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Jacobi polynomial truncations and approximate solutions to classes of nonlinear differential equations Dodd, Ronald Edward 1966

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JACOBI POLYNOMIAL TRUNCATIONS AND APPROXIMATE SOLUTIONS TO CLASSES OE NONLINEAR DIEPERENTIAL EQUATIONS  by  RONALD EDWARD DODD B . A . S c , University of B r i t i s h Columbia, 1964. A THESIS SUBMITTED IN PARTIAL EULEILMENT OE 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 Members of the Committee  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  ,  In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for  an advanced degree at the U n i v e r s i t y of B r i t i s h Columbia, I agree  that the L i b r a r y s h a l l make i t f r e e l y aval]able for reference and study,  I farther agree that permission.for extensive copying of t h i s  thesis for s c h o l a r l y purposes may be granted by the Head of my Department or by his representatives.  It  is understood that copying  or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain shall, not be allowed without my written permission.  Department of  £/&ctrtca*t  The U n i v e r s i t y of B r i t i s h Columbia Vancouver 8, Canada  Date  ABSTRACT Solutions to classes of second-order, nonlinear d i f f e r e n t i a l equations of the form i  +  2&x  + f(x) = 0„  x(0) = 1,  are approximated i n this work.  i(0) = 0  The techniques which are  developed involve the replacement of the characteristic, f ( x ) , i n the nonlinear model hy piecewise-linear or piecewisecubic approximations.  From these, closed-form time solutions i n  terms of the c i r c u l a r trigonometric functions or the Jacobian e l l i p t i c functions may be obtained.  Particular examples i n  which f(x) i s grossly nonlinear and asymmetric  are considered.  The orthogonal Jacobi and shifted Jacobi polynomials are introduced f o r the approximation i n order to s a t i s f y c r i t e r i a which are imposed on the error and on the use of symmetry. Error bounds are then developed which demonstrate that the maximum error "in the • normalized "time- solution: i s bounded, no matter how large the coefficients of the nonXnear terms i n the model become.  Because of these error-bound  r e s u l t s , an h e u r i s t i c measure of the departure from l i n e a r i t y i s defined f o r classes of symmetric  oscillations, and the  weighting of convergence of the Jacobi and shifted.Jacobi polynomial expansions i s set according to t h i s measure. For asymmetric  conservative models,, shifted  Chebychev polynomials are used to obtain near-uniform approximations to the characteristic i n the nonlinear d i f f e r e n t i a l equation. ii  Based on the equivalence of the c l a s s i c a l approximation techniques which i s given for the symmetric, conservative models, extension of the polynomial approximation to classes of,non-conservative models i s considered. Throughout the work, hy comparison with c l a s s i c a l approximation methods, the polynomial approximation techniques are shown to provide an improved, direct and more general attack on the approximation, problem with a decrease i n tedious labor.  iii  TABLE OP CONTENTS Page LIST OF ILLUSTRATIONS LIST OF TABLES  , ,  f  ,  LIST OF PRINCIPAL SYMBOLS  ,  vi ,. . v i i i  ,  ACKNOWLEDGEMENT 1.  xii  INTRODUCTION 1.1 1.2  2.  ix  Description of the Mathematical Model ........... Some Existing Approximation Techniques  1 4  DEVELOPMENT OF THE APPROXIMATION TECHNIQUES 2.1  Considerations of Approximation Range and C r i t e r i a for Closeness of F i t i n the Approximate Time Solutions ,.„,.... 2.1.1 2.1.2 2.1.3  8  Determination of Bounds on the O s c i l l a t i o n and Normalization of the System Equation , ,...« 8 C r i t e r i a f o r Closeness of F i t i n the Time Solution Approximations ......, .......... 12 Examples of Some Existing Approximation Techniques Under the Given Error Criteria . . . . . . . . 13 &  2.2 2.3 3.  Use of the Jacohi and Shifted Jacobi Polynomials for Approximation 18 Determination of Upper Bounds on the Error i n the Approximate Time Solution .., 25 t  APPLICATION OF THE APPROXIMATION TECHNIQUES TO SPECIFIC NONLINEAR MODELS 3.1  Techniques f o r a Class of Odd-Symmetric Nonlinear Characteristics ...,, 38 3.1.1 3.1.2  3.2  D e f i n i t i o n of the Nonlinear Factor and Choice of Weighting Functions Some Particular Cases  38 41  Techniques f o r Models with Asymmetric Nonlinear Characteristics . . . . , . . . . . . , . . . . . . . , 51 t  3.2.1 3.2.2  First-Order or Linear Approximations ,... 52 Cubic Approximations to Nonlinear, Asymmetric Models , 54 iv  Page  4.  3.3  An Extension of Lanczos' Economization to the Transient Response of Lightly-Damped Models .... 58  3.4  Discussion and Possible Extension of the Results  CONCLUSIONS  64 67  APPENDIX A The Closed-Eorm Solution of Second-Order, Conservative System Models with Cubic Characteristics 69 APPENDIX B Derivation of the Jacobi and Shifted Jacobi Polynomials .., .,-...... o . . . . . . . . . . ,  75  APPENDIX C The Krylov-Bogoliubov Approximation; i t s Equivalence with the One-Term Ritz Method, and with a Linear Expansion i n Ultraspherical Chebychev Polynomials ............. 78 REFERENCES  81  v  v  LIST OP ILLUSTRATIONS Pigure  Page  1.1  A Nonlinear LC C i r c u i t  2.1  Normalization Oscillation  2.2  Comparison of the First-Order Ritz Approximate Solution and a Linear, Least-Square Error Approximation  2.3  2  of an Asymmetric, Nonlinear  10  15  Comparison" of Chehychev and Least-Square Error Cubic Approximations to x+x+x3+lOx5 - o  15  2.4  The Approximation of Two Nonlinear Characteristics.  20  2.5 2.6  A Plot of Two Shifted Jacobi Polynomials Saturation of the Maximum and Quarter-Period Errors for Linear Approximations as the I n i t i a l Amplitude of the O s c i l l a t i o n i s Increased  25 34  2.7  Saturation of the Maximum Error and the Error at the Approximate Quarter Period for Models of the Porm *x+nxP = 0 36  3.1  A Graphical Representation of the Measure of Departure from Linearity  39  3.2  A Comparison of the Error i n Two Linear Approximations to *x+ y |xp sgnx = 0  42  3.3  The Relative Error. in the Frequency f o r Linear Approximations to x+x+x-^+lOx-? o, x(0) = 1, x(0) = 0  45  Comparison of Solutions to Models Obtained from Chehychev, Ultraspherical Jacobi and Shifted Jacobi Approximations to x*+x+x3+lOx5 = 0 ......  50  3.5  The Error i n Ritz and Shifted Chehychev Approximations to *x*+x+x+3x3 = 0  53  3.6  The Error D i s t r i b u t i o n i n Shifted Chehychev and Least-Square Error Cubic Approximations to M+x+x3+lOx5 = U(t)  57  Linear Approximations to x+0.4x+x+x^+5x  61  #  =  3.4  3.7  2  vi  =0  Page 3.8  The Piecewise-Linear Approximation of x + 0.2x + tanh(2x) = 0  vii  63  LIST OP TABLES Table  Page  2.  Summary of Approximation Errors ........... <.,... 17  3.1  The Choice of Weighting f o r the Jacobi and Shifted Jacobi Polynomial Approximations  40  3.2  Comparison of the Eirst-Order Ritz Method and Linear Jacobi Polynomial Approximations  43  3.3  Cubic Polynomial Approximation of the Nonlinear Characteristics  47  A  Standard Eorms f o r Jacobian E l l i p t i c Integrals of the E i r s t Kind  74  viii  LIST OP PRINCIPAL SYMBOLS a  =  a constant  a^  =  coefficients of a polynomial characteristic  a(t)  =  amplitude parameter i n Z-B approximation  A^  =  c o e f f i c i e n t of a Jacohi polynomial  h  =  a constant  b^  =  coefficients of an approximate polynomial characteristic  =  a sum of coefficients of Jacohi polynomials  =  constant coefficients  =  capacitance  =  a constant  Cn  =  the Jacohian e l l i p t i c cosine function  Dn  =  a Jacohian e l l i p t i c function  EMAX  =  the maximum value of E(z + e)  E(x)  =  the error i n the approximation to f(x)  E^  =  a constant proportional to the t o t a l energy  f(x)  =  a nonlinear characteristic i n the d i f f e r e n t i a l equation  f(x)  =  an approximation to f(x)  P  =  the nonlinear factor  g  =  a parameter i n Legendre's transformation  G- ^ ^(x)  =  the k  h  =  a parameter i n Legendre s transformation  =  a function of the coefficients of the Jacohi polynomials  =  current through an inductor  B _ n  m  c c  k  C C  Q  a,  k  i-^  t h  shifted Jacobi polynomial 1  ix '  J  =  a mass or moment of i n e r t i a  k  =  the modulus of a Jacohian e l l i p t i c function  =  a constant  k  o  a function of the coefficients of the Jacohi polynomials L  =  inductance  m  = V / ( P - J S K P - ST  M(x)  =  n  =  p  =  a parameter i n the Legendre transformation  P^fx)  =  an u l t r a s p h e r i c a l Jacohi polynomial  PM,  =  the absolute value of the k^ Jacobi or shifted Jacobi polynomial at the point where E(z + e) i s a maximum  q  =  a parameter i n the Legendre transformation  q^  = .the charge on a capacitor  r  = \ / ( P " ±0 (P - *)  R  =  s  = y / ( q - ja) (q - £J  S  =  Sn  =  the Jacohian e l l i p t i c sine function  t  -  time  ^  =  shifted time  T  =  the period of an o s c i l l a t i o n  T  =  an approximation to T  Tn  =  the Jacohian e l l i p t i c  T(9)  =  a mechanical torque or restoring force  U(t)  =  the unit step function x  a mathematical model ^ / ( C L  - ©Kg. - A)  V  h  X  / f ( x ) x " ( l - x)P dx o c  1  _1  O  /f(x)x (l - x ^ d x a  O  tangent function  v(t),w(t),x(t)  =  dependent variables  jc(t)  =  an approximation to x(t)  x  =  a bound on an asymmetric o s c i l l a t i o n  =  a constant  y(t),z(t)  =  dependent variables  a,p  =  scalar parameters which determine the  X  m i n  Q  weighting of the shifted Jacobi polynomials I~\x)  =  the Gamma function  c>  =  the c o e f f i c i e n t of viscous damping  At  =  an increment i n time  e(t) e  = =  x(t) -af(t) the particular integral i n a d i f f e r e n t i a l equation f o r e(t)  Q,\  =  roots of the denominator polynomial to which Legendre's transformation i s applied  \i  =  a scalar parameter which determines the weighting of the ultraspherical Jacohi polynomials  u.  =  a root of the denominator polynomial to which Legendre s transformation i s applied  p  1  JC  =  3.14159. »• radians  jt  =  a root of the denominator polynomial  0  =  flux density i n an inductor  a)  =  angular frequency i n radians per second  xi  ACKNOWLEDGEMENT Grateful acknowledgement i s given to the National Research Council of Canada for the assistance received under Block Term Grant A68 from 1964 to 1966 and to the B r i t i s h Columbia Telephone Company f o r the award of a scholarship i n 1964-1965. The author would l i k e to express appreciation to his supervising professor, Dr. A. C. Soudack, f o r his i n s p i r a t i o n and guidance throughout the course of the work. Thanks i s also given to Dr. J. S. MacDonald f o r reading the manuscript and for making many helpful suggestions. The author i s also indebted to Mr, A. G. Longmuir and Mr. E. Lewis f o r discussion of the manuscript, to Mr. B. Wilbee f o r proof-reading, to Mr. R  ff  Proudlove for photographing  curves and to Miss L. Blaine f o r typing the thesis.  xii  JACOBI POLYNOMIAL TRUNCATIONS AND APPROXIMATE SOLUTIONS TO CLASSES OP NONLINEAR DIFFERENTIAL EQUATIONS 1. 11 %  INTRODUCTION  Description of the Mathematical Model In order to obtain mathematical representations which  approach the natural behaviour of some r e a l systems, nonlinear relations among variables are often required.  Many of these  systems are described by mathematical models f o r which closedform solutions cannot be found i n terms of known functions, and a numerical scheme must be employed to obtain an accurate solution.  There i s a considerable advantage i n having a closed-  form solution to a problem because one achieves insight into changes i n the solution with v a r i a t i o n of particular parameters i n the model.  In engineering, the concern with approximation  techniques i s important because many models derived from the physical world are themselves only approximations.  Therefore,  i t i s often useful to approximate a nonlinear system model by one from which a closed-form solution may be obtained, I n i t i a l consideration i n this study i s given to the second-order d i f f e r e n t i a l equation M(x) = x + f(x) = 0,  x(0) = 1,  x(0) = 0,(1.1)  which describes undamped or conservative o s c i l l a t i o n s with a single degree of freedom.  I t i s not required that f(x) have  zero-point symmetry i n this model.  Thus, the zero on the r i g h t -  hand side of equation ( l . l ) does not necessarily imply that the -  2 o s c i l l a t i o n s are free - a constant or step function driving term i s allowed.  The p r i n c i p l e of superposition does not apply to  general nonlinear systems.  In particular, nonlinear o s c i l l a t i o n s  described by equation ( l . l ) show a dependence of the on the amplitude  of the oscillation^.  frequency  To provide a constant  frame of reference from which the examples to be considered may be compared, the i n i t i a l conditions are normalized shown i n equation ( l . l ) .  