0 . 1 6 5 - 0 . 0 5 5 - 0 . 0 5 5 - 0 . 0 5 5 - 0 . 0 5 5 - 0 . 0 4 3 - 0. 0 6 8 - 0 . 0 5 6 - 0. 0 6 4- 0 . 2 0 - 0 . 1 5 - 0 . 0 2 7 - 0. 2 0 •fO.075 - 0 . 3 8 •fO. 032•••^O.005
E s t i m a t e d f r o m " L e a s t S q u a r e s " F i t o f . E x o e r i m e n t a l D a t a ^3 0. 2 4 5 0 . 2 3 7 0. 2 6 0 0. 2 1 7 ' 0-315 0-287 0 . 2 5 6 "3 0. 1 1 4 0 . 1 3 4 0 . 1 1 6 0 . 0 9 7 0 - 1 4 0 0.121 0-093 0 .1 0 1 0. 1 3 ' t 0.081 0 . 1 0 0 0- 1 0 3 0 . 1 4 9 0.154 - 0 . 0 5 5 - 0 . 0 9 - 0 . 0 5 4 + 0 . 1 9 - 0 . 0 5 9 - 0 . 2 6 - 0 . 0 4 5 0 - 0. 0 6 8 - 0 . 2 2 - 0 . 0 5 6 -f.067 - 0. 0 5 2 •f 0.033
the bow (the bow sections of the Series 60 model have been pared to fine vee forms while the isrofile remains the same), there is some vibration at the boSv i n yawed motion. Such a condition would affect the measured moments.
Mariner Class Hull
The Mariner class hull. Table 5, F i g . 22, reported i n reference [5] is treated here. A f t e r that note was pub-lished, however, i t was found that the calibrations used to reduce the test data were i n error. The data have since been revised, w i t h correct calibrations, and are shown i n Figs. 23-25, f o r the hull without propeller, w i t h rudder amidship.
The tests had been conducted on the rotating arm at Davidson Laboratory i n 1963. The model was r u n at a Froude number around 0.20, so that f o r this model, also, the effects of wave making can be neglected.
The charts show that the theoretical derivatives ob-tained by using equations (13)-(16) fit the experimental data reasonably well. Table 6 gives a comparison of the stability derivatives and indices derived here and the results of reference [17] as obtained by an oscillator technique.
Destroyer Model • The results f o r the DD692 destroyer model. Fig. 26,
Table 7, w i t h t w i n rudders and propellers, tested on the rotating arm at Davidson Laboratory [6] i n 1963 are presented. The coefficients of measured forces and moments at three turning diameters for zero-rudder angle and Froude nmnber of 0.155 are shown i n Figs. 27 and 28.
Equations (13)-(16) are used f o r the theoretical derivatives, and i t is assmned that the effects of the off-center t w i n rudders at zero angle and of the propellers are
M O D E L 8 S E R I E S 60, 0 3 = 0.8
B / H = 2.68, £ / H = l8.75
Fig. 13 Body plan of Series 60 model 8 negligible additions to the effect of the skeg at the stern.
The calculated static rates on a straight course seem reasonable extrapolations of the rotating-arm data. The calculated rotary rates are close to the measured slopes, certainly w i t h i n the experimental error.
Hopper-Dredge Models
The results for the hopper-dredge model. Fig. 29, under two displacement conditions, are presented i n Figs. 30-33. This model was tested on the rotating arm at Davidson Laboratory in 1960 [ 7 ] , at Froude numbers of 0.12 and 0.20 for the heavy-displacement case and 0.155 for the light case.
The theoretical derivatives shown on the charts were calculated f r o m equations ( 5 ) - ( 8 ) for bare hull, modified by equations ( 1 8 ) - ( 2 1 ) for the small skeg-plus rudder at the stern. The pertinent characteristics of the models are given i n Table 8. Table 9 compares the theoretical values of stability index tri w i t h those calculated f r o m the measurements and reported i n reference [7].
Although the theoretical estimates are on the average 18 percent less than the experimental, both indicate an extremely unstable vessel. The theory and results of this report underline the recommendations of reference [ 7 ] , viz., to mcrease the deadwood forward , of the rudder stock and to increase the chord of the rudder a f t for stability.
