Analysis of Turning Ability

CONTROLLABILITY 209

Section 6

210 PRINCIPLES OF N A V A L ARCHITECTURE

TIME t

< THIRD (STEADY) PHASE ( i , = i = O V # O r # o )

TIME t

FIRST P H A S E < THIRD (STEADY) PHASE

( i , = i = O V # O r # o )

SECOND PHASE

( G # o ig o V Z O rzo)

i

Characteristics of transient phases of o turn START OF RUDDER DEFLECTION, t = O

Fig. 21

The first phase starts at the instant that the rudder begins to deflect and may be complete by the time the rudder reaches its full deflection angle. During this period, the rudder force, Y, SR, and rudder moment, N, SR, produce accelerations and are opposed solely by the inertial reaction of the ship because there has not yet been an opportunity for development of hydro- dynamic forces arising from a substantial drift angle, 0, or a rotation, r, to develop. Hence, in this stage, p

= v / V = r = 0. Using the dimensional Equation (10) and introducting the rudder force and moment, the linearized equations of motion in the first phase of turning are:

(1, - N + ) i - NG = N, 6,

The values of the accelerations, l j and i that occur in this phase can be obtained from these equations. I t may be noted in Figs. 20 and 21 that the transverse acceleration, v, is negative or directed to port in this phase, whereas the turn will eventually be to star- board. This is because (for a rudder a t the stern) the rudder force Y, S R is directed to port for a starboard turn.

The accelerations v and i can exist in isolation only momentarily, for they quickly give rise to a drift angle, p, and a rotation, r, of the ship. With the introduction of these parameters, the ship enters the second phase of turning. Here the accelerations of the ship coexist with the velocities and all of the terms of Equation

(10) along with the excitation terms ^Y6 S R and N, 6, are fully operative (12). The crucial event that takes place a t the beginning of the second phase of the turn is the creation of a Y, v-force positively directed to starboard in Fig. 20 towards the center of the turn, resulting from the introduction of the drift angle, p.

The magnitude of this force soon becomes larger than the Y, 6,-force which is directed to port (Fig. 5). As shown in Fig. 21 this causes the acceleration v to cease to grow to port and eventually to be reduced to zero as the inwardly directed Yv v-force comes into balance with the outwardly directed centrifugal force of the ship. However, in the second phase of the turn, the path of the center of gravity of the ship at first re- sponds to the Y, SR-force and tends to port before the Y, v-force grows large enough to enforce the starboard turn. This port offset although visibly protrayed in Fig.

20 is negligible or nonexistent in practice because of the shortness of phase 1, and the quick development of the large Nvv-moment in ship forms.

6.3 Steady Turning Radius. Finally, after some os- cillation (some of which is due to the settling down of the main propulsion machinery and is characteristic of the particular type of machinery and its control sys- tem) the second phase of turning ends with the estab- lishment of the final equilibrium of forces. When this equilibrium is reached, the ship settles down to a turn of constant radius as shown in Fig. 20. This is the third, or steady, phase of the turn. Here v and r have nonzero values, but d and i are zero. Thus, using Equa- tions (10) the linearized equations of motion in a steady turn are:

CONTROLLABILITY 21 1

- Y,v - (Y, - A‘u,)r = Ys6, (251 \ I

-N,v - N,r = N,S,

These two simultaneous equations can be solved for r and v provided that the stability derivatives Yo, Y,, N,, N , and the control derivatives Y, and N , are known.

Noting that r’ G

4

⁼ rL/V and that the steady turning radius R ⁼ V l r , then ^{r ’ =} L / R or the recip- rocal of the ratio of the steady turning radius to the ship length. Solving the nondimensional version of (25) we obtain:

- L Y ’ u ( N ’ 7 ) - N’,( Y , - A’)

= - 6,

[

Y v N ‘ , - N’,Y‘, and

N’,( Y‘, - A’) - Y , N ‘ , Y’,N’, - Nit,( Y ‘ , - A’) v 1 = -0 ⁼ 6,

where p and 6 are in radians and positive R denotes a starboard turn.

Thus, according to the preceding linear theory, the steady turning radius would be proportional to the ship length, L, and inversely proportional to the rudder- deflection angle, 6,, and the drift angle 0 would be directly proportional to 6,.

Solutions (26) and (27) are useful for estimating the steady turning radii and drift angles of stable ships with fairly large diameter turns of about four ship lengths or more. They are used to estimate the turning radii of torpedoes, and are useful for estimating the turning radii of ships at less than maximum rudder angles.

