Frequency response research on steering qualities of ships

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Frequency Response Research on Steering

Qualities of Ships

By

Kensaku NOMOTO

Reprinted from

TECHNOLOGY REPORTS OF THE OSAKA UNIVERSITY Vol. 8 No. 294

Faculty of Engineering Osaka University

Osaka, Japan

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(Received Nov. 12, 1957)

Frequency Response Research on Steering

Qualities of Ships

By

Kensaku NOMOTO (Department of Naval Architecture)

Abstract

Now the frequency response approach is being widel used in

communi-cation and control engineering. It may be also su essfully applied to

steering problem of ships. The present paper relates to fudamentals of

steering research in this scheme.

Preface

It is a desirable quality for a ship that she well-behaves in steering : to

keep her course without any difficulty and to change her heading quickly when

necessary. In consequence,, there have been a number of scholary studies on

steering and turning1,2,3,4,5). They employ equation of motion with several coefficients : some of them indicate an inertia and moment of inertia of a ship ; some, called "resistance derivatives ", indicate resistance of water ; and others indicate forces acting on her rudder. The solution of the equation determines the motion of a ship for a given process of her helm angle. Thus the dynamic

character of a ship in steering is described by the equation of motion and

consequently by a set of the coefficients. Eq. (1), which was written by .Davidson

and Schiff), may be a standard of these equation of motion. Employing the equation, they obtained many useful results on steering problems of ships, and

Schiff and Gimprich developed the treatment to the problem of auto-piloting6).

The method has, however, a practical difficulty. Determination of the

coefficients for a given ship requires lengthy experimental procedures introducing somewhat exceptional instrumentation such as a three component dynamometer,

a rotating-arm apparatus or curved models. This is probably the reason why the studies on steering have largely been limited to fundamental researches in

laboratory.

Then the author attempted to construct a new expression of steering quality

73

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74 Frequency Response Research on Steering Qualities of Ships

which describes dynamic character of a ship in steering just as do the equation of motion, and which may be determined from observations of a given process

of helm angle and the resulting motion of a ship or a free-running, self-propelled

model. This is quite possible in principle : since the process of helm angle is connected to the motion of a ship through her dynamic character in steering, it is natural that the simultaneous observations of the two should provide a good information on her steering quality. This point of view has a close relation to the concept of transfer function or frequency response technics which has been

used successfully in communication and control engineering. In the analytical sense, the present method and the usual method of coefficients are equivalent.

Both of them define the governing equation of motion and consequently provide a complete expression of steering quality of a ship. The former has, however, the advantage that it does not require such experimental procedures as lengthy as the latter.

Another important aim of the present treatment relates to a more generalized investigation of the steering problem considering whole steering system composed

of a ship, her helmsman or auto-pilot and steering gear. As a matter of fact, the steering system is a feedback system as shown in Fig. I. The more complicated is the system, the

External Disturbance _{more difficult} _{is an} _orthodox

I b S

I Y ea.

... I procedure dealing with

simul-S I

Desirrd el Aar

Helmsman 0

iet Shlerin9- i-

:Lip

_especially, _some _{nonlinearities}

taneous differential equations

Gear

a

i6

_{in _autopilots which often have}

I CompiSSI

dominant effects on

course-keep-ing performance offer trouble-Fig. 1. Scheme of Steering System. _{some problems.} _{In such}

circum-stances, the frequency response approach is more advantageous. The present treatment of steering quality just follows this approach to suit the generalized

steering problem.

1. Transfer function expression of steering quality

Steering quality indices

1. 1 Introduction of transfer function in steering :The present work starts

from the modern studies on steering of ships employing equation of motion

and resistance derivatives, the general procedure of which is as follows :

(1) to follow the method of- hydrodynamics dealing with motion of a solid

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(2)

of which is at her C.G., x-axis along her longitudinal direction. and z-axis

vertical

(2) to consider a coupled motion composed of a rotation about the vertical axis (turning angular motion) and a translation along the lateral axis (drifting

motion), neglecting their dependence upon the other motion involving decelera-tion caused by steering ;

(3) to express resistance of water against ship motion by a linear function of drifting velocity and turning angular velocity of the ship.

There are several forms of equation of motion following these general procedure3) and the present approach for steering problem may be introduced from every one of these equations. In other words, a, equation of motion in every form based on the above general procedure would yield quite the same

transfer function expression.

According to Davidson and Schiff) in this case, the equation of motion is

mntr' Czik (m1 Cf)L) =CAS nfe! + Ck(2 Cmlir = C8 S where Q

=(I)e,

e: turning angular velocity,

V V: ship speed, ips : drift angle, : helm angle, 1: ship length

denotes c-i; =(-17) , s=

(7)

t

d 1 d V

ml, m2, n: coefficient of inertia

CZ, Ck, C., C1 coefficient of resistance viz, resistance derivative CA, C,: coefficient of rudder force.

