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Jacobi polynomial truncations and approximate solutions to classes of nonlinear differential equations Dodd, Ronald Edward 1966-09-29

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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 Br i t i sh Columbia, 1964. A THESIS SUBMITTED IN PARTIAL EULEILMENT OE THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in the Department of E lec t r ica 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 lec t r i ca l Engineering THE UNIVERSITY OF BRITISH COLUMBIA AUGUST, 1966 In presenting this thesis in par t i a l fulfi lment of the requirements f o r an advanced degree at the University of Br i t i sh Columbia, I agree that the Library shal l make i t f reely aval]able for reference and study, I farther agree that permission.for extensive copying of this thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. It is understood that copying or publication of this thesis for f inancia l gain shall, not be allowed without my written permission. Department of £/&ctrtca*t The University of B r i t i sh Columbia Vancouver 8, Canada Date ABSTRACT Solutions to classes of second-order, nonlinear differential equations of the form i + 2&x + f(x) = 0„ x(0) = 1, i(0) = 0 are approximated in this work. The techniques which are developed involve the replacement of the characteristic, f(x), in the nonlinear model hy piecewise-linear or piecewise- cubic approximations. From these, closed-form time solutions in terms of the circular trigonometric functions or the Jacobian e l l i p t i c functions may be obtained. Particular examples in which f(x) is grossly nonlinear and asymmetric are considered. The orthogonal Jacobi and shifted Jacobi polynomials are introduced for the approximation in order to satisfy 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: is bounded, no matter how large the coefficients of the non- Xnear terms in the model become. Because of these error-bound results, an heuristic measure of the departure from linearity is defined for classes of symmetric oscillations, and the weighting of convergence of the Jacobi and shifted.Jacobi polynomial expansions is set according to this measure. For asymmetric conservative models,, shifted Chebychev polynomials are used to obtain near-uniform approxi mations to the characteristic in the nonlinear differential equation. i i Based on the equivalence of the classical approxi mation techniques which is given for the symmetric, conserva tive models, extension of the polynomial approximation to classes of,non-conservative models is considered. Throughout the work, hy comparison with classical approximation methods, the polynomial approximation tech niques are shown to provide an improved, direct and more general attack on the approximation, problem with a decrease in tedious labor. i i i TABLE OP CONTENTS Page LIST OF ILLUSTRATIONS , v i LIST OF TABLES f , , ,. . v i i i LIST OF PRINCIPAL SYMBOLS , , ix ACKNOWLEDGEMENT x i i 1. INTRODUCTION 1.1 Description of the Mathematical Model ........... 1 1.2 Some Existing Approximation Techniques 4 2. DEVELOPMENT OF THE APPROXIMATION TECHNIQUES 2.1 Considerations of Approximation Range and Criteria for Closeness of F i t in the Approximate Time Solutions ,.„,.... 8 2.1.1 Determination of Bounds on the Oscillation and Normalization of the System Equation , ,...« 8 2.1.2 Criteria for Closeness of Fit in the Time Solution Approximations ......, .......... 12 2.1.3 Examples of Some Existing Approximation Techniques Under the Given Error Criteria . . . . . . & . . 13 2.2 Use of the Jacohi and Shifted Jacobi Polynomials for Approximation t 18 2.3 Determination of Upper Bounds on the Error in the Approximate Time Solution .., 25 3. APPLICATION OF THE APPROXIMATION TECHNIQUES TO SPECIFIC NONLINEAR MODELS 3.1 Techniques for a Class of Odd-Symmetric Nonlinear Characteristics ...,, 38 3.1.1 Definition of the Nonlinear Factor and Choice of Weighting Functions 38 3.1.2 Some Particular Cases 41 3.2 Techniques for Models with Asymmetric Nonlinear Characteristics ...., t......,....... , 51 3.2.1 First-Order or Linear Approximations ,... 52 3.2.2 Cubic Approximations to Nonlinear, Asymmetric Models , 54 iv Page 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 64 4. CONCLUSIONS 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 Poly nomials .., . , - . . . . . . o . . . . . . . . . . , 75 APPENDIX C The Krylov-Bogoliubov Approximation; i t s Equivalence with the One-Term Ritz Method, and with a Linear Expansion in Ultraspherical Chebychev Polynomials ............. 78 REFERENCES 81 v v LIST OP ILLUSTRATIONS Pigure Page 1.1 A Nonlinear LC Circuit 2 2.1 Normalization of an Asymmetric, Nonlinear Oscillation 10 2.2 Comparison of the First-Order Ritz Approximate Solution and a Linear, Least-Square Error Approxi mation 15 2.3 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 A Plot of Two Shifted Jacobi Polynomials 25 2.6 Saturation of the Maximum and Quarter-Period Errors for Linear Approximations as the I n i t i a l Amplitude of the Oscillation i s Increased 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 De parture from Linearity 39 3.2 A Comparison of the Error in Two Linear Approxima tions to *x+ y |xp sgnx = 0 42 3.3 The Relative Error. #in the Frequency for Linear Approximations to x+x+x-^ +lOx-? = o, x(0) = 1, x(0) =0 45 3.4 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 in Ritz and Shifted Chehychev Approxi mations to *x*+x+x2+3x3 =0 53 3.6 The Error Distribution in Shifted Chehychev and Least-Square Error Cubic Approximations to M+x+x3+lOx5 = U(t) 57 3.7 Linear Approximations to x+0.4x+x+x^+5x =0 61 v i Page 3.8 The Piecewise-Linear Approximation of x + 0.2x + tanh(2x) = 0 63 v i i LIST OP TABLES Table Page 2. Summary of Approximation Errors ........... <.,... 17 3.1 The Choice of Weighting for 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 for Jacobian E l l i p t i c Integrals of the Eirst Kind 74 v i i i LIST OP PRINCIPAL SYMBOLS a = a constant a^ = coefficients of a polynomial characteristic a(t) = amplitude parameter in Z-B approximation A ^ = coefficient of a Jacohi polynomial h = a constant b^ = coefficients of an approximate polynomial characteristic B n_ m = a sum of coefficients of Jacohi polynomials c c k = constant coefficients C = capacitance C Q = 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 in the approximation to f(x) E^ = a constant proportional to the total energy f(x) = a nonlinear characteristic in the differen t i a l equation f(x) = an approximation to f(x) P = the nonlinear factor g = a parameter in Legendre's transformation G- k^ a ,^(x) = the k t h shifted Jacobi polynomial h = a parameter in Legendre1s transformation = a function of the coefficients of the Jacohi polynomials i - ^ = current through an inductor ix ' J = a mass or moment of inertia k = the modulus of a Jacohian e l l i p t i c function k = a constant o L = inductance m a function of the coefficients of the Jacohi polynomials = V / ( P -JSKP - ST M(x) = a mathematical model n = ^ / ( C L - ©Kg. - A) V p = a parameter in the Legendre transformation P^fx) = an ultraspherical Jacohi polynomial PM, = the absolute value of the k^ h Jacobi or shifted Jacobi polynomial at the point where E(z + e) i s a maximum q = a parameter in the Legendre transformation q^ = .the charge on a capacitor r = \ / ( P " ±0 (P - *) X R = / f ( x ) x o c " 1 ( l - x)P _ 1dx O s = y/(q - ja) (q - £J S = / f ( x ) x a ( l - x ^ d x O Sn = the Jacohian e l l i p t i c sine function t - time ^ = shifted time T = the period of an oscillation T = an approximation to T Tn = the Jacohian e l l i p t i c tangent function T ( 9 ) = a mechanical torque or restoring force U(t) = the unit step function x v(t),w(t),x(t) = dependent variables jc(t) = an approximation to x(t) x m i n = a bound on an asymmetric oscillation X Q = a constant y(t),z(t) = dependent variables a,p = scalar parameters which determine the weighting of the shifted Jacobi polynomials I~\x) = the Gamma function c> = the coefficient of viscous damping At = an increment in time e(t) = x(t) -af(t) e = the particular integral in a differential p equation for e(t) Q,\ = roots of the denominator polynomial to which Legendre's transformation is applied \i = a scalar parameter which determines the weighting of the ultraspherical Jacohi polynomials u. = a root of the denominator polynomial to which Legendre1s transformation is applied JC = 3.14159. »• radians jt = a root of the denominator polynomial 0 = flux density in an inductor a) = angular frequency in radians per second xi ACKNOWLEDGEMENT Grateful acknowledgement is given to the National Research Council of Canada for the assistance received under Block Term Grant A68 from 1964 to 1966 and to the British Columbia Telephone Company for the award of a scholarship in 1964-1965. The author would like to express appreciation to his supervising professor, Dr. A. C. Soudack, for his inspiration and guidance throughout the course of the work. Thanks is also given to Dr. J. S. MacDonald for reading the manuscript and for making many helpful suggestions. The author i s also indebted to Mr, A. G. Longmuir and Mr. E. Lewis for discussion of the manuscript, to Mr. B. Wilbee for proof-reading, to Mr. Rff Proudlove for photographing curves and to Miss L. Blaine for typing the thesis. x i i JACOBI POLYNOMIAL TRUNCATIONS AND APPROXIMATE SOLUTIONS TO CLASSES OP NONLINEAR DIFFERENTIAL EQUATIONS 1. INTRODUCTION 1%1 Description of the Mathematical Model In order to obtain mathematical representations which approach the natural behaviour of some real systems, nonlinear relations among variables are often required. Many of these systems are described by mathematical models for which closed- form solutions cannot be found in terms of known functions, and a numerical scheme must be employed to obtain an accurate so lution. There is a considerable advantage in having a closed- form solution to a problem because one achieves insight into changes in the solution with variation of particular parameters in the model. In engineering, the concern with approximation techniques is important because many models derived from the physical world are themselves only approximations. Therefore, i t is 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 in this study is given to the second-order differential equation M(x) = x + f(x) = 0, x(0) = 1, x(0) = 0,(1.1) which describes undamped or conservative oscillations with a single degree of freedom. It is not required that f(x) have zero-point symmetry in this model. Thus, the zero on the right- hand side of equation ( l . l ) does-not necessarily imply that the 2 oscillations are free - a constant or step function driving term is allowed. The principle of superposition does not apply to general nonlinear systems. In particular, nonlinear oscillations described by equation ( l . l ) show a dependence of the frequency on the amplitude of the oscillation^. 