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Active-passive motion compensation systems for marine towing 1975

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ACTIVE - PASSIVE MOTION COMPENSATION SYSTEMS FOR MARINE TO WING by PETER ANDREW STRICKER B. Eng. M c G i l l U n i v e r s i t y M o n t r e a l , 1971 . A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER CF APPLIED SCIENCE i n the Department of M e c h a n i c a l E n g i n e e r i n g We accept t h i s t h e s i s as con f o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH CCLUMEIA VANCOUVER CANADA January 1 9 7 5 In presenting th i s thes i s in pa r t i a l f u l f i lmen t of the requirements ft an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree tha the L ib ra ry sha l l make i t f r ee l y ava i l ab le for reference and study. I fu r ther agree that permission for extens ive copying of t h i s thes i s fo r s cho la r l y purposes may be granted by the Head of my Department or by his representat ives . It is understood that copying or pub l i c a t i on of th i s thes i s fo r f i nanc i a l gain sha l l not be allowed without my wr i t ten permiss ion. Department of MEGUAmCAL Bd£ldEEg.^& The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date APg.lL 2 , l<?7S~- i i ABSTRACT The dynamic behaviour of an a c t i v e - p a s s i v e motion compensation system f o r handling towed marine v e h i c l e s i s examined, and a mathematical model developed. In the a n a l y s i s , the passive system considered i s pneumatic, while the a c t i v e system i s e l e c t r o - h y d r a u l i c . The towed body i s assumed to be a point mass subjected to uydrodynamic drag, and attached to the motion compensator by means of a l i n e a r s p r i n g r e p r e s e n t i n g the cable. I t i s not intended, i n t h i s p r o j e c t , to mcdel the tewed body i n greater d e t a i l . The equations of the passive, a c t i v e , and towed tody systems are derived, and l i n e a r i z e d t c permit a r e l a t i v e l y simple frequency-domain s o l u t i o n . A time s i m u l a t i o n based on the nonlinear equations, i n c l u d i n g Coulomb f r i c t i o n i n the compensator, i s developed for use on an IBM Systeis/370 computer. A l a b o r a t o r y model i s used to conduct experiments at three f r e q u e n c i e s , and the r e s u l t s i n d i c a t e good agreement ketween the l i n e a r , s i m u l a t i o n , and r e a l models. Extension of the equations to cover multi-frequency inputs, two-dircensiona1 tewing c a b l e s , and slow-acting servovalves i s a l s o discussed to f a c i l i t a t e a p p l i c a t i o n to marine systems. i i i TABLE OF CONTENTS Chapter I - I n t r o d u c t i o n ............................. 1 1.1 Problem D e s c r i p t i o n ....................... 1 1.2 S t a t e of the A r t .......................... 6 1.3 O b j e c t i v e s and Scope of P r o j e c t ........... 13 Chapter II - T h e o r e t i c a l A n a l y s i s 14 2.1 T y p i c a l System 14 2.2 E q u i v a l e n t Model 17 2.3 The Passive System 20 2.4 The A c t i v e System ......................... 30 2.5 The A c t i v e - P a s s i v e System 37 2.6 The C o n t r o l System ........................ 42 2.7 Computer S i m u l a t i o n 46 Chapter I I I - L i n e a r A n a l y s i s 47 3.1 L i n e a r i z e d Passive System 47 3.2 L i n e a r i z e d A c t i v e System 55 3.3 L i n e a r i z e d A c t i v e - P a s s i v e System .......... 58 3.4 Performance A n a l y s i s and o p t i m i z a t i o n ..... 61 Chapter IV - The Laboratory Model 70 4.1 General D e s c r i p t i o n 70 4.2 Performance P r e d i c t i o n and E v a l u a t i o n ..... 77 Chapter V - A p p l i c a t i o n 81 5.1 Input C o n d i t i o n s .......................... 81 5.2 Two-Dimensional Cable Model 85 5.3 Servovalve Model Extension ................ 87 5.4 C o n t r o l System C o n s i d e r a t i o n s 90 Chapter VI - C o n c l u s i o n s ............................. 93 References 94 Appendices 96 i v LIST OF ILLUSTRATIONS Fi g u r e Page 1.1.1 Motion Compensation Systems 4 1.2.1 I s o l a t i o n and Absorption 7 1.2.2 Performance C h a r a c t e r i s t i c s 7 1.2.3 P a s s i v e Pneumatic System 9 1.2.4 Tuned Bam Tensioner 12 2.1.1 A c t i v e / P a s s i v e Bam Tensioner 15 2.2.1 E q u i v a l e n t System 18 2.3.1 P a s s i v e System 21 2.3.2 Block Diagram of Tank Dynamics ....................... 23 2.3.3 Block Diagram o f C y l i n d e r Dynamics 23 2.3.4 Block Diagram of Valve Dynamics ...................... 27 2.3.5 Block Diagram of P a s s i v e System Dynamics ............. 29 2.4.1 A c t i v e System 31 2.4.2 Ser v o v a l v e C h a r a c t e r i s t i c s , 31 2.4.3 Block Diagram of A c t i v e System ....................... 36 2.5.1 Cable/Body Model 38 2.5.2 Block Diagram of Cable/Body Dynamics 38 2.5.3 Block Diagram of A c t i v e / P a s s i v e System ............... 41 2.6.1 A c t i v e System with C o n t r o l B l o c k s 43 3.1.1 Pressure-Flow Curve f o r T h r o t t l i n g Valves ............ 51 3.1.2 L i n e a r i z e d P a s s i v e System T r a n s f e r F u n c t i o n .......... 51 3.2.1 L i n e a r i z e d Servovalve C h a r a c t e r i s t i c s 57 3.2.2 L i n e a r i z e d A c t i v e System T r a n s f e r Function ........... 57 3.3.1 L i n e a r i z e d Cable/Body T r a n s f e r F u n c t i o n 60 3.3.2 B. D. of L i n e a r i z e d A c t i v e / P a s s i v e System ............ 60 3.4.1 Bam C e n t e r i n g Network 65 4.1.1 Laboratory Apparatus 71 4.1.1 Laboratory Apparatus - Schematic 72 4.1.3 Motion Generator Arrangement 73 4.1.4 C o n t r o l System 75 4.2.1 T h e o r e t i c a l and Experimental B e s u l t s 80 5.1.1 Sea State S p e c t r a l Density Function ... 83 5.1.2 Ship Heave Response .................................. 83 5.1.3 S h i p Heave S p e c t r a l Density F u n c t i o n ................. 83 5.2.1 Two-Dimensional Cable Model 86 5.3.1 T y p i c a l Servovalve Response 89 5.4.1 Motion Compensation T r a n s f e r Function ................ 91 5.4.2 T y p i c a l Feedback Network 91 V LIST OF SYMBOLS SYMBOL MEANING A area of p i s t o n A A area o f a c t i v e c y l i n d e r p i s t o n A P area of passive c y l i n d e r p i s t o n Co t h r o t t l i n g v a l v e c o e f f i c i e n t C R c a p i l l a r y c o e f f i c i e n t C H hydrodynamic drag f a c t o r C L l i n e a r drag f a c t o r CSV s e r v o v a l v e flow c o e f f i c i e n t D d i f f e r e n t i a l operator d/dt f f r i c t i o n f o r c e i n ram FA a c t i v e c y l i n d e r f o r c e FMBT c a b l e t e n s i o n F P passive c y l i n d e r f o r c e FBAM t o t a l ram f o r c e F B ( S ) body t r a n s f e r f u n c t i o n F C (S) c a b l e t r a n s f e r f u n c t i o n G P ( S ) passive system t r a n s f e r f u n c t i o n H (s) open loop t r a n s f e r f u n c t i o n feedback loop t r a n s f e r f u n c t i o n H^tS) feedforward loop t r a n s f e r f u n c t i o n H s v(s) serv o v a l v e t r a n s f e r f u n c t i o n Hy (S) ram c e n t e r i n g loop t r a n s f e r f u n c t i o n K, displacement feedback g a i n K 2 v e l o c i t y feedback gain K 3 a c c e l e r a t i o n feedback g a i n K F F feedforward s t a t i c gain KMA mechanical advantage K P p a s s i v e system s t a t i c gain Ks gas s p r i n g s t i f f n e s s K s v s e r v o - a m p l i f i e r gain m mass of gas i n passive c y l i n d e r H mass of towed body N pa s s i v e system volume r a t i o P p r e s s u r e Po i n i t i a l pressure i n p a s s i v e system P* . P* p a s s i v e c y l i n d e r p r e s s u r e s P. pressure downstream t h r o t t l i n g valve PM pressure upstream t h r o t t l i n g valve Ps supply pressure i n a c t i v e system AP pressure drop i n a c t i v e c y l i n d e r Q A o i l flow i n t o a c t i v e c y l i n d e r Q L leakage flow Qv s e r v o v a l v e flow r low l e v e l servo s i g n a l R gas c o n s t a n t f o r n i t r o g e n 5 Laplace v a r i a b l e t time T abs o l u t e gas temperature u d i s t u r b a n c e i n p u t displacement V volume Vc,, V C 2 passive c y l i n d e r volumes p a s s i v e tank volumes v i W power consumption x body displacement xt tow point displacement y piston displacement Yc tow pt displacement r e l a t i v e to ship r servovalve actuating current V r a t i o of s p e c i f i c heats 8 small piston displacement 5 passive system c r i t i c a l damping r a t i o 5 servovalve c r i t i c a l damping r a t i o \, servovalve flow gain "Xz servovalve flow-pressure c o e f f i c i e n t X3 bypass valve flow c o e f f i c i e n t tr, t, passive system time constants •c w servovalve time constant u30 design frequency oin passive system natural frequency co<,v servovalve natural frequency v i i ACKNOWLEDGEMENT I am most g r a t e f u l f o r the p a t i e n t help and encouragement of my s u p e r v i s o r , Dr. R. BcKechnie. S p e c i a l thanks i s due to Dr. Keefer of B. C. Research C o u n c i l who i n i t i a l l y gave me the idea f o r t h i s p r o j e c t , and f o r h i s subsequent a s s i s t a n c e . Thanks are a l s o due to Mr. Johnson of F l e c k B r o t h e r s f o r donating a se r v o v a l v e and a m p l i f i e r ; to.Dr. V i c k e r s f o r making the v a l v e work; to Messrs. Hoar and Hurren and t h e i r crew f o r b u i l d i n g the apparatus and l e n d i n g equipment; t o the Department of E l e c t r i c a l E n g i n e e r i n g f o r the use of t h e i r analogue computer; and t o Miss Wendy A l l e n f o r keypunching the manuscript. A l l computing was done a t the U n i v e r s i t y o f B r i t i s h Columbia Computing Centre. The p r o j e c t was funded by the Nat i o n a l Research C o u n c i l of Canada under Grant #67-8183. 