The great majority of merchant ships have turning diameters of from two to four ship lengths a t full rudder angle, and many ships have turning diameters of two ship lengths or less. Such tight turns introduce strong nonlinearities that tend to reduce the validity of the linear equations of motion. Procedures for pre- dicting the maneuvers of tight-turning ships are dis- cussed in Section 8.

Relationship Between Steady Turning Radius and the Hydrodynamic Derivatives. Equation (26) devel- oped from linear theory may be used for stable ships to predict the effect of changes in the hydrodynamic derivatives on the turning radius. In slightly modified form, Equation (26) is:

6.4

It is seen that the numerator is identically the sta- bility criterion, C, Equation (14b) of Section 4.2; it was shown in Section 4.2 that the value of the numerator is independent of the choice of origin. If the relation- ships

N’, =

+

( Y , ) x ‘ a and

N‘, = ( N ’ , ) a

+

Y ’ , x ’ m

are substituted in the denominator, it reduces to Y , ( N ‘ , ) a - Y s ( N ’ a ) a which is also independent of the choice of origin, thus if the ship is stable the nu- merator is positive and if the ship is unstable the nu- merator is negative. The sign of the denominator is always positive for the following reasons:

Y’, is always negative and N‘, is always neg- ative for rudders located at the stern (Fig. 22). In the figure 6, is negative following the definition of these 6’s given in Section 3.5 and following the sign con- vention given in the nomenclature at the end of this chapter. The moment N resulting from negative 6, however, is positive according to the same sign con- vention. Similarly, if S were positive, N would be neg- ative. Hence, the derivative N, is always negative for rudders at the stern. In Fig. 22, the force, Y , arising from the negative 6 is also negative, if 6 were positive, Ywould be positive; hence, the derivative, Y,, is always positive hence, their product is positive.

Y ,

is always positive, N ‘ , is almost always neg- ative; hence, subtracting their product will add posi- tively to Y , N ’ s .

(c)) If is positive (it is rarely so), its magnitude (a)

( b )

C STRAIGHTENING INFLUENCE OF H U L L AND PROPELLER ON THE F L O W TO THE RUDDER

a ANGLE OF ATTACK ON THE RUDDER

“.p,

ACTUAL DRIFT ANGLE AT THE RUDDER + € GEOMETRIC DRIFT ANGLE AT THE RUDDER

CENTER OF THE STEADY TURN

w

Fig. 22 Orientation of ship and rudder in a steady turn to starboard

212 PRINCIPLES OF N A V A L ARCHITECTURE

z AXIS ^I AXIS

I

y AXIS-

-

_{- -}

I

STARBOARD TURN

AFT (6) _3rd. PHASE ( A ) 1st

I

PHASE

Fig. 23 Disposition of forces in yz plane in a turn

is bound to be small so that the larger positive Y',N', product will determine the sign of the d e n ~ m i n a t o r . ~

It may therefore be concluded that if a ship is stable and its rudder is located a t the stern, a positive (star- board) R will always result from a negative 6 and vice versa. However, if the ship is unstable, then the nu- merator of the right-hand side of Equation (26) is neg- ative and R will have the same sign as 6 . Physically this means that a ship will turn against its rudder, which is in accord with the behavior of an unstable ship. Since Equation (26) deals only with the slope of the R versus 6 curve a t 6 = 0, which is a region of unstable equilibrium for unstable ships, it cannot be used to predict the turning radii of unstable ships.

For stable ships, Equation (26) may be used to ex- amine the effect of changes in the individual deriva- tives on the turning radius. Equation (26) shows that the effect of changes in Y ' , on R depends on the relative magnitude of

(N,)

compared to N ' , . If N ' , has a greater magnitude than

(N,)

then increasing the mag- nitude of

Y ,

would decrease the radius of turn. On the other hand, if the magnitude of (N',) is larger than N ' 6 , then an increase in the magnitude of

Y ,

will usually increase the radius of the turn. Since for stable ships, ( N f T ) is usually much more negative than N'!, the usual effect of increasing the magnitude of

Y ,

is to increase the radius of the turn. Thus, while the Y,v- force is responsible for the initiation of a turn in the desired direction, an increase in the magnitude of Y, does not necessarily reduce the steady turning radius.

The effect of N ' , on R is readily predictable. If

N ,

is negative, increasing its magnitude will decrease the positive value of the numerator of the right-hand side of (26) and increase the positive value of the denomi-

For rudders located a t the bow, N6 will always be positive and R will always have the same sign as tiR for stable ships. The sign of Y, is always positive whether the rudder(s) are located a t the bow or a t the stern.

nator, hence R will decrease on two counts. On the other hand, if

N ,

were positive, increasing its mag- nitude would increase the numerator and decrease the denominator, and hence R would be increased.

