A non-linear frequency domain simulation for SWATH craft

TEZHNISCH Schee0

Me141weg''

DAVID W. TAYLOR NAVAL

SHir5-RESEARCH AND DEVELOPMENT CENTER

Bethesda, Md. 20084

GENERALIZED NONLINEAR TIME DOMAIN MOTION PREDICTOR FOR SWATH

by WALTER LIVTNGSTON E-1 CO

0

pt. E-1 114

APPROVED FOR PUBLIC RELEASE: DISTRIBUTION UNLIMITED

SHIP PERFORMANCE DEPARTMENT

0

rT1

JULY 1979 DTNSRDC /S PD

0857

-01

\e,

MAJOR DTNSRDC ORGANIZATIONAL COMPONENTS

OFFICER-IN-CHARGE CARDE ROCK 05 SHIP PERFORMANCE DEPARTMENT 15 STRUCTURES DEPARTMENT 17 SHIP ACOUSTICS DEPARTMENT 19 DTNSRDC COMMANDER 00 TECHNICAL DIRECTOR 01 OFFICER-IN-CHARGE ANNAPOLIS 04 AVIATION AND SURFACE EFFECTS DEPARTMENT 16 COMPUTATION, MATHEMATICS AND LOGISTICS DEPARTMENT 18 PROPULSION AND AUXILIARY SYSTEMS DEPARTMENT 27 CENTRAL INSTRUMENTATION DEPARTMENT 29 SYSTEMS DEVELOPMENT DEPARTMENT 11 SHIP MATERIALS ENGINEERING DEPARTMENT 28

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I. REPORT NUMBER

DTNSRDOSPD-0857-01

2. GOVT ACCESSION NO. 3. RECIPIENT'S,CATALOG

NUMBER-4: TITLE (end Subtitle) ' '

Generalized Nonlinear Time Domain Motion Predictor for SWATH Craft

_

S. TYPE OF REPORT & PERIOD COVERED

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6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(a)

Walter Livingston

B. CONTRACT OR GRANT

NUMBER(a)-,

9. PERFORMING ORGANIZATION NAME AND ADDRESS

David W. Taylor Naval Ship R&D Center Bethesda, Maryland 20084

10. PROGRAM ELEMENT. PROJECT, TASK AREAS WORK UNIT NUMBERS

Task Area ZF43-421

Work Order No. 1,150C-103-74

IL CONTROLLING OFFICE NAME AND ADDRESS

"Naval Material Command (08T)

12. REPORT DATE July 1979

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82

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Nonlinear, Time Domain, Simulation, Arbitrary Inputs, SWATH Craft .

20. ABSTRACT (Continue on re eeeeaaide if rilietteetry end identify by block number)

A generalized nonlinear time domain simulation model development for SWATH

craft is described. The model, as developed, considers motion in the

pitch-heave, degrees of freedom only. There is no technical impediment to expanding this to the full six degrees of freedom of rigid body motion. As contrasted With a frequency domain simulation currently used for predicting motions of

SWATH Oraft, the generalized nonlinear time domain simulation will accept any non-pathological function of time as a forcing function. Also frequency

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motion predictions made with both the frequency domain simulation and the generalized time domain simulation show sensibly identical agreement. Where the generalized time domain simulation is used in a manner that reveals the full effects of nonlinearities (the frequency domain simulation linearizes), there, as would be expected, are differences between the predictions made with the two simulations.

UNCLASSIFIED

TABLE OF CONTENTS

Page

LIST OF FIGURES

iv

LIST OF TABLES

vi

NOTATION

viii

ABSTRACT 1

AD MIN I STRATI VE IN FORMATION 1

INTRODUCTION 1

BACK; ROUND 3

COORDINATE SY STEM-FREQU EN a RESPONSE FUNCTION 6

APPROXIMATION MODELING 11

RATION AL POLYNOMIAL APPROXIMATION 11

TRANSLATING TO THE TIME DOMAIN 29

FREQUEN CY INDEPENDENT NONL IN EARITI ES 44

NONLINEAR HULL AMP ING

44

FIN LIFT AND D RA; MODELING 47

SEAWAY DESCRIPTION 48

DISCUSSION OF RESULTS AND CONCLUSIONS 52

SCENARIO NO. 1 53

SCENARIO NO . 2

59

SCENARIO NO . 3 62

ACKNOWLEDGMENT

62

RE FEREN CES 65

APPENDIX - SYNOPSIS OF THE MODELING DEVELOPED IN

67

REFERENCE 1

LIST OF FIGURES

Page

Figure 1 - Frequency Response,

H33(jw),

at. 10 Knots 12 Figure 2 Frequency Response, H35(jw), at 10 Knots 13

Figure 3 - Frequency Response,

H53(jw),

at 10 Knots 14 Figure 4 - Frequency Response, 1155(jw), at 10 Knots 15

Figure 5 Poles-Zeros of a 6th Order Numerator/8th 17 Order Denominator Rational Polynomial

Approximation to

H33(J(0)

Figure 6 - Poles-Zeros of

a

10th Order-Numerator/12th 18 Order Denominator Rational Polynomial

Approximation to

H33(jw)

Figure 7 Poles-Zeros of a 6th Order Numerator/8th 20

Order Denominator Rational Polynomial

Approximation to

H35(iw)

Figure 8 - Poles--Zeros of

a

10th Order Numerator/12th 21

Order Denominator Rational Polynomial

Approximation to

H35(jw)

. .

Figure 9 - Poles-Zeros of a 6th

Order

Numerator/8th 23

Order Denominator Rational Polynomial

Approximation to

H53(jw)

Figure 10 - Poles-Zeros of a 10th Order Numerator/12th 24

Order Denominator Rational Polynomial

Approximation to

H5300

Figure 11 - Poles-Zeros of a 7th Order Numerator/9th 26 Order Denominator Rational Polynomial

Approximation to

_H55(jw)

Figure 12 Poles-Zeros of a 11th Order Numerator/13th 27

Order Denomintor Rational Polynomial

Approximation to

H5500

Figure 13 - Velocity Components that Make Up the Total 46

Local Velocity Vector, V

Figure 14 - Pierson-Moskowitz Power Spectrum for a 49 Sea State 6 (15 ft Significant Wave Height)

LIST OF FIGURES (Cont)

Figure 15 Pierson-Moskowitz Power Spectrum for a Sea

State 6 (15 ft Significant Wave Height) Where Encounter Frequencies Correspond to 4 10 Knot Ship Speed in Head Seas

'Figure 16 - Comparison of Heave Responses Obtained with the Frequency Domain Simulation and the

Generalized Nonlinear Time Domain Simulation

for the SWATH 6A Proceeding at 10 Knots

into

Regular Heed Waves of Unit Amplitude.' No Viscous Lift or Drag

Figure 17 -,- Comparison of Pitdh Responses Obtained with the Frequency Domain Simulation and the

Generalized Nonlinear Time Domain Simulation

for the SWATH 6A Proceeding at 10 Knots into

Regular Head Waves of Unit AmplitUde. 10

Viscous Lift or Drag

Figure 18 - COmpariscin Of Pitch Responses Obtained with the Frequency Domain Simulation and the

Generalized Nonlinear Time Domain

for

the SWATH 6A Proceeding at 10 Knots into an Artificial Head Sees Includes Viscous

Lift Or Drag

Figure 19 _{Comparison of Pitch Responses Obtained with} the Frequency Domain Simulation and the

Generalized Nonlinear Time Domain Simulation

for the SWATH 6A 'Proceeding at 10 Knots into an Artificial Head Sea. Includes Viscous

Lift

or

.Drag

Figure 20 - Heave Response for the SWATH 6A Proceeding into a Head Sea at 10 Knots (Modeling includes all Nonlinearities)

Figure 21 - Pitch' Response for the SWATH 6A Proceeding into a Head Sea at 10 Knots (Modeling includes all NOnlinearitieS)

Figure 22 - Heave Response for the SWATH 6A Directed into a Head Sea at Zero Knots

Figure 23 - Pitch Response for the SWATH 6A Directed into a Head Sea at Zero Knots

LIST OF TABLES

Page

Table 1 Selected Values of the Unit Impulse Responses 19 Computed from Two Different Rational Polynomial

Approximations to the Frequency Response

Function

H33

(jm)

-Table 2 Selected Values of the Unit Impulse Responses 22 Computed from Two Different Rational Polynomial

Approximations to the Frequency Response

Function H35

(jw)

.

Table 3 - Selected Values of the Unit Impulse Responses 25 Computed from Two Different Rational Polynomial

Approximations to the Frequency Response Function H

ow)

53

Table.