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 as shown in equation ( l . l ) . Conversion of models with arbitrary i n i t i a l conditions to this form i s outlined in paragraph (2.1,1). in the description of electrical and mechanical oscillating systems. Suppose, for example, that i T ( 0 ) = g(0) where 0 is Figure 1.1 A Nonlinear LC Circuit If the inductor alone is nonlinear, by Kirchhoff's current law one obtains Cv + g(0) = 0. Now v = N0 for an N-turn c o i l from Faraday's Law. Hence, Common examples of the model in equation ( l . l ) occur the flux density for the nonlinear inductor in the LC circuit shown in Figure ( l . l ) . C 0 + W 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 which the c a p a c i t o r a l o n e i s n o n l i n e a r , one o b t a i n s 'q. +1 h(q) = 0, (1.3) where q i s the charge on the c a p a c i t o r and the v o l t a g e a c r o s s the c a p a c i t o r i s d e f i n e d by the r e l a t i o n v ^ ( q ) = h ( q ) . H a y a s h i ^ 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 which the c u r r e n t - f l u x r e l a t i o n s h i p has the form i^(0) = c-j.0 + c^0 + c^0 + c^0 , and f o r which the c o e f f i c i e n t s Cr- and c„ dominate over c n and c , 5 7 1 3 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 i n t e r e s t . 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 . Suppose t h a t such a non l i n e a r f o r c e i s g i v e n by T(9) = g(9) where 9 i s a d i s p l a c e m e n t . By d'Alembert's p r i n c i p l e f o r a r e s t r a i n e d element of mass or moment of i n e r t i a , J , 9 + j T(9) = 0 i s o b t a i n e d as the system e q u a t i o n . F o r example, i f the motion of a body i n a c e n t r a l f o r c e f i e l d depends o n l y on the 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 the e q u a t i o n of motion i s S-| + f ( r ) - = . O f r ( 0 ) = 1, r ( 0 ) = 0, (1.4) d\ where X i s the 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 . G o l d - ( 2 ) s t e i n v ' shows t h a t d e t e r m i n a t i o n of the motion of a system con s i s t i n g of two i n t e r a c t i n g p a r t i c l e s may be reduced t o the p r o  blem of d e t e r m i n i n g the motion of a s i n g l e p a r t i c l e i n an e x t e r -nal field, such that the motion is governed hy equation ( 1 . 4 ) . (3) Pipes gives a model of a mechanical system executing free 2 2 asymmetric oscillations. The model i s x + n x + h x = 0, and i t is of importance in the theory of seismic vibrations. As a fi n a l " k-l example, the application of models of the form x + cx|x| = 0 to the description of motion in principal modes of certain classes of nonlinear systems having many degrees of freedom has been shown by Rosenberg^. The electrical and mechanical examples quoted above a l l have the form of the generic model in equation ( l . l ) . In Chap ter 3 an extension is made to the case in which the model in equation ( l . l ) assumes light, viscous damping. Primary consi deration is given to the conservative system, however, so that the approximation of the nonlinear amplitude-frequency relation ship may he studied with variation of the characteristic, f(x), alone. 1.2 Some Existing Approximation Techniques Classical first-order approximation techniques, such as (5) the perturbation method w / and the averaging method of Krylov and Bogoliubov^ (Z-B method), require an explicit linear term which must dominate over the nonlinear terms in the model. These techniques are unsuitable for the models in this work because the "quasi-linear" nature of the model is undefined. The Ritz- (7) Galerkin v ; averaging method or the Principle of Harmonic (8) Balance , makes no such restrictions on the deviation of the model from linearity, but i t w i l l be shown that this method f a i l s to yield practical approximate solutions when the deviation 5 from l i n e a r i t y becomes a p p r e c i a b l e . Moreover, i n Appendix C the one-term R i t z method i s shown t o g i v e the same f i r s t - o r d e r approximate s o l u t i o n as the K-B a v e r a g i n g t e c h n i q u e . As an example, c o n s i d e r the n o n l i n e a r model • • • M(x) = x + f ( x ) = 0, x(0) = 1, x(0) = 0, i n wh 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 + A cos wt i s assumed f o r t h i s model. The R i t z o c o n d i t i o n s p l u s a c o n s t r a i n t from the i n i t i a l c o n d i t i o n s d e t e r  mine the parameters X q , A and co i n t h i s assumed approximate s o l u t i o n . These c o n d i t i o n s a re X Q '+ A = 1, 2% M(x)d(wt) = 0 0 and 2at M(x y)cos(wt)d(wt) = 0. 0 S a t i s f a c t i o n o f t h e s e f u l l - p e r i o d a v e r a g i n g i n t e g r a l s on the r e s i d u a l , M ( x ) , hears no d i r e c t r e l a t i o n t o the e r r o r i n the approximate time s o l u t i o n . I n t h i s work the a b s o l u t e e r r o r e ( t ) = x ( t ) - x ( t ) i s c o n s i d e r e d . Another i n d i r e c t p r o p e r t y of the R i t z method i s t h a t the form of the approximate s o l u t i o n must he assumed. Hence any f e a t u r e s not assumed i n the approximate s o l u t i o n w i l l n o t be found. A l s o , the R i t z 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 f o r which the 6 solution i s d i f f i c u l t . If the ahove f i r s t - o r d e r solution i s refined to the form x(t) = X Q + Acoswt + Bcos3oot, then four non l i n e a r equations i n X Q, A, B and co are obtained from the Ritz conditions. (Q) In a recent paper K , Denman and Lui considered the • • -i approximate solution of the equation x + ax + bx"^  =0. The nonlinear c h a r a c t e r i s t i c was expanded to a l i n e a r polynomial i n terms of ul t r a s p h e r i c a l polynomials. The techniques given i n Appendix A allow a closed-form solution to be written for this cubic equation by inspection. Soudack^ 1^ has given techniques for the approximation- of a nonlinear model i n the form of equation ( l . l ) , i n which the nonlinear c h a r a c t e r i s t i c i s a polynomial with odd symmetry. The techniques replace the nonlinear polynomial by a cubic one and the closed-form solution 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 using the Jacohian e l l i p t i c functions. Approximate models obtained from a Chehychev expansion of the nonlinear c h a r a c t e r i s t i c were found to give much better approxi mate time solutions than models with cubic cha r a c t e r i s t i c s ob tained by a least-square error or Legendre polynomial f i t to the nonlinear c h a r a c t e r i s t i c s . This work investigates piecewise-linear and piecewise- cubic approximations to nonlinear c h a r a c t e r i s t i c s . A dire c t approach toward making the error i n the approximate closed- form time solutions "small" i s undertaken. In Chapter 2, c r i t e r i a f or the closeness of the approximate time solutions are given,and some ex i s t i n g approximation techniques are 7 investigated under these c r i t e r i a . The derivation of hounds on the approximation error is then made. In Chapter 3 new approxi mation techniques which employ the Jacobi and shifted Jacohi polynomials are introduced. The restriction that the nonlinear oscillation have a small amplitude or that the nonlinear charac t e r i s t i c be quasi^-linear is not imposed. 8 2. DEVELOPMENT OP THE APPROXIMATION TECHNIQUES 2.1 Considerations of Approximation Range and Criteria for  Closeness of Pit in the Approximate Time Solutions. 2.1.1 Determination of Bounds on the Oscillation and Norma lization 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 for 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 t (2.3) for two real roots, (x = 1, x = x . ), determines the turning ' * mm 7 9 points.(l, x . ), of a bounded oscillation for a normalized 7 mm differential equation with an integrable nonlinear character i s t i c , f(x). The above discussion assumes that there are at least two real roots of equation ( 2 . 3 ) over the range of interest in the dependent variable, and that the solution to equation ( 2 . 1 ) is 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 in the phase plane from extrema of the potential function, Y(x). A singularity at a relative minimum of V(x) i s a centre point and the motion is locally bounded. This "potential well" which exhibits a local minimum is the most common one encountered in this study. A singularity at a local maximum of V(x) i s a saddle point and the motion i s locally unstable. At a point of inflection of V(x), the local behaviour of the singularity is like that of both a centre point and a saddle point and is thus unstable. In attempting to solve a nonlinear problem, some knowledge of the kind of solution to be expected is almost essential. Using the potential function thus provides useful information about the solution in various regions of x even before a solution of the system equation is attempted. No generality has been lost by fixing the i n i t i a l conditions as in equation ( 2 . 1 ) . Consideration i s given to bounded, periodic oscillations in this work. These oscillations * may be started, arbitrarily, at a point where x(t) i s zero hy making a shift in the independent variable. One special case of interest serves to illustrate the normalization which is carried out. Consider a system which is i n i t i a l l y at rest to be acted upon by a step input at t = 0. That i s , + h(y) = kU(t), y(0) = y(0) = 0. (2.4) dt^ The bounds on the oscillation are obtained by finding the real roots of potential function The constant, C , is obtained from the i n i t i a l conditions and o 7 the f i r s t integral. In this case C Q must be chosen so that V(0) = 0S and i t is evident that y = 0 is one bound on the oscillation. If the real root of V(y) closest to zero is y = \ and a minimum of V(y) is enclosed on the interval (0,\), then a bounded oscillation limited hy zero and \ i s obtained. V(y) = /(h(y) - k)dy +CQ . y(t) 0 t 0 s Pigure 2.1 Normalization of an Asymmetric, Nonlinear Oscillation 11 The transformation y(t) = Xx(^) along with a shift in the inde pendent variable, ^ = t - ^  , may he used to transform equation s (2.4) to one of the form 2 + rh(\x) -r = 0 ac(S=0) = 1, i(T=0) = 0. d? 2 X X .