On the motions of high speed planing craft

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research in hydrodynamics

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ON THE MOTIONS OF HIGH SPEED PLANING CRAFT

By

C. C. Hau

May 1967

TECHNICAL HEPORT 603-1

DISTRIBUTION OF THIS DOCUMENT IS UNLIMITED

Prepared Under

ivai Ship Research and Development Center

Department of the Navy

Contract No. Nonr 5153(00)

r.

'w

t1

_e

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TABLE OF CONTENTS

Page

ABSTRACT

_i

INTRODUCTION

i

HYDRODYNAMIC CHARACTISTICS ₂

TURNING AND STABILII? ₁₅

CONCLUDING R1ARKS

₂₅

-ii-LIST OF FIGURES

Figure 1 - Orientation of Body Axes Relative to Fixed

Wind Axes

-

Comparison of Calculated and Experimental Lift

Coefficient for Rectangular Flat-Plate Planing Surface

- Comparison of Calculated and Experimental

Longi-tudinal Center of Pressure for Rectangular Flat-Plate Planing Surface

- Comparison of Calculated and Experimental Lift

Coefficients for a Planing Surface Having a 20 Angle of Dead-Rise

- Comparison of Calculated and Experimental Side

Force Coefficients for a Planing Surface Having a 200 Angle of Dead-Rise

- Comparison of Calculated and Experimental Pitching

Moment Coefficients of a Planing Surface Having a

200 Angle of Dead-Rise

Comparison of Calculated and Experimental Rolling Moment Coefficients of a Planing Surface Having

a 20 Angle of Dead-Rise

Comparison of Calculated and Experimental Yawing Moment Coefficients of a Planing Surface having a 20° Angle of Dead-Rise

Variation of Flow Coefficient with Angle of Dead-Rise Figure 2 Figure 3 Figure '4 Figure 5 Figure 6 Figure 7 FIgure 8 Figure ₉ A B B o b C C CZb K -pV3 b3

C Pitching moment coefficient, M

V2b3

N

C Yawing moment coefficient,

pV2b3 CKb C D, e C d E' FN g h NOTATION beam Aspect ratio,

mean wetted length Buoyancy force

Bobyleff's flow coefficient

Beam of planing surface

Pabst's aspect-ratio correction

Side force coefficient,

bY'

b'

Lift coefficient, Z

bVaba

Rolling moment coefficient,

Cross-flow drag coefficient

Skin friction coefficient

Draft of planing surface semi -per Imet er

b cam Normal force gravitatIonal acceleration PerturbatIor in elevation Y HYDRONAUTICS, Incorporated HYDRONAUTICS, Incorporated

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-lv-.

K

_{Rolling morflent, positive to starboard}

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Turning rate

Chine wetted length

Keel wetted length

h_xR

,h

,F

yR zR

Angular momentum components of the rotor or

propeller relative to x-,y-,z-axes.

Mearwettè

lengtli

I 1 1

X y z

Moments of Inertia of craft about x-,y-,z-

axes

M

_{Pitching moment, positive bow}

up

I

zx

Product of inertia of craft with respect to

_zx-axes

Si

vn

a

Mass of craft

Two-dimensional deflected fluid

_{mass in transverse}

plane

X,Y,..etc.

Force stability derivatives wltn respect to

u

...

etc.

N

_{Yawing moment, positive to starboard}

p,q,r

- -

Angular velocity components about

x,y,z axis

respectively

K ,M , .. .

u u

Moment stability derivatives with respect to

u

K M

etc.

Time rate of change of

_p,q,r

R

_turn

S

V

Turnir

_-

radIus

Wetted area of planing surface

Constant free stream speer'..

Subscripts

a

o

Quantities due to acceleration

Quantities due to cross-flcw

u,v,w

_{Translational velocit}

_{components in x-,y-,z-axls}

I

Ir.ltial reference value

respectively

p

_{Quantitywlth respect to port panel}

X,Y,Z

Time rate of change of u,v,w

Force components in x-,y-,z

_{directions respectively}

S

t

_{Quantity with respect to traIling edge}

Quantity wIth respect to starboard panel

-x,y,z

_{Right-handed body axes}

c.p.

Center cf pressure

k

r

Angle of dead-rise

Dead-rise function

Trivn angle

Pitch angle

Roll angle

Yaw angle

3lde-sllp angle

HYDRONAUTICS, Incorporated

-1-ABSTRACT

In this report the hydrodynamics and dynamics of high

speed planing craft are studied. The first part of this study

presents a simple method, based on the known results of airfoil theory, of estimating the hydrodynamic characteristics of high

speed planing craft. Comparisons between present calculations

and experIments for flat and prismatic surfaces in steady

trim-ming, rolling and yawing attitudes are made. The calculations

seem generally to be in good agreement with the data. The

sec-ond part of this report discusses the turning and stability of

planing craft. The governing equations, based on the kinematics

and dynamics of rigid bodies are in many respects sirnhjar to

that given for hydrofoil boats. Once the hydrodynamic

charac-teristics are known the stability and turning performance of the craft can be assessed with the aid of these equations.

I.

INTRODUCTION

The hydrcdynamlc planing of boats and ships has attracted

much attention in the last thirty years. The literature on

steady symmetric planing problems is quite extensive. Among the

most noteworthy

IS

that of Wagner's airfoil analogy (i).

How-ever, the information on unsymmetric and unsteady motion, which is very Important in the practical design of planing craft, seems

ratner fragmentary. It Is of great Interest to seek a basic

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-2-In the present report an attempt is made to extend the

airfoil analogy to planing surfaces in arbitrary motions.

The

first part of this Study presents

_{a simple method, based on the}

known results of airfoil theôry, of

_{estimating the hydrodynamic}

characteristics of a high speed syinmetrlcal

_{craft planing at}

constant speed undergoing small perturbations

_{on an initially}

flat free water surface.

_{Comparisons between present}

calcula-tions and experimental data of Savitsky,

_{et. al. (2) for flat}

and prismatic planing surfaces in steady trim, roll and yaw

at-titudes are made.

_{The calculations seem generally to}

_{be in good}

agreement with the data.