4 Selected Values of the Unit ImOulse Responses 28

Computed from Two Different Rational Polynomial Approximations to the Frequency Response

Function H (jw)

55

Table 5 Time Histories of Heave and Pitch Obtained Using 30

Two Different Sets of Fits to the Frequency

Responses 1133(j ),

H3500,

H53(jm),

H55(10)

for

the SWATH 6A Proceeding into a. Seaway at 10 KnotS

Table 6 - Tabulation of Frequency Response Values for the 32

Time Histories for Heave and Pitch Tabulated in

Table 5

Table 7 - Comparison of Values

for

the Frequency Response, 34

H3304,

and the Values Computed Using a 6th

Order Numerator Over an 8th Order Denominator

Rational Polynomial Approximation to H33.014 for SWATH 6A Moving With a Forward Speed of 10 Knots at a Depth of 19.17 feet (5.84 meters) to

the Centerline of the Hulls

Table 8 - Comparison of ValueS for

the

Frequency Response 35 H33(10)), and Values Computed Using a 10th Order

Numerator Over

a

12th Order Denominator Polynomial Approximation to

H33(10

for the SWATH 6A Moving

tgith.a

Forward Speed of 10 Knots at a Depth of

19.17 feet (5.84 meters) to the Centerline of the Hulls

LIST OF TABLES (CONT)

Page

Table 9 - Comparison of Values for the Frequency Response 36

H35(N3), and Values Computed Using a 6th Order Numerator Over an 8th Order Denominator Rational

Polynomial Approximation to

1115(j

to) for the SWATH

6A Moving with a Forward Speed"Of 10 Knots at a

Depth of 19.17 feet (5.84 meters) to the Center=

line of the Hulls

Table 10 Comparison of Values for the Frequency Response 37

H35(j(13), and Values Computed Using a 10th Order

NumeratOr Over a 12th Order Denominator Rational

Polynomial Approximation to H15(j(13) for the SWATH 6A Moving with a ForWard-Speed'of 10 Knots at a

Depth of 19.17 feet (5.84 Meters) to the Center-line of the Hulls

Table 11 - Comparison of Values for the Frequency Response 38

H5s

410), and Values Computed Using a 6th Order

Numerator Over a 8th Order Denominator Rational

Polynomial Approximation to

lisp (13)

for the

SWATH 6A Moving with a Forward' Speed of 10 Knots at a Depth of 19.17 feet (5.84 meters) to the Centerline of the Hulls

Table 12 - Comparison of Values for the Frequency Response 39 H53(jw), and Values Computed Using. a 10th Order

Numerator Over an 12th Order Denominator Rational Polynomial Approximation to H5390 for the SWATH

6A Moving with a Forward Speed of 10 Knots at a

Depth of 19.17 feet (5.84 meters) to the Centerline

of the Hulls

Table 13 - Comparison 'Of Values for the Frequency Response 40

H55(10;

and Values Computed Using a 7th Order

Numerator Over an 9th Order Denominator Rational

Polynomial Approximation to

H55.00

for the SWATH

6A Moving with a Forward Speed of 10 Knots at a

Depth of 19.17 feet (5.84 meters) to the

Center-line

of

the

Hulls

Table 14 -Comparison of Values for the frequency ReSponse 41

H55(ID), and Valdes Computed Using a 11th Order

Numerator Over an 13th Order Denominator Rational

Polynomial Approximation to

H5500

for the SWATH

6A Moving, with a Forward Speed of 10 Knots at a Depth Of 19.17 feet (5.84 meters) to the Center-line of. the hulls

NOTATION

A(w.) Amplitude of a Frequency Response Function

1

at w, (f)

A Fin Area

A. Coefficients in Numerator of Rational Polynomial

1

A Projected Area of Hull

P

A (w.) Wave Amplitude at w. .

w

I

1

A_wp Waterplane Area

A Frequency Dependent Coefficient

33

A Frequency Dependent Coefficient

35

A Frequency Dependent Coefficient

53

A Frequency Dependent Coefficient

55

a constant Used in formula for viscous damping

on hulls

a(t) Time History of Wave Amplitude

a Added Mass of Hull Per Unit Length

33

a3(3f)

Added Mass of the ith Fin

Frequency Dependent Coefficient

33

35

Frequency Dependent Coefficient Frequency Dependent Coefficient

53

Frequency Dependent Coefficient 55

Coefficients in Denominator of of Rational Polynomial

Vertical Distance Between Center of

Buoyancy and the Center of Gravity

When the Ship is at its Mean Position

Hull Wave Damping Coefficient per

33

_{Unit Length}

C33 C35

C53

Frequency Independent Coefficient

55

D(t) Cross Flow Drag of Fin

F(6) Amplitude of Wave Exciting Force

in

Heave

Fe)

_{Amplitude of Wave Exciting Moment in 'Pitch}

5

Acceleration due to Gravity

H 33(j w) Frequency Response Function

H (juj)

35

- . Frequency Response Function

H.5300

Frequency Response Function

H55

(j)

Frequency Response Function

113

Significant Wave Height

15 Mass Moment of Inertia of Craft about

the Y - AxisQ

Square Root of -1

Ko Wave Number ("t2m/A)

L Length Of Hull

NOTATION (CONT)

Cross Flow Drag Coefficient for Hull

Cross Flow Drag Coefficient for Fin

Lift Coefficieht of Fin

Frequency Independent Coefficient Frequency Independent Coefficient Frequency ,Independent Coefficient

NOTATION (NT)

Length of Ith Fin

Inertial Mass of the Craft

Waterplane Area Moment About the

YITAxis (f)

Inertial Mass of Ith Fin

P(a Ordinate of the Energy Spectrum

Representation of a Seaway

Real part of

Laplace Variable

Time

TEMP _{Temporary Variable}

Constant FOrward Velocity of

Ship

V _{Local Total Fluid Velocity relative}

to Craft

Mind Speed

in

Knots

X Y Z o o o Z OA 2 033 35 cc(t)

Inertial COordinate System Moving at the Mean (Constant) Forward Velocity

of the Craft. Motions of the Craft .

are Pia-Slimed to be Perturbations abOut this Axis System

Amplitude of Heave Response of Craft to

a

Sinusoidal Input

-Contribution to Heave Displacement Due to Force

Contribution to Heave Displacement Due to Moment

Local Angle of Attack

Seaway Heading B = 0 for Following Seas.; B = 180 for Head Seas

(t) CX0

Zy0(t)

(t)

CZ0 NOTATION (CONT)

Phase- Angle. Selected from a Uniform

Distribution (Random) at Frequency wi Pitch Angle

Amplitude of Pitch Response of Craft to

a

Sinusoidal Input

53(0

Contribution to Pitch Angle Due to Force

055 (t) Constribution to Pitch Angle Due to Moment

0 _{Velocity Potential of Incoting Sinusoidal}

Wave System

S(D. )

Phase

of

a Frequency Response Function at

w, 1 Phase Angle Phase Angle Phase Angle Encounter Frequency

The ith Wave Angular Frequency.

Wave Frequency

Absolute Value.

First Ordinary Derivative of a Quantity with Respect to

Time

Second Ordinary Derivative of a Quantity with Respect to Time

MASS Density

of

Seawater Wave Length

Wave Orbital Velocity In the X Direction

Wave Orbital Velocity in the Y Direction

Wave Orbital Velocity in the Z Direction

X1 80.02

ABSTRACT

- A generalized nonlinear time domain simulation

model development for SWATH craft is described. The

model, as developed, considers motion in the pitch-heave

degrees of freedom only. There is no technical impediment to expanding this to the full six degrees of freedom of rigid body motion. As contrasted with a frequency domain simulation currently used for predicting motions of SWATH

craft, the generalized ncinlinear time domain simulating

wil

accept any non-pathological function of time as a forcing

function. Also frequency independent nonlinearities are easily accommodated. For linear situations, motion

predictions made with both the frequency domain simulation and the generalized time domain simulation show sensibly

identical agreement. Where the generalized time domain

simulation is used in a manner that reveals the full effects

of nonlinearities (the frequency domain simulation

linearizes), there, as would be expected, are differences between the predictions made with the two simulations.

ADMINISTRATIVE INFORMATION

The work has been authorized by the

Naval

Material Command, (08T),

funded under the Ships, Subs, and Boats Program Task Area ZF43-421, and

administered by the Ship Performance Department High Performance Vehicle

Program Office (1512).

INTRODUCTION

A system modeling that is usually employed for the characterization

of surface ship motions and, in particular, is used

at

the David W.

Taylor Naval Ship R&D Center CDTMSRDC) for predicting motions of

Small-Waterplane,-Area-Twin-Hull (SWATH) craft consists of a system of.

equations often referred to as differential equations with frequency dependent coefficients.- As a matter of fact, these equations are not

differential equations in the usually understood sense of.the Words.

1

They are (although somewhat disguised) algebraic expressions for

computing the steady state frequency response of the system they are

meant to represent. In the interest of broadening the range of

understanding of the behavior of SWATH craft via simulation (a

frequency response function carries only a limited characterization of the dynamics of a craft), an effort was undertaken at the DTNSRDC to develop a genearlized nonlinear time domain simulation model. The first phases of this effort have been completed. This report describes accomplishments accrued to date and includes discussions

of the various techniques developed and employed in realizing these accomplishments.

