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Active-passive motion compensation systems for marine towing Stricker, Peter Andrew 1975-12-31

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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 Montreal, 1971.  A THESIS SUBMITTED I N PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER CF APPLIED SCIENCE i n t h e Department of Mechanical  We a c c e p t  this thesis required  Engineering  as c o n f o r m i n g  to the  standard  THE UNIVERSITY OF B R I T I S H VANCOUVER CANADA January 1 9 7 5  CCLUMEIA  In  presenting this  thesis  in p a r t i a l  fulfilment of  an advanced degree at the U n i v e r s i t y of B r i t i s h the L i b r a r y s h a l l I  make i t  freely available  f u r t h e r agree t h a t p e r m i s s i o n  for  Columbia,  I agree  r e f e r e n c e and  for extensive copying of  this  tha  study. thesis  s c h o l a r l y purposes may be granted by the Head o f my Department o r  by h i s of  for  the requirements f t  this  representatives. thesis  It  i s understood that  for financial  gain shall  not  be a l l o w e d without my  written permission.  Department of  MEGUAmCAL  The U n i v e r s i t y o f B r i t i s h Vancouver 8, Canada  Date  APg.lL  2 , l<?7S~-  Bd£ldEEg.^&  Columbia  copying or p u b l i c a t i o n  ii  ABSTRACT The  dynamic  compensation  behaviour  system  of  f o r handling  examined, and a mathematical the  passive  an  system  active-passive  towed  marine  model developed.  considered  In  motion  vehicles i s the  analysis,  i s pneumatic, w h i l e t h e 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 t o be point  mass  s u b j e c t e d t o uydrodynamic d r a g , and a t t a c h e d t o t h e  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 cable.  a  the  I t i s not i n t e n d e d , i n t h i s p r o j e c t , t o mcdel the tewed  body i n g r e a t e r d e t a i l . The e q u a t i o n s systems  of  the  passive,  are derived,  and  linearized  active, tc  and  towed  tody  permit a r e l a t i v e l y  s i m p l e frequency-domain s o l u t i o n . A time s i m u l a t i o n based on t h e nonlinear  equations,  including  compensator, i s developed A  Coulomb  friction  in  the  f o r use on an IBM Systeis/370 computer.  l a b o r a t o r y model i s used t o conduct experiments  a t 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 t h e l i n e a r , s i m u l a t i o n , and r e a l models. E x t e n s i o n of the to  cover m u l t i - f r e q u e n c y  i n p u t s , two-dircensiona1  and s l o w - a c t i n g s e r v o v a l v e s  i s also  a p p l i c a t i o n t o marine systems.  discussed  equations  tewing c a b l e s , to  facilitate  iii  TABLE OF Chapter I - I n t r o d u c t i o n  CONTENTS  .............................  1  1.1 1.2  Problem D e s c r i p t i o n ....................... S t a t e of the A r t ..........................  1 6  1.3  Objectives  Chapter  Scope o f P r o j e c t  ...........  II - Theoretical Analysis  2.1 2.2 2.3 2.4 2.5 2.6 2.7 3.1 3.2 3.3 3.4  Analysis  IV - The  Laboratory  47  Model  General  4.2  P e r f o r m a n c e P r e d i c t i o n and E v a l u a t i o n  70  Description  70 .....  V - Application  5.1 5.2 5.3 5.4 Chapter  14 17 20 30 37 42 46  L i n e a r i z e d P a s s i v e System 47 L i n e a r i z e d A c t i v e System 55 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 P e r f o r m a n c e A n a l y s i s and o p t i m i z a t i o n ..... 61  4.1  Chapter  13 14  T y p i c a l System E q u i v a l e n t Model The P a s s i v e System The A c t i v e System . . . . . . . . . . . . . . . . . . . . . . . . . The A c t i v e - P a s s i v e System The C o n t r o l System . . . . . . . . . . . . . . . . . . . . . . . . Computer S i m u l a t i o n  Chapter I I I - Linear  Chapter  and  81  Input C o n d i t i o n s .......................... Two-Dimensional Cable Model S e r v o v a l v e Model E x t e n s i o n . . . . . . . . . . . . . . . . C o n t r o l System C o n s i d e r a t i o n s VI - C o n c l u s i o n s  77  .............................  81 85 87 90 93  References  94  Appendices  96  iv  L I S T OF  ILLUSTRATIONS  Figure  Page  1.1.1 1.2.1 1.2.2 1.2.3 1.2.4  Motion Compensation Systems I s o l a t i o n and A b s o r p t i o n Performance C h a r a c t e r i s t i c s P a s s i v e P n e u m a t i c System Tuned Bam T e n s i o n e r  4 7 7 9 12  2.1.1 2.2.1 2.3.1 2.3.2 2.3.3 2.3.4 2.3.5 2.4.1 2.4.2 2.4.3 2.5.1 2.5.2 2.5.3 2.6.1  A c t i v e / P a s s i v e Bam T e n s i o n e r E q u i v a l e n t System P a s s i v e System B l o c k Diagram o f Tank Dynamics . . . . . . . . . . . . . . . . . . . . . . . B l o c k Diagram o f C y l i n d e r Dynamics B l o c k D i a g r a m o f V a l v e Dynamics . . . . . . . . . . . . . . . . . . . . . . B l o c k Diagram o f P a s s i v e S y s t e m D y n a m i c s . . . . . . . . . . . . . A c t i v e System Servovalve C h a r a c t e r i s t i c s , B l o c k D i a g r a m o f A c t i v e System . . . . . . . . . . . . . . . . . . . . . . . Cable/Body Model B l o c k D i a g r a m o f C a b l e / B o d y Dynamics B l o c k Diagram o f A c t i v e / P a s s i v e System . . . . . . . . . . . . . . . A c t i v e System with C o n t r o l B l o c k s  15 18 21 23 23 27 29 31 31 36 38 38 41 43  3.1.1 3.1.2 3.2.1 3.2.2 3.3.1 3.3.2 3.4.1  P r e s s u r e - F l o w C u r v e f o r T h r o t t l i n g V a l v e s . . . . . . . . . . . . 51 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 Linearized Servovalve C h a r a c t e r i s t i c s 57 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 F u n c t i o n . . . . . . . . . . . 57 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 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 Bam C e n t e r i n g Network 65  4.1.1 4.1.1 4.1.3 4.1.4 4.2.1  Laboratory Apparatus Laboratory Apparatus - Schematic Motion G e n e r a t o r Arrangement C o n t r o l System T h e o r e t i c a l and E x p e r i m e n t a l B e s u l t s  71 72 73 75 80  5.1.1 5.1.2 5.1.3 5.2.1 5.3.1 5.4.1 5.4.2  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 ... S h i p Heave R e s p o n s e . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S h i p Heave S p e c t r a l D e n s i t y F u n c t i o n . . . . . . . . . . . . . . . . . T w o - D i m e n s i o n a l C a b l e Model T y p i c a l S e r v o v a l v e Response Motion Compensation T r a n s f e r F u n c t i o n ................ T y p i c a l F e e d b a c k Network  83 83 83 86 89 91 91  V  LIST OF SYMBOLS SYMBOL A AA  A  P  Co C  R  C  H  C CSV L  D f  FA FMBT  F  P  FBAM  F (S) F (S) B  C  GP(S)  H (s) H^tS)  H (s) Hy (S) K, K K sv  2  3  KFF KMA  K Ks K m  P  sv  H  N P Po P* . P.  P*  PM  Ps AP QA QL  Qv r R 5  t T u V Vc,,  V  C2  MEANING area of p i s t o n area o f a c t i v e c y l i n d e r p i s t o n area of passive c y l i n d e r p i s t o n t h r o t t l i n g valve c o e f f i c i e n t capillary coefficient hydrodynamic drag f a c t o r l i n e a r drag f a c t o r s e r v o v a l v e flow c o e f f i c i e n t d i f f e r e n t i a l operator d/dt f r i c t i o n f o r c e i n ram active cylinder force cable tension passive c y l i n d e r force t o t a l ram f o r c e body t r a n s f e r f u n c t i o n cable t r a n s f e r function p a s s i v e system t r a n s f e r f u n c t i o n open l o o p 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 feedforward loop t r a n s f e r function servovalve transfer function ram c e n t e r i n g l o o p t r a n s f e r f u n c t i o n displacement feedback g a i n v e l o c i t y feedback gain a c c e l e r a t i o n feedback gain feedforward s t a t i c gain mechanical advantage p a s s i v e system s t a t i c g a i n gas s p r i n g s t i f f n e s s s e r v o - a m p l i f i e r gain mass o f g a s i n p a s s i v e c y l i n d e r mass o f towed body p a s s i v e s y s t e m volume r a t i o pressure i n i t i a l p r e s s u r e i n p a s s i v e system passive c y l i n d e r pressures p r e s s u r e downstream t h r o t t l i n g v a l v e p r e s s u r e upstream t h r o t t l i n g valve s u p p l y p r e s s u r e i n a c t i v e system p r e s s u r e drop i n a c t i v e c y l i n d e r o i l flow i n t o a c t i v e c y l i n d e r l e a k a g e flow servovalve flow low l e v e l s e r v o s i g n a l gas c o n s t a n t f o r n i t r o g e n Laplace v a r i a b l e time a b s o l u t e gas t e m p e r a t u r e disturbance input displacement volume p a s s i v e c y l i n d e r volumes p a s s i v e tank volumes  vi  W x x y Yc r  power consumption body displacement tow point displacement p i s t o n displacement tow pt displacement r e l a t i v e to s h i p servovalve actuating current  t  V 8 5 5 \, "Xz X tr, t, •c u3 oi co<, 3  w  0  n  v  r a t i o of s p e c i f i c heats s m a l l p i s t o n displacement passive system c r i t i c a l damping r a t i o s e r v o v a l v e c r i t i c a l damping r a t i o servovalve flow gain s e r v o v a l v e flow-pressure c o e f f i c i e n t bypass valve flow c o e f f i c i e n t p a s s i v e system time constants servovalve time c o n s t a n t design frequency passive system n a t u r a l frequency s e r v o v a l v e n a t u r a l frequency  vii  ACKNOWLEDGEMENT I  am  most g r a t e f u l f o r t h e p a t i e n t h e l p  o f my s u p e r v i s o r , Keefer  Dr. R. B c K e c h n i e .  o f B. C. R e s e a r c h C o u n c i l  for  this  also  due t o  servovalve  and  Johnson amplifier;  of  who i n i t i a l l y  Fleck  gave  Brothers  to.Dr. Vickers  t o t h e Department  Wendy  Allen  f o r the use o f t h e i r for  keypunching  analogue  project under  was f u n d e d by t h e N a t i o n a l  Grant  #67-8183.  crew f o r b u i l d i n g t h e of E l e c t r i c a l  computer;  and t o  the m a n u s c r i p t . A l l computing  done a t t h e U n i v e r s i t y o f B r i t i s h  a  f o r making t h e v a l v e  apparatus  Engineering  idea  f o r donating  and t h e i r  equipment;  me t h e  a s s i s t a n c e . Thanks a r e  work; t o M e s s r s . Hoar and H u r r e n and l e n d i n g  encouragement  S p e c i a l t h a n k s i s due t o D r .  p r o j e c t , and f o r h i s s u b s e q u e n t Mr.  and  Columbia Research  Computing Council  Miss was  C e n t r e . The of  Canada  1  CHAPTER I I8TB0DUCTI0H  1.1  Problem The  safety  depend,  to  Description and p e r f o r m a n c e o f  a large extent,  t o d e c o u p l e wave i n d u c e d towing motion  cable.  compensation  vibration  is  to  The  is  from  gear  from  the  accomplished  by a of  a  an  in  external  are  force  automobile  isolation  features. F i r s t ,  low,  i n s y s t e m s where a  the  suspensions,  problem  the frequency  order  may be i n t h e o r d e r  o f 0.1  of  Hz, w h i l e  of fifteen  of  superposition  of  and Thus,  stern  several  frequencies the  input  of  harmonic as  the  ship  disturbance  ship  the a s s o c i a t e d  f e e t . Second, the  functions  defined  three  surface  i n c l u d i n g t h e water i t e n t r a i n s ,  the  cushions.  has  i n e x c e s s o f 30, 000 pounds-mass. F i n a l l y ,  displacement  c r motion  and r o c k e t - b o r n e i n s t r u m e n t a t i o n towing  of t h e s u b m e r s i b l e ,  function.  ship  a special class  a r e employed  Some common e x a m p l e s  distinguishing  amplitudes  usually  essentially  devices  isolated  marine  typically  i s  vehicles  of the handling  motions o f t h e s u r f a c e  system —  earthquake absorbers,  amplitude  on t h e a b i l i t y  decoupling  isolating be  disturbance.  motion  submersible  isolator.  Vibration mass  This  towed  mass  i s large  —  the v e r t i c a l  consists of  of  the  different  by a s p e c t r a l d e n s i t y is  somewhat  more  2  predictable  than t h e forms  of v i b r a t i o n  p r e s e n t i n the p r e v i o u s  examples.  The  marine  designed  to  requirement  towing  maintain i s met,  submersible w i l l eliminated.  