Course Stability Dependence on Hull Geometry
A n analysis of the assumed expressions for ship lateral-force and yawing-moment derivatives w i l l be made i n the
Fig. 14 Series 60, stern profile o f models 1, 2, 3, 4, 7, 8 w i t h design rudder
Table 4 Pertinent Characteristics o f Extreme Vee Modifica-tion of Series 6 0 Model 1
Table 6 Stability Derivatives for Mariner Class Model (Without Propeller) Model 9,1,1 L e n g t h I , f t 5.0 Beam B , f t 0.667 D r a f t H , f t 0.267 D i s p l a c e m e n t A , l b 33.10 P r i s m a t i c c o e f f i c i e n t C , = 2x / x P ° B l o c k c o e f f i c i e n t 0 . 6 1 4 0.6 L C c / i , f r o m bow 0 . 5 I I B / H 2.50 I/B 7.5 i / H 18.75 Rudder s p a n , f t 0.200 Rudder c h o r d , f t 0.105 9 . 0 , 0 m' - l a t e r a l added-mass ^ c o e f f i c i e n t Theoret i c a I Estimate" 0.310 . 0.068 0.065 -0.059 0.136 0 0 Lamb's C o e f f i c i e n t s o f A c c e s s i o n t o I n e r t i a f o r E q u i v a l e n t E l l i p s o i d Minor a x i s / m a j o r a x i s , 2 H / / , 0 , 1 0 7 • k l ( l o n g i t u d i n a l ) 0 . 0 2 2 • • kg ( l a t e r a l ) O.957 . , k ' ( r o t a t i o n a l ) O.875 • O t h e r P h y s i c a l C h a r a c t e r i s t i c s m ' , mass c o e f f i c i e n t m^', l o n g i t u d i n a l a d d e d - m a s s c o e f f i c i e n t m^, l a t e r a l a d d e d - m a s s c o e f f i c i e n t m^, r o t a t i o n a l a d d e d - m a s s c o e f f i c i e n t n ^ , v i r t u a l m o m e n t - o f - i n e r t i a c o e f f i c i e n t x / e , CG o f l a t e r a l added mass f r o m LCG X p A , c e n t e r o f a r e a o f p r o f i l e f r o m LCG 0 ' ( e s t i m a t e d drag c o e f f i c i e n t a t
added moment-of- O.OO8 i n e r t i a c o e f f i c i e n t Experimental _Ranqe (Ref.17) 0.295 to 0.218 0.066 to 0.122 0.066 to 0.055 -0.050 t o -0.037 0.11^4 t o 0.151 0.007 -0.tt2 (best) - 0 . 1 6 (average) = 0 ) c a l c u l a t e d from t h e o r e t i c a l d e r i v a t i v e s c a l c u l a t e d from e x p e r i m e n t a l r a t e s 0.159 C.003 0.156 0.11(3 0 . 0 1 9 0.022 0.027 0.017 0.62 0.55
A l s o , as shown on the c h a r t s , a reasonable f i t to the Davidson Laboratory rotatingarm e x p e r i -mental data.
O s c i l l a t o r r e s u l t s nondimensionalized according to the convention adopted in the present paper.
-0.17 -0.17
Table 7 Pertinent Characteristics of DD692 Destroyer M o d e l
f t
Table 5 Pertinent Characteristics o f Mariner Class Model
Length t , f t (LWL) Beam B , f t D r a f t H , f t (mean) Displacement A , lb P r i s m a t i c c o e f f i c i e n t , C = 2x /JB P o B l o c k c o e f f i c i e n t LCG/£ , from bow B/H i/B UH 5 . 0 0.731 0.236 32.6 0.620 0.607 0.524 3 . 1 0 6 . 8 4 21.19 L e n g t h t Beam B , f t D r a f t H , f t D i s p l a c e m e n t A , l b P r i s m a t i c c o e f f i c i e n t B l o c k c o e f f i c i e n t , C„ b L C G / 1 , f r o m bow B / H l/B UH C - l-f. It P ° 5.710 0 . 6 0 2 0 . 2 0 8 25.22 0.566 0.522 2.90 9.45 27.40 Lamb's C o e f f i c i e n t s o f A c c e s s i o n to I n e r t i a f o r E q u i v a l e n t E l l i p s o i d Minor a x i s / m a j o r a x i s , 2H/X . 0.094 ( l o n g i t u d i n a l ) 0.02 ( l a t e r a l ) 0 . 9 6 k' ( r o t a t i o n a l ) 0 . 