Along with the capabilities provided by frequency response modeling, the generalized nonlinear time domain simulation modeling brings to the SWATH design process additions such as:

the capability for generating time histories for any motion variable* during simulated transient as well as steady state behavior,

the capability for generating time histories of the simulated force and moment systems acting on the craft during transient as well as steady state behavior,

the possibility of designing, developing, or assessing feedback control systems for motion attenuation or for

course-keeping in the time domain,

*At the present time these are restricted to heave and pitch. However, there is no technical impediment to expanding these to motion variables in all -six degrees of freedom of rigid body motion.

the possibility for simulating mission oriented Scenarios, the highly desirable capability of easily incorporating into the modeling any frequency independent nonlinearity. This is especially important

in

_{control system development since the simulation}

of realistic control systems usually _{means handling a number of}

non-linearitieS,

the easy incorporation of nonlinearities that represent coupling forces amongst various degrees of freedom, and

the possibility of applying System Identification/Parameter

Estimation technology to the Process of refining the _{simulation model}

per

se,

As presented in this repOrt, the techniques described _{are developed}

for the SWATH 6A Craft. This does

not

_{reflect a constraint. These} techniques can be adapted to other SWATH designs _{as well as to any Craft} for Which steady state frequency response representations exist. Such craft might include, among others, air cushion vehicles (AcVs), surface effect Ships (SESS), and conventional monohull _{surface ships.}

BACKGROUND

Before proceeding with the development in _{this report, there are}

two points of position that need stating. _{First, it is not a purpose} of this report to assess or to attempt to validate the modeling of the dynamics of the SWATH 6A craft. _{Where motion predictions are included,}

they are included for purposes of demonstrating _{the capabilities of the}

generalized nonlinear time domain simulation. _{There is no commentary}

related to the dynamic. behavior of the craft as might be interpreted through such predictions. A .second point addresses the distinction

between

a

common interpretation of "time domain simulation' and the

interpretation given it in this report. Fiat the former, a frequency response is computed for some mode of motion such as heave, pitch,. Toll, etc. Using the linearity assumption,

a

time history for that model

is

constructed according to the following formulation.

V

a(t)

= A(mi)Aw(wi)cos-Nit+O(mi)Wwi))

Here the time history, a(t), is a synthesis of a finite number, N, of cosines, A(w) is the magnitude of the frequency response function at

wi- ; A(w) is the wave amplitude at wi determined from a wave energy.

spectrum; cp(wi) is the phase of the frequency response function at

wi;

and B(wi) is a phase angle also at wi selected randomly from a

uniform distribution Over the range 0 to TR radians. The wave

amplitude, Aw(wi) is determined as follows. Let P(0) be the energy

.spectrum for the particular sea state chosen (see sketch below)

.

-L-

UGLE

rizEclusicv oF

14

POWER PCTRUFi

FOR 114E SEASTATE

SCLECTED

Further, let the time history be synthesized with, gay, ten cosines,. The range of teciVer. Which P(w) has values larger than one!-hundredth

its peak values, for example, is divided into ten

thth

equal divisions. The:

frequency of the i cosine is chosen, at the center of the 1 subdivision. The amplitude, A (Pi ) is chosen as twice the square, root of the area

th

of the rectangle. Each rectangle is contructed SO as to approximate

the area under the power spectrum for the particular A0 in question. The

basic point to be made concerning this type of 'time domain simulation'

is that

the

simulation is carried out

in

the frequency domain. The 'time domain' display is actually only one of an infinite number of

functions resulting from different sets of random phase shifts..

The interpretation given 'time domain simulation' in this report

differs from the one described above. In the interpretation here,

frequency response function (which may incorporate frequency dependent

coefficients) is generated and this response is approximated with a

rational polynomial in. the Laplace variable s.- Having obtained this

polynomial to the degree of approximation desired, a translation to

the time domain results in an ordinary differential equation with

constant coefficients. It is this equation which is solved as an initial

value problem for any arbitrary function of time as input. A desirable

feature of this 'generalized nonlinear time domain simulation' is the

facility with which frequency independent. nonlinear Modeling can be accommodated. The complete development of the modeling that permits the

computation of the frequency response functions which serve as the starting point for the development of the generalized nonlinear time

domainsimulation model Is given In Reference 1.* Those portions of

this reference upon which

the

development of. the generalized non-linear time domain Simulation model draws most heavily are given in, the appendix Of this report for ready reference.

COORDINATE SYSTE14-FREQUENCY RESPONSE FUNCTION

Only heave and pitch are treated here. Nothing is lost by this

reatrictiOn except unnecessary complication, since the combination, sway-yaw-roll, is assumed to be completely uncoupled from the

combination, pitch-heave, and further Surge is assumed to be completely

uncoupled from any other degree of freedom. These assumptions Were taken for the development

in

Reference 1 and carry over to the

development in this report.

MotiOns.and forceS are referenced to a frame Which is a right-handed

orthogonal Cartesian coordinate system. frame does not notate but translates with constant velocity, V , and, thus, is an inertial frame. The origin of this frame lies in the plane

of

the mean free surface and the plane of the mean position of the longitudinal center. plane of the draft. The X 0 Y plane coincides with the plane of the

000

mean free surface; the X-axis points toward the bow of the craft.; the

Yo-,axis points toward port; and the Z0-axis points in the direction of decreasing water depth. The craft makes small motions

*A complete listing of references is given on Page 65.

(i.e. perturbations) about this coordinate system. The .reference

point on the craft is that point which, at calm Water

coincides with the origin of the Coordinate system.

The springboard for the development of the nonlinear time domain simulation model in pitch and heave is the formulation of the steady state frequency responses for the SWATH 6A given below as EquationS

(7) and (8).

The equation of motion for heave is given in the appendiX as

(M+A3.5-)Z0+B 3i3 0+C33 Zo +A3 OfB35 6+C35

-0=

F(e)3 e-jwt,

and for 'pitch as

( ) -jut

(I +A )19 B 6+C ELI-A +B Z C Z =

55

55

55

5.3-0

53 o+ 53

5 e

. Equations

(1)

and

(2)

are steady state equations and it their use it

is assumed that the steady state can be reached. (M+A

(0), B

33

-

33

B35(w), (I5+A55(0),

85500,

A53(w), and B53(w) are

_usuallY

functions of frequency. c33, c35, c53, and c55

are

frequency independent.

The amplitudes of the forcing functions (usually _{wave induced) are}

complex to allow these functions to exhibit phase relationships _With

respect to the Wave. Equations (1) and (2),are assumed Valid for

sinusoidal inputs and, because they are linear steady-state equations,

generate responses that are necessarily sinusoidal; i.e.,

j wt

Z0(t) = ZoAe ₍₃₎

jwt

Here both zu and eA are complex to allow for phase relationships With

tespect to the forcing functions.

Equations (1) and (2) by using (3) and (4) can be Written

as

algebraic equations, i.e.,

F(e)-

(e) Z - 3 F5 OA-H

(jw =-K

35 7

.2

(jw) =-K3

(jw) .=

ce)

.F5 )

i.(03.33+C3

- jOB

. 35

85 jw1353-1-c53

K=w2(I

+A )=.103 +C - 5 55 55 55 Setting

(e)

F -3

(5)

(6)

/1:1

:3

d

4 IC1

/I

K IC1t21 3 4 1/1:13 41

ri

K2 3 4

//I

8

( 8)

A where

K =

=w2(14+A K2 = =W2A35 K -3 = -w2A

53

H3

) =

gives the following for (5) and (6)

Z-OA = H33(jw)F(e)3 + 35(jw)F.(e)5

eA

=-H

53

(jw)1J°3 +

H55(jw)F(e)5

Equations (9) and (10) can be further manipulated, i.e. One can set

(e) Z

=113

0

)F3

053

(e) Z

= H

(jw)F

.0

-35

5

35

_(e)

e H

(jw)F

53 53 3

55

7 H55(iw)F(e), 5

Since F(e) mAy be complex it Can be Written as

F(e)

A F(e)1 6,10-1(w)

3 '

Equation (11) now becomes

-zo

Ii1300

F(e)

f.J01

(4))

33

'

3

le

or and

Z3

e-J4) (w) = (Jw) IF(e)1 0 33 3 H33.

(j

)

033

.-',14) (w) =

--.3

Z , itself, may be complex and can be Written as 033

IZ e-J4)2(w)

-033

033

Substituting in (15), One has

H33(j)

20

w) = --33

ei° (

)

where

(1)3(w) =1(w) + 4)2(w)

(18)

Equation (17) is the defining relation for a steady state frequency

response function. Similar developments can be carried out for

H35003),

H53(jw), and

H55(10.

The frequency response functions, H33(jw), H35(p), H53(jw), and

H55(jw) are functions of 1(1, K2, K3, and K4 and the Ks are functions of frequency dependent coefficients (see Equation (7)). Tbe frequency dependent coefficients involved are identified in the appendix and are

computed using the program described in Reference 2..

The steady state frequency response function initiates the

develop-ment of the nonlinear time domainsimulation model. This development is covered in detail in subsequent sections of this report. Suffice it to say that an important step in developing the time domain model is to approximate the frequency response function by a functional form (here, a rational polynomial in the Laplace variable S) which is easily trans-latable to the time domain. Once this translation is made, the

simulation is treated as a constant coefficient ordinary differential equation initial value problem.

