NPS6277002
IWuunr
TECHNICAL REPORT SSCTIOK
POSTGRADUATE SCHOOt
NAVAL POSTGRADUATE SCHOOL
Monterey, California
A METHOD FOR SCALING THE HEAVE MOTION
EQUATIONS OF THE C.A.B. 6D.O.F. LOADS AND
MOTION PROGRAM FROM MODEL TO FULLSIZE
CRAFT
Alex Gerba,
Jr . and George J .
Thaler
December
1977
Progress
Report for Period
September 1977
Ending
PreDared for:
1 Sea Systems Command (PMS304)
FEDDOCS ice Effect Ship Project Office
D208.14/2:NPS6277O02 . Box 34401
as da, Maryland 200 34
NAVAL POSTGRADUATE SCHOOL
Monterey, California
Rear Admiral Isham Linder
Superintendent
Jack R. Borsting
Provost
The work reported herein was supported by funds provided
by the Naval Sea Systems Command, Surface Effects Project
Office. Reproduction of all or part of this report is autho
rized.
This report was prepared by:
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SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered)
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READ INSTRUCTIONS
BEFORE COMPLETING FORM
I. REPORT NUMBER
NPS6277002
2. GOVT ACCESSION NO,
3. RECIPIENT'S CATALOG NUMBER
4. TITLE (and Subtitle)
A Method for Scaling the Heave Motion
Equations of the C.A.B. 6D.O.F. Loads
and Motion Program from Model to Full
Size Craft
5. TYPE OF REPORT & PERIOD COVERED
Project Report for Periojd
Ending September 1977
6. PERFORMING ORG. REPORT NUMBER
7. AUTHORS
Alex Gerba, Jr. and George J. Thaler
3. CON TRACT OR GRANT NUM8ERr»;
. PERFORMING ORGANIZATION NAME ANO ADDRESS
Naval Postgraduate School
Code 62
Monterey, California 93940
10. PROGRAM ELEMEN"". PROJECT, TASK
AREA 4 WORK UNIT NUMBERS
II. CONTROLLING OFFICE NAME ANO AOORESS
Naval Sea Systems Command (PMS304)
Surface Effect Ship Project Office
P.O. Box 34401; Bethesda, MD 20034
12. REPORT DATE
December 19 77
'3. NUMSER OF PAGES
42
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18. SUPPLEMENTARY NOTES
19. KEY WORDS (Continue on reverse aide It neceaaary and Identity by block number)
Heave Equation Scaling
Surface Effect Ship
CA3 Loads and Motion Program
Performance Prediction Techniques
20. A8STRACT (Continue on reverae aide If neceaaary and Identity by block number)
A method has been developed for predicting the fullscale perfor
mance of the heave motion characteristics of the C.A.B. Surface
Effect Ship under certain operating conditions. A simplified
heaveonly model is used to demonstrate the procedure of scaling
model dimensions to fullsized craft and is validated using the
6D.0.F. Leads and Motion Program for the 100B craft. The
results of scaling the heaveonly model to a 3 K ton craft is
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SUMMARY
A method has been developed for predicting the fullscale
performance of the heave motion characteristics of the C.A.B.
Surface Effect Ship under certain operating conditions . A
simplified heaveonly model is used to demonstrate the procedure
of scaling model dimensions to fullsized craft and is validated
using the 6D.O.F. Loads and Motion Program for the 100B craft.
The result of scaling the heaveonly model to a 3 K ton craft
is also presented.
TABLE OF CONTENTS
I. Introduction
II. Description of HeaveOnly Model
III. Scaling Method with 6D.O.F. Model Validation
IV. Step Weight Transient Response
V. Conclusions
VI. Recommendations
Appendix A  DSL Program of Heave Motion Equations ■
Appendix B  Linear System Equations
I. INTRODUCTION
Before construction is started on a new ship design, it is
standard procedure to build and test models. Towing tank models
supply specific data enabling the projection of small craft
dynamic behavior to the larger craft through appropriate analyt
ical and experimental techniques. In the case of the CAB type
Surface Effect Ship, there are special considerations that must
be given to the bubble of air that provides the majority of the
lift force and, therefore, strongly affects the heave motion
dynamic characteristics.