as  Conversion of models with arbitrary  i n i t i a l conditions to this form i s outlined i n paragraph (2.1,1). Common examples of the model i n equation ( l . l )  occur  in the description of e l e c t r i c a l and mechanical o s c i l l a t i n g systems.  Suppose, for example, that i ( 0 ) = g(0) where 0 i s T  the flux density for the nonlinear inductor i n the LC c i r c u i t shown i n Figure  (l.l).  C  Figure 1.1  A Nonlinear LC C i r c u i t  If the inductor alone i s nonlinear, by Kirchhoff's current law one obtains Cv + g(0) = 0. Faraday's Law. 0 + W  Now  v = N0 for an N-turn c o i l from  Hence,  B(0) =  0.  (1.2)  C a r r y i n g out a d u a l a n a l y s i s f o r the case i n w h i c h the a l o n e i s n o n l i n e a r , one  'q.  +1  capacitor  obtains  h ( q ) = 0,  (1.3)  where q i s t h e c h a r g e on t h e c a p a c i t o r and t h e v o l t a g e a c r o s s t h e c a p a c i t o r i s d e f i n e d by t h e r e l a t i o n v ^ ( q ) = h ( q ) .  Hayashi^  g i v e s p r a c t i c a l examples of i n d u c t o r s f o r w h i c h the  current-  f l u x r e l a t i o n s h i p h a s t h e f o r m i ^ ( 0 ) = c-j.0 + c^0  + c^0  and f o r w h i c h t h e c o e f f i c i e n t s Cr- and c„ d o m i n a t e  over c  5  + c^0 , n  and  c,  1  3  7  f o r the l a r g e r v a l u e s of 0 on.the i n t e r v a l of  interest.  In m e c h a n i c a l systems the n o n l i n e a r c h a r a c t e r i s t i c r e p r e s e n t s a r e s t o r i n g f o r c e or t o r q u e . l i n e a r f o r c e i s g i v e n by T(9)  = g(9)  Suppose t h a t s u c h a n o n -  where 9 i s a d i s p l a c e m e n t .  By d ' A l e m b e r t ' s p r i n c i p l e f o r a r e s t r a i n e d e l e m e n t moment o f i n e r t i a ,  J,  9 + j T(9)  is  o f mass o r  = 0  o b t a i n e d as t h e s y s t e m e q u a t i o n .  F o r example, i f the motion  o f a body i n a c e n t r a l f o r c e f i e l d d e p e n d s o n l y on t h e d i s t a n c e , r , from- some f i x e d p o i n t , t h e n t h e e q u a t i o n o f m o t i o n i s  S-| + f ( r ) - = . O  f  r ( 0 ) = 1,  r ( 0 ) = 0,  (1.4)  d\ where X i s t h e a z i m u t h a l a n g l e i n s p h e r i c a l c o o r d i n a t e s .  Gold-  (2)  stein  v  ' shows t h a t d e t e r m i n a t i o n o f t h e m o t i o n o f a s y s t e m  s i s t i n g o f two i n t e r a c t i n g p a r t i c l e s may  con-  be r e d u c e d t o t h e p r o -  b l e m o f d e t e r m i n i n g t h e m o t i o n o f a s i n g l e p a r t i c l e i n an  exter-  nal f i e l d , such that the motion i s governed hy equation ( 1 . 4 ) . (3)  Pipes  gives a model of a mechanical system executing free 2 2 asymmetric o s c i l l a t i o n s . The model i s x + n x + h x = 0, and i t i s of importance i n the theory of seismic vibrations. As a f i n a l " k-l example, the application of models of the form x + cx|x|  = 0  to the description of motion i n p r i n c i p a l modes of certain classes of nonlinear systems having many degrees of freedom has been shown by  Rosenberg^. The e l e c t r i c a l and mechanical examples quoted above a l l  have the form of the generic model i n equation ( l . l ) .  In Chap-  ter 3 an extension i s made to the case i n which the model i n equation ( l . l ) assumes l i g h t , viscous damping.  Primary consi-  deration i s given to the conservative system, however, so that the approximation of the nonlinear amplitude-frequency r e l a t i o n ship may he studied with variation of the characteristic, f ( x ) , alone. 1.2  Some Existing Approximation Techniques C l a s s i c a l f i r s t - o r d e r approximation techniques, such as (5)  the perturbation m e t h o d and B o g o l i u b o v ^  w/  and the averaging method of Krylov  (Z-B method), require an e x p l i c i t l i n e a r term  which must dominate over the nonlinear terms i n the model.  These  techniques are unsuitable f o r the models i n this work because the "quasi-linear" nature of the model i s undefined.  The R i t z -  (7)  Galerkin  v  ;  averaging method or the Principle of Harmonic  (8) Balance  , makes no such r e s t r i c t i o n s on the deviation of the  model from l i n e a r i t y , but i t w i l l be  shown that this method f a i l s  to y i e l d p r a c t i c a l approximate solutions when the deviation  5 f r o m l i n e a r i t y becomes a p p r e c i a b l e . the  Moreover, i n Appendix  C  o n e - t e r m R i t z method i s shown t o g i v e t h e same f i r s t - o r d e r  approximate s o l u t i o n  a s t h e K-B a v e r a g i n g  technique.  As a n e x a m p l e , c o n s i d e r t h e n o n l i n e a r m o d e l  ••  •  M ( x ) = x + f ( x ) = 0, x(0) = 1, x(0) = 0, i n w h i c h f ( x ) may be asymmetric, or b i a s s e d . A f i r s t - o r d e r R i t z approximate s o l u t i o n *x(t) = X  o  + A c o s wt i s assumed f o r t h i s m o d e l .  c o n d i t i o n s plus a c o n s t r a i n t from the i n i t i a l  solution.  Ritz  conditions  m i n e t h e p a r a m e t e r s X , A and co i n t h i s assumed q  The  deter-  approximate  These c o n d i t i o n s a r e X  Q  '+ A = 1,  2% M(x)d(wt) = 0  0 and 2at  M ( x ) c o s ( w t ) d ( w t ) = 0. y  0 Satisfaction  of these f u l l - p e r i o d averaging  integrals  on t h e  r e s i d u a l , M ( x ) , h e a r s no d i r e c t r e l a t i o n t o t h e e r r o r i n t h e approximate time s o l u t i o n . e(t)  I n t h i s work t h e a b s o l u t e  = x ( t ) - x ( t ) i s considered.  Another i n d i r e c t  error property  o f t h e R i t z method i s t h a t t h e f o r m o f t h e a p p r o x i m a t e solution  must he assumed.  Hence a n y f e a t u r e s n o t assumed i n  t h e a p p r o x i m a t e s o l u t i o n w i l l n o t be f o u n d . method y i e l d s n o n l i n e a r a l g e b r a i c e q u a t i o n s  Also, the R i t z f o r which the  6 solution i s d i f f i c u l t .  I f the ahove f i r s t - o r d e r s o l u t i o n i s  r e f i n e d t o the form x ( t ) = X  Q  + Acoswt + Bcos3oot,  then f o u r non-  l i n e a r equations i n X , A, B and co are obtained from the R i t z Q  conditions. In a recent paper  (Q) K  , Denman and L u i considered the • •  -i  approximate s o l u t i o n of the equation x + ax + bx"^ = 0 . n o n l i n e a r c h a r a c t e r i s t i c was expanded t o a l i n e a r i n terms of u l t r a s p h e r i c a l polynomials.  The  polynomial  The techniques g i v e n  i n Appendix A allow a closed-form s o l u t i o n to be w r i t t e n f o r t h i s c u b i c equation by i n s p e c t i o n . S o u d a c k ^ ^ has g i v e n techniques f o r the approximation1  of a n o n l i n e a r model i n the form of equation ( l . l ) , i n which the n o n l i n e a r c h a r a c t e r i s t i c i s a polynomial w i t h odd symmetry. The techniques r e p l a c e the n o n l i n e a r polynomial by a c u b i c one and the closed-form s o l u t i o n of the r e s u l t i n g d i f f e r e n t i a l equation i s obtained u s i n g the Jacohian e l l i p t i c f u n c t i o n s . Approximate models obtained from a Chehychev expansion  of the  n o n l i n e a r c h a r a c t e r i s t i c were found t o g i v e much b e t t e r a p p r o x i mate time s o l u t i o n s than models w i t h c u b i c c h a r a c t e r i s t i c s obt a i n e d by a l e a s t - s q u a r e e r r o r or Legendre polynomial f i t t o the n o n l i n e a r c h a r a c t e r i s t i c s . This work i n v e s t i g a t e s p i e c e w i s e - l i n e a r and p i e c e w i s e cubic approximations  to nonlinear c h a r a c t e r i s t i c s .  A direct  approach toward making the e r r o r i n the approximate c l o s e d form time s o l u t i o n s " s m a l l " i s undertaken.  In Chapter 2,  c r i t e r i a f o r the closeness of the approximate time are given,and  some e x i s t i n g approximation  solutions  techniques are  7 investigated under these c r i t e r i a .  The derivation of hounds on  the approximation error i s then made.  In Chapter 3 new approxi-  mation techniques which employ the Jacobi and shifted Jacohi polynomials are introduced.  The r e s t r i c t i o n that the nonlinear  o s c i l l a t i o n have a small amplitude or that the nonlinear t e r i s t i c be quasi^-linear i s not imposed.  charac-  8 2. 2.1  DEVELOPMENT OP THE APPROXIMATION TECHNIQUES  Considerations of Approximation Range and C r i t e r i a f o r Closeness of P i t i n the Approximate Time Solutions.  2.1.1  Determination of Bounds on the O s c i l l a t i o n and Normal i z a t i o n of the System Equation. The f i r s t integral of equation ( l . l ) i s a statement  of the law of conservation of energy f o r a conservative system with a single degree of freedom.  Writing equation ( l . l ) i n  the form x | | + f(x) = 0  x(0) = 1,  i ( 0 ) = 0,  (2.1)  and integrating, one obtains |^ +  ff(x)dx = 0  o  ,  or '2 2 - + V(x) = E..  (2.2)  The integration constant associated with this f i r s t  integral  may be evaluated from the i n i t i a l conditions given to equation (2.1).  In equation (2.2) the constant, E^, i s equal to the  i n i t i a l value of the potential function, V(l)«  The solution  of V(x) = E for  (2.3)  t  two r e a l roots, (x = 1, x = x . ), determines the turning '  *  mm  7  9  p o i n t s . ( l , x mm . ), of a bounded o s c i l l a t i o n for a normalized 7  d i f f e r e n t i a l equation with an integrable nonlinear characteristic, f(x). The above discussion assumes that there are at least two r e a l roots of equation ( 2 . 3 ) over the range of interest i n the  dependent variable, and that the solution to equation ( 2 . 1 )  i s bounded.  C u n n i n g h a m h a s given a technique which determines  the position and nature of singular points i n the phase plane from extrema of the potential function, Y(x).  A singularity  at a r e l a t i v e minimum of V(x) i s a centre point and the motion i s l o c a l l y bounded.  This "potential well" which exhibits a  l o c a l minimum i s the most common one encountered i n this study. A s i n g u l a r i t y at a l o c a l maximum of V(x) i s a saddle point and the motion i s l o c a l l y unstable.  At a point of i n f l e c t i o n of  V(x), the l o c a l behaviour of the s i n g u l a r i t y i s l i k e that of both a centre point and a saddle point and i s thus unstable. In attempting to solve a nonlinear problem, some knowledge of the kind of solution to be expected i s almost essential.  Using  the potential function thus provides useful information about the solution i n various regions of x even before a solution of the system equation i s attempted. No generality has been l o s t by f i x i n g the i n i t i a l conditions as i n equation ( 2 . 1 ) .  Consideration i s given to  bounded, periodic o s c i l l a t i o n s i n this work.  These o s c i l l a t i o n s *  may be started, a r b i t r a r i l y , at a point where x(t) i s zero hy making a s h i f t i n the independent v a r i a b l e .  One special case  of interest serves to i l l u s t r a t e the normalization which i s  carried out.  Consider a system which i s i n i t i a l l y  be acted upon by a step input at t = 0. + h(y) = kU(t),  at rest to  That i s ,  y(0) = y(0) = 0.  (2.4)  dt^ The bounds on the o s c i l l a t i o n are obtained by finding the r e a l roots of potential function V(y) =  / ( h ( y ) - k)dy +C . Q  The constant, C , i s obtained from the i n i t i a l conditions and o 7  the f i r s t i n t e g r a l .  In this case C  Q  must be chosen so that  V(0) = 0 and i t i s evident that y = 0 i s one bound on the S  oscillation. y  If the r e a l root of V(y) closest to zero i s  = \ and a minimum of V(y) i s enclosed on the i n t e r v a l (0,\),  then a bounded o s c i l l a t i o n limited hy zero and \ i s obtained.  y(t)  0  t  0  s Pigure 2.1  Normalization of an Asymmetric, Nonlinear O s c i l l a t i o n  11 The transformation y(t) = Xx(^)  along with a s h i f t i n the inde-  pendent variable, ^ = t - ^ , may he used to transform equation s (2.4) to one of the form 2 d?  2  + h(\x) -r = 0 r X  X  ac(S=0) = 1,  i(T=0) = 0. .(2.5)  Pigure (2.1) i l l u s t r a t e s how this normalization i s carried out. Equation (2.5) may now he written as ,2  + f(x) = 0  x(0) = 1,  which i s the form of equation ( l . l ) .  x(0) = 0,  (2.6)  Normalization to (0,1)  i s useful hoth for comparison of mathematical models on (0,1) and for comparison of errors i n the approximate time solutions. If the normalized f(x) i n equation (2.6) has odd symmetry on (-1, l ) and i s monotonically increasing on (0, l ) , then the o s c i l l a t o r y time solution i s symmetric with respect to the f i r s t quarter period ahout the x(t) = 0 axis.  It w i l l  he shown i n section (2.3) that the maximum error i n the approximate time solution over a fixed time i n t e r v a l depends on the maximum error, E(x) tion (2.6).  , i n the approximation to f(x) i n equa-  Because of the ahove symmetry i n the normalized  time solution, i t i s only necessary to make approximations to f(x)  on (x = 0, x = 1) for the f i r s t quarter period.  Taking  advantage of this symmetry i n f(x) thus seems an obvious choice because i t i s i n t u i t i v e that E(x)  _ i s smaller for a smaller nicix  approximation i n t e r v a l i n the x-f(x) plane.  