1 CHAPTER I I8TB0DUCTI0H 1.1 Problem D e s c r i p t i o n The s a f e t y and performance of towed submersible v e h i c l e s depend, to a l a r g e e x t e n t , on the a b i l i t y o f the ha n d l i n g gear to decouple wave induced motions of the su r f a c e s h i p from the towing c a b l e . T h i s decoupling i s u s u a l l y accomplished by a motion compensation system — e s s e n t i a l l y a s p e c i a l c l a s s of a v i b r a t i o n i s o l a t o r . V i b r a t i o n i s o l a t i n g d evices are employed i n systems where a mass i s to be i s o l a t e d from an e x t e r n a l f o r c e c r motion d i s t u r b a n c e . Some common examples are automobile suspensions, earthquake a b s o r b e r s , and rocket-borne i n s t r u m e n t a t i o n cushions. The marine towing i s o l a t i o n problem has three d i s t i n g u i s h i n g f e a t u r e s . F i r s t , the frequency o f s u r f a c e s h i p motion i s low, i n the order of 0.1 Hz, while the a s s o c i a t e d amplitude may be i n the order of f i f t e e n f e e t . Second, the mass of the submersible, i n c l u d i n g the water i t e n t r a i n s , i s l a r g e — t y p i c a l l y i n excess of 30, 000 pounds-mass. F i n a l l y , the v e r t i c a l displacement of the s t e r n of the sh i p c o n s i s t s of the s u p e r p o s i t i o n of s e v e r a l harmonic f u n c t i o n s of d i f f e r e n t amplitudes and f r e q u e n c i e s as de f i n e d by a s p e c t r a l d e n s i t y f u n c t i o n . Thus, the i n p u t d i s t u r b a n c e i s somewhat more 2 p r e d i c t a b l e than the forms of v i b r a t i o n present i n the previous examples. The marine towing motion compensation system i s u s u a l l y designed to maintain constant towing cable t e n s i o n . Once t h i s requirement i s met, i t then f o l l o w s that the a c c e l e r a t i o n of the submersible w i l l be zero, and a l l u n d e s i r a b l e motion w i l l be e l i m i n a t e d . To provide t h i s c o n s t a n t cable t e n s i o n , a l l systems attempt to pay out or haul i n c a b l e as the s h i p moves up or down, thus decoupling the motion of the s h i p from the c a b l e . The index of performance o f such a motion compensation system i s the r a t i o of the amplitude of towed body t o s h i p displacement. I t i s g e n e r a l l y s t i p u l a t e d t h at the performance index must be l e s s than a given value at the frequency which c o n t a i n s the g r e a t e s t energy as taken from the sea s t a t e s p e c t r a l d e n s i t y f u n c t i o n . T h i s i s then c o n s i d e r e d the primary design frequency. In a d d i t i o n , the minimum ac c e p t a b l e index of performance f o r other f r e q u e n c i e s w i t h i n the spectrum may be s p e c i f i e d . Once the o v e r a l l p h y s i c a l c o n s t r a i n t s are met (e.g.. weights, geometries, e t c . , over which the d e s i g n e r has l i t t l e or no c o n t r o l ) the problem then becomes that of determining the p h y s i c a l c h a r a c t e r i s t i c s of the system. Some of the design c o n s i d e r a t i o n s i n c l u d e s i m p l i c i t y , r e l i a b i l i t y , i n i t i a l c o s t , and power consumption. Three of the more popular motion compensation systems are 3 shewn i n F i g . 1.1.1. although very d i f f e r e n t i n appearance, each employs a pneumatic sp r i n g i n the form of a gas accumulator to operate a h y d r a u l i c a l l y actuated p o s i t i o n e r . With systems <a) and (b), the p o s i t i o n e r i s a c y l i n d e r which c o n t r o l s the v e r t i c a l displacement of a sheave over which the cable i s reeved. System (c) uses a h y d r a u l i c winch to haul i n and pay out cable. In each case, the cable tension i s balanced by a p a s s i v e l y - a c t i n g pneumatic s p r i n g 1 . As w i l l be seen i n Section 1 . 2 , such systems, under c e r t a i n c o n d i t i o n s , can be tuned to perform adequately over a narrow frequency range. It i s p o s s i b l e to design a purely a c t i v e system, i n which a s i g n i f i c a n t amount of energy i s expended to achieve the s t a b i l i z a t i o n e f f e c t . In such a system, a transducer monitors the motion of the load and a f t e r s u i t a b l e s i g n a l processing, c o n t r o l s the flow of o i l to a h y d r a u l i c actuator. Active systems are s u p e r i o r to passive ones i n t h a t they are capable of good motion i s o l a t i o n over a wider frequency range. However, because they r e q u i r e a bulky power source and consume a l a r g e amount of energy when c o n t r o l l i n g a massive load, they are not s u i t a b l e f o r marine a p p l i c a t i o n s . To improve the performance of a passive system without a large expenditure of power, a hybrid a c t i v e - p a s s i v e system i s 1 In a passive system, the sum of the p o t e n t i a l energy i n the s p r i n g and k i n e t i c energy of the load i s conserved, apart from some d i s s i p a t i o n due to damping and f r i c t i o n . Thus, no e x t e r n a l energy i s required to operate the system. 4 2U MUTATOR ACTUATOR TOWING CABLE (o) RAM TENSIONER ACCUMULATOR (b) BOOM BOBBER ACCUM. HYDRAULIC MOTOR REEL Cc) TENSIONING WlklCM FIG. 1.1.1 MOTION COMPENSATION SYSTEMS 5 proposed. Such systems have been s u c c e s s f u l l y used to i s o l a t e small components from an environment of severe shock and v i b r a t i o n . However, as o u t l i n e d i n the next s e c t i o n , no work was found r e l a t e d to the a p p l i c a t i o n of such systems to the marine towing f i e l d . 6 1.2 State of the Art V i b r a t i o n i s o l a t i o n systems, both a c t i v e and p a s s i v e , have been widely i n v e s t i g a t e d i n the past decade. In g e n e r a l , i t appears t h a t there were two d i s t i n c t methods of d e a l i n g with the problem; one being h i g h l y t h e o r e t i c a l , and the ether being the a n a l y s i s of a p a r t i c u l a r problem. The l a t t e r method, e s p e c i a l l y i n the f i e l d of ocean e n g i n e e r i n g , has been very e m p i r i c a l i n nature, with l i t t l e or no mathematical j u s t i f i c a t i o n of the ideas presented. This s e c t i o n w i l l d i s c u s s some of the r e l e v a n t work that has been done, f i r s t , i n v i b r a t i o n i s o l a t i o n , and second, i n the f i e l d of ocean e n g i n e e r i n g a p p l i c a t i o n s . In g e n e r a l , there are two d i s t i n c t v i b r a t i o n r e d u c t i o n methods a v a i l a b l e : a b s o r p t i o n and i s o l a t i o n . 1 I s o l a t i o n i n v o l v e s p l a c i n g a r e s i l i e n t m a t e r i a l between the d i s t u r b a n c e source and the r e c e i v e r (the system to be p r o t e c t e d ) , whereas a b s o r p t i o n i n v o l v e s the attachment of an energy absorbing device to e i t h e r the source or r e c e i v e r ( F i g . .1.2.1). I s o l a t i o n can be achieved e i t h e r a c t i v e l y or p a s s i v e l y , and can be made e f f e c t i v e over a wide range of f r e q u e n c i e s . A b s o r p t i o n i s g e n e r a l l y achieved p a s s i v e l y using a spring-mass system which i s i n resonance with the source and r e c e i v e r at one p a r t i c u l a r frequency. At that frequency, the r e c e i v e r e x p e r i e n c e s no input at a l l , but t h i s 1 See Ref. (15) a RECEIV£R RECEIVER ISOLATOR S O U R C E 4— Cc) ISOLATION! ft>) ABSORPTION! ABSORBER RECEIVER SOURCE ^ FIG. 1.2.1 VIBRATION ISOLATION 6, ABSORPTION! (a) ISOLATION (b) ABSORPTION F I G - 1.2-2 PERFORMANCE C H A R A C T E R I S T I C S 8 e f f e c t i s confined to a very narrow frequency band. A l s o , u n desirable resonant peaks occur at two f r e q u e n c i e s , corresponding to the separate n a t u r a l frequencies of the r e c e i v e r and absorber. F i g . 1.2.2 i l l u s t r a t e s the performance of i s o l a t o r s and absorbers. A passive pneumatic i s o l a t o r as shown i n F i g . 1.2.3 (a) has been examined by Cavanaugh l. He solved the l i n e a r i z e d t h i r d order system equations i n the freguency domain, and found the optimum c r i t i c a l damping r a t i o i n terms of the tank to c y l i n d e r volume r a t i o . F i g . 1.2.3(b) shows the frequency response of the system, and F i g 1.2.3(c) shows the c r i t i c a l damping r a t i o f u n c t i o n which y i e l d s the smallest maximum amplitude r a t i o . Another passive i s o l a t i o n system d i r e c t l y a p p l i c a b l e to automobile suspensions has been examined by Thompson2. He considered a two-dimensional l i n e a r system with four degrees of freedom, and developed an optimum performance index based cn r i d e comfort and road-holding a b i l i t y . A more general approach to op t i m i z i n g passive suspensions has been presented by Hedrick f o r use i n the design of high speed tracked v e h i c l e s 3 . An optimum passive shock i s o l a t o r , which uses a v a r i a b l e f r i c t i o n element to d i s s i p a t e energy has 1 See Ref. (5) 2 See Refs. (20) and (22) 3 See Ref. (7) * See Ref. (11) 9 LOAD G.AS TANKS PNEUMATIC- CYLINDER A - U Ca) CONFIGURATION! o i FREQUENCY (b) PERFORMANCE (c) OPTIMUM CRITICAL DAMPING RATIO N = TANK VOLUME CYLINDER VOLUME FIG. 7.23 PASS IVE P N E U M A T I C SYSTGLM 10 been proposed by Mercer and Rees*. A c t i v e systems f o r shock and v i b r a t i o n i s o l a t i o n have a l s o been examined. Soliman proposed a s e r v o v a l v e c o n t r o l l e d pneumatic system u s i n g displacement and v e l o c i t y feedback to c o n t r o l the s e r v o v a l v e 1 . Thompson c o n s i d e r e d a c t i v e systems f o r automobile s u s p e n s i o n s 2 . P o r t e r , Athans, and Karnop a l l presented h i g h l y mathematical methods f o r d e a l i n g with l i n e a r a c t i v e systems 3, K r i e b e l developed an a c t i v e system for shock i s o l a t i o n * . None of the above-mentioned work i s i n a form which i s r e a d i l y a p p l i c a b l e to the problem of motion compensation i n the marine environment. The t h e o r e t i c a l s o l u t i o n s r e l a t e mostly to l i n e a r systems, while the more p r a c t i c a l s o l u t i o n s are too s p e c i f i c and r e q u i r e much m o d i f i c a t i o n t c make them u s e f u l f o r other design purposes. There are a number of marine motion compensation systems (mostly passive) o p e r a t i o n a l around the world, but l i t t l e documentation e x i s t s t o help p r e d i c t a system's performance before i t i s b u i l t . Most systems c o n s i s t of a pneumatic s p r i n g , as d e s c r i b e d i n S e c t i o n 1.1, and are c l a s s e d as v i b r a t i o n i s o l a t o r s . Keefer proposed a simple manner i n which an i s o l a t o r 1 See Refs. (17) and (18) 2 see Ref. (21) a See Refs. (13), (1) and (8) * see Ref. (10) 11 can be made i n t o an absorber by making the d i r e c t i o n of moticn of the compensator mass orthogonal to the i n p u t d i s t u r b a n c e 1 . F i g . 1.2.4(a) i l l u s t r a t e s a ram t e n s i o n e r i n such a c o n f i g u r a t i o n . Such a system i s tuned so t h a t the anti-resonance occurs at the freguency which c o n t a i n s the dominant amplitude of v i b r a t i o n . F i g 1.2.4(b) shows the t y p i c a l performance of a tuned system. Note t h a t damping i n c r e a s e s bandwidth at the expense of the system's a t t e n u a t i o n . Buck 2 and S u t h e r l a n d 3 suggested the use of a c t i v e systems, but n e i t h e r has developed a complete a n a l y s i s of such a system, nor suggested a method of p r e d i c t i n g the performance of a r e a l , n o n l i n e a r system. In a d d i t i o n , they have not r e c o g n i z e d the f a c t t h a t power consumption can be reduced by i n c o r p o r a t i n g a passive system to support the s t a t i c weight of the load while the a c t i v e system i s used s o l e l y f o r motion compensation. 1 See Bef. (9) 2 See Ref. (3) 3 See Ref. (19) 12 (cO GENERAL ARRANGEMENT F R E Q U E N C Y (b) PERF ORMANCE FIG. 1.2.4 TUNED RAM T E N S I O N E R 13 1.3 Objectives and Scope of Project The objectives of t h i s project are f i r s t , to study the dynamics of an active-passive motion compensation system for marine towing, and second, to use the results of this study to develop guidelines f o r use in designing r e a l systems. These objectives are accomplished by proceeding i n six steps: 1. Representing a t y p i c a l system i n a form which closely approximates r e a l i t y , yet which lends i t s e l f to mathematical analysis and simulation. 2. Developing the mathematical model, including such no n l i n e a r i t i e s as hydrodynamic drag and dry f r i c t i o n . 