motion  compensation  constant towing cable  i t then f o l l o w s  be z e r o ,  and  a l l  that  index  i s the r a t i o  displacement. index  of  must be  contains spectral design  It  is  less  the  than a g i v e n  Once  of  value  energy  as  at taken  In a d d i t i o n ,  t h e minimum  the  overall  within physical  (e.g.. weights, g e o m e t r i e s , e t c . , o v e r little  or  no  determining the  cost,  Three  the  problem  then  considerations and  power  include  or  to  ship  performance  frequency the the  which  sea  state  primary  a c c e p t a b l e index of spectrum  constraints  which  the p h y s i c a l c h a r a c t e r i s t i c s  design  initial  control)  the  up  compensation  the  from  be  cable.  body  that the  will  a l l systems  the  towed  this  of the  moves  such a motion  stipulated  for other frequencies  specified.  ship  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  frequency.  performance  the  amplitude  generally  greatest  density  of  Once  motion  o f the s h i p from  performance of the  as  usually  the a c c e l e r a t i o n  To p r o v i d e t h i s c o n s t a n t c a b l e t e n s i o n ,  down, t h u s d e c o u p l i n g t h e m o t i o n  system  is  tension.  undesirable  a t t e m p t t o pay o u t o r h a u l i n c a b l e  The  system  the  may  be  are  met  designer  has  becomes  that  of  o f t h e s y s t e m . Some  of  simplicity,  reliability,  consumption.  o f t h e more p o p u l a r m o t i o n  compensation  systems  are  3  shewn i n F i g . 1.1.1. a l t h o u g h very d i f f e r e n t i n appearance, employs  a  pneumatic s p r i n g i n the form of a gas accumulator t o  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 . and  (b),  the  positioner  v e r t i c a l d i s p l a c e m e n t of r e e v e d . System cable.  In  is a  a  With  cylinder  sheave  over  systems  which which  <a)  c o n t r o l s the the  cable  is  (c) uses a h y d r a u l i c winch t o h a u l i n and pay out  each  passively-acting 1.2,  each  case,  the  pneumatic  cable  tension  is  balanced  by a  s p r i n g . As w i l l be seen i n S e c t i o n 1  such systems, under c e r t a i n c o n d i t i o n s ,  can  be  tuned  to  p e r f o r m a d e q u a t e l y over a narrow f r e q u e n c y range. I t i s p o s s i b l e t o d e s i g n a p u r e l y a c t i v e system, i n which a significant  amount  of  energy  is  expended  s t a b i l i z a t i o n e f f e c t . In such a system, the  motion  of  the  a  to  achieve  transducer  l o a d and a f t e r s u i t a b l e s i g n a l  the  monitors  processing,  c o n t r o l s the f l o w of o i l t o a h y d r a u l i c a c t u a t o r . A c t i v e systems a r e s u p e r i o r to p a s s i v e ones i n t h a t they are motion  capable  of  good  i s o l a t i o n over a wider f r e q u e n c y range. However, because  they r e q u i r e a b u l k y power s o u r c e and consume a l a r g e amount energy  when  f o r marine  controlling  a massive l o a d , they are not s u i t a b l e  applications.  To improve t h e performance of a p a s s i v e large  of  expenditure  of  system  without  a  power, a h y b r i d a c t i v e - p a s s i v e system i s  In a p a s s i v e 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 l o a d i s c o n s e r v e d , a p a r t from some d i s s i p a t i o n due t o damping and f r i c t i o n . Thus, no external energy i s r e q u i r e d t o o p e r a t e t h e system. 1  4  TOWING CABLE  ACTUATOR  2U MUTATOR  (o) RAM TENSIONER  ACCUMULATOR  (b) BOOM BOBBER ACCUM.  HYDRAULIC MOTOR  Cc) TENSIONING  F I G . 1.1.1  MOTION  REEL  WlklCM  COMPENSATION  SYSTEMS  5  proposed. Such systems have been s u c c e s s f u l l y small  components  from  an  environment  v i b r a t i o n . However, as o u t l i n e d  of  used severe  field.  isolate  shock  i n the next s e c t i o n , no work  found r e l a t e d t o the a p p l i c a t i o n of such systems to towing  to  the  and was  marine  6  1.2  State  of  Vibration been  widely  the  isolation  problem;  one  analysis  of a  in  field  nature, ideas  with  being  particular of  section  no  has  general,  i t  d e a l i n g with  the  ether being  the  the  latter  method,  been v e r y  mathematical  have  especially empirical in  justification  of  the  will  d i s c u s s some of  in vibration  the  relevant  isolation,  and  work t h a t  second, i n  the  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 . there  methods a v a i l a b l e : placing  involves  either  o f an  or r e c e i v e r  the source  and  Ref.  the (15)  ( F i g . .1.2.1). I s o l a t i o n  frequencies.  using a  whereas  energy absorbing  can  r e c e i v e r a t one receiver  is  be  to  no  and  generally  either  achieved over  frequency.  a  achieved  which i s i n r e s o n a n c e  particular  experiences  source  absorption  device can  involves  be made e f f e c t i v e  Absorption  spring-mass system  reduction  Isolation  1  disturbance  protected),  o r p a s s i v e l y , and  passively  vibration  isolation.  m a t e r i a l between t h e  attachment  wide r a n g e o f  distinct  and  t o be  actively  frequency,  two  (the s y s t e m  the  source  are  absorption  a resilient  the r e c e i v e r  See  p r o b l e m . The  and  passive,  In  methods o f  ocean e n g i n e e r i n g ,  l i t t l e or  In g e n e r a l ,  1  distinct  decade.  highly theoretical,  been done, f i r s t ,  field  the  were two  past  a c t i v e and  presented. This  has  systems, both  i n v e s t i g a t e d i n the  appears that there  the  Art  At  input at a l l , but  with that this  RECEIVER a  RECEIV£R ISOLATOR SOURCE  Cc)  4—  ISOLATION! ABSORBER RECEIVER SOURCE  ^  ft>) ABSORPTION! FIG. 1.2.1  (a)  VIBRATION  ISOLATION  ISOLATION  FIG-  1.2-2  6, A B S O R P T I O N !  (b)  PERFORMANCE  ABSORPTION  CHARACTERISTICS  8  e f f e c t i s confined undesirable  to  a  resonant  corresponding  to  very peaks  the  narrow  frequency  band.  occur  at  frequencies,  separate  natural  two  frequencies  Also,  of  the  r e c e i v e r and a b s o r b e r . F i g . 1.2.2 i l l u s t r a t e s the performance isolators A  and a b s o r b e r s . i s o l a t o r as shown i n F i g . 1.2.3  p a s s i v e pneumatic  been examined by Cavanaugh .  He  l  order  system  equations  optimum c r i t i c a l volume  of  solved  the  (a) has  linearized  third  i n the f r e g u e n c y domain, and found the  damping r a t i o i n terms of the tank to  cylinder  r a t i o . F i g . 1.2.3(b) shows the frequency response of the  system, and  Fig  1.2.3(c)  shows  the  critical  damping  ratio  f u n c t i o n which y i e l d s the s m a l l e s t maximum a m p l i t u d e r a t i o . Another  passive  isolation has  system  been  d i r e c t l y a p p l i c a b l e to  automobile  suspensions  considered  a t w o - d i m e n s i o n a l l i n e a r system  freedom, and developed an optimum  examined  by  Thompson . 2  He  with f o u r degrees of  performance  index  based  cn  r i d e comfort and r o a d - h o l d i n g a b i l i t y . A  more  g e n e r a l approach  to o p t i m i z i n g passive suspensions  has been p r e s e n t e d by Hedrick f o r use speed  tracked  vehicles . 3  An  optimum  which  uses a v a r i a b l e f r i c t i o n element  See 2 See 3 See * See  Ref. Refs. Ref. Ref.  1  (5) (20) and (7)  (11)  (22)  in  the  design  p a s s i v e shock to d i s s i p a t e  of  high  isolator,  energy  has  9  LOAD  G.AS  PNEUMATIC-  TANKS  CYLINDER A  Ca)  - U  CONFIGURATION!  oi FREQUENCY  (b)  PERFORMANCE  (c)  OPTIMUM  CRITICAL  DAMPING RATIO  N =  FIG. 7.23  TANK VOLUME C Y L I N D E R VOLUME  PASSIVE PNEUMATIC  SYSTGLM  10  been  proposed Active  been  by M e r c e r systems  examined.  pneumatic control  and  f o r shock  Soliman  system  1  presented active  a  Porter,  Kriebel  3  and  dealing  d e v e l o p e d an a c t i v e  also  feedback to  a c t i v e systems  Athans,  m a t h e m a t i c a l methods f o r  have  controlled  and v e l o c i t y  Thompson c o n s i d e r e d 2  systems ,  isolation  servovalve  displacement  suspensions . highly  and v i b r a t i o n  proposed  using  the s e r v o v a l v e .  automobile  Rees*.  for  Karnop a l l with  system  linear  f o r shock  isolation*. None o f t h e a b o v e - m e n t i o n e d readily marine  applicable  systems,  specific other  are  before  more  form  which  compensation  solutions practical  much m o d i f i c a t i o n  isolators.  number o f m a r i n e operational  exists  i t i s built.  described  See see a See * see  a  passive)  documentation  2  the  a  relate  t c make them  i n the  mostly  solutions  i s  to  are too  useful  for  design purposes.  (mostly  1  while  in  o f motion  The t h e o r e t i c a l  and r e q u i r e  There  as  t o t h e problem  environment.  linear  work i s  in  to  help  Most s y s t e m s Section  1.1,  motion  around  the  predict consist  world,  a system's  but  little  performance  o f a pneumatic  are  systems  spring,  classed  as  vibration  K e e f e r p r o p o s e d a s i m p l e manner i n which  an  isolator  R e f s . (17) and (18) R e f . (21) R e f s . ( 1 3 ) , (1) and (8) R e f . (10)  and  compensation  11  c a n be of  made i n t o  the  Fig.  compensator  1.2.4(a)  Such  a system  a t t h e f r e g u e n c y which  vibration. system.  Note  Buck  2  a  ram  tensioner  i s t u n e d so t h a t contains  that  that  has d e v e l o p e d a c o m p l e t e  power c o n s u m p t i o n  3  performance  of a tuned  a t t h e expense  analysis  of  systems,  o f such a  the p e r f o r m a n c e  c a n be r e d u c e d  i s used  Bef. Ref. Ref.  a  amplitude of  (9) (3) (19)  solely  weight  f o r motion  by  incorporating  o f the l o a d  compensation.  system,  of a  s y s t e m . I n a d d i t i o n , t h e y have not r e c o g n i z e d  system  See See See  such  3  to support the s t a t i c  1  in  S u t h e r l a n d s u g g e s t e d t h e use o f a c t i v e  system  2  1  the a n t i - r e s o n a n c e  bandwidth  n o r s u g g e s t e d a method o f p r e d i c t i n g nonlinear  moticn  attenuation.  and  neither  damping i n c r e a s e s  of  disturbance .  the dominant  F i g 1.2.4(b) shows t h e t y p i c a l  the system's  but  mass o r t h o g o n a l t o the i n p u t  illustrates  configuration. occurs  an a b s o r b e r by making t h e d i r e c t i o n  real,  the a  while the  fact  passive active  12  (cO GENERAL  ARRANGEMENT  FREQUENCY  (b) FIG.  1.2.4  P E R F ORMANCE  TUNED  RAM  TENSIONER  13  1.3  O b j e c t i v e s and Scope of P r o j e c t The o b j e c t i v e s of t h i s p r o j e c t  dynamics  of  an  active-passive  are  first,  to  study  motion compensation system f o r  marine towing, and second, t o use the r e s u l t s of t h i s develop  guidelines  for  use  in  study  Representing a t y p i c a l closely itself  2.  system  approximates  in  reality,  a  yet  form  which  which  lends  to mathematical a n a l y s i s and s i m u l a t i o n .  Developing the mathematical model, i n c l u d i n g nonlinearities  as  hydrodynamic  drag  such  and  dry  friction. 3.  Linearizing  the  mathematical  equations  and  conducting a frequency-domain a n a l y s i s t o o b t a i n a first  approximation  of  the  important  system  parameters. 4.  Developing a d i g i t a l computer which  will  validate  simulation  program  and o p t i m i z e the parameters  derived i n step #3, 5.  C o n s t r u c t i n g a s m a l l working model validity  of  the  mathematical  to and  check  the  simulation  models. 6.  R e l a t i n g the  results  design o f r e a l systems.  of  to  d e s i g n i n g r e a l systems. These  o b j e c t i v e s are accomplished by proceeding i n s i x s t e p s :  1.  the  the  foregoing  to  the  14  CHAPTER I I THEORETICAL  2 .1  Typical Some  systems  of  System  the  commonly  system  t h e p a s s i v e one. bobber,  this  tensionig  winch,  adding  adding a  configurations  shewn  mathematically.  The  passive  motion  It i s possible  by a d d i n g an a c t i v e  In t h e c a s e  means  used  1.1.1.  were shown i n F i g .  active-passive  ANALYSIS  of a  the  second in  similarity  cylinder,  1.1.1  between  are' (a) and  the mechanical advantages of the reeving,  (a),  o r t h e boom, i n t h e c a s e o f the  winch,  cylinders single  connected  cylinder.  