In this report, a method has been developed for scaling the
heave motion equations of the CAB, 6DOF Loads and Motion Program,
from model dimensions to fullsized craft and thereby obtain a
prediction of the large craft heave motion characteristics. A
simplified heaveonly model (Reference 1) is used to demonstrate
the procedure. The scaled equation results are validated to a
good approximation using the Oceanics 6DOF L & M Program for
the 100B craft. In addition, the results of further scaling
to the 3 K ton craft is also presented.
II. DESCRIPTION OF THE HEAVEONLY MODEL
A simplified heaveonly model of the XR3 craft was developed
and validated in Reference 1 for the purpose of obtaining a
better understanding of the vertical motion characteristics of
the CABSES. In this model, the pitch variations are reduced
to zero by assuming 1) the center of pressure (CP) directly
under the center of gravity (CG) , 2) the sidewall symmetrical
about the CG with uniform rectangular crosssection from bow to
stern, and 3) the pitch moments of the seals and aerodynamics
cancel each other. The lift force of plenum pressure and side
wall buoyancy oppose the craft weight. All other lift and drag
forces are neglected since constant speed conditions are assumed
In addition, it was assumed that the rear seal maintains a
constant leakage area, that is, the seal follows the water level
at a fixed separation.
Figure 1 shows a top and sideview of the simulated CAB craft
All dimensions are chosen to approximate those of the XR3
craft and were used to compare the results of this analysis to
the L & M program solution under similar operating conditions.
The equations of motion for this system are given below.
Orifice Leakage Rate, q
out
C A„
n I
V
2P,
cu. ft
sec .
(1)
Fan Map Input Rate, q. = n
Q.  k? u
x i b
cu. ft
sec .
(2)
K
Absolute Plenum Pressure, P,
(Adiabatic Process)
= P
a \ V, p
Ira
Psf.
(3)
Plenum Volume, V, = (V  A, I ,) cu. ft
D n Da
(4)
Plenum Air Flow Rate, M fe = P a (q in  % ut )
slugs
sec .
W
Heave Acceleration, Z =
H H
F
■'+)
ft
sec
Heave Velocity, Z =
/:
Z dt
ft
sec.
(5)
(6)
(7)
Heave,
Z =
/
Z dt 4 Z(o)
ft
(8)
Plenum Pressure Lift Force, F = A, P, lbs.
p b b
(9)
Buoyancy Lift Force, F =
2<A 4 £ d )pg
lbs
(10)
Plenum Gage Pressure, F, = (P,  P ) Psf.
& b b a. y
(ID
Draft,
In = Z + Zs feet
(12)
The system parameters and constants are listed below
Adiabatic process coefficient, y = 14
Leakage area, A = Y, Z =2.50
Leakage orifice coefficient, C =0.90
o n
Air density,
Atmospheric pressure,
Plenum area,
p o = .002378
a
P o = 2116.
a
A, = 200.
b
co
x
3
o
Q.
CJ
c
w.
CO
ro
I
or
x
0)
2 £
3
■o
CO
T3
C
o
CL
o
(n
K3*
CO
N
V*
Z (0)
I
Z(0)
W/
m
*2
m
2^
Fe
m
2 A s P g
m
_P
m
m (0)
b
o
m
Z (Heave) ^n^ Zs
m,
t^Q^t
0
£^ (Draft)
u
Q+r\
>
m
v P
b <3
( )
JL
oi —
in
out
< — o^i — F&U t
U C A
I n i
M £ f *~t
m
Figure 2. Schematic Diagram of Signal Flow.
V = 383.
n
Empty plenum volume,
Craft weight,
Keel line area of sidewall,
Draft, initial condition,
Density of water,
Gravity acceleration,
Center gravity location,
Number of fans ,
Steady state fan output,
Fan map slope,
The schematic diagram of signal flow is shown in Figure 2.
The coordinate system and equilibrium conditions for heave
and draft are shown in Figure 3 .
(x> C.G.
W = 6720.
A A = 75/4
£ d (o) = 0.36
p = 1.99
g = 32.17
Zs = 2.5
n = 8.
Q ± = 64.3
k = 0.693
A
/f\
Zs = 2.5 ft
0.5 ft.
_ >k
Z(o) = 2.14 ft.
plenum roof
water level
7\
V
£ d (o) = 0.36 ft
£k
X
i
keel
Figure 3. Coordinate and System and Initial Condition of Heave
and Draft.
It should be noted that the range of heave motion is
2.5 < Z < 0.5 since 2.5 feet would put the water level at the
keel line and 0.5 feet would put the water level at the plenum
roof. Draft is measured from the keel and would have the range
10
< I, < 2.0. Also note that if the craft "drops" into the
water, I, increases and Z becomes less negative, that is, Z
increases in the positive (downward) direction.