The former equation corresponds to lateral translation and the latter corres-ponds to turning angular motion.

Although this nondimensional expression has a very fine form, it is a

equation of mlition on time, base that describes directly the motion of a ship. Moreover, this expression is often desirable in design problems and treatments

-of the whole steering system involving the other elements besides a shier. Then

rewriting the equation of motion (1) on time base,

(0m.2dtdt +C1*(-17)(miCf)e =CA8

(-bYnc-cilt-MCk0. = C,8

Taking the Laplace transforms of both sides of the equations,

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-17) M2*(0-) + (-t17-)m2P

+clvr(p)

(-) me (p)=Cx8(p)

(T17-)2n6+0-)+I(T17)2np + ()clip) cm* (p)= c (p)

where

then

0. (p) I V \ CMCA ± CICI, + ( 171 )CMCM1,2+CaCgP

\ 1 )CiCk - mC,i±( 1\ 2122Ck + nCi ( l )2 ' m2n p2

\V ICzCk-mCm, \V I CiCk-mCm

c.

(i 1 \ 2 m2n 1 m2Ck+nCi z

l\2

ni2n

-1k v )cick -mC.P+

(V)

mdu (0-) ± (

CiCk - TT) cc k - n2C2(°-)

1+ t 1\ m2Ck+nCz ±t 1 \ 2 M272 p2

V )CICk-mC, k 17) CiCk-mC. Unifying the coefficents, and factoring the denominator

K(1+ T?p) {TiT,p+ (Ti+T2)}0.0)+TiT2eCo)

(1+ TIP) (1+ T 2P) 8 (P) ± (1+ TIP) (1+ T2P) (3)

where( VI

C.Cx+ CiC TiT2=l 1\ 2 In2n

(1 \ m2Ck+nCz

K-

_{k 1} _CzCk-mC.' _{V) CiCk- mC.'(T1+ T2) - V )CyCk -111Cm '}

T3- (I

V)CmCMA2C+ICA

The first term of the right side corresponds to a ship motion excited by a steering 8(t), and

beginning of the

Ys(p) describes

called the transfer function of a ship in steering.

Since a transfer function of a system is generally introduced from. the governing differential equations through the Laplace transformation, it may be

retransformed to a differential equation when required. The operation is simply to

.d

substitute p by

' as is generally known., In .the present case, the

retrans-formed equation of motion is

(P) ik(t)e-ntdt, 6(P) 8(p)=- 8 (t)e-Ptdt

and m=mi-C1

The time origin of the transform 0- is selected at the beginning of steering. Eliminating ik(t) from these simultaneous equations, and also eliminating

1./f(o_) by the following condition at the beginning of steering,

rt.(-1-17.)2e(0--)C+1°6-17 d(0-) C vOlt (0-) = °

wo obtain

the second corresponds to a memory of her imotion at the

K(1+ T3P)

steering. Therefore, a rational function _{(1+ Tip) (1+ T2p)}

a response character of a ship in steering, which may be

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d28 do

Ti T2 Trt:3 + + T2)-dt + . (4)

This equation describes, steering motion of a ship just as do the original &equation of motion (2) so far her turning angular motion is concerned. On tithe, other hand, this equation does not provide any information about her drifting motion. A drift angle of a ship in usual steering is, however, the order of few -degrees and its effect on her path is practically negligible. Therefore, the equation of motion (4) or the transfer function Ys(p) is sufficient to describe steering motion of a ship.

The coefficieknts of the original equation of motion (2) must be determined

usually through experimental procedure, except the coefficients of inertia. .01_211aLe.

-towing tests with a three-component dynamometer give Cz, Cm and probably Cx and C. Turning towing tests with a rotating arm carriage or oblique towing tests for curved models provide Ck and Cf. It is possible, of course, to define \ the transfer function through determining all the coefficients of the original equation along these procedure. There is another way, however, to define the transfer function and then the equation of motion (4) : that is, to record a given process of helm angle and the resulting motion of a model or actual

20 cib

-30

-50

-2 0 -

-/0

Fig. 2. Steering Transfer Functions of Two Ships.

S4'l \-t-_t

.143 .0596

Chain lines represent first-order simulation (See Part 2) A FREIGHTER IY51,o)1 in. db SHIP A

Ilibilibseiw)li"

db 20* 440.

6

. 4fr 60° so° AP, .

