Maneuvering of Large Tankers
Willem B. van Berlekom/ Visitor, and Thomas A. Goddard," Associate Member
This paper p r e s e n t s t i i e results of a series of t e s t s i n v e s t i g a t i n g t h e m a n e u v e r a b i l i t y of large t a n k e r s w h i c h w e r e c o n d u c t e d f o r Esso I n t e r n a t i o n a l o n t h e s t e e r i n g a n d m a n e u v e r i n g s i m u -lator a t t h e Swedish State S h i p b u i l d i n g E x p e r i m e n t a l T a n k (SSPA). T h e s i m u l a t i o n s w e r e based u p o n t h e r e s u l t s of large c a p t i v e - m o d e l t e s t s w h i c h were c o m p a r e d w i t h available full-scale trial results t o insure t h e v a l i d i t y of t h i s a p p r o a c h . T h e s e t e s t s i n d i c a t e d t h a t t h e m a j o r c o n s i d e r a t i o n s i n v o l v e d in t h e m a n e u v e r i n g of large t a n k e r s are in m a n y w a y s d i f f e r e n t f r o m t h o s e a f f e c t i n g s m a l l e r s h i p s . As a r e s u l t , it a p p e a r s t h a t t h e a p p l i c a b i l i t y o f s o m e of t h e s t a n d a r d m a n e u v e r i n g trials c o m m o n l y u s e d f o r s m a l l e r s h i p s are o f l i m i t e d v a l u e f o r large t a n k e r s . S o m e revised m a n e u v e r i n g t e s t s for large t a n k e r s are s u g g e s t e d . T h e investigation also i n d i c a t e d t h e i m p o r t a n c e o f t h e s h i p ' s c o n t r o l s y s t e m ( w h e t h e r a u t o p i l o t or h e l m s -m a n a n d -m a s t e r ) t o t h e -m a n e u v e r a b i l i t y o f large t a n k e r s . It w a s f o u n d t h a t t h e c o n t r o l sys-t e m has a m o r e s i g n i f i c a n sys-t e f f e c sys-t o n m a n e u v e r a b i l i sys-t y sys-t h a n large c h a n g e s in d y n a m i c s sys-t a b i l i sys-t y or a w i d e range o f s h i p p r o p o r t i o n s .
Introduction
T H E E X T R E M E L Y R A P I D G R O W T H i n tanker size t h a t has occurred i n recent years has necessitated the development and application of new method.s of analysis i n many areas of the design process. One of the areas of greate.st importance f o r very large tankers is their maneuverability. I n the con-text of this paper, maneuverability is considered t o involve all the areas of ship control w i t h the exception of stopping. While the stopping question is certainly of importance, i t has not been considered here. This paper describes a series of tests undertaken for Esso International on the steering and maneuvering simulator at the Swedish State Shipbuilding Experimental T a n k (SSPA) i n Goteborg, Sweden. The tests were conducted i n two separate parts.
T h e first program investigated the maneuverabilitj^ of a proposed new tanker design b y comparing its performance w i t h t h a t of au existing 190,000-dwt tanker. T o insure the v a l i d i t y of such an approach i t was necessary to investigate the correlation between the simulator predictions of maneu-verability for the 190,000-dwt shiji and the available full-scale t r i a l results for that class of ship. I t was also necessary to determine meaningful points of comparison between the two designs. This led to an evaluation of the applicability of the existing so-called standard maneuvers to the actual maneuver-ing a b i l i t y of large tankers. Further, i t was necessary to consider the importance of the ship control system, whether automatic or manual (autojjilot or the combination of helms-man and master/pilot), i n evaluating the perforhelms-mance of the ship under consideration. A limited investigation of the
1 Principal Scientific Officer, Swedish State Shipbuilding E.x-perimental Tank, Goteborg, Sweden.
2 Research Project Engineer, Esso International Division, Exxon Corporation, New York, N . Y .
The opinions expressed herein are those of the authors and do not necessarily reflect those of the Swedish State Shipbuilding Experimental Tank or the Exxon Corporation.
Presented at the Annual Meeting, New York, N . Y., Novem-ber 16 and 17, 1972, of T H E S O C I E T Y O F N . W A T . A R C H I T E C T S A N D M A R I N E , E N G I N E E R S .
effect of shallow water was also made. I t is believed that these points are of more general interest than the performance of a particular proposed new design. Accordingly, they w i l l be discussed i n terms of the performance of the existing 190,000-dwt ship only.
The second series of tests w^as intended to examine more generally the impact on maneuverabilitj' of v a r y i n g difïerent ship parameters such as ship size, L/B ratio, and rudder area. Because of the results of the first test series, this program i n -cluded predictions of several additional maneuvers which were intended to be more representative of actual maneuver-ing situations. A n investigation of the effect of dynamic iustabiUty on maneuverability was also conducted.
M e t h o d s of investigation
There are several ways of investigating the maneuvera b i l i t y of ships. One would immaneuveragine t h maneuvera t i f extensive f u l l -scale trials were economically feasible, these would yield the best material for the analysis of a ship's maneuverability. This is, however, only p a r t l y true. W e can agree t h a t the results of such tests give the most accurate information on the ship's response to specific inpuls and disturbances, b u t the analysis cannot be carried out to any refined degree because the situation is analogous to the black-box problem. Thus the inputs are known and the outputs (i.e., the ship's response) are known b u t the connections are, i n many cases, not clear. Full-scale trials also have other disadvantages. Weather conditions encountered duruig such trials can have a significant impact on t r i a l results, distorting them and making them difficult to analyze. Perhaps most siguificant, i t may well be too late to correct any problems discovered i n f u l l -scale trials, k technique which permits analysis during the design phase is far more useful.
Another obvious way of investigating a shiji's maneuver-ability is to use free-sailing scale models. The advantages compared w i t h full-scale tests are better control of external disturbances, lower costs and, of course, the ability t o conduct model te.sts during the design phase. There are, however,
Fig. 1 C o o r d i n a t e s y s t e m s
several drawbacks, the principal ones being scale effects on propulsion and rudder characteristics and the reduced time scale.
M o s t model testing is carried out according to Froude's law. This implies that the resistance of the model is rela-t i v e l y higher rela-than rela-tharela-t of rela-the full-scale ship. Thus f o r a free-sailing model the propeller(s) must be more highly loaded t h a n f o r the actual ship. The rudder is normally working i n the propeller slipstream and thus the rudder forces are highly dependent on the propeller load condition. Also, the model time scale is inversely proportional to the square root of the ship-to-model length scale. Thus the model w i l l re-spond much more rapidly than the actual ship, which ob-scures one of the most significant characteristics of large tankers; namely, the length of time i t takes them to respond to forces acting upon them.
For investigations of the maneuverability of large tankers, real-time simulations provide an accurate and versatile tool. T h e prerequisite is, however, t h a t the ship's motions are accurately described b y a mathematical model. T h i s is the principal point and also one of the main difficulties. A great deal of effort has been devoted to the construction of mathematical models describing the motions of aircraft and, more lately, those of ships; see N o r r b i n [ 1 ] ' for a survey and additional references on the subject. Real-time simulation is the method of investigating ship's maneuverability t h a t is now used at SSPA and elsewhere.
M a t h e m a t i c a l model for large t a n k e r s
For describing the maneuverability of large tankers i t can be assumed t h a t the following three equations are sufficient:
X = 0 (axial force) (1) Y = 0 (transverse force) (2) N = 0 (yawing moment) (3)
' Numbens in brackets designate References at end of paper.
