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ON THE RESONANCE PROPERTIES OF QUASI-LINEAR SECOND-ORDER DIFFERENTIAL-DIFFERENCE EQUATIONS  by  ROBERT ALLAN ANDERSON B . S c . ( E . E . ) , University of Manitoba, 1963. i  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF  MASTER OF APPLIED SCIENCE  i n the Department of E l e c t r i c a l Engineering  We accept this thesis as conforming to the required standard  Research Supervisor  ..  H  O  Members of the Committee  o  e  a  o  o  a  e  e  s  O  a  O  o  e  o  a  o o o o * a e a o » a o a o a » s o  o  o  o  «  o  o  «  a  n  o  o  o  o  Head of the Department . Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA August, 1966  o  o  o  o  a  a  s  In presenting  this thesis  i n p a r t i a l f u l f i l m e n t o f the  requirements  f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the study.  L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e  I f u r t h e r agree t h a t permission., f o r e x t e n s i v e c o p y i n g o f t h i s  t h e s i s f o r s c h o l a r l y purposes may  be  Department o r by h i s r e p r e s e n t a t i v e s .  g r a n t e d by the Head o f  w i t h o u t my  written  Department o f  permission.  ^e*L<^g rve/tf  The U n i v e r s i t y o f B r i t i s h Vancouver 8 , Canada.  ^*s<5rS*/<^£r^ <S-  Columbia  my  I t i s understood that  or p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not  Date  and  be  copying allowed  ABSTRACT Since very l i t t l e has appeared i n the l i t e r a t u r e regarding solutions of driven nonlinear d i f f e r e n t i a l - d i f f e r e n c e equations,  i t has been the purpose of this investigation to •  obtain approximate solutions to these equations and to investigate  their resonance properties.  considered are second-order quasi-linear difference  The equations differential-  equations. S t a b i l i t y c r i t e r i a are presented for equations  having delayed damping' and for equations having a delayed restoring force. Application of the Ritz method leads to general equations which determine the constants i n the assumed s o l u t i o n . The general equations for systems with odd n o n l i n e a r i t i e s are used to obtain the resonance  properties  for several s p e c i f i c examples. Unusual jump resonance phenomena are obtained when the input frequency i s v a r i e d . Regions of the response curve occur which are not connected to each other. The approximate solution is v e r i f i e d by an analog computer simulation employing track and store techniques to enable automatic plotting of the response curves. The R i t z method results simulation  compare favourably with the analog-  results.  ii  TABLE OF CONTENTS Page ABSTIiAOT •  o  »  o  6  e  0  *  *  0  TAB XlR OF CONTENTS  O  O  0  O  0  O  O  O  O  LIST OF ILLUSTRATIONS  ~  O  O  O  O  •  •  «  O  O  0  O  O  O  «  o  0  »  O  o  0  O  B  0  O  *  0  O  «  9  O  o  0  O  o  O  O  o  0  0  O  e  0  O  O  »  o  0  0  0  «  0  O  O  0  0  0  O  O  O  0  9  9  *  O  A  O  O  O  O  O  O  O  O  O  O  O  O  O  *  «  «  «  »  *  *  «  -  e  <  i  «  *  »  «  *  XX H  v  *  LIST OF SYMBOLS  .  vi  ACKNOWLEDGEMENT 1•  2  o  vii  INTRODUCTION 1.1  2.1  2.2  1  ooooeooooooooooooooooooooeoooooooa***  Scope of the Present Work . „ . » . . . . . . . . . . . . . . .  STABILITY  2 4  o o o o o e o o o o o o o o o o o f t o o o o o o o o o D o o e o f f o o o o o o *  Some Theorems and Definitions Concerning Lyapunov's Second Method for Equations with T XLH© D©l£iy o o o o o o o o o a o s f t o o o o o o o o o o o o a o o o o s o o o  5  S t a b i l i t y of Systems with Time Delay i n BclI~fcXC"U.l£tX* C&.SGS >  O  O  *  O O O O O O O O O O O v f t O O O O  •  0 0 0 0  0  10  0 *  2.2.1  Systems with Delayed Damping  10  2.2.2  Systems with Delayed Restoring Force . . . . . .  12  3. APPROXIMATE ANALYTICAL SOLUTIONS  18  3«1 3.2  Ttl© Ri."fcZ M©"fcilO(3. o e a o e o o o o o o o o o o o t o o o o o o o o o b o o 19 Application of the Ritz Method to Steady-State Oscillations i n Nonlinear Systems with Delay.. 21  3.3  I l l u s t r a t i v e Examples and Comparison of the Ritz-Method Results to the Analog-Simulation R© STXl"fc S 0  4.  X  0  9  0  9  0  0  0  0  0  0  0  0  0  0  0  0  0  0  O O 0 0 O 9 O 0 O 0 9  9  0  0  0  0  9  25  0  VERIFICATION OF THE APPROXIMATE SOLUTION BY ANALOG S IiyRJIAT ION  O  S  0  9  O  O  O  0  0  O  O  0  O  O  O  O  0  0  0  0  O  O  0  0  0  O  O  O  O  0  0  O  0  O  O  •  9"  0  5 • CONCLUSIONS APPENDIX A SOME DEFINITIONS AND PROPERTIES PERTAINING TO QUADRATIC FORMS o o o o o o o o o o o . o o o o o o o o . o o O  O  O  O  O  O  O  O  "  9  0  0  0  0  0  iii  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  0  9  0  0  0  0  0  4" V 54" 56  APPENDIX B  ON THE OPERATION OP SOME ANALOG- COMPUTER COMPONENTS  REFERENCES  •  » » e o ( t » o o t » o * Y o » 8 o o B O « B o o e f t o o » t  e * s  5T  B.l  Integrator Mode Control , „  57  B.2  Electronic Comparators  57  B.3  AND Gates . . . . . . « « « • . . . . . . . . . . . . . . • > .  57  o  58  .  o  .  i  >  .  .  .  .  s  *  .  .  s  .  .  .  iv  .  .  D  o  .  a  o  i  >  o  «  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  LIST OF ILLUSTRATIONS Figure  2.1 3-1  Page Region of S t a b i l i t y )J  — L  •  «  e  a  e  «  *  o  «  s  o  o  o  a  0  ( X  3|  — Xi  ( X  )I  ^  L  4s 0 a o <> O O O O O O O Q O O O O O O 0 *  O t t O O B 8 O O O 0 0 Q o a O <»  o  a  o  s  o  a  a  a  a  o  s  a  o  o  a  O ©  System with o l» o o o o a a o o o o o o o a o o  System with  3 0 O 0 tl i  Amplitude Response Curve for | f  ( X  ) | .^5 L  0  8  0  0  0  0  0  0  0  0  0  0  0  0  0  0  System with  Amplitude Response Curve for [ f*  3 A  ( X  1> »  17  o o i> * o o o o e e o o o * o o o o o  Frequency Response Curve for | f  3.3 _  e  Frequency Response Curve for | f  3.2  9  e o a o o o o o o e o o o o o a  System with  o a B I O O 0 O O O 0 0 O 0 0 0 O O 0 0  0 0  - 1  1  . . . 28 . . . 28 . . . 28 . . . 28  o o a o o e o o e o a o a o  . . . 31  Response Curves, Example 2 . a. o  o o o a s a o o o o o o a a o  . . . 36  3-7  Response Curves, Example 3  a  o  a  0  a  o  a  a  o  a  o  0  «  e  «  3.8  Response Curves, Example k ..  o  o  o  o  a  o  o  o  a  o  o  u  a  o  o  o  ...  3.9  Response Curves, Example 5  o  o  o  a  a  o  o  a  o  o  o  a  o  o  o  o  ... U3  3-10  Response Curves, Example 6 . , }  o  o  o  o  e  o  0  o  o  o  a  o  a  «  o  o  h.l  Nonlinear O s c i l l a t o r  > O 0 0 (3  0  0  0  0  0  0  0  0  0  0  0  0  0  0  *  0  k.2  Control C i r c u i t  h.3  3.5  . Response Curves, Example 1 .l ,o o  3.6  :  >  a  l  a  . . . 39 hi  . . . 45 0 .  49  ...  h9  Analog Simulation of a System with Delay . . . . . . .  51  .........  3  v  0 O O I> 0 O 0 O Q O O O 0 0 O O 0 O 0 O  .  LIST OP SYMBOLS -  a constant  =  coefficients  h  =  a positive constant  k  -  a constant  L  =  a positive constant  s  =  complex frequency  =  the largest value i n the  C  sup(  X-  n  )  (  t  | 9 0 0 0 ,  |X^  x^  of damping  set,  J )  =  time  =  a time-constant  i(t)  = =  an approximation to x(t) the f i r s t derivative of x(t) with respect to t  x(t)  =  the second derivative of x(t) with respect to t  a  =  a dummy variable of integration  P(x)  =  the Gamma function of x  e  =  a positive constant  =  a variable which i s r e s t r i c t e d to the  T x(t)  ^  i n t e r v a l , - h < i9- <L 0 u-  =  a constant  £  =  a dummy variable of integration  0  =  <P 00 2  to n  a variable which i s r e s t r i c t e d to the i n t e r v a l , - h <[0 < 0 = a function = angular frequency =  coefficient of restoring force vi  ACKNOWLEDGEMENT  Grateful acknowledgement i s given to the National Research Council of Canada for f i n a n c i a l support received under the Block Term Grant A68 i n 1964-1965 and 1965-1966. The author wishes to express his appreciation to his supervising professor,  Dr* A . C . Soudack, for his invaluable  suggestions and guidance throughout the course of this work. The author wishes to thank Dr. E . Y . Bonn for reading the manuscript and for h i s useful comments and suggestions* Thanks are also given to Mr. H. Kohne for i n s t a l l i n g the trunk lines to the tape recorder, Mr. R. Proudlove f o r photographing several of the i l l u s t r a t i o n s , Mr. J . E * Lewis and Mr. A . D . Martin for t h e i r proof reading, and to Miss H. Klassen for typing the t h e s i s .  vii  1 1.  INTRODUCTION  Ordinary d i f f e r e n t i a l equations equations) have'been and s t i l l are a.useful  (or systems of t o o l i n the ana-  l y s i s of a wide variety of physical phenomena*  It must be  kept i n mind, however, that i n using ordinary d i f f e r e n t i a l equations to' describe phenomena, i t i s assumed that the future behavior of the system depends only on the present state and i s independent of the past.  