_{The second part of this report}

dis-cusses the stability and turning of planing

_craft.

_The

govern-ing equations, based on the kinematics

_{and dynamics of rigid}

bodies are in many respects similar

_{to that given for hydrofoil}

boats.

_{Once the hydrodynarnic characteristics are known the}

sta-bility and turning performance of the craft can be found with

the aid of these equations.

The present analysis, with assumptions of

_{small disturbance}

and no wave disturbances, is not intended to be exact and

rigor-ous, but it Is to be hoped that lt will help

_{given an}

under-standing of the principle factors

_{governing the behaviour of}

plsni.g craft in motion.

II.

HYDRODYNAMIC CHARACTERISTICS

ir. constrast to normal ships, which are supported on the

surface of the water by buoyancy,

_{a rapidly moving planing craft}

is supported on the free water surface

_{mainly by a dynamic lift}

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-3-force, resulting from the upward reaction of the fluid

on the

moving body.

_{At sufficiently high speed the effect of gravity}

on the flow may be assumed to be negligibly small.

_{Except in}

the immediate neighborhood of the leading edge, the planing

problem in this case resembles closely that of the motion of

a

wing.

_{At the leading edge of the wing the fluid velocity is}

infinite which results in the presence of a suction force.

_The

absence of this suction force ori a planing surface results in

the presence of a spray plume.

_{The planing problem can then be}

solved readily within the framework of the classical thin

wing

theory.

The force acting on the wetted bottom surface

may be

di-vided Into components normal and tangential to the surface.

The tangential component represents the frictional force

and

is determined by the fluid motion in the boundary layer.

The

normal component depends on the pressure distribution of the

bottom surface.

It has been shown, by Wagner (i), that, for

a

flat planing surface of infinitespan, the normal

force,

may be approximated by

cosi

\L)

2 w

If the angle o

attack,

T,

_{is sufficiently small.}

_and

are tne lift and suction force respectively of

a wing of the

same planform.

_{Expression [il is also expected to hold}

approxi-matel

_{for a planing surface of finite span.}

A span _{is the aspect ratio of the wing}

- chord

E _{is the semiperirneter to the span}

S is the planform area of the _wing

V _{is the uniform free stream speed}

p is the fluid density

For a wing of large or moderate aspect ratio the _{effect of finite}

span on the lift expression is derived from Jones' modified

lift-ing line theory (3). _{For a low aspect ratic wing the}

three-dimensional correction is calculated according _{to the lifting}

surface theory of Krienes (n).

Combining Fuations [i) and C2] the normal force _{on flat}

planing surface at trim angle T can easily be _{shown to be}

f

irA

FN p\ 3

[ 2(1 + EA/2) sin T cos

For a planing surface having constant dead-rise angle _{ß it is}

necessary to take into account the effect of the deflected fluid

motion due to dead-rise. _{The appropriate expression for the}

force normal to the keel, F, becomes

irA

= pVS

[2(1 + EA/2) T cos where -

Ç-(i

\a 'tan p\S ) ) for small

is the dead-rise function which may be deduced from the work of Wagner (i).

Ir.

the foregoing analysis, the effect of transverse flow,

proportional to sin T, is assumed to be nligible. _{As the value}

of T increases, the cross-flow effect becomes _appreciable,

espe-cially for low aspect ratio planing surfaces. It is expedient,

in practice, to include the part due to cross-flow _{in the normal}

farce calculation. The normal force for planing surfaces witn

-4-The lift of a thin flat

_wing

_{with angle of attack T is}

well-where

-5-known and can

L

_i

be shown to be

rA

Wing of infinite span

For large and moderate aspect

A =

b

is the aspect ratio of the planing surface

is the beam wetted semi-perimeter

for A i

l+EA/2smnT

b earn SW irA T ratio, A 1 t2

_{E' =}

2

for A<l

respectively,

Low aspect ratio wing,

A < 1, say

m S

is the mean wetted length

is the wetted bottom area _

i

+ A sin

where HYDRONAUTICS, Incorporated

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-6-small constant dead-rise angle _{in steady trim motion may then}

be expressed as

vs[

2(1 + E'A/2)A Sin T cos T]

x(ß)

+ CDC

p(V sin T cos

_t

5

where CD is the cross-flow

_drag

_{coefficient and generally of}

O(i). uation 5) Is very similar to that obtained by

Shuford (5). His results amounts to assuming that Et _{2 in}

all cases; furthermore, the values of ti(o) and the force

com-ponent due to cross-flow were determined by him empirically to be approximated by the following expressions

=

1 -

Sin ß

Nk,c p115S sin2T cos3T cos

ß,

CD

for dead-rise angles up to 500.

Trie normal force for a prismatic planing surface in roll

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-7-F

c.pVS

irA w

-Ns 2

s2(l+E'A/2) ()

cos

i

Ii(o)

irA

2(lE'A/2)

s y V sin p cos

i

+

!.

_sinß

-V

)CO5

p + F c Np 'w +

CDc

cos _ß

pv2spÇ

irA 'w'

2(l+E'A/2)

Ivi

cos irA sinn cos 2(l+E'A/2) y a.'

+ CDC (

-

sin

)

j

respectively, where

-

The

subscripts "s" and "p" denote the quantities with

respect to the "starboard" and "port" panels

respec-tively.

1,

is the aspect ratio of the port panel

is the aspect ratio of the starboard

panel;

is the keel wetted length

6]

is the mean wetted length of the port panel

is the mean wetted length of the

starboard panel

and yaw is estimattd here in a manner s1mIla- to that used for

an aeroplane wing with dihedral. _{The orientation of the}

_body

axes x,y,z relative to the fixed axes, x,y,z in terms

_of

trIm (6),

roll (qi), and yaw

_($)

is

illùstrated in Figure

_{1. For}

the case where

A p A a = = b 2I Cos b 2

.cos

d(draft)

x-axis is parallel to the keel

6

and

T

are the

same.