At steadystate condition, the plenum pressure assumed is
P b (o) = 29.27 PSF producing a F (o) = 5854 lbs. For the draft
of £j(o) =0.36 feet, the buoyancy force F»(o) = 866 lbs.
The air mass in the plenum for the above steadystate was
M, (o) = 0.612359 slugs and the air flow rate was q. (o) = 353.
cu. ft. /sec.
11
III. SCALING METHOD WITH 6D.O.F. MODEL VALIDATION
The scaling method used in this report has the underlying
assumption that the mass density of the model and the fullsize
craft are the same and that the two ships are geometrically
similar. Speed dependent effects such as frictional resistance
are not considered in this model but again the assumption is
that the ships are operated at corresponding speeds as given by
Froude (Reference 2) .
The idea then is to scale the model linear dimensions by the
scale factor, X = L r /L where L^ is the fullscale ship linear
' rs m rs r
dimensions and L_ is the model linear dimensions. Thus, the full
m '
scale craft linear dimensions, including draft, are obtained by
multiplying model dimension by the scale factor X. All model
areas, including the leakage area, would be increased by X 2 . The
volumes and the weight of the model would be increased by X 3 .
The scaling of ambient pressure is not required since the
effects of ambient pressure on the vertical plane motion have beei
shown in Reference 4 to have negligible effects on both the pitch
and heave motion at standard atmospheric pressure. It is importa:
to note that in the heave motion equations, the plenum pressure
is determined by the adiabatic process, that is, P, = P (M,/V,p )
r ' b a v b' b M a'
However, for a given model operating at a specific draft, the
required plenum pressure, F, , is obtained by subtracting the
required buoyancy force from the weight and dividing the result
by the crosssectional area of the plenum. Once this value of
12
P, = (P,  P ) is obtained, it is necessary (as stated in
Reference 1) to initialize the air mass , M, in the computer
program, in order to obtain a balance of the adiabatic process
equation at the desired operating point.
In addition to the geometrical scaling of the model dimen
sions, it was also found necessary to apply the scale factor to
the slope of the fan map curve. In order to obtain similar ship
characteristics, the slope of the fan map curve at the operating
point must be scaled by the square of the scale factor X. The
reason for using X z can be obtained from the linear model char
acteristic equation developed in Reference 1 and included in this
report in Appendix B. The linear model characteristic equation
is third order; but as shown in Reference 1 and in Appendix B,
the second order approximation 6 2  a 33 i  a zl = yields good
results for the step weight transient response. The damping
coefficient a 3 3 is directly related to the sum of two terms
where one term is the fan map slope, k , and the other term is
the leakage area, A». Therefore, to get similar ship character
istics, the same procedure used for scaling area (X 2 ) must be
used on the fan map slope.
Equations 1 through 12 were programmed on the IBM 360/6 7
using the IBM program, "Digital Simulation Language," (DSL). The
complete listing of the programmed equations is given in Appendix
A, where SF represents the scale factor, A. It should be noted
that the leakage area used for the threeton model in this report
does not represent the XR 3 model presently in use at the Naval
Postgraduate School (Reference 3). It was necessary to increase
13
the leakage area to a large value in order to validate the scalin
method by direct comparison with the Oceanic ' s 6D.O.F. model of
the 100B craft.
Using a scale factor of 3.15, the threeton craft was geo
metrically scaled to the 100ton size craft. As stated earlier,
an important consideration in obtaining similar ship response in
heave motion is the slope of the fan map characteristics. It is
necessary to change the slope of the fan characteristic curve by
the square of the scale factor X in order to obtain the desired
heave response. Figure 4 shows the bubble fan map of the 100B
craft. The straight line drawn on the curve is the fan charac
teristic of the 100ton craft obtained by scaling the threeton
fan slope with a scale factor A 2 = (3.15) 2 .
The validation procedure used in this report was a 10 percent
step weight transient. This type of disturbance was also used to
compare the results of the scaling procedure as given in the next
section .
The response of the 100ton craft is shown in Figure 5 and
the 100B craft in Figure 6. Table I shows a comparison of the
two craft for the same step weight disturbance. It can be seen
from the data that the CG acceleration and increment of draft
change for the five second transient are within reasonable
agreement .