In order to more accurately  describe physical systems, therefore,  i t i s necessary to con-  sider the fact that t h e i r rate of change depends not only on their present state, but also on their past history.  In place  of the ordinary d i f f e r e n t i a l equation , ff  = f (x,t)  ,  x(t Q .) = Cr<  (1*1)  we must write | |  = f  [ x ( t ) , x(#),  t]  ,  (1.2)  where x9- ranges over a set of values less than t, and the i n i t i a l f u n c t i o n , x ( t # ) , i s specified for the r a n g e , l e s s than the i n i t i a l time, t . simplest  i s a d i f f e r e n t i a l - difference ~  where  Of the many equations of this type, perhaps the  = f  [ x ( t ) , x(t  equation ,  - t ) , . o o , x ( t - t^), 1  t ] , (1.3).  0 < t-^ < tg < * . . < t^.  In this case^ only a certain f i n i t e i n t e r v a l of the immediate past history of the system i s involved i n the determination of the present (instead of the whole differential  equation).  past as i n an integro-  The study of differential-difference begun by John Bernoulli  equations was  i n 1728? and theory regarding these  equations has been developed i n several hundred papers since then.  Due to t h e i r application to control theory, this study  has been greatly intensified i n recent years. - Many Russian • (3) authors, such as Krasovskii v(2)'and Khalanai , have considered  the problem of s t a b i l i t y .  Bellman and C o o k e ^ have developed  theorems analogous to those for ordinary d i f f e r e n t i a l Pinney  y  equations.  has also contributed to the general theory, and  bibliographies by Choksy^^and Weiss  give reference to  356 separate items a l l dealing with time delay systems and/or associated mathematics.,  Abundant references  are also given i n  the previous works c i t e d . Differential-difference of e l a s t i c i t y  equations arise i n the theory  , mathematical economics^^population in sti studies  .(ID the study of combustion i n l i q u i d - f u e l rocket motors (12) mathematical biophysics ,the theory of automatic, control (13),(14), (15), (16) , , . ' ' ' ' and i n a number of other areas involving time delays. 1.1  Scope of the Present Work Some theorems developed by Krasovskii pertaining to  the s t a b i l i t y of systems with delay are given i n Chapter 2. These theorems are then applied to discuss the s t a b i l i t y of'two rather general classes of second—order nonlinear systems with delay. Although much of the theory pertaining to uniqueness, and s t a b i l i t y of solutions to  existence,  differential-difference  3 equations has been developed, very l i t t l e has been done to (17) obtain solutions due to t h e i r complexity  .  Because of this  complexity i t i s desirable to obtain an approximate solution. This approximate solution must approach the actual solution with some reasonable degree of accuracy and must also exhibit any qualitative behaviour characteristic of the system, such as jump resonance.  Therefore, approximate solutions have been (2  developed i n Chapter 3 by employing the well known Ritz methodv The general equations for the amplitude and phase of a oneterm approximation to the solution are obtained and applied to several particular examples.  The results thus obtained  are then compared to those obtained by an analog computer simulation discussed i n Chapter 4.  Both sets of results show  that rather unusual jump resonance phenomena are often obtained. In Chapter 5 some general conclusions and ideas for future research are given.  4  2.  STABILITY  Before investigating the solution of any d i f f e r e n t i a l equation, i t i s useful to determine beforehand which values of the coefficients solutions.  of the equation ( i f any) lead to stable  Then,only systems which are inherently stable i n  the undriven case need be studied to determine t h e i r resonance properties. The s t a b i l i t y of a l i n e a r  differential-difference  equation can be determined by examining the roots of i t s associated characteristic e q u a t i o n '  .  If these roots have negative  r e a l parts, the solution of the equation i s stable*  Because  of i t s transcendental nature, the characteristic equation, i n general, possesses an i n f i n i t e number of roots.  The r e l a t i o n  between the location of these roots and the coefficients of the differential-difference  equation i s necessarily much more  complicated than for an ordinary d i f f e r e n t i a l equation.  A  graphical technique sometimes provides more useful information than a direct examination of the roots of the characteristic equation.  One such technique, employing a dual Nyquist diagram,  is given by J o n e s ^ ^ ,  The algebraic part and the transcendental  part of the characteristic equation can be plotted separately and the s t a b i l i t y determined from t h e i r intersection points. The effect  of the delay on the s t a b i l i t y can then be easily  determined. For nonlinear systems with delay, the s t a b i l i t y  is  most easily determined by applying Lyapunov's second (or direct)  5  method.  T h i s method i s d e s c r i b e d  vanced c o n t r o l theory.  i n most modern books on ad-  I t s great merit  i s that i t can be a p p l i e d  to s t a b i l i t y d i s c u s s i o n s f o r very general  equations without  finding e x p l i c i t solutions* 2.1  Some Theorems and. D e f i n i t i o n s Concerning Lyapunov's Second Method f o r Equations w i t h Time Delay The Russian author, K r a s o v s k i i , has done a great  of work concerning  deal  the s t a b i l i t y of systems with time delay.  Some of h i s theorems and d e f i n i t i o n s are g i v e n i n t h i s s e c t i o n ; (2) the i n t e r e s t e d reader i s r e f e r r e d t o K r a s o v s k i i ' s book ' f o r v  the proofs and f u r t h e r i n f o r m a t i o n  on the s u b j e c t .  We are i n t e r e s t e d i n equations o f the form, dx. dt^  =  iL i ( -  X  x  t  l  l  i i ) » •- •>  ^^"^in^'  ^ ( t ) , . . . , x ( t ) , t j (2.1) n  (i = l....,n; 0<h (t)<h), where the r i g h t hand memb er s, X_^ ( y ^ y o • • t y ^ J x - ^ , o , . , \ are continuous f u n c t i o n s of t h e i r arguments and a r e defined f o r  ( i = 1,..., n),  y | <H  < H  ±  where H i s a constant  (or i n f i n i t y ) .  (2*2)  We suppose f u r t h e r that  these f u n c t i o n s s a t i s f y a l i p s c h i t z c o n d i t i o n with respect t o (2.2), i . e . ,  the x. and y . i n the r e g i o n J  j  II  II  II  It  I  T  t  T  Xj^(y-^ .», y^; x-^,..., x ^ j t ) — X^(y-j_»«»•» y^$ x^,,.., n ' ^ ^ x  yo  •n <  L(  .n |xj-x'|  d=i  +  £ .i=i  ).  It w i l l also be assumed that  \  [x (0),t] = 0 3  on the entire a r c . x.(0) = 0 ;  i = 1 , . . . , n;  j = 1 , . , . , n)  J  -h<0£O. The following notation i s used: XI  = sup(J x^ 2 —  ^  poo  tr 9  l ° * °' "^"n^  x|| | = sup(|x (0)| ) i  0 L  -h  o  1  i o  y  n '-  X (0 ) = [x (0 )] q  for -h<0<O  Q  L  J  ( i =1,..., n ; -h<#o*0).  To emphasize the dependence of x(t) on the i n i t i a l curve and the i n i t i a l value of t , we s h a l l denote  {j*-^^)] by  x [x ( 0 ) , t , t ] q  Q  D e f i n i t i o n 1.  (a) The solution,x =' 0,of equation (2.1) i s called  stable i f f o r every positive number, e >0,we can find a positive number,6>0,such that whenever the inequality,  kteyll  h  i* •  is s a t i s f i e d , the r e l a t i o n , < e holds for t > t . o (b) I f , whenever condition (a) i s s a t i s f i e d , the conditions.  7  lim  = 0  t-»- oo x [ x ( 0 ), t , t ] | | o  are satisfied  <  H  for a l l t > t 0 ( H 1 =  for a l l i n i t i a l curves, X ( 0 ).,satisfying Q  const.),  the i n -  equality ,  then the solution,x = 0, of equation (2,1) stable and the region (2.3)  is  asymptotically  l i e s i n the region of a t t r a c t i o n of  the unperturbed motion,Suppose there i s a f u n c t i o n a l , v[x(0), t  Theorem 1.  J  ,  that  s a t i s f i e s the conditions, ||x(0)|| ) + W2(||x(0)||  |V [x(0) f t ] | <W1( 7 [x(0), t ] > t ^  m  0  +  W  - P ( f )  |) ,  [||x(0)||], =  -f[||x(0)||],  (2.4)  where W^(r) and Wg(r) are functions that are continous and monotonic for r > 0 y and W.^0) = W2(0) =0; continous and positive for r ^ 0. of equation (2.1)  w(r) i s a function that  is  Then the n u l l s o l u t i o n } x = 0,  i s asymptotically stable.  If X^ and b-^.j(t)  a  r  e  periodic functions of the time,t,  a l l with p e r i o d , © , (or i f X:. and h. . are independent of the t i m e , t ) , then condition (2.4) may be replaced by the following weaker hypothesis: A sufficient  condition that lim . sup (-fr)should be nonpositive A t t—0 + along a trajectory i s that the equation ,  8  lim be v a l i d for a l l t ^ t D e f i n i t i o n 2,  sup(^) = 0 ,  only along the trajectory, x = 0.  