5.1 General. The controls-fixed stability indexes discussed in the preceding section constitute one of the important elements of path keeping at sea. Because the practical problem of path keeping involves re- peated instances of path correction, its basic elements tend to merge with those of path changing. These basic elements are shown on the control loop of Fig. 1. Path keeping and path changing ability of a ship depends on:
( a ) The magnitude and frequency of any yawing moments and sway forces acting to disturb the ship from the desired path.
( b ) The character of the response of the ship with controls fixed to these disturbances. This response will be reflected in changes in the ship’s path shown at the extreme right of Fig. 1.
(c) The rapidity with which the error between the ship’s path and the desired path can be detected, and with which corrective action can be initiated.
(d) The rate at which the corrective action is trans- lated into movement of the rudder. This is a function
of the play between the third and fourth elements of the control loop and the rate a t which the steering gear can deflect the rudder in the fourth box of Fig.
1.
( e ) The magnitude of the control force and moment applied to the ship by the rudder.
Of these five elements, only the second is dependent on the controls-fixed stability of the ship. This is an important element, but so are all the others. Usually, deficiencies in any single element of the control loop can be compensated for by improvements in other ele- ments. For example, it is shown in Section 11.2 that the use of properly designed automatic controls in element (c) can correct for controls-fixed instability in element (b). Often it is assumed that increases in rud- der size, element (e), or in the rate of rudder deflection, element (d), can correct for deficiencies in the path keeping or path-changing ability of a ship. The latter view is shown in Section 17 to be incorrect. Although minor degrees of controls-fixed instability are com- monplace in ships, the best design is likely to be that Previous Page
206 PRINCIPLES OF N A V A L ARCHITECTURE
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which possesses minimum deficiencies in each element of the control loop.
5.2 Definitive Maneuvers. The naval architect is mainly concerned with elements (b),
(4,
and (e) of the path-keeping and path-changing problems. Therefore, certain definitive maneuvers have been devised to dem- onstrate the efficacy of these elements of the control loop and to exclude as much as possible the influence of element (c). Essentially, these maneuvers establish the basic stability and control characteristics of a ship independent of its helmsman or autopilot:(a) Direct or reversed spiral (see Section 4.3) (b) Zigzag, Z, or Kempf overshoot (see Section 5.3) (c) Turning (see Section 6.1)
The spiral maneuver as described earlier serves mainly to determine stability characteristics, whereas the zig- zag maneuver is to determine control characteristics.
The turning maneuver denotes turning qualities. All three maneuvers are important for both merchant and naval ships. Specific performance criteria and other related trials are discussed in Section 14.
5.3 Zigzag Maneuver. Second to the spiral ma- neuver in importance is the zigzag maneuver, also known as the Kempf overshoot or "Z" maneuver (Kempf, 1944).
The results of this maneuver are indicative of the ability of a ship's rudder to control the ship. However, just as the results of the spiral maneuver give some indication of control effectiveness (yaw-angle rate ver- sus rudder angle), so do the results of the zigzag test depend somewhat on the stability characteristics of the ship as well as on the effectiveness of the rudder.
The typical procedure for conducting the test is as follows (Gertler, 1959):
( a ) Steady the ship as in step (a) of the spiral ma- neuver. (See Section 4.3).
( b ) Deflect the rudder a t maximum rate to a pre- selected angle, say 20 deg, and hold until a preselected change of heading angle, say 20 deg, is reached.
(c) At this point, deflect the rudder at maximum rate to an opposite (checking) angle of 20 deg and hold until the execute change of heading angle on the op- posite side is reached. This completes the overshoot test.
(d) If a zigzag test is to be completed, again deflect the rudder at maximum rate to the same angle in the first direction. This cycle can be repeated through the third, fourth, or more executes although characteris- tics through the first overshoot are most important as discussed in Section 15.
Fig. 18 shows the results of a zigzag maneuver car- ried through five executes. The results shown are those that can be readily obtained with a controlled model in a towing tank or with a well-instrumented ship at sea. With ordinary ship navigational aids, only the rudder angle and yaw-angle curves are readily obtain- able.