14
» ■ ■ t
I I
u
as
LTI
tn
UJ
a:
Q.
3u
Z
SI
>
u
OS
a
j
as
E
X
iff
S
g
?
s
hi
to
S
S3
I*.
in
0.
u
OS
La
u
as
a.
LU
J
H
2 3
3
C 323 IX ) 5J} 31Hd MQ1JNI
15
NDNLIHtRH ^Vjltr
HULLL WD
lnc
_ /
L
nv
FIGURE 5A.
100Ton Craft
CG Accsl vs Time
<■ q r n    n a n
16
C '_ :]
pyfsj
pi rr
r I
HOHLINERR 5VSTEM QRR'T US TIME
\
\
\
\
FIGURE 5B
100Ton Craft
Draft vs Time
>j L
i n
UNIT5IKCH
J [T5.^INCH
pis
17
run sia. i
NQhLINERR SVSTEM D REo5URE US TIME
i
13
/
/
FIGURE 5C. 100Ton Craft
Pressure vs Time
;i. ::
•"■".! 1
. i
UNITS^IhCH
ul iITS> INCH
13
n L
JZ2
i:.
FIGURE 6A.
100B Craft
CG Accel vs Time
i r Lf
 i nnf+nv i n r t q r n r M
 i« L'w.. ■ jLj Uln .J 1 i i U M '
• p <. —
 nrc nn IJK1TTC: IKru
: : I^
~"> n UTQ 1 ! T DC" M PI i 1 i
r U n ! o HI nLnu^HL
 h d i n
U '"1 1 LI
St
15 CG HERUcfiCC
LG05 CRRFT
TTM
19
_.. ..
■
N.
V
\
s, 1

*~v^^
•
FIGUR
E 63. 100
Dra
B Craft
ft vs Time
i<:i Pi =
n?  _ .,
'NITS INCH
; n u t d
r q. nnnr
i_ _: L) !~1 n !
5 INCH
! " r r ~ ki H : P
h ! h _ ' ; U k r_
■inn
n n
■■ . !_ n J U J
ki
20
c::
u:i
/
/
f
•i h
jL
/
]••.•
p:;
TTs
c:v
/
/
S
FIGURE 6C
1003 Craft
Plenum Pressure vs Time
SCRL£>d.00E+30 UNITS INCH.
SCRLr^i 2Z~^2Z UNITS INCH.
1QK
SOT
~J
UT
^8
^i
T REMOvr
JRL
RLE
kll IM KPc
f 1 or
HI"
21
FOR IGHR CRR
UERSU.S TTHF
"
TABLE I. Verification of Scaling
Procedure Using Step
Weight Disturbance
CRAFT
100 TON
100B
Weight (K lbs)
210
210
Cushion length (ft)
65.31
65.31
Scale factor, X = L f /L
J_ O ill
3.15
3.15
Leakage area (ft 2 )
24.80
25.52
Fan map slope (CFS/PSF)
6.88
6.88
Plenum pressure (PSF)
92.2
92.3
Number of fans, n
8
8
C.G. t_ (sec)
0.675
0.51
Acceleration
7o overshoot
6.10
5.1
Draft increment, (ft)
0.82
0.90
(05) second
It should be pointed out that in Reference 1, it was shown
that the simplified heaveonly model could be made to closely
approximate the 6D.O.F. equation results for the step weight
disturbances even though the effects of pitch motion on the
total leakage area are not considered in the simplified model.
Due to time constraints, no great effort was expended in obtain
ing a closer agreement in the 100ton craft response for the
above reason.
22
IV. STEP WEIGHT TRANSIENT RESPONSE
The use of a steptype disturbance as a rapid check on the
transient characteristics of a system is a well established
practice. A step weight removal was chosen because of its ease
of application and also because of its possible utilization as a
practical method for the validation of the XR3 6D.O.F. equa
tions. The procedure is to start with the system in steady
state operation at the heavier weight and at the initial time
of the transient, a 10 percent weight is removed and the
response calculated for the first five seconds of the transient.
The response of a three ton craft with dimensions approximate
to those of the XR3 craft is shown in Figure 7. The three ton
craft was then scaled (X = 3.15) to a 100ton size with the
transient response shown earlier in Figure 5. The threeton
craft was further scaled (A = 9.63) to the dimensions of a
3Kton craft and had the response characteristics as shown in
Figure 8.
A comparison of the characteristics of the threeton, 100
ton and the 3000 ton ships is shown in Table II.
