8.1 Nonlinear Equations of Motion. Captive model tests and associated simulation studies using nonlinear equations are the most powerful and flexible means available today for predicting controllability. This ap- proach may be initially more costly than a free-running model test program, but once the required hydrody- namic coefficients are determined from the model test data, a wide variety of more accurately predicted ma- neuvers and ship operations can be rapidly and eco- nomically simulated, with the effects of environment, control systems, and external forces readily studied.
Linear theory as discussed in prior sections is useful for analyzing the influence of ship features on controls- fixed stability and on the turning ability of directionally stable ships in the linear range. Captive model tests
can be used to develop coefficients for these equations.
However, as previously noted, linear theory fails to predict accurately the characteristics of the tight ma- neuvers that most ships are capable of performing, and it cannot predict the maneuvers of directionally unstable ships.
There is no completely analytical procedure available to this date (1989) for predicting the characteristics of these nonlinear maneuvers. As a result, current com- puter-aided techniques utilize the experimental results from captive model tests, with the equations of motion expanded to include significant nonlinear and coupled terms.
A variety of different approaches to developing a set of nonlinear equations of motion exist, ranging Previous Page
218 PRINCIPLES OF NAVAL ARCHITECTURE
from using wing theory to applying a Taylor's series expansion to force and moment parameters. A "Cat- alog of Existing Mathematical Models" (Hagen, 1983) provides a primer on nonlinear models and reporting of coefficients and data. The works of Abkowitz (1964, 1965), Strom-Tejsen (1965), Eda and Crane (1965), No- rrbin (1971)) Goodman and Gertler (1976), and others should also be reviewed. Fedyayevsky (1964) devel- oped a modular system based on physical relationships and the use of wing theory, and more recently, the Japanese Mathematical Modeling Group (MMG) through Kose (1982) and others has developed a similar modular model (Section 16 addresses these develop- ments and the modular approach).
The Abkowitz and Strom-Tejsen Taylor's expansion approach is herewith presented to give an understand- ing of the development of this popular non-modular model. The approach is based on a restatement of Equation (6) to include rudder angle as follows:
Y
= f(u, V , r,U,
V , +, 6,) (30) N"1
I t is assumed in Equation (30) that the only impor- tant forces and moments acting on the ship induced by the rudder are those due to rudder deflection, 6, and that the forces and moments produced on the ship as a result of 6, and
8,
are negligible.The complete Taylor expansion of Equation (30) with terms up to the third order is as follows for X with similar expressions for
Y
and N .(As noted in Section 6.7, tight maneuvers involve large speed losses; hence, consideration of the X-equa- tion constitutes a vital part of this section whereas it could legitimately be neglected in the earlier consid- erations of linear maneuvers):
X = X o
+
[X,6U+
X,v+
X,r+ X,U
+
$ [X7Luu6u3+ xuvv?13 +
. . . .. +
X,,,SR3+
6XU,,6uvr+
6Xu,,6uvU+
. . .+
6Xi,,,V+6,]where X o is the force in the x-direction at the equilib- rium condition, that is, u1 = V
IF X INCREASED
0
Fig. 30 Three possible relationships between X and v
Terms higher than third order are not included in Equation (31) because experience has shown that ac- curacy is not significantly improved by their inclusion.
Furthermore, practical limitations of measurement techniques and the state of refinement of present the- ory do not justify the inclusion of higher order terms.
As a consequence of the geometrical symmetry of ships about the xx-plane, the relationship between X and v, for example, must correspond in general form to one of the three relationships shown in Fig. 30. The feature common to all three relationships is that they are symmetrical about the ordinate X . If the relation- ship between X and v as depicted by curves 1 or 3 of Fig. 30 is to be expressed as an expansion in powers of v beyond the first power, then only the even powers of v can appear in the expression and the coefficients of the odd powers must be zero. That is, X is an even function of v which takes the form:
X ( v ) = azv,
+
a4v4+
asv6+
. . . (32) wherea, = XX,,; u4 = 1/24X,,,, etc., from Equation (31) Again, as a result of symmetry about the xx-plane, Abkowitz (1964) shows that X i s also an even function of r ,
a,,
d and i , that is,X ( r ) = b2r2
+
b 4 r 4+
b,r6+
. . .X(6,) = C 2 6 R 2
+
c48R4+
c 6 6 R 6+ . .