The normal force for starboard

and

port surfaces may

-sin

T

be approximated

as _p = +

HYDRONAUTICS, Incorporated =

sln1 ()

-8-b tan(fi-fp)cos(fi+w)

-

7

tan r

C08 fi

b tan(fi-ç)cos(fi-)

c,s

k

ir

tan T cos

fi b

S-

_p

_2cosfi

b

S

is the mean wetted area of the starboard

s

2cosfi

panel

rn

is the mean wetted length of the planing surface

y = V(sln T cos $

sin

- sin $ cosq') is the velocity

component perpendicular to plane of

Symmetry

w = V(sinrcos* cosqli-sin$ sincp) is the velocity component

norial to the keel and in the plane of

symmetry

is the mean wette

is the estimated chine

wetted length of the

port panel

is the estimated chine

wetted length of the

starboard panel

d area of the port panel

is the side slip angle, positive for rotation to

°

_{the right}

T

is angle of keel with horizontsi plane

The hydrodynamic forces and moments for straight steady

no-tion in the body axes can easily be shown to be approximated by

the following relationships:

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z = z +z

sp

= - (FNS + F

_Np

) cos

fi

Y = y

_sp

-i-y = - (FNS_

_{FN)Sinß+CfPVV(S+S)}

X X i-X

sp

= - CfpV(S + s)

K = (y Z -i-y Z

Ss pp

) -

(z Y -i-z Y

_{s pp}

M = (z

s sp p

'X -i-z 'X )

-(x Z -i-x z

Ss pp

N=(xY+xY)_(y X-s-y 'X)

_{Ss pp}

_s

_{s pp}

where

pressed as

-9-Cf

is the friction coefficient,

x_s

and x

_p, y_s

and y ,

z

and z

are the distances in

p s _p

x-, y-, z- coordinates between the reference point and

the center of pressure of starboard and port panels

respectively,

y_s

and y ',

_p z_s

and z

_p

are the distances in y- and

z-coordinates between the reference point and the

center of friction force of starboard and port panels

(taken generally at the centroid of the wetted surface

area) respectively.

The values of x

_s

and x ,

_p y_s

and y ,

_p z_s

and z

_p

can be

ex-X =x

s

ts

-x,

r

x =x

p

tp

-x

r

= z

_c.p.s

- z

_r

, z

= z

- z

p

c.p.p

r

(Z-force)

(Y-force)

(X-force)

(rolling mcanent)

(pitching moment)

(yawing moment)

r'

p =

_{.p.p -}

8

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-lo-.

with _{as the location of the reference}

point measured

from the trailing edge at _{the keel. In Equations}

[8]

x is the

distance forward from the trailing edge of the planing surface

to the lor.gitudinal center of _{pressure, analogous to that given}

in airfoil theory and may be approximated as

{(z -Z

s s s,c

)+1z liz

2 s s,cJ, s

(x

[tz -z )+!z

_1A"z

tp p p,c

2 pp,cJ/

p

where Z and Z

are the Z-force components due to cross-flow.

B.0 p,C

y ard z _{are the lateral and vertical}

coordinates of the

c.p. c.p.

center of pressure and will be taken,for a first approximation, as h + 2. y c_p.s c.p.st + k c,s 'i k y

c.p.p

c.p.p

l

¿

+ I k c,pj

relative to the plane of symmetry; and

c.p.a c.p.s c.p.a

z z

_-y

_tanß

zc.p.p zc.p.p yc.p.p

tana

relative tc tr.e keel.

C 9]

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-11-Numerical calculations, based on Equations [6] _{- (ii), of}

the forces and moments in straight steady flight, _{for a}

rectangu-lar flat plate planing surface and a prismatic _{planing surface}

having a 200 angle of dead-rise have been nade and are compared

with the experimental data cf Savitsky, et. al. (2) assumIng

CD ./3 and () = (1 - sin ) as determined empirically cy

Shuford (5). _{For the fiat plate case, the side force in the body}

axes system is, due only to the frictional force, _{quite small and}

not considered. In Figures 2 and _{3 the calculated and measured}

non-dimensional vertIcal force (CZb _{= Z/(pV2b2) and longitudinal}

center of pressure x1/L _{for various loadings, trim, roll and yaw}

angles are shown. _{It is clear from the figures that the}

agree-ment between calculatLons and experiagree-ments is quite good. For

the planIng surface having 200 angle _{of dead-rise comparison}

between calculated and experimental non-dimensional vertical

force (c), side force (Cyb = Y/(ç'b3 )) and pitching moment

(c

_{= ii/(4pvt)) atout the trailing edge are shown in Figure}

4, 5 and 5 rerpecively.

The results are, in general, In good

agree-ment.

The experimental data on rolling and yawing moments are

Somewhat scattered especially at heavy _{loadings. Calculated and}

experImental non-dimensIonal roiling moment

Kb = K/( Va

t3))

about tne keel and yawing moment (C _{= N/(*pV2b3 ) about the}

trailIng edge, for a few typical _{cases, are also shown Ir}

Fig-ure 7 and 8 respectIvely. _{Scatter in the data, however, makes}

Comparisons difficult.

Ir. trie foregolna discussions

the planing craft is assumed to

be operated at hign speed. _{The gravity effect upon the}

hydro-dy.ar.Lc cnaractenIstIc _{fcr planing craft at lower speed operatiors}

Eio]

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-12-may be taken into account by adding the buoyancy force due to the volumetric displacement which may be expressed aa

B c Tìpg

i

b sin )

ni

where = + L) is dependent upon the values of , , d and

; g is the gravitational acceleration; 1 is a constant which,

based on some preliminary analysis of planing craft test data

given by Clement and Blount

(6)

may be tentatively taken as 0.7.

Pending further comparisons with data, the center of buoyancy is assumed to be located at the center of effective hydrostatic pressure on the wetted surface as determined from the equilibrium water surface.