3. Linearizing the mathematical equations and conducting a frequency-domain analysis to obtain a f i r s t approximation of the important system parameters. 4. Developing a d i g i t a l computer simulation program which w i l l validate and optimize the parameters derived in step #3, 5. Constructing a small working model to check the v a l i d i t y of the mathematical and simulation models. 6. Relating the re s u l t s of the foregoing to the design of r e a l systems. 14 CHAPTER I I THEORETICAL ANALYSIS 2 . 1 T y p i c a l System Some of the commonly used passive motion compensation systems were shown i n F i g . 1.1.1. I t i s p o s s i b l e to devise an a c t i v e - p a s s i v e system by adding an a c t i v e a c t u a t o r p a r a l l e l to the p a s s i v e one. In the case of the ram t e n s i o n e r and boom bobber, t h i s means adding a second c y l i n d e r , while f o r the t e n s i o n i g winch, adding a second h y d r a u l i c motor. The three c o n f i g u r a t i o n s shewn i n F i g 1.1.1 are' very s i m i l a r mathematically. The s i m i l a r i t y between (a) and (b) i s obvious — only the mechanical advantages of the reeving, i n the case of (a), or the boom, i n the case of ( b ) , are d i f f e r e n t . In the case of the winch, the h y d r a u l i c motor i s e q u i v a l e n t to a number of c y l i n d e r s connected i n p a r a l l e l , and hence can be modelled as a s i n g l e c y l i n d e r . Thus, f o r the purpose of t h i s p r o j e c t the ram t e n s i o n e r i s s e l e c t e d as a t y p i c a l system. The o v e r a l l c o n f i g u r a t i o n of the t y p i c a l system i s shown i n F i g . 2.1.1. The p a s s i v e subsystem i s the same as before, while the a c t i v e subsystem c o n s i s t s cf a h y d r a u l i c c y l i n d e r c o n t r o l l e d by an e l e c t r o h y d r a u l i c s e r v o v a l v e . The c o n t r o l system c o n s i s t s of a c c e lerometers mounted on the towing and ram sheaves, whose s i g n a l s are processed and fed to the s e r v o v a l v e . The s i g n a l p r o c e s s i n g network i s the most v i t a l component of the system, FIG. 2,1.1 A C T I V E / P A S S I V E R A M T E N S I O N E R 01 16 and w i l l be the s u b j e c t of thorough a n a l y s i s . As pointed out e a r l i e r , the l o a d can be a d i v i n g b e l l suspended from a s t a t i o n a r y s h i p , a submerged body towed at high speed (in the order of 10 knots), or a s u r f a c e v e s s e l such as a barge. For the case of a submersible supported from a shi p which i s not moving h o r i z o n t a l l y with r e s p e c t to the water, the c a b l e can be r e p r e s e n t e d as a one-dimensional e l a s t i c l i n k whose l o n g i t u d i n a l a x i s i s v e r t i c a l . In the case of a moving s h i p towing a barge or submersible, the cable w i l l assume a complex t h r e e - d i m e n s i o n a l curve. Since t h i s p r o j e c t i s concerned p r i m a r i l y with the behaviour of the motion compensation system, the t y p i c a l system c o n s i d e r e d w i l l i n c l u d e the one-dimensicnal c a b l e . A p p l i c a t i o n of the approach to the case of t h r e e - dimensional c a b l e , as developed by Walton and Polachek, i s d i s c u s s e d i n Chapter V. 1 7 2.2 The Equivalent Model To f a c i l i t a t e the analysis of the ram tensioner described i n Section 2.1, the following s i m p l i f i c a t i o n s w i l l be made: 1 . The s t a t i c tension in the cable due to the submersible's weight i s not considered. This s i m p l i f i c a t i o n does not af f e c t the dynamics of the motion compensator. 2. The cable i s considered to be a one-dimensional e l a s t i c l i n k for the reasons set forth in Section 2.1. 3 . The passive subsystem i s considered to be purely pneumatic. This r e s t r i c t i o n actually increases the complexity of the problem, but i s included to demonstrate the method of application of the compressible f l u i d flow eguations. In many applications, the passive system would actually be an " a i r - o v e r - o i l " , or hydropneumatic system, as shown in Figure 1 . 1 . 1 . Using these s i m p l i f i c a t i o n s , i t i s possible to model the chosen system as shown i n Figure 2.2,1. The form shown i n Fi g . 2.2.1 was devised to f a c i l i t a t e mathematical analysis and test- model construction. The active and passive motion compensation SERVO V A L V E G A S B O T T L E S X DAMPING VALVES o n i l l X BYPASS VALVE MM $•3 C O N T R O L S Y S T E M r —f-y- j l _ ACCELEROMETERS CABLE PASSIVE CYLINDER ACTIVE CYLINDER U FIG. 2.2.1 E Q U I V A L E N T S Y S T E M 19 cylinders are placed horizontally on a carriage, with t h e i r piston rods connected sc as tc function i n p a r a l l e l . The carriage i s driven horizontally i n a sinusoidal manner, with the desired frequency and amplitude, simulating the v e r t i c a l motion of the ship. The load i s modelled by a second carriage containing the desired mass, connected to the motion compensation piston rod by means of a spring which i s assumed to model the cable. The entire system i s similar tc the r e a l case of F i g . 2.1.1 except that a l l motion i s horizontal instead of v e r t i c a l . As a r e s u l t , the s t a t i c weight of the towed body i s not considered. Therefore, i n modelling the spring c h a r a c t e r i s t i c of the passive system, i t i s necessary to pressurize both sides of the passive cylinder, such that the net s t a t i c force at the piston red i s zero. In general, t h i s equivalent system accurately models the motion compensation system, but does not f u l l y consider the dynamics of the cable and towed body. However, the design method developed here i s f l e x i b l e enough to accomodate these additions i f the necessary parameters are available to the designer. 20 2. 3 The Passive System The passive side of the system under consideration consists of a pneumatic ram, with each end connected via a t h r o t t l i n g valve to a receiving tank (Fig. 2.3.1). The t h r o t t l i n g , valves are used to introduce damping into the system. The mass flow to and from a tank or cylinder i s derived i n Appendix A, and i s given by R i s the gas constant for the particular gas used, T i s the absolute temperature, V i s the tank or cylinder volume, P i s the absolute pressure, and i i s the r a t i o of s p e c i f i c heats. Because the receiving tanks have fixed volumes, and the mass flows are proportional to the negative of the pressure changes, we can write (2. 3. 1) where m i s the mass flow rate, (2.3. 2)  22 and m oc - Pt (2.3.3) where Vt i s the tank volume and Pt i s the tank pressure. Substituting (2.3.2) and (2.3.3) into (2.3.1) gives , VM p " 7RTT7 *' (2.3.4) ™ - - V t * P where the subscripts 1 and 2 refer to the l e f t and right hand sides of the passive system, respectively. In general, the volume of the receiving tanks, and hence the volume of f l u i d contained, i s large compared to the mass flow in and out of the tanks; thus the temperature of the gas can be considered constant, 1 i . e . 1 For an actual system having an e f f e c t i v e piston area of 13.46 square inches, a displacement of f i v e feet as measured at the tow point causes a change in absolute temperature of only 1.9%, assuming adiabatic compression or expansion (worst case). 23 r R To • i n g . 2 .3 .2 BLOCK DIAGRAM of TANK DYNAMICS FIG. 2 . 3 . 3 BLOCK D I A G R A M of CYLINDER DYNAMICS 24 Ti = = T„ (2.3.5) F i g . 2.3.2 i s a block diagram of the tank flow equations. The mass flow to and from the cylinders i s also expressed by (2.3.1), except that: V z - V C 2 - A2^/ (2.3.6) V, = Atj V 2 = -A2j (2. 3.7) VV1, oC P, (2.3.8) where Vn and V2 are the volumes of the l e f t and right sides of the c y l i n d e r , VC| and V C 2 are the i n i t i a l values of V r and V z, flt and A 2 are the e f f e c t i v e piston areas, and y i s the piston displacement. 25 Substituting (2.3.6), (2.3.7) and (2.3.8) into (2.3.1) gives I Z ' 1 1 J J K-l, (2. 3.9) I £ T 2 Due to the r e l a t i v e l y small volume of the cylinder and the large variation in pressure, temperature change i s nc longer n e g l i g i b l e . Using the equation of state for an i d e a l gas, p. y. T; = — 1 — L ( i * l , 2 ) (2.3.10) where m-{ i s the mass of gas i n the i - t h side of the cyl i n d e r . Substituting (2.3.10) into (2.3.9) gives r ± A + A , i ' _ m f I Pz A z . "j (2.3.11) Fig.. 2.3.3 shows the block diagram of the the cylinder equations. The mass flow through each t h r o t t l i n g valve i s derived in 26 Appendix A and i s given by (2.3.12) where C 0 i s the valve constant, P u i s the upstream pressure, Pd i s the downstream pressure, and T u i s the upstream temperature. I t i s observed that the d i r e c t i o n of mass flow i s from high to low pressure; that i s , P u, the upstream pressure, i s the greater of P t t r V A . By the convention shown i n F i g . 2.3.1, i f P^ > Pt then ml> 0 ; conversely, when Pti < Pt , < 0. Consequently, i n s o l v i n g (2.3.12) the upstream end must f i r s t be determined. Then the c o r r e c t a l g e b r a i c sign can be assigned to m-. F i g . 2.3.4 shows the block diagram of the valve eguaticns. The equations (2.3.4), (2.3.9) and (2.3.12) must be solved simultaneously to y i e l d P, and P z given a piston displacement y. The f o r c e generated i n the ram can then be found: (2.3. 13) where Fp i s the ram f o r c e . F I G . 2 .3 .4 . B L O C K D I A G R A M of V A L V E D Y N A M I C S 28 The block diagram of the p a s s i v e system dynamics i s shown i n F i g . 2.3.5. Si n c e i t i s d i f f i c u l t to s c l v e t h i s system of equations a n a l y t i c a l l y , a numerical s o l u t i o n i s now developed. The s t r a t e g y used i n the numerical s o l u t i o n i s as f o l l o w s : 1. C a l c u l a t e P t i as shown i n F i g . 2.3.2 2. C a l c u l a t e P-, as shown i n F i g . 2.3.3 3. C a l c u l a t e m ;as shown i n F i g . 2.3.4. 4. Repeat steps 1 t o 3 with the new value o f m; u n t i l Pt. and P-, no longer change from i t e r a t i o n to i t e r a t i o n . 5. C a l c u l a t e the ram f o r c e from (2.3.13). Mi T A N K . * 1 Pt, C Y U M D E R . S I D E * I F IG. 2.3.5 B L O C K D I A G R A M of P A S S I V E S Y S T E M D Y N A M I C S 30 2.4 The Active System The active side of the system under consideration consists of a positive displacement hydraulic pump, r e l i e f valve, gas accumulator, servo-valve, and hydraulic cylinder. (Fig. 2.4.1) It i s assumed that the pump discharge flow rate always exceeds the system requirement, hence maintaining the pressure in the accumulator equal to the r e l i e f valve setting. In f a c t , the accumulator pressure w i l l vary s l i g h t l y with changes i n flow rate due to f r i c t i o n losses i n the hydraulic l i n e s , but t h i s f l u c t u t i o n i s ne g l i g i b l e compared to the working pressure. Therefore, the supply pressure i s considered constant. The compressibility e f f e c t of the hydraulic f l u i d i s examined in Appendix C, and,is found tc contribute an error i n flow of only 2.6% for a t y p i c a l f u l l - s c a l e system. Therefore, the compressibility of the f l u i d i s not considered. The flow-pressure r e l a t i o n s h i p for the servo-valve i s given by the manufacturer for selected values of actuating signal, z (Fig. 2.4.2). As shown i n Appendix B, thi s r e l a t i o n s h i p , for a zero-lapped v a l v e 1 , can be accurately modelled by 1 See Ref. (6) for the equations of under- and over-lapped servo-valves. GAS OIL FROM PUMP -> ACCUMULATOR TO TANK C - > — 11 ACTUATING SERVO V A L V E . CURRENT" FA FIG. 2.4.1 ACTIVE S Y S T E M AP F1 Gr 2.4. 