(33) (34) It follows from the previous analysis that cross- coupled terms in Equation (31) such as Xvuv6u,CONTROLLABILITY 219
X m r 6 u , X,,6,6~, Xvvvvv3Su, XrrrUr3Su, and so on, in- volving odd powers of v, r, and 6 , are also zero. How- ever, cross-coupled terms such as XV,,,v2 6u, XrrUr2 6u, X S G u S R 2 8 ~ , and so on, are nonzero because they involve even powers of v, r, and 6,. Also, terms such as X,,vr, X,,V,,, XT,r6R, X,,vr6u, and XTa,r6R6u are nonzero because they involve even-powered products of v, r, and 6, (see also Section 8.6).
In contrast to X , the expressions for Y and N are odd functions of v, r , 6, V, and r ; that is, only the coefficients of the terms in the expansion with odd powers are non-zero; those with even powers are zero.
Odd functions are like those shown in Figs. 6, 8, 11, and 13 where in all cases the graph of the function is reflected about the origin. The expansion of Y or N as a function of v, r, S,,
V,
or .i. is typically as follows:Y(v) = d,v
+
d3v3+
d5v5+
.. .
Y(6,) = e l S R
+
e3aR3+
e5aR5+ . .
.(35) ( 3 5 4 Although superficially it appears that there should be a correspondence between the relationship of Y to v shown in Equation (35) and the relationship of X to u, in reality they are vastly different, for several rea- sons. One is that the equilibrium value of v, designated vl in Section 3, is taken as zero. (Any asymmetry due to propeller rotation is neglected for this restricted purpose but is taken account of later in Equations (37) and (38).)
The equilibrium value of u, u, is not zero but is equal to the ship velocity, V. Another reason is that the X-force is the component along the x-axis of the difference between two oppositely directed forces, namely, the ship resistance and the propeller thrust, whereas the Y-force is the component of a direct hy- drodynamic force. For these reasons and others, X is neither an odd nor an even function of u but rather its expansion includes all powers of 6u.
Additional terms of the nonlinear equations can be eliminated by considering the nature of acceleration forces. Abkowitz (1964) states that no second or higher order acceleration terms can be expected, on the as- sumption that there is no significant interaction be- tween viscous and inertia properties of the fluid and that acceleration forces calculated from potential the- ory when applied to submerged bodies give linear terms. Hence all terms such as XCCiL2, X&,
Xi+?,
Xii,2i3, and so on, of Equation (31) are taken as zero.
Since X , and X i are also zero because of symmetry, the only acceleration derivative that is not taken as zero in the nonlinear equation of motion for X is X,, which is also the only acceleration derivative that ap- pears in the linear equation for X (10).
Combining the nonlinear Taylor expansion for X , Equation (31) with the dynamic response terms of the X-equation, Equation (5), and taking all of the preced-
ing considerations into account, the equation for X becomes:
where
L(U,
V, r, 6,) = X o+
X,SU+
~X,,,SU'+
iX,,,6u3+
fX,,v2+
fX,,r2+
:X,,SR2The relationship between Y or N a n d Su corresponds to that shown in curve 2 of Fig. 30 for X versus v.
That is, because of symmetry about the xx-plane, Y(u)
= N(u) = 0 and the derivatives Y,, Y,,, Y,,,, Y,, N,, Nu,, N,,,, and Nu are all zero.
As stated earlier, Y and N are odd functions of a,
T ,
a,,
6, andr.