I t can be noted t h a t the roll equation is neglected, as i n this case rolling is slight.
The forces and moment are usually given w i t h reference to a ship-fixed coordinate system; see reference [1] and F i g . 1. Equations (1), (2), and (3) can now be w r i t t e n as:
mill — TV — Xg-r^) = Xhydro + (4)
m(() + ru + Xo-f) = Fhydro + Yiut (5) 7 „ - r + mxoiv + ru) = hydro + A''dist (6) The left-hand sides of equations (4), (5), and (6) are the inertial contributions referred to a ship-fixed coordinate system, while the right-hand sides represent forces or mo-ments acting on the ship. The subscript " h y d r o " stands f o r hydrodynamic forces and moments i n calm water and the subscript " d i s t " f o r disturbance forces and moments due to wind, waves, current, etc. The subdivision into h y d r o -dynamic and disturbance forces and moments might seem artificial, b u t i t has proved to be practical i n the analysis.
The hydrodynamic forces and moments i n calm water are considered to be the sum of forces (and moments) i n calm deep water and effects due to confinements i n the waterway, i.e., shallow water and banks. This subdivision is essen-tially proposed i n [1] and has proved to be useful i n the simulations. T h u s :
.X^hydro = + Xj, oonf (7) deep confinement
water effects etc.
The deep-water terms X/, „, Yi, „. and A'^^ „ can be con-sidered as functions of the ship's motions (velocities and accelerations), rudder angle, and engine settings. Formallj'', this can be expressed as follows:
= F , c i , 5 , M ) (8a)
Yn » = Y„ Ut, V, ^ , 8, (86)
N>,„ = N, „(«, V, w, 5, //) (8c)
where
t = time
V = translatory velocity vector
ÖI = angular velocity vector
5 = rudder angle
/ i = engine setting
For the maneuvering of large tankers, substantial simpli-fications can be introduced. The vector V has t w o com-ponents, u and V (axial and transverse velocity), and the vector CÜ has one component, the yaw rate, !/•, thus:
Xft „ = Zft „(<, u, V, «A, 5, M ) (9) with similar terms f o r F and A''.
I n order to gain some clarity and to express expUcitly the influence of engine maneuvers on the ship's m o t i o n s , « t h r e e auxiliary equations are introduced [1]. These equations are propeller torque, Q, thrust, T, and flow velocity at the rudder, c, which can be f o r m a l l y given as:
T = T{u, n) (10)
Q = Q{u,n,tx) (11)
c = c(u, n) (12)
These equations are discussed i n [1]. I t may be noted t h a t
wake and thrust deduction factors are here assumed to be independent of propeller loading. The flow velocity at the rudder is a mean value over the rudder height [1 ].
The principal equations of motions can thus be given as:
Xh « = V, i/-, T, c, 8) (13)
YHO,= F , „ ( M , V, xjy, T , c, 8) (14)
Nn„== N, „ ( « , V, i^, T, c, 8) (15)
Thus there are six equations and six dependent variables (u, v, 4', c, T, n), t w o i n j j u t variables (5 and n), and one independent variable, time t. The functions X^ „ { ) etc. can be approx-imated b y a polynomial representation; and b y using physical reasoning, symmetry relationships, comi)arisons w i t h results f r o m potential flow, etc., explicit expressions can be derived. A n example is given here:
(»i - Yi)v = Y^u.uv + Y,i,iv/v/
+ {Yui + m) ui' + Y,u„ c/c/8
+ Y T - T + . . . (16) where Yi, F „ „ etc. are the so-called hydrodynamic derivatives
and w i l l be discu.ssed i n the following.
One feature of the SSPA mathematical model is that princi-pally i t uses second-order terms. M o s t mathematical models in common use instead include third-order terms f o r the non-linearities. One main reason f o r using third-order terms instead of second-order terms appears to be that the mathe-matics are less complicated, as no absolute value terms are necessary. F r o m physical reasoning [1], second-order terms appear to be more ajipropriate.
The representation of confinement effects is extensively
discussed i n [ 1 ] . Shallow-water effects are introduced by a depth parameter f = T/{h - T), where T is ship's d r a f t and
h the water depth. I n a similar way, bank effects are i n
-cluded through a bank clearance parameter i) = L/%JB, where L is the ship's length and VB the distance f r o m the bank. The hydrodynamic coeiBcients
The hj'drodynamic coefficients can p a r t l y be calculated u.sing results f r o m potential theory and semi-empirical formulas; see [1], However, f o r the investigations reported liere, data were obtained using the Planar M o t i o n Mechanism ( P M M ) at the Hydro-og Aerodynamisk L a b o r a t i u m ( H y A ) i n Lyngby, Denmark. While the P M M technique has been treated extensively elsewhere [2], a short recapitulation seems appropriate. The planar motion mechanism is r i g i d l y attached t o the towing carriage i n a conventionally jiropor-tioned towing tank (rather than the square basins used for rotating-arm testing). For the testing described i n this paper, models of appro.ximately seven meters i n length were used. As the model is moved down the t a n k b y the carriage i t is forced to perform precisely controlled movements.
There are two modes of operation, "static" and " d y n a m i c . " I n the static mode the model is constrained to travel along a .straight j j a t h down the towing tank w i t h constant velocity and constant d r i f t angle. I n the dynamic mode the model is forced t o oscillate i n a sinusoidal w^ay so t h a t various com-binations of sway and yaw m o t i o n are obtained. I n b o t h modes the rudder angle can be varied. Thus i t is possible to obtain the forces and moments due to sway {v, v), y a w {ip, ^z), rudder angle (5), and cross-coupling effects between sway motion, y a w motion, and rudder angle. T h e evaluation of these forces and moments is comprehensively described i n [ 2 ] . The original data were given b y H y A i n the usual cubic fit
-Nomenclature.
A = advance Ar rudder area
B = ship beam
CB = block coefficient u
c flow velocity at rudder
Ct = slope of reverse spiral at 5 = O V D = tactical diameter
h water depth V
= moment of inertia with respect AV
to 2-axis X
k = nadder efficiency coefficient Xa L = ship length
Xa
1/ = point of application of yaw damping force
w = point of application of sway damping force
m = ship mass Y
N = yaw moment
VB )3
n propeller revolutions per minute VB )3
Q
=
propeller torque 7R. = steady turning radius 8
r yaw rate = i/^ '8
S|! = smallest root i n stability equa- 8*
tion Sc
T = ship draft
T = thrust
i = time
= time to second execute—zigzag test
= time to maximum overshoot—• zigzag test
= first half period—zigzag test = component of ship speed i n
x-direction
•• component of ship .speed i n ij-direotion
• translatory velocity vector mean value of speed loss force i n x-direetion
distance to interception with new course-course change maneuver
distance of ship center of grav-i t y from orgrav-iggrav-in of coordgrav-inate system
force in y-direction distance from canal bank d r i f t angle
yaw gain constant rudder angle
rms value of rudder angle ordered rudder angle counterrudder
water depth parameter [ f = T/(h - T)]
bank clearance parameter
(.V = L/yB)
engine setting
cr = yaw rate gain constant
\p = heading angle of ship
(J-o = desired oourse
^0 = overshoot angle in course change maneuver
= overshoot angle i n zigzag test = rms value of course deviation
\p = yaw rate
tp = rnis value of yaw rate
w = angular velocity vector V = ship volume of displacement
N'u,} = hydrodynamic coefficients etc.