The effect of N', and N ' , on R' is equally clear.

According to Equation (18), an increase in the mag- nitude of N ' , will increase R while an increase in the magnitude of N , will decrease. This result is in accord with an intuitive examination of the question.

The effects of the remaining derivatives on R depend on the sign of

N a ,

and can be deduced from Equation (26) when the sign of the derivative is known (see Table 1).

While use of the rudder is intended to produce motions only in the yaw (xy) plane, motions are also induced by cross coupling into the pitch (xx) and roll (yx) planes. The unwanted mo-

6.5 Heel Angle in a Turn.

Table 1-Effect of Changes in the Derivatives on Steady Turning Radius (for Stable Ships with Rudder(s) at the Stern)

Sign of the derivatives

Derivative for ships

always negative always negative always negative either positive

or negative either positive

or negative always positive always positive NOTES:

1. A ' is not a derivative but is included here for conven- ience.

2. Signs in italics refer to the signs of the derivatives for usual ship forms.

3. For typical values of the foregoing derivatives for a wide range of merchant ship forms (nondimensionalized on the basis of p, L, T and V), see Table 6.

CONTROLLABILITY 213

tions in the roll plane, particularly, are likely to be large enough to be of significance.* The magnitude of the heel angles induced by the rudder can be estimated by considering the heeling moments arising from the vertical disposition of the forces described in the pre- ceding section. That disposition for the first phase of a starboard turn, is shown in Fig. 23(a). The direction of most of the forces may be obtained from the first of Equations (24) if all terms are gathered on one side and equated t o zero as follows:

Y,S,

+

^{Y , 6}

+

^{Y , P - AV = 0} (28) Since Y, is always positive and 6, is negative for a starboard turn, Y,S, is negative or directed to port.

Since Y , is always negative and v is negative in the first phase of a starboard turn, Y, v is positive or directed to starboard. Since Y , may be either positive or negative, the sign of Y , i is not predictable from (28). In any event, Y , i is very small compared to Y , V . Finally, since V is negative and A is positive, (- Av) is positive or directed to starboard.

The approximate angle of heel,

4,

may be obtained by equating the resultant heeling moment, which is the sum of the moments of each of the forces in the yx-plane, Fig. 23(a), to the hydrostatic righting mo- ment. A graphical solution of this equation is described in Section 7 of Chapter 11. For this purpose Y , Y , may be taken to be acting a t half draft, Y, S, a t the vertical center of the rudder and AV and Ax,r a t the center of gravity of the ship. If moments are taken about the half draft, it is obvious from Fig. 28(a) that the heel angle,

4,

will be to the starboard (positive) in the first phase of a starboard turn.

The forces acting in the yx-plane taken from Equa- tion (25) for the third phase of a starboard turn are shown in Fig. 23(b). If moments are taken about the center of gravity of the ship, it is seen that the heel angle.

4.

is likely to be to port (negative) since Y,v

+

Y,r must be much larger than Y,S in order to enforce the starboard turn. Thus, between the first and third phase of a turn, the heel angle of a surface ship changes sign. The heel-angle time record of ship with a large turning heel angle is shown for a starboard turn in Fig. 24. It is seen that the amplitude of the initial heel to starboard in the first phase of the turn is small compared to the amplitude of the second heel to port. This second heel involves a large overshoot angle beyond the equilibrium value computed in ac- cordance with Fig. 23(b). However, eventually the port heel settles down to a fairly steady value correspond- ing to the computed value for the final phase of the turn.

From an operational point of view, a potentially dan-

For submerged submarines, the pitch angles induced in turning by the rudder as well as by hull asymmetries are also frequently large enough to be of concern.

2 0 I

4 0 1

0

I

^{I I}

#-

dE IN SECONDS

Fig. 24 Roll-angle time records for a starboard turn

gerous situation exists just prior to the completion of the first large heel to port. A helmsman, fearing too large a heel to port, might at this instant decide to return the rudder quickly to amidships. This would eliminate the Y,S, force and the heel to port would be aggravated rather than alleviated. The only safe action to take in such a situation is to immediately, but slowly and cautiously, reduce the rudder angle and at the same time reduce speed as quickly as possible.

In the case of a submarine turning, submerged, the heel angle is inboard (starboard heel for a starboard turn) throughout all phases of a turn. The reason for this is that the positions of both the Y,V

+

^{Y + i force}

of Fig. 23(a) and the Y,v

+

Y,r force of Fig. 23(b) are considerably higher relative to the center of gravity on a submerged submarine than on a surface ship. In particular, the bridge fairwater existing on practically all submarines is an effective lifting device and con- tributes heavily to both the magnitude and the height of the Y,v-force of Fig. 23(b), increasing rolling mo- ment,

K,v.

