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An analysis of the Fraser River tidal control Kay, Donald Hughie James 1951

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AN ANALYSIS OF THE FRASER RIVER MODEL TIDAL CONTROL by DONALD HUG-HIE JAMES KAY  A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLI-ED SCIENCE i n t h e Department of •ELECTRICAL ENGINEERING  We a c c e p t t h i s t h e s i s as conforming t o t h e s t a n d a r d r e q u i r e d from c a n d i d a t e s f o r t h e degree o f MASTER OF APPLIED SCIENCE  Members o f t h e Department o f ELECTRICAL ENGINEERING  THE UNIVERSITY OF BRITISH COLUMBIA A p r i l , 1951  Abstract  The  development  o f an a u t o m a t i c c o n t r o l  i n v o l v e s a c o n s i d e r a t i o n of t h e problems response o f such a system. o u t l i n e these problems of water l e v e l  o f t h e s t a b i l i t y and  The p u r p o s e o f t h i s t h e s i s i s t o  as t h e y appear i n t h e a u t o m a t i c  i n a t i d a l basin  stressing three points  (2) D e s i g n t h e n a t u r a l  f r e q u e n c y w e l l above t h e o p e r a t i n g f r e q u e n c y .  The  theory i s  (1) D e s i g n t h e system  so t h a t o s c i l l a t o r y c o n d i t i o n s p r e v a i l .  necessary introduce s t a b i l i z i n g  control  model.  A s y n o p s i s o f g e n e r a l servomechanism briefly outlined  system  ( 3 ) Where  networks.  r e s u l t s o b t a i n e d by t e s t s r e v e a l e d one t h i n g ;  t h e i n t r o d u c t i o n o f e x t e r n a l c i r c u i t s had v e r y l i t t l e on t h e c o n t i n u o u s o p e r a t i o n o f t h e t i d a l m o d e l .  An  effect examin-  a t i o n o f t h e t h e o r e t i c a l a n a l y s i s o f t h e c o n t r o l s y s t e m however brought out t h r e e reasons f o r l i m i t i n g t h e v a l u e o f t h e constant K . 4  The c o n s t a n t K 4 c o r r e s p o n d s t o t h e moment o f  i n e r t i a o f a mechanical system.  The c o n s t a n t K  4  i s related  to t h e parameters o f t h e system, w h i c h a r e t h e a r e a o f t h e b a s i n , t h e l e n g t h o f t h e w e i r s , a n d t h e pump d i s c h a r g e .  i Table of  Contents  page Introduction  1  General Theory  2  Description  of Tidal Control  Equipment  8  Theoretical Analysis  11  T e s t s and  24  Results  Discussion  and  Conclusions  26  Acknowledgments  29  Bibliography  30  List  of  Photographs  following  page  P h o t o #1 West P o r t i o n o f R i v e r B a s i n  7  P h o t o #2  Centre P o r t i o n of River  7  P h o t o #3  East P o r t i o n of River Basin  7  P h o t o #4  Tidal Basin  7  P h o t o #5  Electronic Control  P h o t o #6  Hydraulic  The  Set  Equipment  8  Amplifier  8  Jack  8  of Weirs  8  P h o t o #7 H y d r a u l i c P h o t o #8  Basin  ii List  of  Illustrations  page Fig.  #1 B l o c k D i a g r a m o f a S i m p l e S e r v o m e c h a n i s m  Fig.  #2 E r r o r - t i m e C u r v e s f o r V i s c o u s - c l a m p e d  2  Servomechanism Fig.  following  #3 R e s o n a n c e C u r v e s o f S e r v o m e c h a n i s m  with following 6  V i s c o u s O u t p u t Damping Fig.  #4 B l o c k D i a g r a m o f t h e C h a r t  Fig.  #5 B l o c k D i a g r a m o f T i d a l  Fig.  #6 S c h e m a t i c D i a g r a m o f T i d a l  Fig.  #7 B l o c k D i a g r a m o f M o d i f i e d T i d a l  List  6  following 9  Reader  Control  13  System  13  Basin Control  System  16  o f Graphs  following  page  G r a p h #1 O u t p u t V e c t o r L o c u s f o r C o n t r o l S y s t e m  25  G r a p h #2 S e r v o R e s p o n s e w i t h I n t e r n a l F e e d b a c k C i r c u i t  25  G r a p h #3 S e r v o R e s p o n s e w i t h o u t I n t e r n a l  Feedback  Circuit  25  G r a p h #4 S e r v o R e s p o n s e w i t h o u t D i f f e r e n t i a t i n g G r a p h #5 S e r v o R e s p o n s e w i t h D i f f e r e n t i a t i n g G r a p h #6 S t e p F u n c t i o n R e s p o n s e Differentiating  Circuit  Circuit  '  25 25  without  G r a p h #7 S t e p F u n c t i o n R e s p o n s e w i t h Circuit  Circuit  25 Differentiating 25  Introduction  The d e v e l o p m e n t  o f an a u t o m a t i c c o n t r o l  system  i n v o l v e s a c o n s i d e r a t i o n of t h e problems o f t h e s t a b i l i t y response of such a system.  Under s t a b i l i t y ,  t h e s y s t e m must  n e i t h e r i n d u c e n o r support any harmonic o s c i l l a t i o n s . any o s c i l l a t i o n s  and  Further,  t h a t a r e i n d u c e d by o u t s i d e d i s t u r b a n c e s must  be a t t e n u a t e d t o a n e g l i g i b l e v a l u e i n a v e r y f e w c y c l e s . U n d e r r e s p o n s e , t h e s y s t e m must f o l l o w t h e i n p u t s i g n a l as a c c u r a t e l y a s p o s s i b l e a n d w i t h a minimum t i m e d e l a y . The p u r p o s e o f t h i s  thesis i s to outline these  p r o b l e m s as t h e y a p p e a r i n t h e a u t o m a t i c c o n t r o l o f w a t e r i n a t i d a l b a s i n model.  The s e r v o - m e c h a n i s m  level  i t s e l f i s not  u n l i k e t h a t u s e d i n many o t h e r s y s t e m s b u t t h e e q u a t i o n s i n t r o d u c e d b y t h e h y d r a u l i c s o f t h e s y s t e m make t h e p r o b l e m u n i q u e . One o f t h e f a c t o r s i n v o l v e d i s t h e n o n - l i n e a r i t y o f t h e w e i r discharge equation.  Another i s the l a g i n the water  level  r e s p o n s e t o w e i r movement c a u s e d b y t h e f i n i t e v e l o c i t y wave i n w a t e r .  1  of a  z G e n e r a l Theory-  Control of t h e closed  s y s t e m s may be e i t h e r o f t h e o p e n c y c l e o r  cycle types.  I n an open c y c l e s y s t e m t h e s i g n a l  t h a t operates t h e c o n t r o l l e r i s independent o f t h e output. a closed and  In  c y c l e system a percentage o f t h e output i s f e d back  compared w i t h t h e i n p u t  signal.  