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Full text of "A method for scaling the heave motion equations of the C.A.B. 6-D.O.F. loads and motion program from model to full-size craft"

NPS62-77-002 



IWuunr 

TECHNICAL REPORT SSCTIOK 

POSTGRADUATE SCHOOt 



NAVAL POSTGRADUATE SCHOOL 

Monterey, California 




A METHOD FOR SCALING THE HEAVE MOTION 
EQUATIONS OF THE C.A.B. 6-D.O.F. LOADS AND 
MOTION PROGRAM FROM MODEL TO FULL-SIZE 

CRAFT 


Alex Gerba, 


Jr . and George J . 


Thaler 




December 


1977 




Progress 


Report for Period 
September 1977 


Ending 



PreDared for: 

1 Sea Systems Command (PMS-304) 
FEDDOCS ice Effect Ship Project Office 

D208.14/2:NPS-62-77-O02 . 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: 



UNCLASSIFIED 



SECURITY CLASSIFICATION OF THIS PAGE (When Data Entered) 



REPORT DOCUMENTATION PAGE 



READ INSTRUCTIONS 
BEFORE COMPLETING FORM 



I. REPORT NUMBER 



NPS62-77-002 



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. 6-D.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 (PMS-304) 
Surface Effect Ship Project Office 
P.O. Box 34401; Bethesda, MD 20034 



12. REPORT DATE 

December 19 77 



'3. NUMSER OF PAGES 

42 



14. MONITORING AGENCY NAME a AOORESS^// different from Controlling Office) 



15. SECURITY CLAS3. (of thia report) 



Unclassified 



15«. DECLASSIFICATION/ DOWNGRADING 
SCHEDULE 



16. DISTRIBUTION ST ATEMEN T (of thia Report) 



17. DISTRIBUTION STATEMENT (of the abatract entered In Block 20, If different from Report) 



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 full-scale perfor- 
mance of the heave motion characteristics of the C.A.B. Surface 
Effect Ship under certain operating conditions. A simplified 
heave-only model is used to demonstrate the procedure of scaling 
model dimensions to full-sized craft and is validated using the 
6-D.0.F. Leads and Motion Program for the 100-B craft. The 
results of scaling the heave-only model to a 3 K ton craft is 



DO ( JAN 73 1473 EOITION OF 1 NOV 65 IS OBSOLETE 



S/N 102-014- 6601 



1 



UNCLASSIFIED 



SECURITY CLASSIFICATION OF THIS PAGE (Whan Data Sntared) 



UNCLASSIFIED 



.L.C .UW1TY CLASSIFICATION OF THIS PAGEfWhw. Data Entered) 

also presented. 



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SUMMARY 

A method has been developed for predicting the full-scale 
performance of the heave motion characteristics of the C.A.B. 
Surface Effect Ship under certain operating conditions . A 
simplified heave-only model is used to demonstrate the procedure 
of scaling model dimensions to full-sized craft and is validated 
using the 6-D.O.F. Loads and Motion Program for the 100-B craft. 
The result of scaling the heave-only model to a 3 K ton craft 
is also presented. 



TABLE OF CONTENTS 

I. Introduction 

II. Description of Heave-Only Model 

III. Scaling Method with 6-D.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, 6-DOF Loads and Motion Program, 
from model dimensions to full-sized craft and thereby obtain a 
prediction of the large craft heave motion characteristics. A 
simplified heave-only model (Reference 1) is used to demonstrate 
the procedure. The scaled equation results are validated to a 
good approximation using the Oceanics 6-DOF L & M Program for 
the 100-B craft. In addition, the results of further scaling 
to the 3 K ton craft is also presented. 



II. DESCRIPTION OF THE HEAVE-ONLY MODEL 



A simplified heave-only model of the XR-3 craft was developed 
and validated in Reference 1 for the purpose of obtaining a 
better understanding of the vertical motion characteristics of 
the CAB-SES. 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 cross-section 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 XR-3 
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 = 1-4 

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 



K-3-* 



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 steady-state 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 steady-state was 
M, (o) = 0.612359 slugs and the air flow rate was q. (o) = 353. 
cu. ft. /sec. 



11 



III. SCALING METHOD WITH 6-D.O.F. MODEL VALIDATION 

The scaling method used in this report has the underlying 
assumption that the mass density of the model and the full-size 
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 full-scale 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 cross-sectional 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 three-ton 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 6-D.O.F. model of 
the 100-B craft. 

Using a scale factor of 3.15, the three-ton craft was geo- 
metrically scaled to the 100-ton 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 100-B 
craft. The straight line drawn on the curve is the fan charac- 
teristic of the 100-ton craft obtained by scaling the three-ton 
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 100-ton craft is shown in Figure 5 and 
the 100-B 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. 