(2.5) Pigure (2.1) illustrates how this normalization is carried out. Equation (2.5) may now he written as ,2 + f(x) = 0 x(0) = 1, x(0) = 0, (2.6) which i s the form of equation ( l . l ) . Normalization to (0,1) is useful hoth for comparison of mathematical models on (0,1) and for comparison of errors in the approximate time solutions. If the normalized f(x) in equation (2.6) has odd symmetry on (-1, l ) and is monotonically increasing on (0, l ) , then the oscillatory 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 in section (2.3) that the maximum error in the approxi mate time solution over a fixed time interval depends on the maximum error, E(x) , in the approximation to f(x) in equa- tion (2.6). Because of the ahove symmetry in the normalized time solution, i t is 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 in f(x) thus seems an obvious choice because i t is intuitive that E(x) _ is smaller for a smaller nicix approximation interval in the x-f(x) plane. The Ritz and 12 ultraspherical polynomial truncation techniques make approxi mations on (-1, l) for this symmetric case. The smallest interval of symmetry for an oscillation derived from a model with an asymmetric, nonlinear characteristic is one half period* A piecewise approximation to f(x) may therefore he made on the range ( l , x m i n ) , and the remainder of the approximate time solution may be obtained by symmetry with this half period approximate solution, Pigure (2,4) in section (2.2) shows linear approximations to symmetric and asymmetric f(x). 2.1.2 Criteria for Closeness of Fit in the Time Solution  Approximations Por a model normalized to (0,1) the error in the time solution approximation i s defined by c(t) = x(t) - x^t). This error function i s not, in general, obtainable because x(t) cannot always be found in closed form. An IBM 7040 di g i t a l computer was employed to obtain numerical approximate solutions to the nonlinear differential equations. - The numerical so lutions 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 six decimal digits. A technique suggested by « (12) Probergv ; was also employed to check the accuracy«i Using this technique, a change in the numerical integration step size from 0.01 sec. to 0,001 sec. produced no change in the f i r s t five significant digits of the solutions for the time 13 intervals which, were considered. A goal of this work is to make the approximation error, e(t) s as small as possible with a minimum amount of mathematical labor. The choice of the smallest intervals of time symmetry for the approximations in section (2.1.1) is directed toward this goal. The principal error criterion chosen is that the error at the ends of ah interval,of approxi mation be small compared to the maximum error over the approximation interval. Since the numerical solution and the approximate closed-form solution are matched at t =0, the object is to have the error small at the end of the f i r s t approximation interval. This i s important because the so lution is constructed by matching each successive, partial 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 in the approximate time solution for a given model has an upper bound. To justify 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 is extended in time. Also, the frequency of the nonlinear oscillation is determined exactly. 2.1.3' Examples of Some Existing Approximation Techniques Under  the Given Error Criteria — — — Approximate solutions to the nonlinear differential 14 equation, x + x + + 10x5 = 0, x(0) = 1, x(0) = 0, (2 . 7 ) are carried out throughout the remainder of this chapter. The characteristic in this equation i s grossly nonlinear for the larger values of |x| on (-1, l)» The following analysis indicates the importance of the error c r i t e r i a in section (2.1.2): (a) First-order Ritz and Linear Approximations The first-order Ritz approximate solution to equation (2.7) is 'x(t) = oosj"^^, A linear, least-square error f i t to the characteristic in this model over the interval (0,1) yields the approximate equation 3E" + 9.043*' - 2.105 = 0, 'x(O) = 1, x^ (0) = 0. Solution of this equation gives 'x'(t) = 0 . 7 6 7 cos(3.007t) + 0.233 which i s valid for the f i r s t quarter cycle of the oscillation. In Figure (2.2) these first-order approximations are plotted with the numerical solution for the f i r s t quarter cycle. (h) Cuhic Polynomial Truncations The characteristic in 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 in Appendix A, solution of the Chehychev model gives x(t) = Cn(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 is the Jacohian 15 xit),5«t> Figure 2.3 Comparison of Chebychev and Least-Square Error Cubic • • •* c Approximations to i'+ x.+ x' + lOx-' = 0 16 e l l i p t i c cosine function. Curves of the error, e(t), over the quarter period for these closed-form approximate solutions are shown in Pigure (2.3). (c) A Two-term Ritz Approximation If a two-term Ritz approximation in the form x(t) = A coscot +B cos3<jot is assumed, then the Ritz conditions give three algehraic equations in A, B and co. These ares A + B = 1, -co2A + A + |A5 + ^ A 5 = 0 (2.8) and -9co2B + B + |B3 + ^ °rB5 + ^  + ^ |A5 = 0. Solution of these equations is d i f f i c u l t without- some knowledge of values of A and B to he expected. The error curves in Pigure (2.3) show the Jacohian e l l i p t i c cosine to he a close (13) approximation to the nonlinear oscillation. Soudackv suggested a device which facilitates the solution of equations (2.8) for 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 in 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 is found to he ~ ( t ) = 0.955 cos (2.624t) + 0.045 cos (7.872t). 17 (d) Summary Approximation e <° -I}max £(f} £ 0-T) v 'max Frequency Error Ritz First-Order -0.06 +0.049 -0.25 3.9% Linear Least-Squares -0.12 -0.10 +0.33 -8.2% Cubic Chebychev 0.0073 0.0036 0.014 0.3% Cubic Least-Squares -0.014 -0.014 0.05 -1.1% Two-term Ritz -0.045 -0.045 +0.14 -3.65% Table 2. Summary of Approximation Errors The results are summarized in 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 relative 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 oscillation. 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 is 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 relative error in 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 is obtained from the two-term Ritz method than from either of the cubic polynomial approximations. Also, the labor for the two-term Ritz method is considerably greater. The approximation techniques outlined in Table 2 do not satisfy the error c r i t e r i a which have been imposed because the error at the quarter period is not small compared to the maximum error over the quarter period. In Appendix C, re placement of -an odd-symmetric, monotonically increasing charac t e r i s t i c in equation ( 2 . 1 ) by an ultraspherical, Chebychev linear approximation is shown to give the same solution as the first-order Ritz method. This equivalence of the Ritz average over time with an orthogonal polynomial approximation in the x - f(x) plane provides motivation for 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 for  Approximation" "~" " The object of the remainder of this chapter is to give insight into the relation between the approximation of the nonlinear characteristic in the x versus f(x) plane and the re sulting error in the approximate time solution. The goal is then to obtain approximate time solutions which w i l l satisfy the error c r i t e r i a imposed in paragraph ( 2 . 1 . 2 ) . To achieve these ends, the Jacobi and shifted Jacohi polynomials are chosen for the piecewise approximation of the characteristic in 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 is absolutely 19 i n t e g r a b l e ^ 1 ^ i n terms of a set of polynomials, P^(x), ortho gonal with weight function, 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]°kPk(x)> ( 2' 9 ) k=0 where h £ f(x)P,(x)W(x)dx c,_ = ^r; — — (2.10) f [ P k ( x ) J W(x)dx a The shifted Jacohi polynomials are orthogonal on (0,1) with respect to the weighting function W(x) = (l-x)^ -"*" xa-"^. For the u l t r a s p h e r i c a l Jacohi polynomials the weighting function i s ( l - x 2 ) ^ - 1 and the i n t e r v a l of orthogonality i s (-1,1). Thus normalization of the o s c i l l a t i o n s to a maximum amplitude of unity i s necessary for the expansion of f(x) i n terms of these polynomials. Derivations and closed—form expressions for these polynomials are given i n Appendix B. Only the shifted Jacohi polynomials, G r ^ a , ^ ( x ) , a r e considered for the l i n e a r or f i r s t - o r d e r approximation of the models i n this work. Figure (2.4) shows how these approximations are carried out. For the symmetric ch a r a c t e r i s t i c i n Figure (2.4a), 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 ( 0 , 1 ) . In contrast, Ritz and ult r a s p h e r i c a l polynomial approximations are on the in t e r v a l (-1,1) for th i s symmetric case. For hoth asymmetric and symmetric nonlinear characteristics an asymmetric l i n e a r 20 differential equation of the form x + b^x' + b Q = 0, 'x(O) = 1, x(0) = 0, is obtained on (0,1). The solution to this equation is (a) Symmetric (h) Asymmetric Pigure 2.4 The Approximation 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 odd- symmetric characteristics. This approximation produces the equation x + t^x + b^x^ = 0. From this cubic model with zero- point symmetry, the closed—form solutions may be written by inspection using the techniques given in Appendix A. 21 Approximation in terms of the shifted Jacohi poly nomials yields differential equations with asymmetric, cuhic characteristics of the form x + b,x^ + b 0x + 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 characteristic normalized to (0,1). Distinct from the linear case, the asymmetric cuhic differential equation requires somewhat more labor to obtain the closed form approximate solution than does the odd-symmetric, cubic equation. Nevertheless, the techniques in Appendix A may be applied directly, and the labor is less than that required for the two-term Ritz method shown in paragraph (2.1.3) for an odd-symmetric characteristic. Examples of the closed-form solution of differential equations with cubic characteristics are given in Chapter 3. Erom the expressions for an orthogonal expansion given in equations (2.9) and (2.10), a linear approximation to a given f(x) in terms of the shifted Jacobi polynomials may be written *<x) = f f e f ^ x [ E + [P<« V * ( 3 - f *>] ([fl + ftL _ Sk) { a x 8 ; | ~ \ B ) is the gamma function. Also in this expansion 1 0 (2.11) 22 and 1 f(x) x a ( l - x)?" 