= first derivative w i t h respect to time
= second derivative w i t h respect to time
S u b s c r i p t s
hydro = hydrodynamic forces and moments
dist = outer disturbances on forces and moments
h„ = hydrodynamics forces and
moments i n deep water
h conf = confinement effects on
hydro-dynamic forces and mo-ments
as coefficients f o r the X-, Y- and iV-equatioiis. These data were recalculated at SSPA i n order to get a second-order f i t . Coefficients for the quadratic fit are given i n Appendix 1. The hydrodynamic coefficients and their derivation w i l l not be f u r t h e r discussed here, as the subject is not w i t h i n the
scope of this paper.
Disturbance efTecfs
The basic mathematical model describes the ship's motions in calm deep water, i.e.
7n.{u - rv - Xg r^) = X,, „(w, i;, i/-, T, c, 8) etc.
The effects of shallow water and other confinements i n the waterway, and the effects of current, wind, waves, etc., can be accounted f o r b y introduchig additional terms i n the equations. Thus these effects are all regarded as distur-bancas superposed on the calm, deep-Avater case. This manner of introducing shallow water, wave.?, etc. has jjroved to be very efficient and provides a basis for a systematic development of such effects. Confinedevelopment effects w i l l directly h i -flueuce the hydrodynamic forces (and moments), w'hich im])lies t h a t hydrodynamic coefficients (derivatives) are changed. On the other hand, wind, waves, etc. will intro-duce outer forces acting on the ship.
The effects of confinements i n the waterway can be calcu-lated theoretically only for some simple ship forms. Thus i n most cases experimental values and empirical formulas must be used; f o r a survej' see N o r r b i n [1] and others. D a t a used in this report are largely based on a series of .shallow-water and canal tests on the P M M at H y A .
The effect of wind can be directly calculated if tlie forces and moments due to wind are known. Such data may be obtained f r o m wind tunnel tests.
The effects of current can be calculated. I n this context i t should be remembered t h a t constant current does not i n principle introduce any siguificant maneuvering problems. On the other hand, changing currents (especially crass cur-rents) have a considerable influence on the maueuveriug of large tankers. When simulatuig changing cross currents we have used simplified situations as a stepwise changing current or a linearly changing current.
Simulating the influence of waves on large tankers at present imposes great difficulties. The problem can be split into two parts, followhig the usual approach when calculating the influence of waves. One part is the frequencj' response, i.e., the side force and yawing moment as functions of the frequency of the wave. Thi.s subject has been treated b y Salvesen et al. [ 3 ] . The second part is the sea spectrum, which gives the distribution of the wave amplitudes as functions of the frequencies. Extensive research has been carried out on sea spectra [4] and a number of mathematical appro.Kimations have been proposed. There is, however, one severe drawback to most models; t h a t is, the description of the low-frequency range, which is very uncertain. On the other hand, this part of the spectrum will be the most i m -portant part for large tankers (say above 100,000 d w t ) .
A t SSPA two ways of simulating the influence of waves have been used. The first method uses a pseudoratidom noise sjiectrum f o r the yawing moment. This spectrum is essentially a white noise, b u t single low-pass filtered w i t h
T = 125 sec. The second method uses the common
super-position principle f o r calculating the ship's response to wave loads. I t must be recognized, however, t h a t data on the side force and yaw moment respon.se to regular waves are scarce, whether theoretically calculated or experimentally deter-mined. General results f r o m the simulator investigations
indicate t h a t the influence of waves on large tankers is rather small. However, i t can be concluded that more research i n this field is needed.
S S P A steering and maneuvering simulator The SSPA Steering and Maneuvering Simulator consists ot an analog computer, recording and display devices, and a bridge mock-up. The analog computer [EM 680) solves the etiuations of motion and generates signals f o r the recording and display devices.
Recording devices include a multichannel recorder and
x-y plotters. On the multichannel recorder, quantities
such as course angle, yaw rate, rudder angle, and vessel speed can be recorded as functions of time. The x-y plot is used to record the ship's track.
Display devices consist of oscilloscopes, which are used to display the ship's track. On the oscilloscope picture of the ship's track a transparent copy of a sea chart can be applied and thus this picture will be rather like the picture on a radar PPI(plane position indicator)-screen.
The bridge mockup consists of a steering stand and an i n -strument jianel showing vessel's speed, course, yaw rate, pro-peller r p m , applied rudder angle and actual rudder angle. On a T V screen i n f r o n t of the helmsman a simpMfied picture of the surroundings as seen f r o m the bridge is transmitted f r o m an oscilloscope. This picture is angularly true and thus Umited to about ± 10 deg f r o m the ship centerline. ( I t may be added t h a t the SSPA simulator now includes five T V screens, which has extended the range of sight to ± 5 0 deg f r o m the ship centerline.)
General procedure for simulator investigations
A t SSPA several simulator investigations of the maneuver-ing abilities of large tankers have been carried out. I n most cases these studies have been performed using a standard procedure a-s follows:
(o) Calculation of coefficients i n the mathematical model. (&) Adapting the equations for the analog computer (i.e., scahng etc.).
(c) the "phasing-in" stage.
(d) predictions of standard maneuvers.
(e) special tests.
I t seems appropriate here to include some discussion of the so-called "phasing-in" stage. When the coefficients i n the equations of motion have been calculated [stage (a) above], the equations are rescaled t o fit the analog computer [stage (6)]. Care is taken to obtain maximum accuracy f o r all the variables concerned. The phasing-in stage consists of calcu-lations h i the analog computer of some standard maneuver te.sts and comparison of these results w i t h values f r o m f u l l -scale trials or expected values. One d i f f i c u l t y is t h a t most sea trial data have been obtained f o r approach speed of about 16 knots, t h a t is, for full-speed ahead. For analysis of the maneuvering abilities i t is of great interest to have f u l l -scale data for lower speeds, which are used when maneuvering i n t o ports, etc. A n additional d i f f i c u l t y occurs because f u l l -scale maneuvering trials t3'picall}' consist of turning circles at maximum rudder angle, a crash stop, and occasionally a zig-zag and spiral test, a l l usually conducted i n the f u l l y loaded condition only. W i t h the po.ssible exception of the zigzag test, these maneuvers provide l i t t l e i n f o r m a t i o n concerning the transient, rather than the steady-state, behavior of the ship. As will be discussed i n tlie following, f o r the large tankers considered i n this paper i t is the transient conditions t h a t are of most significance.
I n the phasing-iu stage the ability of the simidated ship to
perform a l l of the maneuvers required i n the simulation is checked. This includes extreme maneuvers such as respond-ing to rudder movements f r o m hard-over to hard-over.
T a n k e r simulation study
The first study performed at SSPA f o r Esso was begun i n the autumn of 1969 and is referred to herein as the "Tanker Simulation Study." The purpose of this study was t o evaluate the maneuvering characteristics of a proposed new large-tanker design. A t t h a t time i t was felt t h a t there were no adequate criteria available for evaluating the maneu-verability of a new design, and so the evaluation was made by comparing the performance of the new design w i t h t h a t of an existing ship. I n this case one of Esso's 190,000-dwt ships was chosen as a basis of comparison, for the following reasons:
• B y the end of 1969 five ships of this class were i n opera-tion, and their maneuvering characteristics were well docu-mented by full-scale trials. I n addition, the overall handfing of these ships was generally considered to be excellent b y their masters.
• A considerable amount of model data and maneuvering predictions based on these data existed.