The d i f f e r e n c e o r e r r o r  t h e n o p e r a t e s t h e c o n t r o l l e r so a s t o r e d u c e t h e e r r o r . open c y c l e system i s f a i r l y  simple  and w i l l  n o t be  The  considered  in this thesis. The  closed  c y c l e s y s t e m c a n be f u r t h e r  subdivided  i n t o a u t o m a t i c c o n t r o l o r r e g u l a t o r s y s t e m s and s e r v o m e c h a n i s m s . The  f u n d a m e n t a l d i f f e r e n c e i n t h e two s y s t e m s i s i n t h e i r  a p p l i c a t i o n , rather than the p r i n c i p l e s involved. matic regulator i s designed t o maintain  the output close t o  some f i x e d i n p u t , a s f o r e x a m p l e , i n a v o l t a g e  regulator.  servomechanism i s designed t o m a i n t a i n  t h e output  c l o s e t o some i n p u t  time,  which varies with  automatic p o s i t i o n c o n t r o l l e r . may b e c o n t i n u o u s o r  The i n p u t  Controller  1 Block  The  arbitrarily  as f o r example, an  i n the l a t t e r  discontinuous.  K  Fig.  The a u t o -  Load  Diagram o f a Simple Servomechanism  case  3 Fig.  1 i s a diagram  c o n t r o l system. and  S  o f an elementary  represents the input signal,  the difference or error.  i n p u t and o u t p u t controller.  closed cycle the output  The d i f f e r e n t i a l  compares t h e  s i g n a l and i n j e c t s t h e d i f f e r e n c e i n t o t h e  The c o n t r o l l e r w i l l  a m p l i f y t h e e r r o r and w i l l  produce such a change i n t h e o u t p u t  as w i l l  tend t o reduce t h e  error. C o n s i d e r f o r example a s i m p l e m e c h a n i c a l system w i t h case w i l l "F".  &?  and So  angular positions.  be a moment o f i n e r t i a  positioning  The l o a d i n t h i s  " J " and a f r i c t i o n  component  L e t t h e t o r q u e a p p l i e d t o t h e l o a d be p r o p o r t i o n a l t o  Then t h e b a s i c e q u a t i o n s w i l l  Gj  be  - Go - e  T « KS c7t  W  x  Where J = moment o f i n e r t i a o f s y s t e m F s viscous T - output  friction torque of c o n t r o l l e r  Combining t h e three  or  6>  0  =  K  e  equations  p  7  This yields a Characteristic a second  order d i f f e r e n t i a l  i n the design.  jL  l i n e a r equation that i s  e q u a t i o n J p 4- Fp •+• K.  constants a r e i n g e n e r a l independent adjusted  ,  2  The t h r e e  a n d c a n be i n d i v i d u a l l y  4 There i s one b a s i c system.  t h i s simple  cause a v e l o c i t y e r r o r i n t h e system, o r t h e e r r o r  be p r o p o r t i o n a l  t o t h e r a t e o f change o f p o s i t i o n .  v e l o c i t y error plus  t h e i n e r t i a o f t h e system w i l l  cause o s c i l l a t i o n .  Increasing  increase  This  tend t o  the f r i c t i o n factor w i l l  the time o f o s c i l l a t i o n b u t i t w i l l the  servo  A n e r r o r must e x i s t b e f o r e a n y c o r r e c t i o n i s a p p l i e d .  This w i l l will  fault with  decrease  t h e magnitude o f  velocity error. Where more r i g i d  d e s i g n r e q u i r e m e n t s have t o be met,  s t a b i l i z i n g c i r c u i t s must b e i n t r o d u c e d  t o reduce t h e magnitude  o f t h e v e l o c i t y e r r o r and r e d u c e t h e t i m e o f o s c i l l a t i o n . the  In  c o n t r o l l e r o f the elementary system t h e output i s propor-  t i o n a l t o the error, that equals a  i sthe controller transfer  function  constant. T.F. = K The s t a b i l i z i n g c i r c u i t s may b e i n t r o d u c e d  into the  c o n t r o l l e r so t h e t r a n s f e r f u n c t i o n may b e a d i f f e r e n t i a l equation o f t h e type T.F. « A + B + C p - H D p + ... p 2  The c i r c u i t s c a n b e a d j u s t e d constants o f any desired determine the various  value.  t o make a n y o f t h e  The d e s i g n r e q u i r e m e n t s  will  values.  The t e r m A r e p r e s e n t s P component.  P = d_ dt  the integral factor or reset  Increase i n t h e value o f t h i s term decreases the  amplitude o f the v e l o c i t y error.  The d i s a d v a n t a g e o f t h i s i s  that as t h e v e l o c i t y e r r o r approaches zero t h e system approaches  instability.  I f t h e v e l o c i t y e r r o r e q u a l s z e r o any  oscillation will  be s u s t a i n e d .  I f t h e v e l o c i t y e r r o r becomes •  negative then the o s c i l l a t i o n s w i l l The  terms  i n v o l v i n g p,p , 2  b u i l d up  In  frequency increases. to  i s to improve  most p r a c t i c a l s y s t e m s The  The  primary reason f o r  the frequency  the reponse w i l l  fall  response.  o f f as  the  d e r i v a t i v e f a c t o r s c a n be a d j u s t e d  increase the cut o f f frequency.  c a t i o n s t h e f i r s t o r second is  indefinitely.  e t c . represent the v a r i o u s  powers o f t h e d e r i v a t i v e o f t h e e r r o r . i n t r o d u c i n g t h e s e terms  induced  I n most p r a c t i c a l  appli-  order of derivative i s a l l that  used. R e f e r r i n g to the simple servo system a g a i n the  characteristic  e q u a t i o n has  -I_XFJS - K  P=  Z  2 J f\ZZj  This w i l l magnitude of the  J  permit t h r e e s o l u t i o n s depending  (^Tj)  (2)  /LEL^f  >  "j-  the  ^-E—J  overdamped  =  critically  damped  J  \ Z J J  (3)  on  parameters.  (1)  In  i t s r o o t s as  2  <  u n d e r damped  t h e f i r s t c a s e p i s r e a l w i t h two  surd  roots.  The  transient  may  be e x p r e s s e d i n h y p e r b o l i c f u n c t i o n s w i t h an e x p o n e n t i a l  decay term.  s o l u t i o n of d i f f e r e n t i a l equations of t h i s  T h i s case i s termed  i s no t e n d e n c y  as t h e overdamped c a s e .  