100-Ton 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 



100-Ton Craft 
Draft vs Time 



>j L 

i n 



UNIT5-IKCH 
J [T5.^INCH 



pis 



17 



run sia. i 



NQhLINERR SVSTEM D REo5URE US TIME 



i 



13 



/ 



/ 



FIGURE 5C. 100-Ton Craft 

Pressure vs Time 



;i. :: 



•"■".! 1 






-. i 



UNITS^IhCH 
ul iITS> INCH 



13 



n L 



JZ2 



i:. 



FIGURE 6A. 



100-B 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 <-. — 


- nr-c 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 



100-3 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 


100-B 


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 


(0-5) second 







It should be pointed out that in Reference 1, it was shown 
that the simplified heave-only model could be made to closely 
approximate the 6-D.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 100-ton craft response for the 
above reason. 



22 



IV. STEP WEIGHT TRANSIENT RESPONSE 

The use of a step-type 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 XR-3 6-D.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 XR-3 craft is shown in Figure 7. The three- ton 
craft was then scaled (X = 3.15) to a 100-ton size with the 
transient response shown earlier in Figure 5. The three-ton 
craft was further scaled (A = 9.63) to the dimensions of a 
3K-ton craft and had the response characteristics as shown in 
Figure 8. 

A comparison of the characteristics of the three-ton, 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 



3-Ton 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 : 



^T-J 



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. 3-Ton 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. 



3000-Ton Craft 
CG Accel vs Time 



27 






' J ! 1 L 1 : 



n 



p h n - 



FIGURE 83 



3000-Ton Craft 
Draft vs Time 



9 ° 



M k ' i 



T I. 



1 U ' ^ L i 



JV3TFM 






/ 



/ 



\ 


I 








\ 


/ 




FIGURE 8C. 3000-Ton Craft 






r 




Pressure vs Time 




\ 


r 










--..'- 


■■■■-' 


!_., : - 





29 






; I 



" 



TABLE II. Comparison of Scaled Craft 
Response to a 10 Percent 
Weight Transient 



CRAFT 


3 -TON 


100-TON 


3000-TON 


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 


(0-5) second 









It can be observed that the larger craft C.G. acceleration 
percent overshoot and time of first ciaxii-ium 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 full-sized 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 three-ton 
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 6-D.O.F. equations. The 
task of scaling the 6-D.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 full-size 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 XR-3 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 XR-3 
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 XR-3 Computer Program 
(6-D.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^ z-T^T^ 

FL i S = 2 . J * •? $ *R HO -* G-M. 0' 
FP':=wT-FL3S 



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 ^ 



TD-PB&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/R-0* 



- 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 



•K-3-4- 



,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 

Z= r M ? GRL I ZlCtW ) 



SAMPL= 



PRPLOr 

S&A-EM 



2 5,^ M^,W0J: , V0,L0, V3.M3DO J , k ^, 5 V 3 t T" , O^r ,V^ 



;^Mg, ? T^ 



,Q 



GMPH 
G**PH 
SiHrW 
GR 4. P H 
GR^PH 



Si 



t 
t f 



Ml 



MS 

MS 
M » 

W3~ 



,P8 
*VR 

fVS 
tTD 



U 



SA 
SA 



Mc 



CA 



,VM 



LL ETRW3TT7I ♦ ;"" ^ f L0) 
LL D3WG( 2t It r7M5*PB4P ) 
It. PP.WG H ,T tV? » VM) 



CA 

C* 
KL 

T^ 



OR 
CALL OP 



W3 
AG 



(*t 

(5, 



1, 7 ^,V° ) 

i, n m -♦ .^no 7 ) 



LL iKHWMPL 



) 



g r o i i 



) C^j 



'JTT 



^a 



CALL 



ST1 

//p 



p 
Lnr 



■X 72 8€fr6 i 



-o+ 



: ? u" p.<tscj' 



DW 



^29 



M29 

fcl 

N29 

-H29- 



Nn 

BO 

80 

! S. 

BQ 

MO 

-66 

NO 



NUT NEAR SYSTEM" DR4F*" VS '2 ML 
X 7 r> G^^ 3 * PLENUM PRESSURE ny\iv?*«* 

LWI { NFSR SY31 :,/r " "MLTSUR" 1 - Vs "> »•<- 
X 73 3 S ? , 3 a PL? N UM D R£ S S US " 



QVV | M ~ C 



YVH &U VOLJM" V j 3l . r MI1M V " 1 L il M i5 

X 70 GgR8* °L5NUM PRESSURE DYM^M'CCS 
NL*NS*R SYSTEM PRESSURE RA7?iH V e -" M c 

X 70 G'°"D\ ? ' _T ,, | J V < s " 5 Hffi 1 HVNn . — q 

NL S NE AS S YS TE M C G 4CC ^L VS ^ 7 MF 



35 



APPENDIX 3. LINEAR SYSTEM EQUATIONS 



The linear system equations are arrived at by application 

(a) 

of the Taylor Series expansion about the steady-state 

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 f-A. ) 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 XR-3 
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