3.1 Definitions of Motion Stability. The concept of path keeping is strongly related to the concept of course stability or stability of direction. A body is said to be stable in any particular state of equilibrium in rest or motion if, when momentarily disturbed by an external force or moment, it tends to return, after release from the disturbing force, to the state of equi- librium existing before the body was disturbed. In the case of path keeping, the most obvious external dis- turbing force would be a wave or a gust of wind. For optimum path keeping, it would be desirable for the ship to resume its original path after passage of the disturbance, with no intervention by the helmsman.
Whether this will happen depends on the kind of motion stability possessed.
The various kinds of motion stability associated with ships are classified by the attributes of their initial state of equilibrium that are retained in the final path of their centers of gravity. For example, in each of the cases in Fig. 3, a ship is initially assumed to be traveling at constant speed along a straight path. In Case I, termed straight-line or dynamic stability, the final path after release from a disturbance retains the straight-line attribute of the initial state of equilib- rium, but not its direction, In Case 11, directional sta- bility, the final path after release from a disturbance retains not only the straight-line attribute of the initial path, but also its direction. Case I11 is similar to Case I1 except that the ship does not oscillate after the disturbance, but passes smoothly t o the same final path as Case 11. The distinction between these two cases is discussed in Section 4. Finally, in Case IV, positional motion stability, the ship returns to the original path, ie: the final path not only has the same direction as the original path, but also its same transverse position relative t o the surface of the earth.
The foregoing kinds of stability form an ascending
hierarchy. Achieving straight-line stability (Case I) is the designer’s usual goal for most ships when steered by hand. The other cases require various degrees of automatic control.
Course Stability With Controls Fixed and Controls Working. All of these kinds of stability have meaning with control surfaces (rudders) fixed at zero, with con- trol surfaces free to swing, or with controls either manually or automatically operated. The first two
3.2
FINAL PATH IS STRAIGHT ELIT DIRECTION CHANGED
- 1 STRAIGHT LINE STABILITY
\
ORIGINAL STRAIGHT
a
FINAL PATH.SAME .LINE PATH DIRECTION AS
1 DIRECTIONAL STABILITY DIFFERENT POSITION
- (WITH COMPLEX STABILITY INDEXES) ORIGINAL STRAIGHT
LINE PATU \ FINAL PATH. SAME AS CASE I
DIRECTIONAL STABILITY WITH REAL STABILTY INDEXES)
ORIGINAL S T R A I G H T S LINE PATH
a
POSITIONAL MOTION STABILITYa
INDICATES INSTANTANEOUS DISTURBANCE Fig. 3 Various kinds of motion stability (Arentzen, 1960)Previous Page
196 PRINCIPLES OF N A V A L ARCHITECTURE
cases involve only the last two elements of the control loop of Fig. 1, whereas the last case involves all of the elements of the control loop. In normal marine usage the term stability usually implies controls-fixed sta- bility; however, the term can also have meaning with the controls working. The following examples indicate distinctions:
(a) A surface ship sailing a calm sea possesses positional motion stability in the vertical plane (and therefore directional and straight-line stability in this plane) with controls fixed. This is a n example of the kind of stability shown by Case IV of Fig. 3. In this case, hydrostatic forces and moments introduce a unique kind of stability which in the absence of these forces could only be introduced either by very sophis- ticated automatic controls or by manual control. The fact t h a t the ship operator and designer can take for granted this remarkable kind of stability does not de- tract from its intrinsic importance.
( b ) In the horizontal plane in the open sea with stern propulsion, a self-propelled ship cannot possess either positional or directional stability with controls fixed because the changes in buoyancy t h a t stabilize in the vertical plane a r e nonexistent in the horizontal plane. However, a ship must possess both of these kinds of stability with controls working either under manual or automatic guidance. Possible exceptions in- clude sailing vessels, some multi-hull ships, and foil or planing craft but not other surface effect ships.
(c) The only kind of motion stability possible with self-propelled ships in the horizontal plane with con- trols fixed is straight-line stability. In fact, many ships do not possess it. In subsequent sections of this chap- ter, with some exceptions, whenever controls-fixed sta- bility is mentioned, the intended meaning is controls- fixed straight-line stability. This kind of stability is desirable, b u t not mandatory.
With each of the kinds of controls-fixed stability, there is associated a numerical index which by its sign designates whether the body is stable or unstable in t h a t particular sense and by its magnitude designates the degree of stability or instability. To show how these indexes a r e determined, one must resort to the differ- ential equations of motion.
3.3 Assumptions of linearity and Simple Addable Parts. In order t o understand the impact of various ship design characteristics and features on ship con- trollability, it is necessary to first become familiar with certain fundamental aspects relating to the concept of stability and to the development and use of the linear equations of motion. The use of non-linear equations for analysis and prediction and the determination of coefficients through captive model tests, use of theo- retical and empirical coefficient determination methods, and systems analysis is introduced in Sec- tions 8 and 9.