f o r o s c i l l a t i o n a t a l l but the time of  form  There  response  6 is has  t o o . l o n g t o be p r a c t i c a l . two  I n t h e second  identical real roots.  case the  T h i s e q u a t i o n has  equation  a solution  of  an e x p o n e n t i a l decay term m u l t i p l y i n g a c o n s t a n t t e r m p l u s a constant times time. damped c a s e .  The  T h i s case i s r e f e r r e d t o as the  damping f a c t o r F  critical  e q u a l s E / K J . A g a i n t h e r e i s no t e n d e n c y  d a m p i n g and  a t i o n but the time o f response, case, i s s t i l l In two  i s known a s t h e  c  t o o l o n g t o be  critically  for oscill-  although s h o r t e r than the  first  practical.  t h e t h i r d and most i m p o r t a n t c a s e t h e e q u a t i o n  r o o t s t h a t a r e complex c o n j u g a t e s .  This y i e l d s the  has  oscill-  a t o r y s o l u t i o n , a n e x p o n e n t i a l f a c t o r m u l t i p l y i n g a s i n e and cosine term. s i n e and  T h i s c a s e i s c a l l e d t h e underdamped c a s e .  c o s i n e terms w i l l  n a t u r a l frequency of the  J  h a v e a f r e q u e n c y r e f e r r e d t o as system  271 v  J  U//  2 i s a graph of servomechanism response  step f u n c t i o n input. d a m p i n g r a t i o s "G".  The The  various curves correspond to damping r a t i o  "C"  CJ^  d a m p i n g and  damping o f  i s the n a t u r a l frequency of the system COj i s t h e a p p l i e d f r e q u e n c y .  dimensionless to permit comparison If  The  of d i f f e r e n t  a l s o be  ment w i l l  The  the  without  systems. sinusoidal  some t i m e p h a s e  d e g r e e o f a m p l i f i c a t i o n and p h a s e  depend on t h e a p p l i e d f r e q u e n c y .  the  sinusoidal with  e i t h e r p o s i t i v e o r n e g a t i v e a m p l i f i c a t i o n and displacement.  different  c u r v e s a r e made  instead of a step f u n c t i o n input a  input i s applied, the output w i l l  to a  i s d e f i n e d as  r a t i o o f t h e a c t u a l damping t o t h e c r i t i c a l system.  the  *J-/£-(£.)*  l> zn  Fig.  The  I f the  displace-  applied  0  5  10 FIG.  Error-time  Curves  for  15  30  0  1 Relative  2  2 Frequency  3  FIG. 3 Viscous-damned  Servomechanism  Resonance With  Curves  Viscous  of  Output  Servomechanism Damping  7 frequency approaches  the n a t u r a l frequency a c o n d i t i o n  very s i m i l a r to t h a t encountered  exists  i n an o s c i l l a t o r t a n k  circuit.  A very s m a l l input s i g n a l w i l l r e s u l t i n a l a r g e output t h e o n l y l i m i t i n g f a c t o r i s t h e damping o r f r i c t i o n T h i s i s v e r y u n d e s i r a b l e i n most s e r v o s y s t e m s .  signal,  coefficient.  The  ideal  s i t u a t i o n e x i s t s when t h e o u t p u t e q u a l s -the i n p u t and  the phase  displacement  response  of  i s a minimum.  F i g . 3 i s a graph of the  a simple servo subject to various frequency  inputs.  T h i s r e v i e w o f t h e servomechanism t h e o r y has r e v e a l e d three points.  (1) D e s i g n t h e s y s t e m  conditions w i l l prevail, t h e underdamped c a s e .  so t h a t  oscillatory  t h a t i s have i t s a t i s f y case I I I ,  (2) D e s i g n t h e n a t u r a l f r e q u e n c y  well  above t h e o p e r a t i n g f r e q u e n c y e s p e c i a l l y i f t h e i n p u t s h o u l d c o n t a i n predominant (3) Where more r i g i d duce networks  harmonics  higher than the  fundamental.  d e s i g n r e q u i r e m e n t s h a v e t o be met,  so a s t o r e d u c e  increase the frequency  the v e l o c i t y e r r o r , or to  response.  intro-  Photo #2 C e n t r e Port-ion o f R i v e r B a s i n  Photo #4 T i d a l B a s i n w i t h Recorder i n Foreground  8 Description of the T i d a l Control  Equipment  The m o d e l c o v e r s a n a r e a o f t h r e e includes the Eraser  R i v e r from M i s s i o n  acres.  It  t o t h e mouth,  Pitt  R i v e r and P i t t L a k e , and a p o r t i o n o f t h e G-ulf o f G e o r g i a i n the v i c i n i t y o f the t i d e f l a t s .  The h o r i z o n t a l s c a l e i s one  i n s i x h u n d r e d , t h e t i m e and v e r t i c a l and t h e v e l o c i t y s c a l e i s one  i n eight point  views o f t h e model a r e i l l u s t r a t e d P h o t o g r a p h s 1,2  i n seventy,  five.  Various  i n p h o t o g r a p h s 1,2,3  and 3, a r e v i e w s o f t h e w e s t , c e n t r e ,  portions of the;river basin. tidal  s c a l e i s one  and  and  4.  east  Photograph 4 i s a view of the  basin. The w a t e r l e v e l  components i l l u s t r a t e d  c o n t r o l equipment c o n s i s t s o f s i x  i n p h o t o g r a p h s 5,6,7  (1) A p h o t o c e l l  reader that provides  p r o p o r t i o n a l to the desired (2)  and  tide  8. a  voltage  level.  a p a i r o f l o a d r e s i s t o r s t h a t a c t s as t h e differential.  (3) An a m p l i f i e r t h a t  c o n s i s t s o f a D.C.  electronic  a m p l i f i e r and a h y d r a u l i c a m p l i f i e r . (4) The h y d r a u l i c  j a c k t h a t r a i s e s and l o w e r s t h e  weirs. (5) The  set of weirs  (6)  t i d e f l o a t potentiometer that produces  The  voltage  proportional to the actual t i d e  There are o t h e r cubic  that c o n t r o l the water  pieces  f o o t p e r s e c o n d pump t h a t  level. a  level.  o f e q u i p m e n t , s u c h a s t h e 20 supplies water to the basin;  The power s u p p l i e s f o r t h e f l o a t p o t e n t i o m e t e r , t h e p h o t o  cell  t o f o l l o w page  Photo-#5 E l e c t r o n i c C o n t r o l Equipment (1) P h o t o C e l l (2) D.C. (3)  Including  Reader  Amplifier  J u n c t i o n b o x c o n t a i n i n g power and  differential  P h o t o #6  Hydraulic  Amplifier  supplies  8  P h o t o #8 The  Set of Weirs  reader,  and t h e D.C.  system t h a t  a m p l i f i e r ; and t h e pump and r e g u l a t i n g  supplies the pressurized  o i l f o r the hydraulic  a m p l i f i e r , that are not a c t u a l l y part o f the closed control  system. The p h o t o c e l l  reader,iwhich  t i g h t b o x , i s shown s c h e m a t i c a l l y  i s encased i n a  i n F i g . 