tensioner  The Fig.  overall  2.1.1.  an  Thus,  The  f o r the purpose  as a t y p i c a l  configuration passive  subsystem  hence  consists  electrohydraulic  are  processing  processed  network  i s the  in  of t h i s  cf a hydraulic  fed  The  three similar  the  case  — of  In t h e c a s e  t o a number o f m o d e l l e d as project  system  i s t h e same as  servovalve.  and  boom  the  a ram  system.  The  of a c c e l e r o m e t e r s mounted on t h e t o w i n g signals  and  (b) i s o b v i o u s  can be  of the t y p i c a l  subsystem  to  w h i l e f o r the  (b), are d i f f e r e n t .  and  parallel  very  motor i s e q u i v a l e n t  in p a r a l l e l ,  i s selected  the a c t i v e by  the h y d r a u l i c  an  motor.  only  of  devise  tensioner  hydraulic  Fig  to  actuator  ram  second  compensation  i s shown i n  before,  cylinder  control and ram  most v i t a l  component  controlled  system  consists  sheaves,  to t h e s e r v o v a l v e . of  while  The  the  whose signal system,  FIG.  2,1.1  ACTIVE/PASSIVE  RAM  TENSIONER 01  16  and  will  be t h e s u b j e c t  As  pointed  suspended  barge. not  c a n be  from a s t a t i o n a r y  towing  analysis.  the  load  c a n be a d i v i n g  s h i p , a submerged  body towed  knots), or a surface  vessel  moving h o r i z o n t a l l y represented axis  a barge  a  one-dimensional  is  vertical.  In  curve.  Since  with the b e h a v i o u r of the  typical  system  cable. Application dimensional discussed  as  elastic  cable,  i n Chapter  of as V.  the  motion  the cable  link  by  to  a  whose  is  the  the  Walton  ship  complex concerned  compensation  include  approach  developed  project  a  which  t h e c a s e o f a moving  this  considered w i l l  at h i g h  ship  t o t h e water,  o r s u b m e r s i b l e , t h e c a b l e w i l l assume  three-dimensional primarily  with r e s p e c t  bell  such a s  For the case o f a s u b m e r s i b l e supported from a  longitudinal  the  earlier,  ( i n t h e o r d e r o f 10  speed  is  out  of thorough  system,  one-dimensicnal case and  of  three-  Polachek,  is  17  2.2  The E q u i v a l e n t  Model  To f a c i l i t a t e the a n a l y s i s of the ram  tensioner  i n S e c t i o n 2 . 1 , the f o l l o w i n g s i m p l i f i c a t i o n s  1.  The  static  tension  submersible's  in  weight  the  i s not  s i m p l i f i c a t i o n does not a f f e c t motion  2.  The  cable  described  w i l l be made:  due  to the  considered.  This  the dynamics  of the  compensator.  cable  i s considered  to be a one-dimensional  e l a s t i c l i n k f o r the reasons s e t f o r t h i n  Section  2.1.  3.  The  passive  subsystem i s considered  pneumatic. T h i s r e s t r i c t i o n complexity  of the  demonstrate compressible  the  a c t u a l l y i n c r e a s e s the  problem, method  fluid  flow  to be p u r e l y  but  of  i s included  application  eguations.  to  of the  In  many  a p p l i c a t i o n s , the p a s s i v e system would a c t u a l l y be an  "air-over-oil",  shown i n Figure Using chosen  or  hydropneumatic  1.1.1.  these s i m p l i f i c a t i o n s , i t i s p o s s i b l e  system  system, as  to  model  as shown i n F i g u r e 2 . 2 , 1 . The form shown i n F i g .  2.2.1 was devised t o f a c i l i t a t e mathematical a n a l y s i s and model  the  construction.  The a c t i v e and passive motion  test-  compensation  CONTROL  SERVO VALVE  —f-y-  GAS BOTTLES  BYPASS VALVE  X  DAMPING VALVES  PASSIVE  r  X MM $•3  ill  o n  S Y S T E M  CYLINDER  ACTIVE  j  ACCELEROMETERS CABLE  CYLINDER  U  FIG.  2.2.1  EQUIVALENT  l_  SYSTEM  19  c y l i n d e r s are placed piston  rods  horizontally  connected  containing  The  the  amplitude, s i m u l a t i n g load  desired  is  modelled  mass,  in  their  the  ship.  function  with  c a r r i a g e i s d r i v e n h o r i z o n t a l l y i n a s i n u s o i d a l manner, with  the  tc  carriage,  The  of  as  a  parallel.  d e s i r e d frequency and  sc  on  the v e r t i c a l  by  a  connected  compensation p i s t o n rod by means of a spring model  the  c a b l e . The  of F i g . 2.1.1 vertical. not  As  Therefore,  of  the  in  passive  p r e s s u r i z e both s i d e s of the p a s s i v e static  f o r c e at the  In g e n e r a l , motion  horizontal  system,  dynamics o f the cable and  instead  of  towed body i s the  is  spring  necessary  to  c y l i n d e r , such that the  net  system  but  does  accurately not  models  f u l l y consider  towed body. However, the design  developed here i s f l e x i b l e enough to accomodate these if  r e a l case  p i s t o n red i s zero.  t h i s equivalent  compensation  it  motion  which i s assumed to  modelling  system,  carriage  the  r e s u l t , the s t a t i c weight of the  considered.  characteristic  to  e n t i r e system i s s i m i l a r t c the  except that a l l motion i s a  second  motion  the necessary parameters are a v a i l a b l e to the  the the  method  additions  designer.  20  The P a s s i v e System  2. 3 The of  a  passive s i d e of the system under c o n s i d e r a t i o n c o n s i s t s  pneumatic  ram,  with each end connected  v a l v e t o a r e c e i v i n g tank  (Fig. 2.3.1).  The  via a throttling  t h r o t t l i n g , valves  are used to i n t r o d u c e damping i n t o the system. The  mass flow to and from  a tank or c y l i n d e r i s d e r i v e d i n  Appendix A, and i s given by  (2. 3. 1)  where m i s the mass flow r a t e , R i s the gas constant f o r the p a r t i c u l a r gas used, T i s the a b s o l u t e  temperature,  V i s the tank or c y l i n d e r volume, P i s the a b s o l u t e and  pressure,  i i s the r a t i o of s p e c i f i c  heats.  Because the r e c e i v i n g tanks have mass  flows  are  proportional  to  fixed  volumes,  and  the  the negative of the pressure  changes, we can w r i t e  (2.3. 2)  22  and  oc  m  -  Pt (2.3.3)  where V and  t  i s the tank volume  Pt i s the tank  Substituting  pressure.  (2.3.2) and  ,  (2.3.3) i n t o  (2.3.1) g i v e s  p  VM  " 7RTT7 *' ™  -  -  V t  *  (2.3.4)  P  where the s u b s c r i p t s 1 and 2 r e f e r to the l e f t  and  right  hand s i d e s of the p a s s i v e system, r e s p e c t i v e l y . In  general,  the volume of f l u i d flow  in  and  the  volume of the r e c e i v i n g tanks,  contained, i s large  out of the tanks;  can be considered c o n s t a n t ,  1  compared  to  and  hence  the  mass  thus the temperature of the  gas  i.e.  For an a c t u a l system having an e f f e c t i v e p i s t o n area of 13.46 square i n c h e s , a displacement of f i v e f e e t as measured at the tow p o i n t causes a change i n absolute temperature of only 1.9%, assuming a d i a b a t i c compression or expansion (worst c a s e ) . 1  23  •  r R To  i 2.3.2  ng. BLOCK DIAGRAM  of  TANK  DYNAMICS  FIG. 2 . 3 . 3 BLOCK D I A G R A M of  CYLINDER  DYNAMICS  24  Ti =  F i g . 2.3.2  (2.3.5)  i s a b l o c k diagram of the tank flow equations.  The by  = T„  mass  flow t o and from the c y l i n d e r s i s also  (2.3.1), except  V  that:  - V  z  expressed  =  V  =  VV1,  (2.3.6)  - A ^/ 2  Aj  V, 2  C2  t  (2. 3.7)  -A j 2  oC  P, (2.3.8)  where V  and V  n  are the volumes  2  of  the  left  and  right  s i d e s of the c y l i n d e r , V  and V  C|  fl and t  and  A  2  C2  are the i n i t i a l  values of V  r  and  are the e f f e c t i v e p i s t o n a r e a s ,  y i s the p i s t o n  displacement.  V, z  25  Substituting  (2.3.6),  I  (2.3.7) and (2.3.8) i n t o  Z  '  1  1  (2.3.1) g i v e s  J J K-l, (2. 3.9)  I £T  Due large  2  to the r e l a t i v e l y small volume o f the c y l i n d e r and the  v a r i a t i o n i n pressure,  temperature  n e g l i g i b l e . Using the equation  change  is  nc  longer  of s t a t e f o r an i d e a l gas,  p. y. T;  =  — — 1  (i*l,2)  L  (2.3.10)  where m-  {  is  the  mass  of  gas  in  the  i - t h side of the  cylinder.  Substituting  (2.3.10) i n t o  (2.3.9) g i v e s  r±A '  Fig..  2.3.3  _  +  m  f I  Pz  shows  the  block  i  A,  A  z  (2.3.11)  . "j  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 i n  26  Appendix A  and i s g i v e n by  (2.3.12)  where C  0  i s the valve  P  u  i s the upstream p r e s s u r e ,  P and  d  T  u  constant,  i s t h e downstream p r e s s u r e , i s the upstream  I t i s observed  temperature.  t h a t t h e d i r e c t i o n of mass f l o w i s from high  t o low p r e s s u r e ; t h a t i s , P , u  greater P^ > P  of  P  ttr  V  then  t  Consequently, determined.  .  A  m> l  By 0;  i n solving Then  m-. F i g . 2.3.4  the  upstream  pressure,  i s the  the c o n v e n t i o n shown i n F i g . 2.3.1, i f conversely,  when  P  ti  < P ,  < 0.  t  (2.3.12) the upstream end must f i r s t  be  the c o r r e c t a l g e b r a i c s i g n can be a s s i g n e d t o  shows t h e b l o c k diagram o f t h e v a l v e e g u a t i c n s .  The e q u a t i o n s  (2.3.4),  (2.3.9) and (2.3.12) must be  s i m u l t a n e o u s l y t o y i e l d P, and P The f o r c e generated  z  solved  given a p i s t o n displacement  i n t h e ram can then be  y.  found:  (2.3. 13)  where Fp i s t h e ram f o r c e .  FIG. BLOCK  DIAGRAM  of  2.3.4. VALVE  DYNAMICS  28  The in  block  F i g . 2.3.5. S i n c e  equations The  diagram  i t is difficult  analytically,  strategy  of the p a s s i v e  Calculate  2.  C a l c u l a t e P-, as shown i n F i g .  3.  Calculate  4.  Repeat s t e p s P . and P-, t  as shown i n F i g .  ti  m as  no  sclve  system  shown i n F i g .  1 t o 3 with longer  2.3.3 2.3.4.  t h e new  change  value  from  C a l c u l a t e t h e ram f o r c e f r o m  of m  ;  until  iteration  (2.3.13).  of  developed.  2.3.2  iteration. 5.  this  s o l u t i o n i s as f o l l o w s :  1.  ;  to  d y n a m i c s i s shown  a n u m e r i c a l s o l u t i o n i s now  used i n t h e numerical  P  system  to  Mi  TANK.  *1  Pt,  CYUMDER. S I D E  * I  FIG. BLOCK  DIAGRAM  of  2.3.5  PASSIVE  SYSTEM  DYNAMICS  30  2.4  The A c t i v e System The  of  a c t i v e s i d e of the system under c o n s i d e r a t i o n c o n s i s t s  a p o s i t i v e displacement  accumulator, It  hydraulic  the  is  assumed  that  accumulator  the accumulator  the  pressure  will  examined  compressibility in  hydraulic  compared  effect  flow of only 2.6% f o r a t y p i c a l  The by  the  (Fig.  pressure  vary s l i g h t l y with changes i n flow  of  Appendix C, and,is found  the c o m p r e s s i b i l i t y  the  valve s e t t i n g . In f a c t ,  to  the  T h e r e f o r e , the supply pressure i s c o n s i d e r e d The  gas  ( F i g . 2.4.1)  hence m a i n t a i n i n g  l o s s e s i n the  i s negligible  valve,  pump discharge flow r a t e always  equal t o the r e l i e f  r a t e due t o f r i c t i o n fluctution  relief  s e r v o - v a l v e , and h y d r a u l i c c y l i n d e r .  exceeds the system requirement, in  pump,  the  lines,  but  this  working  pressure.  constant. hydraulic  fluid  is  t c c o n t r i b u t e an e r r o r i n  full-scale  system.  Therefore,  of the f l u i d i s not c o n s i d e r e d .  f l o w - p r e s s u r e r e l a t i o n s h i p f o r the s e r v o - v a l v e i s g i v e n manufacturer  f o r s e l e c t e d values of a c t u a t i n g s i g n a l , z  2.4.2). As shown i n Appendix B, t h i s r e l a t i o n s h i p ,  for a  zero-lapped v a l v e , can be a c c u r a t e l y modelled by 1  See Ref. servo-valves. 1  (6)  f o r the e q u a t i o n s o f under- and over-lapped  GAS  ACCUMULATOR OIL ACTUATING FROM  CURRENT"  PUMP  ->  SERVO VALVE.  TO  TANK  C - > —  11 FIG. 2.4.1  FA  ACTIVE  SYSTEM  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 v a l v e .  AP i s the pressure drop a c r o s s the l o a d . C  sv  i s the  characteristic  constant  of  the  servo-  valve , P and  s  i s the supply  constant,  z i s the a c t u a t i n g s i g n a l .  Leakage  a c r o s s the c y l i n d e r i s o f t e n u s e f u l i n s t a b i l i z i n g  a servo-system, Leakage  p r e s s u r e , assumed  is  and  provided  is by  therefore  included  in  the  analysis.  