I t was believed t h a t i f the maneuvering characteristics of the proposed new design were similar to t h a t of the 190,000-dwt ship the new shi]) would be acceptable.
I n this section the results f o r the new design are not pre-sented. Instead, the simulator results for the 190,000-dwt ship are treated and comparisons w i t h full-scale t r i a l results are presented.
To perform an evaluation of this nature, i t was necessary to use standard, repeatable maneuvers as the basis for com-parison. T o be meaningful, these maneuvers have to be representative of the full-scale maneuvering situations t h a t the ships would encounter i n actual operation. A t the time the study was undertaken there were three maneuvers i n general use f o r this ])urpose; namely
• The turning circle, w i t h rudder hard-over. • The zigzag test.
• The spiral test.
I n addition to these "standard" maneuvers, other simula-tions were made, including coursekeeping i n a seaway and characteristic port-approach maneuvers. The port-approach maneuvers included shallow-water effects.
The mathematical model
As discussed previously, the 190,000-dwt ship was tested extensively on the P M M at H y A . These tests included the standard deep-water tests i n full-load and ballast condition as well a.s shallow water (three depths) and a limited series i n a canal. The H y A data were recalculated to fit the SSPA mathematical model. I n the phasing-in stage, not only t r i a l data f o r the 190,000-dwt tankers were taken i n t o consider-ation b u t also results f r o m sea trials w i t h other large tankers of about 200,000 d w t . As the P M M equipment used f o r these tests was limited to relatively small d r i f t angles and yaw rates, the nonlinear coefficients should be considered critically. I t must be remembered t h a t prediction of the
parts of the spiral test t h a t (because of these nonlinearities)
contain large rudder angles or the steadj^-statc portion of a turning circle w i t h large rudder angles is somewhat uncertain. This fact w-as not considered to be a seriou.s drawback to the investigation as i t was mainly concerned w i t h the course-keeping abihties i n a disturbed sea, rather small course changes, etc., which .sliould not introduce large nonlinearities.
I t should be noted t h a t H y A has recently installed a new
1 1 1 [ 1 I I I 1 1 I 0 1 2 3 4 5 6 7 8 9 10 Kmin) Fig. 2 190,000-dwt t a n l < e r — s i m u l a t e d s t e p r e s p o n s e for r u d d e r a n g l e 5 = 5 d e g T a b l e 1 S t e p r e s p o n s e — r u d d e r angle = 5 d e g T I M E A F T E R E X E C U T E C O U R S E C H A N G E Y A W RATE
O F R U D D E R , sec A\p, deg 4', deg/sec
0 0 0 30 1.0 0 . 0 6 60 3 . 0 0.105 90 6 . 6 0 . 1 4 5 120 1 2 . 0 0 . 1 8 5 150 1 8 . 0 0 . 2 2 0 180 2 4 . 0 0 . 2 5 0
planar-motion mechanism which is intended to provide the capability of investigating significantly larger d r i f t angles and yaw rates.
Prediction of standard maneuvers in deep water
The simulations included the following standard maneuvers: D i e u d o n n é spiral
Zigzag test T u r n i n g circle
A i i additional test, the step response, was also simulated. This maneuver is basically similar t o the turning circle except j ' a w rate, d r i f t angle, and heading change are recorded i n addition to advance, transfer, etc.
The spiral, zigzag, and turning circle tests are well known and have been described extensively; see f o r example [ 5 ] . I t must, however, be recognized t h a t these tests, as they are usually carried out, essentially give the ship's response i n the steady state and not i n the transient phase, For a large tanker most maneuvering w i l l be contained i n the transient phases. This can be demonstrated b y an example. I t is assumed t h a t a tanker of 200,000 d w t is steaming at 16 knots and a course change of 30 deg is ordered. I f this maneuver is carried out at sea, where no special restrictions have to be observed, i t is most l i k e l y t h a t this course change w i l l be carried out using rather small rudder angles. Thus f o r a course change 30 deg to starboard a rudder angle of 5 deg to starboard is ordered. Assuming f o r simplicity calm weather, etc., the ship's response w i l l be as i n F i g . 2, which is the step-response curve f o r rudder angle 5 deg. Results f r o m this figure are given i n Table 1.
0.7 0.6 0.5 0.4 0.3 0.2 0.1 t=IO0sec t=200sec t-»00
V
///
—/ /— 10 15 20 25 30 8° 35 Fig. 3 190,000-dwt t a n k e r — y a w r a t e a f t e r f sec f r o m r u d d e r e x e c u t e Fig. 4 Z i g z a g t e s t : f i r s t o v e r s t i o o t a n g l e ; to t i m e t o s e c o n d e x e c u t e ; f, t i m e t o m a x i m u m o v e r s i i o o t ; hi f i r s t iialf periodThus i t can be seen the i n i t i a l motion i,s tiuite slow. I t can also be seen i n Fig. 2 that even after ten iniiuites, steady-state conditions have not been reached since speed and yaw rate are both decreasing.
The spiral test (both the D i e u d o n n é and the reverse [6]) is intended to give the relationship between yaw rate and rudder angle i n the steady state. F r o m Table 1 and F i g . 2 i t c m be seen t h a t a large tanker w i l l rarely operate at steady state. Thus, i t becomes impoi'tant to consider whether the sjiiral test is i n a n j ' way representative of the handling of a large tanker. I n F i g . 3 the steady-state yaw response has been comjiared w i t h the yaw rates for 100 and 200 sec after the ruddei- is put over to 5 deg. F r o m this figui'e i t is clear t h a t in the transient phases a very d i f ï e r e n t jncture of the yaw res])onse is obtained compared w i t h the s))iral test as repre-sented b y the curve of i ->- co. I t may also be pointed out that, although the ship is dynamically uii.stable, no auomalous behavior will be obtained for small rudder angles; i.e., a starboard rudder w i l l induce a motion to starboard, etc.
30 20 to 3 5 %ec 300 200 KO
\
O 1 2 3 ^ i '/sec Fig. 5 190,000-dwt t a n k e r — e f f e c t s of r u d d e r t u r n i n g rate o n results f r o m 20/20 d e g z i g z a g t e s tThus i t appears t h a t the D i e u d o n n é spiral does not jn'ovide significant information as to the maneuvering of large tankers. I n general the same is true of the reverse spiral [6] w i t h the exception t h a t the slope of the ^-5 curve through the origin has been u.sed for the determining of autopilot constants.
There are also practical disadvantages to the use of the spiral test. Because of the ver^- long period of time required for a large tanker to i-each steadj'-state conditions, the spiral test requires several hours to conduct properly. I f , as a result of the time pressures which always exist during ship trials, the turning rate is recorded before steady-state condi-tions are reached, the results of the test can be significantly affected. While the technique used i n the reverse spiral test reduces the amount of time required to record t u r n i n g rate at each rudder angle, the test s t i l l requires a considerable amount of time. For example, during the trials of the
Esso Bernicia [7] five reverse spirals were conducted.