(Po

11111114

labb,..

ri

MIL

K - Ti T2 T3 T Ship A .10 5.9 1.1 2.9 4.1. A Freighter .090 45 6.0 10 41

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ship, and to apply the harmonic analysis to the results. Details of the procedure

will be discussed in Part 3. This procedure may be much easier than the one determining the coefficients of the original equation. This is a remarkable advantage of the transfer function approach in steering researches.

Fig. 2 illustrates the transfer functions of representative two ships in Bode's expression which consists of the plots of 20 log II's( jco)I and Arg Ys(jco)

against log co. t(jco) denotes the transfer fuction Yg(p) in that case when p is selected as jco (Refer to 3.1). Ship A was introduced from the "ship A" in the paper of Schiff and Gimpricho by estimating prbbable .values of- her speed

-and length. Her transfer function was constructed, from the coeffieients of equation of motion shown in the paper, which had been probably determined through oblique towing and turning towing tests. The other ship is a

.full-loaded, ocean-going freighter, the transfer function of which was determined. through the frequency response procedure at the Osaka University Tank. "Ship A" is probably a T.B. destroyer and a typical model of quick responsi-bility in steering. The freighter has quite poor quick-responsiresponsi-bility.

1. 2. K, T1, T2 and 7-3 Steering quality indices :

.T1 and T The general -solution of the equation of motion which defines the general character of all possible motions of a ship has ,the form

Ciet/T2

Then it may be said that in respect to T1 and T2

Positive values of both of T1 and T2 assure stability on course and a negative value of either or both indicate instability on course, where a stable ship means such a ship that has an inherent tendency to decrease her turning angular velocity after an external disturbance without any steering. Any

imaginary value of them indicates the presence of an inherent periodic yawing,

as is never the case for usual ships.

After a small disturbance, the disturbed motion of a stable ship with her rudder amidship is damped exponentially with a rapidity dependent on the exponent T1 and T2. The smaller values of them correspond to the quicker

decay of the motion.

When a certain helm angle is set, a stable ship enters into turn exponen-tially with a rapidity dependent on the exponent T1 and T2. The smaller values of them correspond to the quicker build-up of turning.

In a word, T1 and T2 are indices of stability on course and quick-responsibility in steering. These two abilities are quite the same things, because the former

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say T1, is fairly larger than the other for ordinary ships, it may be reasonable

to concentrate on Ti so as Davidson and Schiff. Their stability index pi is

_( 1\ 1

kv)T1"

unify T1, T2 and 7'3 so as .to produce a new effective time constant T. This is

the first-order simulation discussed in Part 2.

if:

When .a certain helm angle is set, a stable ship enters into turn increasing gradually her turning angular velocity and finally she settled in a steady turning with a constant angular velocity. K. is the proportion of the angular .velocity in this steady turning to the helm angle, as is evident from the equation of motion (4).

In a turning with a large helm angle, however, the realized steady turning angular velocity is generally different from the one which a ship approaches during the initial phase of turning, because considerable speed reduction and gradual variations of resistance-derivatives come to nonlinearize the motion.

-Since an usual steering motion may be regarded as a succeding,process of, initial

phase of turning, it is more reasonable to define K with respect to the steady turning which a ship approaches than to define it with respect to the realized steady turning.

It should be noted that circumstances are completely different for an unstable

-ship; she falls into turn towards an infinite turning angular velocity even by an infinitesimal small helm or external disturbance, while her wild motion is

eased soon after by nonlinearity as shown by Davidson and Schiff4). Therefore,

K is rather meaningless for an unstable ship. Analytically, K of an unstable ship has generally a negative value. It seems that, however, the problem is one of rather theoretical interest, because instability is fairly rare for usual -ships, and if it be found, it is corrected in most cases by an additive fin or by

increasing rudder area.

In summary, K is an index of turning ability for a stable ship, relating (D

colsely to the Davidson's index 7 min.

2'3: T3 represents the contribution of steering speed in initiating turning

motion, as is inspected from the equation of motion (4). No ship has, however, so fast steering speed or so large T3 that the effect may become dominant. In addition, 7'3 and T, affect a ship's dynamic character in such a manner that

their effects cancel each other. To unify T1, T2 and 7'3 to an effective time constant is reasonable in these circumstances.

Thus the indices K, T1, T2 and 7'3 describe tlie general character of a ship in steering quite directly. Then the indices may be called steering quality indices equivalent to Another way to simplify in a paractical manner is to

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80 _{Frequency Response Research on Steering Qualiites of Ships} 2. _{First-order simulation in steering problem} _{K, T-expression}

2.2. First-order simulation of response of a ship to steering: Let us

consider-such a steering in which the helm angle increases uniformly up to a certain angle 80, then remains at this value. Every steering process produced by usual steering gears may by considered as a sequence of this type of steering to a fair approximation. The response of a ship is easily obtained from the

equation of motion (4) as follows ;

during where ti is the time when 80 is reached,

0

20 Kio f

tz

T, - T3 2 Z/7;

0(e)

7-2

/- e

-

rz 72- T3 7i2 2

)

I

The result suggests that the motion of a ship

may be simulated by al

30 5'0

_Sec.