The Ritz and  12 u l t r a s p h e r i c a l polynomial truncation techniques make approximations on (-1, l ) f o r this symmetric case. The smallest i n t e r v a l of symmetry f o r an o s c i l l a t i o n derived from a model with an asymmetric, nonlinear characteristic i s one half period*  A piecewise approximation to f(x) may  therefore he made on the range ( l , x the approximate  m i n  ) , and the remainder of  time solution may be obtained by symmetry with  this half period approximate  solution,  Pigure (2,4) i n section  (2.2) shows l i n e a r approximations to symmetric and asymmetric f(x). 2.1.2  C r i t e r i a f o r Closeness of F i t i n the Time Solution Approximations Por a model normalized to (0,1) the error i n the  time solution approximation i s defined by c(t) = x(t) - x ^ t ) . This error function i s not, i n general, obtainable because x(t) cannot always be found i n closed form.  An IBM 7040 d i g i t a l  computer was employed to obtain numerical approximate  solutions  to the nonlinear d i f f e r e n t i a l equations. - The numerical solutions were obtained using a fourth-order Runge-Kutta-Gill integration subroutine.  Direct comparison of the numerical  solutions with closed-form Jacohian e l l i p t i c function solutions for cubic models shows that the numerical solutions are accurate to s i x decimal d i g i t s . «  A technique suggested by  (12)  Proberg  v  ;  was also employed to check the accuracy«i  Using  t h i s technique, a change i n the numerical integration step size from 0.01 sec. to 0,001 sec. produced no change i n the f i r s t f i v e s i g n i f i c a n t d i g i t s of the solutions f o r the time  13  intervals which, were considered. A goal of this work i s to make the approximation error, e(t) as small as possible with a minimum amount of s  mathematical labor.  The choice of the smallest intervals of  time symmetry f o r the approximations i n section directed toward this goal.  (2.1.1)  is  The p r i n c i p a l error c r i t e r i o n  chosen i s that the error at the ends of ah interval,of approximation be small compared to the maximum error over the approximation i n t e r v a l .  Since the numerical solution and  the approximate closed-form solution are matched at t =0, the object i s to have the error small at the end of the f i r s t approximation i n t e r v a l .  This i s important because the so-  l u t i o n i s constructed by matching each successive, p a r t i a l approximate solution with the f i n a l value of the previous approximation. In section (2.3) i t i s shown that the maximum error i n the approximate time solution f o r a given model has an upper bound.  To j u s t i f y the choice of the above error c r i t e r i a ,  suppose that the error i s exactly zero at the end of a symmetric approximation interval, such as the quarter period.  Then, hy  symmetry, the bound on the error does not increase on subsequent approximation intervals as the solution i s extended i n time. Also, the frequency of the nonlinear o s c i l l a t i o n i s determined exactly.  2.1.3' Examples  of Some E x i s t i n g Approximation Techniques Under the Given Error C r i t e r i a — — — Approximate  solutions to the nonlinear d i f f e r e n t i a l  14 equation, x + x +  + 10x = 0,  x(0) = 1,  5  x(0) = 0, (2.7)  are carried out throughout the remainder of this chapter. The c h a r a c t e r i s t i c i n this equation i s grossly nonlinear f o r the larger values of |x| on (-1, l)» indicates the importance (a)  The following analysis  of the error c r i t e r i a i n section (2.1.2):  First-order Ritz and Linear  Approximations  The f i r s t - o r d e r Ritz approximate solution to equation (2.7) i s 'x(t) =  oosj"^^,  A l i n e a r , least-square  error f i t to the c h a r a c t e r i s t i c i n this model over the i n t e r v a l (0,1) yields the approximate equation 3E" + 9.043*' - 2.105 = 0,  'x(O) = 1,  Solution of t h i s equation gives 'x'(t) = 0 . 7 6 7  x^(0) = 0.  cos(3.007t)  + 0.233 which i s v a l i d f o r the f i r s t quarter cycle of the oscillation.  In Figure (2.2) these f i r s t - o r d e r  approximations  are plotted with the numerical solution f o r the f i r s t quarter cycle. (h)  Cuhic Polynomial Truncations The characteristic i n equation (2.7) i s now replaced  by odd-cubic characteristics obtained by a Chehychev polynomial approximation and a least-square error f i t .  Using the procedure  i n Appendix A, solution of the Chehychev model gives x(t)  = C n ( 0 . 7 7 0 3 , 3 . 3 7 3 t ) , and the least-square error model  gives'x(t) = Cn(0.7512, 3.2761), where Cn i s the Jacohian  15  xit),5«t>  F i g u r e 2.3  Comparison of Chebychev and Least-Square E r r o r Cubic ••  Approximations t o i'+ x.+  •*  c  x' + lOx-' = 0  16 elliptic  cosine function.  Curves of the error, e ( t ) , over the  quarter period f o r these closed-form approximate solutions are shown i n Pigure (2.3). (c)  A Two-term Ritz Approximation If a two-term Ritz approximation i n the form x(t) =  A coscot +B cos3<jot i s assumed, then the Ritz conditions give three algehraic equations i n A, B and co.  These ares  A + B = 1, -co A + A + |A 2  + ^A  5  = 0  5  (2.8)  and -9co B + B + |B 2  3  + ^°rB + ^ 5  + ^|A  5  = 0.  Solution of these equations i s d i f f i c u l t of values of A and B to he expected.  without- some knowledge  The error curves i n  Pigure (2.3) show the Jacohian e l l i p t i c  cosine to he a close (13)  approximation to the nonlinear o s c i l l a t i o n .  Soudack  v  suggested a device which f a c i l i t a t e s the solution of equations (2.8) f o r A, B and co.  The Jacohian e l l i p t i c  cosine solution  has approximately 5$ third harmonic, so the choice A = 0.95 and B = 0.05 i s made as a f i r s t guess i n equations (2.8). After a t r i a l and error procedure with various values of A and B, the two-term Ritz approximate solution i s found to he ~ ( t ) = 0.955 cos (2.624t) + 0.045 cos (7.872t).  17 (d)  Summary  Approximation  e  <° -I max }  f  £( }  Frequency £ 0-T) Error 'max v  Ritz First-Order  -0.06  +0.049  -0.25  Linear Least-Squares  -0.12  -0.10  +0.33  -8.2%  0.014  0.3%  -1.1%  Cubic Chebychev  0.0073  0.0036  Cubic Least-Squares  -0.014  -0.014  0.05  Two-term Ritz  -0.045  -0.045  +0.14  Table 2.  3.9%  -3.65%  Summary of Approximation Errors  The results are summarized i n Table 2.  Column three  of this table shows the maximum error over the f i r s t period. For  a l l the approximations considered, the error at the approxi-  mate quarter period, e(T/4), i s not small compared to the maximum r e l a t i v e error over the f i r s t quarter period.  Thus, the  numerical and closed—form solutions show a large phase difference after only one cycle of the o s c i l l a t i o n .  The cubic Chebychev  approximation gives an approximate time solution which comes closest to the error c r i t e r i a discussed above.  S t i l l , the  maximum error over the f i r s t quarter period i s 0.0073 and the maximum error over the f i r s t period grows to 0.014.  Column  four of the table- shows that the magnitude of the r e l a t i v e error i n the frequency of the approximation depends on the amplitude of the error at the approximate quarter period. Under the above error c r i t e r i a a poorer approximate  time  solution i s obtained from the two-term Ritz method than from either of the cubic polynomial approximations.  Also, the  labor for the two-term Ritz method i s considerably greater. The approximation techniques outlined i n Table 2 do not s a t i s f y the error c r i t e r i a which have been imposed because the error at the quarter period i s not small compared to the maximum error over the quarter period. placement  In Appendix C, r e -  of -an odd-symmetric, monotonically increasing charac-  t e r i s t i c i n equation ( 2 . 1 )  by an u l t r a s p h e r i c a l , Chebychev  l i n e a r approximation i s shown to give the same solution as the f i r s t - o r d e r Ritz method.  This equivalence of the Ritz average  over time with an orthogonal polynomial approximation i n the x - f(x) plane provides motivation f o r an investigation of the approximating properties of orthogonal polynomials more general than the ultraspherical Chebyohev polynomials. 2,2  Use of the Jacobi and Shifted Jaoobi Polynomials f o r Approximation" "~" " The object of the remainder of this chapter i s to  give insight into the r e l a t i o n between the approximation of the nonlinear characteristic i n the x versus f(x) plane and the r e s u l t i n g error i n the approximate then to obtain approximate  time solution.  The goal i s  time solutions which w i l l s a t i s f y  the error c r i t e r i a imposed i n paragraph ( 2 . 1 . 2 ) .  To achieve  these ends, the Jacobi and shifted Jacohi polynomials are chosen f o r the piecewise approximation of the characteristic i n the nonlinear model.  Approximations more general than  those obtained from Chebychev and Ritz approximation techniques may be obtained using these,polynomials. The expansion of a function which i s absolutely  19 integrable^ ^  i n terms of a s e t of polynomials, P^(x), o r t h o -  1  gonal with weight  f u n c t i o n , W(x), on an i n t e r v a l  (a,h) to a  polynomial of degree, n, has the form n  ¥ ( x ) =X]°k k > P  (x)  (2  '  9)  k=0 where h  £ f(x)P,(x)W(x)dx c,_ = ^r; — — [P (x)J  f  k  a The  shifted  (2.10)  W(x)dx  J a c o h i polynomials are orthogonal on ( 0 , 1 ) w i t h  r e s p e c t to the weighting f u n c t i o n W(x) = (l-x)^ "*" x "^. -  the u l t r a s p h e r i c a l is  (l-x )^ 2  - 1  a-  For  J a c o h i polynomials the weighting f u n c t i o n  and the i n t e r v a l of o r t h o g o n a l i t y i s ( - 1 , 1 ) .  Thus n o r m a l i z a t i o n of the o s c i l l a t i o n s to a maximum amplitude of u n i t y i s necessary f o r the expansion these polynomials.  of f ( x ) i n terms of  D e r i v a t i o n s and closed—form expressions  f o r these polynomials are g i v e n i n Appendix B. Only the s h i f t e d  J a c o h i polynomials, G r ^  a ,  ^(x),  considered f o r the l i n e a r or f i r s t - o r d e r approximation models i n t h i s work. are c a r r i e d  out.  F i g u r e ( 2 . 4 ) shows how these  F o r the symmetric c h a r a c t e r i s t i c  ( 2 . 4 a ) , the l i n e a r i z a t i o n i s on the i n t e r v a l R i t z and u l t r a s p h e r i c a l interval  (0,1).  polynomial approximations  (-1,1) f o r t h i s symmetric case.  and symmetric n o n l i n e a r c h a r a c t e r i s t i c s  a  r  e  of the  approximations i n Figure In c o n t r a s t , are on the  For hoth asymmetric an asymmetric  linear  20  d i f f e r e n t i a l equation of the form x + b^x' + b x(0) = 0, i s obtained on (0,1).  (a)  Symmetric  Pigure 2.4  = 0, 'x(O) = 1,  The solution to this equation i s  (h)  The Approximation  Q  Asymmetric  of Two Nonlinear Characteristics  For the approximation  of the nonlinear characteristic hy  a cubic characteristic, both the ultraspherical Jacohi and shifted Jacohi polynomials are employed.  The ultraspherical  Jacobi polynomials, P ^ ^ ^ ( x ) , may he used to approximate oddsymmetric c h a r a c t e r i s t i c s . equation x + t^x + b^x^ = 0.  This approximation produces the From t h i s cubic model with zero-  point symmetry, the closed—form solutions may be written by inspection using the techniques given i n Appendix A.  21 Approximation i n terms of the shifted Jacohi polynomials yields d i f f e r e n t i a l equations with asymmetric, cuhic characteristics of the form x + b,x^ + b x 0  + h-,x + t> = 0 .  These models are obtained from the refined approximation on (0,1) of a characteristic with odd symmetry on (-1,1), or from the approximation of an asymmetric (0,1).  characteristic normalized to  Distinct from the l i n e a r case, the asymmetric cuhic  d i f f e r e n t i a l equation requires somewhat more labor to obtain the closed form approximate solution than does the odd-symmetric, cubic equation.  Nevertheless, the techniques i n Appendix A  may be applied directly, and the labor i s less than that required for the two-term Ritz method shown i n paragraph (2.1.3) f o r an odd-symmetric c h a r a c t e r i s t i c .  Examples of the closed-form  solution of d i f f e r e n t i a l equations with cubic characteristics are given i n Chapter 3. Erom the expressions for an orthogonal expansion given i n equations (2.9) and (2.10), a l i n e a r approximation to a given f(x) i n terms of the shifted Jacobi polynomials may  be  written  *<x) = f f e f ^ x [ E [P<« V * +  ([fl {  |~\B)  +  a  ftL _ x  0  3  - f *>]  Sk);  (2.11)  8  i s the gamma function. 1  (  Also i n this expansion  22  and 1 f(x) x ( l - x ) ? " dx. a  1  0 For arbitrary a, 8 and f(x) the integrals R and S do not, i n general,have closed-form solutions*  One special case  has been found, however, i n which the expansion technique i n equation (2.9) i s general. If f (x) = x , q  q + <x> 0,  then R and S reduce to the form 1  0 which i s the integral f o r the Beta function.  This result allows  one to make approximations to classes of hardening and saturating characteristics which have q > l and q < l respectively.  