Nonlinearities which are not frequency dependent (frequency dependent nonlinearities are not addressed in this report) are treated as

contributions to the forcing functions.

When such functional forms can be found, and when nonlinearities can be treated as contributions to the forcing functions, a very flexible and

fertile time domain simulation results. This type of simulation provides

considerable potential for characterizing the motion behavior of the

dynamic system under study.

APPROXIMATION MODELING

The prime purpose for approximating the frequency response functions

with functional forms which are directly and readily translated to a time

domain representation is not to provide deeper insight into the physical

processes involved but, rather, to obtain forms whose time domain

trans-lation can be treated as valid mOtiOn predictors.. Hopefully, the additional information embodied in the motion predictions per se will provide the possibility for probing more deeply the physical processes involved.

RATIONAL POLYNOMIAL APPROXIMATIONS

The rational polynomial in the Laplace Variable, S (= iw),

is

the

functional form selected for approximating the frequency response func tioni.-.The usefulness of this form becomes Apparent when it is realized that its translation to the time domain is an ordinary differential equation with COnstant Coefficients. The four frequency response

functions that are approximated are (see Equations (9) and (0)) H33(jw),

H35(jw), H53(jw), and H55(jw). These are shown plotted in Figures I,

2,

3,

and 4, respectively, for a craft forward speed of 10 knots and When the craft is at at eqUilibrium draft of 19.17 ft (5.48m). The program Which was developed to carry out the approximation to the four frequency. response functions is based on a complex. curve fitting technique. The

6.85

10 --

2

0

0

0.0

0.2

0.4

0.6

0.8

1.0

t2

1.4

1.6

we (ENCOUNTER

FREQUENCY IN RAID/SEC)

-20

-40

-60.

>

-80

cn

-100 z

r

-120

;l

-140

-9

-160

-180

,

at 10 Knots

Figure 1 - Frequency Response,

'H33

0.6

0.8

1.0

1.2

we (ENCOUNTER FREQUENCY IN RAD/SEC).

Figure 2 - Frequency Response,

H3 ( j

), At 10 Knots

1.4 1.6 1.3

-100 m

-150

-200

-7

200

150

100

50

0

-50

-u cn rn

>

emMID

4

9.0

0.2

0.4

2.25

0.0

1.12

150

I

200

02

0.4

0.6

0.8

1.0

1.2

1.4

1.6

.

we (ENCOUNTER

FREQUENCY IN RAD/SEC)

Figure 3 Frequency Response, 'R5 3 OW

at *10 Knots

Magnitude

Phase

4.4.0 IMMO MOD WMIN MEI

200

150

₁₀₀

50

rciT1

0

0

rn

-50

0

-100,

0.0

0.2

0.4

0.6

0.8

1.0

1.2

We (ENCOUNTER FREQUENCY IN RAD/SEC)

Figure 4 - 'Frequency Response,

H55 (,w),

at 10 Knots

1.4

tailored to 'fitting functions' that take the form of arltio4a1

polynomial in the Laplace variable (S jw). Input to the program

consists

of

magnitudes and phases of the frequency response function being fitted, the order of the numerator and the denominator

polynomials of the rational polynomial selected

as

the fitting

function, and a fitting error criterion for stopping the iteration

process. Output from the program comprises the estimated values for

the coefficients in the selected rational polynomial, zeros

of

the

numerator Polynomial, and zeros of the denominator polynomial.

Figures 5 and 6 identify different choices of rational polynomial

(i.e. different numerator/denominator orders) approximations to H33 (jw) with their respective pole-zero values. Table 1 shows Selected values

ofthe unit impulse responses computed from the polynomial approximations

identified in Figures 5 and 6. It is important to note that Very nearly

the same impulse response function

is

obtained from seemingly disparate rational polynomials where each is an approximation to the same

frequency response function. Since each of these approXimations

generates essentially the same

utile

impulse response they must, as will

-be demonstrated, generate similar motion predictions When incorporated

into the time domain Simulation. Figures 7 and 8 ideneify different choices of rational polynomial approximations to

H35010

Table 2 showing tabulated Values of the unit impulse computed from the approximations. Figures 9 and 10 show approximations to H35(jW) with Table 3 showing tabulated values of the

unit

impulse responses, While Figures 11 and-12 Show approximations tO

H5500

with Table 4 showing the corresponding

6

5

4

3

2

A6S +A5S +AO +A3S +A2S +A1S+A0

-8 7

6

5

4 3, 2

B8S +B7S +Be +B5S +BO +B3S +B2S +BiS+Ba

FIGURE 5 -POLES-ZEROS OF A 6TH ,ORDER NUMERTOR/8TH ORDER DENOMINATOR

.RATIONAL POLYNOMINAL APPROXIMATION TO H

33 ROOTS OF NUMBERATOR ROOTS OF DENOMINATOR REAL PART I MAG PART REAL PART IMP PART I

-.35743E-01

.30257E+00

'

-.42677E-01

.29838E+00

-.10980E-01

.77504E+00

-.25458E-01

.66534E+00

-.35743E-01

-.30257E+00

'

-.42677E-01

-.29838E+00

, -.67126E-01

.87107E+00

-.11300E-01

.77426E4-00

-.10980E-01

-.77504E+00

-.25458E-01

-.66534E+00

-.67126E-01

-.87107E+00

-.54822E-01

.87620E+00

-.11300E-01

-.77426E+00

-.54822E-01

-.86720E+00'

A 0

.74544E-05

= .10000E+01

A1= .73243E-05

B

.12372E+0,1

A2= .110375E-03

B

= .16561E+02

.,2

A

= .367'13E-04

B.

.92815E+01

3 3 = .A14

.25760E-03

B

.68044E+02

4

A5 7 .39874E-04

B .= .20918E+02

5 A

= .17512E-03

B

= .104.95E+03

6 6 B

= .14725E+02

7 13,8

= .54841E+02

FIGURE 6 - POLES-ZEROS OF A

10TH ORDER NUMERATOR/12TH ORDER DENOMINATOR

RATIONAL POLYNOMIAL APPROXIMATION TO

H33(jw)

ROOTS OF NUMBERATOR ROOTS OF DENOMINATOR REAL PART I MPL P ART REAL P ART I MPG PART

-.33147E-01

.29396E+00

-.38389E-01

.28872E+00

-.28454E-02

.76478E+00

-.25680E-01

.66542E+00

-.33147E-01

-.29396E+00

-.38389E-01

-.28872E+00

-.23658E+00

.74306E+00

-.27653E-02

.76469E-1-00

-.36971E-01

-.17028E+01

-.25680E-01

-.66542E+00

-.45926E-01

-.85765E+00

-.22082E+00

.74261E+00

-.28454E-02

-.76478E+00

-.35825E-01

-.17047E+01

-.36971E-01

.17028E+01

-.38394E-01

-.85825E+00

-.45926E-01

.85765E+00

-.27653E-02

-.76469E+00

-.23658E+00

-.74306E+00

-.35825E-01

.17047E+01

-.38394E-01

.85825E+00

-.22082E+00

-.74261E+00

A0=

A1=

.73834E-05

.12518E-04

B0=

B =

1

.10000E+01

.18948E+01

A2= .12789E-03

B2= .20221E+02

A3= .13226E-03

B = 3

.24106E+02

A )4= .52406E-03

B4= .11525E+03

A5=

.35244E-03

B 5= .96577E+02

A,6 = .84845E-03

B6= .28917E+03

A7

.32454E-03

B7=

.16414E+03

A8= .55990E-03

B8= .35175E+03

A9 =

.78814E-04

B9= .11755E+03

A10= .11086E-03

B1

.19460E+03

111= *25545E+02

B12= *35296E+02

TABLE 1.

SELECTED VALUES OF THE UNIT IMPULSE RESPONSES COMPUTED FROM TWO DIFFERENT RATIONAL POLYNOMIAL APPROXIMATIONS'

TO THE FREQUENCY RESPONSE FUNCTION H

Ow)

33

TIME SEC

RATIONAL POLYNOMIAL APPROXIMATION TO lk(j(0) 6TH ORDER NUMERATOR 10TH ORDER NUMERATOR

8TH ORDER DENOMINATOR Xl°' 12TH ORDER DENOMINATOR Xl°'

0 .0 -0.0000 0.0000 5.0 ,0.088Z -0.0885 10.0 0.1752 0.1755 15.0 0.2097 -0.2106 20:0 0.2120 0.2127 25.0 -0.2152. -0.2154 30.0 0:2153 0.2147 35.0 -0.2017- -0.2003 40.0 0.1802 0.1798 45.0 -0.1567 -0.1577 50.0 0.1295 0.1304 55.0 (T.1013 -0.1016 60.0 0.0780 0.0788 65.0 -0.0591 -0.0600 70.0 0.0405 0.0401' 75.0 -0.0224 -0.0208 80.0 0.0076 . 0.0061 85.0 0.0038 0.0045 90.0 -0.0132 0.0131 95.0 0.0203 0.0194 100.00 -0.0188 -0.0176'

4 2 A S6 +A

S5 +A S +A S3 +A S +A S+A

6 5 4 3 2

1

0 8 7 6 5 4 3 2

+BTS +B6S +B5S +BO

+B3S +B2S +BiS+130

B7 =

.30459E+03

B 8

= .88263E+03

FIGURE 7 - POLES-ZEROS OF A 6TH ORDER

NUMERATOR/ 8TH ORDER DENOMINATOR

RATIONAL POLYNOMIAL APPROXIMATION TO

H39 (jw)

ROOTS OF NUMBERATOR .