23
NaHLINERR
■^
Li
j u _)
RCCE.L
n
TTM
J. i i
i. 3_
FIGURE 7A
3Ton Craft
CG Accel vs Time
i =
• 't?i
L ' J J
UNIT
3< 1 !^U! ■
3<II >
24
^' I . ; ' ■
NONLIHERR
S VST EM
U n h
i :
^TJ
Q i TTK1F
I !
FIGURE 7B. 3 Ton Craft
Draft vs Time
:■
' ;
 ■  _
25
I
U!
■
ki'" 7 ' !
.r>
I ca k : i t
LiliLnn «J i J ! L, ! i nLOJUhL v J ! i ! ! L_
:•
I 3
i i
/
! IT. *
In /
1/
FIGURE 7C. 3Ton Craft
Pressure vs Time
r p
n i ■>.
:i. a
UNITS INCH
will • »_J ' _ : ' '. J
'A
26
.■ :
H^tSLI
FPS
k !
T T
Mi
FIGURE 8A.
3000Ton Craft
CG Accel vs Time
27
' J ! 1 L 1 :
n
p h n 
FIGURE 83
3000Ton Craft
Draft vs Time
9 °
M k ' i
T I.
1 U ' ^ L i
JV3TFM
/
/
\
I
\
/
FIGURE 8C. 3000Ton Craft
r
Pressure vs Time
\
r
..'
■■■■'
!_., : 
29
; I
"
TABLE II. Comparison of Scaled Craft
Response to a 10 Percent
Weight Transient
CRAFT
3 TON
100TON
3000TON
Weight (K lbs)
6.72
210
6000
Cushion length (ft)
20. 7
65.31
199.3
Scale factor, ^f s /^ m
1.0
3.15
9.63
Leakage area (ft 2 )
2.5
24.8
231.8
Fan map slope (CFS/PSF)
0.693
6.88
64.27
Plenum pressure (PSF)
29.3
92.2
281.9
Number of fans, n
8
8
8
C.G.
tp, sec
0.425
0.675
1. 175
Acceleration
% overshoot
3.62
6.10
11.79
Drafu increment (ft)
0.27
0.82
2.39
(05) second
It can be observed that the larger craft C.G. acceleration
percent overshoot and time of first ciaxiiium increase by a facto
roughly approximated by the square root of the scale factor,
i.e. /A . As shovm in Reference 1, the vertical mocion damping
is the result of the pressure effects acting on the C.G.
acceleration and does noc depend directly on the mass of the
craft as it does for the system which has viscous friction
damping .
30
V. CONCLUSIONS
An important consideration in any scaling method used to
extend model data to fullsized C.A.B. SES craft is to properly
scale leakage area and the fan map characteristic slope. To
obtain similar ship characteristics, the same scaling factor
used for the linear dimensions should be applied to the leak
age area and fan characteristic slope as outlined in Section
III.
It was also observed that when scaling up from a threeton
model that the C.G. acceleration percent overshoot and time
of the first maximum occur at a value roughly approximate
to the square root of the scale factor and do not depend
directly upon the mass as it does in systems with viscous
friction damping.
31
VI. RECOMMENDATIONS
The authors feel confident that the procedure outlined
in Section III is a valid one for scaling the heave motion
equations; however, it remains to be shown that the method will
yield similar results for the scaled 6D.O.F. equations. The
task of scaling the 6D.O.F. program is a lengthy process and
was not carried out in this study due to the time constraints.
Whenever this task is completed, it would be worthwhile to
carry out a frequency response study of the type used in
Reference 4 to observe the effect of the scale factor on the
C.G. acceleration in order to further validate the procedure
used for scaling from model to fullsize craft.
Time constraints also prevented the completion of the
analysis related to the dependence of the C.G. acceleration
characteristics on the square root of the scale factor but
the authors feel confident that the characteristic equation of
the linear model will yield che desired explanation. We are
continuing this study and will include the results obtained
in our summary report.
32
LIST OF REFERENCES
1. Gerba Jr. , A. and Thaler, G. J. , Pressure Ratio Effect on
the Heave Motion Characteristics and Pressure Dynamics of
the XR3 Loads and Motion Program for Step Weight Transients ,
NPS Progress Report to SESPO, January 1977.
2. Comstock, J. P. (Editor), Principles of Naval Architecture ,
Society of Naval Architects and Marine Engineers , New York ,
1967.