The solution,x = 0, of equation (2,1) i s called  uniformly asymptotically stable with respect to the t i m e , t o > 0 ,  (0 ), i n  and with respect to the i n i t i a l curve, x  Q  region (2,3),  i f i t s a t i s f i e s condition (b) of D e f i n i t i o n 1 and also s a t i s f i e s the following conditions? (i)  the number, 5>0, of D e f i n i t i o n 1 (a) may he chosen  independent of t (ii)  >0j  for arbitrary i|>0,  there exists a number,T (r\), such  that  holds for every t > t  + T(r^), independent of the choice of a  piecewise-continuous i n i t i a l curve,x Q (0 ), i n region (2.3)* If the right hand member of equation (2.1) i s a periodic function of time with period,9,(or i s independent of the t i m e , t ) , then the n o i l solution,x = 0,is always uniformly asymptotically stable i n the sense of D e f i n i t i o n 2, Now consider the "perturbed" system of equations , dx, ~  =  X  i ^ ^ C ^ ) , ...fxn(t)j ^[t-h^Ct)] r.'..rxn[t-]iln(t)],  + R± ^ x ( t ) , . . . , x n ( t ) ; x 1 [ t - g i l ( t ) ] , . . 0 , x n x  (i,  j = l,...,n  ; O<h^(t0^h  t-gin(t)  tj,  tj.  ? 0 < g ^ (t) < h ) , ±  where the continuous functions,R^,are not required to reduce to zero for  x. = y . 3  3  =0,  Definition 5. The n u l l solution,x = 0, of the system (2.5) called stable for persistent disturbances i f for every  is  e>0  there exist positive numbers,<SQ? r|, A, such that the s o l u t i o n , x  [ (0 )> X O  Q  ^ 9 "fc]*0^ "the system (2,5)  s a t i s f i e s the i n e q u a l i t y ,  q  l  x  for a l l t £ t Q ,  [xo(0o)? V  * ] II <  '  e  t Q ^ 0, whenever the i n i t i a l curve, X (0 ), s a t i s f i e s Q  Q  the inequality , xo ^ (0 o  )h < <S , 0  and the perturbed time delays, h . a n d the  f unct ions,R.,sat isf y  the i n e q u a l i t i e s , Ri(x1...., for  x <e  xn j y  1  (2.1)  n  ,  t ) | < rj_ , ( i = l , . . . , n ) ,  y || < e, and  | h ± ^ ( t ) - k ^ ( t ) | <A Theorem 2.  y  (i, j = lf...rn).  Suppose that the n u l l solution,x = 0,of equation  i s asymptotically stable uniformly with respect, to time,  t Q ,and the i n i t i a l c u r v e , X ( 0 ) , i n the sense of D e f i n i t i o n 3., q  o  then the n u l l solution,x = 0, i s stable for persistent  disturbances.  10  2.2  S t a b i l i t y of Systems with Time Delay i n Particular Cases Systems with Delayed Damning  2.2.1.  Consider the second-order nonlinear equation, 2  = X [ x ( t ) , y(t)]  ^-f  (y = ^  ! h i s a positive  + ^ [ y ( t - h ) , t]  ,  (2.6)  constant),  where the functions X and ^ s a t i s f y the requirements , X(x,y) - X(x.O) y  <  _  XJ^Ol x  a 9  <  ^  f  o  ^ o, y ^ 0 ,  r x  (2.7)  where a and b are positive constants, and |y(y.t)|  £  l|yj.  (2.8)  We write equation ( 2 . 6 ) i n the equivalent form, dx dt ||  (2.9) = X ( x 0 ) + [X(x.y) - X ( x , 0 ) ] + ^>[y(t-h), t] y  .  (2)  Krasovskii^  has shown the following?  Define the functional V by 0  x  V[x(*), y W ] = -  f  x(£,o)d£  0  + z£  : +  |  /  y e) d£ . 2  (  - b  (2.10)  To estimate the derivative, (2.9),  write  along a trajectory of the system  11  H=  [x(x,y)  - X(x,0)] y + yj(>[y (t-h), t] +  Conditions (2.7) fll ..<; = a £ [ ± )  +  and (2,8) E  y2(t-h).  (2.ii)  the hypotheses of Theorem 1 i f the  right hand member of inequality (2.11) i s a function of the arguments,y(t) satisfied  _ S|i(t-h).  give the estimates ,  |y(t)y(t-h)|  The f u n c t i o n a l , V , s a t i s f i e s  ayJL  and y ( t - h ) .  negatire-definite This condition i s  i f the numbers,a and L,are related by a>L„  (2,12)  Thus,inequality (2.12) i s a condition sufficient for the asymptotic s t a b i l i t y of the n u l l solution,x = y = 0,of the system If the right hand member of equation (2*6)  (2.9)»  i s a periodic  function of time with p e r i o d , © , ( o r i s independent of time), then the n u l l solution i s always uniformly asymptotically stable i n the sense of D e f i n i t i o n 2 and hence the system w i l l be stable for persistent  disturbances.  The s t a b i l i t y c r i t e r i o n expressed by equation (2.12) can also be obtained h e u r i s t i c a l l y .  Due to the delay, h,there  w i l l be some frequencies of o s c i l l a t i o n , co, such that wh i s an odd multiple of rt. At these frequencies,' the delayed damping term, y ( t - h ) , w i l l be out of phase with the damping term, y ( t ) . Since a and L are, e f f e c t i v e l y ,  the coefficients  of the damping  and delayed damping terms, then a must be greater than I i n order that the system should have no negative damping at any frequency.  12 2.2.2  Systems with Delayed Restoring Force Consider the l i n e a r equation with delayed restoring  force , !  if,  dt<  ^  +  a  ^  .  +  b  l  t)-ox(t.h)  (  a  =  0  (a, b, and c are constants;  ,  h i s a positive  constant)  This can Toe written i n the equivalent form , = y  ,  (2.14)  = -bx - ay + cx(t-h). We wish to find  a^^  and a 2 2  v(x,y) = a i ; L x  such that the quadratic form,  + 2a 1 2 xy + a 2 2 y  ,  (2.15)  i s positive-definite and s a t i s f i e s the condition , |J y + &  (-bx - ay) = - x 2 - y 2  .  (2.16)  Substituting equation (2.15) i n equation (2.16), we obtain a  ll  a  10  X d  and  =  & 2  ^  +  ' ,  (2.17)  = 1_ 2b  (2.18)  "22=  '  ( 2  '19)  v(x,y) i s a positive-definite function of the arguments,x(t) 2 and y ( t ) , i f  > 0 and 2.1 22 a  a  >  a  12  ^  s e e  Appendix A ) .  A sufficient  condition for t h i s is that a and b be greater than zero.  13  The functional,V, can be taken i n the form, 0  V [x(<9), y ( d ) ] = a l l X  +'2a  2  1 2  xy + a22y  2  + u  {  x2(£)d£,  ( .20) 2  -h  (\x>0) • The value of the derivative, | | , along a trajectory of the system (2.14) i s lim  =  At-»-0  _ |"(l_^)x2(t)  +  y  2(t)  +  ^x2(t_h) _  2  cc 1 2 cx(t)x(t-h)  +  -2a22cy(t)x(t-h)] .  (2.21)  The right hand member of equation ( 2 . 2 1 ) i s a negative-definite function of the arguments,x(t),  x(t-h)  and y(t), i f  ( 1 - u-) > 0 and  •  ( 1 - u)  (2.22)  (u. - a 2 2 c 2 )  - a22  c >0.  (2.23)  2  Maximizing the l e f t hand member of inequality ( 2 . 2 3 ) with respect to |i,we find • +a  1 * -  2  c  2  ff-  •  (2.24)-  Substituting this value of |i i n inequality ( 2 . 2 3 ) , we obtain 1  "  a  |2°2  j  _a  2  2  2  c2>  0  .  (2.25)  'Since we have chosen b > 0 , then <x^ > 0 (see equation (2.18), and hence inequality (2.25) may be written a22c2< 1 - 2a12|c|  .  (2.26) 2 2.  From inequality (2.22) and equation (2.24),we find that must be less than unity.  Thus i f inequality (2,26) i s  a  2 2  c  satisfied  14 (and hence (2.23)), then inequality (2.22) Is satisfied w e l l , and the right hand member, of equation (2.21) i s definite.  as negative-  The functional,"?, thus s a t i s f i e s the hypotheses of  Theorem 1 and hence inequality (2.26) i s a condition sufficient for the asymptotic s t a b i l i t y of the n u l l solution,x = y = 0, of the system (2.14). Having determined the s t a b i l i t y c r i t e r i o n for the linear equation (2.13), l e t us now consider the nonlinear equation , 2 ^ - | = X [ x ( t ) , y(t)]  +^[x(t-h), t]  (y = | f ; h i s a positive  (2.27) constant),  where the functions X a n d y s a t i s f y the requirements, X(x,y) - X(x t 0)  =  _&f  Xix.01  <  ^  for x ^ 0, y ^ 0, (2.28)  where a and b are positive constants, | ^ [ x ( t - h ) , t ] | <L  |x(t-h)|  and .  (2.29)  The l i n e a r equation (2.13) w i l l now be a special case of the general equation (2.27)*  Equation (2,27) can be written i n  the equivalent form , ff  = y  (2.30)  f  | Z = X(x,0) + [X(x,y) - X(x,0)] + <^[x(t-h), Define the functional ,V, by  t]  .  15  .A.  V [x(#),y(<£)]  = -2 a 2 2  X(<f,o)d£ + ( a ^ - a 2 2 b ) x 2  j 0 0  + 2a 1 2 xy + a 2 2 y 2 + u  x2(£)d£  j  ,  (2.31)  -h (u>0)  .  To estimate the derivative,  along a trajectory of the system  (2.30), we write dV  dT =  [-2a 2 2 X(x.O) + 2 ( a  + 2( a 1 2 x + a  2  1  y)  l x  - a22b)x + 2a12y] y  £x(x,0) + [X(x f y) - X(x,0)]  + ux 2 - ux 2 (t-h)  + jp[x(t-h), t  .  (2.32)  Conditions (2.28) and (2.29) give the estimates,  H  £-2[«  1 2  ^  2  + (« a + a 12  2 2  *  - «n)xy + ( «  2 2  a  -  «  1  2  )y ] 2  J  + 2a 1 2 Lx  I x(t-h)| + 2 a 2 2 l y  + ux 2 - ux 2 (t-h) If and  a  1  1  ?  a  1 2  | x(t-h)|  .  (2.33)  , a 2 2 and u- satisfy equations (2.17),  (2.18),  (2,19)  (2.24) respectively, then the right hand member of i n -  equality (2.33) i s equivalent to the right hand member of equation (2.21) with cx(t-h) replaced by L Jx ( t - h ) | .  Thus,  the right hand member of inequality (2.33) i s a negativedefinite function of the arguments,x(t), x(t-h) 2 a22L  and y ( t ) , i f  2 < 1 - 2a12L  ^  ( 2 o 5 4 )  16 The functional,V, s a t i s f i e s the hypotheses of Theorem 1 and hence inequality (2.34) i s a condition sufficient for the asymptotic s t a b i l i t y of the n u l l solution,x. = y = 0,of the system (2.30). 