ROOTS OF DOMINATOR

REAL PART I MAG PART REAL PART I MAG PART

-.10235E-01

-.50779E-21

-.11331E400

.20470E+00

-.97390E-01

.27378E+00

-.28301E-01

.29407E+00

- .42483E-02

.73,103E+00

- .11331E+00

- .204,70E-1-00

-.97390E-01

- .27378E+00

-.26563E-01

.66535E+00

- .30602E+01

.10444E-13

- .28301E-01

- .29407E+00

-.42483E-02

-.73103E+00

-.43752E-02

.73131E+00

--.26563E-01

- .66535E+00

-.43.752E-O.2

- .73131E+00

A4

-.25138E-05

B=

4

.35661E+05

5

= - .64497E-05

B 5

= .30180E+03

A6 =

-.19701E-05

B 6 = .10160E+04

A0= -.27847E-08

B0= .10000E+01

A1= -.27945E-06

B1= .49245E+01

A2= -.76122E-06

B2= .37190E+02

A3= -.39726E-O5

B3= .83715E+02

4

BS12+B

+8 S10 +B S9 +B,,S8 +B S7 +B.S6 +B S5 +B. S +B S3 +B S2 +13 1S 10 9 o 7

b

4 3 2 1 12

FIGURE 8 - POLES-ZEROS OF A 10TH ORDER NUMERATOR/12TH ORDER DENOMINATOR

RATIONAL POLYNOMIAL APPROXIMATION TO H

Ow)

3 5 ROOTS OF NUMBERATOR ' ROOTS OF DENOMINATOR REAL PART I mAG PART REAL PART I MAG PART 13412E-01 -.15713E-22 -.13301E+00 -.50487E-28 -.24153E+00 .20610E+00) -.32390E-01 29924E+00 -.39267E-01 .33453E+00 -.53160E-01 '. .32850E+00 -.39267E-01 -.33453E+00 -.28475E+00 62641E-15 -.37290E-02 75877E+00 -.32390E-01 , -.29924E+00 -.24153E+00 -.20610E+00 -.25809E-01 .66548E+00 -.23212E+01 77981E-14 .-.53160E-01 , -.32850E+00 .37290E02 --.75877E-1-00 -.34382E-02 , 75886E+00 -.19535E-01

85688E+00

-.25809E-01 -.66548E+00 -.19535E-0)1 -.85688E+00 -.19426E-01 .85849E+00 , -.34382E-02 -.75886E+00 , -.19426E-01 -.85849E+00 A A 1 = 57850E-08 -.39673E-06 = Bo = 1 .10000E+01 .12886E+02 A = -424183E-05 = .73291E+02 2 2 A = -.11560E-04 = .37076E+03 3 3

A

= -.32820E-04 = .12570E+04 4 4 A = -481033E-04 = .34148E+04 5 5 A6 -.10253E-03 = 6 .81915E+04 A = -417234E-03

B=

.12101E+05 7 7 A8 -.11375E-03 B8 = .23342E+05

A

= -.11201E-03

B=

.18065E+05 9 9

A

= -.38413E-04

B=

29825E+05 10 10

B=

95889E+04 11

_{B= .13974E+05}

TABLE 2

SELECTED VALUES OF THE UNIT IMPULSE RESPONSES COMPUTED FROM TWO DIFFERENT RATIONAL POLYNOMIAL APPROXIMATIONS

TO THE FREQUENCY RESPONSE FUNCTION

H35(j)

ITIME SEC 6TH ORDER NUMERATOR 8TH ORDER DENOMINATORMO RATIONAL POLYNOMIAL_APPROXIMATION TO H35(1) 10TH .ORDER NUMERATOR 7. 12TH ORDER DENOMINATOR 3(10

0.0

0.0000

0.0000

5.0

-0.1577

-0.1568

10.0

'0.2216

0.2242

1,5.0

-0.0211

-0.0192

20.0

0..0037

0.0003

25.0

-0.0840

-0.0786

30.0

0.0459

0.0437

35.0

0.0396

0.0437

40.0

-0.0269

-0.0262

45.0

-0.0140

-0.0149

50.0

-0.0173

-0.0195

55.0

0.0543

0.0548

60.0

-0.0307

-0.0286

65.0

0.0090

0.0086

7040

-0.0297

-0.0308

75.0

0.0388.

0.0400

80.0

-0.0162

-0.0175

85.0

0.0069

0.0078

90.0

-0.000

-0.0195

95.0

0.0176

0.0190

100.0

0.0039

-0.0053

A6 S5+A S4+A +A

S2+A S+A

6 5 4 2

1

.0 8 8

7

6 5 4 3 2 B S +B7 S +B6 S +B5 S +B S +B3 S +B S +B1S+A0 4

FIGURE 9 - POLES-ZEROS OF A 6TH ORDER NUMERATOR/8TH ORDER DENOMINATOR

RATIONAL POLYNOMIAL APPROXIMATION TO H53(w)

ROOTS OF NUMBERATOR ROOTS OF DENOMINATOR REAL PART IMPS PART REAL PART IM AG PART -.12033E+00 -.63344E-15 -.67256E-01 .28377E+00 -.46956E-01 .32732E+00 -.27387E-01 .30041E-H)0 -.46956E-01 -.32732E+00 -.67256E-01 -.28377E+00 .18211E+01 .79324E-16 -.25501E-01 .66544E+00 -.61000E-03 .76503E+00 -.27387E-01 -.33041E+00 -.61000E-03 -.76503E+00 -.50354E-03 .76488E+00 -.25501E-01 -.66544E+00 . -.50354E-03 -.76488E+00 = .2696.1E-07 Bo = .10000E+01

A1

= .23247E4-06 B1 .23003E+01

A2

= .34981E-06 B2 = .27919E+02

A3

= .22058E-05 B - .36083E+02 3 -A = -.60327E-06 Bit = .23032E+03 4

A

=. .30869E-05 B5 = .12522E+03 5

A

= -.19225E-05 B6 = .60854E+03 6 BT = .12018E+03

B8

= .49807E+03

10

9 8

7

6 5 4 3 2 A10S

+A S +A S +A_S +A S +A S +A S +A S +A S +A S+A

9 8

-7

6 4 3 2 1 0

12

11

10

9 8 7 6 5 4 3 2 +B S +B S

+B S +BnS +B S +B.S +B S +B,S +B S +B S +B S+B

1312S

11

10

9 o 7 o 5 4 3 2 1 0 A =

.26425E-07

= .10000E+01

Bo

FIGURE 10 - POLES-ZEROS OF A 10TH ORDER NUMERATOR/12TH ORDER DENOMINATOR

RATIONAL POLYNOMIAL APPROXIMATION TO H 53(i to)

ROOTS OF NUMBERATOR ROOTS OF DENOMINATOR REAL PART I MAC PART REAL PART I MAL PART

-.11144E+00

-.75935E-17

-.77480E-01

.28235E+00

-.52661E-01

.32838E+00

-.27087E-01

.29835E+00

-.52661E-01

-.32838E+00

-.77480E-01

-.28235E+00

.21436E+01

.48437E-15

-.25638E-01

.66560E+00

-.86380E-03

.76543E+00

-.27087E-01

-.29835E+00

-.39174E-01

.85668E+00

-.78420E-03

-.66560E+00

-.86380E-03

-.76542E+00

-.25638E-01

-.66560E+00

-.28104E+00

.10312E+01

-.34337E-01

.85429E+00

-.39174E-01

-.85668E+00

-.78420E-03

-.76514E+00

-.28104E+00

.10312E+01

.32048E+00

.98898E+00

-.34337E-01

-.85429E+00

-.32048E+00

-.98898E+00

o

A1 =

26585E-06

B1

.32166E+01

=

.59830E-06

.32233E+02

=

.31791E-05

B

= .65944E+02

3 3 A =

a1095E-05

B4 =

.32921E+03

4 A =

.90208E-05

B

= .41866E+03

5 5

A6 =

.15791E-05

B6 =

.13237F+04

A =

.89858E-05

B

= .11332E+04

7 7

A8 =

.18880E-05

B8 =

.24466E+04

A =

.26104E-05

B

= .13756E+04

9 9 A

= -.20320E-05

B

= .206.86E+04

10

10

_B

= .61542E+03

11

_B12=

.63340E+03

TABLE 3

SELECTED VALUES OF THE UNIT IMPULSE RESPONSES CONFUTED FROM TWO DIFFERENT RATIONAL POLYNOMIAL APPROXIMATIONS

TO THE FREQUENCY RESPONSE FUNCTION H5300

25

RATIONAL POLYNOMIAL APPROXIMATION TO 119/00

TIME SEC

6TH ORDER NUMERATOR 7 10TH ORDER NUMERATOR X 1 0 8TH ORDER DENOMINATOR - 0 12TH ORDER DENOMINATOR 0.0 0.0000 0.0000 5.0 0.2738 0.2698 10.0 -0.2331 -0.2360 15.0 0.0251 0.0266 . 20.0 -0.0986 -0.0991 25.0 0.1836 0.1827 30.0 -0.0554 -0.0550 35.0 =0.0150 -0.0142 40.0 13.0626 '-0.0628 45.0 0..0643 0.0643 50.0 . 0.0239 0.0238 55.0 -0.0316 -0.0324 60.0 -0.0180 -0.0173 65.0 0.0014 0.0020 70.0 0.0433 0.0426 75.0 -0.0305 -0.0305 80.0 - 0.0024 0.0025 85.0 -0.0170 -0.0171 90.0 0.0353 0.0354 95.0 '-0.0192 -0.0189 100.0 0.0040 0.0038