If the motion of the planing craft is unsteady the pressure distribution on the wetted surface does not adjust itself to its

equilibrium value Instantaneously. In general there is a time

lag. Owing to the presence of this time lag, the hydrodynamic

forces and moments which act on the planing craft at any given moment depend on the entire history of the motion and are

diffi-cult to estimate. However for maneuvering and turning problems,

or for motions of long period type, the influences of past

mo-tion are probably negligible. The force system, in these cases,

wouid te determined on the basis of quasi-steady theory; i.e., they wofld mainly depend on the instantaneous state of velocity

and acceleration. The first order quasi-steady normal force due

t the time-dependent velocity perturbations may be shown to be

approximated by

t12)

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-13-L ,y ,z and

and will be assumed to be acting at

s c.p.a _c.p.sI

W1 = fi

-

qxf + PYÇ

Vi = -P i- rx

fi is the velocity due to heaving h in the vertical

fixed axis

p,q, and r are the angular velocity components about the body x-, y- and z-axes respectively

JCfYf and z, for a first approximatIon, are to be

taken as the distances in x-, y- and z- coordinates between the reference point and the traIling edge of the planing surface.

The factor two in the fIrst term of Equations [13) arises from the fact that the longitudinal deflected virtual mass varIeS with both the longitudinal dimensions and depth of submergence of each

sectIon of the planing surface. (For details ses SchnItzer (7)).

The normal force due to accelerations, and -,may be

approximated by t

.L,y

_{c.p.p. c.p.pj}

,z respectively where

F . Ns1 ! pV2S 2 s rA wAS V1 sin [2(l+E'A/2) 2(l+E'A/2) irA

t 13]

F Np1

i

2 p 1A 2w1 P

_iL

[2(l+EA/2) V - 2(l+EIA/2) y

F '

Nsa

pb2

_I

c(A)

[Tr(ß) + B()

tan

î_

tan lrpL b2

+ c(A

_s)

16

tang

'Trtb2

FN

c(A)

[7fl)

+ )( tan ¶ - tan ) ] ii

-114--

The

quasi-stéady hydrodynamic forces and moments of the

planing craft may then be computed according to Equations [7],

£13],

Ei*)

and [15]. Once the hydrodynarnic characteristics are

-15-known, the behaviour of the planing craft in static and dynamic

motIon can be assessed. _{In the following the problem of turning}

and stability will be discussed In some detail.

III. TURNING AND STABILITY

-= -

_{i si:}

9 + y c-os 8

sin

+ w cos 8 cos

I-7rpb2

- C(À)

' tan

at the centroids of the wetted Bobyleff's flow coefficient the three-dimensional Correction, factor may be used, i.e.,

' 0.1425

areas.

and is Pabst's

If the planing craft Is assumed to be rigid the

equations

of motion in body axes with origin at the c.g.

craft, xz plane as plane of symmetry

can be derived Without

diffIculty, as shown In (8):

X - m,g sin e = m(

+ qw - rv)

y +

c

e sin

= m(

+ ru -

pw)

z +

e _{= m(}

_{+ pv - qu)}

=

-Izx i+i.)

+ (I_I)r +

-

hr

M =

I

+(i- i)rp + I(p2_

_r)

_{+ h}

_{r - h}

_p

xR zR Euierian of any

[16)

1

and are assumed to act

where B() is the so-called

given by Figure

_9.

For

empirical correction

c(A)=

i i- A3

(1

c(A s

c(A)

p 1 I 0.1425 [15) p q

r

= = =

r t + i

z ZX

(qr-) +

-

Sir,

9

9 cs

+

ces e

co

9 coz

- - O (i,_ I)pq

-Sn

sin

hRP

1 + A2

s

i

1

-A s A s il 0.1425

lA

(

p'

Ap A HYTiRONAUTICS, Incorporated HYDRONAUTICS, Incorporated

e, and $ are the pitch, roll and yaw angles (see Figure i)

re Is the mas8 of craft

and 1 are the angular acceleration components of craft about x-, y- and z-axea respectively

X,Y and Z are the hydrodynamic force components

(including buoyancy and propulsive forces) of. craft in x-, y- and z-axes respectively

K,M and N are the hydrodynamic moment components

- o

(including moment due to buoyancy) of craft dt

about x-, y- and z-axes respectively

I 1 and I

are the moments of inertia of craft about

X

_y

Z

X-,

y- and z-axes respectively

- la the product of inertia of craft with respect

to zx-axes

h ,h and h are the angular momentum components of rotor

xR yR zR

or propeller relative to x-, y- and z-axes

X - mg sin 8 = O

K=M=N=O

respectively, so-called G factor, defined as

Equations [17] are the kinematic relations of rigid body motion.

Equations

[16]

represent the components of dynamic equilibrium

In each of the Ix degrees of freedom. Equations [16] and [17]

characterize completely the rigid craft otlon and are sufficient

to determine its response to an arbitrary set of time-dependent forces and moments.

A problem of particular interest concerning craft motions

is that of steady turning maneuvers. In steady turn the values of

are all assumed to be identically zero. By

ne-glecting the higher order terms and the effect due to rotor the governing equations, combining Equations [16] and [l7],may be shown to be

It is to be noted that the expressions X,Y,Z,K,M,N, th Equa-tions [18] also consists of' forces and moments due to the action of control surfaces.

The turning characteristics can be best described by the

mR (l

G=

turn Y

tan -i- - sec 9 sec

mg gR mg

turn

where

= (tan + sec e sec is the turning rate

V

R=

turn --- Is the turning radius

The lar-ser the value of G the smaller the turning radius for a

gi;er speed. For true-banked turn, in which Y = O, the value of

G Is snply gR va t urn tan _(20] [19] Y + mg cos 8 Z + mg 8 sin cos m$V cos e cos q = -m$V cos 8 sin (18] HYDRONAUTICS, Incorporated HYDRONAUTICS. Incorporated -16-

-17-HYDHONAIITICS1 Incorporated

-18-which determines the ideal bank angle p for a given V and

_Rtn

3lrLce the maximum bank angle in the operation of planing _craft

depends very much on the seaway and is genrally small _{the value}

of G in a true-banked turn is therefore quite limited. _The

turning characteristics may be greatly improved if large latèral acceleration can be accepted during the turn.

Another problem of fundamental interest in regards to

mo-tiori studies Is the stability of the craft. Customarily in

sta-bility studies the craft is initially assumed to be in steady equilibrium flight; it is desired to determine the motion caused

by a disturbance of very short duration. _{The problem is in}

gen-eral extremely difficult to solve. _{For a preliminary study the}

following simplified assumptions are made:

The disturbances are infinitesimal

The deviations from a steady state are either of the long period (quasi-steady) or exponential

type.