4.  i s r e f l e c t e d o f f the galvanometer m i r r o r transparent  The l i g h t  source of  onto a photo  The o u t p u t o f t h e t u b e i s c o n n e c t e d i n s e r i e s  t h e g a l v a n o m e t e r and one o f t h e l o a d r e s i s t o r s i n t h e  differential. is  light  through a piece  p a p e r and a p a i r o f c o n v e x l e n s e s  m u l t i p l i e r tube. with  cycle  The g a l v a n o m e t e r w i l l  deflect u n t i l the light  i n t e r r u p t e d by t h e t i d e c y c l e , w h i c h i s a b l a c k  on t h e t r a n s p a r e n t  paper.  Time s c a l e e x t e n d s a l o n g  and  water l e v e l  the  l i g h t by s y n c h r o n o u s m o t o r .  photo c e l l  scale across.  The t i d e c h a r t  i s driven  the voltage  t i d e l e v e l from t h e photo c e l l  p o t e n t i o m e t e r and h a s a n o u t p u t v o l t a g e The c i r c u i t  past of the  level.  proportional  r e a d e r and t h a t  p r o p o r t i o n a l t o a c t u a l t i d e l e v e l from the t i d e  float  equal t o the d i f f e r e n c e .  c o n s i s t s o f two l o a d r e s i s t o r s c o n n e c t e d i n s e r i e s  with the input voltages  impressed across  the r e s i s t o r s i n  polarity. The a m p l i f i e r s y s t e m c o n s i s t s o f a D.C.  a m p l i f i e r that actuates system.  the paper  reader i s proportional to the desired t i d e  to the desired  painted  Thus t h e o u t p u t v o l t a g e  The d i f f e r e n t i a l r e c e i v e s  opposite  line  a balanced o i l valve  electronic  i n the hydraulic  The d i s p l a c e m e n t o f t h e b a l a n c e d o i l v a l v e  i s pro-  p o r t i o n a l t o t h e m a g n i t u d e and p o l a r i t y o f t h e e r r o r o r difference voltage.  The o u t p u t o f t h e h y d r a u l i c a m p l i f i e r i s  no.  \  -LOG:-:  DIAGRAM  OF  TH:,  CHART  READER  10 i n t u r n p r o p o r t i o n a l to the displacement of the balanced o i l valve.  hydraulic The  The'output  o f t h e h y d r a u l i c a m p l i f i e r moves t h e  jack which  i s m e c h a n i c a l l y connected  hydraulic  j a c k speed  to the weirs.  and h e n c e t h e w e i r s p e e d  p o r t i o n a l t o the hydraulic a m p l i f i e r output.  i s pro-  W a t e r i s pumped  c o n t i n u o u s l y a t a r a t e o f 20 c u b i c f e e t p e r s e c o n d into the t i d a l basin.  The s u r p l u s w a t e r  w e i r s b a c k i n t o t h e sump. thus r e s u l t  f r o m a sump  i s s p i l l e d over the  Any v a r i a t i o n i n w e i r l e v e l  i n a s i m i l a r v a r i a t i o n i n the water  will  level.  A voltage proportional to the actual tide l e v e l i s o b t a i n e d b y means o f t h e t i d e f l o a t p o t e n t i o m e t e r . iometer i s m e c h a n i c a l l y connected  to a f l o a t .  The p o t e n t -  The o u t p u t o f  the potentiometer i s f e d into the other load r e s i s t o r i nthe differential. D u r i n g o p e r a t i o n t h e t i d e c h a r t , which has t h e sequence o f t i d e s f o r an e n t i r e y e a r p l o t t e d on i t , continuously past the l i g h t  source.  sequence o f o p e r a t i o n s f o r f i v e days. to have t h e system  i s driven  This c a l l s f o r a continuous I t i s highly  desirable  o p e r a t e w i t h o u t a n y b r e a k and t o h a v e t h e  e r r o r i n t h e o u t p u t l e s s t h a n 5%, p r e f e r a b l y l e s s t h a n 3 % . T h i s e r r o r o n l y r e f e r s t o t h e a m p l i t u d e , t h e phase i s r e l a t i v e l y unimportant l a r g e and r e m a i n s  fairly  displacement  as l o n g as i t i s n o t u n r e a s o n a b l y constant.  11 Theoretical  Analysis  I n d e s i g n i n g a s e r v o s y s t e m more t h a n o n e c a n he t a k e n . This approach and  approach  The i n p u t c a n be assumed t o be a s e r i e s o f s t e p s . i m p o s e s v e r y s e v e r e r e q u i r e m e n t s on t h e s y s t e m  i n c e r t a i n c a s e s may n o t be j u s t i f i e d .  Another  approach  c a n be t o assume t h e i n p u t i s a s i n u s o i d a l f u n c t i o n .  The  s y s t e m c a n t h e n be d e s i g n e d t o h a v e t h e a m p l i f i c a t i o n  less  t h a n some p r e d e t e r m i n e d  v a l u e and t h e phase d i s p l a c e m e n t a t  the operating frequencies l e s s than the a l l o w a b l e value.  If  t h e f u n c t i o n i s more c o m p l e x a F o u r i e r a n a l y s i s o f t h e f u n c t i o n c a n b e made a n d t h e h i g h e s t a p p r e c i a b l e h a r m o n i c  c a n b e made  t o c o m p l y w i t h t h e a m p l i t u d e and p h a s e d i s p l a c e m e n t  requirements.  I n a n a l y z i n g t h e t i d a l c o n t r o l system i t i s obvious that requiring  a good s t e p f u n c t i o n r e s p o n s e , i s i m p o s i n g  c o n d i t i o n s t h a t a r e more s e v e r e t h a n n e c e s s a r y .  The n o n  r e p e t e t i v e n a t u r e o f t h e t i d e c y c l e does n o t p e r m i t t h e u s u a l Fourier analysis of the cycle.  Harmonic a n a l y s e s o f t i d e  c y c l e s h a v e b e e n made t o f a c i l i t a t e t i d e p r e d i c t i o n .  This  a n a l y s i s r e l a t e s t h e v a r i o u s components o f t h e t i d e c y c l e t o the behaviour o f d i f f e r e n t  c e l e s t i a l bodies.  components e x i s t i n a d e t a i l e d  Some 170  a n a l y s i s but the f o u r major  c o m p o n e n t s a r e t h e p u l l o f t h e moon, t h e p u l l o f t h e s u n , and two  c o m p o n e n t s due t o moons d e c l i n a t i o n .  