2 SERVOVALVE CHAR ACT ERISTICS 32 (2. 4. 1) w her 6 Q v i s the volume flow through the valve. AP i s the pressure drop across the load. C s v i s the c h a r a c t e r i s t i c constant of the servo- valve , Ps i s the supply pressure, assumed constant, and z i s the actuating s i g n a l . Leakage across the cylinder i s often useful in s t a b i l i z i n g a servo-system, and i s therefore included in the analysis. Leakage i s provided by means of an au x i l i a r y path around the piston, and controlled by means of a valve. The leakage flow, QL, i s given by C v i s the c h a r a c t e r i s t i c constant of the valve. The t o t a l flow into the ram, QA, i s the difference between the flow through the servo-valve and the leakage flow: (2.4. 2) where 33 The sign convention i s such that Q A i s positive when i t causes the piston to move to the right. Substituting (2.4.1) and (2.4.2) into (2.4.3) gives The velocity of the piston with respect to the cylinder, y, can now be expressed as where a A i s the e f f e c t i v e area of the piston. The force available to do work at the end of the piston rod, F A, i s given by FA - A A A P (2.4.6) Equations (2.4.5) and (2.4.6) can now be combined to y i e l d 34 y d i r e c t l y as a f u n c t i o n of F : (2.4.7) In computing (2,4.7) i t i s necessary to i n t r o d u c e an a r t i f i c i a l s i g n assignment to avoid negative values w i t h i n the surds. T h i s i s done by n o t i n g t h a t 1 . In the case of flow through the s e r v o - v a l v e , P s must take the s i g n of z. The surd then becomes 2. In the case of the leakage flow, the surd takes the s i g n of F A , which i s a c t u a l l y the d i r e c t i o n of pressure drop. The surd thus becomes S ; 9 n ( F A ) y | F * / ^ | ( 2 - " - 9 ) S u b s t i t u t i n g the modified e x p r e s s i o n s (2.4.8) and (2.4.9) i n t o (2.4,7) gives the equation f o r computing y from F A c o n s i s t e n t with the s i g n convention: 35 a- ct "/Ms-*/* I - ,2-<-,o) Fig. 2.4.3 shows the block diagram of the active system. 2 >• SICM ( 5 ABS ABS — 3 * - AA x: x < SI6M FIG. 2 . 4 . 3 BLOCK DIAGRAM of A C T I V E S Y S T E M 37 2.5 Active-Passive System The t o t a l ram force i s the sum of the forces exerted by the passive and active cylinders, less f r i c t i o n : F t o B - FP + F . - f ' ( 2 . 5.D where FR*M i s t n e r a m force, F A i s the active cylinder force, F p i s the passive cylinder force, and f i s the f r i c t i o n f c r c e , as discussed i n Appendix E. The force f e l t by the cable i s d i r e c t l y proportional to F R A M , where the constant of proportionality i s the rec i p r o c a l of the mechanical advantage of the reeving: F = — Fo <2'5* 2) "NET J/ »RAM where KMfl i s the mechanical advantage of the reeving, and F^ i s the force acting on the cable. The towed body can be represented by a mass K, subjected to hydrodynamic drag and towing cable tension. (Fig. 2.5.1) The cable i s assumed to be a massless li n e a r spring. The compensator force causes an elongation of the cable 3 8 = = D > | k / V A BODY CABLE —*- Xi > % DRAG FIG. 2.5.1 CABLE /BODY MODEL FIG. 2.5.2 BLOCK DIAGRAM of CABLE. /BODY DYNAMICS 39 according to the r e l a t i o n fwT = ICC-X, - 7c) (2.5.3) The cable then applies the same force to the body, whose motion can be described by: FKET = M X + C % 2 (2.5.4) where M i s the mass of the towed body, and C H i s the hydrodynamic drag factor. Equating (2.5.3) and (2.5.4) and rearranging, gives the nonlinear d i f f e r e n t i a l equation of motion of the body: * + % + ^ X = W ' X ' ( 2 . 5 . 5 ) where 60c i s the cable-mass natural frequency, ^ Ke/M" . Once (2.5.5) i s solved, i t i s possible to find F M 6 1 by direct application of (2.5.4) or (2.5.3). The block diagram of the towed body and cable system i s shown in F i g . 2.5.2. The absolute displacement of the shipboard end of the 40 cable, x„ i s the sum of the input, u, and the displacement of the end of the cable with respect to the input, y c: X, = U + vjc (2.5.6) For the case where the actuator acts on the cable through a mechanical advantage (e.g., the ram tensioner of F i g . 2.1.1), the motion of the cable with respect to the ship's stern can be expressed as: y c = I ^ A y <2-5-7> w here y i s the extension of the actuator. Equations (2.5.6) and (2.5.7) are combined with the block diagrams of the passive, active, and cable-mass systems (Figs. 2.3.5, 2.4.3, and 2.5.2) tc give the block diagram of the entire system, as shown i n F i g . 2.5.3. CABLE/BODY FIG. 2.5-2 K MA PASSIVE SYST. FIG. 2.3.5 ACTIVE SYST- FIG. 2-4.3 CONTROL SYST. FIG- 2-6 2 - — FIG. 2.5-3 BLOCK DIAGRAM of ACTIVE-PASSIVE SYSTEM 42 2.6 The Control System The control system generates the signal tc operate the active actuator by means of monitoring and processing cer t a i n variables. The controlled variable can be considered as either x O R F»IET r and Figure 2.5.2 suggests that both can be controlled simultaneously because they are l i n e a r l y dependent. This means that the index of performance, as discussed in Chapter I , can te either F H E T /u or x/u, and both must be made tc f a l l below certai n specified l i m i t s for acceptable operation. In addition, the control system must ensure that the long term average motion of the actuator piston does not d r i f t from the centre of the cyli n d e r . The controlled variable (F N C T or x) can be used to generate the primary actuating signal for the servovalve. This constitutes a simple feedback control system where the reference input is,zero, as shown in Fig. 2.6.1. (The passive system i s omitted from Fig. 2.6.1 for c l a r i t y . ) The "Control Elements" block may contain f i l t e r s , integrators, etc., as required for best operation. In addition tc feedback, i t may be desirable tc include a portion of the disturbance input, u ( t ) , as indicated by the feedforward loop in Fig. 2.6 .1. The piston centering control i s a miner loop used to restore the piston to the centre of the actuator slowly with 4 3 FEEDFORWARD CONTROL ELEMENTS TRANSDUCER SERVO V A L V E ACTUATOR SHIP MOTION U(i) CONTROL ELEMENTS TRANSDUCER L O A D CENTERING CONTROL ELEMENTS TRANSDUCER FEEDBACK FIG. 2.6.1 ACTIVE SYSTEM WITH CONTROL BLOCKS U(s) HFFfe) 1 V(s) X(s) K SY 2(s) F I G . 2 . 6 . 2 BLOCK D I A G R A M of CONTROL S Y S T E M 44 respect to the forcing freguency. That i s , the time constant of the loop i s at least one order of magnitude greater than the r e c i p r o c a l of the input freguency. The choice between using F N t T and x as the controlled variable i s made based on the a v a i l a b i l i t y and s u i t a b i l i t y of transducers. In either case, however, i t i s necessary to provide a physical path in the towing cable for the feedback s i g n a l to reach the surface vessel. If t h i s i s not possible, then the absolute motion of the shipboard end of the cable, x t, can be used as the feedback s i g n a l . Such a strategy i s e a s i l y feasible i n the case of the ram tensioner and boom-bobber (Fig. 1.1.1 (a) , (b)) , but net so for the constant tension winch (Fig. 1.1.1(c)). Fcr demonstration purposes, i t i s assumed that x i s the feedback variable. The control system elements are now lumped into three blocks, as shown in F i g . 2.6.2. The t o t a l control voltage, B (s) i s given by (in Laplace notation): R(Y)= M fBCs)X(s) + WFFCs)\J(s) + U,(s)Y(s) (2.6.1) where Hpj,(s) represents the feedback element, Hpp(s) represents the feedforward element, and (s) represents the piston centering element. The servovalve actuating current i s obtained by passing the control voltage through a power amplifier Z(s) = J<svR(s) where Z (s) i s the actuator current, and K w i s the amplifier gain. 46 2.7 Computer Simulation The active-passive system as depicted i n F i g . 2.5.3 has been modelled by means of a "Continuous Systems Modelling Program" (CSMP), an IBM product for use on their System/370. The programming language consists of a number of functional blocks such as integration, d i f f e r e n t i a t i o n , etc., in addition to a l l the usual mathematical functions available in Fortran. These blocks are assembled in much the same manner as an analogue computer network, but without the inconvenience of scaling variables. Integration can be performed using any of f i v e d i f f e r e n t b u i l t - i n routines; the one employed in t h i s project i s fourth order Bunge-Kutta with fixed integration i n t e r v a l . This method was selected because i t i s found to be the least expensive for the degree of accuracy required. The l o g i c flow of the program i s exactly as depicted in Fig. 2.5.3. The l i s t i n g i s shown in Appendix F.1. 47 CHAPTER I I I LINEAR ANALYSIS The equations of the a c t i v e / p a s s i v e motion compensation system developed i n Chapter I I are d i f f i c u l t to handle without extensive use of computers. In order . that the designer can q u i c k l y gain a f e e l f o r the problem and thereby e s t a b l i s h f i r s t approximations f o r important parameters, I have s i m p l i f i e d the equations to permit a f a s t approximate s o l u t i o n . The s i m p l i f i e d approach uses l i n e a r i z e d equations and a frequency-domain s o l u t i o n . 3.1 L i n e a r i z e d Passive System In l i n e a r i z i n g the passive pneumatic system, I have assumed tha t the changes i n pressure w i t h i n the c y l i n d e r and tanks are l i n e a r with respect to the p i s t o n displacement, y, and that y i s too s m a l l to a f f e c t the temperature i n the system. This leads to the f o l l o w i n g : 1 . S=Ay: i . e . , the p e r t u r b a t i o n i n p i s t o n displacement i s s m a l l , and denoted by S. 2. P 1+P 2=2P 0; i . e . , the average pressure i n the c y l i n d e r i s constant, and equal to the quiescent pressure P 0 . 3. P1:=-Pa; i . e . , the r a t e of increase of pressure on one s i d e of the c y l i n d e r i s equal to the rat e of 48 decrease on the other. 4. The temperature throughout the passive system i s constant and equal to T<>. Furthermore, the geometry of the system i s assumed to be symmetric, which leads to the following: 5. A t+A Z=2A P; i . e . , the e f f e c t i v e piston areas are averaged to a constant Ap. 6. Vc, + V C i=2V c; i . e . , the tank volumes are averaged tc a constant Vc . The gas flow equations are li n e a r i z e d by f i r s t considering the mass flow through the t h r o t t l i n g valve as given in ( 2 . 2 . 1 2 ) , and restated here: where (3.1.1) Considering the upstream pressure, P w , as constant and equal to the guiescent pressure P c, and small variations in pressure drop (Pe -Pa ), equation (3.1.1) can be linearized into 4 9 the form: A w - # Cr A ( P . - P a ) (3. 1. 2) where 9w- 5>v̂  3Cfl/P.) S>CPJ/PO) 3 ( P o - ^ ) (3. 