8 9 Other P h y s i c a l C h a r a c t e r i s t i c s m^, mass c o e f f i c i e n t 0.177 m' l o n g i t u d i n a l added-mass c o e f f i c i e n t 0.003 m' l a t e r a l added-mass c o e f f i c i e n t 0.136 a m^ r o t a t i o n a l added-mass c o e f f i c i e n t 0.126 v i r t u a l m o m e n t - o f - i n e r t i a c o e f f i c i e n t 0.019 x / X , CG o f l a t e r a l added mass from LCG 0.066 X p / £ , c e n t e r of a r e a of p r o f i l e from LCG 0.058 D' ( e s t i m a t e d drag c o e f f i c i e n t a t P = 0) 0 . 0 1 4 L a m b ' s C o e f f i c i e n t s o f A c c e s s i o n t o I n e r t l b f o r E q u i v a l e n t E l l i p s o i d M i n o r a x i s / m a j o r a x i s , 2 H / i 0.073 \ ( l o n g i t u d i n a l ) 0 . 0 1 5 k . ( l a t e r a l ) ' • 0.972 k ' ( r o t a t i o n a l ) 0.920 O t h e r P h y s i c a l C h a r a c t e r i s t i c s mass c o e f f i c i e n t l o n g i t u d i n a l a d d e d - m a s s c o e f f i c i e n t l a t e r a l a d d e d - m a s s c o e f f i c i e n t r o t a t i o n a l a d d e d - m a s s c o e f f i c i e n t v i r t u a l m o m e n t - o f - 1 n e r 1 1 a c o e f f i c i e n t x / X , CG o f l a t e r a l a d d e d m a s s f r o m LCG X / i , c e n t e r o f a r e a o f p r o f i l e f r o m LCG P • 0.119 0.002 0.087 0.082 0.0122 0.070 0.054 D ^ , d r a g c o e f f i c i e n t a t 8 = 0, U = 2.1 f t / s e c 0.017 CT s t a b l 1 i t y i n d e x - 0 . 7 6
W I T H O U T RUDDER A N D P R O P E L L E R cr - 1 . 0 0 . 2 5 . H/ j e= 0 . 0 5 3 4 F R O M T H E O R Y O A O F R 0 M " L E A S T S Q U A R E S " F I T O F E X P E R I M E N T A L D A T A ' O H/ J ? = 0 . 0 5 3 4 A = 0 . 0 6 9 0 • = 0 . 0 4 3 5 W I T H RUDDER A N D P R O P E L L E R
Fig. 15 Stability index n f o r Series 60 hulls
light of the results presented, to discover the major f o r m parameters on which course stability depends. I t is well known, and further proof has been added here, that low aspect-ratio skegs or deadwood at the afterbody are es-sential for minimum stability. I t has also been demon-strated that increasing skeg area a f t by widening the chord of skeg or rudder improves the stability, and that removing area w i t h fin effect at the stern lowers the stabil-i t y . B u t asstabil-ide f r o m such fstabil-in areas, how c an one teU by the dimensions and body lines of a ship whether the design w i l l lead to greater or less stability? The answer lies i n the make-up of the various terms involved i n equations (5)-(16). These w i l l be examined now.
Por practical ships, the longitudiual coefficient of ac-cession to inertia ki is close to zero, so that the longitudi-nal added-mass coefhcient m i ' is negligible. The lateral and rotational coefficients of accession to inertia, ki and k', are approximately equal, and therefore nii' can be substituted for m/. The v i r t u a l moment-of-inertia co-efficient is close to (mo' + mi')/!^ for the variety of h u l l forms treated here. I n general, variations i n Xp and x
(the centers of pressure of the l i f t and lateral added mass, respectively) and i n position of longitudinal center of g r a v i t y are minor i n their influence on the stability
Table 8 Pertinent Characteristics of Hopper-Dredge M o d e l
Condi t ion Heavy Light
Length X, f t (LUL) "1.333 4 . 1 6 7 Beam B. f t .(,0.721 0.721 Draft H, f t 0.299 0 . 2 0 8 Displacement A, lb 111.65 3 2 . 0 8 Prismatic c o e f f i c i e n t , C = 2x / j ï P o 0 . 7 2 7 0.8110 Block c o e f f i c i e n t C^ 0.717 0 . 8 Z O L C G / X, from bow 0.500 0.512 B/H 2.I1I5 3.461 i/s 6.00 5.78 l/» 111.51 20.00
