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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. St si :: us UNCLASSIFIED SECURITY CLASSIFICATION OF THIS P*3£(1Wi«n Date £«•« 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