These took f r o m 66 m i u to 2 hr and 24 m i u to conduct. The zigzag test is used to investigate both the a b i h t y of a ship to initiate a t u r n (as do the turning circle and step resjionse) and also to check a t u r n . Figure 4 defines the v a r i -ables which are ordinarily recorded and on which the analysis is based. These include overshoot angle, A ^ , and various measures of time response, ;o, li, k- The analysis of zigzag tests has been treated extensively i n [ 8 ] and others, and i t is possible to extract information on the ship's hydrodynamic coefficients. I t must be remembered, however, t h a t the results of a zigzag test, particularly overshoot angle, A ^ , are to some extent de]3endent on the rudder t u r n i n g rate, as
T a b l e 2 T u r n i n g - c i r c l e tests—190,000-dwt tanl<er S T E A D Y T U R N I N G R U D D K R A D V A N C E T A C T I C A L R A D I U S A N G L K (w) DiA (m) {m) C O M M E N T S 20° p t 1260 1260 460 Simulator 20° sthd 1290 1300 450 Simulator 35° p t 1010 990 300 Simulator 35° p t C1245 938 352 Full scale—Esso 35° p t Norway (1130 1020 422 Full scale—Esso Malaysia 35° stbd 1020 1010 300 Simulator n263 981 340 Full scale—Esso 35° stbd Norway 1.1060 950 422 Full scale—i^sso Malaysia
shown i n F i g . 5; see also [ 9 ] . When comparing sea t r i a l )-esults w i t h simulations etc., there is i n many cases some uncertainty as to the accuracy of the indication of rudder angles f o r the ship. Errors of the order of several degrees can occur i f special calibration procedures are not adopted. I t also appears t h a t overshoot angle is quite dependent on wind conditions. A comparison of the results of the simu-lated zigzag tests and sea t r i a l re.sults for the Esso Bernicia is presented i n F i g . 6. The results appear to agree reasonably well. A f u r t h e r discussion of the zigzag test ap]3eai\s later i n this paper.
Results f r o m simulations of turning-circle tests should be treated criticallj^, at any rate for large rudder angles where nonlinearities w i l l be predominant. I t must also be realized that turning circles carried out at sea are under the influence of wind, etc. Simidator results are comjiared i n Table 2 w i t h some sea t i i a l results for two identical 190,000-dwt tankers, the Esso Norway and the Esso Malaysia.
Correlation between the simulator and full-scale t r i a l results is not as good as for the zigzag test, b u t nevertheless i t appeared satisfactory for the purj30ses of this project. I t is interesting to note the relatively large differences t h a t occur i n the full-scale t r i a l results of two identical ships. The step response test was simulated for several rudder angles and some results are given i n Figs. 2 and 7.
t (min) Fig. 7 190,000-dwt t a n k e r — s i m u l a t e d s t e p r e s p o n s e f o r r u d d e r
angle 3 = 20 d e g
Coursekeeping abilities in a seaway disturbance
Investigations of the coursekeeping abilities i n a seaway disturbance were carried out using both autopilot and manual steering. Several of Esso's e.xperienced shipmasters partici-jiated i n the manual steering as well as SSPA personnel. The seaway was simulated as a distui'bance i n the yawing moment, which can be described as a pseudorandom noise. The main features were t h a t the noise spectrum was cut off at a frequency equal to 0.05 rad/sec and t h a t i t had a period of about ten minutes. (The period is obtained f r o m the genera-tion process of the pseudorandom noise.) Thus, this seaway distui-bance contains no high frequencies, although an ordinary sea spectrum w i l l i n general contain frequencies up to 0.8-1.0 rad/sec.
When the ship was i n autopilot control a sinijilified auto-pilot equation was used
2h -2
/
n manual sfscrinqf
creasir 9(
A I I I \ 1 1 1 O 2 < G 8 to 12Fig. 8 190,000-dwt tanl<er—rms v a l u e s of course d e v i a t i o n in s i m u l a t i o n s of c o u r s e k e e p i n g t e s t s o.i r oos O ->ing I I I 1 1 1 1 O 2 A 6 3 10 12 1'
Fig. 9 190,000dwt t a n k e r — r m s v a l u e s o f yaw rate in s i m u l a -t i o n s o f c o u r s e k e e p i n g -t e s -t s
Ï - 2
i increi
Q I \—: 1 1 1 1 1
O 2 -< s a lO 12
Fig. 10 190,000dwt t a n k e r — m e a n value o f s p e e d loss in s i m u -l a t i o n s o f c o u r s e k e e p i n g t e s t s
S* = y{ip - ^{.„) + a^P
S* = rudder angle ordered b y autopilot
^0 = desired course
7 , 0- = constants of proportionality
The rudder is assumed to turn at a constant rate of 2.33 deg/ sec and there is a time lag i n the steering engine of 0.4 sec.
The simulations have shown t h a t the control system has a predominant influence on the ship's coursekeeping behavior. Thus, b y varying the constants 7 and <r appreciable differ-ences can be obtained. I n order to analyze the results f r o m the coursekeeping tests, root-meati-square (rms) values of course deviation ( A ^ ) , j ' a w rate (ip) and rudder angle (5) w^ere calculated as well as mean value of speed loss A F . These calculated values of A ^ , \p and A F have been plotted agauLst S w i t h the constants of proportionality ( 7 and cr) as parameters; see Figs. 8, 9, and 10. Results f r o m the autopilot steering tests show quite clearly that i t is possible to minimize the ship's course deviation w i t h o u t using excep-tionally large rudder angles. Thus, there exists some k i n d of
X Yow rote indicalor operotive 0 Yow rote indicator inoperative
) ^— 5/ 3(si. n i I \ I 1 1 1 1 1 1 1 "1 2 3 4 5 6 7 8 9 I0^„ II
Fig. 11 Effects o f yaw r a t e i n d i c a t o r o n c o u r s e k e e p i n g s i m u l a -t i o n s
optimum condition which can be defined by properly weight-ing course deviation, rudder angle, and speed loss. This problem of defining the oijtunuin conditions is not as sim])le as finding minimum speed loss, because this condition i n general does not imply m i n i m u m rudder motions.
I t is interesting to compare the autopilot steering tests w i t h the manual steering tests. I n these tests the bridge mock-up described earlier was used. For the coursekeeping tests the ship's forecastle and the horizon line were presented together w i t h three dots fixed i n space which were intended to provide the helmsman w i t h an indication of when the ship yawed. The gyrocompass could also be used f o r this purpose. For a number of runs a yaw rate (rate of turn) indicator was also provided. Each test r u n lasted 30 m i n w i t h the same seaway disturbance used as for the autopilot tests. The results f r o m the manual steering tests were analyzed i n the same way as the autopilot tests; see F i g . 8.
I t is interesting to note t h a t the manual steering tests show i n general greater values of rudder angle S and smaller values of course deviation A ^ than the autopilot steering. This indicates t h a t the helmsmen are using large values of gain on course deviation, 7 , and/or y a w rate, o-. The large amount of scatter i n the manual steering tests is not unex-jjected. N o t only were there significant differences i n the steering techniques used b y the individual helmsmen, there was also a significant "learning curve" effect apparent i n the different runs made by one helmsman. These factors a l l tend to obscure the test results and make analysis very difficult. I n general i t was found t h a t a manual steering test only re-flects a particular helmsman's ability rather than the ma-neuvering characteristics of the .ship. The conclusion f r o m this is t h a t i f comparisons of the relative maneuvering char-acteristics of several ships are t o be made, autopilot steering will provide more significant results. M a n u a l steering .should i n principle be used only for training purposes. The present practice at SSPA is t o use only autopilot steering when i n -vestigating the maneuvering qualities of ships. Another significant advantage to using autopilot control i n studies of this k i n d is t h a t tests can be run on the computer 100 times as fast as real time. This is obviously not possible w i t h manual steering.