Lamb's C o e f f i c i e n t s of Accession to I n e r t i a for Equivalent E l l 1psoi ds Minor axis/major a x i s , 2HA 0.138 0 . 1 0 0
k^ ( l o n g i t u d i n a l ) 0.033 0.020
kg ( l a t e r a l ) 0 . 9 3 6 0.960
k' ( r o t a t i o n a l ) 0 . B I 5 0.885
Other Physical Character i s t ics
m'. mass c o e f f i c i e n t 0 . 2 3 9 0 . 2 8 4
m^', longitudinal added-mass c o e f f i c i e n t 0 . 0 0 8 0 . 0 0 6 mS l a t e r a l added-mass c o e f f i c i e n t 0 . 2 3 7 0H71
ra^, r o t a t i o n a l added-mass c o e f f i c i e n t - 0.207 0.158
n^, v i r t u a l moment of i n e r t i a 0.0315 0.0311 x / ^ , CG of l a t e r a l added mass from LCG 0.021 . 0 . 0 1 3
X p A . center of area of p r o f i l e from LCG 0 . 0 1 6 0 . 0 1 4
(drag c o e f f i c i e n t at p = 0) 0 . 0 2 5 0 . 0 2 8
I -1.0 0 . 4 OS 0.1 F R O M L E A S T S Q U A R E S F I T O F E X P E R I M E N T A L D A T A F R O M T H E O R Y O - N l 0.01 o . o a 0 . 0 3 RUDDER A R E A , S Q . F T . 0 . 0 4
Fig. 16 Comparison o f calculated and experimental stability derivatives and indices f o r Series 60, model 2 w i t h rudders of different chord
derivatives of hulls w i t h deadwood aft. The l i f t coef-ficient varies inversely w i t h length-draft ratio. The less important drag coefficient depends, as is known, on block coefhcient and beam-draft ratio, and hence on ??io', which
is a f u n c t i o n of block coefficient and beam-length ratio, and on l/R. The major factors influencing the deriva-tives and stability index tri are thus seen to be m^' , and l/E.
The dependence of <7i on x^/l, which under Albring's assumption is equal to half the prismatic coefficient, is implicit i n its dependence on ma and m-i'. The ship-mass coefficient is
B mo = 2CB J The lateral added-mass coefficient formula
(25)
« 1 2
r
•J XsCJihlx
w i t h ki ~ 1, and h = H, maximum draft, for almost the entire length of commercial and naval vessels, can be written approximately as im' - ^ a (26) where _ 1 r " I Jx. Cdx
an average sectional-inertia coefficient for the hull. Cs is a function of section beam-draft ratio and section-area coefficient, depending more heavily on the latter.
.125 W . L . . 0 6 2 W . L , B . L .
0 . 1 0 0
0 . 0 7 5
Fig. 17 Body plan of extreme vee modification of Series 60 (model 9)
0 . 0 5 0 F A I R E D r' = 0 - C A L C U L A T E D Y ' ^ - 0 , 0 5 0 E X P E R I M E N T A L D A T A /3=0 • + 2 ° O - 2 °
•
2 4 6 /3, D E G 8 10 0.1 0 . 2 0 . 3 r'= £ / RFig. 18 Series 60, model 9,1,1 total lateral-force coefficient
.0.050 EXPERIMENTAL DATA ;S=0 • + 2.5° D - 2 5 ° A ' 0.1 0.2 0.3 r ' = i / R
Fig. 19 Series 60, model 9,1,1 yawing-moment coefficient
F A I R E D r' = 0 y' - C A L C U L A T E D Y ' ^ 4 D E G 8 K)
0.02 0.01 O -0.01 -0.02 ESTIM ATEO > ==ft-U -O (^/J=0 -0.031¬ -0.6 -0.4 -0.2 O 0.2 0.4 r ' = i / R ^ 1 -CALC JLATE 1 w,a=o
d
>-— EXPERIMENTAL DATAd
>-— •d
>-— • — + 2» O — - 2 °•
1 1 -0.6 -0.4 -0.2 O 0.2 r'= i / R 0.4 0.6 -0.6 -0.4Fie. 23 Mariner class force and moment coefficients versus r' = l/R
Table 9 Stability Index o-j for the Hopper-Dredge M o d e l
Condition Heavy-Light Model speed, fps 2.39 1.40 1.80 theoretical estimate -^0.83 -fO.83 -1-0.60 Calculated f r o m measurements [7] -i-0.89 - f l . 0 4 . -1-0.82
Fig. 3. Tlius prismatic, Avliicli is the quotient of bloclc coefhcient by midship-section-area coefficient, is involved i n both mo' and 1 ) 1 2 ' .