I t follows that all the cross-coupled terms in the complete Taylor expansion of Y and N involving even powers or even-powered products of v, r, 6,, V , and r a r e zero. Thus Y2,t,Uv26~, Y,,,r28u, Y,,vr, Y,,v~R, YrarSR, Y,,vT~u, Y,,,vS,~U, etc., and similar terms for N , are all zero.The Y-force and N-moment induced by the rotation of a single propeller or by unirotating multiple pro- pellers, at v = 6, = 0, identified as Yo and NO in Equation (26) must, of course, be included in the non- linear equations for Y and N. In addition, since Yo and NO are likely to be speed dependent, the following terms are also taken as nonzero in the Taylor expan- sion:
Y," 6u N," su Y,,"(~u)~ N , , 0 ( 6 ~ ) ~
The Y-force and N-moment induced by propeller ro- tation a t v # 0, also discussed in Section 17.9 (see Fig.
253), as well as their speed dependency, are included in the followng nonzero terms in the Taylor expansion:
Following Abkowitz's reasoning as noted in the dis- cussion of Equation (31) the only acceleration deriva- tives not taken as zero in the nonlinear expansion of Y and N are those appearing in the linear equations of motion. These are Yc, Y+, Ni,, and N+.
Combining the third-order Taylor expansions for Y and N [similar to that shown for X in (31)], with the dynamic response terms of the Y and N-equations of Equation ( 5 ) and taking all of the preceding consid-
220 PRINCIPLES OF NAVAL ARCHITECTURE
erations into account, the nonlinear equations of mo- tion for Y and N are as follows: Y-Equation:
N-Equation:
- NCV
+
(I, - N + ) i = & ( u , V, r, 6R) (38)An equation similar to Equation (38) could also be developed for the roll moment, K , which could be used to solve for the heel angle,
4,
as a function of time.Equations (36), (37) and (38) can be solved simulta- neously for the accelerations u, V, and r, as follows:
These solutions can be rewritten in the form:
where u ( t ) , v ( t ) , r ( t ) , and SR(t) are the instantaneous values of u, v, r, and 6, at any time, t.
Equation (40) is a set of three first-order differential equations for which approximate numerical solutions are readily obtained on a digital computer. The key to the numerical solution is that values of u, v, and r at time t
+
S t are obtained from knowledge of the values of u, v, r, and 6, a t time t using a simple first-order expansion; that is,u ( t
+
6 t ) = u ( t )+
6 t i L ( t )v(t
+
6 t ) = v(t)+
Gtiqt) (41) r(t+
S t ) = r ( t )+
G t i ( t )This method is found to give adequate accuracy for the present type of differential equations because the accelerations u, V , and r vary but slowly with time, owing to the large mass or inertia of a ship compared to the relatively small forces and moments produced by its control surface. Any desired accuracy of the solutions can be obtained with a computer by using smaller time intervals 6 t .
The mathematical model has been developed in di- mensional form. The equations are equally valid in nondimensional form with the stipulation that the ve- locity used for nondimensionalization should be the velocity at any time, t, but not the initial velocity. For further simplification, the nondimensionalizing veloc- ity in the nonlinear equations is taken as u ( t ) rather than V ( t )
One reason cited in this chapter for non-dimension- alization is that the nondimensional derivatives are independent of speed. The extent to which this as- sumption is not true for nonlinear maneuvers is taken account of in Equations (36), (37), and (38) by the in- clusion of such terms as Y:,, YL,,, YL, YL,, YA,, YAU,, and so on, which represent the changes in the nondimensional derivatives Y:, Y;, Y s with speed.