4  By use of the simple transformation,t^ = b t , i n equation (2.27)  w e  y replace b by unity,, a by  m a  r  and  1  by L  = k  ,  (k>0)  .  .  The s t a b i l i t y c r i t e r i o n (inequality (2.34)) then becomes (2D.,) 2 * 1  k2 1 - k  .  (2.35)  The stable region defined by inequality (2.35) i s indicated i n Fig.  2.1. Again, i f the right hand member of equation (2.27) i s  a periodic function of time with p e r i o d , © , ( o r i s independent of time), then the n u l l solution i s always uniformly asymptotically stable i n the sense of Definition 2,and hence the system w i l l be stable for  persistentdisturbances.  Having determined the s t a b i l i t y c r i t e r i a for the two equations  (2.6) and (2.27), we may now proceed with the development  of the approximate solutions.  Stable Region  /  •O  1.0  2.0  3.0 2 D  1  Figure 2.1 Region of S t a b i l i t y  4.0  5.0  18  3.  APPROXIMATE ANALYTICAL SOLUTIONS  In the study of l i n e a r systems i t i s convenient to deal with sinusoidal inputs and the r e s u l t i n g sinusoidal outputs.  The r a t i o between the complex amplitudes of output and  input i s known as the "transfer function".  Although the i n -  formation represented by these transfer functions seems to be very s p e c i f i c , the property of superposition, inherent i n linear systems, makes these functions the basis for a complete description of system behavior. In nonlinear systems' the property of superpostion does not hold.  The o u t p u t s , . i n general, are no longer sinusoidal  and the response to a sinusoidal input does not permit the response to an input of any other type to be foretold  exactly.  Nevertheless, the sinusoidal input functions are a convenient method for investigating certain representative  features of (22) v system behaviour, such as the phenomena of jump resonance '* (23) .  The problem then,is to determine, with some reasonable  degree of accuracy, the amplitude and phase of the system output corresponding to any amplitude and frequency of the sinusoidal input.  The curves obtained for varying input frequency or  amplitude w i l l be called response curves.  This problem has been  solved by B l o t t e r ^ f o r quasi-linear systems with no delay by employing the Ritz method.  This method can also be applied to  systems with delay as i s done i n section 3.2.  It i s  interesting  to note that i f the delay i s reduced to zero, the equations developed reduce to those developed by Blotter(as indeed they  19  should). 3.1  The -Ritz Method Although the Ritz method i s described i n the  (22) l i t e r a t u r e , a b r i e f discussion of i t i s given here. v  The Ritz method postulates the existence of a function, F(x,x,t),such that the Euler-Lagrange equation,obtained from the minimization of  , f t I = J E ( x , i , t ) dt , a i s the nonlinear equation we wish to solve, i . e . , 6E  - _d dt  ax  = 0  = E(xO  *  (3.D  (3.2)  dx  Consider then the minimization of the right hand member of equation (3.1) given oo  x(t) = £  a ^ (t) , k  k=6  k  where x(t) i s an exact solution to E(x) = 0. i f the ^.("O a complete, l i n e a r l y independent  set.  form  We seek an approximate  solution , n x(t) = where n i s a r b i t r a r y .  a ^ (t) , k  (3.3)  k  The larger the n, the more accurate i s the  solution and the more work involved.  Substituting equation (3.3)  in equation (3.1), we obtain I  =  J  E(a  Q  fa  + a  x  ^  + ...+  a  n  fo*  Q  fa  +  &  ^  ±  +...+  t a ^ ,t)dt n  n  . (3.4)  20 Since the s e t , ^ k , i s chosen beforehand  (i.e.,  trigonometric  functions i f E(x) yields an o s c i l l a t o r y s o l u t i o n ) , must be minimized with respect to a^.  (3-^)  dl = d a. yields d a  1  k  IF da  t  k  Setting  0  b  dt =  equation  dt L  dx  6%  =  0  o  .  -I  which f i n a l l y becomes tb  = jf• \M I dF -- JL d lM)]  JLI  6a  k  t  L  dx  dt V d x '  a' If we now specify that  ^  dt •  J U ^  dx  J  ^k(ta)  ^^b^  =  °?  =  periodic with period ( t ^ - t ) , equation (3>5)  r*  r  di = ) da k  [_*£ - JL /jiE\l  t  L  dx  dt  Since F ( x , x , t ) was a so postulated  V  o r  t n a t  ^  l  s  becomes dt  =  (3.6)  0  that  - _d_ V&F \ = dt V dx / »x  E(x)  f i n a l l y becomes t  J_I £a Equation (3-7)  ^  dx/J  bF  equation (3-6)  (3.5)  k  b  = J"  x E(x) dt  =  (3-7)  0  i s known as the Ritz averaging i n t e g r a l and may  be taken to mean that we are trying to s a t i s f y the  differential  equation (3«2) i n some "weighted" average. Due to the orthogonality of the trigonometric functions with unity weighting, for o s c i l l a t o r y systems the Ritz method i s equivalent to the P r i n c i p l e of Harmonic  21 Balance  (22)  (Ik)  . Cunningham  uses this method i n his attack on  nonlinear d i f f e r e n t i a l - d i f f e r e n c e 3.2  equations.  Application of the Ritz Method to Steady-State O s c i l l a t i o n s i n Nonlinear Systems with Delay The Ritz method w i l l now be applied to the  differential-difference  equation ,  E(x) = x ( t ) ' + 2D 1 o) n g 1 [x(t)J + + koo2if2 [x(t-h)J Equation ( 3 - 8 )  2D2w'n g 2 [x(t-h)] + co | f ] _ - G s i n wt = 0 .  (  3  [x(t)] -  8  )  i s the equation of motion of a f a i r l y general  nonlinear system with delay, subjected force. The functions,f^ and f 2 , delayed restoring forces,  to a harmonic driving  describing the restoring and  and the functions, g-^ and g 2 ,  describing the damping and delayed damping forces,  are a l l  assumed to be single-valued and integrable functions of their respective  arguments. If f^, f 2 ,  respective  g-^  arguments,  f  l  ["^t)]  =  "  g  l  [-xt^]  =  f  and g 2 are odd functions of their  that;is  i [x(t)]  ,  f 2 [-x(t-h)] = - f 2  [x(t-h)]" (  - S i [x(t)]  ,  3  -  9  )  g 2 [-x(t-h)] = - g 2 [x(t-h)],_  the resulting motion has the mean value zero. If only terms with frequency,oo, are considered, an appropriate  assumption for the approximate solution i s x(t)  The Ritz conditions are  = X sin(wt - 0)  .  (3-10)  22  j*  2* to  E [ x ( t ) ] sin wt dt = 0  0 3t to  2  and  E [ x ( t ) ] cos to t dt = 0  j 0 0  When equations  .  (3*9) are applied, the Ritz conditions become 2  [ l " I 2D rj_G 2 sin(r|to n h) + k F 2 cos(r|0) n h)] [ 2 D 1 r | G 1 + 2 D 2 ^ G 2 cos(r^to n h) - k F 2 s i n ( r p n h f | 2  F  +  2  + = [s]  (3-11)  and 2 D  tan  0.=  F  ia l G  l " I  2  +  +  2D 2 ^[G 2 c p s ^ o ^ h )  - kF2 a i n Q ^ h )  ?  ( 3 > 1 2 )  2 D r | G 2 sin(r|_tonh) + k F 2 cos(r|to n h) 2  where 2  2  F-, = J+_  *  x  \  f., (Xsina)sina da = J+_  o  «  1  \ 0  X  2 F9  = J±_ *x  2 f Q (Xsina)sina  \  J 0  da = J+_ * x  o  2 G-, =  )  f 0 (XCOSQ;)cos a da, (3 . l h ) 2  2 g (toXsina)sina da =  k \ wtox o  f, (Xcosa)cosa da, (3.13)  h  1  *  w  X  2 g 9 (toXsina)sin a da =  •7 =  to  ,  V  \ o  g, (wXcosa)cosa da, (3-15)  2 \ g (tdXcos a)cos a da , (3 .16) 9  s = _G_ 2  .  (3.17)  23 If the driving term i n equation (3-8)  has the form,  G cos tot, ins tead of G s i n out, the assumed solution would be x(t)  = X cos (tot - 0)  instead of equation (3-10). The resulting equations (3.17)5 however, are unchanged by these I f f-p f^,  g-^  (3.11) to  replacements.  and g 2 are non-odd f u n c t i o n s t h a t  they do not s a t i s f y equations  (3»9)  5  is  the resulting motion does  not have zero mean value. Therefore, a mean value,M,must be included i n the assumption for x. Equation (3.10) i s replaced by x = M + X s i n (tot - 0) = M + A. sintot - B costot where  A = X cos 0  and  ,  B = X s i n 0.  Consequently, there w i l l be three Ritz conditions for determining the three constants,M, A  and B or M, X  and 0. These conditions  are 2%  to j  " E [x(t)] s i n tot dt = 0  0 2%  to E [ x ( t ) ] cos tot dt = 0  j  0 2% 00  \  E [x(t)] d t = 0  0 If we l e t  2«  ^ to  00 F  01  j  0  fx[x(t)]dt,  G Q 1 = _1_ j W  n .0  g l  [x(t)]dt,  2k  FQ2  2jc  2jt  CO  CO  = j  f [x(t)]dt,  G Q 2 = JL_ j  2  0  n  w  2  0 co  OJ  FS1  g [x(t)]dt,  = 1 j  ^[xCt^sin  tot dt,  G g l = _1_ [ g [ x ( t ) ] sinco t dt, x  * 0  • % * 0  2%  " F S2  =  to  - j %  f 2 [ x ( t ) ] s i n tot dt,  j  Gg2 =  0  2jt co  g [ f ( t ) ] s i n cot dt, 2  n" 0  W  to Fcl  = 1 j %  CO  f [ x ( t ) ] cos tot dt,  G C 1 . = ._1_ J"  x  0  "n  2x  to = 1 j  1  0  51  2JL  FC2  g [x > (t)]cos tot dt,  f [x(t)]cos  tot dt,  2  GC2  co = _1_ ^ g [ x ( t ) ] c o s cot dt, 2  the Ritz conditions become  and  F  01  +  kF  02  +  2D  1 01  +  2D  2G02  F  S1  +  kF  S2  +  2D  1GS1  +  2D  2 G S2 "  G  =  0  F C 1 + k F C 2 + 2D G c l + l | B = 0 2  In general, equations.(3.18),  3'V. (  » '"  S  =  0  1 8  (3-19)  .  (3.19)  (3.20) and (3.20)  represent a system of nonlinear algebraic equations for the three unknowns,M, A. and B. The application is thus tedious, and hence only systems with odd n o n l i n e a r i t i e s are considered hereafter.  