1 dx. 0 For arbitrary a, 8 and f(x) the integrals R and S do not, in general,have closed-form solutions* One special case has been found, however, in which the expansion technique in equation (2.9) is general. If f (x) = x q, q + <x> 0, then R and S reduce to the form 1 0 which is the integral for 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 is quite general because q need not be an integer. When f(x) is i t s e l f a polynomial, a truncation or an expansion in terms of the shifted Jacobi polynomials may he made using Lanczos' Economization^1-^. Lanozos^ 1^ shows that this truncation technique gives identical coefficients to the expansion determined by equations (2.8) and (2.9) when ortho gonal polynomials are used for the truncation. To illustrate the procedure, consider the truncation of f(x) = x + 3x 5 (2.12) to a linear polynomial on (0,1) using the shifted Chehychev polynomials, G^®'^' ^*^\x). For simplicity, the superscript are dropped and the complete expansion of (2.12) may he written f(x) = c QG o(x) + c 1G 1(x) + c 2G 2(x) + c 5& 3(x). A linear polynomial i s desired so G^(x) and G2(x) are used successively to obtain expressions for the cubic and the quadratic in terms of a linear polynomial. From G^(x) = -1 2 3 + 18x - 48x + 32x , one obtains x3 = ^| (1 - 18x + 48x2) + ^ | G 3 ( X ) J therefore, x + 3x3 = ^ - z=j*x + |x 2 + ^ | G^(x), Similarly, x 2 = |(-1 + 8x) + |G 2(X) ; hence, + 3x 5 = - || + f|x + T | G 2 ( X ) + 3§ G3 ( x )- ( 2 , 1 3 ) For a linear truncation &2(x) and G^(x) are set equal to zero. Hence, ?(x) = - i | + § x . 24 Carrying through the orthogonal polynomials to equation (2.13) in the expansion makes the distribution of the error in the function approximation explicit. 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) is E __(x) = 9/l6 + 3/32 = 21/32, and i t occurs at x = 1 where the polynomials have a maximum oscillation. The distribution of the error in a convergent expansion can be predicted roughly from the f i r s t term neglected. The f i r s t term neglected in (2.13)» c 2G 2(x), oscillates with six times the amplitude of c^G-^x). Therefore, the error on (0,1) behaves like &2(x) with three, near-equal error maxima. The two shifted Jacohi polynomials in Figure (2.5) oscillate with a larger value near x = 1 than near x = 0 on their range of orthogonality. Thus, the distribution of the error in the x - f(x) plane for a linear approximation behaves like (n ^ n ft "\ &2 * (x), and the distribution of the error in a cubic polynomial approximation behaves like 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, heuristic arguments justify variation of the error in the approximation of the characteristic near x = 1 in order to obtain an improvement in the time solution at some point such as the approximate quarter period. Change in the weighting function of the shifted Jacobi polynomials allows this variation in distribution of the error over the orthogonal range (0,1) in the x - f(x) plane. 25 1 . 0 0 . 5 - 0 . 0 - - 0 . 5 - - 1 . 0 x 0 . 0 0 . 5 1 . 0 Pigure 2 . 5 A Plot of Two. Shifted Jacohi Polynomials 2 . 3 Determination of Upper Bounds on the Error in the Approxi  mate Time Solution rough upper hound on the error in the time solution approxi mation from the error in a cuhic polynomial function approxi mation to a polynomial of higher degree in the model. A similar approach i s used here to determine the upper hound when a linear approximation i s made to a higher degree polynomial. For simplicity replace 'x hy z, then the approximate equation may he written as and the original differential equation may he written in the form A method was developed hy Soudack ( 1 7 ) which gives a z +f(z) = 0 z ( 0 ) = 1 , z ( 0 ) = 0 , 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) is the error in the normalized time solution (which is not in general obtainable in closed form) and E(x) is the error in the function approximation. Now, * . WW WW W W X = Z + £ = - f ( z ) + £ = - f"(z + e) - E(z + e); hencej £ = 1 (z) - *±(z + e) - E(z + e), For the linear 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 differential equation for the error i s thus "E + b 1£ = -E(z + E) E ( 0 ) = E ( 0 ) = 0 . ( 2 , 1 4 ) 27 The i n i t i a l conditions are obtained as shown because the nu merical solution and approximate solution are matched i n i t i a l l y . The solution to equation ( 2 , 1 4 ) has the form e(t) = e p - e p cos(y"b~^ t ) . ( 2 , 1 5 ) The particular integral, E , is determined from equation ( 2 , 1 4 ) (18) ^ using convolution v ;, that is EP " b-, 1 [ B * 0 ) ] sin(t - 8) d6, where E (t) is the function obtained by replacing z and £ by their respective time functions. Since a bound on the error is of interest here, consider EP ~ lb 1 E (8) sin(t - 8) dB. 0 Now E (8) is bounded because f(x) and f(z) are bounded on (0,1). L e t | B * ( t ) | m a 3 c = |E ( Z + e ) | m a x = EMAX, EMAX can be found from f(x) and f'(x). Thus, the bound on the particular integral now becomes EMAX J 0 sin (t - 0) dp. Since the sine function is less than or equal to one, for a time 28 interval At, we have Iep I - fiffAt* ^ Specifically, the time interval, At, w i l l he an approximate quarter period or an approximate half period depending on the interval over which the truncation is made. From equation (2,15) one obtains | e ( t ) | ^ | e p | - | l - cosj^ ^- 2 EMAX At. This relation determines an upper hound on the error in a fi r s t - order approximate time solution from the error in the x - f(x) plane matching of the characteristic. Using a similar method for a cubic polynomial truncation f (z) = t>0 + b^z + b 2z + b^z . (19) Soudack^ has shown that a rough upper bound for the error in the time solution approximation i s These results are for extreme upper hounds because the above approach i s a pessimistic one in 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 linear, least-square error approximation to equation (2*6) in paragraph (2.1.3). The upper hound predicted for the f i r s t quarter-period by- equation (2.16) is 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 is e(t) = -0*12. The value of the above bounds on the error for approxi mations 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 poly nomial using the shifted Chebychev polynomials. Below, this result is extended to the expansion of a polynomial in terms of the orthogonal Jacobi or shifted Jacobi polynomials to either a linear or a cubic polynomial. Let P n W = " Aon " ^ n * +• • - V * * V - D n ^ + V** ( 2 ' 1 8 ) represent an orthogonal Jacobi or shifted Jacobi polynomial as given in Appendix B. The coefficient A ^ i s positive i f n-m is even; the signs have been chosen as shown for convenience. Also, for convenience B n - 2 - A ( n - 2 ) n ^ ( n - 1 ) ( n - l ) 3.0 and A(n-2)n A(n-m)(n-l) B ( n - 2 ) A ( n - 3 ) ( n - 2 ) T A ' A T • • • B n-m (n-m)n A (n-l)(n-l) '(n-2) (n-2) B (n-m+1)A (n-m) (n-m+1)  A(n-m+1)(n-m+1) are defined. Application of Lanczos' Economization to the non linear characteristic f(x) = y | Si**1 v i e l d s p-1 *(x) = V ^ ' where n k=p A., + n-p E m=l B(k-m)A.i(k-m)  A(k-m)(k-m) ( 2 . 1 9 ) The error in this approximation'is E(x) = f(x) - f(x). -In terms of the nonlinear characteristic and the orthogonal polynomials n n E a k ^  Akn Pk ( x) A kk P ( 2 . 2 0 ) In.both ( 2 . 1 9 ) and ( 2 . 2 0 ) , p = 2 for a linear approximation and p = 4 for a cuhic polynomial approximation. Let PM^  he the absolute value of the k^ *1 Jacohi or 31 shifted Jacohi polynomial at the point xffi where E(x) assumes a maximum oscillation. EMAX. Erom equations (2.16), (2.17), (2.19) and (2.20) n n 2 A t Hxi PM I—I 7 — L D A k k Z i 7 — k v . k k J max n a^ + E *kk *11 n-2 E m=l B ( k - m ) A l ( k - m )  A ( k - m ) ( k - m ) Expanding the summations one obtains |e(t)| ^ 2At I v 'I max a n H l ^ A q r ^ + a n - l H 2 ^ A q r ^ + > * * + a p H n - - p + l ^ A q r ^  a l + a n K l ^ A q r ^ + a n - l K 2 ^ A ( l r ) +• • • + a p K n - p + l ^ A q r ^ (2.21) This error bound depends only on the coefficients, a^ ., in the polynomial which is being approximated,the coefficients, Aqr, of the orthogonal Jacobi polynomials and the time interval over which the approximation is made. If a^ gets very large in the polynomial being approximated, then from equation (2,21) *<*>|»«-~ 2 A t which is constant. If more than one of the a^ such as a^ and a k - l a r e : m c r e a s e d , then £ ^ l m a x — 2 A t ' ^ A - k + i ^ ^ ^ ^ k - A - k ^ ^ ^ >+* • - • ^ n - k + l ( A * r } + a k - l K n - k + 2 { A * r ) + ' ' ' 32 2At a k - l V k + l ( A * r ) - ^ f " Hn-k+2(Aqr) and this error hound i s also a constant as a^ . and ak_]_ get very large. These results apply to both linear 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 trun cation of f(x) by Lanczos' Economization and the maximum error in the truncation have the same dependence on the coefficients, a^ .. Thus, when the quotient i s taken in equation (2,21), the dependence on the a^ . is removed, in the limit, as the a^ . grow large» Prom these results i t follows that i f the maximum error in the approximate time solution over a given time interval is 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 in this work agree with these error bounds. As an example, consider the nonlinear model which is given in equation (2.7), that is x + x + x? + 10x5 = 0, x(0) - 1, x(0) = 0, (2,7) If the amplitude of the oscillation is now doubled, then the transformation x = 2y normalizes this equation to unity 33 amplitude in the form y + y + 4 y 3 + l60y 5 = 0, y(0) - 1, y(0) - 0. Similarly, i f the amplitude of the oscillation in (2.7) is halved, the normalization yields w + w + 0.25MJ + 0.156w9 = 0, w(0) = 1 , w(0) = 0, where x = 0*5w. Clearly, these normalizations show considerable differences in the characteristics as x(0) is varied. Other normalizations similar to these have been made on equation (2.7), and then first-order 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 k ^ u o ' u o ; (x), G k w and G k ( 0 ' 5 ' ° ' 6 ) ( x ) . 