1.3) The flow through the valve w i l l be positive for the half cycle when the piston moves one way, and negative for the other half. Therefore, (P0"P<i) c a n assume positive or negative values. It i s thus necessary to select m0= (P e-P d ) =0 as the equilibrium point about which perturbations are considered. However, t h i s w i l l y ield i n f i n i t e value for C r, since the slope of the m vs (P0 -Pa ) curve, as shown in Fig. 3.1.1, i s v e r t i c a l at P o-P d=0. It i s therefore more reasonable to select values of Va, Pj which coincide with some average operating condition, for example the root-mean-sguare value. The equilibrium point, however, i s s t i l l the o r i g i n . Renaming P e and Pd to correspond to the tank and cylinder pressures, the l i n e a r i z e d mass flow equations are: 50 = C r f Pt,- P.) (3. 1.4) Equation (3.1.4.) i s plotted together with i t s nonlinear form in F i g . 3.1.1. The cylinder flow equations are considered next. Incorporating the s i m p l i f i c a t i o n s of geometry as discussed above, equation (2.2.9) becomes: v '+ A ' s p, + P. (3. 1. 5) The net mass flow into the cylinder i s given by (3. 1. 6) Incorporating the approximations of pressure as described abcve, and introducing the d i f f e r e n t i a l operator B= ~ t, equation (3.1.6) can be rewritten as: FIG 3,1.1 PRESSURE - FLOW CURVE {or TMROTTLIKJG V A L V E S 1 + 2 s/u)A 5 F P N+l L 1 + ' N + t S J F IG. 3.1.2 LINEARIZED PASSIVE SYSTEM TRANSFER FUNCTION! 52 (3. 1.7) F i n a l l y , the flow out of the receiving tanks i s considered. Equation (2.2.4) can be rewritten, in d i f f e r e n t i a l operator notation, as: DP*. Vi (3.1.8) Solving (3.1.4.) for Ptl and p t i and substituting into equation (3.1.8), the net flow into the cylinder can be expressed as: /. , x • v t r PCP.-PQ 1 (3. 1.9) Equating (3.1.9) and (3.1.6) and solving for P-P gives P. - P 2Y A P P» (3.1.10) I + Vt /Vc v t — D The force exerted by the ram i s then given by 53 FP = Ap ( p , _ P z ) (3.1.11) and the o v e r a l l transfer function of ram force tc piston displacement i s ^ _ 2 y P o A P (3.1.12) L 1 T TC R R T . U J Lumping parameters and introducing the Laplace operator, s, in place of the d i f f e r e n t i a l operator, D, the ov e r a l l transfer function becomes N + 1 1 + 2 s 1 + (3. 1. 13) where K * = 2 V y f c A ' = Static s t i f f n e s s , N=Vt/Vt= tank to cylinder volume r a t i o , A N D 2 S / C J N = Y C V R T . The parameters w„and 3 are the natural freguency and c r i t i c a l damping r a t i o , respectively, and are ultimately a function of the mass which the system must control. In pa r t i c u l a r , 5% [W7 W „ - 7 H , J "" 2 r c V E T 0 ( 3 . 1 . 1 4 ) The linearized transfer function Gp(s) i s now extended to cover the f u l l range of piston displacement, y, such that F P = G p G O i , ( 3 .1 .15 ) The transfer function i s shown i n block diagram form i n Fig. 3 . 1 . 2 . Introducing the time constants f , = 2 % * ; T a » — ( 3 . 1 . 1 6 ) the transfer function may be restated as Cf (i) ( 3 . 1 . 1 7 , 55 3.2 Linearized Active System The equation of the active system, as presented in Section 2.3, i s restated here: 9 = ̂ z v̂  -  ? a / a a ~ x V /̂A* (3.2.D Equation (3.2.1) i s to be l i n e a r i z e d about some operating point (Yo »z 0 ,FA(>) . Perturbations about that point can be represented, by A-notation, as Ay = (A., A * T- A^Ap;) + A 3 AFA (3.2.2) where (3. 2. 3) ^* S f )SETV6 ~ ^ A ^ r T T 7=—71—1 (3.2.4) ..Cy ^ S - U F J ^ ~ ZAZ JKJA* (3,2'5) The constants A , and X 2 are termed the "flew gain" and > "flow-presure c o e f f i c i e n t " of the servovalve, respectively, and 56 ^ 3 the "flow c o e f f i c i e n t " of the bypass valve. F r i c t i o n i s net considered in the l i n e a r analysis because of the discontinuity a t y= 0. Equation (3.2.2) i s v a l i d only for perturbations about the operating point (y e ,z 0 ,FAo) . However, during normal operation of the valve the operating point can t r a v e l in a band spanning both the negative and positive regions of j , z , and F A. The l i n e a r i z e d equation then becomes inappropriate i n i t s present form. It i s proposed that the perturbations y, z, and FA be centred about the o r i g i n , i . e . , y, =z„ =FAo=0, but that ">3 be calculated about a root-mean-square point using equations (3.2.3) to (3.2.5.). Equation (3.2.2) can thus be rewritten as y = + F A (3.2.6) and (Yo# zof FAo) c a n ^e considered as the BHS operating point. The e f f e c t of l i n e a r i z i n g the servovalve equation i s to change the family of parabolae to one of straight l i n e s , as indicated in F i g . 3.2.1. The l i n e a r system thus developed i s found to model the active system adeguately over the entire operating range. The transfer function i s shown in block diagram form in Fig. 3.2.2. 57 ~FI6.. 3.2.1 LINEARIZED SERVOVALVE C H A R A C T E R I S T I C S BYPASS VALVE r- CYLINDER • A —————— A FORCE -* SERVO-ACTUATING SWJWAL A , S E R V O V A L V E PISTON VELOCITY FIG, 3.2,2 LINEARIZED ACTIVE SYSTEM TRANSFER FUNCTION! 58 3.3 L i n e a r i z e d A c t i v e - P a s s i v e System The l i n e a r t r a n s f e r f u n c t i o n s of the a c t i v e and passive systems are combined i n the same manner as i n S e c t i o n 2.4. It w i l l be necessary, however, to f i r s t l i n e a r i z e the body dynamics t r a n s f e r f u n c t i o n . R e c a l l i n g equation (2.5.4) F N & T * M X + C w X (3.3 .1) i t i s seen t h a t o n l y the damping term i s n o n l i n e a r . T h i s can be l i n e a r i z e d about i=0, g i v i n g F N e 1 - M * + C t X (3.3.2) where CL=Crt | | 7 = 2 C H i e and x, i s taken as the root-mean-sguare v e l o c i t y . C L i s then the l i n e a r i z e d drag c o e f f i c i e n t . The d i f f e r e n t i a l equation of (2.6,4.) can now fee l i n e a r i z e d to g i v e - X + ^ "X + C0c X = X , (3.3.3) The t r a n s f e r f u n c t i o n of the compensator motion x 1 # t c body motion x can then be expressed, i n Laplace n o t a t i o n , as 59 *. 1 + 2 % » . s + s7«2 <3-3-'" where i c 2-MtOt and F c (s) i s the cable transfer function. The transfer function of body displacement, x, tc cable tension, F H f r t , i s derived from (3.3.2) FB(s) = % = M s S C L S 0.3.5) The block diagram of the l i n e a r i z e d cable and body dynamics i s shown i n Fig. 3.3.1. The block diagram of the linearized motion compensation system i s shown in Fig 3.3.2. F 16, 3.3.1 LINEARIZED CABLE/BODY TRANSFER F UMCTION Ny(s) + Ksv z X i FIG. 3.3-2 BLOCK DIAGRAM of L I N E A R I Z E D A C T I V E - P A S S I V E S Y S T E M 61 3.4 Performance Analysis and Optimization As a f i r s t approximation to system performance, the linear model derived i n Section 3.3 w i l l be examined in the freguency domain. It w i l l be most convenient to use the r a t i o of body to surface ship displacement, x/u, as the c r i t e r i o n of performance. Due to the l i n e a r i t y of the system, a minimum x/u i s equivalent to minimum variation in FNE.T /U. Solving the block diagram of Fig 3.3.2 yields the o v e r a l l transfer function: X6) _ FcCs) Uts) | + W(s) (3'a*1) where H (s) i s the open-loop transfer function, given by: In designing an active-passive system, the absolute value of the closed loop transfer function, |TJ(S)|, i s t c be minimized within the range of operating frequencies. This in turn yields H (s), the closed loop transfer function, a maximum. In general, the designer has no control over the cable and body dynamics, which are represented by F c (s) and F 6 ( s ) . Furthermore, he has very l i t t l e control over the passive system. 62 Gp.(s) , since the primary design c r i t e r i o n for that system i s to be able to carry the s t a t i c weight of the towed body.1 Thus, in maximizing H (s) , i t i s necessary to design H F 8 ( s ) , H F F (s) , Hy(s), and to select a suitable servo-valve and hydraulic cylinder as represented by A| and 3.3.1 The Feedforward Element The f i r s t step in maximizing H{s) i s to minimize i t s denominator. Ideally, i t i s set to zero, which yields = " ^ u  S ^ ^ X ^ W l ^ V Uy£0l (3.4.3) As pointed out e a r l i e r , Hy (s) i s the ram centering network. Because i t i s r e l a t i v e l y slow-acting i t has l i t t l e or no effect at operating frequencies. Thus, i t can fce deleted from (3.4.3). The feedforward compensator can now be given as H f ^ ) - - r r j — , [s + ( V W o ] (3-'-'" 1 The s t i f f n e s s of the passive system i s usually specified i n order that the active-passive system he capable of operating i n a purely passive mode when working i n a low sea state, or in case of a power f a i l u r e i n the active system. 6 3 3.4.2 The Feedback Element The second step i n maximizing H (s) i s t c maximize the numerator. An examination of (3.4.2) r e v e a l s t h a t t h i s can be done by s e t t i n g the gain of H r a ( £ ) as l a r g e as p o s s i b l e ; however, the s e r v o v a l v e s a t u r a t e s when the a c t u a t i n g c u r r e n t , z, exceeds a c r i t i c a l value, z 0 . T h e r e f o r e , H F B (s) must be optimized with r e s p e c t to the above c o n s t r a i n t . As i s customary i n p o s i t i o n s e r v o s , the feedback element H F B (s) i s assumed to be a combination of displacement, v e l o c i t y , and a c c e l e r a t i o n feedback. Thus, K, + I4S + SZ (3.4.5) where K j , K z and K3 are c o n s t a n t s . T h i s y i e l d s a s e r v o - a c t u a t i n g c u r r e n t z, given by ZM = KSV[HFF6)U(S)T- WpB(OX(sT>] (3.4.6) (The ram c e n t e r i n g network, Hy(s), i s again ignored because i t has n e g l i g i b l e e f f e c t at o p e r a t i n g f r e q u e n c i e s ) . The amplitude of the a c t u a t i n g c u r r e n t , |z<s)|, i s now c o n s t r a i n e d to be l e s s than or equal to the c r i t i c a l v alue, z e : 64 |Z(s)| - Ksv| HFF(s)U(s) + WFB(s)X(s)| ^ (3.4.7) For convenience, (3.4.7) i s rewritten i n terms of r a t i o s : Z(s) - 1/ where HFF(s)+ H, Is (3. 4. 8 ) u„ i s the amplitude of the input, ju (s)^ , and — (s) i s the closed loop transfer function as given by (3.4 . 1 ) . The optimum values of K,, Kz and K 3 are fcund using a d i g i t a l computer program l i s t e d in Appendix T.3. The optimization process i s carried out at one frequency only, that being the "design frequency" which contains the greatest energy. 3.4.3 Ram Centering The ram centering network i s required to maintain the long term average ram displacement zero. Dnder s t a t i c conditions, t h i s i s equivalent to returning the ram tc centre position in a given (long) time a f t e r receiving a step input. Considering the centering loop as shown in Fig 3.4.1, the active ram force F A can be set to zero under s t a t i c conditions. Q ^ r Z + ——* "FIG. 3.4.1 R A M CENTER ING N E T W O R K 66 This gives a closed loop transfer function of Y(s) Letting Hy(*) = Ky = constant gives (3.4.9) (3.4. 10) where It now remains to select Kv such that where 0)„ i s the design frequency. 3.4.4 System S t a b i l i t y The s t a b i l i t y of the system i s determined by solving the c h a r a c t e r i s t i c equation 1 + U(s) = O (3.4.11) and examining the root locus as the feedback gains K, , Kj_ and K 3 are varied. In general, the c h a r a c t e r i s t i c function i s cumbersome to deal with, and the following s i m p l i f i c a t i o n s are therefore assumed: 67 1. Let Gp (s) = -Kp i . e . , assume the p a s s i v e system behaves as a s i m p l e s p r i n g (where Hccke's Law a p p l i e s ) . 