3. Menzel, R. F., Study of the Roll and Pitch Transients in
Calm Water Using the Simulated Performance or the XR3
Surface Effect Ship Loads and Motions Computer Program ,
NPS Master Thesis , December 1975. '
4. Gerba Jr., A. and Thaler, G. J., Frequency Response Studies
of Ambient Pressure Effects on the XR3 Computer Program
(6D.O.F.) , NPS Progress Report to SESPO, September 1977.
3a
APPENDIX A. DSL PROGRAM OF HEAVE MOTION EQ .
fTfLf HHlVl MOTION DYNAMICS r F
P4RAM KL=12» 79
I d c a r <
"PC
ST NPLQT=2
"2^
M? t WD ' jr, vP.,LDt V3,MBO
:> q i i
mr
tr*r
KMsfS. '
SF=9.63
RH0=1..99
b=* 3 ^ . i /
ZS=2o5^3F
7=l  R63*SF
Z^ r = =l ►365*SF
Z53=2«L+*'F
W"=672U.(3 Ci >3.)
M=WT/G
^3 = 2 ; »*( ".= **2. )
VN=383 .*( E=**3 . )
RG*Msl»/l»4
FM = 3 .
U=2„50?* (*F**2o )
icrc^ zT^T^
FL i S = 2 . J * •? $ *R HO * GM. 0'
FP':=wTFL3S
35*3* :=fpica3
PB ,;> C= P^ *° 3^ R? C
? n 7 r a" 1
*5 3 3Tr"
VB >' Z  VM= * B*L 1 C +* M * " D ' 3
PRI C=PBIC/pJi
M3 ' : = ( P5 j r**R s ■: h ) *v3. .;* sh ";>
Q31T ! C*CM** L *$ OR T ( 2 .J*PB4R? C /RH n * J
KQ = o S°3*L£ F**7„? 1 _____
Q.< =(QCU
/f n +P3' r
*KQ
H 3 ^
TDPB&RTC
P5 }=29.2
"0=2^,0
D5R mil V 5
1. 0=7+ 7S
Ft M=s ( 2 •? * * 5*RH J* G* ! . D ) / M
VM=MB/R0*
 V3
PB"=
(AVRJ
R = PS
= A R * p
7M/VB
**GaM*S.XGfN( l»f VM)*S"*GN(1. » V9 1
»r»wr/
N"GRi
wor
Q ?> 4 = rVT(Q
M= PH' M
•K34
,P9
g "s u
* * = A 8
w=g ?m
\L*SQRT( 2,
Q'3U*
pc
,/RH
 » »
M?0
= f?.H0i£ kFtrSW
34
M B 7 . ' rG*L ( '  '3 7 : t'  igon
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35
APPENDIX 3. LINEAR SYSTEM EQUATIONS
The linear system equations are arrived at by application
(a)
of the Taylor Series expansion about the steadystate
operating point. The development is shown below for the two
differential equations .
Equation (1) , let W/M = W ; F /M = F m
n m p pm
and F/M = F«
C cm
then from equation 6 in Section II,
2(0) + 6z = W/M  (F (o) + 6F pm )  (F tm (o) + S? lm )
or,
5 z =  o F  5 F ,
pm im
(la)
Equation (2) , let m/o = V for convenience
n s a m
then, from equation 5 in Section II,
V m (o) + SV m = n Qi  nk q (P b (0) + oP b )
" C A,, 2F b<°\ *
n £(— )
2Pu(o) % ?
 C A, %( — ^ ) (— ) 5P,
n % 2 v p v o b
M a ' a
or,
(a) Steady state and incremental variations are noted as follow
using heave as an example.
z = z (o) + 6 z
36
Next, the auxiliary algebraic equations are linearized.
F Co) + 6F p = A b (P b (o) + 5P b )
5F p " h 5F b
(3a)
V o) + 6F * = 2 Vw§ U d W+si d )
6F l ' < 2 Vw^ Sl d
(4a)
Following the above procedure, equations 1 and 2 from Sec
tion II are linearized with the results given in equations
(5a) and (6a) .
Sq.  ~(nk ) 5P K
(5a)
and
For equation 3 in the report, the equation becomes,
P b (o) + 5P b)
V°>V ♦ Y / v  (0) '
V (o)
v b (o)
V b (o)
(V (O))
6 V _ m 5 V,
[V (o)] 2 b
m
 1
3P b = V (
V (o
m
V
m
V (o) SV.
m b
V,_(o) V (o) V, (o) V (o
m
b
P, = (P, (o)
o b
1 SV.
V (o)
m
m
V, (o)
b
5V, (7a)
37
and linearized equations 4, 11 and 12 follow.