The  method i s quite general because q need not be an integer. When f(x) i s i t s e l f a polynomial, a truncation or an expansion i n terms of the shifted Jacobi polynomials may he made using Lanczos' Economization^ -^. 1  Lanozos^ ^ 1  shows that  this truncation technique gives i d e n t i c a l coefficients to the expansion determined by equations (2.8) and (2.9) when orthogonal polynomials are used f o r the truncation. To i l l u s t r a t e the procedure, consider the truncation of  f(x) = x + 3x  (2.12)  5  to a l i n e a r polynomial on (0,1) using the shifted Chehychev polynomials, G^®'^'  For simplicity, the superscript  ^*^\x).  are dropped and the complete expansion of (2.12) may  he  written f(x) = c G ( x ) + c G ( x ) + c G ( x ) + c & (x). Q  o  1  1  2  2  5  3  A l i n e a r polynomial i s desired so G^(x) and G (x) are used 2  successively to obtain expressions f o r the cubic and the quadratic i n terms of a l i n e a r polynomial. 2 + 18x - 48x  From G^(x) = -1  3 + 32x , one obtains  x  3  = ^ | (1 - 18x  + 48x ) + 2  ^|G (X)J 3  therefore, x + 3x  3  = ^  - z=j*x + |x  + ^ | G^(x),  2  Similarly, x  2  = |(-1  + 8x)  + |G (X) 2  ;  hence, + 3x  5  = - | | + f|x +  For a l i n e a r truncation & (x) 2  Hence, ?(x)  =  - i |  +  § x .  T|  G  2  ( X )  +  3§ 3 G  ( x )  -  ( 2 , 1 3 )  and G^(x) are set equal to zero.  24  Carrying through the orthogonal polynomials to equation (2.13) i n the expansion makes the d i s t r i b u t i o n of the error i n the function approximation e x p l i c i t .  Since the maximum value of  G-g(x) or (^(x) i s . standardized to be unity, the maximum error produced by this truncation on (0,1) i s E __(x) = 9/l6 +  3/32  = 21/32, and i t occurs at x = 1 where the polynomials have a maximum o s c i l l a t i o n . The d i s t r i b u t i o n of the error i n a convergent  expansion  can be predicted roughly from the f i r s t term neglected.  The  f i r s t term neglected i n (2.13)» c G ( x ) , o s c i l l a t e s with six 2  times the amplitude of c^G-^x).  2  Therefore, the error on  behaves l i k e & (x) with three, near-equal error maxima.  (0,1) The  2  two shifted Jacohi polynomials i n Figure (2.5) o s c i l l a t e with a larger value near x = 1 than near x = 0 on t h e i r range of orthogonality.  Thus, the d i s t r i b u t i o n of the error i n the  x - f(x) plane f o r a l i n e a r approximation behaves l i k e (n ^ n ft "\ &2  *  (x), and the d i s t r i b u t i o n of the error i n a cubic  polynomial approximation behaves l i k e Gr^®'^' ^ * ^ ( x ) with four, unequal error maxima.  Since the exact and approximate  time solutions are matched i n i t i a l l y where x = 1 and x = 0, h e u r i s t i c arguments j u s t i f y v a r i a t i o n of the error i n the approximation of the characteristic near x = 1 i n order to obtain an improvement i n the time solution at some point such as the approximate quarter period.  Change i n the weighting  function of the shifted Jacobi polynomials allows this v a r i a t i o n i n d i s t r i b u t i o n of the error over the orthogonal range i n the x - f(x) plane.  (0,1)  25  1.0  0.5-  0.0-  -0.5-  x  -1.0 0.0  1.0  0.5  Pigure 2 . 5 A Plot of Two. Shifted Jacohi Polynomials 2.3  Determination of Upper Bounds on the Error i n the Approximate Time Solution A method was developed hy Soudack ( 1 7 )  which gives a  rough upper hound on the error i n the time solution approximation from the error i n a cuhic polynomial function approximation to a polynomial of higher degree i n the model.  A similar  approach i s used here to determine the upper hound when a l i n e a r approximation i s made to a higher degree polynomial. For simplicity replace 'x hy z, then the approximate equation may he written as z +f(z)  = 0  z(0) = 1,  z(0) = 0,  and the o r i g i n a l d i f f e r e n t i a l equation may he written i n the form  26  x + f(x) + B(x) = 0  x(0)  = l  r  x(0) = 0,  where x(t) = z(t) + e(t) and E(x) = f(x) - f ( x ) . Note that e(t) i s the error i n the normalized time solution (which i s not i n general obtainable i n closed form) and E(x) i s the error i n the function approximation. * .  WW  Now,  WW  WW  X = Z + £ = - f(z) + £ = - f"(z + e) - E ( z + e);  hencej  £ = 1 (z)  - *±(z + e) - E(z + e),  For the l i n e a r approximation "f (z) = b  Q  + b-^,  we have E  = b  Q  + b^z - b  Q  - b.^(z + e) - E(z + e)  = - b ^ - E(z + e). The d i f f e r e n t i a l equation f o r the error i s thus "E + b £ 1  = -E(z  + E)  E(0) = E(0) = 0 .  (2,14)  27  The i n i t i a l conditions are obtained as shown because the numerical solution and approximate solution are matched  initially.  The solution to equation ( 2 , 1 4 ) has the form e(t)  = e - e cos(y"b~^ t ) . p  (2,15)  p  The p a r t i c u l a r i n t e g r a l , E , i s determined from equation ( 2 , 1 4 ) using convolution (18), that^i s v  ;  1 E  P  [B*0)]  " b-,  s i n ( t - 8) d6,  where E (t) i s the function obtained by replacing z and £ by their respective time functions.  Since a bound on the error  i s of interest here, consider  1 E  P  ~  E  lb  s i n ( t - 8) dB.  (8)  0  Now  E  (0,1).  (8)  i s bounded because f(x) and f ( z ) are bounded on  Let|B*(t)|  m a 3 c  = |E(Z  found from f(x) and f'(x).  + e)|  m a x  = EMAX,  EMAX  can be  Thus, the bound on the particular  i n t e g r a l now becomes  EMAX  J  sin  (t - 0) dp.  0  Since the sine function i s less than or equal to one, for a time  28 i n t e r v a l At, we have  Ip I - fiff * e  ^  At  S p e c i f i c a l l y , the time i n t e r v a l , At, w i l l he an approximate quarter period or an approximate half period depending on the i n t e r v a l over which the truncation i s made. From equation (2,15) one obtains |e(t)|^|e |-|l -  cosj^  p  ^- 2 EMAX  At.  This r e l a t i o n determines an upper hound on the error i n a f i r s t order approximate time solution from the error i n the x - f(x) plane matching of the c h a r a c t e r i s t i c . Using a similar method for a cubic polynomial truncation f (z) = t> + b^z + b z 0  2  + b^z .  (19) Soudack^  has shown that a rough upper bound f o r the error i n  the time solution approximation i s  These results are f o r extreme upper hounds because the above approach i s a pessimistic one i n which maximum or worst  possible errors .are considered.  In practice, the actual  error i s much smaller than that predicted by equations (2,16)  29  and (2*17).  As an example, consider the l i n e a r , least-square  error approximation to equation (2*6) i n paragraph  (2.1.3).  The upper hound predicted for the f i r s t quarter-period byequation (2.16) i s E M J U C  =  A t  0.70.  Prom Table 2 the actual maximum error obtained over the f i r s t quarter period i s e(t)  = -0*12.  The value of the above bounds on the error for approximations to polynomial characteristics arises from a proof by Soudack^ ^. 2  It was  shown that |e(t)| i s bounded when the I I max  polynomial' characteristic i s truncated down to a cubic polynomial using the shifted Chebychev polynomials.  Below, this  r e s u l t i s extended to the expansion of a polynomial i n terms of the orthogonal Jacobi or shifted Jacobi polynomials a l i n e a r or a cubic polynomial. P  nW  = " on " ^ n * A  +•  •  -  V  *  *  to either  Let  V-Dn^  +  V**  ( 2  '  1 8 )  represent an orthogonal Jacobi or shifted Jacobi polynomial given i n Appendix B.  The c o e f f i c i e n t A ^  i s positive i f n-m  even; the signs have been chosen as shown for convenience. for  B  n-2  convenience  -  A  (n-2)n  ^(n-1)  (n-l)  as is Also,  3.0  and  A  B  n-m  (n-m)n B  T  (n-2)n (n-m)(n-l) AA (n-l)(n-l) A  B  (n-2) (n-3)(n-2) A  '  A  '(n-2) (n-2)  T  •  •  •  (n-m+1) (n-m) (n-m+1) A  A  (n-m+1)(n-m+1)  are defined. Application of Lanczos' Economization to the nonl i n e a r c h a r a c t e r i s t i c f(x) =  y | Si**  1  v  i  e  l  d  s  p-1  V ^ '  *(x) =  where n-p  n  B  A., +  E  k=p  (k-m) .i(k-m) A  (2.19)  (k-m)(k-m)  A  m=l  The error i n this approximation'is E(x) = f(x) - f ( x ) . -In terms of the nonlinear c h a r a c t e r i s t i c and the orthogonal polynomials  n E  A  ak  n ^  A  kn k ) P  ( x  (2.20)  kk P  In.both ( 2 . 1 9 ) and ( 2 . 2 0 ) ,  p = 2 f o r a l i n e a r approximation and  p = 4 f o r a cuhic polynomial approximation. Let PM^ he the absolute value of the k^* Jacohi or 1  31 shifted Jacohi polynomial at the point x  ffi  maximum o s c i l l a t i o n . EMAX.  where E(x) assumes a  Erom equations (2.16), (2.17),  (2.19) and (2.20) n  n  2At  L  n  max  E  a^ +  Z  7—  I—I D  A  kk  i  PM  v n-2. k k B  E  *11  *kk  Hxi  7—  k  J (k-m) l(k-m) A  A  m=l  (k-m)(k-m)  Expanding the summations one obtains  |e(t)| ^ I 'I max  a  2At  v  a  l  n l ^ H  +  a  A q r  n l ^ K  ^  + a  A q r  n-l 2 ^  ^  H  + a  A q r  ^ * * p n--p+l ^ + >  + a  H  n - l 2 ^ l ) +• • • + K  A (  r  a  p  K  A q r  ^  n-p+l ^  A q r  ^  (2.21) This error bound depends only on the c o e f f i c i e n t s , a^., i n the polynomial which i s being approximated,the c o e f f i c i e n t s , Aqr, of the orthogonal Jacobi polynomials and the time i n t e r v a l over which the approximation i s made.  If a^ gets very large i n  the polynomial being approximated, then from equation (2,21)  *<*>|»«-~ 2 A t which i s constant. a  k-l  £  a r e  : m c r e a s e  ^lmax—  d,  If more than one of the a^ such as a^ and then ' ^ A - k + i ^ ^ ^ ^ k - A - k ^ ^ ^ >+* • -  2  A  t  •^n-k+l  ( A  *  r }  + a  k-l n-k 2 K  +  { A  *  r ) +  '''  32  a  2At  k-l  Vk+l * -^f" ( A  r )  H  n-k+2(Aqr)  and this error hound i s also a constant as a^. and a _]_ get very k  large.  These results apply to both l i n e a r and cuhic approxi-  mations to f(x),  and they may he extended inductively to the  case where more than two of the a^. grow large.  The form of  these error hounds depends on the fact that hoth the truncation of f(x) by Lanczos' Economization and the maximum error in  the truncation have the same dependence on the c o e f f i c i e n t s ,  a^..  Thus, when the quotient i s taken i n equation (2,21), the  dependence on the a^. i s removed, i n the l i m i t , as the a^. grow large» Prom these results i t follows that i f the maximum error i n the approximate time solution over a given time i n t e r v a l i s bounded, then the error at any point on the approximate time solution, such as at the approximate quarter period, must also be bounded.  The approximate time solutions  which have been obtained i n this work agree with these error bounds.  As an example, consider the nonlinear model which i s  given i n equation (2.7), that i s x + x + x?  + 10x  5  = 0,  x(0) - 1,  x(0) = 0, (2,7)  If  the amplitude of the o s c i l l a t i o n i s now doubled, then the  transformation x = 2y normalizes this equation to unity  33  amplitude i n the form y + y + 4 y + l60y 3  5  = 0,  y(0) - 1,  y(0) - 0.  Similarly, i f the amplitude of the o s c i l l a t i o n i n (2.7) i s halved, the normalization yields w + w + 0.25MJ  where x = 0*5w.  + 0.156w = 0, 9  w(0) = 1 ,  w(0)  = 0,  Clearly, these normalizations show considerable  differences i n the characteristics as x(0) i s varied.  Other  normalizations similar to these have been made on equation (2.7), and then f i r s t - o r d e r approximations to the models obtained have been carried out using the one-term Ritz method and the ro c ,n R> (0,5?0»56) / x shifted Jacobi polynomials G ^ ' (x), G u o  k  and G  ( 0 k  u o ;  w  k  ' '°' (x). 5  6 )  Figure (2.6) shows that both the maximum errors, e(t)  m a x  , over the f i r s t quarter period and the errors, e(T/4),  at the approximate quarter period are bounded as the amplitude of the nonlinear terms i s increased hy increasing x(0). The saturation of the error at the approximate quarter period occurs at a lower value f o r the nonsymmetrically-weighted,  shifted  Jacobi polynomial approximations than f o r the u l t r a s p h e r i c a l , shifted Chebychev or one-term Ritz method approximations.  When  x(0) = 1, as i n equation (2.7), f o r example, the errors at the approximate quarter period are 0.049 and -0.004 f o r the one(Cl  term Ritz and  R  0 f)  ' * '(x) approximations respectively.  Extension of these solutions to a f u l l period by symmetry yields the maximum error of 0.25 f o r the one-term Ritz solution,  34  • (t) max  .04  h  (i) (ii) (iii) (iv)  Ritz method a = B = 0.5 a = 0.5, P = 0.56 a = 0.5, P = 0.6  35 and 0.08 f o r the (j ( ° " ' 5  0 , 6  )( ) x  approximation.  A numerical so-  l u t i o n of equation (2.7) gives a derivative at the quarter period of -2.1981.  The r e l a t i v e errors i n the derivative at the  approximate quarter periods are - 2 9 % f o r the f i r s t - o r d e r R i t z approximation and -13% f o r the Gt^®*-**®*^ (x) approximate tion.  solu-  Also, from a numerical solution a quarter period of  0*5777 seconds i s obtained.  The approximate  quarter periods ares  0,553 s e c , a r e l a t i v e error of 3 - 9 % and 0,5728 sec,, a r e l a t i v e (o R n f) error of 0.8% from the Ritz and mations, respectively.. Figure ( 2 . 6 )  ? r  ;  ( x ) polynomial approxi-  The improvement over the Ritz method i n  i s mainly i n the phase or frequency.  