25  3•3  Illustrative Results  Examples and Comparison of the Ritz-Method  to the Analog-Simulation Results  There are numerous p a r t i c u l a r examples leading to equations with delay, e . g . ,  differential-difference  which arise from many p a r t i c u l a r f i e l d s  equations,  of i n t e r e s t , a s  mentioned i n the Introduction. A simple example i s the equation describing: the (rh) thickness of a sheet of metal coming from a r o l l i n g m i l l : x(t") = -k[x(t-h) - x 0 ] where x i s  ,  the thickness at any time ,t, XQ i s the desired  thickness, k i s a constant v determined by the control system and h i s the delay due to the separation of the r o l l s and the measurement point. Studies i n the f i e l d of.population  growthlead  to the equation, rx(t) "l - x(t-h) x the population at any s time,t, r i s the reproduction  kit)  where x i s rate, x  -  is. the steady-state population ultimately reached (or  the average value thereof),  and h. i s  the delay due to the  that the population does not react immediately to i t s  fact  increasing  number. An example from the f i e l d of economics i s Goodwin's (9)  nonlinear model of the business cycle fry(t+0) + (1 - a) where y i s  \  y(t+0) = 0[y(t)]  ,  the income at time,t, 0 is the delay between  26  investment decisions and corresponding outlays, # i s the time constant of the income-consumption r e l a t i o n s h i p , a i s the change i n consumption per unit change i n income, and 0(y) i s  the  nonlinear induced investment. The analysis  of a microphone, amplifier  combination with acoustic feedback^ I(t)  + R I(t)  + _1_ I(t) LC  L  + Ak I C  and speaker  leads to the equation, Bk3L2  t-2$  c  I3  where I i s the plabe current at time,t, A. and B are constants related to the tube c h a r a c t e r i s t i c s , § i s the distance constant,  c is the v e l o c i t y of sound,  to the r e f l e c t i n g object,  and R, L  k i s an amplification  and C relate to the c i r c u i t elements.  A system with distributed parameters may sometimes be treated by approximating i t s  transfer  G =  function by one of the form,  -hs Ts + 1 -hs  i . e . , the transfer element.  function of a delay (e  The analysis to equations  and  of control systems  ) and a time-constant J  sometimes  leads  of the form, x(t)  + a^ x(t)  +  x(t)  + ax i(t)  + a'Q x(t)  & 1  x(t-h) + aQ x(t)  =. 0  + aQ xCt-h) = 0  (3-21) .  (3.22)  Equation (3.21) contains a natural damping term,a x x(t),as well as a delayed damping term, a-^x(t-h). Equations of this type arise when an a r t i f i c i a l l y produced damping i s added to increase an i n s u f f i c i e n t natural damping, as i n the s t a b i l i z a t i o n of a rolling s h i p ^ ' .  Equation (3-22) contains a natural  27 restoring force, aQx(t), and a delayed restoring  force,SQX(t-h).  Equations of this type may a r i s e , for example, i n the guidance of an a i r c r a f t .  The delay,h,could be due to the computation time  of a computer i n an autopilot or to a human operator who controls the rudder p o s i t i o n and, therefore, The simple equations  the restoring  force.  (3.21) and (3.22) (at  least  simple i n appearance) may be further complicated by nonlinear terms and the presence of a driving term or i n p u t , f ( t ) . The nonlinear terms may be due to hysteresis;  backlash i n gears;  mechanical stops; clamping c i r c u i t s ; f r i c t i o n ; i n amplifiers,  inductors  and capacitors;  saturating  effects  and a multitude of  other sources. Equations of this type with f ( t )  taken to be  G cos cot or G s i n cot are considered i n this section using the approximate  technique described i n the previous section. The  results are compared with the results obtained by  an analog  simulation. Before the approximate technique i s applied to specific examples,  however, a b r i e f description of the  phenomenon known as "jump resonance"  '  w i l l be given.  OJ^K  The jump resonance phenomenon i s peculiar to systems having a nonlinear restoring force, f(x), and a sinusoidal input. The n o n l i n e a r i t y , f ( x ) , is assumed to be an odd function i n the following discussion.  If the input amplitude is held constant  and the input frequency,co, i s increased,  the response curves,  ABODE,are obtained (see F i g s . 3.1 and 3.2); frequency i s decreased,  i f the input  the response curves,EDFBA,are obtained.  If the damping i s decreased,  the resonant effect  i s more  pronounced and the separation of the jump points,to  and to p., i s  28 c B +t 2 -?P  \  F  /  CD T3  tt  2  -P  i H  -P r-f  P*  O  a  SS  E  •  B •  A  -p p* d) P. w • p rt  2 .a  F  o a,  Figure  *^£  i  to. to 1  .v  c \^  to,  to  to  to.  to  CU  Li  to  to  to  i  2  •p  ID  - —  1  E  3«1 Frequency Response Curve f o r System with |f (x)| St L  - p rt o a,  E  F i g u r e 3«2 Frequency Response Curve f o r System with |f (x)| :s L  0 ti 2  2  -P  -P r H  O  B  A_^-^!3  G G  -P  P" 0)  rt P^ o a. -p  -  Figure  A  B  r x  l  G G  G  2  2 D  E  C  3.3 Amplitude Response Curve f o r System with |f (x)| £ L  F i g u r e 3»+ 1  Amplitude Response Curve f o r System with |f (x)| & L  2  9  increased. If the damping i s increased, the resonant effect  is  less pronounced and the separation of the jump points is decreased u n t i l some c r i t i c a l damping i s reached beyond ; which (29) no jumps are obtained^  If the input frequency i s held  constant and the input amplitude, G, i s increased, the response curves, ABCDE, are obtained (see F i g s . 3 = 3 and 3 « ^ ) ; i f the input amplitude i s decreased,  the response curves,EDFBA,are obtained.  The resonance phenomenon previously described  pertains  to systems having no delay. Since no previous work on the jump resonance phenomenon for systems with delay has appeared i n the literature,  i t i s useful to apply the Ritz method to these  systems and determine the effect  of the delay on the response  curves. This i s done i n the following examples using equations which stem from the important equations  (3.21) and (3.22).  Example 1. Consider the equation, x(t)  + 2D1a)n k(t)  + 2D2ton x(t-h) + to 2 [x(t) + \x x 3 ( t ) ] 2  - G sin tot = 0  ,  (3-23)  which has a delayed damping term and a nonlinear restoring force of the type referred to i n F i g s . 3.1 and 3.3» This equation i s of the same type as equation (2.6)  and i s , ' t h e r e f o r e ,  stable when D-^> |^2i * Applying equations F  n  (3«13) to (3.16), we obtain = 1 + 3 u  G1 = G2 = 1  2  X  2  ,  (3.2*f)  (3-25)  30 and  F2 = 0  After substituting equations  .  (3-26)  (3 • 2^1-), (3.25)  and (3.26) into  equation (3.11), we obtain  A 3 + a 2 A 2 + a A + aQ = 0 ±  ,  (3-2?)  o  where  A= X  ,  a 2 = 2(1 - y^ + 2D2r£ sinrjw n h) 2  a  and  l  ^ " I  =  2  +  ,  2D 2 r^sinr|W n h) 2 + (21)-^ + 2D r|_cos t| ^ h ) 2 2  aQ = - S  The quantities, X and S, have been replaced by dimensionless quantities. X = X and S = S,where L L L equations  (3.25)  (3.2^+),  / rIn-1  . After  substituting  and (3.26) into equation (3.12), we  obtain tan 0 =  2  M  +  2D2rlcosyiconh  ^  (  3  >  2  ^  A + 1 - r^2 + 2 D 2 ^ s i n r | c o n h The response  curves can now be obtained by solving for the  positive r e a l roots of the cubic equation (3.27) i n A and then substituting these-values of A into equation  (3-28).  This  computation has been done using an IBM 70^0 d i g i t a l computer which has a plotter available for recording output data. The response curves for any desired values of the coefficients  can  then be quickly.obtained using a r e l a t i v e l y simple computer program. Typical response curves  are  shown i n  F i g „ 3°5 »  These approximate curves are to be compared to those obtained by an analog simulation,where the amplitude, X , i s taken to be the peak value of the output waveform and the phase,0,to be the  31  Figure 3 * 5 Response Curves, Example 1.  32 difference  (in radians) between the zero crossings of the output  and" input waveforms. As i s the case for systems without delay,  the R i t z -  method results show that i n some regions the output can exist i n three states (corresponding to the cases where three positive r e a l roots of equation (3.27) exist), whereas the analogsimulation results show only two states. This is due to the fact that the two extreme states are stable and the middle state i s unstable  and, therefore,  could never be obtained  experimentally. It i s evident that the approximate results are quite close to the analog-simulation results output amplitudes where the effect  (especially for low-  of the nonlinearity i s  small),  and also that the presence of the delayed damping produces isolated regions of the response curve when the input frequency i s varied, whereas the response curve for varying input amplitude i s similar to F i g . 