A note of caution is needed on the exclusive use of auto-matic control for evaluating ship designs. I n actual opera-tion, while autopilot control w i l l be used for most open sea steaming, maneuvering i n port approaches, restricted w^aters, etc. is ordinarily done under the control of a helmsman and the master or pilot. For this reason, even though automatic
control is used to simulate such maneuvers, i t is important that the helmsman be able to iierform the maneuvers. I n the test program discussed i n the paper, i t was found t h a t the helmsmen could respond w i t h constants of proportionality similar to those used i n the autopilot equations which pro-vided satisfactory performance for the various designs con-sidered. However, i f the automatically controlled simula-tions of a particular ship design indicated t h a t very unusual constants of proportionality were required to maintain ade-quate control, this could indicate t h a t a helmsman might have difficulty w i t h such a ship. I n this case, i n spite of the d i f f i -culties involved, use of manually controlled simulations would be necessary to provide a complete picture of the over-all controllability of t h a t design.
As mentioned previously, during the manual runs a yaw rate indicator was provided on some of the runs for each helmsman. As shown i n Fig. 11, the presence of this indicator has a significant effect on results. The coimected points indicate cases where t w o runs were made on one shij) w i t h one helmsman, where f o r one run the helmsman was provided w i t h a measure of yaw rate. On the other r u n the yaw rate indicator was inoperative. I n all cases e.xcept one, presence of the yaw I'ate indicator has imjjroved coursekeeping ability by reducing both course deviation and rudder motion. I n the one excejition, course deviation was reduced b u t w i t h some increase i n rudder motion.
Prediction of standard maneuvers in shallow water
Predictions of the spiral test, the zigzag test, the step-response test, and the turning-circle test have been calculated for a nominal approach speed of 8 knots. Thus, i t is assumed that the engine settings are for 8 knots i n deep water; i n shallow water a .speed reduction is obtained due to the greater resistance. This speed loss has been jilotted i n F i g . 12 together with full-scale results for the 210,000-dwt tanker
Magdala [10], A few results are given for an actual approach
speed of 6 knots i n order to compare the SSPA predictions with the earlier H y A predictions.
Results of the spiral tests are shown i n F i g . 13. As the water depth decreases the vessel will become more stable and the steady-state turning ability decreases for small rudder angles. H y A and SSPA predictions are compared i n F i g . 14. The agreement is quite good, although H y A coefficients pre-dict t h a t the vessel is stable for h/T = 2.0 and SSPA values
IO MAGDALA " -Esso I90 000 DWT SIMULA no N . 1 , 20 35 JO 35 ^ t,[m]
Fig. 12 E f f e c t of shallow water on s h i p ' s s p e e d
give a slightly unstable ship. This difference is also re-flected in the diagram of "dynamic stability lever," I'r - l \ , which difference directly indicates i f a ship is stable on course or not. Thus
Vr - I'v > 0 f o r stable ships I'r — I' V < 0 for unstable ships
where N" " ~ V " •r uc x"o - N",r 1 Y"
I n F i g . 15, values of I'r - I'„ are given as calculated f r o m H y A coefficients, SSPA coefficients, and, f o r compar-ison, results f r o m F u j i n o [11]. Although the SSPA co-efficients are calculated f r o m the same test data as the H y A coefficients, the differences are due to the curve-fitting using the shallow-water parameter ^ = T/{h — T), &Q the H y A test data are not exactly used.
Predictions of zigzag tests are shown i n F i g . 16. The trends here are t h a t for decreasing water depths overshoot angle decreases, time to second rudder execute increases, as
* C/s) 0 2 Oli -0.1; 0.2 ' A h 1 h/T=00 =2.0 =1.63 =1.2 h/T=00 =2.0 =1.63 =1.2 i s \ \ \ \
\
V ""^ V (knots) 8 H6 10 5 0 - 5 -10 -15 - 2 0 - 2 5 - 3 0 -35 8' Fig. 13 190,000-dwt t a n k e r — s i m u l a t o r r e s u l t s o f spiral t e s t swell as time t o check the i n i t i a l motion and first half period. The increases i n time are, however, rather small for d r a f t t o -depth ratios larger than about 1.50.
Results of turning-circle tests are shown i n F i g . 17. The effects of shallow water are small for draft-to-depth ratios greater t h a n about 1.50. For still smaller values of hjT advance, tactical diameter and turning radius increase quite substantially for rudder angle S = 20 deg. For rudder angle 5 = 35 deg there seems to be only a small influence of water depth. These conclusions seem to be verified by full-scale trials; compare f o r instance the Magdala tests [12]. I t should be remembered, however, t h a t there is considerable uncertainty i n these predictions, as they are heavily de-Ijendent on the nonlinear hydrodynamic coefficients.
The step-response tests are shown i n Figs. 18, 19, and 20 for three rudder angles (3 = 5 deg, 20 deg, 35 deg) and for three water depths. Only y a w rate and ship's .speed ( F ) are given. I t can be seen t h a t decreasing water dejith re.sults i n a more sluggish response of the ship. I t is also interesting to note the influence on the ship's speed that is obtained during a t u r n i n shallow water. As water depth decreases, the speed loss (compared w i t h actual approach speed) decreases. This is very pronounced for the case of h/T = 1.2, where the ship's .speed at times greater t h a n 400 sec after rudder e.xecute is greater at h/T = 1.2 t h a n i n deeper water. This apjiarently occurs because of the ver}' low yaw rates and d r i f t angles (/3) t h a t are obtained at h/T = 1.2. W i t h regard to the yaw rate i t can be concluded t h a t i n shallow water larger rudder angles are necessary to obtain the same t u r n i n g rate as i n deep Waaler. This is i n accordance w i t h full-scale exiierience f r o m maneuvering large tankers i n shallow water, I t must, however, also be borne i n mind t h a t these eft-ects are quite
o l 1 1 ' > 1 O / 2 3 ^ 1 , I I 1 1 1 1 C O 2 IS 133 125 h/T Fig. 17 190,000-dwt tanl<er—simulator r e s u l t s o f t u r n i n g - c i r c l e t e s t s
small, except for very shallow water; t h a t is depth-to-draft ratios of the order of 1.2 and less.
I t was n o t possible t o conduct a detailed comparison of these simulations wdth full-scale shallow-water t r i a l data, which are limited for large tankers. A t this time i t ap-pears only t h a t the simulations are representative of the trends encountered i n full-scale tests. Further analysis is needed i n this area.
Conclusions from (he tanker simulation study
Based on the Tanker Simulation Study, the following con-clusions can be d r a w n :
1. Computer-generated simulations based on large cap-tive-model test data appear sufficiently representative of the full-scale maneuvering of large tankers to permit the use of such simulations as a method for evaluating the maneuver-a b i l i t y of these shi})s.
2. The three standard maneuvering tests, namely the spiral, zigzag, and turning-circle tests, appear to be of limited value i n estabhshing the maneuvering qualities of large tankers. This is particularly true of the spiral test, which represents stead.y-state conditions while large tankers rarely, if ever, reach steadj'-state conditions while maneuvering.