means o f an a u x i l i a r y path around the  p i s t o n , and c o n t r o l l e d by means o f a v a l v e .  The  leakage  flow,  Q , i s given by L  (2.4. 2)  where C The  v  i s the c h a r a c t e r i s t i c constant of the v a l v e .  t o t a l flow i n t o  the flow through  the ram, Q , i s the d i f f e r e n c e between A  the s e r v o - v a l v e and the leakage  flow:  33  The s i g n convention i s such t h a t Q causes the p i s t o n (2.4.2) i n t o  A  is  positive  to move to the r i g h t . S u b s t i t u t i n g  (2.4.3)  when i t  (2.4.1) and  gives  The v e l o c i t y o f the p i s t o n with respect  to the c y l i n d e r , y,  can now be expressed as  where  a i s the e f f e c t i v e area o f the p i s t o n . A  The rod,  force  available  to  do work at the end of the p i s t o n  F , i s given by A  FA -  Equations  A AP A  (2.4.5) and (2.4.6) can now be combined to  (2.4.6)  yield  34  y directly  as a f u n c t i o n o f  F :  (2.4.7)  In  computing  sign is  (2,4.7) i t i s n e c e s s a r y  assignment  done by  1 .  In  to avoid  the  In t h e  case  of  the  sign  case  of F ,  pressure  drop.  consistent  w i t h i n the  surds.  with  o f z. The  leakage  the s e r v o - v a l v e ,  surd  flow,  which i s a c t u a l l y  A  The  surd thus  then  becomes  the  surd  This  9  the d i r e c t i o n  gives the  modified the  sign  s  of  becomes  (2-"-9)  A  the  P  takes  ; n(F )y|F*/^|  Substituting (2.4,7)  flow through  of the  the sign  S  into  values  artificial  noting that  must t a k e  2.  negative  t o i n t r o d u c e an  expressions  equation convention:  for  (2.4.8) and  computing  y  (2.4.9) from  F  A  35  a- t "/Ms-*/* I -  -<-  c  Fig.  2.4.3 shows the block diagram of the a c t i v e  ,2  system.  ,o)  2  >•  SICM  (5 ABS  ABS  — 3 * -  AA  x: x  SI6M  FIG. 2 . 4 . 3  BLOCK DIAGRAM  of A C T I V E  SYSTEM  <  37  2.5  A c t i v e - P a s s i v e System The  t o t a l ram f o r c e i s the sum of the f o r c e s exerted  p a s s i v e and a c t i v e c y l i n d e r s , l e s s  F  t o B  -  F  P +  by the  friction:  F.-f  '  (2.5.D  where R*M i  F  and The  s  t  n  e  r  a  m  force,  F  A  i s the a c t i v e c y l i n d e r f o r c e ,  F  p  i s the passive c y l i n d e r f o r c e ,  f i s the f r i c t i o n  force  felt  f c r c e , as d i s c u s s e d i n Appendix E.  by the c a b l e i s d i r e c t l y p r o p o r t i o n a l to F  where the constant  of p r o p o r t i o n a l i t y i s the r e c i p r o c a l  of  ,  R A M  the  mechanical advantage of the r e e v i n g :  F"NET  = — J/  Fo »RAM  <'* ) 2 5  2  where K and The  Mfl  i s the mechanical advantage of the r e e v i n g ,  F ^ i s the f o r c e a c t i n g on the c a b l e . towed body can be r e p r e s e n t e d  hydrodynamic  drag  and  towing  by a mass K, subjected to  cable tension.  ( F i g . 2.5.1) The  c a b l e i s assumed to be a massless l i n e a r s p r i n g . The  compensator f o r c e causes an  elongation  of  the  cable  3 8  ==D>|  k/VA  BODY  CABLE — * - Xi  FIG. 2.5.1  FIG. 2.5.2  DRAG > %  CABLE /BODY  BLOCK DIAGRAM  MODEL  of CABLE. /BODY DYNAMICS  39  according  t o the r e l a t i o n  fw = ICC-X, - 7c)  (2.5.3)  T  The  cable then a p p l i e s the same f o r c e to the body, whose motion  can be d e s c r i b e d by:  FKET =  M X + C %  (2.5.4)  2  where M i s the mass o f the towed body, and  C  Equating  i s the hydrodynamic drag  H  (2.5.3)  and  (2.5.4)  n o n l i n e a r d i f f e r e n t i a l equation  *  +  %  +  ^  X  factor.  and  rearranging,  gives  the  of motion of the body:  =  '  W  X  '  (2.5.5)  where 60  Once  c  i s the cable-mass n a t u r a l frequency,  (2.5.5)  is  solved,  ^ K /M" . e  i t i s p o s s i b l e to f i n d  a p p l i c a t i o n o f (2.5.4) or (2.5.3).  The  block  F  by d i r e c t  M 6 1  diagram  o f the  towed body and c a b l e system i s shown i n F i g . 2.5.2. The  absolute  displacement  of  the  shipboard  end of the  40  c a b l e , x„ i s the sum of the i n p u t , u, and  the  displacement  of  the end of the c a b l e with respect to the i n p u t , y : c  For  the  case  X, =  U + vjc  where  the  mechanical advantage the  (2.5.6)  actuator  acts on the cable through a  (e.g., the ram t e n s i o n e r  of  Fig.  2.1.1),  motion of the c a b l e with r e s p e c t to the s h i p ' s s t e r n can be  expressed  as:  y  c  =  I^A y  <--> 2  5  7  w here y i s the extension Equations diagrams  of  of the a c t u a t o r .  (2.5.6) and (2.5.7)  are combined with  the  block  the p a s s i v e , a c t i v e , and cable-mass systems  (Figs.  2.3.5, 2.4.3, and 2.5.2) t c give the block diagram of the e n t i r e system, as shown i n F i g . 2.5.3.  CABLE/BODY FIG. 2.5-2  KMA  PASSIVE SYST. FIG. 2.3.5  -  —  ACTIVE SYSTFIG. 2-4.3  CONTROL SYST. FIG- 2-6 2  FIG. 2.5-3  BLOCK DIAGRAM  of ACTIVE-PASSIVE  SYSTEM  42  2.6  The The  active  C o n t r o l System  c o n t r o l system generates actuator  v a r i a b l e s . The O  R  and  r  F»IET  F  certain  because  and  tc  operate  processing  suggests they  t h a t both can  be  specified  and  the  certain  as e i t h e r x controlled  are l i n e a r l y dependent. T h i s means  of performance, as d i s c u s s e d i n Chapter  /u or x/u,  H E T  means of monitoring  Figure 2.5.2  t h a t the index  signal  c o n t r o l l e d v a r i a b l e can be considered  simultaneously  either  by  the  both  must  be  l i m i t s f o r acceptable  made  drift  fall  te  below  o p e r a t i o n . In a d d i t i o n ,  the c o n t r o l system must ensure that the long of the a c t u a t o r p i s t o n does not  tc  I , can  from  term average motion the  centre  of  the  cylinder. The the  controlled variable  primary  actuating  c o n s t i t u t e s a simple  (F  N C T  or x) can  signal  for  as shown i n F i g . 2 . 6 . 1 .  omitted  Fig.  block best  may  servovalve.  feedback c o n t r o l system where the  input i s , z e r o , from  the  be used t o generate  2.6.1  (The  for c l a r i t y . )  The  contain f i l t e r s , i n t e g r a t o r s , etc.,  passive  This  reference system  is  " C o n t r o l Elements" as  required  for  operation. In  addition  t c feedback, i t may  p o r t i o n of the d i s t u r b a n c e feedforward The  input,  be d e s i r a b l e t c i n c l u d e a  u(t),  as  indicated  by  the  loop i n F i g . 2 . 6 . 1 .  piston  centering  control  r e s t o r e the p i s t o n to the c e n t r e of  is the  a  miner  actuator  loop used to slowly  with  43  SHIP FEEDFORWARD  CONTROL ELEMENTS  MOTION  TRANSDUCER  SERVO  U(i)  LOAD  ACTUATOR  VALVE  CONTROL ELEMENTS  TRANSDUCER  CONTROL ELEMENTS  CENTERING  TRANSDUCER  FIG. 2.6.1 ACTIVE SYSTEM WITH CONTROL  U(s)  FEEDBACK  BLOCKS  H fe) FF  1  V(s)  K SY  2(s)  X(s)  FIG.  2 . 6 . 2 BLOCK  D I A G R A M of C O N T R O L  SYSTEM  44  respect  to the f o r c i n g freguency.  the loop i s a t l e a s t one  That  order of  magnitude  r e c i p r o c a l of the i n p u t freguency. and  x  as  the  availability however,  controlled  and  it  towing c a b l e f o r  necessary the  The  end  of  to  provide  feedback  the  signal  cable,  x , t  s i g n a l . Such a s t r a t e g y i s e a s i l y ram  tensioner  for  the  and  constant  demonstration  boom-bobber tension  purposes,  it  is  made  transducers.  v e s s e l . I f t h i s i s not p o s s i b l e , then shipboard  greater  than  based  on  the  a p h y s i c a l path  i n the  reach  the  surface  the a b s o l u t e motion of can be used as the  (Fig.  the  feedback  case  of  ( F i g . 1.1.1 (a) , (b)) , but winch  N t T  case,  f e a s i b l e i n the  is  F  either  to  In  of the  c h o i c e between using  variable  s u i t a b i l i t y of  is  i s , the time constant  the  net  1.1.1(c)).  so Fcr  assumed that x i s the feedback  variable. The blocks,  c o n t r o l system  elements  are  as shown i n F i g . 2.6.2. The  i s given by  R(Y)=  now  lumped  into  three  t o t a l c o n t r o l v o l t a g e , B (s)  (in Laplace n o t a t i o n ) :  M Cs)X(s) + W Cs)\J(s) + U,(s)Y(s) fB  (2.6.1)  FF  where Hpj,(s) r e p r e s e n t s the feedback element, Hpp(s) r e p r e s e n t s the feedforward and The  element,  (s) r e p r e s e n t s the p i s t o n c e n t e r i n g element. s e r v o v a l v e a c t u a t i n g c u r r e n t i s obtained  by p a s s i n g  the  control  voltage through a power  Z(s)  amplifier  = J< R(s) sv  where Z (s) i s the a c t u a t o r and  K  w  i s the a m p l i f i e r  current, gain.  46  2.7  Computer The  been  Simulation  a c t i v e - p a s s i v e system as d e p i c t e d  modelled  by  Program" (CSMP), an  means IBM  of  a  in  "Continuous  product f o r use  as  2.5.3  Systems  has  Modelling  on t h e i r System/370.  programming language c o n s i s t s of a number of such  Fig.  functional  The  blocks  i n t e g r a t i o n , d i f f e r e n t i a t i o n , e t c . , i n a d d i t i o n to a l l  the usual mathematical f u n c t i o n s  available  blocks  the same manner as an analogue  are  assembled  computer network, variables.  but  Integration  different built-in  in  much  without can  Bunge-Kutta with  method  selected  because  fixed it  l o g i c flow  F i g . 2.5.3. The  integration is  of the program i s  listing  using  These  of  scaling  any  of f i v e  employed i n t h i s p r o j e c t i s  expensive f o r the degree of accuracy The  Fortran.  inconvenience  performed  r o u t i n e s ; the one  f o u r t h order was  be  the  in  found  interval. to  be  the  This least  required. exactly  i s shown i n Appendix  as  F.1.  depicted  in  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  system developed i n Chapter I I a r e d i f f i c u l t t o extensive  use  of  computers.  In  q u i c k l y g a i n a f e e l f o r the problem approximations  handle  without  o r d e r . t h a t t h e d e s i g n e r can and thereby e s t a b l i s h  first  f o r i m p o r t a n t parameters, I have s i m p l i f i e d the  e q u a t i o n s t o permit a f a s t approximate approach  compensation  uses  linearized  equations  s o l u t i o n . The and  a  simplified  frequency-domain  solution.  3.1  L i n e a r i z e d P a s s i v e System In l i n e a r i z i n g the p a s s i v e pneumatic system, I have assumed  t h a t t h e changes i n p r e s s u r e w i t h i n t h e c y l i n d e r and  tanks  are  l i n e a r w i t h r e s p e c t t o t h e p i s t o n d i s p l a c e m e n t , y, and t h a t y i s t o o s m a l l t o a f f e c t t h e temperature  i n t h e system. T h i s l e a d s t o  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 i s s m a l l , and denoted  2.  P +P =2P ; 1  2  by S.  i . e . , the average  0  pressure  c y l i n d e r i s c o n s t a n t , and e q u a l t o pressure P 3.  P =-P ; :  1  a  0  displacement  the  i n the quiescent  .  i . e . , t h e r a t e o f i n c r e a s e of p r e s s u r e on  one s i d e of t h e c y l i n d e r i s e q u a l t o t h e r a t e  of  48  decrease 4.  The  on the other.  temperature  throughout  the p a s s i v e system i s  constant and equal t o T<>. Furthermore, the geometry of the system i s assumed  to be  symmetric, which leads t o the f o l l o w i n g :  5.  t  Z  P  averaged 6.  i.e.,  A +A =2A ;  i . e . , the tank volumes are averaged  c  constant The  tca  V . c  gas flow equations  the mass flow through and r e s t a t e d  e f f e c t i v e p i s t o n areas are  t o a constant Ap.  Vc, + V = 2 V ; C i  the  are l i n e a r i z e d by f i r s t  the t h r o t t l i n g  considering  valve as given i n  (2.2.12),  here:  (3.1.1)  where  Considering  the  upstream  equal to the guiescent pressure pressure  drop  (P -P ), equation e  a  pressure, P , c  and  (3.1.1)  P , w  small  as constant and variations  can be l i n e a r i z e d  in into  49  the form:  #  A w  (3. 1. 2)  C A (P.-Pa)  -  r  where  5>v^  3Cfl/P.)  S>CPJ/PO)  3(Po-^)  9w-  (3. 1.3)  The flow through cycle  the valve w i l l be p o s i t i v e  for  the  half  when the p i s t o n moves one way, and negative f o r the other  h a l f . Therefore,  (P "P<i)  c  a  0  n  assume p o s i t i v e or negative values.  