This i s  because the errors at the approximate quarter periods saturate at considerably smaller values f o r the weighted  approximations.  The saturation of the maximum error over the f i r s t  approximate  quarter-period does not show such.large differences. In Figure ( 2 . 6 ) ,  the near-uniform, shifted Chebychev  approximation i n curve ( i i ) shows the effect of an approximation on the i n t e r v a l (0,1).  The near-uniform Ritz or Chebychev  approximation on (-1,1) does not account f o r the symmetry i n the quarter-period. Thus, E(x) obtained from the match to ^ ' max f(x) hy the Ritz or Chebychev approximation on (-1,1) i s larger than E(x)  from the, shifted Chebychev approximation on (0,1),  As a r e s u l t , both e(t)  and e(%/4) i n curve ( i ) are-larger  for the Chebychev approximation than f o r the shifted Chebychev approximation shown i n curve ( i i ) .  The improvement obtained  using the nonsymmetrically-weighted  shifted Jacobi polynomials  shows, however, that near-uniform or Chebychev matching i n the  Je(t)|  max P = 7,  o = 0.5,  P - 0.63 ,  (0.084)  p = 5,  g = 0.5,  p  (0.07)  p = 3,  g = 0.5,  p r= 0.56,  (0.052)  .02  P = k,  o = 0.5,  P = 0.44,  (0.041)  .00  H  .10 a  0.6,  .08  .06  .04  —1  1  10  o.i  h  10  1-  s  10  5  (a)  :(T/4)  p = 7,  a = 0.5,  P = 0.63,  (0.084)  p = ^  g = 0.5,  p - 0.44,  (0.011)  p = 5,  o = 0.5,  P «= 0 . 6 , .  (0.06)  p = 3,  a s 0.5,  0 = 0.6,  (0.03)  .01  .0075  .005  .0025  -t-  ,000 0.1  10  t  1  P-  (b)  F i g u r e 2.7 ' S a t u r a t i o n o f the Maximum and Quartey P e r i o d E r r o r f o r L i n e a r Approximations t o x + nxP = 0 x(0)  =  1,  x(0) =  0.  37 x - f(x)  plane does not provide the hest approximation to the  nonlinear amplitude-frequency relationship under the error c r i t e r i a which have been imposed. Now consider maximum and quarter-period values of  e(t)  for l i n e a r approximations over the f i r s t quarter period to equations of the form  x + nx  = 0,  p  In this case, x(0)  x(0) = 1,  x(0) = 0.  i s fixed at unity and different values of n  and p are used. In Pigure {2.1),  curves showing the errors for various  values of p and for different l i n e a r shifted Jacohi polynomial approximations are plotted versus an increase i n n i n the characteristic.  For c l a r i t y , the errors i n the Ritz approxi-  mations, which are much larger than those plotted i n Figure (2.7) 9 have been placed i n brackets on the curves.  The quarter-period  amplitude errors are 0.010, 0.03, 0.06 and 0.09 for the Ritz approximation to models with p equal to one-third, three, and seven, respectively.  five  These error curve's do not show the  same increase toward saturation as those i n Figure (2.6). is because the characteristic i n equation (2.7)  This  has a l i n e a r  term present which becomes dominant when the effect  of the non-  l i n e a r terms i s made small by decreasing the i n i t i a l amplitude. As predicted by equation (2*21), however, the curves i n Figure (2.7)  demonstrate that the error i s bounded as the  of the nonlinear terms increase.  coefficients  38 3.  APPLICATION OP THE APPROXIMATION TECHNIQUES TO SPECIFIC NONLINEAR MODELS  3.1  Techniques for a Class of Odd-Symmetric Nonlinear Characteristics  3.1.1  D e f i n i t i o n of the Nonlinear Factor and Choice of Weighting Functions Characteristics with odd symmetry on (-1,1) and with  a monotonic increasing property on (0, 1) are often called hardening when f"(x)>0 and softening when f"(x)<0. of characteristics i s approximated  i n this section.  This class For p a r t i -  cular cases, the variable weighting of the error i n the x-f(x) plane afforded hy shifted Jacobi polynomial expansions has been used i n the previous chapter.  These weighted polynomial  approximations have yielded approximate small error i n the quarter period.  time solutions with a  Extension of this weighting  property to more general characteristics i n this class requires a means for specifying the deviation from l i n e a r i t y of d i f f e r e n t , normalized characteristics on the i n t e r v a l (0,1).  For this  purpose, the nonlinear factor  1 f  2  p h  1  f (x)dx  2  f(l) i s defined.  (3.1)  Pigure (3.1) shows graphically that F i s the  difference between the area under the curve and the area under the chord on (0,1), r e l a t i v e to the area under the chord.  In  value, the nonlinear factor l i e s between -1 and +1 on (0,1). For hardening characteristics F> 0 and for softening charac-  39 t e r i s t i c s P<0.  fU) f(l)  hardening softening  0  1  x  Pigure 3 « 1 A Graphical Representation of the Measure of Departure from Linearity Now consider v a r i a t i o n of the parameters a,8 and u i n the weighting functions W 1 (x).=  x^a-x)^" ) 1  (5.2)  and W (x) = ( 1 - x ) ^ - , 2  (3.3)  1  2  according to the value of P.  A derivation hy Denman and L u i  v  shows that Chehychev approximation to "quasi-linear" f(x) yields approximate time solutions with a small r e l a t i v e error i n the period.  Chebychev approximations are obtained when  a = 6 = |i = 0.5.  Thus, the weighting has been varied starting  from this case with "quasi-linear" characteristics.  Heuris-  t i c a l l y , the i n i t i a l matching of the exact and approximate time solutions at x(0) = 1 and x(0) = 0 allows variation of the  40 error i n the approximation to f(x) near x = 1.  What i s more  important, as the amplitude of the nonlinear terms  increases,  the deviation from l i n e a r i t y i s most severe for the larger values of x on the i n t e r v a l (0,1).  Prom equation  tion of B i s seen to have the greatest effect near x = 1.  (3.2),  varia-  on the weighting  Setting a = 0.5 makes the weighting close to that  obtained from a shifted Chebychev truncation for the small values of x on the i n t e r v a l (0,1),.  This choice i s  reasonable  because Chebychev approximations i n the x-f(x) plane give the closest approximate time solutions for small values of F or for "quasi-linear"  characteristics.  Particular values of B and \i have been chosen empiric a l l y from error saturation curves similar to those i n Figures ( 2 . 6 ) and ( 2 , 7 ) but with the nonlinear factor, P, as the abscissa. Approximation with different weighting using the shifted Jacobi poly-  Nonlinear Measure p£  -0.40  0.37  0.35  -0.25  0.44  0.42  -0.25 ^ P ^ +0.25  0.50  0.50  ^ P^  0.60  0.56  0.58  F ^  0.60  0.63  0.65  - ..40^ P ^  0.25  Table 3.1  3  The Choice of Weighting for the Jacohi and Shifted Jacobi Polynomial Approximations  nomials has been shown to produce large differences  1  i n the error  41 at the approximate quarter period.  Hencej considerahle improve-  ment i n the amplitude-frequency relationship has been found possible by choosing the weighting according to the value of E given i n Table (3°l)°  Prom a model with a saturating or s o f t -  ening characteristic, the frequency decreases as the i n i t i a l amplitude increases.  In contrast, from a model with a hardening  characteristic, the frequency increases with the amplitude of the o s c i l l a t i o n .  Variation of the weighting i n a l i n e a r approxi-  mation varies the slope and thus the approximate frequency changes.  Hence, variations of p or [i are opposite for models  with softening and hardening c h a r a c t e r i s t i c s . 3.1.2  Some Particular Cases The nonlinear factor i s now applied to weighting the  error i n Jacobi and shifted Jacobi polynomial expansions.  In  Tables (3.2) and (3«3) the maximum error over the f i r s t quarter period and the error at the quarter period are given for approximate solutions to the equations x + f(x) = 0, x(0) = 1, x(0) = 0.  The nonlinear functions i n Table (3.2).have a l l been  truncated down to an asymmetric l i n e a r polynomial on (0,1) using Lanczos' Economization and the shifted Jacobi polynomials. I n i t i a l l y , either a five-term Taylor series expansion to a ninth degree polynomial or a four-term Legendre polynomial expansion to a seventh degree polynomial i s used to obtain polynomial approximations to the functions i n Table (3.2). The Legendre approximation i s carried out for the grossly nonlinear functions marked by asterisks because the Taylor expansion converges slowly for these functions.  The maximum error i n these pre-  42 liminary Taylor and Legendre polynomial expansions was kept below 0.5% of the value of the function at x = 1.  Under this  c r i t e r i o n , the l i n e a r shifted Jacohi polynomial truncations carried out i n the x - f(x) plane agree with corresponding l i n e a r expansions obtained by numerical integration of equations (2.9) to three s i g n i f i c a n t figures.  Pigure 3.2  A Comparison of the Error i n Two Linear Approxi-  ^  **  I  mations to x + /|x|  sgn x = 0  The models with characteristics of'the form x  1//q  -,  and with q equal to 2,3,5, and 7 are given to show the behaviour of approximations to models with softening c h a r a c t e r i s t i c s . Rigorously, these models are not Lipschitz at x = 0, hut this non-Lipschitz character at x = 0 w i l l not be present i n any  43  Table 3.2  Comparison of the First-Order Ritz Method and Linear Jacobi Polynomial Approximations Ritz  Nonlinear F  Characteristic  e  Shifted Jacobi  Method  ^ m a x e(T/4)  8  e(t)  m a x e(T/4)  0.17 -0.0170 +0.0027  0.50  -0.015  0.0005  0.39  -0.0390 +0.0160  0.56  -0.041  -0.0063  0.55  -0.0560 +0.0380  0.56  -0.060  +0.0040  3 5 x+x^+10x  0.59  -0.0610 +0.0500  0.56  -0.065  +0.0100  3 5 x+4x +I60x^  0.66  -0.0680 +0.0660  0.63  -0.090  -0.0080  x+9x^+810x  0.66  -0.0700 +0.0670  0.63  -0.090  -0.0080  x+0.25x +0.31x +0.l6x  0.26  -0.0250 +0.0064  0.50  -0.022  +0.0015  3 5+x7 x+x-'+x  0.48  -0.0480 +0.0270  0.56  -0.051  -0.0016  0.67  -0.0680 +0.0690  0.63  -0.090  -0.0040  0.73  +0.0970 +0.0970  0.63  -0.100  +0.0060  0.11  -0.0120 +0.0011  0.50  -0.009  -0.0010  x+0.75x  0.21  -0.0220 +0.0040  0.50  -0.018  -0.0003  x+3x  0.38  -0.0380 +0.0140  0.56  -0.039  -0.0070  0.46  -0.0480 +0.0260  0.56  -0.049  -0.0040  x+1000x  0.50  -0.0520 +0.0300  0.56  -0.053  -0.0010  x  0.34  -0.0340 +0.0100  0.56  -0.032  -0.0070  0.43  -0.0440  0.0190  0.56  -0.044  -0.0050  0.50  -0.0520 +0.0300  0.56  -0.053  -0.0010  0.67  -0.0700 +0.0600  0.63  -0.090  -0.0080  0.75  +0.0840 +0.0840  0.63  -0.100  +0.0090  sinh (x)  0*10  0.0073 +0.0005  0.50  -0.006  -0.0009  sinh (2x)  0.24  -0.0240 +0.0060  0.50  -0.020  +0.0004  x+0.l6x +0.256x 5  3 5 x+x^+x "5 5 x+x+5x-^ y  5  5  5  5  7  x+x +10x +10x 5  5  7  x+x +10x +100x 5  5  X+0.3X  5  5  3  x+12x  3  5  2  x  sgn x 2 e  ^ sgn x  7  44  Table 3.2  (Continued)  Nonlinear Characteristic  Ritz F  £(t)  Method  m a x e(T?4)  Shifted Jacobi 0  e(t)  max  £  (T/4)  sinh  (Ax)**  0.48  -0.0520 +0.0320  0.56  -0.0540 +0.00050  sinh  (Ax)**  0.67  -0.0660 +0.0640  0.63  +0.0900 -0.00900  tan (x)  0.21  -O.0180 +0.0033  0.50  -0.0174 -0.00006  tan  (1.3*)**  0.45  -0.0410 +0.0210  0.56  -0.0440 -0.00500  tan  (l.5x)**  0.62  +0.0840 +0.0840  0.63  -0.0900 +0.00100  -0.11  +0.0100 +0.0050  0.50  +0.0060 +0.00100  -0.33  +0.0280 +0.0060  0.44  +0.0120 +0.00300  -0.33  +0.0290 +0.0060  0.44  +0.0120 +0.00300  -0.50  +0.0410 +0*0120  0.44  +0.0150 +0.00500  ^0.67  +0.0520 +0.0170  0.37  +0.0100 +0.00015  tanh (x)  -0.10  +0.0140 +0.0016  0.50  +0.0110 +0.00300  tanh (2x)**  -0.38  +0.0380 +0.0080  0.44  +0.0230 +0.00400  2x-0.5x  -0.17  +0.0150 +0.0020  0.50  +0.0130 +0.00300  -0.27  +0.0250 +0.0040  0.44  +0.0170 +0.00140  -0.26  +0.0230 +0.0040  0.44  +0.0150 +0.00030  |x|°' sgn x 8  yjxT sgn x 2y|x| sgn x x  5  p 2x-0.9x . sgn- x sin(l.5x)**  45 physical problems or i n any d i g i t a l simulation. • As an example from Table (3.2), the f i r s t - o r d e r approximation of x sgn x = 0- i s shown i n Pigure (3.2).  +7|x|  The error over the f i r s t  period i s shown f o r the one-term Ritz approximation and the shifted Jacobi polynomial expansion, Q (0*5,0.44)(x). Approximation to the c l a s s i c a l pendulum model, x + sin(l.5x) =.0, i s the f i n a l example i n Table (3.2).  An ana-  l y t i c a l solution to this model may be obtained i n terms of the Jacohian e l l i p t i c functions but the l i n e a r i z a t i o n i n Table (3*2) i s given to show an example of the improvement i n the shifted Jacobi truncation technique over the Ritz averaging method. Pigure (3.3) compares results f o r the r e l a t i v e error i n the frequency or period.  One-term Ritz and l i n e a r shifted  Jacohi approximations to the equation x + x + xr + lOx x(0) =1, x(0) = 0, are shown i n this figure.  = 0,  The normalization  with changes i n x(0) has been carried out as described i n paragraph (2.3). 0.06 "  0.04 "  0.02 -  :(0)  -0.01 Figure 3«3  The Relative Error i n the Frequency f o r Linear Approximations to x + 1 + x x(0) = 0  ;  + lOx  = 0, x(0) = 1,  46 In Pigure (3«3)  the G ^ ' ^ a  (x)  approximations have a  smaller error than the Ritz method approximations.  