T a b l e #1 g i v e s t h e  r e l a t i v e magnitude and p e r i o d o f t h e v a r i o u s components.  The  m a g n i t u d e o f t h e moons component i s t a k e n a s u n i t y . These f o u r major the fundamental  components a r e n e a r o r l e s s  frequency, which  than  i s taken as t h e frequency  12 \  Component Moons  Relative  pull  Amplitude  Period  1.0  12.4 h r s .  0.280  12,0 h r s .  d e c l i n a t i o n #1  0.415  24  Moons d e c l i n a t i o n #2  0.258  25.9 h r s .  Suns  pull  Moons  Table  hrs.  #1  T a b l e o f p e r i o d s and r e l a t i v e a m p l i t u d e s o f m a j o r c o m p o n e n t s o f t h e harmonic c o n s t i t u e n t s o f t h e t i d a l o c y c l e o f t h e moons a t t r a c t i v e component.  There a r e components o f  h i g h e r f r e q u e n c y , i n s h a l l o w w a t e r t i d e s p e r i o d s may be a s low as t h r e e hours, but t h e r e l a t i v e magnitudes o f these c o m p o n e n t s a r e l e s s t h a n 0.01.  W i t h t h e s e f a c t o r s i n mind  t h e d e s i g n f r e q u e n c y was t a k e n t o be t h e f r e q u e n c y o f t h e moons attractive  component. The s t e p f u n c t i o n r e s p o n s e  importance response  c a n n o t be c o m p l e t e l y i g n o r e d .  level  i t i s d e s i r a b l e t o have t h e  s e t t l e to the s t a r t i n g t i d e l e v e l  length of time. level  A reasonable  t o a s t e p f u n c t i o n i s d e s i r a b l e f o r two r e a s o n s .  One, when s t a r t i n g t h e s y s t e m water  although o f secondary  Two, i f i n t h e m i d d l e  i n a reasonable  of the run the water  i s either i n t e n t i o n a l l y or accidentally shifted o f f the  curve i t w i l l  be d e s i r a b l e t o h a v e i t r e t u r n a s q u i c k l y a s  p o s s i b l e and w i t h t h e minimum amount o f o s c i l l a t i o n . A f t e r c o n s i d e r i n g these p r o p e r t i e s of the t i d a l cycle,  i t was d e c i d e d t o a n a l y z e t h e s y s t e m  using a design  f r e q u e n c y e q u a l t o t h e f r e q u e n c y o f t h e moons a t t r a c t i v e ponent.  The s t e p f u n c t i o n a n a l y s i s w i l l  be u s e d a s a  com-  check.  Amplifier  TM Basin  F i g . 5 B l o c k Diagram of T i d a l C o n t r o l System  Fig.  6 Schematic  Diagram o f T i d a l  Basin  Definitions Oj  - input  0  =. o u t p u t  o  £^  signal signal  = e r r o r between  and  O  A  £ , - e r r o r b e t w e e n €, and Kg dy_ * dt K]_ =. a m p l i f i e r c o n s t a n t y  s. w e i r  level  A  = a r e a o f t i d a l b a s i n = 12,000 s q . f t .  14 H = head over w e i r Kg - f e e d hack c o n s t a n t Q, =pump d i s c h a r g e = 2 0 c . f . s . b  = w e i r l e n g t h = 40 f t .  A b l o c k diagram o f t h e system as i t o r i g i n a l l y e x i s t e d i s i l l u s t r a t e d i n F i g . 5 and a diagram o f t h e t i d a l b a s i n a t t h e w e i r s i s shown i n F i g . 6.  The b a s i c r e l a t i o n s h i p s t a k e n o f f  t h e s e diagrams a r e  €  fjf  -  ea  =  -  £*  (  H + y  2  )  (*)  The w a t e r f l o w i n g i n t o t h e b a s i n minus the w a t e r f l o w i n g o u t over t h e w e i r s w i l l equal t h e change i n volume. The w a t e r f l o w i n g over the w e i r c a n be c a l c u l a t e d by t h e w e i r discharge formula. Weir discharge ft  with  .".  -  3.33  Q - K,H% o r  °  K H* S  = J.J  3 h  "  K  3  b  =  dXz' cit  = Q -  Ad&  ( 5 )  dt  E q u a t i o n f i v e i s a n o n - l i n e a r e q u a t i o n but i t can be expressed b y a MacLaurens S e r i e s .  »• ®fo - i * /ia )YJ  -*4&  T h e r e f o r e by power  -  expansion  iff ±:f&  & f -  -0  I f £d$g\s  s m a l l , a l l but t h e f i r s t two terms may be  Q at  neglected.  Here t h e maximum v a l u e i s o f t h e o r d e r o f 0.5 so i  reduces t h e v a l i d i t y o f t h e a p p r o x i m a t i o n b u t t h e t h i r d term i l e s s than one t e n t h o f t h e second term,  so (6)  From e q u a t i o n s 3,4 and 6  d t  where  1/  1  7 ?  c>  = J?  a l s o from e q u a t i o n s 2 and 3  £  =  X  -  €j  (8)  from e q u a t i o n s 1,7 and 8  l-rK K t  K J+K Kz t  E q u a t i o n (9.)  TV-  dt  Cl 2  +  + K,£&-d& dt  7  dt*  + K  dt  7  f  d ^  (  1  0  )  cit-  and (10) r e l a t e t h e e r r o r , i n p u t , and  o u t p u t o f a servo system. any g i v e n i n p u t c o n d i t i o n s .  The e q u a t i o n s c o u l d be s o l v e d f o r I n i t i a l l y the i n v e s t i g a t i o n w i l l  be c a r r i e d o u t w i t h o u t t h e i n t e r n a l feedback l o o p , see F i g . 7.  16  Fig.  7 Block Diagram of Modified T i d a l Control System  The basic equations now are  (6)  Combining these equations give  Kre>+-H?+Ki&-#+«*1k  1131  Comparing equations (9) and (12) one can see that the two equations are i d e n t i c a l i f K 5 the only effect  =  .:ffl . 1 KXK2  i n other words  of this internal feedback loop i s the reduction  of the apparent amplification of the amplifier.  This  effect  can be more simply accomplished by the use of an amplifier of  17 lower gain.  S i n c e i t i s d e s i r a b l e t o have t h e system work a s  a c c u r a t e l y a s p o s s i b l e a n d y e t be a s s i m p l e a n d t r o u b l e f r e e , so the f i r s t m o d i f i c a t i o n suggested feedback  i s removal o f t h e i n t e r n a l  loop. The  s o l u t i o n o f equations  considered f o ra frequency  (12) and (13) w i l l  i n p u t and f o r a step i n p u t .  a t r a n s f e r f u n c t i o n a n a l y s i s o f equation  (12) w i l l  If the error i s applied a r b i t r a r i l y of time o f constant amplitude, then the output w i l l same f r e q u e n c y amplitude.  