"Applied Science, Faculty of"@en . "Mechanical Engineering, Department of"@en . "DSpace"@en . "UBCV"@en . "Stricker, Peter Andrew"@en . "2010-01-29T17:49:21Z"@en . "1975"@en . "Master of Applied Science - MASc"@en . "University of British Columbia"@en . "The dynamic behaviour of an active-passive motion compensation system for handling towed marine vehicles is examined, and a mathematical model developed. In the analysis, the passive system considered is pneumatic, while the active system is electro-hydraulic. The towed body is assumed to be a point mass subjected to hydrodynamic drag, and attached to the motion compensator by means of a linear spring representing the cable. It is not intended, in this project, to model the towed body in greater detail.\r\nThe equations of the passive, active, and towed body systems are derived, and linearized to permit a relatively simple frequency-domain solution. A time simulation based on the nonlinear equations, including Coulomb friction in the compensator, is developed for use on an IBM Systeis/370 computer.\r\nA laboratory model is used to conduct experiments at three frequencies, and the results indicate good agreement between the linear, simulation, and real models. Extension of the equations to cover multi-frequency inputs, two-dircensiona1 towing cables, and slow-acting servovalves is also discussed to facilitate application to marine systems."@en . "https://circle.library.ubc.ca/rest/handle/2429/19326?expand=metadata"@en . "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\u00C2\u00A3ldEEg.^& The Un ivers i ty of B r i t i s h Columbia Vancouver 8, Canada Date APg.lL 2 , l) 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 \u00E2\u0080\u0094 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 $\u00E2\u0080\u00A23 C O N T R O L S Y S T E M r \u00E2\u0080\u0094f-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) \u00E2\u0084\u00A2 - - 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 \u00E2\u0080\u00A2 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\u00E2\u0080\u009E (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 \u00C2\u00A3 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; = \u00E2\u0080\u0094 1 \u00E2\u0080\u0094 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 \u00C2\u00B1 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 - > \u00E2\u0080\u0094 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 >\u00E2\u0080\u00A2 SICM ( 5 ABS ABS \u00E2\u0080\u0094 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 = \u00E2\u0080\u0094 Fo <2'5* 2) \"NET J/ \u00C2\u00BBRAM 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 \u00E2\u0080\u0094*- 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\u00E2\u0080\u009E 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 - \u00E2\u0080\u0094 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\u00C2\u00BBIET 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. 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>v^ 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) . 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=\u00E2\u0080\u009471\u00E2\u0080\u00941 (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\u00E2\u0080\u009E =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 \u00E2\u0080\u00A2 A \u00E2\u0080\u0094\u00E2\u0080\u0094\u00E2\u0080\u0094\u00E2\u0080\u0094\u00E2\u0080\u0094\u00E2\u0080\u0094 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 % \u00C2\u00BB . s + s7\u00C2\u00AB2 <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\u00C2\u00A30l (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 \u00E2\u0080\u0094 , [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 ( \u00C2\u00A3 ) 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 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\u00C2\u00BB=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\u00C2\u00A3A 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 \u00E2\u0080\u00A2\u00E2\u0080\u00A2 J S(T)\u00C2\u00ABIT 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\u00E2\u0080\u009E 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. ' ~ . \" . ~~~ '\u00E2\u0080\u00A2 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\u00C2\u00A3Eiication 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 /\u00E2\u0080\u00A2 FLOIAI ll^TO A CLOSES VOLUME FIG- A - i . . R A T S OFCU-AAJG E OF^EMTN-ALPY IA/ITM*) COUTfLOLVOL UM\u00C2\u00A3 .