This means  that the shifted Jacohi polynomial approximate solutions do not go out of phase as rapidly as do the Ritz approximations. In Table (3.3), the.functions to which Lanczos' Economization does not apply have been expanded to cubic polynomials by numerical integration of equation (2.9). the  This allows  behaviour of the amplitude-frequency approximations to non-  l i n e a r models with characteristics more general than polynomials to be studied under cubic, shifted Jacobi polynomial Pigure (3.4)  expansions.  gives the r e l a t i v e errors over the  f i r s t period f o r three cubic polynomial truncations to the model x + x + x? + lOx the  = 0, plotted f o r the f i r s t period.  Under  error c r i t e r i a of this work, the time solutions obtained  from the ultraspherical, Jacobi approximations P^^'^^x) and p^(0.58)^ ^ x  Q n  (.^i)  a r e  poorer than the refined, shifted  Jacobi, G, (0.5,0.56)^ closed-form time solution.  Also,  Table (2.1) shows that over the f i r s t period f o r this same model, the maximum error i n a two-term Ritz approximation i s 0.14, and the error i n a cubic, least-square error approximation i s 0.05.  Por the cubic (j (0.5,0.56) ^  a  pp  r o x  imation in  Pigure (3.4), the maximum error i s -0*0028 over the f i r s t period.  This l a t t e r approximation also shows that the error  i n the extended solution does not grow large with respect to the  error over the f i r s t quarter period because the error at the  approximate quarter period i s +0.0005. As predicted by the r e l a t i o n obtained between e(t)  Table 3.3  Cubic Polynomial Approximation of the Nonlinear Characteristics  Nonlinear  Chebychev P  Characteristics + 0.25x + 0.3125X  Ultraspherical Jacobi U ^  e(t) 'max K  Shifted Jacobi  e(T/4)  P  E  (t)  m a 2  E  (T/4)  0.21  +0.00080  +0.00030  0.50  0.00080  +0.00030  0.50  -0.00020  .0.00000  5  0.39  +0.00190  +0.00090  0.58  -0.00200  -0.00009  0.56  -0.00050  +0.00000  + 0.562x  0.51  +0.00500  +0.00240  0.58  -0.00400  +0.00060  0.56  -0.00100  +0.00020  0.58  -0.00750  +0.00360  0.58  -0.00500  +0.00100  0.56  -0.00120  +0.00050  0.66  +0.01000  0.00470  0.65  -0.00700  +0.00200  0.63  -0.00180  +0.00006  0.66  +0.01000  0.00480  0.65  -0.00720  -0.00010  0.63  -0.00180  +0.00013  0.25  +0.00140  +0.00600  0.50  +0.00140  +0.00600  0.50  -0.00050  +0.00018  0.48  +0.00480  +0.00250  0.58  -0.00400  +0.00040  0.56  -0.00130  +0.00020  0.67  +0.01300  +0.00700  0.65  -0.00900  +0.00030  0.63  -0.00300  +0.00040  X  5  5  X  X  3 +  X^  +  XT  + 3.1641X  5  5  X  + x  + 10x  X  + 4x  X  3 (5 + 9x + 81 Ox  5  5  + I60x  5  5  3  x + 0.25x + 0.31x +0.15x 5  5  7  3  X  +  X  + x  x  y  5  +  X  5  +  X  7  + 10x + 10x 5  7  Table 3.3  (Continued) Chebychev  Nonlinear P  Characteristics  Shifted Jacobi  Ultraspherical Jacobi U ^  e( t) 'max v  e(T/4)  B  £  ^max  0.73  +0.02100  +0.01000  0.65  -0.0130  +0.0040  0.63  -0.0043  +0.00100  0.67  +0.01000  +0.00450  0.65  -0.0074  -0.0002  0.63  -0.0019  +0.00015  0.71  +0.01600  +0.00740  0.65  -0.0100  +0.0007  0.63  -0.0033  +0.00080  0.75  +0.02400  0.01100  0.65  -0.0137  +0.0024  0.63  -0.0047  +0.00180  sinh(x)  0.10  -0.00003  0.00000  0.50  -0.00003  0.00000  0.50  0.0000  0.00000  sinh(2x)  0.24  -0.00400  +0.00013  0.50  -0.00400  +0.00013  0.50  -0.0001  +0.00003  sinh(4x)  0.48  +0.00440  +0.00240  0.58  -0.00360  +0.00050  0.56  -0.0010  +0.00028  tan(x)  0.21  -0.00070  +0.00020  0.50  -0.00700  +0.00020  0.50  -0.0002  +0.00004  tan(l.3x)  0.45  -0.00430  +0.00350  0.58  -0.00570  +0.00250  0.56  -0.0023  +0.00040  tan(l.5x)  0.62  +0.03800  +0.02900  0.65  -0.02100  +0.00700  0.63  -0.0100  +0.00700  -0.00800  -0.00310  0.42  -0.00540  +0.00200  0.44  +0.0009  -0.00040  x + x x  5  5  x^ x  + 10x + lOOx  5  +  x  7  7  x 1/3 '^  7  -0.50  Table 3.3  (Continued) Chebychev  Nonlinear Characteristics  F  e(t)  Ultraspherical Jacobi  e(*A)  H  +0.00004  e(t)  eCT/4)  0.50  max -0.00022  +0.00004  Shifted Jacobi 0  e(t)  e(T?4)  tanh(x)  -0.10  max -0.00020  tanh(2x)  -0.38  -0.00340  -0.00130  0.42  -0.00240  +0.00090  0.44  -0.00030  -0.00012  -0.67  -0.01100  -0.00500  0.42  -0.00760  +0.00180  0.44  -0.00100  -0.00050  X  V5  0.50  max -0.00009  -0.00002  Figure 3 . 4  Comparison of Solutions to Models Obtained from Chebychev, U l t r a s p h e r i c a l Jacobi and Shifted Jacobi Approximations to x + x + TL + 1 0 x J  5  = 0  51 and E(x)  _ i n equations (2.16) and (2.17), the time solution  errors f o r the cuhic function approximations i n Tahle (3.3) are much smaller than errors from the l i n e a r approximations i n Tahle (3.2).  This i s because the cubic i s always a closer f i t  to f(x) than i s the straight l i n e i n the x - f(x) plane. Weighting the convergence i n the x - f(x) plane according to the value of P i n Table (3.1) has resulted i n improvement over the c l a s s i c a l Ritz-averaging and Chebychev polynomial approximations.  Particular examination of the refined, shifted  Jacobi approximations i n Tables (3.2) and (3.3) shows that the  error at" the approximate quarter period i s less than 10%  of the maximum error over the f i r s t quarter period f o r most of the examples considered. 3.2  Techniques for Models with Asymmetric Nonlinear Characteristics When the nonlinear characteristic does not have the  symmetry possessed by those considered i n paragraph (3.1), the o s c i l l a t i o n often has a large dc component.  The procedure  given i n paragraph (2.1.1) permits determination of the range of o s c i l l a t i o n of the dependent variable and normalization of the i n i t i a l conditions to x(0) = 1 and x(0) = 0. formation x = (l-x . )w + x . mm  (l, x  m i n  The trans-  i s used to transform the range-  mm  °  ) to the range (w = 1, w = 0).  Otherwise, the weighting  of the error i n a shifted Jacohi polynomial approximation i n the x - f(x) plane w i l l not be controlled.  The nonlinear charac-  t e r i s t i c now has the form g(w) = f ( | ~ l - z . ~|w + x . ) . An °  •«  L  m  mmj  mm  approximate solution to the model w + g(w) = 0 i s v a l i d f o r the  52  f i r s t half period* Approximations to the characteristics i n this asymmetric class have heen attempted using the shifted Jacobi polynomials.  Some models y i e l d approximate solutions which  show improvement over the near-uniform, shifted Chebychev approximations.  Notably, the characteristics i n these models  have terms of even symmetry which are small compared to the amplitude of the odd-symmetric terms. , For other characteristics which have s i g n i f i c a n t asymmetries,  a common property which would  allow nonsymmetrically-weighted approximations to be used has not been found.  Thus, near-uniform or near equal-ripple shifted  Chebychev l i n e a r and cubic expansions are considered f o r models with asymmetric, nonlinear c h a r a c t e r i s t i c s . 3.2.1  First-Order or Linear Approximations The Ritz-Chebychev equivalence has been shown f o r  models with symmetric  characteristics.  Linear, shifted Cheby-  chev polynomial approximations to equations with  asymmetric  characteristics normalized to (0,1) give similar results to the two-^term Ritz method discussed l n paragraph ( 1 . 2 ) .  The d i s -  t r i b u t i o n of the error over the f u l l period shown i n Figure (3.5)  compares these two techniques.  i n this case i s x + x + x  Our asymmetric  example  + 3x^ = 0, and i t s range of o s c i l -  l a t i o n i s (x = 1, x = - 1 , 1 7 6 ) .  The two-term Ritz approximation  is  ¥ + 3.542x + 0.335 = 0 ,  53  Figure 3.5  Error i n the Ritz and Shifted Chehychev Approxi*2 3 mations to x + x + x + 3x = 0  while direct truncation of the nonlinear c h a r a c t e r i s t i c i n terms of the shifted Chehychev polynomials gives \  + 3.512x' + 0.328 = 0.  In general, the shifted Chehychev polynomial truncation technique i s preferred over the two-term Ritz approximation because the predetermination of x j_ makes the Chebychev approximate m;  n  solution closer to the numerical half-period than does the Ritz approximation technique. Figure (3.5).  This property may he observed i n  Also, the Ritz method requires the solution of  nonlinear, algebraic equations to determine the parameters i n the assumed solution.  54 3.2.2  Cubic Approximations to Asymmetric Models When the normalized characteristic, g(w), i s approxi-  mated hy a cubic polynomial on (0,1), the techniques given i n Appendix A may be used to obtain a olosed-form time solution. As an example, oonsider the approximate step response of a system which i s i n i t i a l l y at rest and i s described by the model x + 10x + x 5  + x = U(t),  3  U(t) i s the unit step function. and the f i r s t  x(0) = x(0) = 0.  (3.4)  Prom the i n i t i a l conditions  integral  |- + 1.6667X + 0.25x + 0.5x - x = 0, 5  5  2  one obtains (x = 0, x = 0.7825) as the range of the o s c i l l a t i o n . The transformation x = 0.7825 w gives the normalized model w + 3.7492w +' 0.6l23w + w — 1.278 = 0, 5  3  w(0) = w(0) = 0,  with (w = 0, w = l ) as the range of the dependent variable. Lanczos  1  Economization  of the characteristic i n this equation  using the shifted Chebychev polynomials on (0,1) plus a s h i f t in the independent variable, defined by t = t + t , produces the approximate equation 2/v  + 11.1569w - 8.7872w + 2.9771w - 1.3490 = 0, 3  d^  $(0) = 0,  2  w(0) = 1,  (3.5)  55 The f i r s t integral of (3.5) i s + 2.7892w - 2,9290w + 1.4885w - 1.3A90v = 0.  h—) d£ 2  4  3  2  d  Integration of this f i r s t integral gives 5  w dt = -  I  0  1  dw. y~w(5.5784w - 5.8582w + 2.9771w - 2.6980) 5  2  hence,, ¥  t  T  £ T + T/2 = ' A  I  Q  dw y-w(5.5784w - 5.8582w + 2.9771^-2.6980) 5  2  (3.6) In general t " £ t, hut the shifted time, t , i s equal to the r e a l time, t , i n this special case because the o s c i l l a t i o n starts one half period before the f i r s t maximum and thus, t  s  = T/2. The procedure described i n Appendix A may be used  to transform the integrand i n equation (3.6) to the form of the E l l i p t i c  Integral of the F i r s t Kind.  Then a closed-form  expression f o r 'w(t) may be obtained from Table A according to the parameters i n this e l l i p t i c  integral.  In t h i s example,  both w = 0 and w = 1 are turning points of the motion,so both zero and one are roots of the polynomial i n equation (3.6). After the roots are found, p = 0.3673 and q = -1.3843 are obtained from Appendix A.  The transformation w = (p + qy)/  yj applied to equation (3.6) gives  56  1 . 3 4 8 4 dt = 0 . 3 3 1 2 -  Choosing h  2  I4.2029y )(i + 2  = 1 4 . 2 0 4 9 and g  4.i273y )  = 4 . 1 2 7 3 so that c  2  2  N  2  = g /h < 1 , 2  2  and making the transformation v = hy, gives the form dv  1.6828 dt =  (3.7)  « (1 - v ) ( l  + 0.2906v )  2  2  A  which corresponds to entry II i n Table A.  Prom equation ( 3 . 7 )  and Table A, the modulus of the e l l i p t i c i n t e g r a l i s k = ./c / ( l + c ) ' / 2^ is k  =J  1 - k  = 0 . 4 7 4 5 , and the complementary modulus  = 0.8802.  In the form of the e l l i p t i c  inte-  g r a l , equation ( 3 . 7 ) becomes 0  1.6828 t =  j  0.8802 | | •  0 In this case, to obtain a closed-form time solution, the important function i s 0(t) rather than t ( 0 ) .  Under the trans-  formation v = cos 0 f o r inversion of the e l l i p t i c  integral  given i n Table A, the Jacohian e l l i p t i c cosine function, Cn, i s defined.  That' i s ,  1.6828t 0.8802 = 1.9117 t = C n ( v ) . _1  Thus, v(t) = Cn(k,cot) = Cn(0.4745, 1.9117t).  Retracing through  the transformations which have been made, the closed-form solution for  i s tf(t) =  P  i  57 0.5675 L l •- CnLQ.4745. 1.9117t)1 (3.8)  1 + 0.2653 Cn(0.4745, 1.9117t)  Figure (3.6) compares the closed-form, shifted Chehychev polynomial approximate solution given i n equation (3.8) with an .010  6(t)  .005 .000  /\  2.0  -.005 Least Square  Error—-  -.010 -  -.015 -  \  -.020  Figure 3.6  The Error Distribution i n Shifted Chehychev and Least-Square Error Cubic Approximation to x + x + x  5  + lOx  5  = U(t)  approximate solution obtained from a cubic least-square error approximation to the characteristic i n this same model.  The  d i s t r i b u t i o n of the error over the f i r s t period of the o s c i l l a t i o n i s shown i n this figure.  In comparison, a l i n e a r shifted  Chebychev approximation to this model gives a maximum error of 0.16  over the f i r s t half-period and this grows to -0.26  over the  58 f i r s t period. The o r i g i n a l normalization, x = 0.7825w, was  performed  on equation (3.1) to change the range of o s c i l l a t i o n to (0,1). Therefore, from equation (3.8), 3  (  t  )  0.2874 [ l - Cn(0.4745. 1.9117t)1  =  1 + 0.2653Cn:(.0.4745, 1.9117t) i s the shifted Chebychev, cubic approximate solution to x + x + x? + lOx  = U(t), x(0) = x(0) = 0.  This approximate  o s c i l l a t i o n i s on the range (0, 0.7824). 3.3  An Extension of lanczos Economization to the Transient Response of lightly-Damped Models 1  It has been shown that the amplitude-frequency r e l a t i o n i n a conservative, nonlinear o s c i l l a t i o n need only be determined over an i n t e r v a l of symmetry.  Nonlinear, non-  conservative o s c i l l a t i o n s show a more interesting, continuous change i n frequency as the amplitude of the o s c i l l a t i o n i s damped.  