Fig. 3. Transient Phase of Turning Motion for two Ships. Chain lines represent first-order simulation.

-7" 3 - T2 _r1 43.7" 73 722

(

0

- T2

ti

6(

t)

SHIP A

fi---

)

. _

agel after ,'he Hine. t1,

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response of

a first-order system to the same o(t).

The first-order system generally means a system with a governing differential equation of first-order, that is in this case,

Tdi +6=--Ko_dt (5)

The response of the system to the o (t) now considered may be obtained by

substituting T1,---=-.T and T2,-- T3=0 into the above equations expressing the

response of a ship. Then selecting T as equals T1 + T2- T3 and taking the same K, the responses of a ship and the simulating first-order system coinside except some disagreement in transient phase.

Fig. 3 shows results of numerical calculation for the representative two ships, viz. "ship A and a freighter presented in Part 1. As is shown in the figure,

the first-order simulation

is more satisfactory for a ship with high

quick-responsibility, i.e. small T. In a practical sense, however, the simulation may

be consid.ered sufficient even for a ship with poor quick-responsibility.

Since the simulation is sufficient for the steering type now considered, and

since every steering produced hy usual steering gears is a sequence of this type of steering, the simulation is expected to be successful for all usual steering motions.

2. 2. Analyses of z-steering tests :Z steering test (the standard manoeuvring

test) proposed by Kempf7) may be utilized for examining the reliability of the

first-order simulation. The procedure is tci construct a first-order simulating system by selecting proper values of K and T so that its response to measured 8(t) may simulate measured zig-tag motion of a ship as closely as possible, and to examine the agreement between the measured ship motion and the calulated one for the simulating system. In fact, such an examination may be also a touchstone for the linear treatment of steering problem.

From this Point of view, Osaka University Tank carried out the analyses of Z-steering tests for about fifty ships including whalers, freighters and tankers

ballasted or loaded. Fig. 4 and 5 show two typical results of the analyses. The concerned ships are a ballasted freighter and a fulhloaded tanker. Obviously.

the first-order simulation is quite satisfactory for the ballasted freighter.

Refering to a number of results for ballasted freighters and tankers, it may be

said that a ship in ballasted condition is generally fit for the first-order simulation

because of her small T. The illustrated result for a tanker is less favourable,

while it may be still within a permissible tolerance practically. Such a disagree-ment is a common feature for a number of full-loaded super-tankers and sometimes

it is found for full-loaded freighters. It may depend upon the speed reduction .-caused by steering and other nonlinear effects, because the disagreethent can

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82 _{Frequency Response Research on Steering Qualities of Ships} 20 -20° o' LI -20° / 0 0 200 sec.

Fig. 4. Z-Steering Test Result for a Ballasted Freighter.

/00 200 300 sec

Fig. 5. Z-Steering Test Result for a Full-Loaded Tanker.

not be reduced by advancing the order of linear treatment by recalling 7'2 and T. In other words, it suggests -a limitation of the linear treatment. Adaptablity

of the first-order simulation for usual freighters half and full loaded seems to be between the ones for ballasted ships and full loaded super tanker

...:, .,____ a

..2t,

t

./..

1

II ,

/

./ °

I

2 + 2t1

t.

t--2t,

K T (I+) =,=,=- PLOT OF MEASURED 19 8' .5

t;

CALCULATION OS.; 73 .072 e OF CHECK

imra

1111"

11.111R,

MAE

a

'

FA MI

11 III

MIN

ill

01.

,,i,

Him

immli.

1

-o= opc

(12)

In summary, it may be said that the analyses of Z-steering tests assure

the first-order simulation on the whole.

2. 3. _{First-order transfer function :Another way for assuring the first-order}

simulation may be provided by expanding the transfer function into a power

series. That is,

K(1+ T3P)

K11 (Ti+ T2 - Top

178(P)_{=-(1+ Tip) (1+ T2p)}

-L (TO + T22 4 Ti T2 - T2 T3 - T3 TO p2...}

This expansion is valid for _{TIP I <1.} On the other hand, also expanding the

first-order simulating transfer function, we obtain

1+ 1 - Tp + T2p2 _ _

Tp

ThiS expansion is valid for Tp

<1.