On substitLiting the approximations noted i n equation (23), i t can be shown that tn is some f u n c t i o n of the i n -verse of {mo mï)llE. A graph of o-i versus (mo' - f
1112)1/H for the stable huUs, w i t h low aspect-ratio skegs,
or deadwood plus rudder, to the stern, shows clearly that the major f o r m factors have been well explored. The experimentally derived values for hulls tested in recent years at Davidson Laboratory w i t h consistent
experi-mental techniques are plotted m Fig. 4. The hulls vary in block coefhcient f r o m 0.50 to 0.80, i n length-draft ratio f r o m 14.5 to 27.40, i n length-beam ratio f r o m 6 to 9.45, and in^beam-draft ratio f r o n i 2.50 to 3.28. The average sectional-inertia coefhcients Cj are tabulated i n Table 10.
Table 10 Model CB 842 0.50 0.83 CO. 60 -^0.70 (O.8O 1.02 CO. 60 -^0.70 (O.8O 1.07 CO. 60 -^0.70 (O.8O 1.16 0.60 0.93 0.61 0.92 . 0.57 0.76
The curve i n Fig. 4 is represented b y the formula
LO"o + mi')l_
and is seen to fit the data very well. B y making use of equations (25) and (26), course stability is shown to vary inversely as
' H
2CB + y c . 2 . 5
which may be computed easily f r o m the ship lines and w i t h the aid of Fig. 3. This relationship shows that sta-b i l i t y w i l l sta-be increased for huhs w i t h low aspect-ratio • skegs to the stern by decreasing one or more of these three f o r m factors: Block coefficient, beam-draft ratio, and average sectional-fnertia coefficient.
Conclusion
The analytical method of reference [1] for estimating force and moment rates i n yawing motion and stabhity on course, which combines Albring's empirical modifica-tions of simplified flow theory w i t h low aspect-ratio wing theory, has been extended here to take into consideration the efi'ects on course stability of higher aspect-ratio fins. The method had been applied i n reference [1] to eight 840 Series huhs, of 0.5 block coefficient and varying beam, draft, and displacement. The huhs were Taylor Standard Series forms w i t h the after deadwood removed, but three of the hulls had also been tested w i t h low aspect-ratio flat-plate skegs i n the place of the removed dead-wood. The extended method has now been applied to 12 other ships: Six Series 60 forms of 0.6 block coefficient and varying beam, draft, and displacement; two Series 60 forms of 0.7 and 0.8 block; an extreme vee modifica-t i o n of a Series 60, 0.6 block f o r m ; and modifica-three omodifica-ther widely different forms—a Mariner class ship, a destroyer, and a hopper dredge at two displacements. A l l had large areas of deadwood aft, except the hopper dredge, which had a smah skeg at the stern. The Series 60 cases without rudders, and the case of one model Avith rudders
- 0 . 1 5 I
O 2
/9, DEG.
1 0
Fig. 24 Mariner class total lateral-force coefficient versus yaw angle /3
a i o
0 . 0 5
- 0 . 0 5
- 8 0
/S, DEG.
Fig. 25 Mariner class yawing-moment coefficient versus yaw angle /S SEPTEMBER 1966
Fig. 26 Body plan of DD692 destroyer model EXPERIMENTAL DATA 0. 178 O 0.28 6 • 0.381 A CALCULATED N / 3 -/ 3
CALCULATED N',.. r ' A T / 3 = 0
6 FRAMES = 2"
Fig. 29 Body plan f o r hopper-dredge model
SEPTEMBER 1966
1 O 1. • l . 1 = 2.39 ft/sec •.' = I.IOft/sec y CAI C U L A T E D Y' ;3, r' y y y' • \>y \ = 0 . 1 3 5 , 8 ^ - 6 - 4 - 2 0 2 4 6 8 13, DEG E X P E R I M E N T A L DATA ^ = 0 0 + 2 ° O + 4 ° , - 2 ° ,
•
- 0 10 O r'= i / R 0.05 0.10 0.15Fig. 30 Hopper dredge, lieavy displacement, total lateral-force coefficient
of larger and smaller chord, have also been treated, mak-ing 24 cases i n all.