Based on the Ritz - K-B equivalence for symmetric,  conservative models and the improvement i n phase obtained from shifted Jacobi polynomial truncations, an extension of these nonsymmetrically-weighted x + 2$x i s now  approximations to models of the form  + f(x) = 0,  x(0) = 1,  i(0) = 0  undertaken. The K-B averaging method requires f(x) to have an  e x p l i c i t l i n e a r term and imposes c r i t e r i a f o r the "lightness" of damping and f o r the "quasi-linear" nature "of the c h a r a c t e r i s t i c .  59  The parameters a(t) and 9 ( t ) , given i n Appendix C, f o r the change i n amplitude and phase, respectively, the f i r s t cycle of the o s c i l l a t i o n .  are assumed constant over In t h i s work, the exponential  decay predicted hy the f i r s t - o r d e r approximate solution i n equation ( C l ) of Appendix C i s assumed, and a direct  piecewise-  l i n e a r i z a t i o n of the characteristic i n terms of the shifted Jacohi polynomials i s carried out.  No r e s t r i c t i o n i s placed  on the change i n frequency of the nonlinear o s c i l l a t i o n over the f i r s t cycle or on the presence of a l i n e a r term i n f ( x ) .  Only  f(x) with an odd-symmetric, monotonic—increasing property are considered for the approximation hecause the nonsymmetricallyweighted approximations have been applied  only to this class.  Also, the K-B approximation of the form x'(t) = a(t)cos (w t + 9(t)),. applicable  to f(x) = w x.+ ^ig(x), would not be expected to f i t  an asymmetric o s c i l l a t i o n . To show the application consider approximate solutions x + 0.4x  + x + x  + 5x  5  5  of the l i n e a r i z a t i o n procedure,  to the equation  = 0,  x(0) = 1,  x(0) = 0. (3.9)  The K-B approximate solution to this equation i s x'(t) = e - ° *  2 t  cos (^4.875 ' t ) .  (3.10)  This solution i s v a l i d f o r the f i r s t cycle of the o s c i l l a t i o n . As shown i n Table (3.2), the characteristic i n equation (3.9) has a nonlinear factor F = 0.55.  A Lanczos  1  Economization of  this characteristic i n terms of the shifted Jacohi polynomials,  60 &  (0.5,0.56)( ) x  %±  + 0A\  f  o  n  ( 0 , 1 ) yields the l i n e a r  + 5.782x - 0.900 = 0,  approximation x^O) = 1,  1  x^O) = 0.  Hence the approximate solution, v a l i d f o r the f i r s t quarter-cycle, is x ^ t ) = 0.847e~"°'  2t  cos (2.40t -0.083) + 0.153. (3.11)  Prom equation (3.10) an approximation to the f i r s t minimum i s x . = -0.76. mm (3.9)  A truncation of the characteristic i n equation ^  on (0, -0.76) i n terms of the G ^ * ' * ) (x) polynomials 0  5  0  56  yields the approximation ¥  + 0.4x  2  2  + 2.64X2 + 0.340 = 0,  x (0.78) = 0, 2  •x (0.78) = -1.66. 2  The i n i t i a l conditions f o r this equation are obtained from equation (3.11) when "x^(t) = 0. The approximate solution f o r the next half-period on (0, -0.76) i s thus ~ (t) 2  = 1.19e"  0,2t  cos ( l . 6 l t - 2.70) - 0.13. (3.12)  The approximate solutions 'x^(t) and 3f (t) are v a l i d f o r the 2  range 0 ^ t ^ 2 . 9 3 sec. and this i s three quarters of the f i r s t cycle.  Using the amplitude decay predicted by the K-B averaging  method, the approximate amplitude after one cycle i s x  = 0.56.  The normalization of equation (3*9) from the range (0, 0.56) to the range (0,1), under the transformation x = 0.56y, gives  -1.00  Pigure 3.7  linear Approximations to x  + 0.4x + x +  +  = 0  62  y + 0.4y + y + 0 . 3 l 4 y  + 0.492y  3  5  (3.13)  = 0.  As the o s c i l l a t i o n i s damped, the amplitude of the nonlinear terms decreases and the value of F, whioh has been d e f i n e d on  (0,1), a l s o decreases. (3.13) has P = 0.28.  The normalized characteristic i n equation A f t e r trunoating t h i s oharaoteristic on  (0,1) i n terms of the G ^ ' ' " ) (y) polynomials, and then 0  5  0  56  changing back to the range (x = 0, x = 0.56), we obtain ¥  5  + Q.4x  3  + 1.08x - 0.091 = 0, 5  x^(2.93) = 0,  x ( 2 . 9 3 ) = 1.00 5  as the third piecewise approximation, v a l i d f o r the next half cycle.  The K-B solution and the above piecewise-linear solution  for equation (3.9) are compared i n Pigure ( 3 . 7 ) . In Pigure ( 3 . 8 ) , piecewise-linear approximations to e•  o  x + 0.2x + tanh(2x) = 0,  o  x(0) = 1,  x(0) = 0,  have been carried out i n terms of the shifted Jacobi polynomials, 0^(0,5,0.44)(x).  The characteristic tanh(x) arises physically as  a model f o r nonlinear barium t i t i n a t e capacitor characteristics (22) '.  For this equation the K-B method does not apply because an  e x p l i c i t l i n e a r term i s not present.  Instead, lanczos' Economi-  zation of a least-square error, seventh-degree polynomial approximation to tanh(2x) i s carried out f o r each l i n e a r i z a t i o n .  To get  an approximation to the amplitude at each half cycle of this damped o s c i l l a t i o n , a l i n e a r Chebychev polynomial truncation i s made on (-1,1), and then the solution of the resulting  x(t),x(t) I . 00  0.75 Numerical Solution-  0.50  0.25  -I—  0.00 1.0  2.0,  •0.25  5.Q  4.0  /  5.0  Piecewlse-Linear  0.50  Approximat ion  0.75 •1.00  Figure 3.8  The Piecewise-linear Approximation of •• • x + 0.2x + tanh(2x) = 0  6.0  t  64  approximation i s used to determine x  m i n  .  This technique gives  an approximate amplitude of x ^ = -0,717 after one half-cycle m  and an amplitude of x  n  = 0.538 after a f u l l cycle.  Linear  truncations i n terms of the shifted Jacohi polynomials, (0.5, 4 4 ) ( ) ^ are then made f o r the f i r s t quarter-cycle 0 ,  G  x  and f o r each successive half cycle. as f o r the example i n equation (3.9). the numerical and approximate  The procedure i s the same Figure (3-8) compares  solution.  In the above two examples, the f i r s t piecewise-linearization i s over only the f i r s t quarter cycle of the o s c i l l a t i o n . This i s useful because the nonlinear terms i n the characteristic have a large magnitude where the amplitude of the o s c i l l a t i o n i s greatest, and thus a small approximation i n t e r v a l i s important. The piecewise-linearization also allows d i f f e r e n t , weighted approximations to be made as the o s c i l l a t i o n damps out. The p o s s i b i l i t y of using cubic truncations to improve the approximation has not been explored.  Only a f i r s t - o r d e r  approximation to the decay i n amplitude of the nonlinear o s c i l l a t i o n has been found. are  Therefore the Jacohian e l l i p t i c functions  not considered i n the approximation of non-conservative  oscillations. 3.4  Discussion and Possible Extension of the Results An advantage of the orthogonal polynomial truncation  techniques given i n this chapter i s that they are generally applicable to a wide range of both conservative and nonconservative models f o r common physical systems.  Also, the  65  refinement of the approximate model from a l i n e a r one t o a cubic one i s direct, and the same techniques are employed for symmetric,  asymmetric and non-conservative models.  In contrast,  *he c l a s s i c a l averaging techniques refer i n d i r e c t l y to the residual error obtained when the assumed solution le substituted into the model.  Refined solutions are also d i f f i c u l t to obtain  from the c l a s s i c a l techniques. Application of polynomial truncation to f i r s t - o r d e r d i f f e r e n t i a l equations was considered. Jacobi approximate  solutions, improved  In some cases, weighted, over those obtained by  near-uniform expansion of the characteristic, have been found tut no r e l a t i o n to the weighting of the truncated approximation to a second-order model has been noted.  This i s because the  same arguments f o r a small error at the end of an i n t e r v a l of symmetry do not apply to f i r s t - o r d e r systems.  For this reason,  approximation of f i r s t - o r d e r systems has not been studied i n this work. Extension of this work to second-order models driven by harmonic time-functions i s also possible.  Results obtained  (23)  by Klotter^ '' show that the backbone of the nonlinear amplitudefrequency response curves i s obtained by setting the driving term to zero. improved  Hence, approximations to the backbone curve,  over the one-term Ritz method, may be obtained d i r e c t l y  from the results of this work.  The arguments for the use of  symmetry and the insight gained into weighting the approximations ln this work could be investigated for the symmetric, state response.  steady-  Also, the error c r i t e r i o n on the error at the  66  end  o f a n i n t e r v a l o f symmetry i s i m p o r t a n t f o r t h e  state approximation.  N e a r r e s o n a n c e , where t h e a m p l i t u d e  the o s c i l l a t i o n i s l a r g e , important.  the e f f e c t  approaches  of the  the backbone curve near r e -  Thus, a n i m p r o v e m e n t o v e r t h e c l a s s i c a l o n e - t e r m  method c o u l d , p e r h a p s ,  of  of the n o n l i n e a r terms i s  Por l i g h t l y - d a m p e d models, the amplitude  steady-state response sonance.  steady-  be s i g n i f i c a n t u s i n g d i f f e r e n t  i n Jacobi polynomial approximations.  Ritz  weighting  67  4.  CONCLUSIONS  Piecewise-linear and piecewise-cubic approximations, from which a n a l y t i c a l solutions to classes of second-order nonl i n e a r d i f f e r e n t i a l equations may be obtained, have been developed i n this work.  I n i t i a l l y , c r i t e r i a for the error i n the approxi-  mate time solutions were imposed.  The introduction of Jacohi  and shifted Jacobi polynomials has given a f l e x i b i l i t y to the approximation techniques which i s not possessed by the c l a s s i c a l approximation methods. Error bounds were then given which prove that the maximum r e l a t i v e error i n the solution to an approximate Jacobi or shifted Jacobi model i s hounded, no matter how large the coefficients i n the o r i g i n a l nonlinear model become. An empirical measure of the departure from l i n e a r i t y , based on these error bound results, permitted approximations to be made using the shifted Jacobi polynomials, G r ^ * ^ ' ^ ( x ) . Approxi0  mate time solutions f o r which the r e l a t i v e error at the approximate quarter period i s of the order of 10% of the maximum r e l a t i v e error over the quarter period have been obtained. Thus, i t has been possible to obtain quantitative approximations to the nonlinear amplitude-frequency relationship.  Unrefined  approximations using the ultraspherioal Jacobi polynomials allowed closed-form solutions to be written f o r symmetric, cubic models by inspection. To provide a general approach f o r the approximation of asymmetric,  conservative models the shifted Chebychev  polynomials, G-,^*-' ^'-^(x), were employed. ,  This approximation  68  technique was found to provide improved solutions over the one-term Ritz method i n the f i r s t - o r d e r approximation case, and over the two-term Ritz method and least-square error approximation i n the cubic case. Improvement i n the amplitude-frequency approximation obtained from the K-B method has been shown possible f o r second-order nonlinear models with l i g h t , viscous damping. Again the G r ^ ^ " ' ^ (x) polynomials have been used f o r a direct, piecewise—linearization of the characteristic i n the nonlinear model. The polynomial truncation techniques have provided a direct attack on the approximation problem.  The results  which have been obtained using these techniques show improvement over the c l a s s i c a l averaging methods with a decrease i n tedious labor.  69 APPENDIX A The  Closed Form Solution of Second-Order, Conservative System Models with Cuhic, Nonlinear Characteristics For the special case i n which the cuhic characteristic  has zero-point symmetry, Soudack^ ^ has derived 24  closed-form  solutions i n terms of the Jacohian e l l i p t i c functions.  The  cases of interest are: (i)  x + ax - bx The  = 0,  5  x(0)  = X ,  x(0) = 0,  a,h>0.  solution i s x(t) = X Sn(k,cot + K(k)) for  X  Q  where co = a - 0.5  hX  2  and k  2  «7  SO  ( i i ) x + ax + bx-  = hX /(2a -  ^ ,  hX ).  2  x(0) = X ,  = 0,  5  2  Q  2  •  x(0) = 0,  a,b<0. 2  The and k  2  2  solution i s x(t) = X Cn(k,cot), where co = a + Q  = bX /2(a +  bX  Q  bX ).  2  2  ( i i i ) x - ax + bx-  x(0) = X ,  = 0,  5  x(0) = 0,  a,b>  0.  An o s c i l l a t o r y solution, symmetric i n the origin, i s obtained i n the form of case ( i i ) when a i s small.  For  a<bX^/2, the solution i s x(t) = X Cn(k,cot), where co = -a + o  and k  o  o  o  = bX^/2(-a + bX^).  An o s c i l l a t o r y solution for larger p  a has the form x(t) = X Dn(k,Lot), where Q  co  = bX /2. Q  0<X <JlsJ^ Q  The  conditions  and X  bX  k  ?  = 2(1 - a/bX ) and Q  on the i n i t i a l amplitude are  £Ja/b.  1  Q  When the nonlinear c h a r a c t e r i s t i c does not have the above symmetry the closed-form solution may  not be written  by  Q  70 inspection.  The system model now has the form  2  M f + b + b,w rg2 o 1  + b w 2  rt  + b,w 3  2  = 0,  3  0  d  w(0) = 1,  w(0) = 0. ( A . l )  The range of o s c i l l a t i o n of w i n t h i s model w i l l always he (0,1) a f t e r the n o r m a l i z a t i o n s desoribed i n paragraphs and (3.2) are c a r r i e d out.  