3«3- At certain frequencies  the  delayed damping w i l l be i n phase with the natural damping , at certain other frequencies  the delayed damping w i l l be out of  phase with the natural damping, while at intermediate  frequencies  the delayed damping w i l l have a component i n or out of phase with the natural damping and a component i n or out of phase with the restoring force. Consequently, as the input frequency i s increased or decreased  the effective damping o s c i l l a t e s between two  extremes. If the upper extreme i s larger than the c r i t i c a l damping necessary for jump resonance,  then isolated regions of  the response curves are obtained as i n F i g . 3>5-  33 These isolated regions can also be explained by considering the roots of equation (3-27). The nature of the roots depends on the quantity, (q+r), where q = S 2 a ( a 2 + 9b )  and  2  27  r = 2 b 2 ( a 2 + b 2 ) 2 + 3^ ,  ~h~  27  2  where  a = 1 - X\ + 2 D r ^ sinrjw h  and  b = 20-^ + 2 D r£ cosrjto n h ' .  2  \ 2  There are two complex conjugate  roots and one r e a l root,  r e a l unequal roots, or three r e a l roots depending on whether (q+r)  Is  three  (two of which are  p o s i t i v e , negative,  equal),  or zero  respectively. When there i s damping but no delayed damping (D-^XD, D 2 =0), q decreases monotonically from some positive value and f i n a l l y becomes.negative,while.r increases some positive value as of for  in r, (q+r)  (q+r)  increases  monotonically from  from zero. Due to the presence  i s positive for large r^. Thus i t i s  possible  to decrease from some positive value to some negative  minimum and then i n c r e a s e , f i n a l l y becoming positive as from zero. Thus, as ^ i n c r e a s e s ,  increases  the number of r e a l roots of equation ,  (3.27) w i l l be one, then three, and f i n a l l y , o n e . When there i s a delay present,q.and r are no longer monotonic due to the presence of the terms, s i n r | ^ a ) r i n cosr^co^h. For large or small values of  , (q+r)  a n  ^  w i l l be  p o s i t i v e . For intermediate values of 1^, however, (q+r) oscillate  about zero as r| increases  o s c i l l a t i o n s occur). Thus,, as  (the larger 10 n h ,  increases,  can  the more  the number of r e a l  roots of equation (3-2'7) w i l l be one, then varying between one  3^  and three, and f i n a l l y , one. Isolated regions of the response curves w i l l then appear i n the frequency range where the number of r e a l roots of e q u a t i o n i ( 3 . 2 7 ) varies between one and three. With the frequency i n the proper i n t e r v a l , the isolated  regions  can be obtained by giving the system a s u f f i c i e n t l y large i n i t i a l condition or by increasing the input amplitude u n t i l a jump i s obtained- and then decreasing the input amplitude to i t s o r i g i n a l value. The jumps- associated with the isolated  regions  are always downward jumps i n amplitude. Consequently, the isolated regions cannot.be obtained simply by varying the input frequency. The Ritz-method results show two isolated regions of the response curve for varying frequency, whereas the analogsimulation results show only one. The amplitude for the unstable portion of the isolated region i s close to that for the stable portion. Thus the stable portion "of the isolated region i s probably unstable for small fluctuations would also explain the difference  i n amplitude. This  between the Ritz-method results  and the analog-simulation results for the isolated region that was obtained i n the analog simulation. The response curve for varying input amplitude similar to that for a system without delay, because the  is  effective  damping remains constant i f the frequency remains constant. the frequency remains constant and the input amplitude  is  varied, the quantities,q and r, mentioned previously,are monotonic even with delay present and, therefore, region with three r e a l roots is  possible.  only one  If  35 Example 2. Consider the equation , x(t)  + 2 D 1 w n x(t)  + 2D co 2  n  x(t-h) + w 2 [ x ( t ) + u 2 x 1 / 3 ( t ) ]  - G s i n wt = 0 ,  (3-29)  which has a delayed damping term and a nonlinear  restoring  force of the type referred to i n Figs. 3.2 and 3 A . Although 1/3  the nonlinear function,x  , is integrable, enabling the Ritz  method to be applied, i t does not satisfy a L i p s c h i t z condition at the o r i g i n because of the i n f i n i t e slope at this point. In any physical system, the slope of the• nonlinearity could be large but never i n f i n i t e . Since equation (3.29) is the: mathematical model of some physical system, we may consider  it  to be an accurate model everywhere except for a small neighbourhood about the point,x : = 0,where we assume the slope of the nonlinearity to be large but not i n f i n i t e . The s t a b i l i t y c r i t e r i o n is then the same as for equation (3.23), i . e . ,  D-^>J|  Applying equations (3.13) to (3.1*6),we obtain  Fx = 1 + u 2 G and  1  2  1  /  , r(l/3) 3  [r(2/3)]  =G =1  2  X~2/3 ,  (3.30)  •  2  F2 = 0 ,  (3-31)  -  (3-32)  where P is a Gamma function. After substituting  (3.30), (3-31)  equations  and (3*32) into equation (3.11) we obtain the  cubic equation (3.27), where now A = X  2 / 3  ,  a 1 = [(1 -  Y[  2  +  2D ^sin»|W h) 2  n  2  + (21>fli-  2 D * ^ cos 2  irjw h) J~ n  2  36  Frequency  '"Put Amplitude  §  '"Put Amplitude 0.2 0.3  § 0.4  Analoo-Simulation Results Ritz-Method Results  0.0  1.0  Frequency 2.0  ^  3.0  4.0  0.0  0.1  Figure 3.6 Response Curves, Example 2.  0.5  37 1  and  a,•o  The quantities, X and S, have been replaced by dimensionless q u a n t i t i e s r X = X and 3 = S,where L L L =r PCI/3) _2l/3 [p(2/3)]  ]3/2 2 .  After substituting equations  j i 3 = 1 . 2 ^ 8 7 )i-  (3«30),  (3.3D  (3.33)  and (3.32) into  equation (3.12), we obtain tan 0 =  2D 1 ^ + 2 D 2 ^ cosr^id h  The response curves can now be obtained as i n Example 1. The curves shown i n F i g . 3.6 are similar to those for a system without delay (see Figs. 3 « 2 and 3-^)  except that  isolated  regions are obtained when the frequency i s varied as i n Example The response curve for varying input amplitude i s  again  similar to that for a system without delay. The approximate results  are close to the analog-simulation results  low-output amplitudes where the approximate results accurate,and the analog-simulation results  except for are least  are inaccurate 1/3  because of the technique used to obtain the function,X  .  Example 3. Consider the equation , . G s i n tot = 0  (3.3*0  38 which has a delayed restoring force and a nonlinear restoring force of the type referred to i n Figs. 3.1 and 3«3- This equation i s the same type as equation (2.27) and i s , t h e r e f o r e , stable when (2Dn)2>  k2 i - |k|  .  Applying equations  (3*13) to (3.16), we obtain  F]_ = 1 + 3. l^ X 2 2  1  ^  +  (3.35)  ,  (3.36)  G = F2 = 1  (3-37)  G2 = 0  (3-38)  1  and  .  After substituting equations  (3-36), (3.37)  and (3«38) into  equation (3.11), we obtain the cubic equation (3.27), where now A = X2  ,  a 2 = 2(1 -r r^2 + k cosr^) n h) a and  1  ,  = (1 - IT]2 + k cos^o) n h) 2 + (20-^ - k s i n r ^ h ) 2  aQ = - 5  2  ,  where the dimensionless quantities,X and S, described i n Example 1 have been used. After substituting equations  (3.36), (3-37)  (3.38) into equation (3.12), we obtain  tan 0 =  l1-  2D  1 + A -..  rn  ksin  h  + k cosrju^h  The response curves can now be obtained as i n Example 1. The response curves shown i n F i g . 3-7 are similar to those for a system without delay (see Figs. 3-1 and 3*3), except that isolated regions are again obtained when the frequency i s  and  39  Analog-Simulation Results Ritz- Method Results Frequency  r\_  Input Amplitude  Figure 3*7 Response Curves, Example 3«  §  ho. varied. The" effective  damping i n this case changes with  frequency because of the component of the delayed restoring force,which i s i n or out of phase-with the natural damping. In other respects this example is similar to Example 1. Example h. Consider the equation, x(t)  + 2D oo x(t) 1  + to 2 [x(t) +\i 2 x 1 / 3 ( t ) ] + kto2 x(t-h)  n  - G s i n tot = 0 ,  (3-39)  which has a delayed restoring force and a nonlinear restoring force of the type referred to i n F i g s . 3*2 and 3>*h. If we 1/3 treat the n o n l i n e a r i t y , x  , as i n Example 2, the s t a b i l i t y  c r i t e r i o n for this equation i s given by equation (3-35)* Applying equations  (3.13) to (3.16),we obtain  equation (3-30) for F^,  and  G]_ ='F 2  = 1  (3 AO)  G2 = 0  .  (3^1)  After substituting equations  (3-30), ( 3 A 0 )  and ( 3 A D into  equation (3.11), we obtain the cubic equation (3.27), where now A .