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\u00C2\u00A3 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 \u00E2\u0080\u00A2\u00E2\u0080\u00A2 ~ \u00E2\u0080\u00A2 r dt \u00E2\u0080\u00A2\u00E2\u0080\u00A2 \u00E2\u0080\u00A2 , SU&STi TUTltJG, (A-2) /WD CA'3) /A/TO (A>I ) S0L\JtA/6 FoQ. m AtiD Su&STFTUi IfJG, \u00C2\u00B1 + J- - \u00C2\u00B1 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?/\u00C2\u00A3 KIEEOLC vAL\J\u00C2\u00A3, 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\u00C2\u00A3. 77V/S C&rJ BE MODELLGb 8/ A CPAJVS^^F^T tJoZZLZ. APPLYIK]& THB B*J&\u00C2\u00A3\u00C2\u00A3Y &2UAT/O/J ~* Liu. = H/ + f Z 'J CA-l) \rifr9&\u00C2\u00A3 .sir- i$ THE AE-D IAJ iTR Av, SUCH 7UAT j Co = CzAv (A 14) : Wtt\u00C2\u00A3(2\u00C2\u00A3 Q IS A FVK)CTIosJ OF fJEEbLE POSITIVES-Q is pEredHifjeb EXPF&IMSAJTALLYJ A*ir> USUALLY PUBLISUFD BY MAdUFACTUHZas FOP. WBIR. VALVES. .. .\u00C2\u00A3&UATIOAJ (A-13) CAU JUVS 6\u00E2\u0082\u00AC ^/P/ZFSSFO AS; i_ Nore ' T H A T . - (A-IS) IS OIJLV VALID FO/Z i.\u00C2\u00A3.} 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 \u00C2\u00A3 UYDgAULIC SErZxlO-VALVES TYPES QF VALVES: VALUES c&fJSipeizeD tfE&e AUB *FOOQ.-WAY SPOOL VALV^ GDA)St\u00C2\u00A3Tl>J6 OF ..^ H&T\u00C2\u00A3tllh}$ ORlFlC\u00C2\u00A3\u00C2\u00A3, FI6 B . I . . . SUPPLY BMAOST MZDM/To LOAD . TO/FROM LOAD FIG. B. I SPOOL VALVE 'THIS A\u00C2\u00A3&filJ6&M&JT CAfiJ B\u00E2\u0082\u00AC 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 U/2\u00C2\u00A3b, hJOfJLIMBAe. SC.R.VOVALVES WAY BE UAJPE FLAPPED, OVE^LAPPED , \u00C2\u00B0 & \"ZERO - LAPPED AS Sht-ovJ^ /A) FlC-j. S 3 . r l \ t ZERO - LAPPED 1 \u00E2\u0080\u0094 1 n 0V\u00C2\u00A3.g- LAPPED . r i .utJbefL-LAPfen F\L(Fl\u00C2\u00A3D 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....- -\u00E2\u0080\u00A2 ' \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 - . ASSuHthJG THAT OdiFtcE AQEA IS PgoPoZT ION A L To DISPLACEMENT, X,- ME. WAVE THE FAMILIAR ORIFICE FLOvS .EQUATIONS - .- \u00E2\u0080\u0094 ;~. - \u00E2\u0080\u00A2 \u00E2\u0080\u00A2 .>' \u00E2\u0080\u00A2 -i W H E Z E P&\u00E2\u0080\u009E KX /.Fs - P, ..= QL) .. ICY Tr\ = QL (B.2) SUPPLY. peessueE A COKlSTA-fJj $ QL .\u00C2\u00BB '. LOAD FLOW COM&/Nlid<$ 77-/\u00C2\u00A3 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\u00E2\u0082\u00AC. EQUATION)'. . APPENDIX C COMPRESSIBILITY EFFECT OF MDRAULIL FLU ID CoUSWtd TUB CYLlMPStZ. SPorili! iN F/G- CJ \u00E2\u0080\u00A2 0; k Q0 -1 . . 1 , p, p2 1 1 \u00E2\u0080\u0094 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. .\u00E2\u0080\u009E pa ESS UZE .... isJ Tti\u00E2\u0082\u00AC CiLWlDER \u00E2\u0080\u00A2 .' .. .'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\u00C2\u00A3 TH15 Gives (Qc)mx = $ J ^ 0 0 \u00C2\u00AB 75-00 = 0,37/ / V / \u00C2\u00AB c \u00C2\u00A30\u00C2\u00A3 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 \u00E2\u0080\u00A2 y =\u00E2\u0080\u00A2 : _: *=..59. in3/sec. THIS -iwass. . tfc = <*> ...; 3- PASSiy/E ''SYSTEM' L l -P0 - 0 psig - X5psia. Aj ~ 314 inz . . . . . . / 4 2 = 2.95 in* . . . . . . : L : . ; ... Av. AREA \u00E2\u0080\u00A2\u00E2\u0080\u00A2 Ap - \u00E2\u0080\u00A2 3 - C 4 i / i * ' . : 1: V c = ZO /n* ' \u00E2\u0080\u00A2 ,! . : . . . . . . j \u00E2\u0080\u00A2 v .4 : \u00E2\u0080\u00A2 V* = . 