B

- .34789E+06

9 . 6 5 4 3

2.

A

+A S +A S +.AS +A S +A S +A

7 6

-5

4 3 2

1+ 0

9 8 7 6 5 4 3 2

B S +B S +B

+B S +R S +B ,S +B S +B S +B S+B

. 8 7 6 5 4 3

1

0

FIGURE 11 - POLES-ZEROS OF A 7TH ORDER NUMERATOR/1TH ORDER DENIOMINATOR

RATIONAL POLYNOMIAL APPROXIMATION TO H

(jw)

55 ROOTS OF NUMBERATOR. ROOTS OF DENOMINATOR REAL, PART

ImiG PART

REAL PART IMAG, PART

-.43423E-01

0

-.25445E-01

.14873E,49

.64294E-01

.31149E+00

-.72299E-01

.17849E+00

-.62605E01

.18918E+00 -.59422E-01

.26716E+00

-.62605E-01

-.18918E-00

-.72299E-01

-.17849E+00

-.64294E-01

-.311.49E+00

-.23946E-01

.36,101E+00

-.33947E-01 .

.64872E-00

-.59422E-01

-.26716E+00

-.33947E-01

-.64872E+00

-;23946E-01,

i

-.29680E-01

-.30101E+00-,

.66715E+00

-.29680E-01

-.66715E+00

Ao =

.21487E707

Bo=

.10000E+01

A

= .59336E-06

1 B

= .45444E+02,

A2 =

.31748E-05

B

= .10484E+03

A

= .22580E-04

B

= .26922E+04

3 3

A4 =

.47122E-04

B4 =

.89752E+04

A5 =

.17812E-03

B5

= .44156E+05

A6 =

.10658E-03

B6 =

.75098E+05

A = .29142E-03

B

= .24591E+06

7

B8 =

.13781E+06

5 4 3

2

A11 +A10 S10, +A9 S9 +A88B +A_S7 +A6

S +

+A.S +A S +A S. +A S+A

-T

4 3 2 1 0

13

12

11

10

9

7

6

5

4 3 2 B S +B S +B11S +B

+B S +B S

S +B,S +B S +B,_S-+B S +B S

S+B

13

10

9

8 7o5 4

3

210

FIGURE 12 - POLES-ZEROS OF AN 11TH ORDER NUMERATOR/13TH ORDER DENOMENATOR

H55 (163)

RATIONAL POLYNOMIAL APPROXIMATION TO

ROOTS OF NUMBERATOR ROOTS OF DENOMINATOR REAL PART I MAG PART REAL PART I MAG PART

-.37122E-01

-.36856E-26

-.21436E-01

-.37684E-18

-.58645E-01

.19500E+00

-.73037E-01

.19045E+00

=.73006E-01

.32572E+00

-.25356E-01

.30087E+00

1

-.58645E-01

-.19500E+00

,

-.72283E-01

.26722E+00

-.17870E-01

.36333E+00

-.73037E-01

,-.19045E+00

-.73006E-01

-.32572E+00

' -.72283E-01

-.26722E+00

-.30921E-01

.64969E+00

-.17043E-01

.36430E+00

-.17870E-01

-.38333E+00

-.25356E-01

-.30087E+00

-.17122E+00

-.12969E+01

-.27374E-01

.6.6641E+00

-.30921E-01

-.64969E+00

-.17043E-01

-.36430E+00

-.17722E+00

.12969E+01

-.18774E+00

-.12898E+01

-.27374E-01 ,

-.18774E+00

-.66641E+00

.12898E+01

A

= .22473E-07

130

= .10000E+01

1

A2 =

.7124,1E-06,

.40069E-05

B1 = .53204E+02.

B2 = .37748E+03

3

= .33151E-04

.133 - .36216E+04

A 4

= .92842E-04

B4 = .14453E+05

= .43639E-03

R5 =

.78460E+05

A6 = .67911E-03

B6 = .19584E+06

A7

.21416E-02

B7

-.72289E+06

A-8 =

.16223E-02

8

= .10872E+07

A9 =

.35810E-02

BQ = -.29095E+07

A10,

A

.10280D-02

.13663E02

B10=

.22404E+07

.43639E+07;

B11

B12=

.13207E+07

B13=

.15968E+07

TABLE 4

SELECTED VALUES OF THE UNIT INpuLsg RESPONSES COMPUTED

FROM TWO DIFFERENT RATIONAL POLYNOMIAL APPROXIMATIONS

TO THE FREQUENCY RESPONSE FUNCTION

H55000

28

RATIONAL POLYNOMIAL APPROXIMATION TO H55(1 ) TIME

SEC.

6TH ORDER NUMERATOR _X108 10TH ORDER NUMERATOR

X108 9TH ORDER DENOMINATOR 13TH ORDER DENOMINATOR

0.0 0.0000 0.0000 5.0 . 0:2688 0.2718 10.0 0.1448 0.1474 I5.0 -0.1220 -0.1212 20.0

-01173

-0.1153 ' 25.0 0.0814 0.0845 30.0 0.1307 0.1320 35.0 -0.0287 -0.0279 40.0 ,-0.0938 -0.0896 45.0 0.0054 .0.0123 50.0 0.0817 0.0879 55.0 0.0097 0.0015 60.0 -0.0055 -0.0050 65.0 -0.0017 -0.0012 70.0 0.0041 0.0045 75.0 0.0019 0.0022 80.0 -0.0029 -0.0025 85,0 -0.0018 -0.0014 90.0 _ 0.0020 0.0024 95.0 0.0017 .0.0018 100.0 -0.0012 -0.0011

unit impulse responses. Table 5 Shows time histories Obtained using

the time domain simulation including nonlinear modeling (see section .

entitled FREQUENCY INDEPENDENT NCNLINEARITIES) for the rational

polynomial-choices approximating

B3(103) H8-5(jW),

Ii5(jw),, and

H.55 (jui) The simulated run consists of the SWATH 6A moving at

equilibrium depth into A State 6 head sea at a constant speed of

10 knots, It is' apparentby a comparison of the figures, that the

motion predictions (and related harmonic analyses shown in Table 6)

for the indicated two sets of choices for approximating polynomials

are sensibly the same. the point being Made here it., of course, that

the closeness of fit of the rational polynomial approximation to the

frequency response being approximated rather than the doeffiCientt or the poles' and zeros of the rational polynomial determine its

appropriateness at a valid motion predictor for

a

particular frequency

response.

Because of the closeness of fit between the two sets of choices

for ratiOnal polynomial approximation and the frequency response being

approximated, it was not informative to demonstrate the quality of fit in plot form. Rather, the quality of fit can be attested from the

tabulations

given in

Table 7 through 14 inclusive.

TRANSLATING TO THE TIME DOMAIN

Given a stable rational polynomial approximation to, for example,

the frequency response function,

H55Ow),

translation to the time

domain is carried out as follows:

A S + A + ---- A So m-1

H55(j

)

= m

.1111S Bn=1S BoSo 29 (19)

TABLE 5

TIME HISTORIES OF HEAVE AND PITCH OBTAINED USING TWO DIFFERENT SETS

OF PITS TO THE FREQUENCY RESPONSES 13.3300, 1135(j

w), 111530w

,

H55 Ow )

FOR THE SWATH 6A PROCEEDING INTO A SEAWAY

AT 10

KNOTS.