Prom equation  (2.1), the  (2.1.1)  first  i n t e g r a l of the motion i s  1(^)2  =  E  _  +  k E  t  (  v  +  b  w£  i  £  ^  +  +  ^ wi  )  ( A > 2 )  - V(w).  The normalized  initial  c o n d i t i o n s of ( A . l ) g i v e E^ = V(l), and  hence & = -/2 d£ from (A.2).  [ V ( l ) - V(w)] ^  (A.3)  The minus s i g n i s chosen from p h y s i c a l c o n s i d e r a -  t i o n s because w decreases as t i n c r e a s e s from zero f o r t h i s bounded o s c i l l a t i o n .  1  I n t e g r a t i o n of (A.3) g i v e s  dw  w  2 0  1  or  [ V ( l ) - V(w)]  = =E + At =  (A.4)  * w  t'  O^w^l x  J  ,  -  dw  _>>  I 2 [ V ( l ) - V(w)]  V  O^w^l, '  .5)  \k.\  71 where  r = -  1  At  0  dw  .  ^2  [V(l) - V(wf)  •  N  (A.6)  The quantity, At, obtained by dividing the range of integration in equation (A.4), i s either one quarter period or one half period depending on the symmetry of the problem being considered, Equation (A.6) i s a complete e l l i p t i c  integral of the f i r s t  (25)  kind. in  A technique given by Hancock  v  allows the integrand  (A,5) to be transformed to the Jacohian e l l i p t i c Integral  of the f i r s t kind using Legendre's transformation, and then a r  closed-form expression for w(t ) maybe obtained. of  Evaluation  (A.6) i s not necessary because the quarter period or half  period may be obtained from w(t ). The integrand i n (A.5) may be written i n factored form as  dw  dw  (A.7) J'  ^  [ ( x - 9 ) ( x - \ ) (x-u.) (x-rcfj^  It i s clear from (A.2) and (A.3) that x = 1 i s a root of the denominator polynomial i n (A.7).  Hence, this polynomial can'  always be reduced to a cubic and the roots may be found. roots i n (A.7) are ordered so that 9 >A> roots ordered f i r s t .  where  u> x with the r e a l  Legendre's transformation i s  w = (p + q y ) / ( l + y),  The  72 9\(u_ + JC) - urt(9 + \ ) p  q  Under this  9 + X  =  -  JJ  -  £  transformation dw _ (g - p) _  ^ ~  dy  "  sH"  ^  Q  )  ±nV)(«2 ±eV)'  i s ohtainedj where m  2  r  = (p - 9 ) (p - \),  2 = (p -  n  = (q - 9 ) (q - X),  2  2 = (q - £)(q -  \x) (p - «),  s  K).  For the special case i n which 9 + X = LI + it the transformation w = y + ( 9 + X,)/2 = y + (u + rc)/2 i s used.  It i s shown hy  Hancock that p + q and pq are always r e a l .  Prom equation  (A.7), equation (A.5) may now he written i n the form dt  (q  = mv  J  - p)dy  .  ±(1 ± g y ) d ± h y ) 2  2  2  2  On the right hand side of this expression,  V  h and g are defined  so that h> g and then the integrand may he further reduced to dt' =  (q - p ) mrh/ +(1 + ^ 1 N  d v  v )(1 + v ) 2  dv; f  2  :  v  (A.9)  '  +(1 + c v ) ( l + v ) 2  2 where v = hy and c  2  2  2 2 = g /h <1.  Hancock shows that N i s always 2  '^2 \  rof e a lsign . It i s also of the eight possible combinations under the rshown a d i c a lthat i n equation (A.9), /-(1 + v ) ( l + c'"v )  73  may be neglected, since W, which i s positive for some of the o r i g i n a l w, cannot be transformed to a function which i s always negative by a r e a l substitution. Using Table A below, closed-form expressions may be obtained for v(t ) according to each of the seven possible sign combinations j  i n equation (A.9).  ~2—\  In this table  A0 = /1 - k s i n 0 , where k and 0 are defined as the modulus and the amplitude  of t  respectively.  The trigonometric sub-  s t i t u t i o n s i n Table A define the Jacohian e l l i p t i c functions.. The quantity k i s defined as the modulus of the e l l i p t i c function, and 0 i s defined as the amplitude of the e l l i p t i c function.  For example, the substitution v = tan 0 ^ Tn(Nt )  = Tn(k,Nt ) for entry I i n Table A defines the Jacohian e l l i p t i c tangent function from v t' _ i ~ N  f  y  & (i + v ) d + 2  AV  The complementary modulus i s defined to b e k  =  Entries VI and V i a i n Table A have the same form. v ^ l , while i n VIa,v^^;  In VI,  The e l l i p t i c integrals of the f i r s t  kind and the e l l i p t i c functions are obtainable from works such as the "Smithsonian E l l i p t i c Function Tables".^ ^ 26  74 TABLE A Standard Forms f o r Jacohian E l l i p t i c  Integrals  of the F i r s t Kind _ dj \ ~ AG  dv 2  2  dv  II  . •* d0  2  dv  III  2  v = cos 0  A0  y (l-v )(l+cW  k^ = 1-c'  v = tan 0  y(l+v )(l+o v ) 2  _ kdj  v = sec 0  k  c 1+c' £  =  1+c'  y(v -l)(l+c v ) 2  2  2  _ z^dj  dv  IV  v -  cos 0 c  V  sec 0 c  y(l+v )(l-c v ) 2  2  2  _ k d0  dv  V J  2  c  =  v = sin 0  \ ~ AC  k  1 1+c' 2 c 1+c'  2  2  dv  VI y  =  k  (l+v )(c v -l) 2  1  =  2  (l-v )(l-c v ) 2  2  2  dv  Via  y(v -l)(c v -l) 2  2  y(v -l)(l-c v ) 2  A  •\  Ai  2 2 k^ = c^  2  dv  VII  \"  v = c sin 0  2  v^ = s i n  2  2  i2  k  , 2 = 1-c  0+cos 0 2 2  k  2  = 1-c  2  75 APPENDIX B Derivation of the Jacohi and Shifted Jacohi Polynomials (27) Lanczos gives a development of the Jacohi polynomials from the hypergeometric © a-x) ^ ' ' F(\ A  w  - 1 + ^x  ~  a , X j  + +  a  series  Va(a  X(  '  x  -  1)Q(Q + 1) + l)»l-2  x  2  \(\ + + 2 ) 9 ( 3 + l)(© + 2) a(a + 1 ) ( a + 2) • 1 - 2 • 3  This series terminates with the power x choice o f © = n + a + 8 - l  3  x  & eo•  i f X = -n.  The  yields the set of orthogonal  polynomials G  ^(oc,3)(x)  =  p  (_n>  n  +  which are orthogonal on (0,1) factor ¥(x) = x ^ d  a  +  p  _  l f  a  .  x  )f  with respect to the weight  - x)^" .  In this study the shifted  1  Jacohi polynomials have been formed hy standardizing the above  - 1.  polynomials so that G r ^ ^ ( l ) a ,  n  9  + cc + 8 -  =n  l , 8  = 8 + n  Defining the quantities  and a  = a + n, the f i r s t  f i v e shifted Jacobi polynomials may be written = 1  G (x) 0  Cr.  i ( x )  G  2  ( x )  a [.!  =  aa.,i \ r = 88 ! L  G (x) = 3  p p  9  -i ]  +  x  1  .  29 n  !!a a  x  ©_©  ^  + a  2*1 T - l + lP2 L  a  «^  x  J  i 3  9  a  2i  ^ a  X  4  i  2  +  X a a  i 2 a  3  1 J  76 aa G  4  r i-t  i"2"i  = Pfi P P ^  ( x )  1  j  2  x +^  x 2  ao^  a  4  9  ^  6  6 x  ?  w ^  +  aa-^  ?x*l  9  aa-^a^  J'  Some special cases of the above shifted Jacobi polynomials are: a = 3 = 0.5, shifted Chehychev polynomials; a = 3 = 1.0, shifted Legendre polynomials. The ultraspherical Jacobi polynomials, orthogonal on (1,-1) with respect to the weight factor ( l - x )^~ , are obtained from the hyper geometric series.. P ^ ^ ( x ) = P(-n, n + 2u - 1, u; (  l  ^  J  ).  k  The f i r s t eight ultraspherical Jacohi polynomials, standardized so that P ^ ( l ) = 1, are: P (x) = 1 Q  P (x) = x  P (x)  X  [X (2LI  =  2  2  P (x) =  [(2LI + 3 ) x - 3x] 3  3  P  P  P  4  =  ( x )  5  ( x )  =  + 1) - l ]  A[i(\i +  l)  [  4LI(H + 1)  [  1  1  ( 2  ^  ( 2  +  ^  +  6< > = 8n(tx + l ) ( n + 2) x  7  5  )  )  (  (  2  ^  2  ^  +  +  +  5  9  3  )  x  )  5  ^  x  "  4  "  6  (  2  1 0 ( 2 | i  +  7  ^  ^  +  +  +  5  3  )  5  )  )  x  x  x  2  0  +  3 +  1  5  X  ]  6  "  1  5  (  2  ^  +  7  )  (2u + 5 ) x + 45(2LI + 5)x?- 15 4  P  7  ( x )  =  S|a(p. + l ) ( i i + 2)  [.(2li + 11-K21. + 9)(2n + 7 ) x ,T 7  21(2LI + 9)(2LI + 7 ) x + 105(2LI + 7 ) x 5  - 105 x j .  3  Some special cases of these ultraspherical Jacohi polynomials are:  LI = 0,5, Chehychev polynomials; LI = 1.0, Legendre poly-  nomials; and LI = 1.5, Chehychev polynomials of the second kind.  78 APPENDIX C The Krylov-Bogoliubov Approximation} i t s Equivalence with the One-Term R i t z Method and with a L i n e a r Expansion  in Ultra-  s p h e r i c a l Chehychev Polynomials. The method of K r y l o v and Bogoliuhov approximation  a p p l i e s t o the second  or the  K-B  order model x + to  x  •  + |if(x, x) = 0, where u i s a " s m a l l " parameter. x + to x + LI j^2&x + g(x)]] ~ 0, the K-B  For the model  approximation may  2  be  assumed i n the form x^t) - a ( t ) cos (to t + 9 ( t ) ) I a ( t ) cos a ( t ) . (28) Cunningham " ' shows the e v a l u a t i o n of a ( t ) and 9 ( t ) hy averaging v  over one c y c l e of the o s c i l l a t i o n .  The r e s u l t i s  a(t) = A e " ^  (Cl)  and 2%  ® ^  ~ 2rcto a o u  f J  ^  s  a  c  o  °^  s  c  o  s  a  (0.2)  d c u  0 For the case i n which the o s c i l l a t i o n s are damped, the  K-B  method a p p l i e s only to models with an e x p l i c i t l i n e a r term. Consider the s p e c i a l case i n which the damping term i s zero and the model takes the form x f to x + ug(x) ~ 0. Q  Then  2%  ff  ~ o w  +  2TCCO  a  I  g  ^  a  c  o  s  °^  c  o  s  a  d  a  = to(a).  0 I n t e g r a t i n g t h i s e x p r e s s i o n f o r the phase, one cc(t) = to(a)t + C.  I f x(0)  (C3) obtains  0 i n the model, then C = 0.  79 Prom (0,3) we have 2JC  c (a) = co2  +  2  o  3ta  J  g(acosa) cos ada,  0 where second-order terms i n LI are neglected. 2%  co2 (a) = - ~  coa  L  2%  cos ada + LI  2  rta  g(acosa)cosada  2  J  0  Thus,  J  0  J  0  2% =  wa  ^ 0  t o w  a c o s c t  +  t-"S(  acosa  )J  cosada  2ic •JJJ^  J°  P(acosa) cosada,  0 where P(x) = to x + ug(x) Q  This expression thus determines the approximate frequency, to(a), for a nonlinear model x + P(x) = 0. For the case i n which f(x) has odd-symmetry i n x + f(x) = 0, x(0) = 1, x(0) = 0, Soudaclc^ ^ has shown that 2  a one-term Ritz approximation i s equivalent to a l i n e a r expansion i n terms of the Chehychev polynomials.  The Ritz  conditions give 2z .2 1 co" = =  f(cos 9)cos ©d©.  'o  (C.5)  80  Similarly a Chebychev polynomial expansion gives  1 x f(x)(l -  xV-°'  5) dx  2%  0  (0.6)  f(cos 9)cos ©d9  for an odd f(x) under the substitution x = cos 9.,  The  one-  term Ritz and Chebychev approximation techniques are applicable to models more general than those r e s t r i c t e d to have an e x p l i c i t l i n e a r term by the K-B approximation. pansions i n (C,4), ( C o 5 ) and (C,6)  Nevertheless, the ex-  show that, f o r conservative  models to which the K-B method applies, the same approximate solution i s obtained from the K-B method and the Ritz method.  c  81 REFERENCES 1.  Hayashi, C,H,, Nonlinear Oscillations i n Physical Systems , McGraw-Hill Book Co., New York, 1964, PP° 158-160,  2.  Goldstein, H., C l a s s i c a l Mechanics , Addison-Wesley Publishing Co,, Reading, Mass., 1950, pp, 58-59.  3.  Pipes, L.A., Operational Methods i n Nonlinear Mechanics , Dover Publications, New York, 1965, p. 7.  4.  Rosenberg, R.M,, "The Ateb (H) - Functions and Their Properties", Quarterly of Applied Mathematics, Y o l . XXI, No. 1, A p r i l , 1963.  5.  Cunningham, W.J., Introduction to Nonlinear Analysis , McGraw-Hill Book Co., New York, 1958, pp. 123-124.  6.  Minorsky, N., Introduction to Nonlinear Mechanics , J.W. Edwards, Ann Arbor, Michigan, 1947, pp. 186-190.  7.  Cunningham, W.J,, op. c i t . , p. 157.  8.  Cunningham, W.J., op. c i t . , p. 164.  9.  Denman, H.H., and King L u i , Y., "Applications of U l t r a spherical Polynomials to Nonlinear Oscillations I I . Free O s c i l l a t i o n s " , Quarterly of Applied Mathematics. Vol. XXII, No, 4, January, 1965.  10.  Soudack, A.C, "Jacohian E l l i p t i c and Other Functions as Approximate Solutions to a Class of Grossly Nonlinear D i f f e r e n t i a l Equations", Tech. Report No. 2054-1. Stanford Electronics Laboratories. Stanford University, Stanford, C a l i f o r n i a , A p r i l 24, 1961.  11.  Cunningham, W.J., op. c i t . , pp. 114-117.  12.  Froberg, C.E., Introduction to Numerical Analysis , Addison-Wesley Publishing Co. Inc., 1965, p. 245.  13.  Soudack, A.C, op. c i t . , p. 41.  14.  Lanczos, C , Applied Analysis , Prentice H a l l Inc., Englewood C l i f f s , N.J., 1961, p. 451.  15.  Davis, P.J., Interpolation and Approximation , B l a i s d e l l Publishing Co., New York, 1963, p. 174.  16.  Lanczos, C , op. c i t . , pp. 457-463.  17.  Soudack, A.C, op. c i t . , pp. 50-52,  82 18.  R a i n v i l l e , E., Elementary D i f f e r e n t i a l Equations , The Macmillan Co., New York, 1958, p. 169.  19.  Soudaok, A.C,  op. c i t . , p. 52.  20.  Soudack, A.C,  op. c i t . , p. 53-55.  21.  Denman, H.H.,  22.  Weber, E., "Introduction, Nonlinear Physical Phenomena," Proceedings of the Symposium on Nonlinear C i r c u i t Analysis. Vol. II, Polytechnic Press of the Polytechnic Institute of Brooklyn, New York, A p r i l 23, 1953.  23.  Klotter, K., "Steady State Vibrations i n Systems Having Arbitrary Restoring and Arbitrary Damping Porces", Proceedings of the Symposium on Nonlinear C i r c u i t Analysis. Vol. I I , Polytechnic Press of the Polytechnic Institute of Brooklyn, New York, A p r i l 23, 1953.  and Zing L u i , Y., op. c i t .  1  24.  Soudack, A.C, "Some Nonlinear D i f f e r e n t i a l Equations S a t i s f i e d by the Jacohian E l l i p t i c Functions", Mathematics Magazine. Vol. 37, No. 3, May, 1964.  25«  Hancock, H., E l l i p t i c Integrals , Dover Publications Inc., New York, 1958, pp. 10-13.  26.  Spenceley G.W., and Spenceley, R.M., Smithsonian E l l i p t i c Function Tables , Smithsonian Institution, Washington, D.C, 1947.  27.  Lanozos, C ,  28.  Cunningham, W.J.,  29.  Soudack, A.C, "Equivalence of the One-term Ritz Approximation and Linearization i n a Chebychev Sense", American Mathematical Monthly. V o l . 72, No. 2, February, 1965.  op. c i t . , pp. 367-370. op. c i t . , pp. 135-137.  

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