Comparing the two expansions, it is inspected that the first-order simulation by putting T= T1+ T2- T3 means the first approximation of Ys(p) which is valid

for TiP I <1. According to the interpretation of the transfer function in the aspect of harmonic analysis, the fact means that response of a ship to low

frequency sinusoidal steging may be well expressed by the first-order simulation.

Considering that every actual steering may be composed of infinitely, large numbers of sinusoidal steerings, the first-order simulation may be adequate for every steering involving mainly low frequency elements. The low frequency

1

means in this case a frequency smaller than 1 In other words, is a rea, sonable unit for measuring the frequencies.

As a matter of fact, an usual ship has not so large T1 and steering process realizable with an usual steering gear is not so hasty. Therefore, the dominant

freqencies in actual steering may be considered relatively low. And in addition, the invalidity of the simulation in higher frequency range does not induce

so considerable difficulty, because the magnitude of the transfer function of a ship

decreases generally with increase of frequency. Thus the simulation may be

successful for steering problems.

As the conclusion of Part 2, it may be said that steering motion of a ship is substantially first-order phenomenon, and consequently steering quality of a ship may be described in brIef by the two fundamental indices, namely K, an index of turning ability, and T, an index of quick-responsibility in steering and dynamic stability on course. In other words, an actual steering motion of

a. §hit) may be well described by the firstorder equation of motion,

-d

(5)

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K. NOMOTO 85

numerical integrations of 0"(j co) and 8( j co) . This procedure means to excite a ship by a steering composed of numerous sinusoidal steerings, and to obtain her frequency response for each sinusoidal steer. ing at one stroke through

resolving the excited motion into sinusoidal elements.

In carrying out these transient response procedure, such a o(t) that finishes its whole process during a relatively short time is .preferable, because it does not require so large a test basin for model tests and for application to actual ships it saves much testing time. Furthermore, it provides a good

information on Yx(jco) up to high frequency range, since such an impulsive o(t)

involves sinusoidal elements distributing up to high frequencies. _{On the other}

hand, over steep o(t) must be avoided because its large amplitude and convulsive

manner would disturb the motion beyond the_usual linear range. Then the optimum width of the impulsive steering may be about the reciprocal of the

highest frequency up to which the determination of Y9( j co) is required.

The selection of a detailed form of the impulsive ô(t) may be arbitrary, that is, trapezoidal, trigonometric or parabolic. For instance, the "sine impulse that is in practice at the Osaka University Tank is defined as follows,

6(0 =o0 sin for

(t) =0 for otherwise.

As a matter of fact, it is fairly difficult to realize so exact

impulse as to yield a formal integration of ô (jco). Then the actual 8 (t) and to obtain 8(jco) through a numerical int

performed applying the Simpson's method for each of the real and imaginary part of the integration, that is

sine or other formal it is better to record

egration. It may be

8 (j co) = 8 (aces cotdt j 8(1) sin cot dt, ₍₇₎

where 0 0 and t1 denote the beginning and finish of steering respectively.

The integration of 0 (jco) for a measured 0 (t) may be tarried out in two steps,

into which the integration is devided at a time 4. tb is selected so large that the motion 0 (t) has already become a single exponential decay defined by T1 at

the time. It must become so, because (t) after the finish of (t) is a free motion

of the form

CieTt/T1+ C2et/T2

and T1> T2. According to experience, tb e ual to twice T seems sufficient

for satisfactory results.

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inter-y1log e

Now summing up the two integrals (8) and (9), 0(jco) may be obtained. Then dividing the 0 (jco) by 8( jco), viz. (7), Y(j0) is obtained.

-As a matter of practice, there are some difficulties to operate a self-propelled model so that her disturbed motion vanishes completely before a test, particularly

in a- small test basin. Then sometimes the above procedure must be followed by a correction to separate out the effect of the disturbed motion. According

to Eq. (3), the correction term to be subtracted from 61(jco) obtained from

measured a (t) along the above procedure is

{ ( T1+ T2) -Hico 2-2} 0.(o) + 7'2b-(0) (1 +jco TO (1 +jco T2)

60_, and 0(0) are an angular velocity and its time rate measured at 0. T2

for the correction may be determined in similar way to Ti : viz, plotting log (0(t)Cie- 1/Ti) against

t,

the gradient of its asymptote equals 1 log e .

-12

Cie- 07'1 may be obtained easily because log Cie- 1/Ti is represented by the straight

line which log e(t) approaches in the above logarithmic plot.

The reliability of the transient response method must decrease with increase

of frequency; because the magnitude of Yg(jco) decreases generally with increase

of co, and because actual impulsive steering involves 'a smaller portion of the higher frequency elements.Tt_ exi.ority f the fteqnQy,..,,ponse method

in the higher frequeLyic range results from these circumstances.

