DTIC ADA262570: A Complete Set of Field Equations for the Dynamic Simulation of a Towed Cable System

39 Pages·0.99 MB·English

by  Defense Technical Information Center

I AD-A262 570
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NUWC-NL Technical Report 10,231
11 January 1993
A Complete Set of Field Equations
For the Dynamic Simulation
| Of a Towed Cable System
3
A. A. Ruffa
N. Toplosky
Antisubmarine Warfare Systems Department
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IDTIC
MAR 3 0 1993
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Naval Undersea Warfare Center Detachment
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New London, Connecticut
DISTRIBUTION STATEMENT A. Approved for public rilease; distribution is unlimited.
9 90 97 93-06446
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m m
PREFACE
This report was funded under NUWC Project No. B25209, "Dual Tow Data Analysis,"
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principal investigator P. E. Seaman (Code 3321), program manager C. W. Nawrocki
(Code 33B). The sponsoring activity is the Program Executive Office for Undersea Warfare,
Captain G. Nifontoff (PMO41 1).
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The technical reviewers for this report were Dr. M. D. Ricciuti (Code 332) and
P. E. Seaman (Code 3321).
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The authors gratefully acknowledge the enthusiastic support of Dr. E. Y. T. Kuo
(Code 3321). Appreciation is also expressed to L. Turner and A. Vuono of Analysis &
Technology, Inc., for their assistance in preparing this report.
REVIEWED AND APPROVED: 11 Jnuary 19M93
C. W. Nawcdd
He, Aantisubmadne Warfar Systens Depmbne&.
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I".-7..
\(cid:127)
Fairm Approved
REPORT DOCUMENTATION PAGE
OMB No. 0704-0188
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1. AGENCY USE ONLY (Leavebak 2 REPORT DATE ~ 3.R EPORT IYPE AND DATES COVERED
I 1I January 1993 IFinal
4. TILE AND SUBTITLE 5. FUNDING NUMBERS
A omplete Set of Field Equations for the Dynamic Simulation of a Towed Cable
System
6.A UTHOR(S) B25209
Dr. A. A. Ruffa
Dr. N. Toplosky
7.P ERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) S. FERFORMING ORGANIZATION
Naval Undersea Warfare Center Detachment REPORT NUMBER
New Lmndon, CT 06320-5594
TR 10,231
9.S PONSORING / MONITORING AGENCY NAME(S) AND) ADDRESS(ES) 10. SPONSORING / MONITORING
Department of the Navy AECRPRNME
Undersea Warfare
Washington, D.C. 20362-5169
11. SUPPLEMENTARY NOTES
12aL DISTRIBUTION / AVAILABILITY STATEMVENT 112b. DISTRIBUTION CODE
Distribution Statement A. Approved for public release: distribution is -unlimited.
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13. ABSTRACT (A~kXnMtm 200 wods
The development of a complete set of field equations describing the dynamic behavior of a towed cable is presented. The model
includes inertial effects, as well as bending, torsional, and extensional rigidity. Th~e resulting field equations are a "generic" set
of 13 coupled, nonlinear, partial differential equations in time and unstretched cable scope. Hydrodynamic loading and "added
IFiel14d. SU BJECT TERMS 15. NUMBER OF PAGES
equations Differentia equations 38
Towed cable effects 16. PRICE CODE
17. SECURITY CLASSIFICATION 18. SECURITY CLASS'F ICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF ABSTRACT
OF RPOTRITSO PFG E O ABSRAC
NSN 740-0-2805500Standard Form 298 (Rev. 2-89)
Ptelotbed by ANS, Std. 239-18
20& 102
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TR 10,231
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TABLE OF CONTENTS
Page
UIS T OF ILLUSTRATIONS ........................................................................ ii
ULS T OF SYMBOLS .................................................................................. iii
FOREWORD .................................................... vii
E C OORDINATE SYSTEMS .................................... ...... 1
The Body-Fixed Coordinate System (1,2 ,3) ........ .......................................... 1
3
'- Computation of I', 2',and 3' ................................................................... 4
The Flow Based Coordinate System (fi, t)....................... 77,
The Cur; rare-Based Coordinate System (1N, A, t)) .......................................... 9
THE DYNAMIC EQUATIONS - TRANSLATION ............................................. 11
THE DYNAMIC EQUATIONS - ROTATION ................................................. 15
CONSTITUTIVE AND KINEMATIC RELATIONS ............................................ 18
Constitutive Equation for the Twisting Moment ............................................... 18
Constitutive Equation for the Tension ........................................................... 18
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Constitutive Equations for the Bending Moments ............................................. 19
Kinematic Relations ............................................................... 21
SUMMARY.....................................................22
REFERENCE ......................................................................................... 27
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Acesion For
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cDTIC TAB
Unannounced
Justification .................
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BY ..........
Distribution I
(cid:127) ....... Availability Codes
Avail andjIor
Dist Special
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TR 10,231
LIST OF ILLUSTRATIONS
Figure Page
I The Inertial and Body-Fixed Systems ................................ 2
2 The (p Rotation .............................................. 3