frequency  a l s o be a c o s i n e f u n c t i o n o f time o f t h e  - C  =  J  +X)  e  Then e q u a t i o n  Ce  c a n then be r e p r e s e n t e d b y  jCOt  j(cot Q  as a c o s i n e f u n c t i o n  d i s p l a c e d b y some p h a s e a n g l e o f a d i f f e r e n t  =e  e  First  be c o n s i d e r e d .  s a y u n i t y , and v a r y i n g  The i n p u t a n d o u t p u t Si  now be  ( 1 2 ) becomes  *>  (14)  w h i c h may b e r e p r e s e n t e d a s a v e c t o r o f m a g n i t u d e  and  K The  o f phase  ~  /80°  displacement  + ore tan—1-7? Cu A 4  (16)  locus o r Nyquist plot o f these•vectors f o r e x i s t i n g  c o n d i t i o n s i s shown i n g r a p h #1.  A l l the necessary  information  18 for  a frequency study i s contained i n t h i s graph.  A v e c t o r from  t h e o r i g i n t o t h e c u r v e i s t h e output o f t h e system, f r o m t h e o r i g i n t o (-1,0) i s t h e e r r o r , curve i s the input s i g n a l .  and f r o m  a vector  (-1,0) t o t h e  The a n g l e b e t w e e n t h e i n p u t and  output v e c t o r s i s t h e phase displacement. As h a s a l r e a d y b e e n m e n t i o n e d  t h e a i m i s t o have t h e  i n p u t and o u t p u t v e c t o r s o f e q u a l magnitude. accomplished  T h i s c o u l d be  i f t h e l o c u s c o u l d be swung c o u n t e r - c l o c k w i s e u n t i l  the p o i n t on t h e l o c u s c o r r e s p o n d i n g t o t h e  operating frequency  i s on t h e l i n e ,  To a c c o m p l i s h  reals  by a d j u s t i n g K 5 would  e q u a l m i n u s one h a l f .  n o t be p r a c t i c a l . A d j u s t m e n t  of K 5 w i l l  o n l y change t h e l e n g t h o f t h e o u t p u t v e c t o r f o r a g i v e n vector, If  i t will  n o t a f f e c t t h e phase a n g l e .  t h e v e c t o r ( 0 ; ^ ) were reduced  sects the l i n e reals  error  R e f e r t o , g r a p h #1.  t o t h e p o i n t where i t i n t e r -  e q u a l m i n u s one h a l f , t h e n i t w o u l d  be  s h o r t e r t h a n t h e e r r o r v e c t o r and t h e a n g l e b e t w e e n ( 0 ; ^ ) (-1,0;C*^) w o u l d  this  and  b e g r e a t e r t h a n 90 d e g r e e s .  C o n s i d e r now e q u a t i o n ( 1 4 ) , i f t h e j t e r m h a d a m u l t i p l y i n g c o n s t a n t g r e a t e r than u n i t y , t h e curve would moved c l o s e r t o t h e n e g a t i v e r e a l a x i s . from t h e f i r s t  derivative of the output.  This j term  be  originates  H e r e i t c a n be  seen  how d e s i r a b l e i t i s t o h a v e a f l e x i b l e c o n s t a n t m u l t i p l y i n g  this  term. Now a s t e p f u n c t i o n a n a l y s i s o f e q u a t i o n ( 1 3 ) w i l l c a r r i e d o u t . The c o n d i t i o n s w i l l  be  t < o  be  19  Pe  =  t  o  t  Equation  ( 1 3 ) becomes  \Kf P  -I-? -I- Ks) £ = (kff  Z  =  £  =  e  e  H r ,  +-f>) Qt  2  Cos/%  large.  I")  (17) i s an e x p r e s s i o n f o r t h e e r r o r o f t h e This i s a t r a n s i e n t term  t h a t a p p r o a c h e s z e r o a s t i m e becomes l a r g e .  possible.  =o  f  system s u b j e c t t o a step i n p u t .  keep t h i s term  >o  )foj  f-—i  e  Equation servo  f > * e  =  i  I t i s desirable to  s m a l l and h a v e i t a p p r o a c h z e r o a s s o o n a s  T h i s may b e a c c o m p l i s h e d  Making K  4  small w i l l  by making K 4 s m a l l o r K  5  cause t h e e x p o n e n t i a l term t o d i e  away f a s t e r a n d i n t h e r a n g e o f v a l u e s c o n s i d e r e d t e n d t o increase the value o f the r a d i c a l . p r a c t i c a l purposes f i x e d ,  4  i sfor a l l  i t represents the c o n f i g u r a t i o n o f  the model a l r e a d y c o n s t r u c t e d . amplifier gain.  Unfortunately K  This leaves K 5which i s the  Any m a n i p u l a t i o n o f t h i s  q u a n t i t y does n o t  r e d u c e t h e t i m e o f d i e away. Since the time  s c a l e i s 1:70, t h e p e r i o d o f t h e compon-  e n t o f t h e moon i s - 1 0 . 6 3 m i n u t e s . angular  T h i s component t h e n h a s a n  velocity;  •  T  6 0 0  The n a t u r a l p e r i o d o f t h e s y s t e m ;  '  20  Kj-  Amplifier  =  = 0.2  ^  7  Constant  of Q foot /sec  (3.33  /foot  of  e r r o r  ky%  Taking  CO  a  Cu  -  = /  n  Q*  / Equation  =2>98xi0~*rds/sec  / —  (18)  A *  A (18) i n d i c a t e s t h e r e l a t i o n s h i p between t h e  p a r a m e t e r s and t h e n a t u r a l f r e q u e n c y o f t h e system. The  n e x t p o i n t o f i n t e r e s t on t h e s e e q u a t i o n s i s t h e  coefficient of thef i r s t ient of thef i r s t If  i t were zero  d e r i v a t i v e o f the output.  d e r i v a t i v e i s t h e f r i c t i o n o r damping  The s y s t e m  may b e i m p r o v e d b y i n c r e a s i n g t h i s f a c t o r b u t t h i s because o f h y d r a u l i c r e l a t i o n s h i p s . duce t h e magnitude o f K  system.  4  4  and K 5 .  until  operation  i s impossible  Although K 5 i s f l e x i b l e any  would i n v o l v e extensive  modifications f o r some  of the stabil-  t h a t would improve t h e o p e r a t i o n .  From t h e c o n s i d e r a t i o n o f e q u a t i o n s (17)  coefficient,  One a l t e r n a t i v e i s t o r e -  A n o t h e r a l t e r n a t i v e w o u l d be t o l o o k  izing circuit  coeffic-  a n y i n d u c e d o s c i l l a t i o n w o u l d be s u s t a i n e d  some e x t e r n a l f o r c e a p p e a r e d t o a l t e r i t .  adjustment o f K  This  ( 1 4 ) , ( 1 5 ) , ( 1 6 ) and  i t h a s b e e n seen- how d e s i r a b l e i t i s t o h a v e s o m e t h i n g  to increase response.  t h e f l e x i b i l i t y o f t h e s y s t e m and improve t h e To t h i s  end a c o n s i d e r a t i o n o f d e r i v a t i v e c o n t r o l  else  21 will  be s t u d i e d .  system w i l l will  The only, d i f f e r e n c e i n t h e e q u a t i o n s o f t h e  be i n t h e t r a n s f e r f u n c t i o n o f t h e a m p l i f i e r .  