2. L e t F c (s)=1 — i . e . , n e g l e c t the c a b l e dynamics. T h i s i s r e a s o n a b l e because the e f f e c t of the c a b l e cn the motion compensation system i s r e l a t i v e l y s m a l l , e x c e p t a t the n a t u r a l frequency of the tewed system. T h i s f r e q u e n c y s h o u l d l i e o u t s i d e t h e range c f c p e r a t i c n . 3. N e g l e c t Hy(s) as e x p l a i n e d e a r l i e r . With the above a s s u m p t i o n s , (3.4.11) can be e x p r e s s e d a s : (3.4.12) The a p p l i c a t i o n of (3.4.12) t o the l a b o r a t o r y rcodel i s d i s c u s s e d i n Chapter IV. 3.4.5 Power Consumption The power consumed by t h e a c t i v e system i s the SUIT c f the power d i s s i p a t e d i n the s e r v o v a l v e and the power r e q u i r e d to d r i v e the l o a d . T h i s i s e q u a l t c the product of the volume r a t e of o i l f l o w i n t o the v a l v e and the s u p p l y p r e s s u r e : 68 W = A A ? s \ i ) (3.4.13) where W i s the instantaneous power consumption. Note that although y changes i n d i r e c t i o n , W i s always p o s i t i v e . This i s due to the f a c t that the d i r e c t i o n s w itching occurs i n the valve i t s e l f , but the o i l flow to the valve i s u n i d i r e c t i o n a l . Thus, the average power consumed i s the r o c t - mean-sguare of the amplitude of W: and W i s the average power consumption. I t i s convenient to express "w i n the freguency domain as a r a t i o to input displacement U(s): (3.4.14) w her e 0.707 i s the rms f a c t o r f o r a sine wave, Ps i s the supply pressure, A A i s the ram area. onoi AA R I6,A FcCs) uCs) (3.4.15) 69 where v 2-(s) i s the closed loop transfer function given i n (3.4.1.) The reason for the inclusion of an accumulator in the active system now becomes obvious: the hydraulic power supply needs only to provide power at the steady rate W, while the accumulator supplies (or absorbs) the difference between the average and instantaneous values. To avoid wasting power when the accumulator i s f u l l y charged, i t i s necessary to use either an unloading valve or a pressure compensated pump. This way, the pump does not discharge through the r e l i e f valve (at high pressure) when there i s no flow demand. 70 CHAPTER IV THE LABORATORY MODEL 4,1 General Description A laboratory model, closely resembling the equivalent system discussed in Chapter II was constructed to validate the mathematical and simulation models developed, and as a prototype small scale mechanical simulator for designing and evaluating f u l l scale systems. The apparatus consists of a hydraulic power supply, active and passive cylinders, a variable frequency displacement generator, and various pieces cf electronic monitoring, processing, and display equipment. See Figures 4.1.1 and 4.1.2. The power supply consists of a variable displacement pump of f i v e gallons per minute capacity driven by a 5 BP e l e c t r i c motor. A pressure r e l i e f valve i s provided on the discharge of the pump, which can be set to between 100 and 2000 psi. The entire unit i s mounted on top of a 15 gallon o i l reservoir. The motion compensator consists of a pair of cylinders mounted i n tandem on a carriage, Fig. 4.1.3. The carriage i s given an approximate sinusoidal displacement cf variable frequency and amplitude by means of a crank mechanism driven ty a 1-1/2 HP DC motor. The piston rods cf the two cylinders are pinned together, so that they act in p a r a l l e l , rather than in  0 GAS STORAGE <2 5 ) X ^ 4 A C C U M U L A T O R 9 i X l - SERVO VALVE ® 2 PASSIVE ACTUATOR ACTIVE ACTUATOE I RELIEF a_VALVE PUMP FIG: 4.1.2 RYDR.AU LlC SUPPLY LABORATORY APPARATUS - SCHEMATIC PASSIVE. CYLINDER ACTIVE CYLIMPER 7a series as the arrangement might f i r s t suggest. The other end of the double-ended piston rod i s pinned to a second carriage which contains weights to represent the mass whose motion i s tc be i s o l a t e d . The passive system i s purely pneumatic, consisting of a pair of gas bottles, one connected to each end of the larger of the two cylinders described above. A flow control valve on each gas bottle i s used to adjust the damping of the passive system. The active system consists of a ga s / o i l accumulator which feeds o i l to a servovalve through a f i l t e r . The servovalve controls the flow of o i l into the two ports of the smaller of the two cylinders mounted on the carriage... A gas l i n e tc the top of the accumulator controls the charge pressure of the system. The control system monitors body and ram motions by a seismic velocity transducer mounted on the mass carriage, and a displacement potentiometer cn the ram, carriage. The signal processing function i s achieved by an analogue computer which integrates, amplifies, and sums the two signals. A f i l t e r i s also included to remove high frequency noise, such as the type generated by the wheels of the carriage. The processed signal i s fed to a power amplifier which drives the servovalve. An oscilloscope and chart recorder are used to monitor any twc of mass velocity,mass displacement, input displacement, or low l e v e l servo-actuating voltage. (Fig. 4.1.4) DISPLACEMENT TRANSDUCER OSCILLOSCOPE VELOCITY TRANSDUCER BAMP- PASS FILTER DIFFERENTIATOR Ttt) TWO-CHANNEL CHART RECORDER SE.RYO- A M P L I F I E R SIGNAL T O SERVOVALVE FIG.. 4. 1.4 CONTROL SYSTEM 76 The apparatus as constructed does not model the spring (i.e. cable) nor the hydrodynamic drag of the body. a l l other aspects of the equivalent system are included. The exclusion of these two parameters does not affect the model of the motion compensation system because they are both properties of the towed system. 77 4.2 Performance Prediction and Evaluation The physical properties of the laboratory model are given in Appendix D. The design i s not based on any p a r t i c u l a r reguirement, but employs hardware which was rea d i l y available. The input amplitude was set to 1.5 inches, and the design frequency to 1 Hz. The velocity transducer used to measure the motion of the body was guite unsuitable at the low frequencies at which the system could operate, and as a result some electronic signal processing was necessary tc obtain a useable s i g n a l . As a r e s u l t of t h i s , only acceleration feedback was available for use in the control system. The s t a b i l i t y equation (3.4.12) can now be expressed using acceleration feedback: As 2 + B S + C = O ,n o -n where B = O z +^3) C u - 1 C ^ C ^ + ^ 3 ) kip By making the acceleration feedback K 3 negative, the c o e f f i c i e n t s A, B and C are always negative (note that and > 3 are negative, while Kp i s p o s i t i v e ) . Therefore, the system is 78 stable for a l l negative values of K 3. A point to note i s that the feedback constant, K 3, has the ef f e c t of increasing the apparent mass of the system, as shown in (4.2. 1). Because only one feedback variable i s considered, i t i s not necessary to use the optimization technique outlined in Section 3.4. a feedback constant of K 5 VK 3= -5 ma/(ft/sec 2) was used i n the experiment. The experiment was conducted at three frequencies: 0.5, 1.0, and 2.0 Hz. In each case, the system was f i r s t run i n the passive mode (K3 = 0) , then with Ksv K5=-5. The computer simulation was then conducted under the same conditions, and the re s u l t s of both are shown in Appendix G. The time-domain records are then transformed tc the frequency domain, and plotted as d i s t i n c t points on a Eode plot in Fig. 4.2.1. The linear model response i s also plotted on the same graph for comparison, over a frequency range of 0.1 to 10 Hz. There appears to be good agreement between the mathematical models and the r e a l system. Any discrepancies are due tc the uncertainties involved i n estimating hydraulic and pneumatic t h r o t t l i n g c o e f f i c i e n t s and mechanical f r i c t i o n . However, by 79 designing a r e a l system with variable t h r o t t l i n g valves, the former uncertainty can be removed since the real system can then be matched to the model. F r i c t i o n , on the other hand, can neither be easily predicted nor altered once the system i s operational. O V E R A L L R E S P O N S E X / U 8 0 LIMEAfc - PASSIVE (K»=o) L INEAR-ACT IVE A EXPERIMENTAL - SIMULATION 0.10 0.20 0.50 1.00 2.00 F R E Q U E N C Y RRTI0 5.00 10.00 FIG. 4.2-t THEORETICAL 6s EXPERIMENTAL RESULTS 81 CHAPTER V APPLICATION T h i s chapter i s intended as a guide tc the a p p l i c a t i o n of the mathematical and computer s i m u l a t i o n models t c the design of r e a l systems. 5.1 Input C o n d i t i o n s The i n p u t , as s t a t e d i n Chapter I , can be approximated by a F o u r i e r s e r i e s r e p r e s e n t i n g the v e r t i c a l displacement cf the s u r f a c e of the water at a given p o i n t . The E r e t s c h n e i d e r e g u a t i o n 1 can be used to o b t a i n an estimate of the s p e c t r a l energy d e n s i t y from the average height and pe r i o d cf the seaway f o r a given Sea State: S C T ) = ^ L L ' T - e ^ - ^ T * ] where 2 S (T) i s the s p e c t r a l energy densxty i n f t /sec, T i s the wave p e r i o d i n seconds, "f i s the average wave p e r i o d i n seconds, and h i s the average peak-to-trough wave height i n f e e t . i See Ref. (12) 82 Values for h and T can be r e a d i l y found in most nautical handbooks. Figure 5.1.1 shows a t y p i c a l plot of the spectral density function for Sea State 4 , having mean wave height and period of 4 .9 feet and 5.4 seconds, respectively. A ship subjected to a multi-frequency displacement input w i l l behave as a low pass f i l t e r , and w i l l not respond s i g n i f i c a n t l y to waves whose length i s less than one-half the ship's length, fi t y p i c a l response curve i s shown in F i g . 5.1 . 2 . The motion of the ship i s then the product of the sea state spectral density function and ship response, as shown in Fig. 5.1 . 3 . The wave component which contains the most energy i s found to have a period T 0, and 0)o=27r/To i s used as the primary design frequency. Figure 5.1 .3 can now be used to obtain the c o e f f i c i e n t s A-t of the Fourier Series which can represent the motion of the ship in the time domain. This i s done by dividing the Ship Motion Spectral Tensity Function into n c e l l s spanning the entire range of period, and calcu l a t i n g the energy associated with each c e l l : <5. 1.2) StT) 20 -\ FIG, S.LI 5£A STATE "4" SPECTRAL DENSITY 1 0 FUNCTION o 10 F16. 