3. 3. General procedure for model test and instrumentation used : Considering

mediate phases of d(t), applying the Simpson's method for each of the real and imaginary part of the integral, that is

tb tb

(t)ej'tdt=

ë(t) cos cot dt j (t) sin cot dt (8)

The secondsecond step is the formal integration of Ciet/ becaus C2et/T2 has already vanished at tb selected reasonably large, i. e.

6(t)ei4dt

(tb)T1

1+ ( rD2[(cos totb.co Ti sin cotb) tb

j(sin cab+ (oTi cos cab)] (9)

where (tb)is (t) at to. A fifst aiiproxiinative Nalue of T1 for this calculation may be determined directly from a measured e(t) by means of a logarithmic

plot. Plotting log 0 (t) against t, it approaches asymptotically a straightTlin

log C1c

with

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The tail trutneate,tiOla described here has been

replaced now by a

new one employing HFwireless

control of -a free...model and self-.

contained measuring apparatus.

The new one has be&n described

iri.brief in a list of Japanese facilities on

manoeuvrability

research that is to be presented for LT.T.C.

through S.Motora.

K. NOMOTO 87.

the circumstances discussed

in the

last two sections, the most reasonable procedure for determining the steering quality indices or the steering transfer

function through a free-running, self-propelled model test may be as follows : to obtain Y(jw) for the relatively lower frequencies from the impulsive

steering tests,

to obtain Y,(jc0) for the higher frequency range from sinusoidal steering

tests,

then to determine K, T1, T2 and T3 by solving the algebraic simultaneous

equations from four values of Ys(jco) reasonably distributed over the

frequ-encies. Other values of Ys(jco) and ArgYs(jw) may be used as a check on

the results.

These tests have been carried out at the Osaka University Tank, a result of which is illustrated in Fig. 6 in a nondimensional form. Since the tests require

°gib

-10 db

-20db

-1.0

-.5

Fig. 6. Steering Test Results of a Free Self-Propelled Model.

accurate measurements of turning angular velocities, a rate-gyro with an electric

pick-up may be the most preferable device. In designing and manufacturing the

rate-gyro, enough consideration should be paid on its frequency character as well

as on the stable sensibility. Fig. 7 shows the principal parts of such a rate-gyro

now in use. While a VHF-wireless control of a model ship might be an excellent - means, satisfactory results have been provided by a "fishing _pele " arrangement

ral

shown in Fig. 8.

These tests may be carried out in an usual towing tank of medium size or ..,::. --...' c ._ >11 "74_--.,, TRANSIENT RESPONSE-

---MODEL-I2, FULL LOAD

-V. :- 1-00. 'n/s 59 : 5: 7: leo°

.

:

.

FRE011E ,_. CY RFSPONSE ..., 1.(/ = (-,f)x T.' = (PT : 7.2'

' E)7

: T3 = (i) T3 : 1.53 2.77 .317 . 687 . . o 0

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-Flexture Pivot

Fig. 7. Principal Parts of Rate-Gyro, Showing Upside Down. An electric-driven gyro is hung from

a bed plate through a pair of flexture

pivots. The gyroscopic moment

pro-portional to turning angular velocity

induces slight deflections of the pivots, which may be measured by a differential transformer with an E-shaped core. The damper plate is immersed in an oil pot for filtering out the mechanical vib-rations and other undesirable "noise".

a conventional swimming pool and they are useful for assembling systematic in-formation on the contributions of hull

forms, relative rudder areas, stern

ar-rangements and other factors affecting

the steering quality. In this connection,

it seems that the scale effect on steering

quality is worth examination, particularly

for the usual stern arrangement of most

merchant ships. This is the reason why some specially arranged steering tests using full-size ships are desirable.

3.4. Standard manoeuvring test by Kempf (Z-Steering test) : The frequency

res-ponse and transient response method

require some special instrumentation in applications for actual ships and in most

cases it may be regarded as rather trouble-some and impractical. Under these

cir-cumstnces, the standard manoeuvring

test proposed by Kempf6) followed by an analysis using the first-order simulation may be a recommendable procedure.

As is stated in 2. 2., the procedure is to select proper values of K and T so that the first-order equation of motion defined by the K and T may express a measured ship motion in the test as fairly as possible. The resulted K and T describe in brief the steering quality of the ship through the first-order equation

of motion (5) or the first-order transfer function. Fig. 4 and Fig. 5 are typical two results obtained at the Osaka University Tank.

The procedure of analysing the standard mamoen-I4i

test doseribed here has been replaced now by a new

one which is described in the attached pages at the

end of this report.