30 _HEAVE_ PITCH TIME USING FITS SHOWN IN FMS USING FITS SHOWN IN FIGS USING FITS SHOWN IN FIGS USING FIT SHOWN IN Frls SEC

5,7,904'

6,8,10 12 57j911.; 6,8,40,12

M

Fr DEG. DEG 0.0 3.390 1.033 3.302 1.007 ,1.152 71.145 0.5 2.915 0.888 2.828' 0.862 -1.206 -1.219 '

1.0

2.464 0.751 2.389 0.728 -0.891 =.0.916. 1.5 2.136 0.651 2.082 0.635 -0.235 -0.265 2.0 1.893 0.577 1,868 0.569 0.622 0.594 2.5 1.571 ,0.479 1.580 0.482 1.470.. 1.448 3.0 0.956 0.291

1,003

0.306 2.098 2.079 3.5

-0.102

-0.031

-0.021 -0.007 2.349 2.331 4.0 -1.595 0.486 -1.491 -0.454 2.173 2.153 4.5 -1.325 -1.014 -3.215 -0.980 1.635 1.612 5.0, -4.458 -1.511 -4,865 -1.483 0.890 0.865 5.5 -6.128 1.868 -6.074 -1.851 0.122 0.102 6.0 -6.560 -1.999 =6.558 -1.999 -0.510 ' =0.521 6.5 -6.139 -1.871 -6.190 -1.887 -0.932 =0.928 7.0 4.920 -1,500 5.012 -1.528 -1.158 -1.138 7.5 ' =3.080 -0.939

=3.191

-0.973 -1.247 -1.212 8.0 -0.862 -0.263 -0.968 -0.295 ,-1.245 -1.199 8.5 1.471 0.448 1.390 0.424 -1.174 -1.123 9.0 3.666 1.117, 3.623 1.104 -1.040 . -0.994 .

TABLE 5 (CONTINUED) 31 HEAVE PITCH TIME USING FITS SHOWN IN FIGS

USING FITS

SHOWN IN FIGS

USING FITS

SHOWN IN_FIGS USING

FIT

SHOWN

IN FIGS

SEC 5,7,9,11 6,8,10,12 5,7,9,11 6,8,10,12

Fr

M FT DEG DEG 9.5 5.498 1.676 5.496 1.675 -0.848 -0.815 10.0 6.775 2.065 6.808 2.075 =0.605 -0.592 10.5 7.346 2.239 7.407 2.258 -0.327 -0.339 11.0 7.130 2.173 7.207 2.197 -0.033 -0.066 11.5 6.133 1.869 6.215 1.894 0.262 0.216 12.0 4.467 1.361 4.543 1.384 0.542 0.497 12.5 2.338 0.713 2.396 0.730 0.802 0.774

130

0.001 0.004 0.033 0.010 1.047 1.047 13.5 -2.285 -0.697 -2.286 -0.697 1.280 1.312 14.0 -4.301 -1.311 -4.333 -1.321 1.482 1.540 14.5 -5.873 -1.790 -5.932 -1.808 1.606 1.674 15.0 -6.881 -2.097 -6.955 -2.120 1.587 1.657 15.5 -7.242 -2.207 -7.318 -2.230 1.366 1.418 16.0 -6.918 -2.109 -6.982 -2.128 0.920 0.944 16.5 -5.922 -1.805 -5.964 -1.818 0.281 0.273 17.0 -4.329 ,-1.319 -4.344 -1.324 -0.464 -0.500 17.5 -2.277 -0.690 -2.264 =0.690 -1.190 -1.244 18.0 0.041 0.012 0.077 0.024 -1.759 -1.819 18.5 2.401 0.732 2.455 0.748 -2.050 -2.104 19.0 4.578 1.395 4.641 1.414 -1.994 -2.033 19.5 6.365 1.940 6.430 1.960 -1.570 -1.620 20.0 7.585 2.312 7.645 2.330 -0.943 -0.945

TABLE 6

TABULATION OF FREQUENCY RESPONSE VALUES FOR THE TIME HISTORIES FOR HEAVE AND PITCH

TABULATED IN TABLE 5

32 FREQ

RAD/SEC

- HEAVE

USING FITS SHOWN IN FIGS 5,8,11,14

.

USING FITS SHOWN

IN FI3S 6,9,12,15

AMP PHASE AMP PHASE

FI DEG DEG 0.4675 1.2509 0.381 6.89 1.2469 0.380 6.78 0.5525 1.5480 0.472 24.36 1.5489 0.472 24.14 0.6375 1.4615 0.445 -49.66 1.4629 0.446 -49.61 , 0.7225 1.0451 0.319 -79.48 1.0519 0.321 -76.94 0.8075 0.6668 0.203 -92.01 0.6762 0.206 -90.98 0.8925 0.3906 0.119 -116.80 0.3910 0.119 -118.08 0.9775 0.1957 0.060 -120.01 0.1975 0.060 -120.22 1.0625 0.1333 0.041 -131.63 0.1350 0.041 -132.93 1.1475 0.0890 0.027 -142.85 0.0897 0.027 -144.65 1.2325 0.0784 0.024 -162.35 0.0786 0.024 -164.34 1.3175 0.0745 0.023 -177.40 0.0749 0.023 -179.42 1.4025 0.0735 0.022 +174.43 0.0732 0.022 +172.87 1.4875 0.0765 0.023 +166.22 0.0757 0.023 +164.76 1.5725 0.0773 0.024 +161.22 0.0747 0.023 +160.60 1.6575 0.0760 0.023 +153.82 0.0737 0.022 +154.82

TABLE 6 (CONTINUER)

TABULATION OF FREQUENCY RESPONSE VALUES

FOR THE TIME HISTORIES FOR HEAVE AND PITCH'

TABULATED IN TABLE.5

FREQ RAD/SEC

PITCH

USING FITS SHOWN

. IN FIGS

5,8,11,14

USING IN FIGS FITS SHOWN

6,9,12,15

AMP DEG PHASE DES

A1'

DEG DEG

0.4675

0.455

=146.05

0.447

-146.03

0.5525

0.517

-148.77

0.517

-148.42

0.6375

0.518

-160.65

0.518

-160.57

0.7225

0.378

-166.90

0.373

-167.92

0.8075

0.276

-161.06

0.279

-161.35

0.8925

0.246

-135.06

0.232

.

-132.22

0.9775

0.253

-128.55

0.257

-125.56

1.0625

0.231

-126.31.

0.240

-124.44

1.1475

0.217

-123.30

0.225

-123.39

1.2325

0.181

-123.10

0.185

_-124.32

1.3175

0.151

T.125.55

0.150

-126.33

1.4025

0.113

-129.47

0.112

-129.00

1.4875

0.066

-135.82

0.068

-134.85

1.5725

0.032

-163.02

0.034

-161.38

1.6575

0.030

+118.31

0.037

+124.30

. .

TABLE 7

COMPARISON OF VALUES FOR THE FREQUENCY RESPONSE, H00, AND

VALUES COMPUTED USING A 6TH ORDER NUMERATOR OVER AN 8TH ORDER DENOMINATOR RATIONAL POLYNOMIAL APPROXIMATION TO H3300 FOR

THE SWATH 6A MOVING WITH A FORWARD SPEED OF 1.0 KNOTS AT A

DEPTH OF 19.17 FEET. (5.84 METERS) TO THE CENTERLINE OF THE HULL

FREOUaiCi RESPONSE_ A6/8 RATIONAL POLYNOMIAL FIT

MAGNITUDE PHASE MAGNITUDE PHASE

FREQUENCY RAD/SEC

Fr/LB

5 M/N

6.

D E

Fr/LB

5 M/N 6 D

X10_

X10 X10 _X10

0.042

0.740

0.507

-4.24

0.749

0.513

,0.63

0.127

0.774

0.530

,1.74

0.775

0.531

-2.16

0.212

0.830'

0.569

-'4.73

0.827 '

0.567

-5.15

0.297

0.763

0.523

-7.78

0.7?7

0.532

8.10

0.382

1.009

0.691

70.23

1.019

0.698

+0.44

0.467

1.391

0.953

-2.20

1.402

0.961

-2.51

0.552

2.290

1.569

-7.91

2.288

1.568

-8.19

0.637

6.670

4.570

-37.81

6.666

4.568

-37.74

0.722

3.898

2.671

-151.27

3.911

2.680

.451.01

0.807

1.667

1.142

164.65

1.668

1,143

,164.06

0.892

1.030

0.706

=178.99

1.038

0.711

-179.01

0.977

0.642

0.440

-179.57

0.631

0:432

-179.66

1.062

0.469

0.321

-179.38

0.465.