5,1.2 SHIP HEAVE RESPONSE SHIP ME AVE SbCT) ft'Aec FIG. 5.1.3 4oH SPECTRAL DENSITY FUNCTION 20 o - i o 84 AS •• J S(T)«IT H Z (5.1.3) where &S t i s the energy associated with the i - t h c e l l , T i s the central period of the i - t h c e l l , and AT i s the c e l l width. The c o e f f i c i e n t s A^ can be expressed as The approximate ship displacement as given by (5.1.2) can then be used i n the simulation model to give a r e a l i s t i c input condition. It i s not necessary to use more than three to f i v e terms in the series to give a good wave p r o f i l e . (5.1.4) 85 5.2 Two-Dimensional Cable Model when dealing with long cable lengths (over 2000 feet) or horizontal motion with respect to the water, the cable assumes a catenary shape which can no longer be assumed one-dimensicna1. Walton and Polachek 1 developed a program tc compute the shape and tension of a cable subject to a displacement boundary condition at one end and hydrodynamic drag along i t s entire length. In essence, the continuous cable i s modelled as a number of e l a s t i c links pinned together end-tc-end. Each link has masses concentrated at i t s two ends, as well as longitudinal and transverse drag c o e f f i c i e n t s . (Fig. 5.2.1) The towed body i s represented as the l a s t l i n k of the cable, given the appropriate values for mass and drag c o e f f i c i e n t . The program uses a displacement input at the top (i.e. surface) end of the cable, and calculates the displacements and a x i a l elongations of a l l the l i n k s . This in turn yields the cable tension in each l i n k . The model of the motion compensation system developed here can use t h i s cable/body model by supplying the variable as the boundary condition, and receiving F N E T, the cable tension at the surface tow point. I f desired, the compensator motion can be resolved into horizontal and v e r t i c a l components to increase the model's realism. i See Ref. (23) 8 6 HEAVE SUR&E t (b) C A B L E S E G M E N T 87 5.3 Servo-valve Hodel Extension The s e r v o - v a l v e considered here i s assumed to operate i n s t a n t l y upon a p p l i c a t i o n of a c o n t r o l s i g n a l . However, i n the case of l a r g e v a l v e s which are u s u a l l y m u l t i - s t a g e , there i s c o n s i d e r a b l e time l a g even at low f r e q u e n c i e s . The dynamic response of such v a l v e s can be c o n s i d e r e d as a f i r s t cr second order system, depending on the accuracy d e s i r e d . The v a l v e equation then becomes (5.3.1) where H s v(s) i s the dynamic c h a r a c t e r i s t i c c f the v a l v e . In the case c f a f i r s t - o r d e r v a l v e , Hsv(s) = (5.3. 2) 1 + tsvS and f o r a second-order. c (5.3. 3) where C s v i s the valve constant (as before) , T$y i s the f i r s t - o r d e r time cons t a n t , 0)W i s the second-order n a t u r a l freguency, and 5*v i s the second-order damping r a t i o . 88 The parameters t„ V f o) w and S s w are estimated from the response curve supplied by the valve manufacturer, Fig. 5.3.1, They are chosen such that the phase lags of the real valve and the model coincide over the freguency range of inte r e s t . When operating near the resonance of the valve (which i s generally not recommended since servo-valves are usually underdamped) a higher order model may be necessary. 8 9 FIG, 5.3.1 SERVOVALVE RESPONSE 90 5.4 Control System Considerations In setting down the performance reguirements of a r e a l system, the frequency response must be c a r e f u l l y considered. In p a r t i c u l a r , long period.waves generally have larger.amplitudes than short waves, hence i t i s not always possible to compensate for them as e f f e c t i v e l y due to the limited t r a v e l of the compensator. Therefore, i t i s desirable to design for 2 e r o compensation at high wave periods (in the order of 20 tc 50 seconds), increasing to maximum compensation at the design frequency. An acceleration feedback system w i l l inherently behave in t h i s manner. At frequencies above the ship's natural frequency i t i s desirable to decrease compensation since the ship does not respond to such waves. Furthermore, shipboard vibrations due to the engine and propellors may be s i g n i f i c a n t above one Hz. A t y p i c a l frequency response which would give acceptable performance i s shown in F i g . 5.4.1. The low frequency cut-off can be moved to the l e f t by either decreasing the s t i f f n e s s of the passive system, increasing amplifier gain, or increasing the time constant of the ram centering loop. The point of maximum compensation i s set by introducing a second-order low-pass f i l t e r , Fig. 5.4.2. The corner frequency coincides with the design frequency, where motion compensation i s maximum. The c r i t i c a l damping r a t i o determines the bandwidth 91 CO o < UJ 2. -2o4 F R E Q U E N C Y FIG. 5.4. I MOTION C O M P E N S A T I O N T R A N S F E R F U N C T I O N 1 Slfi-KJAL TO f ACCELERATION 1 + 2 S/Us + S*A>2 SERVO-AMPUFlfcTR. ui0 SETS DESl&M FREQUENCY 5 CONTROLS BANDWIDTH FIG. 5.4.2 TYPICAL FEEDBACK NETWORK 92 of the response curve. Such a system to v i r t u a l l y any sea state system i s designed to handle the feature can be used to improve an adding on an active one. f i l t e r can be used tc tune the condition, provided that the corresponding amplitudes. This existing passive system by 93 CHAPTER VI CONCLUSIONS The dynamic behaviour of an a c t i v e - p a s s i v e motion compensation system has been analysed and a mathematical model developed. Experiments performed, on a labo r a t o r y apparatus i n d i c a t e that the system i s adequately described by the equations derived. The mathematical model has been s i m p l i f i e d by l i n e a r i z i n g the equations, and computer programs have been developed which can a s s i s t i n the i n i t i a l design of r e a l systems. In a d d i t i o n , a program which solves the nonlinear equations by s i m u l a t i o n has been w r i t t e n , and can be used to r e f i n e the i n i t i a l design. The programs are f l e x i b l e enough to accomodate a v a r i e t y of system c o n f i g u r a t i o n s . This p r o j e c t , i n essence, has provided a design t o o l , based on mathematical a n a l y s i s , to an area which has t r a d i t i o n a l l y r e l i e d on seat-of-the-pants engineering. 94 REFERENCES 1. Athans, M.: On the Design of a D i g i t a l Computer Program f o r the Design of Feedback Compensators i n Transfer Function"Form NTIS~AccT~#AE-700-4 31 2. Blackburn, J . F.: F l u i d Power C o n t r o l MIT Press, Cambridge, Mass,, 1960. ~ ~ ~ 3. Buck, J. R. 6 S t a l l , H. W.: I n v e s t i g a t i o n of a Method to Provide Motion Synchronization During Submersible R e t r i e v a l l a v a l Eng. J . , Dec. 1969. 4.. Burrows, C. R. : F l u i d Power Seryomechanisms Van Nostrand Reinhold Company, London, 1972. 5. Cavanaugh, R. D.: A i r Suspension and Servo C o n t r o l l e d I s o l a t i o n Systems Shock &. V i b r a t i o n Handbook, Ch. 33, McGraw H i l l , 1961. 6. G u i l l o n , M.: Hydraulic Servosystems A n a l y s i s and Design Butterworth and Company, 1969. 7. Hedrick, J . K.: A Summary of The Optimization Technigues that Can Be Applied to Suspension Systems Design Ari z . " s t a t e T u7 Report~#PB-2205537 8. Karnop, D.: V i b r a t i o n C o n t r o l Using Semi-Active Force Generators ASME~Paper # 73-DET-1227 ' ~~ 9. Keefer, I . G.: Improved Hydropneumatic Tensioning Systems f o r -Marine A p p l i c a t i o n s B, C. Research Council Report, T9727~~~~ 1 0 . K r i e b e l , H.: A Study of the F e a s i b i l i t y of A c t i v e Shock I n g i n i e u r Archiv B e r l i n , V o l . 36 #6, ^968. 11 . Mercer, C. A, & Rees, P. L. : An Optimum Shock I s o l a t o r J. Sound S V i b r . , 18(4) 1971." 12. Myers, J . J . : Handbook of Ocean.and Underwater Engineering McGraw Hill,~19697 ~ . 13. P o r t e r , B. & Bradshaw, A.: Synthesis of A c t i v e C o n t r o l l e r s f o r V i b r a t o r y Systems J . of M. E. S c i . , V. 14~#5, 1972. 14. Raven, F. H. Automatic C o n t r o l Engineering McGraw R i l l , 1968. ' ~ . " . ~~~ '• 15. Ruzicka,.J. E.: Fundamental Concepts of V i b r a t i o n C o n t r o l Tech. I n f . S e r v i c e , AIAA~Doc7 #172-295557 16. Shinners, S. M.: Modern C o n t r o l Systems Theory and A£Eiication Addison Wesley, 1972. 17. Soliman, J . I . 8 T a j e r - A r d a b i l i : Active I s o l a t i o n Systems Using a Nozzle Flapper Valve I n s t . M. E. Proc., V. ^82 #30, 1967." " 18. Soliman, J. I . , & T a j e r - A r d a b i l i : Servovalve C o n t r o l l e d Isolation Systems I n s t . M. E. Proc. , ~ v 7 ~ 185~#107"970. 19. Sutherland, A.: Mechanical Systems f o r Ocean Engineering Naval Eng. j77~Oct."1970." ~ ~ 20. Thompson, A. G.: Quadratic Performance Indices and Optimum Suspension Design I n s t . M. E. P r o c , V. 187, 1973. 21. Thompson, A. G.: Design of A c t i v e Suspensions I n s t . M. E. Proc., V. 185,~1970. 95 22. Thompson, A. G.: Optimum Damping i n a. Bandojl^ Excited Nonlinear Suspension Inst. M. E. P r o c , V. 184, 1969. 23. Walton S Polachek: Calculation of Transient Motion cf Submerged Cables Math. Tables 8 l i d s to Computation, V. 14, 1960. 24. Yeaple, F. D.: Hydraulic and Pneumatic Power and Control McGraw H i l l T 1966?" APPENDIX A & AS FLOW S O U A TlOhJS /• FLOIAI ll^TO A CLOSES VOLUME FIG- A - i . . R A T S OFCU-AAJG E OF^EMTN-ALPY IA/ITM*) COUTfLOLVOL UM£ .OFA-i : 6J4 THE: /zA-re OFcm+JQE. OF I^JTEY^JAL En/ee&Y IS TW= FLOlAi HATS OF GAS MULTIPLIED BY ITS UM/T Ik/TE(LtJPrL g^eZC^: . : 7V£ RATE OF CH-AtJ&E OF GtJER^Y IS WE WOfM DOAJZ OS) THE GAS BY EXPAfiJSiDkJ d(Z CJDHPHE SS(OtJ OF TI4C COKfTR-Ol VOLUWC •• ~ • r dt •• • , SU&STi TUTltJG, (A-2) /WD CA'3) /A/TO (A>I ) S0L\JtA/6 FoQ. m AtiD Su&STFTUi IfJG, ± + J- - ± IAJTO (A-4) ., I r dt + " aj C A ' 2 ) ( A t ) (4*0 2. FUHAJ THROUGH A tJeeo>te VALVG FIG. A - 2 7?/£ KIEEOLC vAL\J£, pl&- A-2. , Co*JSlS>TS OF A VARIABLE A&tA ArJMUUJS, Av , WH-iCH /S C0AJT&OLL&D BY RMSI/J6, O& LoujeZ-uOCj A TA-Pe&.&0 h)&&DL£. 77V/S C&rJ BE MODELLGb 8/ A CPAJVS^^F^T tJoZZLZ. APPLYIK]& THB B*J&££Y &2UAT/O/J ~* Liu. = H/ + f Z 'J CA-l) \rifr9&£ .sir- i$ THE A<JEGAG& VBUoCllY ACROSS Ay. A-SSVtftiUG- TH-£ FLUID lS A PEIZFFCT GAS CpT* =• CpTd + £<r-*. (A-?!) JUS MASS FtOuJ dATE IS, 6(\JFtJ 3Y y * = f A„v- . (A-lo) Su6ST(rUr/iJ6 (A-Z), (A-I) MTO CA'/OJ Givei: SU&STiTVT 1^6, TH£ F$UA7(0AJ or srA-rE (oP _ _ J L ' • •  • IK/TO (A-(0 Gives-- Sl/JCe 7V€ VALVE IS (JoT A+J IDEAL Coi<J\/BRiS,lfJCj A / O Z Z L E " , . ir is TH-edGFc&G juerc^ssAieY ro //JT&ODOCE AA/ £MPlQiCAL 0lSa4A£$ £~ COEFFICIENT , C# I/0 TO LTQUArnotO (A-lZ) . 7H(S GD&FF^ia-G+JT C~A*J TU&O BET CDtigM>E-D IAJ iTR Av, SUCH 7UAT j Co = CzAv (A 14) : Wtt£(2£ Q IS A FVK)CTIosJ OF fJEEbLE POSITIVES- Q is pEredHifjeb EXPF&IMSAJTALLYJ A*ir> USUALLY PUBLISUFD BY MAdUFACTUHZas FOP. WBIR. VALVES. .. .£&UATIOAJ (A-13) CAU JUVS 6€ ^/P/ZFSSFO AS; i_ Nore ' T H A T . - (A-IS) IS OIJLV VALID FO/Z i.£.} Pod poiA/Aisr/ZEAM pzesruze &z&ATFe. TUAA) THAT ae&uieep FOR. CFOK^O FLOU). Foe. hjirnoze/O AA/O Ate , Pc = O.S2g Pu APPENDIX £ UYDgAULIC SErZxlO-VALVES TYPES QF VALVES: VALUES c&fJSipeizeD tfE&e AUB *FOOQ.-WAY SPOOL VALV^ GDA)St£Tl>J6 OF ..^ H&T£tllh}$ ORlFlC££, FI6 B . I . . . SUPPLY BMAOST MZDM/To LOAD . TO/FROM LOAD FIG. B. I SPOOL VALVE 'THIS A£&filJ6&M&JT CAfiJ B€ MODFLteb BY A NYDZAULlC WHEATS TotiE Mlb&E, F-fe. E>1. :~- f=l4. 5.2 WHEATSTUME 0ftD$E TUB HYDRAULIC ZESlSTAiJCeS . A&G ; OF <JZ>U/2£b, hJOfJLIMBAe. SC.R.VOVALVES WAY BE UAJPE FLAPPED, OVE^LAPPED , ° & "ZERO - LAPPED AS Sht-ovJ^ /A) FlC-j. S 3 . r l \ t ZERO - LAPPED 1 — 1 n 0V£.g- LAPPED . r i .utJbefL-LAPfen F\<k. 8-3 VALVE SPOOL DE£f6fiJS 0VF2LAPPED VALUES EXf+lBiT A DEAD Zc?A/£ ABOUT . THBIP. cEUTPE POSIT'OMj X = 0, WH-EtZEAS UMDEHLAPPEb VALVGS AHE FAi&LY LtHEA/P. ItJ THAT HEGIOhJ. UUDE/Z- LAPPltJS, TEMPS TO E^UAtiCE STABILITY. OF WE SEMD SYSTEM AT raF expense OF PDMEQ. LOSS AT X-O. /M wis ANALYSIS , 2ee.o-LAfpeO VALVES AgE LEAF 101 OMITTED IN PAGE NUMBERING. \ o \ d o e s w ? T ~ e x i i f 2 . PERlVAVON OF PZESSUZa-F-LOvJ EQUATIONS _A ZBHO - LAPPED VAL\lE MAY BE MODELLED SY THE SlMf>L(Fl£D Cl&CUtT OF Fl<*. B-4- s o- L O A D P ~ O F I G . 