3. The step-response test appears to be of more value
0.4 0.3 0.2 0.1 V h/ T = 00 = 2.0 = 1.2 V(knots) e O 100 200 300 400 500 600 700 800 t(sec) Fig. 18 190,000-dwt t a n k e r — s i m u l a t o r r e s u l t s o f s t e p - r e s p o n s e t e s t s , 5 5 d e g (Vs) 0.4 r 0.3 0.2 0.1 01-h/T=00 =2.0 =1.2 / / II / T / — ll*
/ / '
^^'^^^ V (knols) 8 O 100 200 300 400 500 600 700 8 0 0 t (sec) Fig. 19 190,000-dwt t a n k e r — s i m u l a t o r r e s u l t s of s t e p - r e s p o n s e t e s t s , 5 = 20 d e g * ( ° / s e c ) 0.4 r 0.3 0.2 0.1 V h/T=00 =2.0 =1.2•
V (knots) 8 O 100 2 0 0 300 400 500 600 700 8 0 0 t(sec)Fig. 20 190,QOO-dwt t a n k e r — s i m u l a t o r results of s t e p - r e s p o n s e t e s t s , 5 35 d e g
since i t yields information i n the transient phases during which most large-tanker maneuvering will take place. B y conducting this test at several rudder angles (say 5 deg and 20 deg as well as hard-over), much of the range of possible maneuvering can be studied.
4. I n evaluating the coursekeeping of large tankers i t must be remembered t h a t the control system used, whether automatic or manual, has a predominant influence on the coursekeeping abilities of these shi])s. Thus i t is important t h a t the control system (s) used be properly suited to the particular ship under consideration and also t h a t the proper information be available as an input to the system. The importance of a yaw rate i n p u t has been shown f o r both automatic and manual control.
5. Based on the hmited data available, i t appears t h a t shallow water will i n general result i n a more sluggish re-sponse of the ship as water depth deci-eases.
Steering criteria study Principal objectives
Results f r o m the "Tanker Sunulation S t u d y " had indicated t h a t the usual standard maneuvering tests were not sufficient to establish f u l l y the maneuvering qualities of large tankers. Thus, i t was thought that other tests m i g h t provide better criteria. Secondly, the influence on the maneuvering abilities of ship size, geometrical proportions, and rudder size is not known very well, and accordingly some investigations i n these areas seemed appropriate. Finally, the permissible amount of dj'iiamic instabihty ou course has been much discussed i n the past. Thus, some investigations i n t o this matter were considered to be desirable.
I t was not expected, however, t h a t any final answ-ers t o these problems would be obtained b u t rather t h a t provisional guidehnes for the design of large tankers and maneuvering
T a b l e 3 Principal d i m e n s i o n s — b a s e 200,000-dwt t a n k e r
Lengtli between perpendiculars (L), m 305
Beam (B), m 4 7 . 5 D r a f t to design waterline, (T), m 1 8 . 5 Displacement ( v ) , cu m 225,000 L/B 6 . 4 B/T 2 . 5 7 Block coefficient (Cj,) 0 . 8 4
Ship speed, knots 16
Propeller rpm 80
tests f o r these vessels would be indicated. This project was begun i n the summer of 1970 and is referred to as the "Steeruig C r i t e r i a " project.
Program
The Steering Criteria investigations concerned both different tanker configurations and different maneuvering tests. I n order to facihtate comparisons a base ship was constructed which had conventional proportions for a 200,000-d w t tanker as given i n Table 3.
This ship was almost identical to the Esso 190,000-dwt tanker investigated previously, e.xcept that = 0.84 instead of 0.83. Using this basic vessel, variations were made of ship's size (geosim series), length-to-beam ratio (L/B-series), rudder size (rudder-size series), and ship's dynamic stabilitj^ on course ( " s t a b i l i t y " series); see Appendix 2. Only the deep-water case was considered for a l l the ships, as i t was assumed that effects of confinements would not i n principle irffiuence the comparisons between the ships. For the reasons discussed previously, only autopilot steering was used i n the coursekeeping and course change maneuvers. T h e simula-tions consisted of a series of maneuvers as follows:
• Standard maneuvering tests; i.e., spiral, zigzag,' t u r n i n g . • Circle and step-response tests.
• Coursekeeping i n a disturbed sea. • Course change maneuver. • A revised zigzag test.
Mathematical model and hydrodynamic coefficients
The mathematical model has been described above. For the base 200,000-dwt ship the same hydrodynamic coefficients as f o r the Esso 190,000-dwt tanker were used, although some minor modifications were included.
For the geosim series, propeller size and propeller revolution rate were chosen so t h a t reahstic values were obtained. For all the ships i n t h i s series f u l l speed was equal t o 16 knots.
The coefficients for the ships i n the L / B series were obtained f r o m interpolation and extrapolation f r o m available P M M tests. Comparisons of the hnear " s t a b i l i t y " derivatives w i t h Norrbin's empirical relations [1] were also made; see Appen-dix 2. The rudder size i n this series was kept constant. For all the ships i n this series i t was assumed t h a t they were propelled b y a steam turbine giving a speed of 16 knots at 80 r p m .
For the rudder-size series ships the hydrodynamic coef-ficients were calculated f r o m P M M tests carried out a t H y A for Esso conceruing different rudder shapes and sizes. I t can be noted t h a t the stability derivatives do not alter signfficantly when rudder size varies quite appreciably; see Appendix 2.
T h e " s t a b i h t y " series consists of three ships w i t h the dynamic stability on course varying f r o m v e r y unstable to marginally stable. This v a r i a t i o n was achieved b y v a r y i n g the stability derivatives F"„„, N"uvi ^'ur, N'uri see Appendix 2. The remaining hydrodynamic coefficients were kept con-stant. Loop width &0 AO -20 O -Loop height c^^r/'-«ir-• Basic ship • L/B series K "Sfabilify" series Q2 O.I o>-• X 3
- O P -O.IS -o.i -oos O
I'r-K
Fig. 21 Loop w i d t h a n d l o o p h e i g h t i n D i e u d o n n é s p i r a l
Predictions of standard maneuvers
The spiral test indicates the stabiHty on course, and results f r o m such tests can be characterized b y the loop w i d t h and loop height. Results have been compiled i n F i g . 21 as
t[mln] -B First overshoot • Basic ship a L/B series
A /judder size series y 'Stability" series
-02 -CIS —OJ -O.OS O C- (Y
Fig. 22 R e s u l t s f r o m s i m u l a t i o n s of 20/20 d e g z i g z a g t e s t s
functions of the dynamic stability lever 1% — I'v Thus, loop widths and heights are correlated to I'r — I'v, although the nonlinear coefKcients have some influence on these quantities.
Results f r o m zigzag tests have been comjuled i n F i g . 22. I t is obvious t h a t there is some correlation between the dy-namic stability as characterized b y the value 1', — I'v and the results of the zigzag test. Thus, as stabihty decreases, overshoot angle increases, but the effects on time to second execute, etc. are small. This normally m i g h t be interpreted to i m p l y that the controllability is degraded when stability decreases, flowever, the following point should be noted. Figure 23 presents, for the L/B series, the yaw rate (i/») achieved at time = <o versus 7v/B ratio (that is, time to second execute). The ships w i t h lower L/B ratio (which have less stability) are tiu'uiug significantly faster a t this time because of their better initial tiu'uing ability (w^hich is also characterized b y their slightly lower values of k; see Fig. 22). As a result, the i n i t i a l conditions for the course-checking phase of the zigzag maneuver are not the same for the various ships i n the L/B series. The fact that the ships with low L/B can check a t u r n of significantly higher yaw rate i n approximately the same time (ti — k) as the narrower ships would apjiear to be an indication of imjiroved controllabilitj' rather than the opposite, noted above. The higher overshoot angles for the lower L/B ships is a direct consequence of their turning more rapidly at the beginning of the course-checking phase of the zigzag test (that is, at time = ^o). I t should be noted also t h a t variations i n rudder force (and yaw-ing moment) of ± 2 0 percent f r o m the basic ship give about the same variations i n k, k — k, and ii as rather sub-stantial variations of dynamic stability; see F i g . 22.