The force components X , Y and the moment com- ponent N in Equation ( 5 ) a r e assumed to be composed
of several parts, some of which are functions of the velocities and accelerations of the ship. In the most general case they also include terms dependent on the orientation of the ship relative to the axis of the earth a s well a s excitation terms such as those arising from the seaway or from use of the rudder, but these will be introduced later. For the present they are assumed t o be composed only of forces and moments arising from motions of the ship which in turn have been excited by disturbances whose details we need not be concerned with here. Expressed functionally X , and N are:
X = FJu, v, u, ij,
4,
$ )Y =
F,(u,
v,u, v , 4,
* ) (6) N = Fu(u, v, u, v ,4, 4)
In order to obtain a numerical index of motion sta- bility, the functional expressions shown in Equation (6) must be reduced to useful mathematical form. This can be done by means of the Taylor expansion of a function of several variables. The Taylor expansion of a function of a single variable states t h a t if the func- tion of a variable, x, Fig. 4, and all its derivatives are continuous a t a particular value of x, say x,, then the value of the function a t a value of x not f a r removed from x, can be expressed as follows:
6 'z d"f(x)
.+
. . .+
--+-- ax3
d 3 f ( x )3! dx3 ' n! dx"
where
f(x) = value of function at a value of x close to
X I
f(x,) = value of function at x = x , s x = x - x,
and
_ _ - d " f ( x ) - n t h derivative of function dx"
evaluated a t x = x,
If the change in the variable, Sx, is made sufficiently small, the higher order terms of 6x in Equation (7a) can be neglected. Equation (7a) then reduces to
It may be seen from Fig. 4 that Equation (76) is a linear approximation to the real functionf(x) a t x =
x,
+
6x and that (7b) becomes increasingly accurate a s 6x is reduced in magnitude. Equation ( 7 b ) is called the linearized form of (7a).The linearized form of the Taylor expansion of a
CONTROLLABILITY 197
-
dKx)dx
t t t ,
X -
Fig. 4 Linearization of Taylor expansion of a function of a single variable,
fk)
function of two variables x and y is a simple sum of three linear terms as follows:
where both Sx and Sy must be small enough so that higher order terms of each can be neglected as well as the product SxSy.
The assumption that renders linearization reason- ably accurate, namely, that the admissible change in variables must be very small, is entirely compatible with an investigation of motion stability. Motion sta- bility determines whether a very small perturbation from an initial equilibrium position is going to increase with time or decay with time. Thus, it is consistent with the physical reality of motion stability to use the linearized Taylor expansions in connection with equa- tion (6). For example, by analogy with Equation ( 7 4 , the linearized Y-force of (6) can now be written as:
where the subscript 1 refers in all cases to the values of the variables a t the initial equilibrium condition and where all of the partial derivatives are evaluated at the equilibrium condition. Since the initial equilibrium condition for an investigation of motion stability is straight-line motion at constant speed, it follows that zi, = zjl = $ 1 =
q1
= 0. Furthermore, since most ships are symmetrical about their xx-plane, they travel in a straight line a t zero angle of attack; therefore vl is also zero but this is not necessarily true on ships with an odd number of propellers or with any number of unirotating propellers, Sections 5.3 and 11.0). Also because of symmetry aY/au = aY/azi = 0 since achange in forward velocity or forward acceleration will produce no transverse force with ship forms that are symmetrical about the xx-plane. Finally, if the ship is in fact in equilibrium in straight-line motion, there can be no Y-force acting on,it in that condition, therefore F,(u1, v1, U 1 , 6 1,
ql)
is also zero. Only u, is not zero but is equal to the resultant velocity, V, in the initial equilibrium condition. With these simplifications, Equation (8) reduces toand similarly the surging force and the yawing mo- ment can be written as:
ay
ax ax ax
a u au av a v
x = - u + - s u + - v + - v
ax
.ax
..+ - q J + - q J
a$
all,where the cross-coupled derivatives aY,
a$,
a u l d $ , aN/av, and aN/a 6 usually have small nonzero values because most ships are not symmetrical about the yx- plane even if that plane is at the midlength of the ship (bow and stern shapes are normally quite different).However, the cross-coupled derivatives a x / & , aX/av, 8x4
$
, andax/ a $ ,
like aY/au
and aY/a u are zero because of symmetry about the xz-plane. Hence, equa- tion (9b) reduces to:ax
.ax
au
aux = - u + - s u ( 9 4
where Su = u - u l .
3.4 Notation of Force and Moment Derivatives. In the simplified derivative notation of SNAME (Nomen- clature, 1952), aY/av = Y,, aN/a$ = N+, and so on.
Also for motions restricted to the horizontal plane
$ E r a n d $
z
i.. Using this notation and substituting (9) into (5), the linear equations of motion with moving axes in the horizontal plane are:-Y,v
+
(A - Y 6 ) ir-X,(U - u , ) C (A - X,)ii = 0
- (Y, - A u , ) ~ - Y + i . = 0 (10)
1
-N,V - N, ir - N,r
+
(I, - N + ) i.Every term of the first two equations of (10) has the dimensions of a force whereas every term in the third equation of (10) has the dimensions of a moment.