= X 2 / 3 a  l  ,  = [(1 - H.  2 +  k  cos^co n h) 2 +  - k sinr^h)2]"1  ,  a 2 = 2(1 - r^2 + k cosv|w n h) a-^ and  aQ = - S 2 a^  ,  where the dimensionless quantities,X and S,described i n Example 2 have been used. After substituting equations (3-!+l) into equation (3.12), we obtain  (3.30), ( 3 A 0 )  and  1.5  1.5  0.10 k . -0.16 o „ h . a.oo  1 1 1 1  IX  S . 0.14  IX  1  1 1  « 1.0  « 1.0  1 1  CL  e <  o. E <  0.5  0.5  l\ 0.0 0.0  j:  ._-L 1.0  0.0 2.0  Frequency  r^  3.0  4.0  0.4  0.0  0.6  Input Amplitude  0.8 S  • Analog-Simulation Results Ritz- Method Results  0.0  1.0  Input Amplitude  Frequency n^  2.0  3.0  4.0  0.0 0.0  0.0 1•  r l f1 1 i  |-2.0 41  tt)  ca •3.0  / \  1 1 1 1  -1.0 m c a  1-2.0 CC  to !-3.0  0.2  nrA /iii/ ii \\  0.4  0.6  i  i  a.  F i g u r e 3.8 Response Curves, Example h.  §  0.8  1.0  k2 tan 0 =  2 D  l1 -  k  s  i  n"n  .  h  A " 1 + 1 - lf|2 + k cosr|a) n h The response curves can now be obtained as i n Example 1. The effective  damping changes  with frequency as i n Example 3,  otherwise the discussion for the response curves- shown i n F i g . 3.8 i s the same as for Example 2. The Ritz method can also, be applied to systems which have nonlinear damping terms. These systems do not exhibit jump resonance, but i t i s of interest  to obtain their response curves.  Example 5. Consider the equation , x(t) + 2D 1 co n [x(t)  + j i _ x 3 ( t ) ] + 2D2con x(t-h) + w 2 to2 n - G s i n tot = 0  ,  x(t)  (3A2)  which has delayed damping and nonlinear damping. This equation i s stable when D X > | D 2 | . Applying equations the results into equations  (3.13) to (3-16) and substituting  (3-11) and (3.12), we obtain for the  cubic equation (3.27)  _? A = X  ,  a 2 = 2I>^\_ + D  a  and  2DV| 2  i  a  cos\q_wnh  3  _ (1 - V^2 + 2 D 2 » | sinvito n h ) 2 +• (20^+  aQ =  - S2  2D2r|_ cos y^oonh)2  >+3  2.5  A  2.0  S 1.5 E <  D, • 0.10 D. - 0.09 o h - 4.79 125 S •  7  —  n  IX)  0.5  OX)  0.0  1.0  2.0 Frequency  3.0  4.0  r\  2.0 3.0 Input Amplitude  4.0 §  Input Amplitude 2.0 3.0  § 4.0  5.0  Analog -Simulation Results ' Ritz - Method Results  1.0  Frequency 2.0  3.0  4.0  0.0  0.0  -1.0 0} c  |-2.0 4>  to  5-3.0 a.  F i g u r e 3«9 R e s p o n s e C u r v e s , E x a m p l e  5'  5.0  and  for. the phasetan 0 = 2 D ^ (1 + Y^k) + 2 0 - ^ cosT[tonh  ,  1  1 - n 2 + 2 D 2 r | s i n r|co n h where the dimensionless quantities,X and 3,described i n Example 1 have been used. Typical response curves are shown i n F i g . 3 « 9 . Since the effect  of the nonlinear damping i s small, the R i t z -  method results and the analog-simulation results are quite close.  The nonlinearity increases the damping as the output  amplitude increases, causing the peak of the curve for varying frequency to be somewhat flattened and the curve for varying input amplitude to be concave down. The delayed damping causes s l i g h t humps i n the frequency response curve due to the.varying effective  damping with frequency. This effect  is  somewhat  diminished due to the nonlinear damping term. Example 6. Consider the equation, x(t) + 2D 1 co n [x(t) + u 2 t o 2 / 3 x 1 / 3 ( t ) ] - G s i n 'tot = 0  -r  2D W 2  n  x(t-h) + w 2 x(t)  ,  (3A3)  which has delayed damping and nonlinear damping. If the difficulties  due to the presence of the cube root term are  treated as i n Example 2, the equation i s stable when Applying equations the results into equations cubic equation (3.27) A = X2/3  ,  £> >|D |. X  2  (3=13) to (3.16) and substituting  (3.11) and (3.12), we obtain for the  ^5  Frequency  Input Amplitude S Analog-Simulation Results Ritz-Method Results  Figure 3'10 Response Curves, Example 6.  k6  a,0  2  - ^ nr  J  a1 and  a2  S  a,  2  and for the phase tan 0 =  2 D 1 r ^ ( l + l/[r]_ 2 / 3 A] ) + 2D 2 »^_cos r|_o)nh  where the dimensionless quantities, X and 3, described i n Example 2 have been used. Typical response curves are shown i n F i g . 3.10. The Ritz-method results and the analog-simulation results again quite close.  are  The nonlinearity decreases the damping as the  output amplitude increases, causing the peak of the curve for varying frequency to be sharply peaked and the curve for varying input amplitude to be concave up. This sharp peaking of the . resonance curve would be useful where a high Q c i r c u i t , i s required. The effect Example 5«  of the delayed damping i s the same as for  47  4.  VERIFICATION OF THE APPROXIMATE SOLUTION BY ANALOG SIMULATION  The v a l i d i t y of the approximate a n a l y t i c a l method mentioned i n section 3 depends on assuming the correct form of the solution.  If the assumed form i s incorrect, the results  obtained by this method are completely meaningless. In view of the quasi-linear nature of the systems considered, i t has been assumed i n section 3 that the response of the system to a sinusoidal input w i l l be approximately sinusoidal and of the same frequency as the input.  This  assumption i s easily v e r i f i e d by simulating the system on the PACE 231R analog computer-,,  In order to compare the E i t z -  method results to the analog results i t i s desirable to measure the amplitude and phase of the fundamental component of the output waveform.  Since the system i s nonlinear, the output  waveform, i n general, deviates somewhat from a true sinusoid. This, deviation^ however, i s not large and, therefore, i t  is  reasonable to base the measurement of the amplitude of the fundamental on the peak value of the output waveform and the phase on the  zero-crossover.  The v e r s a t i l i t y of the PACE 2 3 1 R enables automatic plotting of the system output amplitude and phase versus the frequency or amplitude of the sinusoidal system input,  In  view of the large number of examples considered t i t i s  essential  that.the response curves be obtained automatically, otherwise the computing time and the time to plot the curves would be  48 prohibitive. The sinusoidal system input ( A cosoot or A sintot) is obtained by solving the nonlinear d i f f e r e n t i a l equation , x - e A  -  x  - x  x + to 2 x = 0 ,  2  to  (4.1)  J  ("=51}  which was suggested by Van der Pol and i s discussed by J a c k s o n w / (see P i g . 4 * 1 ) .  Equation ( 4 . 1 ) has the l i m i t cycle s o l u t i o n , x = A cos(wt + ©) ,  (4.2)  which i s - e a s i l y v e r i f i e d by substituting equation ( 4 . 1 ) .  equation ( 4 . 2 ) into  If the i n i t i a l conditions are chosen as  x(0) = A  ,  i(0) = 0  ,  the solution begins at the l i m i t cycle and the term,©, i n equation (4.2) becomes zero.  With e f a i r l y large (say 1 0 ) , the solution  tends rapidly to the l i m i t cycle i f any disturbances occur. Therefore, i f the signals corresponding to to and A are variedreasonably  slowly,, the nonlinear o s c i l l a t o r of P i g . 4 . 1 w i l l  continuously y i e l d the output,A cos tot.  The c i r c u i t described  (32) by Humo  does not function i n t h i s manner and hence his  results are i n error,, A control c i r c u i t (Fig* 4 . 2 ) enables the operator  to  hold A constant and automatically increase or decrease to, or to hold to constant and automatically increase or decrease A . With switches, S^Q,  ^12  a n  ^ ^I3»^n ^  e  ^- ^^ position, to e  i s swept by integrator 26 while A i s set by pot. P59> with the switches i n the right p o s i t i o n , A i s swept by integrator 26 while to i s set by pot P59»  With switch,  , i n the right  k9  A a in wt A COS Ut  Figure h.l  Nonlinear O s c i l l a t o r  Figure K.2 Control C i r c u i t  50 position the output of integrator 26 increasesj i n the l e f t position the output decreases.  The sweep rate i s controlled  by pot. P58. Pig.  4.3 shows the analog simulation for the system  of example 1 (sec.  3.3) and the track and store c i r c u i t s used  to detect the system output amplitude or phase.  The analog  simulations of the other systems are similar to P i g . 4.3 and are therefore not shown.  A twenty segment diode function generator  provides a good approximation to the cube root quantity required for  examples 2  t  4  and 6 of section 3.3.  The i n f i n i t e slope  at the o r i g i n for the cube root function cannot be obtained using the function generator and hence the results obtained for  low amplitude inputs to the function generator are somewhat  i n error. The delay element i s simulated by means of an Ampex tape recorder (Model SP300 P.M. D i r e c t ) . 1 7/8  (+  0.4#),  3 3/4  (+  0.4#),  7 1/2  (+  The tape speeds of 0.2%),  and  15 ( +  0.2%)  inches per second, provide delays of 1455? 728, 364, and 183 milliseconds r e s p e c t i v e l y «  The frequency response at a tape  speed of 15 inches per second i s from 0 to 2500 Hz.  The  maximum frequency i s reduced by a factor of two each time the tape speed i s reduced by a factor of two.  