In

these cases the change In a force or moment brought about _by

smell change In a component of velocity, acceleration, trim or

elevation can conveniently be expressed as _{a fmction of the}

so-called stability derivatives.

Consider a planing craft having the force system (x1,y1,Z1, when the components of velocity are (u1,v1,w,,p1,q1, and the elevation and angular displacements are h1 and

Them let the velocities be increased by the small q.antitIes-(u,v,w,p,q,r) and the elevation and displacement

HYDRONAUTICS, Incorporated

-19-perturbations by h and (9,cp,$). If we let the accelerations be

expressed ty _{then by neglecting the second and}

higher order terms ini the Taylor expansion of the

hydrodynainic

force components relative to the equilibrium, we obtain fora

typical force component

XX +uX +vX +wX +pX +qX +rX

_{i u y}

w p q r

+ CiX. + iX. + IX. + X. + 4X +

_¡X.

u y w p ₄ r.

+ exe + + tx

+

and the typical moment component may be written as

K ' _{Ki +} + vK + wK + pK + qKq + rKr

+ ûK. 'K. + iK. + K. + 4K + i'K

u y w p 4

+ OKe + K + _{+ hKh}

where X = . .

. ..

K =

.. . .,

etc., are the stability

u

_ui

derivatives taken at the initial equilibrium condition which are

estimated from analysis or from experiment. It is to be noted

that the force and moment derivatives de to perturbations in elevation and angular displacements are generally brought Out by the

changes

In wetted surfaces.

(21)

HYDRONAUTICS, Incorporated

-20-On substituting Equations (21) and (22) into Equations (16]

the terms X1 and K cancel out the unperturbed terms on the right

hand side of Equations

(161.

Retaining only the first order

terms the. corresponding typical governing equations may be written as

(X. - m)ii + X. +

X.I + X

u + (X + mr )v + (X - mq )w

u y w . u y i w i

Where equals to 1. if the x-axis taken to be parallel to the

keel, and

i- K. + K. + K u + K y

+ K w + (K.

- I

₎

u V W u y w p x

The remaining equations in the set of Equatiuns [16) and £17) may

be derived in an analogous manner. It car. be seen that the

re-sulting equations are a set of ten homogeneous linear differential equations and ten unknowns witn constant coefficients which can

readily be solved.

For the very important case of straight symmetrical flight with small 8, the derivatives of asymmetrical forces and moments With respect to symmetrical variables (or vice versa) are

HYDRONAUTICS, Incorporated

.21

-generally negligible, the linearized equations of motion can be considerably simplified 1f the stability axes are used (w1 = O

in this case) I.e., the x-axis is taken in the direction of horizontal steady straight flight and may be decomposed into

two independent set of equations

((m-x.

)D-X lu - X w - Cx D +

(x9mg)]8 - Xhh = O

o u w q

-(Z.Di-Z )u + [(m-Z.)D-Z 1w-CZ Da+(Z +mV)D+Z

le-z

h O

+X.f+X.4+X.?+Xp+(X-mw)q+(X+mv)r

u u w w 4 q 8 h

p q r p q i r i

+

(x8mg

COB 81) +

x

+ xs +

Xhh = 0 (23) -(M. )u -

(M.D+M )w +

((I -M )-M D-M

le-M

h = O

(25] u u w w

y4

q e h

= w- ve

and + K44

+ (K.

- Izx)' + (K +1

q )p + [K -

(I -I

)r -h +1 p ]q

[(m_Y)D_Y)v -

- CYj,D+(Yr_mVflr = o p

zxi

q z y i ZR

zxl

+ (K -(I -I

)q1 + hYRIr + K88 + K + K1

-(?+K )v+(E

-K.)D5-lc D-K )

- [(K.+I

)D+K +h Ir = O W + K. h = O [2h) '1 V X _p p r zx r yR y n r z y - C(N+IZX)Da+(Np_hYR)D+N1cP + t(Iz_Ni.)D_Nr)'=3

p=

r=;

where D = d/dt and all the derivatives X....K -. - etc., are all

evaUsted with respect to stability axes. Equations [25)

con-tsi-. ar-dy the longitudinal variables u, w, q, 8, h and describe

te perturted longitudinal motions of the craft. Equations (26]

HYDP.ONAUTICS, Incorporated

are functions of lateral variables

_y,

_{p, r,} , arid govern the

perturbed lateral motions of the craft.

The general solution of a set of homogeneous differential equations witn constant coefficients such as Equations £ 25] and

[26], can generally be found in treatises _{on differential}

eqaa-tions. _{Consider, for instance, the system of m second-order} homogeneousequations

where

as

For motiori _{of exponential type,}

the value of q may be written

(D)q1 + Al2(D)q2+ ...

A2(o)q1 + A22(D)q2

+ ...

A1(D)q1 + A

2(D)q2 -- ...

-22-A,1(D) =a11D5 + b1D +c11

A21(D) = 2l + b21D + C21, etc. At e m HYDRONAUTICS, Incorporated

Equations [21f) then reduces to a set of algebraic equations

-23-A11(X

_l

+ Al2(X

2 +

A]Jfl(L m

= O

(X + A22(X + A2m(k m =

A1(A

_{l +}

_Am2

2 +

A(A

m

These equations are compatible if

(X) =

A11(X ) Al2(X ) .

A1(X

A2(X ) A22(X ) .. .

A(X )

A1. )

A2(X_Y

.. .

A(& )

Equation [29] is the characteristic equation of system

_[27]

and

can be expressed as polynomina]. of order n 2m

- -= ,o'u' er- - I

-=0

(26] [29) A

_lam

q = 0

Aq = O

A q = O mm m (27] where A11 = A21 etc.