8 4 2ERO-LAPPED V A L V E IF COMPRESSIBILITY h/IT(Jl*J THE SYSTEM /5 AJFej LECTEO, TUEM I....- -• ' • • • - . ASSuHthJG THAT OdiFtcE AQEA IS PgoPoZT ION A L To DISPLACEMENT, X,- ME. WAVE THE FAMILIAR ORIFICE FLOvS .EQUATIONS - .- — ;~. - • • .>' • -i W H E Z E P&„ KX /.Fs - P, ..= QL) .. ICY Tr\ = QL (B.2) SUPPLY. peessueE A COKlSTA-fJj $ QL .» '. LOAD FLOW COM&/Nlid<$ 77-/£ TvJQ EQUATIONS OF (&-2) A^D R.EAR.ZAN^IM&,, YIELDS[..:..!. VlUE&E QL ~- | X /P&-.AP AR = P, - P, FOR. THAT N E G A T I V E S f O O L D I S P L A C E M E N T ( X ^ - O ) , N O T E 0, QZ - O (6-4) 103 WIS Y/BLDS . QL = J X J p s + & p ' CVHBll/llJG, ( B . 3 ) Arhib , AND LETTltJG Wi+ichi is Thit C&HZIZAL S&MO VAL\J€. EQUATION)'. . APPENDIX C COMPRESSIBILITY EFFECT OF MDRAULIL FLU ID CoUSWtd TUB CYLlMPStZ. SPorili! iN F/G- CJ • 0; k Q0 -1 . . 1 , p, p2 1 1 — FIG,. C-l TH-e F-LOti' OoT., QQ t. IS. . ErQAf&L TO THE FLOIAI IhJ j Q{ f L6SS THE IZATE OF ....FLUID COMPRESSION IHSlPE, THE "COMPRESSIBILITY FLOW'',.. 6?c. 4 A tfc Y... IS THE \/OLUHE OF TUB CYLlriOEZ, 3 IS THE SULU hiODOLOS OF THE . FLuib, dP/Jt THE RATE OF CHA+IG, E OF X. .„ pa ESS UZE .... isJ Tti€ CiLWlDER • .' .. .'FOB. THE FOUJVALEKJT MODEL DISCUSSED IAJ THE TEXT, V ~ 4-10 m3 . & - 3>5ooo psi 5 P. 2oo(l..+ sin oji) , IAJU-ERE CJ^ ^ 4 Hz. 25 " raJ/sec. _ . mis Gfives ft = 3oo u> coso)t 3 dt VJUEA£ TH15 Gives (Qc)mx = $ J ^ 0 0 « 75-00 = 0,37/ / V / « c £0£ MAXIMUM F-LON CONDITION, - Q i = Ay wH-E-zE A = PISTON AREA - . 0 . 3 9 3 / > 2 4. y = P(ST6>N VELOCITY. COUSIPERHJG FULL 12-iNcu STROKE • y =• <S ?w £jt Cf - & cu cos cot 6j)i*c-=V ^ ' I S O m/scc 7-/̂ /S 6/i/<5S ..... (5/ ....= . 0<393 x /s~£> : _: *=..59. in3/sec. THIS <s,iVES THE HATiO - 0.6°Jo T14E COMPRESSIBILITY FLOU) IS THUS N E G L I G I B L E . CONSIDER. uoiN A FULL-SCALE SYSTEM, TYPICALLY, V - 5O00 in3 (8" D I A - * 8 FT LoNC P " IOOO (I + Sin Ot) WHERE comA = 0.2S' Uz = /.£\7 taj/sec. mis Gives C£)»*X = 1 0 0 0 * /• 57 * isio psi So 00 NENCE O c - /S-70 = 8 3 }/73/s^c 106 mz. iKiter FLON RAI& IS Noui Of = -Aijnw = SO x 48 x },SI (In*/sec) - 3110 in3/sec- miS Cwes, THG RATIO Qc _ _B3_ Q; ZTlO 2 . 2 % THIS IS ALSO iJe&LiG I8LE. APPENDIX 0 LABORATORY MODEL 'SPECIFICATIONS' /• INPUT • ju0 .- .. i. \{ " . = i O.I2Sfh - 2. LP AD M = S o «sLû -s *=* 97 !i>-iwass. . tfc = <*> ...; 3- PASSiy/E ''SYSTEM' L l - P0 - 0 psig - X5psia. Aj ~ 314 inz . . . . . . / 4 2 = 2.95 in* . . . . . . : L : . ; ... Av. AREA •• Ap - • 3 - C 4 i / i * ' . : 1: V c = ZO /n* ' • ,! . : . . . . . . j • v .4 : • V* = . 2.SO m 3 ̂ A/ = £ = /4 . . / i : ' r . .. STIFFNESS: KP - 7̂77 2 r ^ 4 p " = ..T-yXjk/\'»\=\W- Ih/iL 4- ACTIVE SYSTEM £E(Z\/0VALVE- VlOlERS SC4--03 3 US&Pfi] @> EOO psi Jrep C s v = OrOU8 Lin/sex)/ff?) A, Ps = j-7s- foe)/fj^i Ll*l£A£ . APPeoxiMATi'OtJ : LET FAo - /a* . . . ...... - \ . y 0 y ^ n s = C7o7^9 -4 -2= /"«/s<zo -Z„...= A- • • - • ! • Id-* AA _ _ * 0 . 3 93 (0 ma .•.,. ... c"sv. r ^ _ o-o a8 r-—1 I.""....'. = w 0.055. (^b/uc)/^. _ . J ... ' =--o,on. (j*/*«-)//t- , - . 4 , ... : 4 -0.OO14 (fr/s*c)/lh . 2 A ^ / ^ 7 A A • : ' * " " ., ;,f ; g_?^ = - 2 . 4 - Qh/<*t)/ ik 2+(o.5°>3,y{2<t .... - -<D.2 0+/scc.)/ll> / ) P P£A / D / X E FlZICTioN PARAMETER .fRiCTiOti .IS-. CONSIDERED AS AN E?)CTER.hiAL FOQLCS d>W THE SYSTEM , AS SHotJtJ >N £1$. £.[ • 109 F A F I G - E . I RAM THE . MET FORCE IS GiVEti BY ^ A V * ' = =- I~A + Fp - f WUERE '.. f IS THE FZICTioN FO&CE i IS A FUNCTION OF t/ (a) y = 0 : f /5 'EQUAL A7v7> OPPOSITE TO (FA +FP) UNTIL MOTION ...„ BEGitJS r 4 (b) • y * O • Jr. IS COslSTAMT AMD OPPOSJTE IN- SE-tJS£ TO j •(FP + FA) L = 0 5 f ° IIC APPEMDIX F. 1 ONLIN 1 » ^ » a j i » » » a i n i i » ^ 3 » a j t » REMCiNONLIN ******** 2 * 3 ***** NONLINEAR MODEL SIMULATION PROGRAM *** 4 * 5 I N I T I A L 6 CONSTANT CSV=0.0118, UG=0.125, AA=0.393, AP=3.0, ... 7 V0=20., VT = 280., M=3.G, GAM=1.2, PS=50G. ,.«,. 8 K0=6.28 9 PARAMETER K2 = G., K3 = C.O, KSV=1.0, P0=15.0,... 10 ZETA=1.00, CV=3.75, FF0=1., YDCR=0.CCOGGG1, CX=1. 10.25 PARAMETER K1=(0.,5.) 11 PARAMETER RW= 1.0, TCV=0. 025 12 N=VT/VO 13 KS=2.*GAM*P0*AP**2/VT*12. 14 WN=SQRT(KS/M) 15 TC1=2.*ZETA/WN 16 TC2=TC1/(N+1.) 17 KP=N*KS/(N+1.) 18 W' = Rfo#WO 19 DYNAMIC 20 Ul=UO*S INE(0 . f W,0.) 21 U2=RA,MP( 0.0) -RAMP ( l.O) 22 U=U1*U2 23 UD=DERIV<0.,U) 24 UDD=DERIV(0.»UD) 25 X=U+Y 25.25 X1=CMPXPL<Q. ,0. ,0.5,W,XD) 25.5 X0UT=W*X1 26 F1=LEDLAG(TC1,TC2,Y) 27 FP=-KP*F1 28 YD=IMPL( 0 . t 0 . 0 5 i F Y O ) 31 XD=UO+YD 32 YDD=DERIV(0.,YD) 33 XCD=UDD+YDD 34 FNET=M*XDD+CX*XD 35 PROCEDURE FFR = FRIC(YD,FFG ,FNET,YDCR> 36 I F ( A B S ( Y D ) - Y D C R ) 1 0 , 1 0 , 1 1 37 11 I F ( Y D ) 1 , 1 , 3 38 1 FFR=-FFG 39 GO TO 4 40 3 FFR=FF0 41 GO TO 4 42 10 F F R = L I M I T i - F F 0 , F F 0 , F N E T ) 42.25 4 CONTINUE 43.25 ENDPROCEDURE 44 FA=FNET-(FP-FFR) 45 XDDD=-W**2*XD 46 R1=-K1*XDDD*W 47 R=CMPXPL(0.,0.,0.5,W,R1) 48 Z1=KSV*R 49 Z = L I M I T ( - 4 0 . ,40. , Z D 50 SGN=FCNSW (Z, -1 .0,0.0, 1.0 ) 51 SGN2=FCNSK(FA,-1.0,0.0,1.0) 52 YD1=CSV*Z*SQRT(PS-LIMIT(-PS,PS,SGN*FA/AA)) 53 YD2=SGN2*CV*SQRT(ABS(FA/AA)) / / / 54 FVD={ Y U i - V D 2 ) / 1 2 . / ^ A 55 V=1MGRL (0.,YC) 55. 25 NOSORT 55.5 * GO TO 30 55.6 51 IF ( K f c f c P . N E . l ) CC TC 30 55.7 TX=TIME+C.CC1 55.8 IF{AMOD(TX » .0 5 ).GT#0.002) GO TO 30 55.81 WRITE(8,31) TIM£,XCuT,U 55.82 31 FORMAT(3E14.6) 55.83 30 CONTINUE 56 PRINT I , X, Y, XCLT, P, FNET, F P , F L 57 T I T L E ACTIVE/PASSIVE MQTICN COMPENSATION SYSTEM 58 TIMER PRCEL=0.05, FINTIM=iO.» DELT = 0.05 59 METHOD RKSFX 60 END 60.7 PARAMETER RW=0.5 60.8 TIMER DELT = 0.1 60.81 ENC 61 STOP 62 ENDJOB :N0 OF F I L E EC * S K I P 112 A P P E N D I X F. 2. LINSYS 1 c******************** REMC !L INSYS *************************** 2 C 3 C L I N E A P I Z E C MUCEL OF MOTICN CCKPtNSAT ION SYSTcM 4 C 5 £ * * * * * * * * * * * * « * $ * * * * * * * * * ******** **>!« ** v * * * i | - * * * i ; * * * j , ' . j ; t w*4«r.< :*****.• 6 COMPLEX G,F,T1,T2,FFF,FFB,Z1,S,H,PWR .7 REAL KS,Kl,K2,K3,KP,M,N,LCGRh,KFF 8 REA0(5,1,END=99)AA,A,PQ,VC,V T, G AM , M , Z , CV 1 ,C V2,CX,DELAY 9 KS=2.*GAM*PG*A**2/VT*12. 10 l»N = SCRT( KS/M) 11 N=VT/VC 12 IF<Z.EG.G.) Z = S Q R T ( ( N + l . ) * ( N + 2 . ) / ( 8 . * N ) ) 13 TC1=2.*Z/WN 14 TC2=TC1/(N+ 1 . ) 15 KP=N-KS/(N+1.) 18 100 REAC(5,1,END=99) WQ,K1,K2,K3,KFF,CBP 18.1 HCV1=1./(CV2+CBP) 18.2 HCV2=CV1*HCV1 18.25 RWN=WN/WO 19 I F t K F F . E C . C . } KFF = KP*(CV2 + CEP )/CV1 20 WRITE(7,4> P0»VC,VT,N»KS,Z,kN,CX 21 WRITE(7,5) C V l , C V 2 , C b P , K l , K 2 , K 3 , K F F 21.25 fcRITE(7,6) kG,RhN 22 L0GRW=-1. 22.2 5 TCV=T A M DEL AY/180 .* 3 . 14 159 )/W0 22.5 ALPH=SQRT(!.+(TCV*WC)**2I 23 DO 20 1=1,81 24 R^=10.0**L0GRW 25 W=kW*WO 26 S = C M P L X ( C , I* ) 27 G = - K P * ( l . + T C l * S ) / ( l . + T C 2 * S ) 28 F=M*S**2+CX*S 29 HFB = ( K1+K2*S+K3*S**2)*ALPH/ l l . * T C V * S )*I«*S / ( W* *2<-k »S+S**2) 30 HFF=KFF 31 h = -(F/HCV1 + CV1*(HFE + H F F ) ) / ( G / F C V l + C V 1*FFF + D 32 T l = l . / ( 1 . + H ) 33 PHIH=ATAN2(AIMAG(H)»R E A L ( H ) M : 1 8 0 . / 2 . 1 4 _ 5 9 34 Z1=HF6*T1+FFF 35 T 2 = T 1 - C M P L X ( 1 . , 0 . ) 36 T1A=CABS(T1) 37 T2A=CA6S(T2) 38 HA=CA6S(H) 39 P H I l = | A T A N 2 ( A I M A G ( T l » t R E A L ( T l ) ) ) * 1 8 C . / 3 . 14159 40 PHI2=(ATAN2(A I MAG(T2 ).FEAL ( f2 ) ) )* 180•/2.14159 41 DB1=20.*AL0G10{T1A) 42 CB2 = 20 .*AL CG10 (T 2 A ) 43 DBH=20.*ALGG10(HA) 44 Z2=CA6S(Z1) 44.25 PWR=500 . * A A*S*T2 44.5 PWRA=CABS(PfcR) 45 IF (MUD{I,2).EQ.1) WRITE(7,S> Rh ,DB1 , PH 11 ,062 , PHI 2 , 46 * CBH,PHIF,Z2,PWRA 46.25 DBZ=2C.*ALCG10(Z2) 46.5 PH I Z= { AT AN 2 { A IMAGt Z D , REAL ( Z D ) I * 18u. / 2. 141 59 47 WRITE(8,3) LCGRW,DE1,C82,PH11,Ph12 113 48 20 LOGRW=LOGRW+0.025 49 GO TO 100 50 1 FORMAT(12E12.0) 51 3 FORMAT(10F13.4) 52 4 FORMAT {• 1PASSIVE SIDE' / 'OPRESS » = 1 , F 6 . 0, 5X, »C YL • VOL . =*, 53' *F6.0,5X,«TANK VOL. = • , F6.0»5X,«VOL RATIO =',F6.3/ 54 *• STAT.STIFFNESS = •,F8.2,5X, «CRIT.DAMP.RATIO = «,F7.3, 55 *5X, • NATo FREQ • = • , F 6 0 2 / , BODY OR0G COEFF. =«,F6.0) 56 5 FORMAT <'OACTIVE S IDE V CLI NEAR VALVE CCEFFS. CV1 ='» 57 *F9.5,5X,*CV2 =',F9.5,5X, ,CBP=»,F9.5/« FEEDBACK CONSTS. K l =«, 57.25 * F 6 . 0 , 58 *3Xt 'K2 = S F 6 . 0 , 3 X , «K3 = »,F6.1/' FEEDFWD CONST. KFF =«,F6.3/) 58.25 6 FORMAT(•OOPERATING FRECUENCY=• f F 7 . 2 / 58. 5 * 5X,'NAT.FREQ./OP.FREQ.=•,F6.2/*0*) 59 99 STOP 60 END OF F I L E SKI P i lit APPENDIX r . 3 PTIM 1 C ******************* REMC :0PTIM **************** ********** 2 C 3 c **** PROGRAM TO OPTIMIZE PARAMETERS OF CCNTROL SYSTEM *** 4 c 4.5 c ********************************************************* 5 DIMENSION VAR(3,6) 6 EXTERNAL TRANSF,FLO,FH I,FMPL 7 COMPLEX S,HFF,F,G 8 REAL M,N,KP,KS 9 COMMON/PARAM/S,HFF,F,G,KCV1,HCV2 10 READ(5,1) AA, A , PO , VC , V T , G AM, Mt, Z , CV1 , CV2 , CBP 10.25 READ(5,1)W 11 1 FORMAT(12E10.0 ) 13 KS = 2.0*GAM,*P0*A**2/VT*12.0 14 WN=SQRT(KS/M) 15 N=VT/VC 16 I F i Z . E Q . O . ) Z = S G R T ( ( N + l . ) * ( N + 2 . ) / ( 8 . * N )) 17 KP=N*KS/(N+1. ) 18 S=CMPLX(0.,W) 18. 25 TC1=2.*Z/WN 18.5 TC2=TC1/(N + 1. J 18. 6 G = K P * ( l . + T C l * S ) / ( l . + T C 2 * S ) 18.7 F=M*S**2 18.8 HCV1=1./CCV2+CBPJ 18. 81 HCV2=CV1*HCV1 18.82 KFF=0. 18.83 H FF = KF F 19 VAR(1,1)=0.0 20 VAR(2, 1 )=0. 21 V A R ( 3 , 1 ) = 0 . 22 CALL C0MPLX(X,VAR,3,3,6,4,9.9,50,150,2 50,10,0.001,TRANSF,FLO, 23 *FHI,FMPL,£999,£777) 24 STOP 25 999 STOP 9 26 777 STOP 7 27 END 28 FUNCTION TRANSF(T,NN) 29 DIMENSION T ( 1 ) 30 COMPLEX S,G,F,HF8,T1,HFF 31 REAL K1,K2,K3 32 COMMON/PARAM/S,HFF,F,G,HCV1,HCV2 34 K1 = T U ) 35 K2=T(2) 36 K3=T(3) 41 HFB=K1+K2*S+K3*S**2 42 T1=(G-HCV2*HFF-HCV1*S) /{F+G+HCV2*HFB-HCV1*S ) 43 TRANS F=CABS (T 1 ) 44 RETURN 45 END 46 FUNCTION FLO(T, N , J ) 47 DIMENSION T ( l ) 48 GO TO ( 1 , 2 , 3 , 4 ) , J 49 1 FLC=0. 50 RETURN 51 2 FLO=0 . us 52 RETURN 53 3 FL0=-10. 5 4 RETURN 54.25 4 FL0=0. 54.5 RETURN 55 END 56 FUNCTION F H I ( T , N , J ) 57 DIMENSION T { 1 ) 58 GO TO (1 ,2 ,3 ,4) , J 59 1 FHI=.0. 60 PETURN 61 2 FHI=0. 62 RETURN 63 3 FHI=10. 64 RETURN 64.25 4 FHI=128. 64.5 RETURN 65 END 66 FUNCTION FMPL<T,N,J) 67 DIMENSION T U ) 68 COMPLEX S,T1,HFF ,HF8,F,G 6 9 COMMON/PARAM/S•HFF t F,G,HC VI,HCV2 6 9 . 2 5 K1=T(1) 69. 5 K2=T(2) 6 9 . 6 K2=T( 3) 69.7 HF8=K1+K2*S+K3*S**2 69.8 Tl=(G-HCV2*HFF-HCV1*S)/(F+G+HCV2*HFB-HCV1*S) 69.81 FMPL=CABS(HFB*T1+HFF) 71 PETURN 72 END OF F I L E SKI P APPENDIX G EtPER. I MENTAL j SIMULATION R.ESULTS Hz. *3 AM PL. RATIO PHASE LAC FIGURE EY.PT SWUL. &XP 'T 0-5 0 Odb -14 di> 0° 18° 6./ 0.5 ...5 -3db -z.i<u> 41° 27° 1-0 ... .0 -Adh -4T5Ji 54° 36° 6.3. 1.0 5 - 7.3 Ah -i&Jb . 72°. 54° . 1 , : 64- 2.0 0 -ISAh ----- 11° 100° 2-0 5 -/hldl - JOSdh 30° 108° THE7 ABOVE ARE PloTTErD OfJ A BODE DIAGRAM Ik) FIG. 4.2-1- •-- - MOTES, ON . FI&UZCS G.I -..£.4 • - LOWE a 2 CURVES WERE GENERATED BY WE COMPUTER. USING THE SIMULATION PROGRAM OF .. APPENDIX Fj- L -I ; .. •• ; - UPPER 2 CURVES ARE FROM CHART RECORDER. FULL SCALE OEFLECTION IS + i-5 INCHES. . .1 - AMPLITUDE RATIOS WD PHASE SN/FTS IN ERE ESTIMATED ... FROM THESE. CVZVES,   REMC P L O T * . 0 0 7 2 0 0 9 8 . 6)1 n-7 i

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