The inputs and out-  puts to the various channels of the tape recorder are  available  as trunk l i n e terminations at the analog patch panel.  The  resistors necessary to protect the tape recorder from overload and to provide the appropriate signal levels at the tape recorder and the analog patch panel are incorporated i n these trunk l i n e s .  51  Figure ^.3  Analog Simulation of a System with Delay  The delay element could also be simulated by approximating the Laplace shift operator, e"*13, which is the transfer function associated with a pure time delay,h.  This simulation,  however, requires a large number of integrators for as  accurate  an approximation as can be obtained with the tape recorder.  The  tape recorder has the added advantage that the delay can be changed merely by changing the tape speed. With switches,S Q 0 and 3 Q ^ r i n the l e f t p o s i t i o n , the output amplitude i s plotted.  The signal,x(t), i s applied to  comparator M5 and the s i g n a l , - x ( t ) , i s applied to comparator M6. Each signal i s compared to zero v o l t s ,  (The operation of the  comparators-, integrator mode control, and AND gates i s described in Appendix B.)  The normal d i g i t a l output,M5 and the comple7  mentary, output, M6,are applied as inputs to an AND gate Fig.  4*3).  (see  The normal output.(M5.M6) of the AND gate i s thus  at a ONE l e v e l for the f i r s t half of each positive half-period of x(t) and at a ZERO l e v e l for the remainder of the period.' The signal,M5.M&, controls the mode of integrator 10,while the signal,M5»M6, controls the mode of integrator 11,  Integrator  10 thus "tracks" the system output during the f i r s t half of each positive half-period, while integrator 11 "stores" and plots the maximum value of each positive h a l f - p e r i o d .  The system  output amplitude i s thus, obtained. With switches, S Q 0 and S Q ^ , i n the right position, the phase of the output i s obtained.  The system input i s applied  to comparator,M6,while integrators 10 and 11 track and store the output of integrator 00.  The signal,M5»M6, which controls  53 the mode of integrator 0 0 , i s now at a ONE l e v e l each time the system input goes positive u n t i l the time the system output goes positives  Prom the time the system input goes p o s i t i v e ,  therefore, integrator 0 0 integrates at a rate proportional to to for a time equal to the time x(t) reset*  lags the input and i s then  The output of the track and store c i r c u i t is then  proportional to the phase of the output. If the signals corresponding to to and A are varied slowly enough that transient effects are n e g l i g i b l e t h e n continuous plots of the steady-state system response curves are obtained i The results obtained by the analog simulation are given i n sec 3«3, and are compared to the results obtained by the approximate a n a l y t i c a l method  54 5.  CONCLUSIONS  The purpose of this investigation was to obtain approximate a n a l y t i c a l solutions of quasi-linear difference equations  differential-  and to determine their resonance  properties, S t a b i l i t y c r i t e r i a for these equations have been given p r i o r to the approximate a n a l y t i c a l solutions and the determination of the resonance properties.  The s t a b i l i t y  c r i t e r i o n for equations with delayed damping is due to (21) Krasovskii  , the s t a b i l i t y c r i t e r i o n for equations with  delayed restoring force has been developed by the author. Approximate a n a l y t i c a l solutions of a general second-order nonlinear d i f f e r e n t i a l - d i f f e r e n c e  equation have  been obtained by employing the Ritz method. General equations which lead to the determination of the constants i n the assumed solutions have been given for systems with odd n o n l i n e a r i t i e s and for systems with non-odd n o n l i n e a r i t i e s . The general equations for systems with odd n o n l i n e a r i t i e s have been used to obtain the resonance  properties  for several s p e c i f i c examples of such systems. It has been found that the response curves for varying input amplitude are similar to those for systems without delay, whereas the response curves for varying input frequency exhibit a rather peculiar jump phenomenon which i s not obtained for systems without delay. When the input frequency is varied, isolated regions of the response curve occur. It has been found that these regions can be obtained p h y s i c a l l y by giving the system a s u f f i c i e n t l y large  55 i n i t i a l condition,or by increasing the input amplitude s u f f i c i e n t l y and then decreasing i t to i t s  o r i g i n a l value. The  isolated regions are attributed to a frequency-dependent effective  damping caused by the i n t e r a c t i o n of the natural  damping with the delayed damping or the delayed restoring  force.  This peculiar jump resonance phenomenon has not previously been mentioned i n the l i t e r a t u r e . The approximate solutions for the specific examples have been v e r i f i e d by an analog computer simulation. This simulation employs track and store techniques to enable automatic p l o t t i n g of the response curves. The Ritz-method results compare favourably with the analog-simulation r e s u l t s . In view of the success of the Ritz method for the examples considered, i t would be useful to prove t h e o r e t i c a l l y that the Ritz method i s applicable to general nonlinear differential-difference  equations.  It would also be useful  to  extend other approximate techniques available for ordinary nonlinear d i f f e r e n t i a l equations difference  equations.  to nonlinear d i f f e r e n t i a l -  It would then be possible  to  investigate  transient behaviour and such phenomena as entrainment of frequency which occurs when an o s c i l l a t o r i s subjected to a sinusoidal driving force. In conclusion, approximate solutions  to some quasi-  linear differential-difference  equations have been obtained  and their resonance properties  determined.  56 APPENDIX A SOME DEFINITIONS AND PROPERTIES PERTAINING TO QUADRATIC FORMS The following definitions and properties to quadratic forms are given by Ayres  (33) J  J  pertaining  :  A homogeneous polynomial of the type n . n q - X'AX = £ £ a x.x. , ± J  i=l whose c o e f f i c i e n t s , a . . , a r e  o=l  elements of F is called a quadratic  form over F i n the.variables, x ^ , . . . , x . The symmetric matrix,A = [ a i j ] > ( a i j  =  a  ji^  i s  c  a  H  e  d  .the matrix of the quadratic form and the rank of A i s called the rank of the form. If the rank i s r < n called singular;  otherwise,  the quadratic form is  non-singular.  A minor of m a t r i x , A , i s called p r i n c i p a l i f i t  is  obtained by deleting certain rows and the same numbered columns of A. Thus, the diagonal elements of a p r i n c i p a l minor of A are diagonal elements of A. For a symmetric matrix,A = ^a^^.J,over F, define  the  leading p r i n c i p a l minors as PQ  1?  Pi  a  l l ' ^2  a  a  li 21  a  12  a  22  , . = . , p^  |A  A r e a l quadratic form,X'AX,is p o s i t i v e - d e f i n i t e and only i f , i t s positive.  if,  rank is n and a l l leading p r i n c i p a l minors are  57  APPENDIX B ON THE OPERATION OP SOME ANALOG COMPUTER COMPONENTS The following specifications were obtained from the PACE 231R MLG System handbook^ 5 4 ^ B.l  Integrator Mode Control The integrator is' placed i n the electronic switching  (ES) mode by grounding i t s ES termination on the' Memory and Logic Unit (MLU) pre-patch panel.  In the ES mode the integrator  is placed i n " i n i t i a l condition" by applying +5 v o l t s (a ONE level) to one of the MLU panel IC terminations S/R i n P i g . 4.3).  (designated  An input of zero vol t s (a ZERO l e v e l )  switches the integrator to the "operate" mode. B. 2  Electronic Compatators The analog inputs are applied at the analog patch  panel and provide the following d i g i t a l outputs at the MLU pre-patch panels (1)  When the analog input sum i s negative (less than -10 mv.) the normal d i g i t a l output i s at a ZERO l e v e l ;  (2)  When the analog input sum i s positive (greater than +10 mv.) the normal d i g i t a l output is at a ONE l e v e l .  B. 3  AND Gates If a l l of the inputs to an AND gate are at a ONE l e v e l  the normal output i s at a ONE levels i f one or more of the inputs are at a ZERO l e v e l the normal output i s at a ZERO l e v e l .  58 REFERENCES 1.  B e r n o u l l i , J . , "Meditationes. Dechordis v i b r a n t i b i s . . . " , Commentarii Academiae Scientarium Imperialis  Petropolitanae, 3 (1728), 13-28. Collected Works.  V o l . i i i , p. 198. 2.  Krasovskii, N . 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