The new procedure is much

easier to practise than the former and its reliability

is equivalent to the one of the former.

Then the

description about the farmer procedure is eliminated

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K. NOMOTO 89

ROCI

Fig. 8. Fishing 12 ele Arrangement For Steer-ing Test.

A free-running, self-propelled model ship

is connected electrically to a

control-and-measuring station on land through a bundle

of fine vinyl-shielded cords carried on a fishing

inla. Walking along ar)lies.in side, an operator handles the finshing so that the cords may not disturb the motion of the model.

At the station, another conducts the propulsion

and steering of the model and another operates an oscillograph recording the turning

angular velocity, helm angle, revolution of

the rtiVpeller and time marks for each 1 or 1/5

sec. The model is driven by a D.C.motor

with a thymotrol unit employing an induction tacho-generator feed-back for speed regulation.

Steering is performed by a steering gear shown in Fig. 9.

Fig. 9. Steering Gear.

The sinusoidal steering is performed by a

sinusoidal motion mechanism driven by a small induction-synchronous motor containning reduction gears. The steering frequency is variable over a

wide range by means of "friction wheel and rotating disk" mechanism and one step of change gear. This steering gear has also a relay circuit for conducting the sine impulsive steering, a

potentiometer for detecting the helm angle and

a fine adjustment device for the effective neutral helm.

(t). Motion of a ship for this

mulating o(t) is calulated

us g the first-order equation

of otion (5). Then equating

the c lculated e(t)

to zero at

te, te' nd tc" , and ignoring

-some of e exponential terms

that becom negligible through

damping, we obtain the

follow-ing equations, here tc, te' and te"

denote the tim at which the

extreme course viations

ap-pear, and where t1 'enotes the time at which 80 is ached at

first and t2, ti and t6 denote

the time at which the lm is

reversed. That is,

for the case that te t2-1-2t , as

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Tlete/Te (te 4)1 T e(t8t2) T e(tet2--2ti)/71

T[e (te' ---42) I T e (te'

_{e (t8't4)IT}

(re-1-1 2t.I)/ 77]= (10)

T[e(te"t4)/Te (te"--t4-2t1) /

e (toil to)/ T +e (te" to 2ti)/T]

and for the ca _{that tet2+24, as is possible in some cases,} T[1+etel T

(teh) /T e(tet2)I 7.1

=

4 +t2)

T[1+ e (t'e-6)/

_{e(t8't2---2ti) /7"e (t1=4)/71te' (4-1-}

_t4) ₍₁₁₎ T[1+e(18"=t4)/ gli) T (te'' = to) I 7.1= te" _{(t1+ t6)}

DenOting the suCceding _{treme course deviations Which appear at te,} _{te' and te"}

by 0e,.0,' and Be" respectiv y, and equating the calculatcd 0(t) to Be, 08' and Be"

at te, te' and te", we obtain fo the case that 42t2+2t1

1 8 3 K te+2t2+ i t1 1 80 1 K e'(t; + 2t2-2t4

i

89 K

and for the case that t8-__t2+24,

1 oo 1 (ti + to --2-- (to2 + 42+ t22) oet, 1 ao

K+ 4)

+ (4'2+ 62+ 62) + + 2t2) _(I 80 1

ee"t,{ (4+ to to" (te"2 + 42+ to) (t4 t )1

These -equations may be Utilized for determining and T substituting

measured Values off-e;t

iti n

-2, 43 t n...8% we", -o

_{aandti. in th}

Actually, since

the equations (10) and (11) can not be solved formally,

_{" tep by step" or}

" cut and try" method must be used, starting from a first appr ximation of T. A convenient way of estimating this approxirnative T is to eq -ste

_{it to the}

measured Value Of (4-0

Since these equations are calculated for Zsteeting With equal helm ngles on

both sides, it is often necessary to apply a correction for "residual hel _{", that}

means an error of a neutral position of a rudder owing to an incomplete adjus _ent

of a steering gear and telemotor, turning effect of a propeller race in a sin

e-screw ship and other miscellaneous factors. Even if the residual helm seems +26-24 -I- 24+

(12) 3

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In the actual applic ion, each of equations (10) (12) or (11) (13) may

produce somewhat diffe nt K and T. If the differences are not large, and if the resulting simulation of the measured ship motion is satisfactory, the mean values of these K' and T's may be considered K and T of the ship. When the resulting simu ton is not fair, it is necessary to shift the K and T so that the

simulation ay be as satisfactory as possible. Sometimes this "cut and try"

operatio is so tedious that some calculating device for, the purpose may be

desira e. Such a device now in construction at the Osaka Univ. Tank is a

single-pu sosed analog-comsingle-puter with a single integral unit employing a two-phase

torantl-1 I 06 04

2 t6 -t0' t4t0 I measured, because the measured values of

7

K.. NOMOTO 91

be negligible, its effect on 0(t) may become appreciable, because the effe cumulative as a test progresses ; it is as if a slow turning is added to ig-zag

motion of the ship. In fact, analyses of Z-steering tests sometim indicate

some difference between the maximum turning angular velocities to starboard and to port in spite of a same helm angle to both sides.