+b X-i-c

a11X3 11 11

+b

X +c

a21X 21 21 [29 3

+ P1X1 + ... + P1k + P

= O

HYDI0NAU'FICS, Incorporated

2k

-The general solution of the system is

n

Xkt

n

Xkt

q1

e ,

_{q2 =}

_2k

e

K=l

k=1

where

_{Is the k root of Equation [293 and lk'}

_{2k ...}

are

con-stants satisfying the system of Equation [293 in which the root

'k is substituted for X.

The roots X,

are either

real or conjugate complex.

The system is dynamically stable If

none of the real part of these roots is positive, i.e., the

dis-turbànce ultimately becomes vanishingly small.

The criteria for stability, equIvalent to Routh's, can be

ex-pressed in a convenient determinantal form with P

_n

> O as follows:

=P T

> O o

n-1

[301

MI'DRONAUTTCS, Incorporated

-25-The necessary and sufficient conditions for stability is that all

che so-calied test functions T ,

₁

T ,

₂

... T

_n

and P

_o

shall be

posi-tive.

For general motions of long period type, by applying the

teonnique of Laplace transform the governing differential

equa-tions .an be reduced to a set of linear algebraic equaequa-tions if

trie Initial conditions are given.

The problem can be solved

ac-cordingly.

IV.

CONCLUDING REMARKS

In trie foregoing analysis the planing craft is assumed to

be operated at high speed (I.e., very large Freude number) on

cala water of infinite extent.

However, the effects of the

sea-way on the motions of planing craft should In practice, be

properly taker into soccunt.

Nevertheless, it is hoped that the

preser.t stuOy will provide:

(1)

A rational background for the analysis of

realistic conditions.

(.)

A rational basis for the planning of a svtemat.1c

experlmer.tal program.

(3)

hatlonal design criteria.

trie procedires for est imatlog the hydrodynamic

character-istirs of ploning craft, developed lo the present study are quite

1aple aro scrangr.tIorward.

The theoretical predictions seem, in

serpral. to be in good agreement with the existing static

measure-mers.

Unfortunately,

,o systematic dynamic test data exlste,at

T =P

_i

T2 =

T3 =

>

n-1

P

_n-1

p

n-3

p-i

_n P

n-3

P

n-5

O P n P

n-2

p n p

_n-2

P

_n-k

>0

o p

_n-i

P

n-3

>0

-26-

-27-preseñt, which can be used to test the validity of the present REFERENCES

approximations in the dynamic case. However, it is expected that

such data will soon be available.

Wagner, H., "The Phenomena of Impact and Planing on Water,"

NACA Translation 3366, ZAMM Bd 12, Heft

k,

pp.

193-215,

August

1932.

Savitsky, D., Prowse, R. E., and Lueders, D. H.,

"High-Speed }iydrodynamlc Characteristics of a Flat Plate and

200

Dead-Bise Surface in Unsymmetrical Planing ConditIons,"

NACA TN

kl87, 1958.

Jones, R. T., "Correction of the Lifting-Line Theory for

the Effect of Chord,' NACA TN

817, 19k1.

k Krienes, K., "The Elliptical Wing Based on Potential Theory,"

NACA TM

971, 19k1.

Shuford, C. L., Jr., "P. Theoretical and cperiaental Study

of Planing Surfaces Including Effect of Cross Section and

Planform," NACA TN

_{3939, 1957.}

Clement, E. P., and Blount, D. L., "Resistance Tests of a Systematic Serles of Planing Hull Forms," Trans. SNkME,

Vol.

71, 1963.

Scnnitzer, E., "Theory and Procedure for Determining Loads and Motions ir ChIne-Immersed Hydrodynamic Impacts of Prismatic Bodies," NACA Rept, 1152, 1953.

Martin, M., "Eauatlor.s of Motior. for hydrofoil Craft," HYDROMAUTICS, Incorporated Technical Report 001-9, Marcn

1962.

r

T

4 . S ? .1

FIGURE I - ORIENTA11ON OF BODY AXES RELATIVE TO AXED WIND AXES IIYDRONAUTICS, INCORPORATED 4 3 o _CZb

(a) r=12° *1O

FIGURE 2 - COMPARISON OF CALCULATED AND EXPERIMENTAL LIFT COEFFICIENTS FOR RECTANGULAR-FLAT-PLATE PLANING SURFACE

SYMBOL

0.70 0.80 0.60

.0

= 0° 2

I

=12°

b0=10°

POSTULATED EXPERIMENTAL CURVE (SAVITSKY) 0.60 2 o T =12°

*=20°

s=

15° 4 FIGURE 3

-

COMPARISON OF CALCULATED AND EXPERIMENTAL LONGITUDINAL CENTER OF PRESSURE FOR RECTANGULAR FLAT-PLAT PLANING SURFACE

IIYDRONAUTICS. INCORPORATED 0.80 0.70 m 0.60 0.80 X A 7A

-T-

_m 0.60 o FIGURE 3 - (CONCLUDED)

T =18°

10° 2

ilb

T= 18°

0=20°

POSTULATED EXPERIMENTAL CURVE (SAVITSKY)

3 HYDRONAUTICS, INCORPORATED 4 3

1/b

₂ 4 3 2

00

15° O . .2

C

_Z1,

)'T=6° *0=10°

0 =15°

0=

.2

(o) 1=6°

_*0=0°

4 .3 3 2 0o SYMBOL 0 SAVITSKY

1 0

00 øtoI

_Lc

15°

0 =00

.2

(c) T= 6°

₌

20°

FiGURE 4 - COMPARISON OF CALCULATED AND EXPERIMENTAL LIFT COEFFICIENTS FOR A SURFACE HAViNG A 20° ANGLE OF DEAD-RISE

HYDROP4AUTICS. INCORPORATED 4 SYMBOL

-SAVITSKY

f

_0.