It i s thus necessary t o s e l e c t m = ( P - P ) =0 as the e q u i l i b r i u m 0  point  about  which  will yield i n f i n i t e (P -P ) 0  a  curve,  d  p e r t u r b a t i o n s are c o n s i d e r e d . However, t h i s value f o r C , s i n c e the s l o p e r  of  the  m vs  as shown i n F i g . 3.1.1, i s v e r t i c a l at P - P = 0 . o  I t i s t h e r e f o r e more reasonable coincide  e  to s e l e c t  values of V , a  d  Pj which  with some average o p e r a t i n g c o n d i t i o n , f o r example  the  root-mean-sguare v a l u e . The e q u i l i b r i u m p o i n t , however, i s s t i l l the  origin.  Renaming  P  e  and P  d  t o correspond  to the tank and  c y l i n d e r pressures, the l i n e a r i z e d mass flow equations are:  50  C f Pt,- P.)  =  r  (3.  Equation Fig.  1.4)  (3.1.4.) i s p l o t t e d together with i t s n o n l i n e a r form i n  3.1.1.  The  cylinder  Incorporating above, equation  flow  the  equations  simplifications  are of  considered  geometry  as  next.  discussed  (2.2.9) becomes:  v  '+  A  '  s  p, +  P.  (3. 1. 5)  The net mass flow i n t o the c y l i n d e r i s given by  (3. 1. 6)  I n c o r p o r a t i n g the approximations  of pressure as d e s c r i b e d abcve,  and i n t r o d u c i n g the d i f f e r e n t i a l  operator B= ~ , equation  can be r e w r i t t e n a s :  t  (3.1.6)  FIG 3,1.1 PRESSURE - FLOW {or  TMROTTLIKJG  CURVE VALVES  1 + 2 /u) 5  F  s  A  N+l  L  1  FIG. LINEARIZED  +  ' +t N  S  P  J  3.1.2  PASSIVE S Y S T E M TRANSFER FUNCTION!  52  (3. 1.7)  F i n a l l y , the flow out o f the r e c e i v i n g tanks i s c o n s i d e r e d . Equation  (2.2.4) can  be  rewritten,  in  differential  operator  n o t a t i o n , as:  Vi  DP*. (3.1.8)  Solving  (3.1.4.)  for P  tl  and p  ti  and s u b s t i t u t i n g  i n t o equation  (3.1.8), the net flow i n t o the c y l i n d e r can be expressed as:  /.  Equating  ,x  • v  t  (3.1.9) and  r  PCP.-PQ  1  (3.1.6) and s o l v i n g  2Y  P. - P  I +  (3. 1.9)  f o r P-P  A P»  (3.1.10)  P  Vt/Vc v t —  gives  D  The f o r c e exerted by the ram i s then given by  53  F  and  the  P  overall  = Ap ( p , _ P )  (3.1.11)  z  transfer  function  of  ram  force  tc  piston  displacement i s  ^  _  2  y  L  P o  1  A  P  (3.1.12)  TC  T  R  RT.  U  Lumping parameters and i n t r o d u c i n g in  place  of the d i f f e r e n t i a l operator,  J  the Laplace operator,  s,  D, the o v e r a l l t r a n s f e r  f u n c t i o n becomes  1 + 2 N+ 1  s (3. 1. 13)  1 +  where K*=  2 V  y  A f c  '  = Static stiffness,  N=V /V = tank t o c y l i n d e r volume r a t i o , t  A  N  D  2  S  /  C  J  N  t  =  Y C V R T .  The parameters w„and 3 are the n a t u r a l damping  ratio,  respectively,  freguency  and are u l t i m a t e l y  and  critical  a f u n c t i o n of  the mass which the system must c o n t r o l . In p a r t i c u l a r ,  5%  W„  The  -  [W7  7  H  linearized  ,  J  ""  2rcVET  t r a n s f e r f u n c t i o n G (s) i s p  (3.1.14)  0  now  cover the f u l l range of p i s t o n displacement, y, such  F  The  transfer  P  =  GpGOi,  function  is  extended  to  that  (3.1.15)  shown i n block diagram form i n F i g .  3.1.2.  I n t r o d u c i n g the time constants  f,=  2%*  ;  T  a  »  —  (3.1.16)  the t r a n s f e r f u n c t i o n may be r e s t a t e d as  Cf (i)  (3.1.17,  55  3.2  Linearized Active The equation  System  o f the a c t i v e system, as presented  in  Section  2.3, i s r e s t a t e d here:  9 = ^ z v^ Equation  x  V^/A*  (3.2.D  (3.2.1) i s to be l i n e a r i z e d about some o p e r a t i n g  (Yo » z ,F ) . P e r t u r b a t i o n s 0  ~  ? a / a a  about t h a t point can  A(>  be  point  represented,  by A - n o t a t i o n , as  =  Ay  (A., A *  T- A ^ A p ; )  +  A  3  AFA  (3.2.2)  where  (3. 2. 3)  ^* f S  )SETV6 ~  7=—71—  ^A^rTT  1  (3.2.4)  ..Cy  ^ S - U F J ^ The  constants  A,  ZAZ JKJA*  ~  and  X  2  are termed the "flew  (3,2  '  5)  g a i n " and  >  "flow-presure  c o e f f i c i e n t " of the s e r v o v a l v e ,  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  i n the l i n e a r a n a l y s i s because of  the  discontinuity  a t y= 0. Equation  (3.2.2) i s v a l i d  o p e r a t i n g point  only f o r p e r t u r b a t i o n s about  ( y , z ,F ) . However, during e  0  Ao  the  normal o p e r a t i o n  of  the valve the o p e r a t i n g p o i n t can t r a v e l i n a band spanning both the  negative  and  positive  l i n e a r i z e d equation  regions  of  j ,  z ,  then becomes i n a p p r o p r i a t e  and  F .  The  A  in  its  present  y,  z, and  form. It  is  proposed  that  the p e r t u r b a t i o n s  F  c e n t r e d about the o r i g i n , i . e . , y, =z„ =F =0, but t h a t  "> be  Ao  calculated  about  (3.2.3) to  nd  (Yo# of Ao) z  The  root-mean-square  (3.2.5.). Equation  y  a  a  F  =  c  (3.2.2) can  +  a  effect  F  ^e considered  n  of  in  Fig.  3.2.1.  found to model the a c t i v e operating The Fig.  to  The  system  3  using  equations  thus be r e w r i t t e n as  (3.2.6)  A  as the BHS  linearizing  change the f a m i l y of parabolae indicated  point  operating  the servovalve one  be  A  of  point.  equation  straight  i s to  lines,  as  l i n e a r system thus developed i s adeguately  over  the  entire  range. transfer  3.2.2.  function  is  shown i n block  diagram form i n  57  ~FI6.. 3.2.1 LINEARIZED  SERVOVALVE  CHARACTERISTICS  BYPASS VALVE  r-  CYLINDER  •A  — —FORCE ————  A  -*  SERVO-ACTUATING SWJWAL  A,  PISTON VELOCITY  SERVOVALVE  FIG, 3.2,2  LINEARIZED ACTIVE  SYSTEM TRANSFER FUNCTION!  58  3.3  Linearized The  systems will  linear are  transfer  combined  be n e c e s s a r y ,  transfer  function.  F  it  Active-Passive  N & T  linearized  to f i r s t  Recalling  *  X  M  equation  + C  w  t h e damping  active  and  linearize  t h e body  passive 2.4. I t dynamics  (2.5.4)  X  (3.3.1)  term i s n o n l i n e a r .  This  can  be  giving  -  Ne1  the  i n t h e same manner as i n S e c t i o n  about i = 0 ,  F  f u n c t i o n s of  however,  i s seen t h a t o n l y  System  M*  +  C  t  X  (3.3.2)  where C =C L  and C  L  || =2C i 7  rt  H  x, i s t a k e n  e  as t h e r o o t - m e a n - s g u a r e  i s then the l i n e a r i z e d  drag  coefficient.  The d i f f e r e n t i a l e q u a t i o n o f to  (2.6,4.)  can now fee l i n e a r i z e d  give  -X  The  velocity.  transfer  +  ^  "X + C0c  function  of  X  the  =  X ,  compensator  (3.3.3)  motion x  m o t i o n x c a n t h e n be e x p r e s s e d , i n L a p l a c e n o t a t i o n ,  1 #  as  t c body  59  1 + 2 % » . s + 7«2  *.  < - -'"  s  3  3  where i  and  2-MtOt  c  F (s) i s the c a b l e t r a n s f e r f u n c t i o n . c  The t r a n s f e r f u n c t i o n of body displacement, F  Hfrt  , i s d e r i v e d from  =  MsSC S  0.3.5)  L  The block diagram of the l i n e a r i z e d c a b l e and body shown  tension,  (3.3.2)  F (s) = % B  x, t c c a b l e  dynamics  i n F i g . 3.3.1. The block diagram of the l i n e a r i z e d  compensation  system i s shown i n F i g 3.3.2.  is  motion  F 16, LINEARIZED  CABLE/BODY TRANSFER  Ny(s)  +  FIG.  3.3.1  3.3-2  LINEARIZED  Ksv  z  BLOCK  F UMCTION  Xi  DIAGRAM  ACTIVE- PASSIVE  of  S Y S T E M  61  3.4  Performance A n a l y s i s and As a f i r s t  model  approximation t o system performance, the  derived  i n Section  domain. I t w i l l surface Due  Optimization  3.3 w i l l  linear  be examined i n the freguency  be most convenient to use the r a t i o of  body  to  ship displacement, x/u, as the c r i t e r i o n of performance.  t o the l i n e a r i t y  of the system, a minimum x/u i s e q u i v a l e n t  to minimum v a r i a t i o n i n F .T /U. NE  Solving  the block  diagram of F i g 3.3.2 y i e l d s  the  overall  transfer function:  X6) _  Uts)  FcCs)  | + W( )  (3  s  '* a  1)  where H (s) i s the open-loop t r a n s f e r f u n c t i o n , given by:  In  designing  of the c l o s e d within H (s),  the  system, the absolute  loop t r a n s f e r f u n c t i o n , |TJ(S)|, i s t c be range of operating  frequencies.  value  minimized  T h i s i n turn y i e l d s  the c l o s e d loop t r a n s f e r f u n c t i o n , a maximum. In g e n e r a l ,  body  an a c t i v e - p a s s i v e  dynamics,  the designer which  are  has no c o n t r o l over the cable represented  by  F (s) c  and  Furthermore, he has very l i t t l e c o n t r o l over the passive  and  F (s). 6  system.  62  Gp.(s) , s i n c e the primary  design c r i t e r i o n f o r that system i s t o  be a b l e to c a r r y the s t a t i c weight o f the towed body. maximizing  H (s) ,  Hy(s), and  to  i t is  select  a  c y l i n d e r as represented  3.3.1  first  step  denominator. I d e a l l y ,  =  "  ^  to  design H ( s ) , H F 8  servo-valve  and  maximizing  H{s)  H f  (s) ,  hydraulic  i s to minimize i t s  i t i s s e t to z e r o , which  S  u  Hy (s)  i t i s relatively  feedforward  F F  in  A| and  Uy£0l  ^ ^ X ^ W l ^ V  is  the  ram  yields  centering  slow-acting i t has l i t t l e  (3.4.3)  network.  or no e f f e c t  a t o p e r a t i n g f r e q u e n c i e s . Thus, i t can fce deleted from The  Thus,  Element  in  As p o i n t e d out e a r l i e r , Because  suitable  by  The Feedforward The  necessary  1  (3.4.3).  compensator can now be given as  ^) - - rrj—,  [ (VWo] s+  -'-'"  (3  The s t i f f n e s s of the p a s s i v e system i s u s u a l l y s p e c i f i e d i n order t h a t the a c t i v e - p a s s i v e system he capable of o p e r a t i n g i n a purely passive mode when working i n a low sea s t a t e , or i n case of a power f a i l u r e i n the a c t i v e system. 1  6 3  3.4.2  The  The  second  numerator. done  An  by  optimized  As H  F B  a  the  with  of  (3.4.2) of  saturates  value,  respect  to  customary to  maximizing  gain  servovalve critical  is  in  examination  (s) i s assumed  and  Element  step  setting  however, the exceeds  Feedback  z . 0  the  H  r a  H (s)  reveals (£)  when t h e  this  l a r g e as  actuating  Therefore,  H  F B  the  can  be  possible;  current,  z,  must  be  (s)  above c o n s t r a i n t .  a combination  a c c e l e r a t i o n feedback.  that  as  in position servos,  be  i s t c maximize  of  the  feedback  displacement,  element  velocity,  Thus,  K, + I4S  +  S  (3.4.5)  Z  where Kj , K This  z  and  has  are  3  constants.  yields a servo-actuating  ZM = (The  K  ram  KSV[HFF6)U(S)T-  centering  network,  negligible effect  The  amplitude  constrained  t o be  current  at  of  Hy(s),  operating  the  l e s s than or  z, g i v e n  by  Wp(OX(sT>]  (3.4.6)  B  i s again  ignored  because  it  frequencies).  actuating equal  to  current, the  |z<s)|,  critical  is  value,  now z : e  64  |Z(s)| -  K | H (s)U(s) + W (s)X(s)| ^ sv  FF  For convenience,  Z(s)  (3.4.7)  FB  (3.4.7) i s r e w r i t t e n i n terms of  Is  H (s)+ H,  - 1/  ratios:  FF  (3. 4.  8)  where u„ i s the amplitude and  of the input,  ju (s)^ ,  — (s) i s the c l o s e d loop t r a n s f e r f u n c t i o n as given  by  (3.4.1).  The optimum values of K,, digital  computer  program  o p t i m i z a t i o n process being  Ram The  and  K  listed  in  are  fcund  Appendix frequency  using T.3.  only,  a The  that  which c o n t a i n s the g r e a t e s t energy.  ram  c e n t e r i n g network i s r e q u i r e d to maintain displacement  zero.  Dnder  i s e q u i v a l e n t to r e t u r n i n g the ram  given  3  Centering  term average ram this  z  i s c a r r i e d out at one  the "design frequency"  3.4.3  K  static  the  long  conditions,  t c centre p o s i t i o n i n a  (long) time a f t e r r e c e i v i n g a step i n p u t . C o n s i d e r i n g the c e n t e r i n g loop as shown i n F i g  active  ram  force F  A  3.4.1,  the  can be set to zero under s t a t i c c o n d i t i o n s .  Q  ^  +  Z  r  "FIG. RAM  ——*  3.4.1  CENTERING  NETWORK  66  T h i s g i v e s a c l o s e d loop  t r a n s f e r f u n c t i o n of  Y(s)  Letting  (3.4.9)  Hy(*) = Ky = constant  gives  (3.4. 10)  where  I t now  remains to s e l e c t K  where 0)„ i s the  3.4.4  System The  design  frequency.  Stability  +  U(s)  =  O  In  cumbersome to d e a l therefore  solving  the  (3.4.11)  examining the r o o t l o c u s as the varied.  