I t is clear that if the Y,v

+

^{Y,r force is}

raised sufficiently high (on some submarines it is raised to a position above the center of gravity), the heel in the third phase of a turn will be in the same direction as in the first phase of a turn. Thus, the first heel of a submerged turning submarine is an inboard heel of very large amplitude called the snap roll with sub- sequent inboard rolls of diminished mean values. Ac- cording to Arentzen and Mandel (1960), the ratio of the snap roll in the first phase of a turn to the steady heel in the final phase of a turn may be as large as 3% for a submarine with a large bridge fairwater and as large as 5 for a submarine without a fairwater. The latter submarine, however, has a much smaller steady

214 PRINCIPLES OF N A V A L ARCHITECTURE

10, I I I , ^{r t}

1 I 1 I

7/-

-1

- -

'

SHlBA (1960) ^I

w

5 - 1 - * w - I I I I I I I I I I I

I I I I I I I I

-'I uqr/ I I I I I I I I I I I

2 3 4 5 6 7 8 9 10 I1 12 13

TURNING DIAMETER / LENGTH

Fig. 25 Speed reduction as a function of turning diameter and block coefficient

heel angle than a submarine with a fairwater. Thus, the fairwater plays a dual role in turning:

(a) I t increases the roll excitation in a turn because of its large influence on the roll moment due to the transverse velocity

K,v.

(b) I t increases the magnitude of the roll damping moment, Kpp, and hence, dampens the amplitude of the overshoot of the snap roll.

See Section 12 and 15.7 for description of yaw-roll coupling effects in waves and further discussion.

6.6 Part of the rea-

son that the initial snap roll of a submarine is so much larger than subsequent rolls is that the speed of the submarine is rapidly reduced as soon as it develops a substantial drift angle. This is also part of the reason why the first roll to port of the surface ship in Fig. 24 is much larger than subsequent rolls. However, in the case of the surface ship, its speed is more greatly reduced by the time it experiences its largest heel angle than in the case of the submarine. This partially accounts for the fact that the ratio of the value of the first large roll to the steady heel shown in Fig. 24 is not nearly as large as the comparable ratios for sub- marines cited from Arentzen and Mandel (1960).

The magnitude of the speed reduction in a turn is largely a function of the tightness of the turning circle (Davidson, 1944). Fig. 25 shows the empirical relation- ship between the ratio of the speed in a steady turn to the approach speed and of the turning diameter to the ship length, developed by Davidson on the basis of a large number of ship trial and model results. The discounting by Davidson of the differences between full-scale and model results has been shown to be er- roneous by Strom-Tejsen (1965), among others. Su- perimposed on the Davidson results in Fig. 25 are Shiba's (1960) results. Davidson and Shiba concluded that the relatively small scatter of data shown could not be related to rudder angle, approach speed, or rudder-area ratio, which were, of course, among the variables of the tests. The effect of changes in type of ship power plant and in ship configuration on the speed

Reduction of Speed in a Turn.

reduction during maneuvers is discussed in Sections 8.7 and 16.4.

In spite of the increasingly severe speed loss asso- ciated with tighter turns, Davidson showed that by decreasing the tactical diameter to two ship lengths or less, significant operational aspects of turning are improved. For example, Fig. 26 shows that a 122 m (400 ft) long, 20-knot ship with a TD/L = 2.0 achieves a full course reversal and has almost completely re- gained its approach speed in the 1% min that are re- quired for the ship to pass its original execute point headed in the opposite direction. On the other hand, the same ship with a TD/L = 4.5 required 2% min and

0 REPRESENT EQUAL E L A P S E D TIME INTERVALS OF ABOUT 2 4 SECS I 2 SHIP LENGTHS OF T R A V E L AT THE APPROACH SPEED1

Fig. 26 Comparison of 180 deg turn ond speed recovery characteristics of 152-m (400-ft) ship (Davidson, 1944)

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C 0 N T R 0 L LAB I L I TY 215

much more sea room for the same maneuver.

The speed used in the computation of heel angle in the final phase of a turn in accordance with Fig. 23(b) should of course be the reduced speed as determined from Fig. 25 and not the approach speed.

Transient turning and complex maneuvers cannot be predicted by linear theory. Instead one must resort either to non-linear theory in conjunction with captive model tests (Section 7) or to free-running radio-con- trolled model tests (Section 8).

Section 7