00

¿

15° .3 .4 .5 .6

G) T= 12°

=0°

.4

C9

b

(°) T = 12° 0=io°

RGUR 4 - (CONT1NUtD)

.6 HYDRONAUTICS. INCORPORATED 4 3 2 flGURE 4 - (CONTINUED) b

(f) T = 12°

.4

=0

.2 .3

c

¶

(9) T1°

.4 .5 .6

-.5 .6

HYDALONAUTICS. INCORPORATED 3 2 4 7

=0

2 S 6 o

iti1

4- (CONCLUDED)

C

Zb Q1)T=18° 4'o 0°

=0

.2 .3

c

_Zb

()r =180 .20°

.4 .5 .6 HYDRONAUTICS, INCORPORATED 0.12 0.08 0.04

Cyo

-0.04 0.12 0.08 0.04 -0.04 -0.08 -0.12 O CZ 'D = 0°

AL

Ç

o

(b) T'6°

*0=10°

01

02

C

Zb

(a) r =6°

_=o°

0.12 0.08 0.04 C

"b

-0.04 0.3 -0.08 -0.08 -0.12 -0.12 0 01

02

0.3 0

01

SYMBOL'D

SAVITSKY J 0 00

etal

A15°

C

Zb 0.2

(c) i6° =200

FIGURE 5 - COMPARISON OF CALCULATED AND EXPERIMENTAL SIDE FORCE COEFFICIENTS FOR A PLANING SURFACE HAVING A 20° ANGLE OF DEAD-RISE

HYDRONAUTICS. INCORPORATED HYDRONAUTICS. INCORPORATED .04 -.12 o 4'=15° .10 .20 .30

c

_Zb (I)

r = 12° * =20°

o

- .30

Tr 18°

FIGURE 5 - (CONTINUED) .40 SAVITSKYÇ SYMBOL

O

O - 15° A .12 .08 .04 - .04 - .08

-.12

o .10 .20

c

_zb t'=o

I

4' =15°

HYDRONAUTICS, INCORPORATED

.!2

08 .04 C _o .10 - .04 - .08 .10 G') .30

c

_Zb

18° j',= 10°

.20 .30

c

fl) r=j°

=20°

o HYDROI4AUTICS INCORPORATED 1.2 0.8 CM 0.4 o 0 0.1

02

C

Zb

(b) T6°

q, =io°

o 0.1

02

Cz

b

(o)i=o°

03

1.2 0.8 CM b 0.4 0.3 SYMBOL

SAVITSKYIO d3

% o

etal

A15

0 = 00 0.1

02

Cz

(c)

, =6°

q, =20°

0.3

0=0 -I-o

I

to

o.

-.12 1 .2 .08

_{o -0}

0.8

b0

_A

_CMb -.04 0.4 - .08 FIGURE 5 - (CONCLUDED)

FIGURE 6- COMPARISON OF CALCULATED AND EXPRIME NIAL PITCH-MOMENT COEFFICIENTS OF A PLANING SURFACE HAVING A 20° ANGLE OF DEAD-RISE

1.2 o .2

=0

.3

c

_Zb (d)

T =12° *=o

o SYMBOL $

SAVITSKY f o

-etal _{È 15°} I I .4 .5 .6 .2

_C

.3 .4 .5 .6 Zb (e) T 12° FIGURE 6- (CONTiNUED) 1.2 8 CM .4 o o .2 .3

c

_Zb

(f) 1=12° I'=20°

o FIGURE 6 - (CONTINUED)

öi5°

.4 .2 .3

Cz

b

) 1=18°

=0°

HYDRONAUTICS. INCORPORATED 1.2 1.2 .2 .3 CZb s

) '=18°

¿10=10° 2 .3

c

_Zb s G) ¶ = 18°

'0=2o°

FIGURE 6 - (CONCLUDED) HYDRONAUTICS. INCORPORATED 0.04 0.02 0° SYMBOL¿1 SAVITSKYJO 0°

etui lA

is°

o

I

0

15°

-o _{0.1 0.2 0.3}

C

Zb

T=12'

I10°

04

o =0°

0.5 = 15°

04

0.5

FIGURE 7-COMPARISON OF CALCULATED AND EXPERIMENTAL ROLLING MOMENT COEFFICIENTS OF A PLANING SURFACE HAVING A 20° ANGLE OF DEAD-RISE -0.04 -0.02

A

A

A

CK

-006

0.8 0.6 0.4 0.2 -0.02

-004

o

01

0.2 CZ

(a) 1=12°

03

A

HYDRONAUTICS. INCORPORATED 0.12 0.08 0.04 o -0.04 -0.08 0.12o C)

H

SYMBOL 0 SAVITSKy

₀

00

atol

15°

o

0I

02

Zb

(a) i = 12°

O

03

0 = 15°

04

0.1 0.2

03

04

-c

-

-Zb

0)r

120

FiGURE-8- COMPARISON OF CALCULA1W AND EXPERIMENTAL YAWING-MOMENT

2

cflcfl5 OF A PLANING SURFACE HAVING A 20° ANGLE OF

DEAD-RISE -

-'i

o

C

ri

'o -1 O

>

O

"

G)

r

O -'t

o

C)

z

-I

D Ffl X

_G)0

>

_ri

G')

r;

D O 'J, UI O'

o

Ut

o

'n M

o

M 'n

-FLOW COEFFICIENT_{- 80}

o

p

p.

p --e

_p

_o

M (4 _{'n O'}

ø3LVOdeOp4I 'S3I1flVNOOAH

o

o

'o

UNCLASSIFIED

Snt cI,iiElcoticn

DOCUME1T CO4TROL DATA R&D

Çtt, dn.Ifl

al Uil.. badi. .uw nd..o,g ,., - ,i...d .dSI,nSi ,.n I. C..àfI.dJ

I. bAIGlCi&tIil G AttIVI'Y (C.nt. s,1

HYDRONALITICS, Incorporated, Pindell School

Road, Howard County, Laurel, Maryland

a..napa., uncuRl?, C LAIlIFICA 71041 UNCLASSIFIED

I. PÇU1 'SiLt

ON THE MOTIONS OF HIGH SPEED PLANING CRAFT

1- DUCRIPTIV* NOTAS (17p. .1 . d ob.IP. IV.)