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 for the nonsymmetrically-weighted, shifted Jacobi polynomial approximations than for the ultraspherical, shifted Chebychev or one-term Ritz method approximations. When x(0) = 1, as in equation (2.7), for example, the errors at the approximate quarter period are 0.049 and -0.004 for the one- (Cl R 0 f) term Ritz and ' * '(x) approximations respectively. Extension of these solutions to a f u l l period by symmetry yields the maximum error of 0.25 for the one-term Ritz solution, 34 • (t) max .04 h (i) ( i i ) ( i i i ) (iv) Ritz method a = B = 0.5 a = 0.5, P = 0.56 a = 0.5, P = 0.6 35 and 0.08 for the (j (°" 5' 0 , 6) ( x) approximation. A numerical so lution of equation (2.7) gives a derivative at the quarter period of -2.1981. The relative errors in the derivative at the approximate quarter periods are - 2 9 % for the first-order Ritz approximation and -13% for the Gt^®*-**®*^ (x) approximate solu tion. Also, from a numerical solution a quarter period of 0*5777 seconds is obtained. The approximate quarter periods ares 0,553 s e c , a relative error of 3 - 9 % and 0,5728 sec,, a relative (o R n f) error of 0.8% from the Ritz and ? r ;(x) polynomial approxi mations, respectively.. The improvement over the Ritz method in Figure ( 2 . 6 ) is mainly in the phase or frequency. This is because the errors at the approximate quarter periods saturate at considerably smaller values for 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 in curve ( i i ) shows the effect of an approximation on the interval (0,1). The near-uniform Ritz or Chebychev approximation on (-1,1) does not account for the symmetry in 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 result, both e(t) and e(%/4) in curve (i) are-larger for the Chebychev approximation than for the shifted Chebychev approximation shown in curve ( i i ) . The improvement obtained using the nonsymmetrically-weighted shifted Jacobi polynomials shows, however, that near-uniform or Chebychev matching in the .10 .08 .06 .04 .02 .00 J e ( t ) | max P = 7, o = 0.5, P - 0.63 , (0.084) p = 5, g = 0.5, p a 0.6, (0.07) p = 3, g = 0.5, p r= 0.56, (0.052) P = k, o = 0.5, P = 0.44, (0.041) H —1 1 h 1- o.i 10 10s 105 .01 .0075 .005 .0025 ,000 :(T/4) (a) 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) 0.1 -t- 10 t (b) 1 P- Figure 2.7 ' Saturation of the Maximum and Quartey Period Error f o r Linear Approximations to 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  teria which have been imposed. Now consider maximum and quarter-period values of e(t) for l inear approximations over the f i r s t quarter period to equa tions of the form x + nx p = 0, x(0) = 1, x(0) = 0. In this case, x(0) is 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 linear shifted Jacohi polynomial approximations are plotted versus an increase in n in the characteristic. For c lar i ty , the errors in the Ritz approxi mations, which are much larger than those plotted in Figure (2.7)9 have been placed in 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, five and seven, respectively. These error curve's do not show the same increase toward saturation as those in Figure (2.6). This is because the characteristic in equation (2.7) has a l inear term present which becomes dominant when the effect of the non linear terms is made small by decreasing the i n i t i a l amplitude. As predicted by equation (2*21), however, the curves in Figure (2.7) demonstrate that the error is bounded as the coefficients of the nonlinear terms increase. 38 3. APPLICATION OP THE APPROXIMATION TECHNIQUES TO SPECIFIC NONLINEAR MODELS 3.1 Techniques for a Class of Odd-Symmetric Nonlinear Character i s t i c s 3.1.1 Definition 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. This class of characteristics is approximated in this section. For parti cular cases, the variable weighting of the error in the x-f(x) plane afforded hy shifted Jacobi polynomial expansions has been used in the previous chapter. These weighted polynomial approximations have yielded approximate time solutions with a small error in the quarter period. Extension of this weighting property to more general characteristics in this class requires a means for specifying the deviation from linearity of different, normalized characteristics on the interval (0,1). For this purpose, the nonlinear factor 1 f 1 2 f (x)dx p h 2 (3.1) f ( l ) is defined. Pigure (3.1) shows graphically that F is the difference between the area under the curve and the area under the chord on (0,1), relative 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 teristics P<0. fU) f ( l ) hardening softening x 0 1 Pigure 3«1 A Graphical Representation of the Measure of De parture from Linearity Now consider variation of the parameters a,8 and u in the weighting functions according to the value of P. A derivation hy Denman and Lui v shows that Chehychev approximation to "quasi-linear" f(x) yields approximate time solutions with a small relative error in 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 W 1(x).= x ^ a - x ) ^ " 1 ) (5.2) and W2(x) = (1-x 2)^- 1, ( 3 . 3 ) 40 error in the approximation to f(x) near x = 1. What is more important, as the amplitude of the nonlinear terms increases, the deviation from l inear i ty is most severe for the larger values of x on the interval (0,1). Prom equation ( 3 . 2 ) , varia tion of B is seen to have the greatest effect on the weighting near x = 1. Setting a = 0.5 makes the weighting close to that obtained from a shifted Chebychev truncation for the small values of x on the interval (0,1),. This choice is reasonable because Chebychev approximations in 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 empiri cally from error saturation curves similar to those in 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 3 p £ -0.40 0.37 0.35 - ..40^ P ^ -0.25 0.44 0.42 -0.25 ^ P ^ +0.25 0.50 0.50 0.25 ^ P ^ 0.60 0.56 0.58 F ^ 0.60 0.63 0.65 Table 3.1 The Choice of Weighting for the Jacohi and Shifted Jacobi Polynomial Approximations 1 nomials has been shown to produce large differences in the error 41 at the approximate quarter period. Hencej considerahle improve ment in the amplitude-frequency relationship has been found possible by choosing the weighting according to the value of E given in Table (3°l)° Prom a model with a saturating or soft 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 oscillation. Variation of the weighting in a linear 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 characteristics. 3.1.2 Some Particular Cases The nonlinear factor is now applied to weighting the error in 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 in Table (3.2).have a l l been truncated down to an asymmetric linear polynomial on (0,1) using Lanczos' Economization and the shifted Jacobi polynomials. In 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 ex pansion to a seventh degree polynomial i s used to obtain poly nomial approximations to the functions in Table (3.2). The Legendre approximation is carried out for the grossly nonlinear functions marked by asterisks because the Taylor expansion converges slowly for these functions. The maximum error in 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 criterion, the linear shifted Jacohi polynomial truncations carried out in the x - f(x) plane agree with corresponding linear expansions obtained by numerical integration of equations (2.9) to three significant figures. Pigure 3.2 A Comparison of the Error in Two Linear Approxi- ^ * * I mations to x + /|x| sgn x = 0 The models with characteristics of'the form x1//q-, and with q equal to 2,3,5, and 7 are given to show the behaviour of approximations to models with softening characteristics. Rigorously, these models are not Lipschitz at x = 0, hut this non-Lipschitz character at x = 0 w i l l not be present in any 43 Table 3.2 Comparison of the First-Order Ritz Method and Linear Jacobi Polynomial Approximations Nonlinear Characteristic F Ritz e^max Method e(T/4) 8 Shifted e ( t )max Jacobi e(T/4) x+0.l6x5+0.256x5 0.17 -0.0170 +0.0027 0.50 -0.015 0.0005 3 5 x+x^ +x 0.39 -0.0390 +0.0160 0.56 -0.041 -0.0063 "5 5 x+xy+5x-^  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^+810x5 0.66 -0.0700 +0.0670 0.63 -0.090 -0.0080 x+0.25x5+0.31x5 +0.l6x7 0.26 -0.0250 +0.0064 0.50 -0.022 +0.0015 3 5 7 x+x-'+x +x 0.48 -0.0480 +0.0270 0.56 -0.051 -0.0016 x+x5+10x5+10x7 0.67 -0.0680 +0.0690 0.63 -0.090 -0.0040 x+x5+10x5+100x7 0.73 +0.0970 +0.0970 0.63 -0.100 +0.0060 X+0.3X5 0.11 -0.0120 +0.0011 0.50 -0.009 -0.0010 x+0.75x5 0.21 -0.0220 +0.0040 0.50 -0.018 -0.0003 x+3x3 0.38 -0.0380 +0.0140 0.56 -0.039 -0.0070 x+12x3 0.46 -0.0480 +0.0260 0.56 -0.049 -0.0040 x+1000x5 0.50 -0.0520 +0.0300 0.56 -0.053 -0.0010 x 2 sgn x 0.34 -0.0340 +0.0100 0.56 -0.032 -0.0070 x 2 e ^ sgn x 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 44 Table 3.2 (Continued) Nonlinear Ritz Method Shifted Jacobi Characteristic F £ ( t )max e(T?4) 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 |x|°'8sgn x -0.11 +0.0100 +0.0050 0.50 +0.0060 +0.00100 yjxT sgn x -0.33 +0.0280 +0.0060 0.44 +0.0120 +0.00300 2y|x|x sgn x -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-0.5x5 -0.17 +0.0150 +0.0020 0.50 +0.0130 +0.00300 p 2x-0.9x . sgn- x -0.27 +0.0250 +0.0040 0.44 +0.0170 +0.00140 sin(l.5x)** -0.26 +0.0230 +0.0040 0.44 +0.0150 +0.00030 45 physical problems or in any dig i t a l simulation. • As an example from Table (3.2), the first-order approximation of x +7|x| sgn x = 0- is shown in Pigure (3.2). The error over the f i r s t period is shown for the one-term Ritz approximation and the shifted Jacobi polynomial expansion, Q (0*5,0.44)(x). Approximation to the classical pendulum model, x + sin(l.5x) =.0, is the f i n a l example in Table (3.2). An ana l y t i c a l solution to this model may be obtained in terms of the Jacohian e l l i p t i c functions but the linearization in Table (3*2) is given to show an example of the improvement in the shifted Jacobi truncation technique over the Ritz averaging method. Pigure (3.3) compares results for the relative error in the frequency or period. One-term Ritz and linear shifted Jacohi approximations to the equation x + x + xr + lOx = 0, x(0) =1, x(0) = 0, are shown in this figure. The normalization with changes in x(0) has been carried out as described in paragraph (2.3). 