As ming that this

hidden turning is present constantly during a test, its rate 6. y be estimated as

where to, t0' 06 and 04 are own in Fig. 4 and 5:

06 04

and

early equal the

ultimate

t6

rate of the course deviation to starboard an to port respectively including the hidden turning, the difference of the t is twice the hidden turning rate.

Estimating

the rate in this manner, t e residual helm correction may be

applied graphically, as is shown in lg. 5. In some cases, the correction

affects considerably Be's, while it s much less effects on te's even for those

cases. This is natural because e effect of the residual helm on o(t) does not accumlate and consequently d, -s not grew to so cbnsiderable extent.

In carrying out the Z-steering test, it is preferable to record the course deviation and the helm angle continually if possible, because such a continuous record provides more reliable results than intermittent measurements at te's and some other moments as is usual practice. For this purpose, it is convenient to use an electric Den-recorder having several pens. Observing a ship's compass, an operator presses a push-button actuating one of the pens at intervals of some

degrees of heading deviation, e.g. each 10 or 2.5°. Another observes a helm indi-cator and sends similar signals at each some degrees of helm angle, e.g. 5°. Time

marks are supplied to another pen by an electric-contact watch every second. On the other hand, the continuous measurement would be impractical for the test in a sea way conducted by an officer on board. When the usual

inter-mittent observation is employed in these circumstances, an attention must be paid

1

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-to observation of helm angle. _{Since an electro-hydraulic steering gear}

consi-derably decelerates when approaching the desired helm angle, it is not reasonable

to estimate t1 from the time when steering is finished. A recommendable way is to measure a steady angular rate of the helm and to obtain t1 by dividing do by the rate. This measurement may be performed by a stop-watch.

Present conclusion

Although investigations on steering quality along the present scheme are still under way, now it may be concluded that

Steering quality of a ship may be described by a transfer function of the

K(1+ T3p)

form Ys(P)= (1+ Tip) (1+ T2p) or a set of the steering quality indices K, T1,

T2 and 7'3. These indices, coefficients of the transfer function, are functions

of hull form, rudder size and other particulars of a ship. Analytically, they are

assembled from coefficients of the equation of motion usually used in steering

researches.

Steering motion of a ship is substantially a first-order phenomenon, and

consequently it is described in brief by a first-order equation of motion,

dB

Tdt+0=-Ko .

Accordingly, steering quality may be described in brief by two fundamental indices K and T.

Impulsive steering test and sinusoidal steering test of a free-running, self-propelled model is an effective means of approaching the steering quaiity of a given ship. It provides a full description of steering quality in terms of K, T1, T2 and T3 through a procedure not so lengthy as the usual approaca

of resistance derivatives.

Kempf's standard manoeuvring test followed by an analysis employing

the _{first-order equation of motion provides a figure of merit for steering}

quality of an actual ship in terms of K and T.'"

Acknowledgement

The present work owes much to co-operations of the following shipbuilding companies and shipping companies. Acknowledgement is made of these

companies.

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K. NOMOTO 93

Fujinagata Shipbuilding Co., Sanoyasu Dockyard Co., Osaka Shipbuilding Co., Shioyama Dockyard Co., Shinnihon Shipping Co., Kawasaki -Shipping Co., Daido Shipping Co., Iino Shipping Co., Inui Shipping Co., Sawayama Ship-ping Co., Muko ShipShip-ping Co., Hiroumi ShipShip-ping Co.

References

S. Akasaki, Journal of the Society of Naval Architects of Japan (J. S. N. A. J.) 61, 379 (1937) and the successive papers by the author up to J. S. N. A. J. 73, (1951). H. Kuenzel und G. Weinblum, " Ueber die Kursstabilitaet von Schiff en" Schiffbau

39, 181 (1938).

S. Motora, J. S. N. A. J. 77, 69 (1946).

K. Davidson and L. Schiff, Trans. S. N. A. M. E. 54, 152 (1946)

S. Inoue, J. S. N. A. J. 90, 31 (1953) and the preceding papers by the author

since J. S. N. A. J. (1951).

L. Schiff and M. Gimprich, Trans. S. N. A. M. E. 57, 94 (1949) G. Kempf, Trans. S. N. A. M. E. 40, 45 (1932).