The principal numerical measures of control ob- tained from the overshoot maneuver as illustrated in
4 0
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20
10
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W W K 0 - 2 0
e
b"
I C
2c I- 0 a
3C
E X - EXECUTE
Yo -O'fTANCE NORMAL TO L -SHIP LENGTH V - S H I P VELOCITY dR- RUDDER INOLE*
1L -YAWANGLE
'ACCORDING TO NOMENCUTURE (1952) FOR A STBD TURN IN ORDER TO SHOW THE EXECUTEPOINTSMORECLEARLY, d n AND
YARE GIVEN THE SAME SIGN IN THIS FIGURE ORIGINAL PATH d,+HOULDJE NEGATIVE AND POSITIVE
i - T I M E F R O M EXECUTE 1' -NON-DIMENSIONAL
TlMErSHiP LENGTHSOF TRAVEL
t v
.-
. OVERSHOOT OVERSHOOT WIDTH Y A W A N G L E / OF PATH
I
PERIOD I
- 2;
X N
L l k
-
-I I I I I I 1 I
2 4 6 8 I0 12 14 16 18
SHIP LENGTHS OF T R A V E L , t '
1 1 I I I I I
0 I 2 3 4 5 6
T I M E I N M I N U T E S
Fig. 18 Results of an overshoot and zigzag maneuver (Morse and Price, 1961)
Fig. 18 are: ( a ) The time to reach the second execute yaw angle; ( b ) the overshoot yaw angle; and ( G ) the overshoot width of path. All of these are important operational parameters. The first is a direct measure of the ability of a ship to rapidly change course. I t improves with increased rudder effectiveness and with decreased controls-fixed stability (Arentzen and Man- del, 1960). The second and third are numerical mea- sures of countermaneuvering ability and are indicative of the amount of anticipation required of a helmsman while operating in restricted waters. It was shown that the magnitude of the yaw-angle overshoot decreases with increased stability but increases with increased rudder effectiveness. On the other hand, the overshoot width of path decreases with both increased stability and increased rudder effectiveness.
The results of the zigzag maneuver are speed de- pendent. In general, for any given ship the time to reach execute decreases with increasing speed, and the overshoot yaw angle and the overshoot width of path increase with increasing speed. However, the non- dimensional time to reach execute, interpreted in Fig.
18, as ship lengths of travel to execute, increases with increasing speed because of the influence of the rate of rudder deflection, 6 R. When 6 is nondimension- alized, 6 I R = 6.L/V, it may be interpreted as de- grees of rudder deflection per ship length of travel.
At low speeds, this nondimensional rate is much higher than at higher speeds since 6 is essentially indepen-
CONTROLLABILITY 207
dent of speed. Hence, with respect to the ship length of travel scale in Fig. 18, the rudder would be deflected more rapidly a t low speeds than at high. Hence, the rudder exerts its full influence longer a t low speeds, which tends to reduce the nondimensional time to reach execute as speed is reduced. However, in spite of this beneficial effect as speed is decreased, the time to reach execute usually increases with decreasing speed. (The effects of the rudder are reviewed further in Section 17).
In the case of submarines, the overshoot maneuver is employed in both the horizontal and vertical plane and its results are perhaps even more operationally significant in the vertical plane than in the horizontal.
This is true because in the vertical plane submarines must operate within a relatively shallow layer of water, while they usually have ample freedom of mo- tion in the horizontal plane except when they are in restricted or congested waterways. Hence, to the sub- marine operator, overshoot pitch angle and overshoot change of depth are very important parameters.
5.4 The K and T Coursekeeping and Turning In- dexes. This section presents the Nomoto simplified analysis of K and T indexes which can be developed from zigzag trial data. These indexes are widely used, simplified analysis tools developed from the linear equations of motion. They are useful in comparing coursekeeping as well as turning abilities, which will be presented further in Section 6.
While Equations (11) and (12) expressed the linear equation as a pair of simultaneous first order differ- ential equations, where the constant coefficients are the dimensionless acceleration and velocity deriva- tives, it is possible to express these equations in an alternative form. It was first shown by Nomoto (1957) that these equations can be written as a pair of de- coupled second order equations as follows:
TI' T,' B'
+
(TI'+
Tz') P+
r' = K'S,(15)
+
K' T,'6;
TI' T,' ij'
+
(TI'+
Tz') fi'+
v' = K,' 6 ,+
K,' T,' 6 R 'This expression for the coefficients in terms of the time constants TI', Tz', T3' and T,' as well as a system gain K' is consistent with control engineering practice.
Since Equations (15) are a linear system as are (11)) a solution similar to (13) may be derived and it may be seen that the roots of the solutions are related to the time constants as follows:
1 1
uI = -- and cZ = --
TI ' T2'
Returning to the linear yaw and sway (11) and (12), it can be seen that they are coupled only through the terms N,' v' and Y,' r ' , which are typically small, par- ticularly for ships with near fore-and-aft symmetry. If these cross-coupling terms are neglected and sway
velocity or side slip angle thus eliminated, turning de- pends only upon yaw rate, r, and is defined by the simplified non-dimensional yaw equation of motion:
(16) Nomoto (1957,1960, and 1966) noted that this equation could be divided by the yaw damping coefficient, N : and rewritten in the parametric form:
T
r+
r ' = K' 6 , (17)n: i-' - N r r ' = Nb,
aR
where the non-dimensional parameters or indexes T and K' are given by:
T' = n : / N : = ( I : - " , ) I N :
= T',
+
T, - Tg (18) K' = NLIN:In dimensional form the equation is Tr
+
r = KaR,where the non-dimensional parameters are related to the dimensional Nomoto parameters T and K by:
T' = T ( V / L ) K' = K ( L / V )
The indices T and K' represent ratios of non-dimen- sional coefficients from (18):
yaw inertia coefficient yaw damping coefficient turning moment coefficient
yaw damping coefficient T =
K' =
Dividing K' by T shows that the two indexes are related by:
K' - turning moment coefficient T yaw damping coefficient
In practice, Equation (15) can be solved by numerical integration. For the simple case where the rudder is put over suddenly to an angle 6 , and held there, the solution for r is given, in terms of T and K, by:
_ -
r = K So (1 - e-t'T) (19) This shows that the yaw rate r increases exponentially with time but at a declining rate dependent on T and approaches a steady value KS, (or K' Va0 / L ) . A larger K thus provides greater steady-state turning ability, and a smaller value of T provides a quicker initial response to the helm. Quick response implies good course-changing ability and good course-checking abil- ity when a turn (or other maneuver) is completed. Since quick response is obviously valuable in course-keeping (steering), it is thus consistent with a smaller
T.