by  equation  1  are  that  s t a b i l i t y of the system i s determined  characteristic  and  such  v  assumed:  general, with,  and  the the  feedback gains K, , Kj_ and characteristic  following  K  3  function  is  simplifications  are  67  1.  L e t Gp (s) = -Kp  i . e . , assume t h e p a s s i v e  as a s i m p l e 2. L e t F  c  (s)=1  spring —  reasonable  (where H c c k e ' s Law  the  i . e . , n e g l e c t the c a b l e dynamics. T h i s because  natural  frequency 3. N e g l e c t  behaves  applies).  the  effect  of  the  motion compensation system i s r e l a t i v e l y at  system  should  Hy(s)  frequency  of  the  c a b l e cn small,  is the  except  tewed s y s t e m .  This  l i e o u t s i d e the range c f c p e r a t i c n .  as e x p l a i n e d  With the above a s s u m p t i o n s ,  earlier. (3.4.11) c a n  be  expressed  as:  (3.4.12)  The in  application Chapter  3.4.5  (3.4.12) t o t h e l a b o r a t o r y  rcodel  i s discussed  IV.  Power C o n s u m p t i o n The  power  of  power consumed by t h e a c t i v e s y s t e m i s t h e  dissipated  d r i v e the  in  load. This  of o i l f l o w i n t o  the  servovalve  i s equal  t h e v a l v e and  t c the the  and  the  product  supply  SUIT  cf  the  power r e q u i r e d  of the  pressure:  volume  to  rate  68  =  W  A  (3.4.13)  ? \i)  A  s  where W i s t h e i n s t a n t a n e o u s power  consumption.  Note t h a t a l t h o u g h y changes i n d i r e c t i o n , W i s always This the  positive.  i s due t o t h e f a c t t h a t t h e d i r e c t i o n s w i t c h i n g o c c u r s i n valve  itself,  unidirectional.  but  Thus,  the o i l f l o w  to  the  valve  is  t h e average power consumed i s the r o c t -  mean-sguare of t h e a m p l i t u d e o f W:  (3.4.14)  w her e 0.707 i s t h e rms f a c t o r f o r a s i n e wave,  and express  P  s  i s the supply pressure,  A  A  i s t h e ram a r e a .  W i s t h e average "w  power consumption.  i n t h e freguency  domain  as  I t i s convenient to a  ratio  to  input  displacement U(s):  onoi AA R I6,A  FcCs)  uCs)  (3.4.15)  69  where v 2-(s)  is  the  c l o s e d loop t r a n s f e r f u n c t i o n given i n  (3.4.1.) The reason active  f o r the  system  now  inclusion  of  average the  supplies  (or  steady  absorbs)  pressure)  in  the  W,  wasting  while  the  between the power  when  i s f u l l y charged, i t i s necessary t o use e i t h e r  an unloading v a l v e or a pressure compensated pump does not  rate  the d i f f e r e n c e  and i n s t a n t a n e o u s v a l u e s . To avoid  accumulator  accumulator  becomes obvious: the h y d r a u l i c power supply  needs only to provide power a t the accumulator  an  discharge  through  the  when there i s no flow demand.  pump. T h i s way, the  relief  valve  (at  high  70  CHAPTER IV THE  4,1  General A  Description  laboratory  system  LABORATORY MODEL  model,  discussed  closely  resembling  i n Chapter I I was constructed  mathematical and s i m u l a t i o n  the  equivalent  t o v a l i d a t e the  models developed, and as a prototype  small  s c a l e mechanical s i m u l a t o r  full  s c a l e systems. The apparatus c o n s i s t s of a h y d r a u l i c power  supply,  active  displacement monitoring, and  passive  generator,  cylinders, and  various  a  and  variable  pieces  evaluating  frequency  cf e l e c t r o n i c  p r o c e s s i n g , and d i s p l a y equipment. See Figures 4.1.1  4.1.2. The  of  and  f o r designing  power supply  five  gallons  motor. A pressure the  pump,  c o n s i s t s of a v a r i a b l e  per minute c a p a c i t y d r i v e n r e l i e f valve  which  can  be  i s provided  displacement  pump  by a 5 BP e l e c t r i c  on the  discharge  of  s e t t o between 100 and 2000 p s i . The  e n t i r e u n i t i s mounted on top of a 15 g a l l o n o i l r e s e r v o i r . The  motion compensator c o n s i s t s  mounted given  in an  frequency  tandem  on  approximate  of  a  of  cylinders  a c a r r i a g e , F i g . 4.1.3. The c a r r i a g e i s sinusoidal  displacement  cf  variable  and amplitude by means of a crank mechanism driven ty  a 1-1/2 HP DC motor. The p i s t o n rods c f the pinned  pair  together,  so  two  c y l i n d e r s are  that they a c t i n p a r a l l e l , r a t h e r than i n  9  iXlACCUMULATOR  0  SERVO VALVE  ®  2  GAS STORAGE  I  5)  <2  X  RELIEF a_VALVE  ^ 4 PUMP  PASSIVE  ACTUATOR  ACTIVE ACTUATOE RYDR.AU LlC SUPPLY  FIG: 4.1.2 LABORATORY  APPARATUS -  SCHEMATIC  PASSIVE. CYLINDER  ACTIVE CYLIMPER  7a  s e r i e s as the arrangement might f i r s t  suggest. The  the double-ended p i s t o n rod i s pinned  to a second c a r r i a g e which  contains  weights  to  represent  other end  of  the mass whose motion i s t c be  isolated. The pair  passive system i s purely  of gas  the two  b o t t l e s , one  pneumatic,  consisting  connected to each end  of  a  of the l a r g e r of  c y l i n d e r s d e s c r i b e d above. A flow c o n t r o l valve on  each  gas b o t t l e i s used to a d j u s t the damping of the passive system. The  active  feeds o i l to a controls the two  the  system c o n s i s t s of a g a s / o i l accumulator which servovalve  flow  through  a  filter.  of o i l i n t o the two  c y l i n d e r s mounted on the carriage... A gas  The  c o n t r o l system monitors  seismic  v e l o c i t y transducer  displacement  potentiometer  processing  function  is  i n t e g r a t e s , a m p l i f i e s , and included  body  and  to  a  cn  the  achieved  ram,  sums the two  oscilloscope  and  mass v e l o c i t y , m a s s  amplifier  t c the  of the  ram  system.  motions  carriage.  which  The  signals.  A  by  displacement, voltage.  a a  signal  filter  n o i s e , such as the  drives  processed the  input  type  servovalve.  displacement,  ( F i g . 4.1.4)  is  signal is  c h a r t r e c o r d e r are used to monitor any  l e v e l servo-actuating  top  by an analogue computer which  to remove high frequency  power  line  of  mounted on the mass c a r r i a g e , and  generated by the wheels of the c a r r i a g e . The fed  servovalve  p o r t s of the s m a l l e r  of the accumulator c o n t r o l s the charge pressure  also  The  twc or  An of low  DISPLACEMENT  OSCILLOSCOPE  TRANSDUCER  TWO-CHANNEL CHART RECORDER  VELOCITY  BAMP- PASS  TRANSDUCER  FILTER  FIG.. 4. 1.4  DIFFERENTIATOR  CONTROL  Ttt)  SE.RYOAMPLIFIER  SYSTEM  SIGNAL  TO  SERVOVALVE  76  The (i.e.  apparatus  as  constructed  cable) nor the hydrodynamic  aspects  does  not model the s p r i n g  drag of the  body.  a l l other  of the e q u i v a l e n t system are i n c l u d e d . The e x c l u s i o n of  these two parameters does not a f f e c t the  model  compensation  both p r o p e r t i e s of the  towed  system.  system  because  they  are  of  the  motion  77  4.2  Performance P r e d i c t i o n and The  physical  i n Appendix D. reguirement,  p r o p e r t i e s of the  The  design  but  system  not  based  velocity  set  to  1.5  transducer  any  particular  readily available.  inches,  and  used to measure the  g u i t e u n s u i t a b l e a t the low could  processing  was  frequencies  operate,  and  as a r e s u l t  necessary  t c o b t a i n a useable  of t h i s , only a c c e l e r a t i o n feedback was c o n t r o l system. The expressed  on  given  the  design  to 1 Hz.  The body was  is  l a b o r a t o r y model are  employs hardware which was  The i n p u t amplitude was frequency  Evaluation  stability  motion of at  the  which  the  some e l e c t r o n i c s i g n a l s i g n a l . As a r e s u l t  a v a i l a b l e for use  equation  (3.4.12)  can  i n the now  be  using a c c e l e r a t i o n feedback:  As  +  2  B  S  + C  = O  ,n  o  -n  where  By  B  = Oz  C  ^ C  ^  the  acceleration  making  coefficients are n e g a t i v e ,  A, B and while  K  C  +^3) +  ^3)  u  -  1  kip  feedback  C are always negative p  K  3  (note  i s p o s i t i v e ) . Therefore,  negative, that the  and system  the >3 is  78  s t a b l e f o r a l l n e g a t i v e v a l u e s of K . 3  A  point t o note i s that the feedback c o n s t a n t , K , has the 3  e f f e c t of i n c r e a s i n g the apparent mass o f the system, in  as  shown  (4.2. 1). Because only one feedback v a r i a b l e i s c o n s i d e r e d , i t i s not  necessary 3.4.  to use the o p t i m i z a t i o n technique o u t l i n e d  i n Section  a feedback constant of K K = -5 m a / ( f t / s e c ) 2  5 V  3  was used i n the experiment. The experiment 1.0,  and  was conducted  at  three  frequencies:  2.0 Hz. In each case, the system was f i r s t  p a s s i v e mode (K = 0) , then with K 3  sv  0.5,  run i n the  K =-5. The computer s i m u l a t i o n 5  was then conducted under the same c o n d i t i o n s , and the r e s u l t s of both are shown i n Appendix G. The  time-domain  frequency  records  transformed  tc  the  domain, and p l o t t e d as d i s t i n c t p o i n t s on a Eode p l o t  i n F i g . 4.2.1. The l i n e a r same  a r e then  graph  model response i s a l s o p l o t t e d on the  f o r comparison,  over a frequency range of 0.1 to 10  Hz. There appears t o be good agreement between t h e mathematical models and the r e a l system. Any d i s c r e p a n c i e s uncertainties  involved  i n estimating  t h r o t t l i n g c o e f f i c i e n t s and  mechanical  a r e due  t c the  h y d r a u l i c and pneumatic friction.  However,  by  79 designing  a  real  system  with v a r i a b l e t h r o t t l i n g v a l v e s , the  former u n c e r t a i n t y  can be removed s i n c e the r e a l system can then  be matched to the  model.  neither  be  operational.  easily  Friction,  predicted  nor  on  the  altered  other  hand,  can  once the system i s  OVERALL  LIMEAfc - P A S S I V E  RESPONSE  80  X/U  (K»=o)  LINEAR-ACTIVE A  EXPERIMENTAL -  SIMULATION  0.10  0.20  0.50  FREQUENCY  FIG. 4.2-t  1.00  2.00  5.00  RRTI0  THEORETICAL 6s EXPERIMENTAL  RESULTS  10.00  81  CHAPTER V APPLICATION  This the real  chapter  as a guide  m a t h e m a t i c a l and computer s i m u l a t i o n  t c the  application  of  models t c t h e d e s i g n  The Fourier surface eguation  Input  Conditions  input,  as s t a t e d  series of  the can  1  a given  i n Chapter  representing  the  water  given  be  energy d e n s i t y  from  at  used  a  to  I , c a n be a p p r o x i m a t e d vertical  obtain  point.  displacement The  an e s t i m a t e  the average height  by a  c f the  Eretschneider of the s p e c t r a l  and p e r i o d  c f the  seaway  Sea S t a t e :  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 / s e c ,  T i s t h e wave p e r i o d "f i s t h e a v e r a g e and  i  of  systems.  5.1  for  i s intended  h i s the average  See R e f . (12)  i n seconds,  wave p e r i o d  i n seconds,  peak-to-trough  wave h e i g h t  in feet.  82  Values  for  h  and  T  can  be  readily  handbooks. Figure 5.1.1  shows a t y p i c a l  density  Sea  function  period of 4 . 9  for  f e e t and  A s h i p subjected will  behave  as  a  5.4  found i n most n a u t i c a l plot  State 4 , having  spectral  seconds, r e s p e c t i v e l y .  to a  multi-frequency  low  pass  filter,  motion of the s h i p i s then the  spectral  displacement  and  will than  density  function  product  not  input respond  one-half  the  The  wave component which  contains  found  to  have a p e r i o d T ,  0) =27r/T  design  frequency.  0  Figure 5 . 1 . 3 Fourier  can  now  of  the  sea  state  and s h i p response, as shown i n F i g .  5.1.3.  and  o  o  the  most  energy  can  be used to obtain the c o e f f i c i e n t s  Function  by  into  calculating  A-  t  Series  represent  T h i s i s done  is  i s used as the primary  <5.  which  and  length, fi t y p i c a l response curve i s shown i n F i g . 5 . 1 . 2 .  ship's  of the  the  mean wave height  s i g n i f i c a n t l y to waves whose l e n g t h i s l e s s  The  of  1.2)  the motion of the ship i n the time domain.  dividing  the  Ship  Motion  Spectral  Tensity  n c e l l s spanning the e n t i r e range of p e r i o d ,  the energy a s s o c i a t e d with  each  cell:  and  StT)  20 -\  FIG, S.LI 5£A STATE "4" SPECTRAL DENSITY  1 0  FUNCTION o  10  F16. 