Therefore, to nondimensionalize (lo), which is conve- nient for several reasons, the force equations are di- vided through by ( p / 2) L 2 V 2 and the moment equations by (p/2)L3V2. (Note the similarity between (p/2)L2V2 used as a nondimensionalizer in this case and ( p / 2 ) S
V
= 0
198 PRINCIPLES OF NAVAL ARCHITECTURE
used to obtain the resistance coefficients in Chapter V). Further, as in (Nomenclature, 1952) a primed sym- bol will be used to designate the nondimensional form of each of the factors appearing in (10). For example:
Thus, for example, the nondimensional forms of the first couple of terms of the last of Equations (10) are:
etc.
If the surge equation is neglected and if the previous notation is adopted, (10) becomes in nondimensional form:
- Y',v'
+
(A' - Y'&' - (Y', - A')r'(11) - y e + ' = 0
-N' v' - N ' . +' - N'
+ (T2
- NJi.' = 02) v
where the main difference between (10) and (ll), aside from the prime notation, is that u, has disappeared since u, / V z 1 for small perturbations.
Because of the fact that the derivative
Y ,
enters into Equation (11) as an addition to the mass term, it is termed the virtual mass coefficient. (The termY ,
is always negative; i.e., Y acts to oppose positive 6
,
see Section 4.2.) I t is thus identical to the concept of added mass. (The force required to accelerate a body in a fluid is always larger than the product of the actual mass of the body times its acceleration. This fact has given rise to the concept of "entrained" or ('added)) mass. However, this added force should be really in-
terpreted as the hydrodynamic force arising because of the acceleration of the body in the fluid. This is precisely the definition of the Y, v-force in (10)). Sim- ilarly, N ' + . is termed the virtual moment of inertia coefficient. The derivatives
Y + .
and N ' , are termed coupled virtual inertia coefficients. As noted earlier, these derivatives would be zero if ship hulls, including their appendages, were symmetrical about their yx- planes.It is convenient to use a notation that distinguishes the forces and moments according to their origins. For example the notation Yuv will be used to denote the y-component of the hydrodynamic force acting a t the center of gravity of the ship that is developed as a result of an angle of attack, p. As has been shown, Y,v is only a linear approximation to, or linearization of, this Y-force as will be further evident from later examination of Fig. 10. Similar symbols and definitions are included in the nomenclature for other forces and moments.
3.5 Control Forces and Moments. It is important to note that all of the terms of (10) or (11) must include the effect of the ship's rudder held a t zero degrees (on the centerline). On the other hand, if we want to con- sider the path of a ship with controls working, the equations of motion (10) or (11) must include terms on the right-hand side expressing the control forces and moments created by rudder deflection (and any other control devices) as functions of time. The linearized y- component of the force created by rudder deflection acting a t the center of gravity of the ship is Y, 6, (see Fig. 5) and the linearized component of the moment created by rudder deflection about the z-axis of the ship is N, 6, where
Y,Y
Fig. 5 Rudder-induced turning moments
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CONTROLLABILITY 199
6 , = rudder-deflection angle, measured from xx- plane of ship to plane of rudder; positive deflection corresponds to a turn to port for rudder@) located a t stern
Y,, N , = linearized derivatives of
Y
and N with re- spect to rudder-deflection angle 6, The lateral force from the deflected rudder thus cre- ates a moment to turn the ship. This turning action causes the ship to develop an angle of attack with respect to its motion through the water. The lateral forces then generated by the well designed ship (acting as a foil moving in a liquid at an angle of attack) create a moment,N,v,
that greatly augments the rudder moment. The combined moments cause the turning motion as indicated in Fig. 5.For the case of small perturbations, which is the only case where (10) and (11) apply, only small deflec- tions of the rudder are admissible. With this restric- tion, the derivatives such as
YIt,,
NIL,, Y r , andN',
are evaluated a t 6, = 0 and are assumed not to change at other admissible values of 6,. Furthermore, for usual ship configurations,Y'?
z 0 and N ' , =: 0.With these assumptions the equations of motion in- cluding the rudder force and moment, are as follows:
(12) Force:
(Sway) = Y ' , 6, where:
A', i, -
Y',v'
- ( Y ' ? - A) r'n', = I f z - N , E 2 I , A ' , = A ' - Y 6 zz 2 A '
I t will be shown in the next three sections how the linearized equations developed so far can be used to analyze the problem of course stability and steady turning. But to make numerical predictions it is nec- essary to obtain values for some or all of the coeffi- cients or derivatives involved. This is primarily done by means of captive model tests, as discussed in Sec- tions 8, 9 and 16. Theoretical approaches to estimating some coefficients and approaches are also described in Section 16.