0.317

,178.63

1.147

0.365

0.250

.479.33

0.363

0.249

-178.31

1.232

0.294

0.201

-179.29

0.295

0.202

-178.25'

1.317

0.244

0.167

-179.23

0.245

0.168

-178.29

1.402

0.206

0.141

-179.15

0.208

0.143

-178.35

1.487

0.177

0.121

179.02

0.180

6.123

-178.42

1.572

0.154

0.106

-178.75

0.157

0.108

-178.50

1.657

0.135

.,0.09.3

-177.57

0.138

_0_.095_

-,178.57

TABLE 8

COMPARISON OF VALUES FOR THE FREQUENCY RESPONSE,

H3010,

_AND VALUES COMPUTED USING. A 10TH ORDER NUMERATOR OVER AN 12TH ORDER

DENOMINATOR RATIONAL. POLYNOMIAL APPROXIMATION TO 113300 FOR THE SWATH 6A MOVING WITH A FORWARD SPEED OF 10 KNOTS AT A

DEPTH OF 19.17 FEET (5.84 METERS) TO THE CENTERLINE OF THE HULL

FREQUENCY' RAD/SEC 0.042 0.127 0.212 0.297 0.382 0.467' 0.552 0.637 0.722 0.807 0.892 0.977 1.062 1.147 1.232 1.317 1.402 1.487 1.572 1.657 FREQUENCY RESPONSE MAGNITUDE FT/LB. 5 X10 0.740 M/N 6 X10 0.507 0.774 0.530 0.830 0.569 0.763 0.523 1.009 0.691 1,391 0.953, 2.290 1.569 6.670 4.570 3.898 2.671 1.142 0.642 0.440 -179.57 0.469 0.321 -179.38 0.365

_0.250

-179.33 0.294 0.201 0.244 0.167 0.206 0.141 0.177 0.121 0.154 0.106 -178.75 0.135 0.093 -177.57 PHASE 0.24' -4.74 -4.73 7.78 0.23 2.20 -7.91 -37.81 151.27 164.65 0.706 178.99 179.29 179.23 179.15 179.02

A10/12 RATIONAL POLYNOMIAL FIT _MAGNITUDE FT/LB 5 X10 0.715 0.740 0.794 0.891 1.066 1.408 2.274 6.632 3.893 1.732 1.040 0.650 0.468 0.363 0.293 0.243 0.206 0.177 0.154 0.136 0.490 0.507 0.544 0.610 0.730 0.965 1.558 4.544 2.668 1.187 0.713 0.445 0.321 0.249 0.201 0.167 0.141 0.121 0.106 0.093 PHASE 0.50 1.76 7.54 2.38 -7.90 -37.83 151.27 164'59 =179.06 179.59 179.39 179.34 179.32 -179.30 .179.24 1.79.14 178.84 177.51 35 r1/N 6 X10 1.667 1.030

TABLE 9.

COMPARISON OF VALUES FOR THE FREQUENCY RESPONSE,

H.3(j43),. AND

VALUES COMPUTED USING A 6TH ORDER NUMERATOR OVER AN 8TH ORDER DENOMINATOR RATIONAL POLYNOMIAL APPROXIMATION

TO H..4c(jw) FOR

THE SWATH 6A MOVING WITH A FORWARD SPEED OF 10 KNCitS AT A

DEPTH OF 19.17 FEET (5.84 METERS) TO THE CENTERLINE OF

THE HULL

36

FREQUENCf.RESPONSE A6/8 RATIONAL POLYNOMIAL FIT

MAGNITUDE PHASE MAGNITUDE PHASE

FREQUENCf RAD/SEC

1/LB

-7 X10_ 1/N 8 X10 D E

r

_,,

1/LB

7 X10 1/N 8 X10

0.042

0.204

0.459

-93.40

0.122

0.274

-109.24

0.127

0:483

1.086

-120.81

0.432

0.971

115.80

0.212

0.938

2.109

-137.09

0.969

2,178

-138.59

0.297

2.632

5.917

.154,91

2.622

5.894

154.31

0.382

1.095

2.462

107.27

1.080

2.428

104.94

0.467

0.918

2,064

100.93

0.885

1.996

101.22

0.552

1.083

2.434

94.84,

1.080

2.428

95.88

0.637

2.467

5.546

66.66

2.496

5.611

66.06

0.722

1.199

2.695

-44.45

1.195

2.686

-44.55

0.807

0,423

0.951

-57.24

0.447

1.005

-58.04

0.892.

0.243

0.546

-58.58

0.239,

0.537

-61.09

0.977

0.146

0.328-

-60.48

0,151

0.339

-61.93

1X62

0.099.

0.223

-59.20

0.104

0.234

--61.94

1.147

0.072

,

0.162

-57.74

0.076

0.171

-61.56

1.232

0.055

0.124

-56.23

0.057

0.128

-60.95

1.317

0.043

0.097

-54.72

0.045.

' 0.101

-60.21

1.402

0.035

0.079

-53.20

0.036

0.081

-59.38

1.487

0.029

0.065

-51.64

0.030

0.067

-58.51

1.572

0.024

0.054.

..49.90

0.025

0.056

-57.60

1.657

'

0.019

0.043

-45.99

0.021_

0.047,

-56.67

TABLE 10

COMPARISON OF VALUES FOR THE FREQUENCY RESPONSE, H35(jW), AND VALUES COMPUTED USING A 10TH ORDER NUMERATOR OVER AR 12TH ORDER

DENOMINATOR RATIONAL POLYNOMIAL APPROXIMATION TO H35(jw) FOR

THE SWATH 6A MOVING. WITH A FORWARD SPEED OF 10 KNOTS AT 'A

DEPTH OF 19.17 FEET (5.84 METERS) TO THE CENTERLINE OF.THE

HULL

37

FREQUENCY RESPONSE A10/12. RATIONAL POLYNOMIAL FIT

MAGNITUDE PHASE MAGNITUDE -PHASE

FREQUENCY

1/LB

1/N U

1/LB".

1/N RAD/SEC 7 ' 8 7 8 X10 XIO G X10 X10

0.042

0.204

0.459

-93,40

0.186

0.418

-88.65

0.127

.0.483

1.086

-120.81

0.476

1,070

-122.18

0.212

0.938

2.109

-137.09

0.954

.

2.145

-137.99

0.297

2.632

5.917

154.91

2.631

5.915

+154.84

0.382

1.095

2.462

107.27

1.101

2.475

+107.65

0.467

0.918

2.064

100.93

0,931

2.053

+100.55

0,552

1.083

2.434

94.84

1.079

2.426

+95.01

0.637

2,467

5.546

'

66.66

2.471

5.555

+66.56

0.722

1,199

2.695

-44.45

1.195

-44,45

0.807

0.423

0.951

-57.24

0.423

0.951

,57.07

0.892

0.243

0.546

-58.58

0.241

0,542

-58,72

0.977

0.146

0.328

-60.48

0.148

0.333

-59.70

1.062

0.099

0.223

-59.20

0.102

0.229

-59.13

1.147

0.072

0.162

-57.74

0.074

0.166

-58.20

1.232

0.055

0.124

-56.23

0.057

0.128

-57.10

1.317

.0.043

0,097

.,54.72

0.046

0.090

-55.93

1.402

0.035

0.079

-53.20

0.036

0.081

-54.73

1.487

0.029

0.065

-51.64

0.030

0.067

-53.51

1.572

0.024

0.054

-49.90

0.025

0.056

-52.30

1.657

0.019

0.043

5.99

0,021

'0.047

-51.10

TABLE 11

COMPARISON OF VALUES FOR THE FREQUENCY RESPONSE, H5.(jw), AND VALUES COMPUTED USING A 6TH ORDER NUMERATOR OVER AN 8TH ORDER

DENOMINATURRATIONAL POLYNOMIAL APPROXIMATION TO Ilq.;(1w) FOR THE SWATH 6A MOVING WITH A FORWARD SPEED OF 10 KNOtS AT A

DEPTH OF 19.17 FEET (5.84 METERS) TO THE CENTERLINE OF THE HULL

38

'FREQUENCY RESPONSE A6/8 RATIONAL POLYNOMIAL FIT

_MAGNI.TUDE PHASE MIGNITUDE _ PHASE

FREQUEN CY RAD/SEC

1/LB

5 X10 1/N

6

X10 D E

1/LB

5 X10

1/N

. 6 X10 E

0.042

0.151

0.336

.29.93

0.168

0.378

14.54

0.127

0.333

0.749

31.44

0,292

0.657

29.78

0.212

0.703

1.580

19.79

0.720

1.619

19.68

0.297

1.943

4,369

-,66.57

1.941

4.365

-66.65

0.382

0.598

1.345

-106.68

0.604

1.357

-106-.48

.

0.467

0.541

1.217

-109.46

0.544

1.223

-110.19

0.552

0.680

1.529

-118.48

0.680

1.529

-118.80

0.637

1.609

3.617 ..-151.34

1.613

3.626

-151.28

-0.722

'

0.799

1.795

92.12

0.794

1.787

92.65

0.807

0.292

0.655

74.51

0.285

0.640

76.04

0..892

0.145

0.326

60.32

0.154

0.345

70.10

0.977

0.085

0.191

61.56

0.097

0.219

66.23

1.062

0.058

0.130

60.89

.0.068

0,152

63.19

1.147

0.042

0.095

.

59.81

0.050

0.112

60.60

1.232

0.032

0.072

58.65

0.039

0,087

58.28

1.317

0.025

0.057-

57.47

0.030

0.068

56.15

1.402

0.021

0.046

56.32

0.025

0.056

54.19

1.487

0..047

0.038

55.30

0.021

0.046

.

52.35

r

1.572

0.014.

0.032

54.73

0.017

0.038

50.63

1.657

0.013

0.038

.56.39

0.015

0.033

49.00