Assuming that a full set of hydrodynamic coeffi- cients
(X;,
XkU,X;,,, Y,,
etc.) is available, and that the rudder deflection SR is defined as a function of time, the first step in the calculation of the trajectory of a ship would be to set the values of u, v, r, and 6, at time t = 0. In the most usual case u, v, and S R atCONTROLLABILITY 22 1
t = 0 would be zero and u would be equal to u l . Having done this, u, 6, and i can be calculated from Equations (39) and the new velocities a t time t = S t can be ob- tained from Equations (41). The process is then re- peated using the new values for u, v, r, and 6 , in Equations (39) and so on. The values of the velocities at a time t are thus obtained from
t-st
u ( t ) = u(0)
+ c
i L ( T ) S tv ( t ) = v(0)
+ c
?j(T) S tr = O
t - S t
142)
r = o
r = O
where u(O), v(O), and r ( 0 ) are the values of u, r, and v at t = 0 and T represents intermediate values of time (between time 0 and time t - S t ) at which the accelerations iL(t), ?j(t), and + ( t ) are determined.
The instantaneous values of the linear velocities of the ship relative to earth axes (which are needed to plot trajectories) instead of relative to ship axes are obtained from Equation (3) I re-expressed as:
k o o ( t ) = u ( t ) cos + ( t ) - v ( t ) sin + ( t ) i o o ( t ) = u ( t ) sin + ( t )
+
v ( t ) cos + ( t )(43)
where koo(t) and i o o ( t ) are the components of the in- stantaneous resultant velocity of the origin, 0, of the ship along a fixed set of earth axes x, and yo, respec- tively.
The instantaneous coordinates of the path of the origin of the ship xoo(t) and yo,(t) relative to the fixed set of earth axes and the orientation of the ship, +(t), can then be obtained by integration of the last of Equa- tions (42) and (43). These are as follows:
t - f i t
t--Rt
+
[ u ( T )+
u(O)] sin+(?)I
6 t (44)- v ( ~ ) sin + ( T ) ) S t
There remains the question of defining the rudder deflection as a function of time. I t is assumed that the
rudder moves with a constant rate of deflection,
6,,
determined in accordance with condition 2 of Section 9.2, and that there is a fixed time lag between the instant that rudder deflection is ordered and the instant that the rudder begins to move (see item ( d ) of Section 5.1). A rudder deflection up to a certain maximum angle 6 , would be simulated in a computer program as fol- lows:
= until t > tl,,,
+
t othen S R ( t ) = S R ( t o )
+
rate ( t - to - tlag)until 6,(t) =
aconst
then = aeonst
A rudder function of this type gives a very close approximation to the actual time history of a ship's rudder when a maneuver is ordered from the bridge.
8.2 Captive Model Tests. Captive model tests in tanks are now carried out using a planar motion mech- anism (PMM) or a rotating arm. In either case the model is tested over a suitable range of important variables such as drift angle, yaw rate, sway accel- eration, yaw acceleration, propeller RPM and rudder angle, and the results are analyzed to obtain the hy- drodynamic coefficients required in the equations of motion. Development of the linear coefficients will be addressed first.
For design of a control surface, knowledge of the lift, drag and center-of-pressure location as a function of angle of attack, velocity, and control-surface con- figuration as given in Section 14 is adequate for most practical problems. Knowledge of the forces and mo- ments generated by control-surface rate of deflection, 6, and angular acceleration, 8, are only occasionally important to the design of the steering engine that deflects the control surface and rarely, if ever, to the motion of the ship V.S. the control-surface system as a whole. However, it was found in Equation (14a) that to determine whether a ship is stable or unstable in straight-line motion, one must know not only the forces and moments generated by angle of attack on the ship, but also those generated by angular velocity. In ad- dition to these, the forces and moments generated by linear and angular acceleration must be known in order to determine the magnitude of the stability indexes, Equation (14) or to compute the trajectory of a ship from the equations of motion.