It  now b e  fa  +  K  < i i )  The b a s i c e q u a t i o n s now become  -e = £  6i  (i)  0  (19) (4)  u-/Q^/f n 1—1(1  _ 2. _ d 3  Combining these  (e)  Q  c/tj  equations (20)  K £+ s  0 +  A study o f t h i s now b e c a r r i e d o u t . K  s  e  -tj^r^ n  magnitude  +  ( 8 1 )  system w i t h a s i n u s o i d a l i n p u t  Equation  Ce. = * which  - $  ( 2 0 ) becomes - J  J  will  C  c  o  G  -cujftyc  —  a g a i n may b e r e p r e s e n t e d a s a v e c t o r o f  (22)  22 and  of phase displacement  A « The the  C u r v e (2)  Now  £  consider  Kg  stant  as  Equation  (26)  the  in  result.  ratio.  analysis of  equation  (21)  a  input.  < 2 5 )  solution  i s the Here the  factor i s revealed.  practical  increase  +  w h i c h has  of the  (24)  6  i s a p l o t of t h i s  amplification  a step function  =  step function the  r e s u l t s i n an  i n g r a p h #1  K 5 improves the  Increasing  A r c t o n ^ -  +  extra term arctan A.  angle  Arcfan^-r,  / 8 0 ° +  circuit.  and  The  does not  s o l u t i o n of  the  equation to  advantage of the  I t increases  the  a  introduction  attenuation  con-  i n t r o d u c t i o n of t h i s f a c t o r i s  involve  any  extensive  of  quite  modification  of  equipment. However the  natural  i n t r o d u c t i o n o f t h e Kg  frequency of the  a c c e p t e d o r one o f f s e t the  of the  change.  system.  f a c t o r reduces  Either this  the  c o n d i t i o n must  o t h e r p a r a m e t e r s must be. a d j u s t e d  A p r o b a b l e a d j u s t m e n t w o u l d be  to  be  to  increase  K 5 .  The was  p o s s i b i l i t y of  considered  and  v e l o c i t y e r r o r but  discarded. does not  Summarizing the  introducing  an  i n t e g r a l component  I n t e g r a l c o n t r o l reduces improve the  amplification  theory there are  two  things  the  ratio. t o be  done.  One, o b t a i n t h e b e s t r e s u l t s b y m a n i p u l a t i o n o f  a n d K2 a n d  t h e n do t h e same b y m a n i p u l a t i o n o f K  5  t h a t the response  equal o r b e t t e r the former.  Two, d e t e r m i n e without  causing  of the latter w i l l  the value of K overdamping.  6  and show e x p e r i m e n t a l l y  that w i l l  improve the response  24  T e s t s and R e s u l t s  To i m p o s e  t h e most s e v e r e c o n d i t i o n s p o s s i b l e a c u r v e  was p l o t t e d t h a t i n c l u d e d t h e h i g h e s t w a t e r o n r e c o r d , t h e l o w e s t w a t e r on r e c o r d , and t h e g r e a t e s t r a t e o f c h a n g e o f water l e v e l . t i d e because  T h i s was a c t u a l l y more s e v e r e t h a n a n y r e c o r d e d t h e p e r i o d s o f h i g h e s t a n d l o w e s t w a t e r s do n o t  o c c u r o n t h e same d a y .  The e a r l y t e s t s w e r e r u n u s i n g  this  h y p o t h e t i c a l c u r v e as a s t a n d a r d f o r c o m p a r i n g t h e r e s p o n s e t o the  v a r i o u s servomechanism  conditions.  L a t e r t e s t s were r u n  u s i n g t h e r e c o r d o f J a n u a r y 4 t o 12 1947 w h i c h i s t h e week o f most s e v e r e c o n d i t i o n s . G r a p h #2 i s a g r a p h o f one o f t h e b e t t e r obtained w i t h the i n t e r n a l feedback c i r c u i t .  curves  G r a p h #3 i s a  p l o t o f one o f t h e c u r v e s o b t a i n e d w i t h o u t t h e c i r c u i t . c u r v e s a r e compared w i t h t h e i n p u t s i g n a l .  A l t h o u g h graph  h a s b e t t e r a m p l i t u d e c h a r a c t e r i s t i c s t h a n g r a p h #2, a n a t i o n has appeared The  The §3  oscill-  a t t h e low t i d e .  c u r v e d r a w n up f r o m most s e v e r e c o n d i t i o n s i s made  up o f two m a j o r c o m p o n e n t s .  One i s a t t h e f u n d a m e n t a l f r e q u e n c y  and t h e o t h e r i s a t t h e s e c o n d h a r m o n i c  frequency.  The  oscill-  a t i o n a p p e a r i n g a t t h e low t i d e i s caused by t h e presence o f the  second harmonic.  The f r e q u e n c y o f t h e s e c o n d h a r m o n i c i s  c l o s e t o t h e n a t u r a l frequency o f t h e system. monic i s n o t as predominant  The s e c o n d h a r -  i n t h e t i d e c y c l e o f the-most  s e v e r e r e c o r d e d week n o r was i t c o n s i d e r e d i n t h e t h e o r e t i c a l analysis.  25  G r a p h s #4 and #5 compare t h e r e s u l t s and w i t h o u t t h e d i f f e r e n t i a t i n g 4 t o 12 was  circuit.  of tests  with  The week o f J a n u a r y  used f o r t h e s e t e s t s w i t h t h e day o f J a n u a r y 5 used  here f o r comparison.  Some t e n d e n c y f o r o s c i l l a t i o n i s s t i l l  a p p a r e n t b u t n o t n e a r l y a s s e v e r e as f o r t h e h y p o t h e t i c a l c u r v e . In  c o m p a r i n g t h e r e s p o n s e w i t h and w i t h o u t t h e  circuit  very l i t t l e  differentiating  improvement i s o b t a i n e d by t h e  addition.  G r a p h s #6 and #7 show t h e r e s p o n s e o f t h e s y s t e m t o a one and one h a l f i n c h s t e p f u n c t i o n w i t h and w i t h o u t t h e d i f f e r entiating  circuit.  advantage  of the d i f f e r e n t i a t i n g  the to  Here,  as i n t h e a n a l y s i s o f t h e s y s t e m , c i r c u i t i s brought out.  a m p l i t u d e o f t h e o v e r s h o o t and t h e t i m e f o r t h e d i e away a r e r e d u c e d c o n s i d e r a b l y .  this latter result  Both  oscillations  I t i s on t h e s t r e n g t h o f  that the d i f f e r e n t i a t i n g  as a m o d i f i c a t i o n t o the c o n t r o l  the  system.  c i r c u i t i s suggested  JLO. f o l l o w page. Zpi-.  