0.06 " 0.04 " 0.02 - :(0) -0.01 - Figure 3«3 The Relative Error in the Frequency for Linear Approximations to x + 1 + x ; + lOx = 0, x(0) = 1, x(0) = 0 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 poly nomials by numerical integration of equation (2.9). This allows the behaviour of the amplitude-frequency approximations to non linear models with characteristics more general than poly nomials to be studied under cubic, shifted Jacobi polynomial expansions. Pigure (3.4) gives the relative errors over the f i r s t period for three cubic polynomial truncations to the model x + x + x? + lOx = 0, plotted for the f i r s t period. Under the 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 for this same model, the maximum error in a two-term Ritz approximation is 0.14, and the error in a cubic, least-square error approximation is 0.05. Por the cubic (j (0.5,0.56) ^  app r o ximation in Pigure (3.4), the maximum error i s -0*0028 over the f i r s t period. This latter approximation also shows that the error in 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 relation obtained between e(t) Table 3.3 Cubic Polynomial Approximation of the Nonlinear Characteristics Nonlinear Characteristics P Chebychev Ultraspherical U e(t) ^ K 'max Jacobi e(T/4) P Shifted Jacobi E ( t ) m a 2 E(T / 4 ) X + + 0.25x5 0.3125X 5 0.21 +0.00080 +0.00030 0.50 0.00080 +0.00030 0.50 -0.00020 .0.00000 X 3 5 + X ^ + XT 0.39 +0.00190 +0.00090 0.58 -0.00200 -0.00009 0.56 -0.00050 +0.00000 X + + 0.562x5 3.1641X5 0.51 +0.00500 +0.00240 0.58 -0.00400 +0.00060 0.56 -0.00100 +0.00020 X + x 5 + 10x5 0.58 -0.00750 +0.00360 0.58 -0.00500 +0.00100 0.56 -0.00120 +0.00050 X + 4x 5 + I60x 5 0.66 +0.01000 0.00470 0.65 -0.00700 +0.00200 0.63 -0.00180 +0.00006 X 3 (5 + 9x + 81 Ox3 0.66 +0.01000 0.00480 0.65 -0.00720 -0.00010 0.63 -0.00180 +0.00013 x + 0.25x5 + 0.31x5 +0.15x7 0.25 +0.00140 +0.00600 0.50 +0.00140 +0.00600 0.50 -0.00050 +0.00018 X 3 5 7 + x y + X + X 0.48 +0.00480 +0.00250 0.58 -0.00400 +0.00040 0.56 -0.00130 +0.00020 X + x 5 + 10x5 + 10x7 0.67 +0.01300 +0.00700 0.65 -0.00900 +0.00030 0.63 -0.00300 +0.00040 Table 3.3 (Continued) Nonlinear Characteristics P Chebychev Ultraspherical U e( t) ^ v 'max Jacobi e(T/4) B Shifted Jacobi £^max x + x 5 + 10x5 + lOOx7 0.73 +0.02100 +0.01000 0.65 -0.0130 +0.0040 0.63 -0.0043 +0.00100 x 5 0.67 +0.01000 +0.00450 0.65 -0.0074 -0.0002 0.63 -0.0019 +0.00015 x^ + x 7 0.71 +0.01600 +0.00740 0.65 -0.0100 +0.0007 0.63 -0.0033 +0.00080 x 7 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 1/3 x ' ^  -0.50 -0.00800 -0.00310 0.42 -0.00540 +0.00200 0.44 +0.0009 -0.00040 Table 3.3 (Continued) Nonlinear Chebychev Ultraspherical Jacobi Shifted Jacobi Characteristics F e(t) max e ( * A ) H e(t) max eCT/4) 0 e(t) max e(T?4) tanh(x) -0.10 -0.00020 +0.00004 0.50 -0.00022 +0.00004 0.50 -0.00009 -0.00002 tanh(2x) -0.38 -0.00340 -0.00130 0.42 -0.00240 +0.00090 0.44 -0.00030 -0.00012 X V 5 -0.67 -0.01100 -0.00500 0.42 -0.00760 +0.00180 0.44 -0.00100 -0.00050 Figure 3 . 4 Comparison of Solutions to Models Obtained from Chebychev, Ultraspherical Jacobi and Shifted Jacobi Approximations to x + x + TLJ + 1 0 x 5 = 0 51 and E(x) _ in equations (2.16) and (2.17), the time solution errors for the cuhic function approximations in Tahle (3.3) are much smaller than errors from the linear approximations in Tahle (3.2). This is because the cubic is always a closer f i t to f(x) than i s the straight line in the x - f(x) plane. Weighting the convergence in the x - f(x) plane according to the value of P in Table (3.1) has resulted in improvement over the classical Ritz-averaging and Chebychev polynomial approxi mations. Particular examination of the refined, shifted Jacobi approximations in Tables (3.2) and (3.3) shows that the error at" the approximate quarter period is less than 10% of the maximum error over the f i r s t quarter period for most of the examples considered. 3.2 Techniques for Models with Asymmetric Nonlinear Charac  teristics When the nonlinear characteristic does not have the symmetry possessed by those considered in paragraph (3.1), the oscillation often has a large dc component. The procedure given in paragraph (2.1.1) permits determination of the range of oscillation of the dependent variable and normalization of the i n i t i a l conditions to x(0) = 1 and x(0) = 0. The trans formation x = (l-x . )w + x . is used to transform the range- m m m m ° ( l , x m i n ) to the range (w = 1, w = 0). Otherwise, the weighting of the error in a shifted Jacohi polynomial approximation in 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 m. ~|w + x . ) . An ° L m m j m m •« approximate solution to the model w + g(w) = 0 is valid for the 52 f i r s t half period* Approximations to the characteristics in this asym metric class have heen attempted using the shifted Jacobi polynomials. Some models yield approximate solutions which show improvement over the near-uniform, shifted Chebychev approximations. Notably, the characteristics in these models have terms of even symmetry which are small compared to the amplitude of the odd-symmetric terms. , For other characteristics which have significant 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 linear and cubic expansions are considered for models with asymmetric, nonlinear characteristics. 3.2.1 First-Order or Linear Approximations The Ritz-Chebychev equivalence has been shown for 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 ln paragraph (1 .2) . The dis tribution of the error over the f u l l period shown in Figure (3.5) compares these two techniques. Our asymmetric example in this case is x + x + x + 3x^ = 0, and i t s range of o s c i l  lation is (x = 1, x = -1,176). The two-term Ritz approximation is ¥ + 3.542x + 0.335 = 0, 53 Figure 3.5 Error in the Ritz and Shifted Chehychev Approxi- * 2 3 mations to x + x + x + 3x = 0 while direct truncation of the nonlinear characteristic in terms of the shifted Chehychev polynomials gives \ + 3.512x' + 0.328 = 0. In general, the shifted Chehychev polynomial truncation tech nique is preferred over the two-term Ritz approximation because the predetermination of x m ;j_ n makes the Chebychev approximate solution closer to the numerical half-period than does the Ritz approximation technique. This property may he observed in Figure (3.5). Also, the Ritz method requires the solution of nonlinear, algebraic equations to determine the parameters in the assumed solution. 54 3.2.2 Cubic Approximations to Asymmetric Models When the normalized characteristic, g(w), is approxi mated hy a cubic polynomial on (0,1), the techniques given in Appendix A may be used to obtain a olosed-form time solution. As an example, oonsider the approximate step response of a sys tem which is i n i t i a l l y at rest and i s described by the model x + 10x5 + x 3 + x = U(t), x(0) = x(0) = 0. (3.4) U(t) is the unit step function. Prom the i n i t i a l conditions and the f i r s t integral |- + 1.6667X5 + 0.25x5 + 0.5x2 - x = 0, one obtains (x = 0, x = 0.7825) as the range of the oscillation. The transformation x = 0.7825 w gives the normalized model w + 3.7492w5 +' 0.6l23w3 + w — 1.278 = 0, w(0) = w(0) = 0, with (w = 0, w = l) as the range of the dependent variable. Lanczos1 Economization of the characteristic in this equation using the shifted Chebychev polynomials on (0,1) plus a shift in the independent variable, defined by t = t + t , produces the approximate equation 2/v + 11.1569w3 - 8.7872w2 + 2.9771w - 1.3490 = 0, w(0) = 1, d ^ $(0) = 0, ( 3 . 5 ) 55 The f i r s t integral of (3.5) is h—)2 + 2.7892w4 - 2,9290w3 + 1.4885w2 - 1.3A90v = 0. d d£ Integration of this f i r s t integral gives 5 w dt = - I dw. 0 1 y~w(5.5784w5 - 5.8582w2 + 2.9771w - 2.6980) hence,, ¥ ' A T I dw t £ T + T/2 = Q y-w(5.5784w5 - 5.8582w2 + 2.9771^-2.6980) (3.6) In general t " £ t, hut the shifted time, t , i s equal to the real time, t, in this special case because the oscillation starts one half period before the f i r s t maximum and thus, t = T/2. The procedure described in Appendix A may be used s to transform the integrand in equation (3.6) to the form of the E l l i p t i c Integral of the First Kind. Then a closed-form expression for 'w(t) may be obtained from Table A according to the parameters in this e l l i p t i c integral. In this example, both w = 0 and w = 1 are turning points of the motion,so both zero and one are roots of the polynomial in 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 1 . 3 4 8 4 dt = 56 0 . 3 3 1 2 - I 4 . 2 0 2 9 y 2 ) ( i + 4 . i 2 7 3 y 2 ) N Choosing h 2 = 1 4 . 2 0 4 9 and g 2 = 4 . 1 2 7 3 so that c 2 = g 2/h 2< 1 , and making the transformation v = hy, gives the form dv 1.6828 dt = « ( 3 . 7 ) ( 1 - v 2 ) ( l + 0.2906v2)A which corresponds to entry II in Table A. Prom equation ( 3 . 7 ) and Table A, the modulus of the e l l i p t i c integral is k = ./c / ( l + c ) = 0 . 4 7 4 5 , and the complementary modulus ' / 2^  is k =J 1 - k = 0.8802. In the form of the e l l i p t i c inte gral, equation ( 3 . 7 ) becomes 0 1.6828 t = j 0.8802 || • 0 In this case, to obtain a closed-form time solution, the im portant function is 0(t) rather than t(0). Under the trans formation v = cos 0 for inversion of the e l l i p t i c integral given in Table A, the Jacohian e l l i p t i c cosine function, Cn, is defined. That' i s , 1.6828t 0.8802 = 1.9117 t = Cn _ 1(v). Thus, v(t) = Cn(k,cot) = Cn(0.4745, 1.9117t). Retracing through the transformations which have been made, the closed-form solution for is tf(t) = P i 57 0.5675 L l •- CnLQ.4745. 1.9117t)1 1 + 0.2653 Cn(0.4745, 1.9117t) (3.8) Figure (3.6) compares the closed-form, shifted Chehy chev polynomial approximate solution given in equation (3.8) with an .010 .005 .000 -.005 - -.010 - -.015 - -.020 6(t) /\ 2.0 Least Square Error—- - \ Figure 3.6 The Error Distribution in 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 in this same model. The distribution of the error over the f i r s t period of the o s c i l  lation i s shown in this figure. In comparison, a linear 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 original normalization, x = 0.7825w, was performed on equation (3.1) to change the range of oscillation 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) is the shifted Chebychev, cubic approximate solution to x + x + x? + lOx = U(t), x(0) = x(0) = 0. This approximate oscillation is on the range (0, 0.7824). 