Good correlation is shown between the values of sta-b i l i t y derivatives calculated sta-by this method and those based on experimental measurements, despite the variety i n shijD design. I t has been sho^vn that low aspect-ratio skegs or deadwood at the afterbody are essential for m m i m u m stability and that additional skeg area or an extension of rudder area aft increases .stability. For the stable ship w i t h large skegs to the stern, the major f o r m factors influencing course stability are demonstrated to be the coefhcients of ship mass and lateral added mass of entrained water and the length-draft ratio, or, as a corollary, the block coefhcient, beam-draft ratio, and average sectional-inertia coefhcient. The functional re-lationship is exiDressed by the empirical formula
0-1 = — B
The results of this report show t h a t the values of sta-bility derivatives and indices determined by the analyti-cal method are of the right orders of magnitude and indicate correct trends. Application to a variety of ship forms has demonstrated t h a t the method can predict relative effects of changes i n the geometry of the ship form, as well as the effects of changes i n skeg and rudder area. The analytical method has thus been proved an effective tool to be used i n designing ships for greater course stability and in planning an economical program of model testing.
References
1 W . R. Jacobs, " M e t h o d of Predicting Course Stabihty and Turning Qualities of Ships," D L Report 945, M a r c h 1963, published \n Internalional Shipbuilding Progress, vol. 11, no. 121, September 1964.
0 . 0 3 0.02 0.01 N ' O - 0 . 0 2 -0.03 O U= 2. • U= 1 1 39 f t / s e c 4 0 f t / s e c y y y y y y y -y y C A L G U L A T E O N' ^ U r' =0 ^
y.
—y
y i . 135•
< < /9, D E G . - 0 . 0 2 - 0 . 1 0 - 0 . 0 5 O 0 . 0 5 0 . 1 0 0.15 r' = £ / RFig. 31 Hopper dredge, heavy displacement, yawing-moment coefficient
on the Hydrodynamic Coefficients of Surface Ships," D L Report 740, M a y 1959.
3 W. Albring, "Summary Report of Experimental and Mathematical Methods for the Determmation of Co-efficients of Turning of Bodies of Revolution," C O N L A N 2.
4 C. L . Crane, Jr., "Research on Ship Controhabil-i t y , " D L Quarterly Progress Report, June 1, 1964.
5 A. Suarez, "Rotating A n n Experimental Study of a Mariner Class Vessel," D L Note 696, June 1963.
(Measurements revised using correct calibration, A p r i l 1964.)
6 A . Suarez, D L Conhdential Letter Report 1012, February 1964.
7 A. Suarez, "Stabihty Analysis of SS Sandcap-t a i n , " D L LeSandcap-tSandcap-ter ReporSandcap-t 786, March-May, 1960.
8 M . Gimprich and W. R. Jacobs, "The Effect of Fins on the Behavior of Free Bodies," E T T D L Report 361, February 1950, Conhdential.
9 H . Lamb, Hydrodynamics, Dover Publications, Inc., New York, N . Y . , sixth edition, 1945.
10 J. P. Breslin, "Derivation of SlenderBody A p -proximations for Force and Moment Derivatives f r o m Tlnee-Dimensional Singularity Distributions," D L TM 134, January 1963.
11 K . K . Fedyaevsky and G. V. Sobolev, "Applica-t i o n of "Applica-the Resul"Applica-ts of Low Aspec"Applica-t-Ra"Applica-tio W i n g Theory "Applica-to the Solution of Some Steering Problems," Proceedings, Netherlands Ship Model Basin Symposium on the Be-havior of Ships in a Seaway, Wageningen, The Nether-lands, September 1957.
12 M . Gertler, " A Reanalysis of the Original Test D a t a for the Taylor Standard Series," D T M B Report 806, M a r c h 1954.
13 M . M a r t i n , "Analysis of Lateral Force and M o
-ment Caused by Yaw During Ship Turning," D L Report 792, March 1961.
14 C. W . Prohaska, "The Vertical Vibration of Ships," Shipbuilder and Marine Engineer-Builder, Octo-ber-November 1947.
15 K . S. M . Davidson and L . I . Schiff, " T u r n i n g and Course-Keeping Quahties," Trans. SNAME, vol. 54, 1946.
16 W. R. Jacobs, "Estimation of Stabihty Deriva-tives and Indices of Various Ship Forms, and Comparison w i t h Experimental Results," D L Report 1035, Septem-ber 1964.
17 J. R. Paulling and L . W . Wood, "The Dynamic Problem of Two Ships Operating on ParaUel Courses i n Close Proximity," University of California Series No. 189, July 18,1963.