3 The 0 Rotation .............................................. 3
4 The v Rpiation ............................................................................. 3
5 The Flow-Based Coordinate System .................................................... . 8
6 The Curvature-Based Cooi"nate Sys.-m ............................................... 10
7 Force Balance on Cable Segment ......................................................... 11
8 Moment Balance on Cable Segment ....................................................... 15
9 Pure Bending ................................................................................ 19
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TR 10,231
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LIST OF SYMBOLS
1. Engliszh Letr
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A A A A Cable bending and torsional constants, defined in (38) and (39).
1 2 3 4
b Binormal unit vector of flow-based coordinate system, defined in figure 5.
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B Binormal unit vector of curvature-based coordinate system, defined in figure 5.
SD Drag and hydrodynamic loading force per unit length, defined in figure 7
= Dn fi + Db b+Dt.
e Cable stretch, defined in (16) and (17).
F. Effective Young's Modulus of cable.
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G Effective Shear Modulus of cable.
i(cid:127) I I2 I1 Principal moments of inertia of cable per unit length, as defined in
equations (31) and (32).
1
1B Cross-sectional bending moment of inertia, (40).
Ip Cross-sectional twisting polar moment of inertia, (37).
S101,t
Arc length a~ong unstretched and stretched cable.
MI M Mt Bending and twisting moments in cable around the 1, 2, and 3 axes, as shown
2
Iin figure.
Mb Total bending moment in cable owing to cable curvature, as shown in figure 9.
Sfi Flow-based normal unit vector, figure 5.
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Nq Curvature-based normal unit vector, figure 6.
R Local radius of curvature of cable.
T Vector from inertial origin to cable point.
•t Flow-based tangent unit vector (same as 3).
1T
Curvature-based tangent unit vector (same as 3).
T Cable tension.
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TR 10,231
LIST OF SYMBOLS (Cont'd)
1. English Le.tter (Cont'd)
fit= Cable velocity in inertial reference frame in the x, y, and z directions.
Local current.
-
0-Cable velocity with respect to water U = V - Uc.
V,V Shear forces in I and 2 directions.
1 2
We ight aer unit length of the unstretched and stretched cajSblwe.
2. Greek Letters
a Angle between cable tangent vector and flow, figure 5.
E Cable strain e (stretch) = e + 1.
o
Euler's second angle, see figure 3.
gLo IA a Mass per unit length of unstretched and stretched cable, and added mass.
p Local cable curvature = YR, also water density.
o Normal stress.
t Y, Torques in principal axis coordinate system, shear stresses.
T Euler's first angle, see figure 2.
Euler's third angle, see figure 4.
r(cid:127)Aon, gular velocity in principal axis coordinate system.
3. Sbscip
x, y, z Refer to inertial cocrdinates.
1, 2, 3 Refer to body-fixed coordinates, figure 1.
n, b, t Refer to flow-based coordinates, figure 5.
N, B, T Refer to curvature-based coordinates, figure 6.
0 Refers to unstretched lengths.
iv
S.. . . .. t- 7_ - _ - .> --- . .-
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TR 10,231
LIST OF SYMBOLS (Cont'd)
4. Sucscit
4.Derivative oir to to :d
* Derivative SoI) /r tot:*.
I - Vector.
Unit vector.
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SI
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TR 10,231
FOREWORD
The following report will develop a complete set of field equations for the dynamic motions
of a towed cable system. These equations, when solved, constitute a dynamic simulation of a
towed cable system.
During gentle maneuvers, the important forces in the governing force balances on a cable
system are tension and ydrodynamic drag. In more severe maneuvers involving highly dynamic
situations, however, the effects of shear, bending, torsion, extension, rotary inertia, and inertia can
be important, and sometimes even mathematically necessary.
Many dynamic models for simulating the motions of a towed cable system already exist.
Many of these models, however, by ignoring the above-mentioned effects, have iestrictive
assumptions built into them. The model represented by the equations developed here does not
have these restrictions. Thus, the equations can be used e:ther for more accurate simulations of
"highlyd ynamic situations or simply for studying the range of validity of simpler models.
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TR 10,231
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A COMPLETE SET OF FIELD EQI .' TIONS
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(cid:127)FOR THE DYNAMIC SIMULAThIN
I OF A TOWED CABLE SYSTEM
TT HE COORDINATE SYSTEMS
The total field equations describing the dynamic behavior of a towed system involve four
coordinate systems: 1) a fixed inertial coordinate system in which to write the translational
dynamic equations, 2) a body-fixed/principal-axis coordinate system in which to write the
rotational dynamic equations, 3) a flow-based coordinate system in which to write the
hydrodynamic loading, and 4) a coordinate system based on the local cable curvature in which to
express the bending and twisting moments.
THE BODY-FIXED COORDINATE SYSTEM (i, 2, 3)
5,
Consider two orthonormal coordinate systems: a fixed inertial one (i, i), and a body-
fixed one (i, 2, 3). (See figure 1.) Three successive rotations, through the Euler angles (p,9 , and
N'. can be performed to transform (X., i) to (i, , ).