5,1.2 SHIP  HEAVE  RESPONSE  SbCT)  ft'Aec FIG. 5.1.3  4oH  SHIP ME AVE SPECTRAL DENSITY  20  FUNCTION  o io  84  AS  •• J S(T)«IT H  (5.1.3)  Z  where &S T and  t  i s the energy a s s o c i a t e d  with the i - t h c e l l ,  i s the c e n t r a l p e r i o d of the i - t h c e l l ,  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  (5.1.4) The then  be  approximate  s h i p displacement as given by  (5.1.2)  can  used i n the s i m u l a t i o n model t o give a r e a l i s t i c i n p u t  c o n d i t i o n . I t i s not necessary to use more than terms i n the s e r i e s to g i v e a good wave p r o f i l e .  three  to  five  85  5.2  Two-Dimensional Cable Model when  dealing  with  long c a b l e lengths  h o r i z o n t a l motion with r e s p e c t to the catenary  shape which can  Walton and shape  and  Polachek  no longer be assumed one-dimensicna1. developed a  1  end  and  elastic  links  masses concentrated  pinned  represented  a t i t s two  as the l a s t  values f o r mass and  axial  program end  drag uses  of  the v a r i a b l e  end-tc-end.  a  Ref.  (23)  entire  ends, as well as l o n g i t u d i n a l  and  The  towed the  body  is  appropriate  input at the top ( i . e .  c a l c u l a t e s the displacements  the  at  the  model of the motion compensation  use t h i s cable/body model by  the  and  l i n k s . T h i s i n turn y i e l d s  s u r f a c e tow  and  supplying  receiving  F  N E T  ,  point. I f d e s i r e d , the  r e s o l v e d i n t o h o r i z o n t a l and  components to i n c r e a s e the model's r e a l i s m .  See  its  has  displacement  all  compensator motion can be  i  boundary  Each l i n k  as the boundary c o n d i t i o n , tension  the  coefficient.  system developed here can  cable  compute  along  l i n k of the c a b l e , given  of the c a b l e , and  elongations  drag  ( F i g . 5.2.1)  c a b l e t e n s i o n i n each l i n k . The  the  tc  c a b l e i s modelled as a number  together  t r a n s v e r s e drag c o e f f i c i e n t s .  surface)  program  hydrodynamic  l e n g t h . In essence, the continuous  The  water, the cable assumes a  t e n s i o n of a cable s u b j e c t to a displacement  c o n d i t i o n at one  of  (over 2000 f e e t ) or  vertical  86  HEAVE SUR&E  t  (b)  CABLE  SEGMENT  87  5.3  Servo-valve The  Hodel  servo-valve  considered  instantly  upon a p p l i c a t i o n  case  large  of  considerable response order  valves  time l a g  of  such  then  here  of a c o n t r o l  which a r e even  valves  system, depending  equation  Extension  at  can on  signal.  to  accuracy  as  operate  However, i n  frequencies.  considered  the  assumed  usually multi-stage,  low  be  is  desired.  there  The  a first  the is  dynamic cr  second  The  valve  becomes  (5.3.1)  where H (s) i s t h e sv  In  the  dynamic c h a r a c t e r i s t i c  case c f a f i r s t - o r d e r  Hsv(s) for a  valve.  valve,  (5.3. 2)  = 1  and  c f the  +  tsvS  second-order.  c  (5.3. 3)  where  and  C  sv  i s the  valve  constant  (as b e f o r e ) ,  T  $y  i s the  f i r s t - o r d e r time  0)  W  i s the  second-order  5*v  i s the  s e c o n d - o r d e r damping  constant,  natural  freguency, ratio.  88  The  parameters  t„  Vf  o)  w  and  S  sw  are  estimated  response curve s u p p l i e d by the valve manufacturer, They the  are  near  the  resonance of the valve  recommended s i n c e s e r v o - v a l v e s are  higher order model may be necessary.  Fig.  5.3.1,  lags of the r e a l valve and  model c o i n c i d e over the freguency range  operating not  chosen such t h a t the phase  from the  of  interest.  When  (which i s g e n e r a l l y  usually  underdamped)  a  89  FIG, 5.3.1  SERVOVALVE  RESPONSE  90  5.4 In  C o n t r o l System  Considerations  setting  the  down  performance  reguirements of a r e a l  system, the frequency response must be c a r e f u l l y particular,  long  period.waves g e n e r a l l y have  considered.  larger.amplitudes  than s h o r t waves, hence i t i s not always p o s s i b l e for  them  as  compensator.  effectively Therefore, i t  compensation  at  high  seconds), i n c r e a s i n g frequency.  due  An  wave to  is  to  periods  to  limited  desirable  maximum  acceleration  the  In  to  compensate  travel  design  of the  for  2ero  (in the order of 20 t c 50  compensation  feedback  at  system  the  will  design  inherently  behave i n t h i s manner. At f r e q u e n c i e s above the s h i p ' s desirable  to  decrease  compensation  natural since  frequency the  it  s h i p does not  respond to such waves. Furthermore, shipboard v i b r a t i o n s due the  engine  typical  and  propellors  frequency  performance  is  response  shown  may  be s i g n i f i c a n t  which  would  is  above  give  to  one Hz. A acceptable  i n F i g . 5.4.1. The low frequency c u t - o f f  can  be moved to the l e f t  by e i t h e r d e c r e a s i n g the  the  p a s s i v e system, i n c r e a s i n g a m p l i f i e r gain, or i n c r e a s i n g the  time c o n s t a n t of the ram c e n t e r i n g The  point  with  of  loop.  of maximum compensation i s s e t by i n t r o d u c i n g a  second-order low-pass f i l t e r , F i g . 5.4.2. The coincides  stiffness  corner  the design frequency, where motion  i s maximum. The c r i t i c a l  frequency  compensation  damping r a t i o determines the  bandwidth  91  CO  o  < UJ  2.  -2o4  FREQUENCY  FIG. 5.4. I MOTION  COMPENSATION  TRANSFER  Slfi-KJAL TO  1 ACCELERATION  1 + ui 5  FIG. 5.4.2  0  2  FUNCTION  S/Us + S*A>  2  f  SERVO-AMPUFlfcTR.  SETS DESl&M FREQUENCY CONTROLS BANDWIDTH  TYPICAL  FEEDBACK  NETWORK  92  of  the  response  system to v i r t u a l l y system  curve.  Such a f i l t e r  can be used t c tune the  any sea s t a t e c o n d i t i o n , provided  that  the  i s designed to handle the corresponding amplitudes. T h i s  f e a t u r e can be used to improve adding on an a c t i v e  one.  an  existing  passive  system  by  93  CHAPTER V I CONCLUSIONS The  dynamic  compensation developed. indicate  behaviour  of  an  active-passive  system has been a n a l y s e d and a Experiments  that  the  performed, on  system  a  motion  mathematical laboratory  i s adequately  described  model  apparatus by  the  equations derived. The the  mathematical  model has been s i m p l i f i e d  e q u a t i o n s , and computer programs have been  can a s s i s t i n t h e i n i t i a l program  which  by l i n e a r i z i n g  developed  d e s i g n o f r e a l systems. In a d d i t i o n , a  s o l v e s t h e n o n l i n e a r e q u a t i o n s by s i m u l a t i o n has  been w r i t t e n , and can be used t o r e f i n e the i n i t i a l d e s i g n . programs  are  which  The  f l e x i b l e enough t o accomodate a v a r i e t y of system  configurations. T h i s p r o j e c t , i n e s s e n c e , has p r o v i d e d a d e s i g n t o o l , on mathematical a n a l y s i s , t o an  area  which  r e l i e d on s e a t - o f - t h e - p a n t s e n g i n e e r i n g .  has  based  traditionally  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 T r a n s f e r Function"Form NTIS~AccT~#AE-700-4 31 2. B l a c k b u r n , J . F.: F l u i d Power C o n t r o l MIT P r e s s , 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 P r o v i d e Motion S y n c h r o n i z a t i o n During S u b m e r s i b l e 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 R e i n h o l d 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.: H y d r a u l i c Servosystems A n a l y s i s and Design B u t t e r w o r t h and Company, 1969. 7. H e d r i c k , J . K.: A Summary of The O p t i m i z a t i o n Technigues t h a t Can Be A p p l i e d to Suspension Systems Design A r i 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 U s i n g S e m i - A c t i v e Force G e n e r a t o r s ASME~Paper # 73-DET-1227 ' ~~ 9. K e e f e r , I . G.: Improved Hydropneumatic T e n s i o n i n g Systems f o r -Marine A p p l i c a t i o n s B, C. Research C o u n c i l R e p o r t , 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 A r c h i v B e r l i n , V o l . 36 #6, ^968. 1 1 . M e r c e r , 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 E n g i n e e r i n g McGraw H i l l , ~ 1 9 6 9 7 ~ . 13. P o r t e r , B. & Bradshaw, A.: S y n t h e s i s 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 E n g i n e e r i n g McGraw 1968.  '  ~ .  "  . ~~~  '•  Rill,  15. R u z i c k a , . 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. S h i n n e r s , S. M.: Modern C o n t r o l Systems Theory and A£Eiication Addison Wesley, 1972. 17. S o l i m a n , J . I . 8 T a j e r - A r d a b i l i : A c t i v e I s o l a t i o n Systems U s i n g a N o z z l e F l a p p e r V a l v e I n s t . M. E. P r o c . , V. ^82 #30,  1967." "  18. S o l i m a n , J . I . , & T a j e r - A r d a b i l i : S e r v o v a l v e C o n t r o l l e d I s o l a t i o n Systems I n s t . M. E. P r o c . , ~ v 7 ~ 185~#107"970. 19. S u t h e r l a n d , A.: M e c h a n i c a l Systems f o r Ocean E n g i n e e r i n g N a v a l Eng. j77~Oct."1970." ~ ~ 20. Thompson, A. G.: Quadratic Performance I n d i c e s 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. P r o c . , V. 185,~1970.  95  22. Thompson, A. G.: Optimum Damping i n a. B a n d o j l ^ E x c i t e d Nonlinear Suspension I n s t . M. E. P r o c , V. 184, 1969. 23. Walton S Polachek: C a l c u l a t i o n of T r a n s i e n t Motion cf Submerged Cables Math. Tables 8 l i d s t o Computation, V. 14, 1960. 24. Yeaple, F. D.: H y d r a u l i c and Pneumatic Power and C o n t r o l McGraw H i l l T 1966?"  APPENDIX & AS  /•  A  FLOW  FLOIAI  S O U A  ll^TO  TlOhJS  CLOSES  A  VOLUME  FIG-  .  .  R  A  T  OFCU-AAJG  S  E  A-i  OF^EMTN-ALPY  COUTfLOLVOL UM£  IA/ITM*)  .OFA-i  :  6J4  IS  TW=  OFcm+JQE.  /zA-re  THE:  FLOlAi  Ik/TE(LtJPrL  HATS  OF  OF  GAS  g^eZC^:  En/ee&Y  I^JTEY^JAL  MULTIPLIED  BY  ITS  UM/T  . : C A ' 2 )  7V£ RATE OF CH-AtJ&E OF GtJER^Y WE WOfM DOAJZ OS) THE GAS BY EXPAfiJSiDkJ CJDHPHE SS(OtJ OF TI4C COKfTR-Ol VOLUWC •• IS  ~ • SU&STi  TUTltJG,  (A-2)  r  dt  •• •  /WD  CA'3)  , /A/TO  (A>I  )  (At) S0L\JtA/6  FoQ.  m  AtiD ±  IAJTO  (A-4)  +  Su&STFTUi J- -  IfJG,  ±  .,  I  r dt + "  aj  (4*0  2.  THROUGH  FUHAJ  FIG.  tJeeo>te VALVG  A  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. THB  APPLYIK]&  = H/ +  Liu.  \rifr9&£  .sir-  A-SSVtftiUG-  B*J&££Y  f  i$ THE  TH-£  FLUID  &2UAT/O/J  Z  'J  A<JEGAG&  lS  A  CpT* =• CpT  JUS  MASS  FtOuJ *  dATE =  ~*  d  VBUoCllY  PEIZFFCT  ACROSS Ay.  GAS  + £<r-*.  IS, 6(\JFtJ f  CA-l)  A„v-  (A-?!)  3Y  y  .  (A-lo)  (A-Z),  Su6ST(rUr/iJ6  SU&STiTVT  (A-I)  TH£  1^6,  srA-rE  (oP  ••  •  •  Gives--  (A-(0  Sl/JCe  '  Givei:  CA'/OJ  or  F$UA7(0AJ  _ _JL IK/TO  MTO  7V€  VALVE IS (JoT A+J IDEAL Coi<J\/BRiS,lfJCj A / O Z Z L E " , . ir is TH-edGFc&G juerc^ssAieY ro //JT&ODOCE AA/ £MPlQiCAL 0lSa4A£$ £~ 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  (A 14) :  Co = CzAv Wtt£(2£ Q  is  Q  IS  pEredHifjeb  PUBLISUFD  BY  Nore  THAT AA/O  }  Pod  FOP. WBIR.  CAU  - (A-IS)  poiA/Aisr/ZEAM FOR.  6€  JUVS  IS  CFOK^O  VALID  &z&ATFe.  FLOU).  Foe.  Ate , c  = O.S2g  USUALLY  VALVES.  ^/P/ZFSSFO  OIJLV  pzesruze  P  POSITIVES-  A*ir>  EXPF&IMSAJTALLYJ  ' T H A T .  ae&uieep  OF fJEEbLE  MAdUFACTUHZas (A-13)  .. .£&UATIOAJ  i.£.  FVK)CTIosJ  A  P  u  AS;  FO/Z  TUAA)  hjirnoze/O  i_  APPENDIX  £  UYDgAULIC TYPES VALUES  GDA)St£Tl>J6  SErZxlO-VALVES QF  VALVES:  c&fJSipeizeD ..^  OF  tfE&e  H&T£tllh}$  AUB  *FOOQ.-WAY  ORlFlC££,  SUPPLY  MZDM/To  LOAD  FIG. SPOOL  .  FI6  B.I.  .  VALV^ .  BMAOST  TO/FROM  LOAD  B. I VALVE  'THIS A£&filJ6&M&JT CAfiJ B€ MODFLteb WHEATS TotiE Mlb&E, F-fe. E>1.  f=l4. 5.2  SPOOL  WHEATSTUME  BY A  0ftD$E  NYDZAULlC :~-  TUB  HYDRAULIC  ZESlSTAiJCeS  . A&G ;  OF <JZ>U/2£b,  hJOfJLIMBAe.  SC.R.VOVALVES "ZERO  - LAPPED  WAY AS  BE  UAJPE FLAPPED,  Sht-ovJ^  /A)  r  l  FlC-j.  OVE^LAPPED  S3  ,  ° &  .  \  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/ZLAPPltJS, TEMPS TO E^UAtiCE STABILITY. OF WE SEMD SYSTEM /M  AT wis  raF  expense  ANALYSIS  OF  PDMEQ.  , 2ee.o-LAfpeO  LOSS VALVES  AT  X-O. AgE  LEAF 101 OMITTED IN PAGE NUMBERING.  \ o  2.  PERlVAVON  OF  8 4  COMPRESSIBILITY  THAT X,-  h/IT(Jl*J  THE  ICY E  Z  L  V  i  i  THE  E  • -  .  IS PgoPoZT ION A L To FAMILIAR ORIFICE FLOvS  -  P,  .>' •  ..=  Q)  -i ..  L  (B.2)  Tr\  =  QL  E  &  A $  Q  ~- |  L  COKlSTA-fJj  .» '. LOAD  L  COM&/Nlid<$ 77-/£ TvJQ R.EAR.ZAN^IM&,, YIELDS[..:..!.  Q  peessueE  SUPPLY.  P„  FLOW  EQUATIONS  X  OF  (&-2)  A^D  /P -.AP &  VlUE&E  =  AR  FOR.  THAT  N  E  G  A  T  I  V  E  S  f  O  0,  P, - P,  O  L  D  Q  Z  I  S  P  L  - O  f  /5 AJFej LECTEO,  •  •  /.Fs  A  SYSTEM  -• KX  H  SY  V  -• ' •  ;~.  e x  P ~ O  OdiFtcE AQEA ME. WAVE THE  .- —  .EQUATIONS -  W  MODELLED  2ERO-LAPPED  I....-  ASSuHthJG DISPLACEMENT,  w ? T ~  LOAD  o-  F I G .  IF TUEM  d o e s  EQUATIONS  PZESSUZa-F-LOvJ  _A ZBHO - LAPPED VAL\lE MAY BE SlMf>L(Fl£D Cl&CUtT OF Fl<*. B-4-  s  \  A  C  E  M  E  N  T  (  X  ^  -  O  )  ,  N  O  T  E  (6-4)  103  WIS  Y/BLDS QL  . CVHBll/llJG,  Wi+ichi  =  (B.3)  is  Thit  J X  J p  Arhib  C&HZIZAL  s  + & p '  ,  S&MO  AND  VAL\J€.  LETTltJG  EQUATION)'.  .  APPENDIX  C  COMPRESSIBILITY  CoUSWtd  EFFECT  TUB  -1  OF  CYLlMPStZ.  • 0;  MDRAULIL  SPorili!  iN  k Q .  .  p  p,  F/G-  FLU  ID  CJ  0  1 , 2  1  1  —  FIG,. C - l  F-LOti' OoT.,  TH-e  IhJ j  Q{  THE  f  Q  THE  L6SS  "COMPRESSIBILITY  A  VJUEA£  .'FOB.  . ErQAf&L  FLOW'',..  tf  3  c  FOUJVALEKJT  V ~ .& 5  mis  IS.  OF ....FLUID  TO  THE  FLOIAI  COMPRESSION  6? .  IHSlPE, 4  c  dt  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 • .' ..  THE  ,  .  t  Q  IZATE  P.  4-10  3>5ooo  m  CJ^  ft  =  DISCUSSED  IAJ  THE  TEXT,  3  psi  2oo(l..+  IAJU-ERE  Gfives  MODEL  ^  3oo  sin oji) 4  Hz.  u> coso)t  25"  raJ/sec.  _ .  TH15  Gives  (Q )  £0£  MAXIMUM  c  -  $  J^  « 75-00 = 0,37/  0 0  F-LON  Qi  wH-E-zE  =  mx  =  A 4.  y  COUSIPERHJG  CONDITION,  Ay  =  PISTON  =  P(ST6>N  AREA  - .  0 . 3 9 3  />  2  VELOCITY.  12-iNcu  FULL  y =•  /V/«c  STROKE  •  <S ? w £ j t  Cf -  &  cu cos cot  6j)i*c-=V ^  ' I S O  m/scc  7-/^/S 6/i/<5S ..... (5/ ....= . 0<393 x /s~£> : _:  THIS  <s,iVES  *=..59. in /sec. 3  THE  HATiO  -  T14E  COMPRESSIBILITY  FLOU)  A  uoiN  CONSIDER.  V P WHERE  mis  Gives NENCE  co  =  C£)»*  =  c  -  TYPICALLY,  DIA-  * 8 FT  /.£\7  taj/sec.  LoNC  + Sin Ot) Uz  1 0 0 0 *  X  O  (8"  3  0.2S'  N E G L I G I B L E .  SYSTEM,  in  IOOO (I mA  THUS  FULL-SCALE  5O00  "  IS  0.6°Jo  So 00  =  /• 57  * isio  /S-70 =  p  si  8 3 }/7 /s^c 3  106  mz.  iKiter  Of  miS  THIS  FLON  = -Aij  nw  Cwes,  IS  RAI&  ALSO  IS  Noui  =  SO x 48 x },SI  -  3110  THG  RATIO  Qc _ Q;  _B3_ ZTlO  iJe&LiG  I8LE.  in /sec3  2.2%  (In*/sec)  APPENDIX  0  LABORATORY  /•  INPUT ju  -  2.  MODEL  • O.I2Sfh  *=*  97  LP AD =  3-  0  ;  ...... /4 Av. AREA  ••  ''SYSTEM'  -  0  ~  314  Aj  =  2  Ap  A/ =  P  z  in* . . . . . .  ZO  : L :.  .  £  =  s v  =  1:  v  3  . 2  r  ^  4  p  VlOlERS  US&Pfi]  O OU8 r  j•  m ^  /4  £E(Z\/0VALVE-  C  :  ,!  .  " =  /i ' .  ..  ..T-yXjk/\'»\=\W-  Ih/iL  SYSTEM  3  .  . . . . . .  :  - ^777  ACTIVE  '  ; ...  /n* ' •  = . 2.SO  • V*  K  in  X5psia.  • 3 - C 4 i/i*  -  c  STIFFNESS:  Ll -  psig -  2.95  V = :  !i>-iwass.  <*> ...  PASSiy/E P  «sLu^-s  S o  . tfc =  4-  i  .- .. i. \{ " . =  0  M  .4  'SPECIFICATIONS'  SC4--03  @> EOO psi  Jrep  Lin/sex)/ff?)  :  r  A, = j-7s-  foe)/fj^i  Ps  Ll*l£A£  . APPeoxiMATi'OtJ LET  .  ...... - \ .  F  Ao  y  0  -  :  /a*  . .  = C7o7^9-4-2=  y ^ n s  /"«/s<zo  AId-* AA  -Z„...=  (0 c" . r sv  ^  *  0.055.  w  ! •  . 3 93  0  .•.,. ...  o-o a8  r-—1  (^b/uc)/^.  =--o,on. (j*/*«-)//t-  '  -0.OO14  4  . 4 , ... :  _  ma  =  ...  _  _  I.""....'.  _ . J  • •- •  2 A^/^7AA g_?^  • : ' * " " =  - 2 . 4 -  2+(o.5°>3,y{2<t  (fr/s*c)/lh ., ;,f ;  Qh/<*t)/  ik  .... -  -<D.2  0+/scc.)/ll>  ,-  .  109  E  /)PP£A/D/X  FlZICTioN  d>W  PARAMETER  .fRiCTiOti .IS-. CONSIDERED THE SYSTEM , AS SHotJtJ  AS AN >N £1$.  E?)CTER.hiAL £.[ •  RAM  FA  FIG-  THE  . MET  FORCE ^AV*  WUERE  i  IS  A  E . I  IS  ' = =  '..  y = 0  GiVEti + F  f  IS  THE  OF  t/  :  - f  p  f  r  FZICTioN  /5  TO  ...„ (b) • y  BY  I~A  FUNCTION  (a)  FO&CE  'EQUAL  (FA +F ) P  A7v7>  OPPOSITE  UNTIL  MOTION  BEGitJS * O  •  Jr.  IN-  IS  COslSTAMT  SE-tJS£  TO  AMD  j L  4  •(F F ) P+  = 0  FOQLCS  A  5 f  °  OPPOSJTE  IIC  APPEMDIX ONLIN 1 2 3 4 5 6 7 8 9 10 10.25 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 25.25 25.5 26 27 28 31 32 33 34 35 36 37 38 39 40 41 42 42.25 43.25 44 45 46 47 48 49 50 51 52 53  F. 1  »^»aji»»»ainii»^3»ajt»  * *****  *  NONLINEAR  REMCiNONLIN  ********  MODEL S I M U L A T I O N  PROGRAM  ***  INITIAL CONSTANT C S V = 0 . 0 1 1 8 , U G = 0 . 1 2 5 , A A = 0 . 3 9 3 , A P = 3 . 0 , ... V 0 = 2 0 . , VT = 2 8 0 . , M=3.G, GAM=1.2, P S = 5 0 G . ,.«,. K0=6.28 PARAMETER K2 = G., K3 = C.O, K S V = 1 . 0 , P 0 = 1 5 . 0 , . . . Z E T A = 1 . 0 0 , C V = 3 . 7 5 , F F 0 = 1 . , YDCR=0.CCOGGG1, C X = 1 . PARAMETER K 1 = ( 0 . , 5 . ) PARAMETER RW= 1.0, TCV=0. 0 2 5 N=VT/VO KS=2.*GAM*P0*AP**2/VT*12. WN=SQRT(KS/M) TC1=2.*ZETA/WN TC2=TC1/(N+1.) KP=N*KS/(N+1.) W' = Rfo#WO DYNAMIC Ul=UO*SINE(0. W,0.) U2=RA,MP( 0.0) - R A M P ( l . O ) U=U1*U2 UD=DERIV<0.,U) UDD=DERIV(0.»UD) X=U+Y X1=CMPXPL<Q. , 0 . ,0.5,W,XD) X0UT=W*X1 F1=LEDLAG(TC1,TC2,Y) FP=-KP*F1 YD=IMPL( 0 . t 0 . 0 5 i F Y O ) XD=UO+YD YDD=DERIV(0.,YD) XCD=UDD+YDD FNET=M*XDD+CX*XD PROCEDURE FFR = F R I C ( Y D , F F G ,FNET,YDCR> IF(ABS(YD)-YDCR)10,10,11 11 IF(YD)1,1,3 1 FFR=-FFG GO TO 4 3 FFR=FF0 GO TO 4 10 FFR=LIMITi-FF0,FF0,FNET) 4 CONTINUE ENDPROCEDURE FA=FNET-(FP-FFR) XDDD=-W**2*XD R1=-K1*XDDD*W R=CMPXPL(0.,0.,0.5,W,R1) Z1=KSV*R Z = L I M I T ( - 4 0 . ,40. , Z D SGN=FCNSW ( Z , -1 . 0 , 0 . 0 , 1.0 ) SGN2=FCNSK(FA,-1.0,0.0,1.0) YD1=CSV*Z*SQRT(PS-LIMIT(-PS,PS,SGN*FA/AA)) YD2=SGN2*CV*SQRT(ABS(FA/AA)) f  /// 54 55 55. 25 55.5  FVD={ Y U i - V D 2 ) / 1 2 . / ^ A V=1MGRL (0.,YC) NOSORT * GO  TO  30  55.6 51 I F ( K f c f c P . N E . l ) CC TC 30 55.7 TX=TIME+C.CC1 55.8 I F { A M O D ( T X » .0 5 ).GT#0.002) GO TO 30 55.81 W R I T E ( 8 , 3 1 ) TIM£,XCuT,U 55.82 31 FORMAT(3E14.6) 55.83 30 CONTINUE 56 P R I N T I , X, Y, X C L T , P, F N E T , F P , F L 57 T I T L E A C T I V E / P A S S I V E MQTICN C O M P E N S A T I O N 58 T I M E R P R C E L = 0 . 0 5 , FINTIM=iO.» DELT = 0 . 0 5 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  SYSTEM  112  APPENDIX LINSYS 1 2 3 4 5 6 .7 8 9 10 11 12 13 14 15 18 18.1 18.2 18.25 19 20 21 21.25 22 22.2 5 22.5 23 24 25 26 27 28 29 30 31 32 33 34 35  36 37 38 39 40 41 42 43 44 44.25 44.5 45 46 46.25 46.5 47  F. 2.  c * * * * * * * * * * * * * * * * * * * * REMC !L I N S Y S * * * * * * * * * * * * * * * * * * * * * * * * * * * C C L I N E A P I Z E C MUCEL OF MOTICN C C K P t N S A T ION S Y S T c M C £ * * * * * * * * * * * * « * $ * * * * * * * * * ******** **>!« ** v * * * i | - * * * i ; * * * j , ' . j ; t w*4«r.< :*****.• COMPLEX G , F , T 1 , T 2 , F F F , F F B , Z 1 , S , H , P W R REAL K S , K l , K 2 , K 3 , K P , M , N , L C G R h , K F F R E A 0 ( 5 , 1 , E N D = 9 9 ) A A , A , P Q , V C , V T, G AM , M , Z , CV 1 ,C V 2 , C X , D E L A Y KS=2.*GAM*PG*A**2/VT*12. l»N = SCRT( KS/M) N=VT/VC IF<Z.EG.G.) Z = S Q R T ( ( N + l . ) * ( N + 2 . ) / ( 8 . * N ) ) TC1=2.*Z/WN TC2=TC1/(N+1.) KP=N-KS/(N+1.) 100 REAC(5,1,END=99) WQ,K1,K2,K3,KFF,CBP HCV1=1./(CV2+CBP) HCV2=CV1*HCV1 RWN=WN/WO I F t K F F . E C . C . } KFF = K P * ( C V 2 + C E P )/CV1 WRITE(7,4> P0»VC,VT,N»KS,Z,kN,CX WRITE(7,5) CVl,CV2,CbP,Kl,K2,K3,KFF fcRITE(7,6) kG,RhN L0GRW=-1. TCV=T A M DEL A Y / 1 8 0 .* 3 . 14 159 )/W0 ALPH=SQRT(!.+(TCV*WC)**2I DO 20 1=1,81 R^=10.0**L0GRW W=kW*WO S = C M P L X ( C , I* ) G=-KP*(l.+TCl*S)/(l.+TC2*S) F=M*S**2+CX*S HFB = ( K 1 + K 2 * S + K 3 * S * * 2 ) * A L P H / l l . * T C V * S )*I«*S / ( W* *2<-k »S+S**2) HFF=KFF h = - ( F / H C V 1 + C V 1 * ( H F E + H F F ) ) / ( G / F C V l + C V 1*FFF + D Tl=l./(1.+H) PHIH=ATAN2(AIMAG(H)»R E A L ( H ) M 1 8 0 . / 2 . 1 4 _ 5 9 Z1=HF6*T1+FFF :  T2=T1-CMPLX(1.,0.)  T1A=CABS(T1) T2A=CA6S(T2) HA=CA6S(H) 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 P H I 2 = ( A T A N 2 ( A I MAG(T2 ) . F E A L ( f 2 ) ) ) * 1 8 0 • / 2 . 1 4 1 5 9 DB1=20.*AL0G10{T1A) CB2 = 20 .*AL CG10 (T 2 A ) DBH=20.*ALGG10(HA) Z2=CA6S(Z1) PWR=500.*AA*S*T2 PWRA=CABS(PfcR) I F ( M U D { I , 2 ) . E Q . 1 ) W R I T E ( 7 , S > Rh ,DB1 , PH 11 ,062 , PHI 2 , * CBH,PHIF,Z2,PWRA DBZ=2C.*ALCG10(Z2) PH I Z= { AT AN 2 { A IMAGt Z D , REAL ( Z D ) I * 18u. / 2. 1 4 1 59 WRITE(8,3) LCGRW,DE1,C82,PH11,Ph12  113 48 20 49 50 1 51 3 4 52 53' 54 55 56 5 57 57.25 58 6 58.25 58. 5 59 99 60 OF F I L E  LOGRW=LOGRW+0.025 GO TO 100 FORMAT(12E12.0) FORMAT(10F13.4) FORMAT {• 1 P A S S I V E S I D E ' / 'OPRESS » = , F 6 . 0, 5 X , »C YL • VOL . =*, * F 6 . 0 , 5 X , « T A N K VOL. = • , F 6 . 0 » 5 X , « V O L R A T I O = ' , F 6 . 3 / * • S T A T . S T I F F N E S S = • , F 8 . 2 , 5 X , «CRIT.DAMP.RATIO = « , F 7 . 3 , * 5 X , • NATo FREQ • = • , F 6 2 / BODY OR0G C O E F F . = « , F 6 . 0 ) FORMAT <'OACTIVE S IDE V C L I NEAR V A L V E C C E F F S . CV1 ='» * F 9 . 5 , 5 X , * C V 2 = ' , F 9 . 5 , 5 X , C B P = » , F 9 . 5 / « FEEDBACK C O N S T S . K l =«, *F6.0, * 3 X t 'K2 = S F 6 . 0 , 3 X , «K3 = »,F6.1/' FEEDFWD CONST. KFF =«,F6.3/) FORMAT(•OOPERATING FRECUENCY=• F 7 . 2 / * 5X,'NAT.FREQ./OP.FREQ.=•,F6.2/*0*) STOP END 1  ,  0  ,  f  SKI P  i  lit  APPENDIX PTIM 1 2  3  4 4.5 5 6 7 8 9 10 10.25 11 13 14 15 16 17 18 1 8 . 25 18.5 18. 6 18.7 18.8 18. 81 18.82 18.83 19 20 21 22 23 24 25 26 27 28 29 30 31 32 34 35 36 41 42 43 44 45 46 47 48 49 50 51  C C  c c c  1  999 777  1 2  r.3  ******************* ****  REMC : 0 P T I M * * * * * * * * * * * * * * * * * * * * * * * * * *  PROGRAM TO O P T I M I Z E PARAMETERS  OF CCNTROL  SYSTEM  ***  ********************************************************* DIMENSION VAR(3,6) EXTERNAL T R A N S F , F L O , F H I,FMPL COMPLEX S , H F F , F , G R E A L M,N,KP,KS COMMON/PARAM/S,HFF,F,G,KCV1,HCV2 R E A D ( 5 , 1 ) A A , A , PO , VC , V T , G AM, Mt, Z , CV1 , CV2 , CBP READ(5,1)W FORMAT(12E10.0 ) KS = 2.0*GAM,*P0*A**2/VT*12.0 WN=SQRT(KS/M) N=VT/VC I F i Z . E Q . O . ) Z = S G R T ( ( N + l . ) * ( N + 2 . ) / ( 8 . * N )) KP=N*KS/(N+1. ) S=CMPLX(0.,W) TC1=2.*Z/WN T C 2 = T C 1 / ( N + 1. J G=KP*(l.+TCl*S)/(l.+TC2*S) F=M*S**2 HCV1=1./CCV2+CBPJ HCV2=CV1*HCV1 KFF=0. H FF = KF F VAR(1,1)=0.0 VAR(2, 1 )=0. VAR(3,1)=0. CALL C0MPLX(X,VAR,3,3,6,4,9.9,50,150,2 50,10,0.001,TRANSF,FLO, *FHI,FMPL,£999,£777) STOP STOP 9 STOP 7 END FUNCTION TRANSF(T,NN) DIMENSION T ( 1 ) COMPLEX S , G , F , H F 8 , T 1 , H F F REAL K1,K2,K3 COMMON/PARAM/S,HFF,F,G,HCV1,HCV2 K1 = T U ) K2=T(2) K3=T(3) HFB=K1+K2*S+K3*S**2 T 1 = ( G - H C V 2 * H F F - H C V 1 * S ) /{F+G+HCV2*HFB-HCV1*S ) TRANS F=CABS (T 1 ) RETURN END FUNCTION F L O ( T , N , J ) DIMENSION T ( l ) GO TO ( 1 , 2 , 3 , 4 ) , J FLC=0. RETURN FLO=0 .  us 52  53 54 54.25 54.5 55 56 57 58 59 60 61 62 63 64 64.25 64.5 65 66 67 68  3 4  1 2 3  69  69.25  69. 5 69.6  69.7 69.8 69.81 71 72 OF F I L E  SKI P  4  RETURN FL0=-10. RETURN FL0=0. RETURN END FUNCTION F H I ( T , N , J ) DIMENSION T { 1 ) GO TO (1 ,2 ,3 ,4) , J FHI=.0. PETURN FHI=0. RETURN FHI=10. RETURN FHI=128. RETURN END FUNCTION FMPL<T,N,J) DIMENSION T U ) COMPLEX S , T 1 , H F F ,HF8,F,G COMMON/PARAM/S•HFF F,G,HC V I , H C V 2 K1=T(1) K2=T(2) K2=T( 3) HF8=K1+K2*S+K3*S**2 Tl=(G-HCV2*HFF-HCV1*S)/(F+G+HCV2*HFB-HCV1*S) FMPL=CABS(HFB*T1+HFF) PETURN END t  APPENDIX EtPER. I  G  MENTAL  AM PL. *3  Hz.  Odb  R.ESULTS  SIMULATION  FIGURE  LAC  PHASE  RATIO SWUL.  &XP 'T  -14 di>  0°  18°  EY.PT  0  0-5  j  0.5 ...5  -3db  -z.i<u>41°  27°  1-0  -Adh  -4T5Ji  54°  36°  ... .0  -i&Jb  5  - 7.3 Ah  2.0  0  -ISAh  -----  2-0  5  -/hldl  - JOSdh  1.0  THE  ABOVE  7  Ik)  ARE  .  11°  100°  30°  108°  PloTTErD  OfJ  6.3. .1 , :  54°  72°.  6./  A BODE  64-  DIAGRAM  FIG. 4.2-1-  MOTES,  .  ON -  LOWE  •--  FI&UZCS 2  a  COMPUTER.  APPENDIX -  -  G.I  -..£.4 •  CURVES  USING  Fj-  THE  WERE  ESTIMATED  -I  THESE.  BY  PROGRAM  L  RATIOS ... FROM  GENERATED  SIMULATION  UPPER 2 CURVES ARE FULL SCALE OEFLECTION AMPLITUDE  -  ;  FROM CHART IS + i-5 INCHES. WD  PHASE  CVZVES,  SN/FTS  WE  OF  ..  .. •• ; RECORDER.  . .1 IN  ERE  REMC  6)1  PLOT*.00720098.  n-7  i  

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