3.3 An Extension of lanczos 1 Economization to the Transient  Response of lightly-Damped Models It has been shown that the amplitude-frequency re lation in a conservative, nonlinear oscillation need only be determined over an interval of symmetry. Nonlinear, non- conservative oscillations show a more interesting, continuous change in frequency as the amplitude of the oscillation is damped. Based on the Ritz - K-B equivalence for symmetric, conservative models and the improvement in phase obtained from shifted Jacobi polynomial truncations, an extension of these nonsymmetrically-weighted approximations to models of the form x + 2$x + f(x) = 0, x(0) = 1, i(0) = 0 is now undertaken. The K-B averaging method requires f(x) to have an explicit linear term and imposes cr i t e r i a for the "lightness" of damping and for the "quasi-linear" nature "of the characteristic. 59 The parameters a(t) and 9 ( t ) , given in Appendix C, for the change in amplitude and phase, respectively, are assumed constant over the f i r s t cycle of the oscillation. In this work, the exponential decay predicted hy the first-order approximate solution in equation ( C l ) of Appendix C is assumed, and a direct piecewise- linearization of the characteristic in terms of the shifted Jacohi polynomials is carried out. No restriction is placed on the change in frequency of the nonlinear oscillation over the f i r s t cycle or on the presence of a linear term in f(x). Only f(x) with an odd-symmetric, monotonic—increasing property are considered for the approximation hecause the nonsymmetrically- weighted 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 oscillation. To show the application of the linearization procedure, consider approximate solutions to the equation x + 0.4x + x + x 5 + 5x 5 = 0, x(0) = 1, x(0) = 0. (3.9) The K-B approximate solution to this equation is x'(t) = e-°* 2 t cos (^4.875 ' t). (3.10) This solution i s valid for the f i r s t cycle of the oscillation. As shown in Table (3.2), the characteristic in equation (3.9) has a nonlinear factor F = 0.55. A Lanczos1 Economization of this characteristic in terms of the shifted Jacohi polynomials, 60 & (0 .5,0.56)( x) f o n (0,1) yields the linear approximation %± + 0A\ + 5.782x1 - 0.900 = 0, x^O) = 1, x^O) = 0. Hence the approximate solution, valid for 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 is x . = -0.76. A truncation of the characteristic in equation mm ^ (3.9) on (0, -0.76) in terms of the G^ 0* 5' 0* 5 6) (x) polynomials yields the approximation ¥ 2 + 0.4x2 + 2.64X2 + 0.340 = 0, x2(0.78) = 0, •x2(0.78) = -1.66. The i n i t i a l conditions for this equation are obtained from equation (3.11) when "x^(t) = 0. The approximate solution for the next half-period on (0, -0.76) i s thus ~ 2 ( t ) = 1.19e" 0 , 2 t cos (l.6 l t - 2.70) - 0.13. (3.12) The approximate solutions 'x^(t) and 3f 2(t) are valid for the range 0 ^ t ^ 2 . 9 3 sec. and this is 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 is 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 y + 0.4y + y + 0 . 3 l 4 y 3 + 0.492y 5 = 0. 62 (3.13) As the oscillation is damped, the amplitude of the nonlinear terms decreases and the value of F, whioh has been defined on (0,1), also decreases. The normalized characteristic i n equation (3.13) has P = 0.28. After trunoating th i s oharaoteristic on (0,1) in terms of the G^ 0' 5' 0" 5 6) (y) polynomials, and then changing back to the range (x = 0, x = 0.56), we obtain ¥ 5 + Q.4x3 + 1.08x5 - 0.091 = 0, x^(2.93) = 0, x 5(2 .93) = 1.00 as the third piecewise approximation, valid for the next half cycle. The K-B solution and the above piecewise-linear solution for equation (3.9) are compared in Pigure (3 .7) . In Pigure (3 .8) , piecewise-linear approximations to e • o o x + 0.2x + tanh(2x) = 0, x(0) = 1, x(0) = 0, have been carried out in terms of the shifted Jacobi polynomials, 0^(0,5,0.44)(x). The characteristic tanh(x) arises physically as a model for nonlinear barium titinate capacitor characteristics (22) '. For this equation the K-B method does not apply because an explicit linear term is not present. Instead, lanczos' Economi zation of a least-square error, seventh-degree polynomial approxi mation to tanh(2x) is carried out for each linearization. To get an approximation to the amplitude at each half cycle of this damped oscillation, a linear Chebychev polynomial truncation is made on (-1,1), and then the solution of the resulting x(t),x(t) I. 00 0.75 0.50 Numerical Solution- 0.25 0.00 •0.25 0.50 -I— t 6.0 1.0 2.0, 5.Q 4.0 / 5.0 Piecewlse-Linear Approximat ion 0.75 •1.00 Figure 3.8 The Piecewise-linear Approximation of • • • x + 0.2x + tanh(2x) = 0 64 approximation is used to determine x m i n . This technique gives an approximate amplitude of x m^ n = -0,717 after one half-cycle and an amplitude of x = 0.538 after a f u l l cycle. Linear truncations in terms of the shifted Jacohi polynomials, G (0.5, 0 , 4 4 ) ( x ) ^ are then made for the f i r s t quarter-cycle and for each successive half cycle. The procedure i s the same as for the example in equation (3.9). Figure (3-8) compares the numerical and approximate solution. In the above two examples, the f i r s t piecewise-lineari- zation i s over only the f i r s t quarter cycle of the oscillation. This is useful because the nonlinear terms in the characteristic have a large magnitude where the amplitude of the oscillation is greatest, and thus a small approximation interval is important. The piecewise-linearization also allows different, weighted approximations to be made as the oscillation damps out. The possibility of using cubic truncations to improve the approximation has not been explored. Only a first-order approximation to the decay in amplitude of the nonlinear o s c i l  lation has been found. Therefore the Jacohian e l l i p t i c functions are not considered in the approximation of non-conservative oscillations. 3.4 Discussion and Possible Extension of the Results An advantage of the orthogonal polynomial truncation techniques given in this chapter is that they are generally applicable to a wide range of both conservative and non- conservative models for common physical systems. Also, the 65 refinement of the approximate model from a linear one to a cubic one is direct, and the same techniques are employed for symmetric, asymmetric and non-conservative models. In contrast, *he classical averaging techniques refer indirectly 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 classical techniques. Application of polynomial truncation to first-order differential equations was considered. In some cases, weighted, Jacobi approximate solutions, improved over those obtained by near-uniform expansion of the characteristic, have been found tut no relation to the weighting of the truncated approximation to a second-order model has been noted. This is because the same arguments for a small error at the end of an interval of symmetry do not apply to first-order systems. For this reason, approximation of first-order systems has not been studied in this work. Extension of this work to second-order models driven by harmonic time-functions is also possible. Results obtained (23) by Klotter^ '' show that the backbone of the nonlinear amplitude- frequency response curves is obtained by setting the driving term to zero. Hence, approximations to the backbone curve, improved over the one-term Ritz method, may be obtained directly 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, steady- state response. Also, the error criterion on the error at the 66 end of an i n t e r v a l of symmetry i s i m p o r t a n t f o r the s t e a d y - s t a t e a p p r o x i m a t i o n . Near reson a n c e , where the a m p l i t u d e of the o s c i l l a t i o n i s l a r g e , the e f f e c t of the n o n l i n e a r terms i s i m p o r t a n t . Por l i g h t l y - d a m p e d models, the a m p l i t u d e of the s t e a d y - s t a t e response approaches the backbone curve near r e  sonance. Thus, an improvement over the c l a s s i c a l one-term R i t z method c o u l d , perhaps, 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 w e i g h t i n g i n J a c o b i p o l y n o m i a l a p p r o x i m a t i o n s . 67 4. CONCLUSIONS Piecewise-linear and piecewise-cubic approximations, from which analytical solutions to classes of second-order non linear differential equations may be obtained, have been developed in this work. I n i t i a l l y , c r i t e r i a for the error in 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 is not possessed by the classical approximation methods. Error bounds were then given which prove that the maximum relative error in the solution to an approximate Jacobi or shifted Jacobi model is hounded, no matter how large the coefficients in the original nonlinear model become. An empirical measure of the departure from linearity, based on these error bound results, permitted approximations to be made using the shifted Jacobi polynomials, G r ^ 0 * ^ ' ^ ( x ) . Approxi mate time solutions for which the relative error at the approximate quarter period i s of the order of 10% of the maxi mum relative 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 for symmetric, cubic models by inspection. To provide a general approach for 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 in the first-order approximation case, and over the two-term Ritz method and least-square error approximation in the cubic case. Improvement in the amplitude-frequency approximation obtained from the K-B method has been shown possible for second-order nonlinear models with light, viscous damping. Again the G r ^ ^ " ' ^ (x) polynomials have been used for a direct, piecewise—linearization of the characteristic in 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 improve ment over the classical averaging methods with a decrease in tedious labor. 69 APPENDIX A The Closed Form Solution of Second-Order, Conservative System Models with Cuhic, Nonlinear Characteristics For the special case in which the cuhic characteristic has zero-point symmetry, Soudack^24^ has derived closed-form solutions in terms of the Jacohian e l l i p t i c functions. The cases of interest are: (i) x + ax - bx 5 = 0, x(0) = X , x(0) = 0, a,h>0. The solution is x(t) = XQSn(k,cot + K(k)) for X Q ^ , where co2 = a - 0.5 hX2 and k 2 = hX2/(2a - hX 2). S O «7 • ( i i ) x + ax + bx-5 = 0, x(0) = X , x(0) = 0, a,b<0. 