V b (o) + 6V b = V n  A b (£ d (o) + 6i d
6V b = " (A b m d
(8a)
P. (o) + 5P, = p (o) + SP,  P
D D D Da
5P^ = 5P,
b d
9a)
£ , (o) + s£. = z (o) + 5z + Z
da i
6£, = 5z
d
10a
Combining (la) , (3a) , (4a) , (7a) and (8a) as follows
6z
m / b
2A , o g\
ir w ^ \ 6 z
M
recall that o£, = 5z
d
o z =  b
M
Y P (O)
1 5V  1 fA. ) 5z
V (O)
m
m
V b (o)
b
2 Vwj)
M
o z
L \ M
2A t p g\ /',?, (o)A,
5 w 1 + b b
MV fa [ o )
z +
YP, (o)A,
d b
w MV (O) J
m
oV
m
38
Also for equation (2a
6V
m
nk + C n A l
q — : —
2P b (o)
yP b (o)
5V
m
lv (o)
m
5V b
v b <o:
6V =
m
nk + C A. / p.
q n L a
2P b (o)
YP b (Q)
m
5V
m
+ ( b \ 5z
vv o) /
and for equation (7a)
5P b = Y P b (o)
1 5V + b
m
m
V b (o,
6z
Now define the state vector
x A [5z 5z 5V ]
— = m
T
and output variable, y = <SP, with
y A c n X! + c 13 x 3
Then the state equations are
xi  x 2
x 2 = a 21 x 1 +a 23 x 3
X 3 = a 3 ! X ! + a 3 3 X 3
Y P ( O ) A
where c M = b b and c
vP. (o.
i 3  b
v b (o;
m
2 1
2A
P Q
M
MV, (o)
o
39
a 2 3
3 1
^ P b ( ° )A b
MV (O)
v m
3 3
yP b (o)
yP b (o:
nk + n L
a
nk
C A*
n <
2P b (o)
2P b (0)J
A
b
v b (o
m
The characteristic equation for this system is given by
( 5 1  A) where
A =
o 1 o
a 2 i o a 2 3
a 3 1 ° a 3 3
Then (il  A) =
i 1
a 2i i
o
o
a 2 3
6  a
a 31 „ . 33J
and the characteristic equation is
. 2
5 6  a 33 6  a 2 i i + ( a 2 1 a 33 " a 23 a 3 1 ) = °
Using the numerical values given in Reference 1, the
characteristic equation is
•i 3 + 62.5 2 + 2133.6 + 789 = o
This cubic equation will factor into one real root near
the origin and a complex pair far removed from the jw axis.
Thus, a step weight response will have a complex pair of poles
which can be approximated by the quadratic form,
* 2  a 33* " a 21 = °
40
For the given numerical values that approximate the XR3
craft, it can be shown that a rough approximation for the
coefficients are:
a 33 i nk q YP b (o)/V m (o)
"•„ = V Y P b (o)/V b (o)
M
Using the above approximation yields a natural frequency of
w
•a 2 i
A 2
A b
M
P b (o)/V b (o
"n = K n J P b (0)/V b (0)
13
where K = A, \ y/M
n b v
and a damping factor
C =
a
3 3
2w
n
 2 V v 2 P b 2 (o
/ V2 m (o)
A b 2 ~y P b (0) / V b (0)
M
C = K.
P (o) V b (o
V z (o
m
14
where
K f = n k
2A,
M
41
INITIAL DISTRIBUTION LIST
Copies
1. Mr. A. W. Anderson 6
PMS 304 3 LA 1
Surface Effects Ships Project Office
P. 0. Box 34401
Bethesda, Maryland 200 34
2. Library, Code 0142 2
Naval Postgraduate School
Monterey, California 93940
3. Office of Research Administration
Code 012A
Naval Postgraduate School
Monterey, California 9 3940
4. Professor A. Gerba, Jr., Code 62Gz
Department of Electrical Engineering
Naval Postgraduate School
Monterey, California 93940
5. Professor G. J. Thaler, Code 62Tr
Department of Electrical Engineering
Naval Postgraduate School
Monterey, California 93940
42
U 18 08 90
5 6853 01058345 3
U18089