As mentioned before, results f r o m turning-circle tests are highly dependent on the nonlinear hydrodynamic coefficients. These coefficients are rather uncertain, as they have been calculated f r o m P M M tests which did not cover large values of d r i f t angle and yaw rate. Thus, data f r o m the t u r n i n g -circle simulations should be treated w i t h some care.
I n F i g . 24 there is some indication of correlations between steady turiiing radius (R) and advance ( A ) , and the dynamic stability on course. Other factors, however, are s t i l l more important. Thus, rudder size (or rather rudder force and yawing moment) has a large effect on B and A. O f t e n the characteristic values f r o m the t u r n i n g test are made aondimensioiial b y dividing b y the ship's length L. I n Fig. 24, R/L and A /L are given. I t should be observed how
0.41-I 0.41-I s 1 1 1
JO 4 0 50 60 7.0 ao L/B
Fig. 23 L / B series—yaw rate i/-,, a t s e c o n d e x e c u t e
this changes the picture, especiallj' for the L/B series. As
L/B decreases, the values of R and A also decrease, b u t the
values of R/L and A/L increase. Thus, although t u r n i n g characteristics are improved as L/B decreases, the opposite effect may appear if nondimensional values of advance, etc. are considered. This certainly calls f o r some caution when non-dimensional values are presented and analyzed. Effects of fshij) size on the results of tactical diameter (D) and advance ( A ) are shown i n Fig. 25 for the simulations and for a number of full-scale tests. I t should be noted t h a t while the simulations are for a geosim series the full-scale t r i a l data are for tankers of somewhat varying proportions, rudder area ratios, etc. I t is also interesting to note the spread i n t r i a l results that e.xists between the three identical 190,000-dwt tankers.
The results f r o m the spiral, zigzag, and turning-circle tests can to some extent be correlated to dynamic stability as given b y the stability lever I'r — I' c- The best correlation is of course obtained for the spiral test; see F i g . 2 1 . The f a c t t h a t such correlations are not entireljr successful can be attributed to several factors. The values of 1', — I', are o n l j ' dependent on the linear hydrodynamic coefficients while results f r o m turning-circle tests and, to some extent, results f r o m zigzag tests are more dejiendent of the nonlinear coef-ficients. However, the dynamic stability has some influence on the transient phases of the shi])'s motion i n consequence of
2h y • Q • A/L • X y ^ D A • RIL • X A * Basic ship D L/B series
^ Rudder size series X 'StabiUti/ series R , A H ' lOOO BOO 200 O -A • A • X X • A A R X • • 0 A • X -Q2 - Q / S —01 —OOS O i'-C Fig. 24 Results of s i m u l a t e d t u r n i n g - c i r c l e t e s t s (500
T a b l e 4 S t e p r e s p o n s e 5 = 20 d e g — b a s i c 200,000-dwt t a n k e r A [ml I200 lOOO eoo eoo •^oo soo D M I200 lOOO eoo soo •too zoo
•
e SPA prtidicHon•
/
•
• S»a tr iais •SSPA pre, diction
A - ,
•
• Sea trials "-4 1 •X.OO 300 Fig. 25 R e s u l t s of t u r n i n g - c i r c l e t e s t s (S loo Dwt X = 35 deg) T I M E P R O M R U D D E R Y A W C O U R S E E X E C U T E H A T E 4, C H A N G E S P E E D V , sec deg/sec Aif', deg knots0 0 0 16.0 100 0.61 32 58 14.6 150 0.66 32 58 12.7 (max. yaw rate) 300 0 . 4 6 135 8.4 500 0.40 220 0.0 900 0.38 370 6.0
Thus, the ship responds relatively quickly i n the i n i t i a l phase (time less than 150 sec) and then follows a slow adjust-ment to steadj'-state values. As the ship's inertial forces are relatively larger t h a n the resisting forces i n the X-direction, speed loss w i l l thus adjust rather slowly, and this mainly accounts for the long time needed to reach the steady state. I t can be expected t h a t f r o m a maneuvering point of view most interest will be concentrated upon the i n i t i a l part of the step response, i.e., f r o m rudder execute to maximum yaw rate. I n order to find correlations between yaw rate response and ship's characteristics, plots have been made of yaw rate versus stability lever 1% — I' „ and secondly of yaw rate versus the ratio of rudder yawmg moment to ship's moment of i n -ertia ik/L^)nei- The ratio of rudder yawing moment t o ship's moment of inertia can be w r i t t e n as
where fc is a coirstant depending on rudder planform, flow conditions and ship's moment of inertia, rudder area As, ship's length L, and ship's displacement V . For the geosim series this ratio w i l l be essentially proportional to For the L/B series f o r which rudder area and displacement are constant the ratio should vary as k/L. The factor k varies, however, very nearly like 1/L. I t is thus justified to p u t
kAj^L L^
some disturbance (as a change i n rudder angle). Thus i t can be expected that there is some correlation w i t h such quanti-ties, w'hich include transient behavior. Thus, the dimen-sional values of advance and tactical diameter, as well as overshoot angle and times and h — k, seem to be correlated to I'r — I'f F r o m the rudder-size series, however, i t is evident t h a t effects of rudder size are s t i l l more important.
Simulations of the ste]j-response test are carried out to gain information on the ship's behavior i n the transient phase following a step change i n rudder angle. There are of course some difficulties i n analyzing results f r o m step-response tests, as the interest is concentrated on the transients. F r o m the maneuvering point of view i t is of interest to know how fast the ship responds to rudder action. As a measure of the ship's response, the j^aw rate after a preselected time after execute of a rudder angle (from neutral position) can be chosen. The choice of this time and rudder angle is not crucial. A typical step-response curve f o r the yaw rate ex-hibits some characteristic features, such as at first an increase to a ma.ximum value and then a rather slow decrease to the steady-state value. I t is also significant t h a t the ship's speed decreases quite fast i n the first jjhases of the motion and then rather slowly to the steady-state value. For the base 200,000-dwt tanker. Table 4 gives typical values f o r a ste))-respouse test a t 16 knots using 20-deg rudder angle.
where k is equal to 1 f o r the geosim series, L/B series, and stability series, and varies between 0.8 and 1.2 for the rudder-size series. As i n most cases the base 200,000-dwt tanker is used for comparison, and a relative number is used:
( f c / X ^ ) R . i = {k/L')/{k/L^ha..
F r o m F i g . 26 i t is obvious t h a t there is a correlation of yaw rate w i t h 1', — I'v as well as with (A;/I/*)Eei- There is, how-ever, one advantage of using (fc/L^) as a correlation basis; t h a t is, the effects of improving rudder efficiency (increasing rudder size, imjiroving flow conditions) are easily seen, which is n o t the case when using 1', — I't.
Coursekeeping abilities in a seaway disturbance
The seaway disturbances were simulated as sinusoidal side forces and yawing moments using seven discrete frequencies. The amplitudes of these forces and moments were calculated f r o m available data on ship response to regular w-aves [13, 14, 15] and a sea spectrum (point sjiectrum) as given b y Piersou and Moskowitz [ 4 ] . I t must be emphasized, however, t h a t the available data are scarce and t h a t the sea spectra f o r low frequencies are not very well known. A f t e r this investigation had been carried out, better data were published [3, 16].
I n order to compare results f r o m using seaway disturbances as described above with the earlier investigations, some