The above discussion of Equation (19) shows thatT
has no effect at all on steady turning rate, but a small Twould reduce the time required to reach a steady turn. A t the same time, the index T is a reciprocal measure of208 PRINCIPLES OF NAVAL ARCHITECTURE
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course stability, with stability increasing with decreas- ing
T.
However, a negative value indicates an unstable dynamic character. The steering quality indexes K andT
have an immediate relationship to the conventional measures of ship turning. It may be shown that, under the previously stated simplifying assumption that sway can be neglected:1
' T 1 = -- T'
where 'T, is the stability index of Section 4. Thus, T' offers a direct quantitative measure of straight-line stability.
For steady turning at constant rudder angle S,, r = KSRo = K' V S R o / L (20) Steady turning diameter, D o , by definition is:
0 l/T'
and hence non-dimensional turning diameter, D o / L , and K' are related by:
D , / L = 2 V / d = 2 / K ' 6 ~ , (22) This relation can be derived from (16) in Section 5 by neglecting sway (placing N ; = 0). This is in accord with the statement in Section 5.3 that R / L depends on the relative magnitudes of N r and N b R . It shows that with a larger value of K' a smaller rudder angle may be used in achieving a given turning diameter.
The main maneuvering qualities of a ship using lin- ear analysis can thus be characterized using only the indexes
T
and K', where increasing values indicate improving performance:2" Course stability 1 / T'
K' Turning ability
Responsiveness to rudder
A highly maneuverable ship (with high responsive- ness to rudder and both good turning and low course
stability) will have a small value of T' and a large value of K'. In other words, a large ratio K ' l T ' , or Norrbin parameter, P = K ' / 2 T , (Nomoto and Norrbin, 1969) is indicative of good maneuverability.
I t is not a good indicator of course-keeping ability (or good steering) however, because this can be achieved either by high course stability and low responsiveness (high T') or by low or even negative stability and high responsiveness (high 1 / 2") plus superior automatic control. In short, a large ratio K ' I T suggests good overall controllability only if stability is no greater than necessary.
Overshoot angle, which is obtained from the zigzag maneuver (Section 4) has often been used as a measure of controllability. Nomoto (1966) has shown that overshoot angle is, for a given rudder angle, nearly proportional to the product K' T'. Overshoot angle thus has the inherent weakness that it cannot be used to discriminate between a ship with: (a) good turning and fast response or good course stability (large K' and small 2") and (b) poor turning and slow response or course stability (small K' and large 7"). The former with large K' / T , is clearly a far superior ship in over- all controllability. But overshoot angle does indicate turn-checking ability.
For further guidance Nomoto has suggested that:
Turning moment coefficient cc A,/ LT and Yaw inertia coefficient a V / L2 T
where A, is rudder area, and V is displaced volume.
Using these approximations:
(23) K' ARL - ARL
a - -
c ,
7
- T' V
where c, is a constant of proportionality. Fig. 19 sum- marizes results for various ships and rudder angles, and indicates by the straight lines that c1 tends to be independent of ship type and rudder angle. I t is clear from (23) that, since large K' / T' is favorable, a large value of A,L / V is desirable. This simplified linear anal- ysis indicates that ship dimensions (particularly L and V), as well as rudder area, will have a significant effect.
Once overall ship dimensions are established, both as- pects of controllability can be significantly improved by increasing rudder size or effectiveness.
The indexes T' and K' can be calculated numerically using Equation (15) if hydrodynamic and mass coeffi- cients for the ship are known. One advantage of these indexes is that they can be derived from the results of the standard trials or free-running model maneuvers for comparison with calculation. They give physical meaning to the standard trials.
The application of T and K' to determining criteria of controllability is discussed in Section 14.7. The ele- ments of turning performance as separated from coursekeeping and control are introduced and ad- dressed more fully in Section 6.
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CONTROLLABILITY 209