D i s c u s s i o n and C o n c l u s i o n s  The r e s u l t s o b t a i n e d b y t e s t s h a v e r e v e a l e d o n e t h i n g ; the i n t r o d u c t i o n o f e x t e r n a l c i r c u i t s has very l i t t l e the continuous  o p e r a t i o n o f t h e t i d a l model.  out on s e v e r a l s t a b i l i z i n g c i r c u i t s in  T e s t s were  t h i s t h e s i s , b u t i n a l l cases t h e r e s u l t s were p o o r e r  t h e step f u n c t i o n response  carried  t h a t have n o t been m e n t i o n e d  any obta-i-ned b y m e t h o d s d i s c u s s e d h e r e i n . in  e f f e c t on  than  Some i m p r o v e m e n t  c o u l d be o b t a i n e d i n some  cases  but t h e b e s t o v e r a l l r e s u l t s were o b t a i n e d u s i n g t h e d i f f e r entiating circuit Repeating  t h a t has a l r e a d y been d i s c u s s e d . equation  (12) t h e t r a n s f e r f u n c t i o n o f t h e  system i s (12)  K limited  i s t h e o r e t i c a l l y a r b i t r a r y but p r a c t i c a l l y i t i s  5  t o one t e n t h o f a f o o t p e r second  per foot of error.  If  K5 i s l a r g e r t h a n 0 . 1 t h e w e i r s w o u l d r i s e f a s t e r t h a n t h e pump c a n b r i n g up t h e w a t e r through  the water  l e v e l w h i c h means t h e w e i r s w i l l  s u r f a c e and t h e water  level will  break  no l o n g e r  f o l l o w t h e w e i r movement. The c o n s t a n t K of  t h e system.  will  s  i s r e p r e s e n t a t i v e o f the time  constant  The s m a l l e r t h i s t i m e c o n s t a n t i s t h e b e t t e r  be t h e r e s p o n s e  T  4  o f t h e system.  2K4 • 224 s e c o n d s  Repeating  e q u a t i o n (17)  This equation r e v e a l s t h e importance stant.  Any r e d u c t i o n i n K  e n t i a l term t o approach  4  will  zero.  reduce  of the time  con-  t h e time f o r t h e expon-  I f the coefficient  of the f i r s t  d e r i v a t i v e had a c o n s t a n t g r e a t e r than u n i t y i t a l s o would dec r e a s e t h e time c o n s t a n t o f t h e system. R e f e r now t o F i g . ( 3 ) i n t h e g e n e r a l t h e o r y . f i g u r e i s a graph frequency. ratio  This  o f i n p u t , output a m p l i f i c a t i o n t o r e l a t i v e  I t i s h i g h l y d e s i r a b l e t o have t h e a m p l i f i c a t i o n  equal t o u n i t y .  T h e g e n e r a l s h a p e o f t h e c u r v e i s -depend-  ent on t h e damping r a t i o  C - 1 = 6.7 The  "G".  I n t h i s case t h e c r i t i c a l  damping  0.149  resonance  c u r v e when G  =  0.149 r i s e s v e r y s h a r p l y  as t h e a p p l i e d f r e q u e n c y a p p r o a c h e s t h e n a t u r a l f r e q u e n c y . a c t u a l value o f "F" i s f i x e d be  i n c r e a s e d by r e d u c i n g F  in  K . 4  c  The  a t u n i t y , t h e r e f o r e CV. c a n o n l y n  which  again c a l l s f o r a reduction  A good v a l u e f o r "C" W o u l d be 0.6. To k e e p away f r o m t h e r e s o n a n t  frequency  point the natural  s h o u l d be w e l l a b o v e t h e a p p l i e d f r e q u e n c y .  the a p p l i e d frequency n a t u r a l frequency  Since  i s f i x e d by d e s i g n r e q u i r e m e n t s t h e  s h o u l d t h e n be made as l a r g e a s p o s s i b l e .  H e r e a g a i n t h e c o n s t a n t K4 s h o u l d be made a s s m a l l a s possible.  T h i s i n v e s t i g a t i o n has brought  out three reasons  why  28 the  constant  K  4  K  4  should  i s approximately  the w e i r s which is  20 c u b i c  feet  In t h i s practical. components. increase second from 224  4  square  feet,  b i s the length of  and Q, i s t h e pump d i s c h a r g e  particular  which  p r o b l e m a n i n c r e a s e i n b and Q, i s  facilities  will  are available  then be halved.  t o 4.2 x 1 0 -  2  t o double  r a d i a n s p e r second, reduce  both  This modification w i l l  from 2 . 9 6 x 1 0 -  the n a t u r a l frequency  0.149 t o 0.21 and i t w i l l  2  radians per  i t will  t h e time  increase " C "  constant  from  s e c o n d s t o 112 s e c o n d s .  formula  o f any f u t u r e model b a s i n s  (28) may w e l l be u s e d  as a g u i d e .  made as s m a l l a s p o s s i b l e and t h e w e i r should n  of the basin  p e r second.  In t h e d e s i g n  CO  between K 4 and  b a s i n . "A" i s t h e a r e a  12,TOO  i s 40 f e e t ,  In fact K  <28)  (28) shows t h e r e l a t i o n s h i p  components o f t h e t i d a l  which  as low as p o s s i b l e .  "j(J.33til's Q*  =  Equation the  be k e p t  be a s larp:e a s p o s s i b l e .  should  be a t l e a s t  five  of this  The a r e a  s h o u l d be  l e n g t h and pump  discharge  When "C" i s 0.3 o r l e s s ,  times  type  then  29  Acknowledgment  The who  have  author  assisted  especially  wishes  to express  him throughout  t o M r . W.B.  Coulthard  Acknowledgment  i s also  h i s indebtedness  the course  of this  f o r h i s guidance  to those  research;  and  encourage-  ment .  Council  whose  grant  made  this  made  to the National  Research  investigation possible.  D.H.J.Kay  The River tested  Model  servo-control  discussed  mechanism  i n this  t h e s i s was  and fabricated, i n the National  laboratories,  o f the Fraser designed,  Research  Council  Ottawa.  Inserted by request.  L i b r a r y , March 10th., 1952  30 Bibliography  Laver  Henry,  Lesnick  Servomechanism McGraw-Hill,  Brown  G.S. New  Hall  and  R o b e r t , and  F u n d a m e n t a l s . New  C a m p b e l l D.P.,  Schureman of  242,  Paul,  Tides,  Office,  and  E;, London,  Manual  Servomechanisms,  1948.  Journal  pp.279-307,  Theory to the Design  of the Franklin  of  Institute,  October,1946.  o f Harmonic  Washington,  1941.  York  Principles of  "Application of Circuit  Servomechanisms," vol.  Leslie  1947.  York, Wiley,  A.C,  Matson  United  Analysis States  and  Prediction  Government  Printing  

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