2.SO m 3 ^ A/ = \u00C2\u00A3 = /4 . . / i : ' r . .. STIFFNESS: KP - 7^77 2 r ^ 4 p \" = ..T-yXjk/\'\u00C2\u00BB\=\W- Ih/iL 4- ACTIVE SYSTEM \u00C2\u00A3E(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\u00C2\u00A3A\u00C2\u00A3 . APPeoxiMATi'OtJ : LET FAo - /a* . . . ...... - \ . y 0 y ^ n s = C7o7^9 -4 -2= /\"\u00C2\u00AB/s3,y{2 / ) P P\u00C2\u00A3A / D / X E FlZICTioN PARAMETER .fRiCTiOti .IS-. CONSIDERED AS AN E?)CTER.hiAL FOQLCS d>W THE SYSTEM , AS SHotJtJ >N \u00C2\u00A31$. \u00C2\u00A3.[ \u00E2\u0080\u00A2 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 ...\u00E2\u0080\u009E BEGitJS r 4 (b) \u00E2\u0080\u00A2 y * O \u00E2\u0080\u00A2 Jr. IS COslSTAMT AMD OPPOSJTE IN- SE-tJS\u00C2\u00A3 TO j \u00E2\u0080\u00A2(FP + FA) L = 0 5 f \u00C2\u00B0 IIC APPEMDIX F. 1 ONLIN 1 \u00C2\u00BB ^ \u00C2\u00BB a j i \u00C2\u00BB \u00C2\u00BB \u00C2\u00BB a i n i i \u00C2\u00BB ^ 3 \u00C2\u00BB a j t \u00C2\u00BB 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. ,.\u00C2\u00AB,. 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.\u00C2\u00BBUD) 25 X=U+Y 25.25 X1=CMPXPL 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 \u00C2\u00BB .0 5 ).GT#0.002) GO TO 30 55.81 WRITE(8,31) TIM\u00C2\u00A3,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.\u00C2\u00BB 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 \u00C2\u00A3 * * * * * * * * * * * * \u00C2\u00AB * $ * * * * * * * * * ******** **>!\u00C2\u00AB ** v * * * i | - * * * i ; * * * j , ' . j ; t w*4\u00C2\u00ABr.< :*****.\u00E2\u0080\u00A2 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\u00C2\u00BBN = SCRT( KS/M) 11 N=VT/VC 12 IF P0\u00C2\u00BBVC,VT,N\u00C2\u00BBKS,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\u00C2\u00AB*S / ( W* *2<-k \u00C2\u00BBS+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)\u00C2\u00BBR 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 \u00C2\u00BB t R E A L ( T l ) ) ) * 1 8 C . / 3 . 14159 40 PHI2=(ATAN2(A I MAG(T2 ).FEAL ( f2 ) ) )* 180\u00E2\u0080\u00A2/2.14159 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 {\u00E2\u0080\u00A2 1PASSIVE SIDE' / 'OPRESS \u00C2\u00BB = 1 , F 6 . 0, 5X, \u00C2\u00BBC YL \u00E2\u0080\u00A2 VOL . =*, 53' *F6.0,5X,\u00C2\u00ABTANK VOL. = \u00E2\u0080\u00A2 , F6.0\u00C2\u00BB5X,\u00C2\u00ABVOL RATIO =',F6.3/ 54 *\u00E2\u0080\u00A2 STAT.STIFFNESS = \u00E2\u0080\u00A2,F8.2,5X, \u00C2\u00ABCRIT.DAMP.RATIO = \u00C2\u00AB,F7.3, 55 *5X, \u00E2\u0080\u00A2 NATo FREQ \u00E2\u0080\u00A2 = \u00E2\u0080\u00A2 , F 6 0 2 / , BODY OR0G COEFF. =\u00C2\u00AB,F6.0) 56 5 FORMAT <'OACTIVE S IDE V CLI NEAR VALVE CCEFFS. CV1 ='\u00C2\u00BB 57 *F9.5,5X,*CV2 =',F9.5,5X, ,CBP=\u00C2\u00BB,F9.5/\u00C2\u00AB FEEDBACK CONSTS. K l =\u00C2\u00AB, 57.25 * F 6 . 0 , 58 *3Xt 'K2 = S F 6 . 0 , 3 X , \u00C2\u00ABK3 = \u00C2\u00BB,F6.1/' FEEDFWD CONST. KFF =\u00C2\u00AB,F6.3/) 58.25 6 FORMAT(\u00E2\u0080\u00A2OOPERATING FRECUENCY=\u00E2\u0080\u00A2 f F 7 . 2 / 58. 5 * 5X,'NAT.FREQ./OP.FREQ.=\u00E2\u0080\u00A2,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,\u00C2\u00A3999,\u00C2\u00A3777) 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 0\u00C2\u00B0 18\u00C2\u00B0 6./ 0.5 ...5 -3db -z.i 41\u00C2\u00B0 27\u00C2\u00B0 1-0 ... .0 -Adh -4T5Ji 54\u00C2\u00B0 36\u00C2\u00B0 6.3. 1.0 5 - 7.3 Ah -i&Jb . 72\u00C2\u00B0. 54\u00C2\u00B0 . 1 , : 64-2.0 0 -ISAh ----- 11\u00C2\u00B0 100\u00C2\u00B0 2-0 5 -/hldl - JOSdh 30\u00C2\u00B0 108\u00C2\u00B0 THE7 ABOVE ARE PloTTErD OfJ A BODE DIAGRAM Ik) FIG. 4.2-1- \u00E2\u0080\u00A2-- -MOTES, ON . FI&UZCS G.I -..\u00C2\u00A3.4 \u00E2\u0080\u00A2 - LOWE a 2 CURVES WERE GENERATED BY WE COMPUTER. USING THE SIMULATION PROGRAM OF .. APPENDIX Fj- L -I ; .. \u00E2\u0080\u00A2\u00E2\u0080\u00A2 ; - 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 "@en . "Thesis/Dissertation"@en . "10.14288/1.0081010"@en . "eng"@en . "Mechanical Engineering"@en . "Vancouver : University of British Columbia Library"@en . "University of British Columbia"@en . "For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use."@en . "Graduate"@en . "Active-passive motion compensation systems for marine towing"@en . "Text"@en . "http://hdl.handle.net/2429/19326"@en .