Technical Report -_{AUTISOIVI) (Lana} 11 . Hsu, C. C. May ₁₉₆₇ NO. OP 7b HO. OP ntfl 33 8 ISt OR SRAJIY HO. Morir

5133(00)

A I. ¿

. OU1SIATOWS REPON? Nuwnn1) Technical Report

603-1

IA JPON? HO(3J (A., .0411 5 .p b. .o.IRuad IA AVA IL*.i1.ITY!LaMITATION nOTICIA

Qualified requesters may obtain copies of this report from DDC

II. IUPPL&U7ARY NOTES lb. IPONSORINO NILS'ART ACTIVITY

Naval Ship Research and

Develop-ment Center, DepartDevelop-ment of the Navy

IL ARSTIACT

In this rep&rt the hydrodynamics and dynamics of high speed

planing craft are studied. The first part of this study presents a

simple method, based on the known results of airfoil theory, of

estimatir,g the hydrodyriamic characteristics of high speed planing craft. Comparisons between present calculations arid experiments for

flat and prismatic surfaces in steady trimirig, rolling and yawing attitudes are made. The calculations seem generally to be in good

agreement with the data. The second part of this report discusses

the turning r.d stability of planing craft. The governing equations,

based on the kinematics and dynamics of rigid bodies are in many

respects sImilar to that given for hydrofoil boats. Once the

hydro-dynamic characteristics are known the stability and

turning

per-formar.ce of craft caribe assessed with the aid of these equations.

DD

'1473

UNCLASSIFIED

UNCLASSIFIED

ti

KEY NORDS Notions Planing Craft Bydrodynamlc Characteristics Turning and Stability

LINK *

flott flot tLINK S MOLSLINK

I. ORIQI)1AIIRG ACTIVITY'. Lot. lb. N and

.1 II. oo,anto.. ..b.otot.0110. ir.ot... D,.tot of D.

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2. REPORT SEcUWrY C1.A8RIVTCATIO?h Ett., lb. otan.

all .ttttttV olan.ilio.tIoo of 1h. apoll. lndiai. .bsth., 'R.ot,Lcttd Os.' I. loclotd.t tonott. I. to b. I. ..an.d'

an.. .0th apytopotato aCsfty retIllatloon

28. CR011?'. Attototolo do r.dJ.i la .p.oLfld t, DoD Db'

tIflN 1200.10 and A,.,.d Foto.. 1,tiornt,INi titans. Eol lb. g.ot*p ota,ba. Alto. *b.n .ppUc.bl., ho* 1h01 opUontl

O10kiroI Kto. b.ao *aad tot GOOOIP 3 tod Gott? 4 ta .tgh. 5 RORT TITI.E1 tnt lb. tnttt IOR dUt It Nil Capital ¡.11a,o. Titi,, Lt.Oil Cm.abottid b. motsaailitd.

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4. DCR1PTIVE ?)TER I! appropriai., aetor Iba typ. of rtpoet. fl. isotta proo..a, 51010117. NOrois. 00 fitti. DIo. lb. iocLiani,. dai.. ob.. s apsodlo roploUnt potiod lt

Coo--.

S. AUTSR(S Erro., lb. oo..(.) of aotho.(.) a. .h.ane at o. I, ob. r.port. Eolo tant n..., float flan.. loIddi. lnitI.I. U .0iItoty, Nb.. r.r* .od branch of Notoic. it.. floro, of lb. prittipal olbor o. so .b.oiot. .lt.bo,o.. r.qoi.an.oL

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b. 0025g. to thiS rapolt.

9h. Ortiz? REPORT if1JLR8)I if the rapoll han b... SSlIOII 00 Otha, rapo'l ariwbot. (.,llb.r by lila oriditalod

o, by h apon.o,), .1.. ont.. ohl nhot(.).

10. AVAU0AWLITY/LIRITAT1ON NOTICER Rot., a.y l

listions Is;haodit..w1ntt .1 lb. ,.p..t, et... iban that

IIIRTRUCTIONS

t.po.ad by sacUy daa.UlosUott salop Siondstd .l.lan.oI.

.a.b Sot

(i) 'Qa.Wl.d l09055 -? lsb C.pl.. of Ibm

tapo,t flow DOC.'

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II, SUPPLEISENTARY yTE5, U.. lot additional .a.plooa.

ta,y ant..

SPONRDRING IIII.ITARY AC'TIYITYt Lid.. Itt ano, el 1h. dap.,ten,ral projani otilo. a, I.boisto:p apoo.00aq (pup'

led loo) Iba ronaa,cb and dsttiopo..L loclodo adot.... ABSTRAC'I'! Rot., a. ab.lr.cl 1itio1 a brl.f std faciuti .000aly of lb. docuw.ol Indicatir, of b. 1h00, rvdo bhou5h It may alio type,, elarwh.rOIn b. body01_{the Iechoa.t r.'}

pon. II .dditlao.l tpaoe i. raquI,.d, a cObtI,fluat.00 nba. t shalt

b. atlaissd.

It i. hIghly d..lrnbI. th.t Ihr abollad of cl...tfled r.pcoly b. tc,cla..iIl.d. Each I.ra,..ph of b. obt.lr.cb abolI bd arltb

NO thdtcaUaa cl tAN m&llt.,y n.dItI'ltl y i,NOI llvallcfl aI lbo io.

fo.watio. io h. paragrapb. r.pr.a.ot.d as ITS). (S). (Cl. or (f)) nere ,.so ImItation 00th. l.ogth al the abollad Nao.

soor. ti.. nog0..I.d I.ogth n, fr0.. 150 to 223 cord.. la. KEY WORDS. Et, word. ai ragbtIlaliI c1cadI'I;I.I tc.c'i O? Itoct pIt... thaI characlort.. rIpaIt arno moi ho ut..0 .a indos .0mb foe oat. Iogttr Ill. roporl. K word. roost be

aelrcr.d .0 that to incartIy ct.a.ificauo. ¡a roquirod. Lient..

llar., loch IN quLpweaI model d.stn.Iico. lINd, flama, atl,b.ry projlct veda osma, gocgnptcc loc.lIoc. 0.1 ho toed a. kay sorda bot .iil b toltow.4 by an iotdic.blOtt 0f t.chrccat con.

1.01. Th. aaa,5a0.,t 02 liaba, of.., aed w,,1h1. lO aptioo.I.

bD

. 1473 (BACK)

D 3581 _UNCLASSIFIED

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