The experimental techniques necessary to measure the significant forces and moments generated by a ship's hull are much more elaborate and sophisticated than those necessary to measure the significant con- trol-surface forces. Only in the case of the determi- nation of the velocity-dependent derivatives of the hull is the experimental technique similar in principle to that used to determine control-surface forces and mo-
222 PRINCIPLES OF N A V A L ARCHITECTURE
f / / / / / / / / / / / / / / / / / / / / / / / / / / / /
1
+ Y/ / / / / / / / / / / / / / / / / / / / / / / / / / / / Fig. 31 Orientation of model in towing tank to determine Y and N (Abkowitz,
1964)
ments as used for this purpose in a modern wind tunnel (Whicker, 1958), or by Joessel in the river Loire in 1873 (Van Lammeren, et a1 1948).
8.3 Straight-line Tests in a Towing Tank. The ve- locity-dependent derivatives Y, and N, of a ship at any draft and trim can be determined from measurements on a model of the ship, ballasted to a geometrically similar draft and trim, towed in a conventional towing tank a t a constant velocity, V, corresponding to a given ship Froude number, at various angles of attack, p, to the model path. Fig. 31 indicates the orientation of the model in the towing tank. From this orientation it is seen that a transverse velocity component, v, is pro- duced along the y-axis such that:
v = -Vsinj3
where the negative sign arises because of the sign convention adopted in this chapter, Fig. 2.
A dynamometer a t the origin, 0, measures the force Y and the moment N experienced by the model a t each value of p tested. These measurements are then plot- ted as a function of v (Fig. 10) and the slopes of the curves taken a t v = 0, give numerical values for the derivatives Y , and N , for the model. These values can be reduced to nondimensional form by dividing by the proper combination, given in Section 3.4, of model length L , model speed V, and towing-tank water den- sity, p. The dimensional ship values of the derivatives can then be obtained by multiplying the nondimen- sional derivatives by the same respective combinations of ship length, ship speed, and seawater density.
With reference to Equation (lo), it is not really nec- essary that the origin, and hence the dynamometer, be located a t the center of gravity of model. The results are independent of the location of G. Rather it is most convenient that the origin and the dynamometer be located at
a
so that xm = 0. If the dynamometer is not located ata,
the derivative N, should be corrected so that it applies to 0 ata.
As described in Section (17), the propeller will usu- ally exert an important influence on the hydrodynamic derivatives. Therefore, model tests to determine these derivatives should be conducted with propellers op- erating, preferably at the ship propulsion point. Also,
since the undeflected rudder contributes significantly to the derivatives the model tests should also include the rudder in the amidship position.
The technique just described can also be used to determine the control derivatives Y, and N,. If in Fig.
31 the model were oriented with zero angle of attack, j3, to the flow but the model were towed down the tank at various values of rudder angle S,, the dynamometer measurements would determine the force Y and the moment Nas a function of rudder angle. Plots of these against rudder angle would thus indicate the values of the derivatives Y, and N,. In addition to these im- portant data, comparison of the values of Y, and N86, obtained by this means at any given rudder angle with the values obtained from Equation (120) at j3, = 0 using isolated control-surface lift and drag data would indicate the magnitude of the interaction effects aris- ing because of the close promixity of the rudder to the hull.
Straight-line tests in a towing tank can also be used to determine the cross-coupling effect of v on Y, and N, and of 6, on Y, and
Nu.
While such information is inadmissible within the context of the linear theory, it is important for the nonlinear theory. Also for the purpose of this theory, knowledge of the shape of the Y versus 6,, Y versus o, N versus 8, and N versus v curves a t large values of v and 6 , will be of importance.8.4 Rotating-Arm Technique. To measure the ro- tary derivatives Y,. and N,. on a model, a special type of towing tank and apparatus called a rotating-arm facility is occasionally employed. In this facility, an angular velocity is imposed on the model by fixing it to the end of a radial arm and rotating the arm about a vertical axis fixed in the tank as seen in Fig. 32. The model is oriented with its x-axis and z-axis normal to
tr
*v = U l
a = o , v = o
CIRCULAR PATH OF MODEL tr
*v = U l
a = o , v = o
CIRCULAR PATH OF MODEL
Fig. 32 Orientation of model in rotating-arm facility to determine Y, and N (Abkowitz, 1964)