2 2 The solution is x(t) = XQCn(k,cot), where co = a + bXQ and k 2 = bX2/2(a + bX 2). ( i i i ) x - ax + bx-5 = 0, x(0) = X , x(0) = 0, a,b> 0. An oscillatory solution, symmetric in the origin, is obtained in the form of case ( i i ) when a i s small. For a<bX^/2, the solution is x(t) = XoCn(k,cot), where co = -a + bXQ o o o and k = bX^/2(-a + bX^). An oscillatory solution for larger p ? a has the form x(t) = XQDn(k,Lot), where k = 2(1 - a/bXQ) and co = bXQ/2. The conditions on the i n i t i a l amplitude are 0<XQ<JlsJ^ and X Q £Ja/b.1 When the nonlinear characteristic does not have the above symmetry the closed-form solution may not be written by 70 inspection. The system model now has the form 2 M f + b r t + b,w + b 0w 2 + b,w3 = 0, w(0) = 1, w(0) = 0. (A.l) drg2 o 1 2 3 The range of o s c i l l a t i o n of w i n this model w i l l always he (0,1) after the normalizations desoribed i n paragraphs (2.1.1) and (3.2) are carried out. Prom equation (2.1), the f i r s t i n t e g ral of the motion i s 1(^)2 = E + _ ( v + b i w£ + ^ £ + ^ wi ) ( A > 2 ) k E t - V(w). The normalized i n i t i a l conditions of (A.l) give E^ = V(l), and hence & = -/2 [V(l) - V(w)] ^ (A.3) d£ from (A.2). The minus sign i s chosen from physical considera tions because w decreases as t increases from zero for th i s bounded o s c i l l a t i o n . Integration of (A.3) gives 1 w dw 0 1 * 2 [V(l) - V(w)] x O ^ w ^ l (A.4) or w dw t' = =E + At = ' .5) J , - _>> O ^ w ^ l , I 2 [V(l) - V(w)]V \k.\ 71 where 1 r dw At = - . • (A.6) ^2 [V(l) - V(wf) N 0 The quantity, At, obtained by dividing the range of integration in equation (A.4), is either one quarter period or one half period depending on the symmetry of the problem being considered, Equation (A.6) is a complete e l l i p t i c integral of the f i r s t (25) kind. A technique given by Hancockv allows the integrand i n (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. Evaluation of (A.6) is not necessary because the quarter period or half period may be obtained from w(t ). The integrand in (A.5) may be written in factored form as dw dw (A.7) J' ^ [ ( x - 9 ) (x- \ ) (x-u.) (x-rcfj^ It is clear from (A.2) and (A.3) that x = 1 is a root of the denominator polynomial in (A.7). Hence, this polynomial can' always be reduced to a cubic and the roots may be found. The roots in (A.7) are ordered so that 9 >A> u> x with the real roots ordered f i r s t . Legendre's transformation is w = (p + qy)/(l + y), where 72 9\(u_ + JC) - urt(9 + \ ) p q = 9 + X - JJ - £ Under this transformation dw _ (g - p) _ dy ^ Q) ^ ~ sH" " ± n V ) ( « 2 ± e V ) ' is ohtainedj where m2 = (p - 9 ) (p - \), n 2 = (q - 9 ) (q - X), 2 2 r = (p - \x) (p - «), s = (q - £)(q - K ) . For the special case in which 9 + X = LI + it the transformation w = y + ( 9 + X,)/2 = y + (u + rc)/2 is used. It i s shown hy Hancock that p + q and pq are always real. Prom equation (A.7), equation (A.5) may now he written in the form (q - p)dy . dt = mvJ ±(1 ± g 2 y 2 ) d ± h 2 y 2 ) V On the right hand side of this expression, h and g are defined so that h> g and then the integrand may he further reduced to dt' = (q - p ) d v mrh/ +(1 + ^  v 2)(1 + v 2) 1 dv; (A.9) N f : v ' +(1 + c 2 v 2 ) ( l + v 2) 2 2 2 where v = hy and c = g /h <1. Hancock shows that N is always real. It is also shown that of the eight possible combinations 2 '^2 of sign under the radical in equation (A.9), /-(1 + v ) ( l + c'"v ) \ 73 may be neglected, since W, which is positive for some of the original w, cannot be transformed to a function which is always negative by a real substitution. Using Table A below, closed-form expressions may be obtained for v(t ) according to each of the seven possible sign combinations in equation (A.9). In this table j ~2—\ A0 = /1 - k sin 0 , where k and 0 are defined as the modulus and the amplitude of t respectively. The trigonometric sub stitutions in Table A define the Jacohian e l l i p t i c functions.. The quantity k is defined as the modulus of the e l l i p t i c function, and 0 is 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 in Table A defines the Jacohian e l l i p t i c tangent function from v t' _ i f & ~ N y ( i + v 2 ) d + AV The complementary modulus is defined to bek = Entries VI and Via in Table A have the same form. In VI, v ^ l , while in VIa,v^^; 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".^ 2 6^ 74 TABLE A Standard Forms for Jacohian E l l i p t i c Integrals of the First Kind dv y ( l + v 2 ) ( l + o 2 v 2 ) _ dj \ ~ AG II III IV V VI Via VII dv . y (l-v 2)(l+cW dv y ( v 2 - l ) ( l + c 2 v 2 ) dv y ( l + v 2 ) ( l - c 2 v 2 ) dv J (l+v 2)(c 2v 2-l) dv y ( l - v 2 ) ( l - c 2 v 2 ) dv y ( v 2 - l ) ( c 2 v 2 - l ) dv y ( v 2 - l ) ( l - c 2 v 2 ) •* d0 A0 v = v = _ kdj _ z^dj _ k d0 \ ~ AC \ " A •\ Ai v tan 0 cos 0 v - V = cos 0 c sec 0 c k^ = 1-c' 2 c£ v = sec 0 k = k c = v = c sin 0 1+c' 1 1+c' 1 1+c' 2 2 c = sin 0 k = 1+c' 2 2 2 k^ = c^ v^ = s i n 2 0+cos2 0 k 2 = 1-c2 2 i 2 , 2 k = 1-c 75 APPENDIX B Derivation of the Jacohi and Shifted Jacohi Polynomials (27) Lanczos gives a development of the Jacohi polynomials from the hypergeometric series F ( \ © a -x) - 1 + ^ x - X(V 1)Q(Q + 1) x 2 ^ A ' w ' a , X j ~ a x ' a(a + l)»l-2 + \ ( \ + + 2 ) 9 ( 3 + l)(© + 2) x 3 + a(a + 1 ) ( a + 2) • 1 - 2 • 3 & e o • This series terminates with the power x i f X = -n. The choice o f © = n + a + 8 - l yields the set of orthogonal polynomials G^(oc , 3 ) ( x ) = p (_ n > n + a + p _ l f a . x ) f which are orthogonal on (0,1) with respect to the weight factor ¥(x) = x ^ d - x)^" 1. In this study the shifted Jacohi polynomials have been formed hy standardizing the above polynomials so that G r n ^ a , ^ ( l ) - 1. Defining the quantities 9 =n + cc + 8 - l , 8 = 8 + n and a = a + n, the f i r s t five shifted Jacobi polynomials may be written G 0 ( x ) = 1 9 Cr. i ( x ) = a [.! + - i x ] i \ 1 . ! ! a x + ^ x 2 i ! L a a a i J 2*1 T - l + - 3 9 ^ 4 X 2 + X 3 1 l P 2 L « ^ a a i a a i a 2 J aa., r 29 n ©_© G 2 ( x ) = 88 G3(x) = p p 76 aa G 4 ( x ) = Pfi i"2"i r i - t x + ^  x 2 - 4 9 ^ 6 6 x ? + w ^ 9 ? x * l 1 P 2 P ^ j a ao^ a a - ^ a a - ^ 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 ob tained 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 Q(x) = 1 P x(x) = X P 2(x) = [ X 2(2 L I + 1) - l ] P 3(x) = [(2LI + 3)x 3 - 3x] P 4 ( x ) = A [ i ( \ i 1 + l ) [ ( 2 ^ + 5 ) ( 2 ^ + 3 ) x 4 " 6 ( 2 ^ + 3 ) x 2 + 0 P 5 ( x ) = 4LI(H1+ 1) [ ( 2 ^ + 7 ) ( 2 ^ + 5 ) x 5 " 1 0 ( 2 | i + 5 ) x 3 + 1 5 X ] P6<x> = 8n(tx + l) ( n + 2) + 9 ^ + 7 ^ + 5 ) x 6 " 1 5 ( 2 ^ + 7 ) (2u + 5)x 4 + 45(2LI + 5)x?- 15 P 7 ( x ) = S|a(p. + l ) ( i i + 2) [.(2li + 11-K21. + 9)(2n + 7)x 7 ,T 21(2LI + 9)(2LI + 7)x 5 + 105(2LI + 7)x 3 - 105 x j . 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 Ritz Method and with a Linear Expansion i n U l t r a  spherical Chehychev Polynomials. The method of Krylov and Bogoliuhov or the K-B approximation applies to the second order model x + to x • + |if(x, x) = 0, where u i s a "small" parameter. For the model x + to2x + LI j^ 2&x + g(x)]] ~ 0, the K-B approximation may be assumed i n the form x^t) - a(t) cos (to t + 9 ( t ) ) I a(t) cos a(t). (28) Cunninghamv " ' shows the evaluation of a(t) and 9(t) hy averaging over one cycle of the o s c i l l a t i o n . The res u l t i s a(t) = A e " ^ ( C l ) and 2% ® ^ ~ a f s ^ a c o s °^ c o s a d c u (0.2) o J 2rcto 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 applies only to models with an e x p l i c i t l i n e a r term. Consider the special case i n which the damping term i s zero and the model takes the form x f toQx + ug(x) ~ 0. Then 2% f f ~ wo + 2TCCO a I g ^ a c o s °^ c o s a d a = to(a). 0 (C3) Integrating t h i s expression f o r the phase, one obtains cc(t) = to(a)t + C. If x(0) 0 i n the model, then C = 0. 79 Prom (0,3) we have 2JC c 2 (a) = co2 + g(acosa) cos ada, o 3ta J 0 where second-order terms in LI are neglected. Thus, 2% 2% co2 (a) = - ~ co2a cos 2ada + LI g(acosa)cosada rta L 0 J J J 0 0 2% = wa ^ t w o a c o s c t + t-"S( a c o s a) J cosada 0 2ic •JJJ^  J ° P(acosa) cosada, 0 where P(x) = toQx + ug(x) This expression thus determines the approximate frequency, to(a), for a nonlinear model x + P(x) = 0. For the case in which f(x) has odd-symmetry in x + f(x) = 0, x(0) = 1, x(0) = 0, Soudaclc^2^ has shown that a one-term Ritz approximation is equivalent to a linear ex pansion in terms of the Chehychev polynomials. The Ritz conditions give 2z .2 1 'o co" = = f(cos 9)cos ©d©. (C.5) 80 Similarly a Chebychev polynomial expansion gives 1 2% f(cos 9)cos ©d9 (0.6) 0 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 restricted to have an explicit linear term by the K-B approximation. Nevertheless, the ex pansions in (C,4), ( C o 5 ) and (C,6) show that, for 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. x f ( x ) ( l - xV-°' 5) dx c 81 REFERENCES 1. Hayashi, C,H,, Nonlinear Oscillations in Physical Systems , McGraw-Hill Book Co., New York, 1964, PP° 158-160, 2. Goldstein, H., Classical Mechanics , Addison-Wesley Publishing Co,, Reading, Mass., 1950, pp, 58-59. 3. Pipes, L.A., Operational Methods in Nonlinear Mechanics , Dover Publications, New York, 1965, p. 7. 4. Rosenberg, R.M,, "The Ateb (H) - Functions and Their Properties", Quarterly of Applied Mathematics, Yol. XXI, No. 1, April, 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 Lui, Y., "Applications of Ultra spherical Polynomials to Nonlinear Oscillations II. Free Oscillations", 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 Differential Equations", Tech. Report No. 2054-1. Stanford  Electronics Laboratories. Stanford University, Stanford, California, April 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 Hall Inc., Englewood C l i f f s , N.J., 1961, p. 451. 15. Davis, P.J., Interpolation and Approximation , Blaisdell 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. Rainville, E., Elementary Differential 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., and Zing Lui, Y., op. c i t . 22. Weber, E., "Introduction, Nonlinear Physical Phenomena," Proceedings of the Symposium on Nonlinear Circuit Analysis. Vol. II, Polytechnic Press of the Polytechnic Institute of Brooklyn, New York, April 23, 1953. 23. Klotter, K., "Steady State Vibrations in Systems Having Arbitrary Restoring and Arbitrary Damping Porces", Proceedings of the Symposium on Nonlinear Circuit Analysis. Vol. II, Polytechnic Press1 of the Polytechnic Institute of Brooklyn, New York, April 23, 1953. 24. Soudack, A.C, "Some Nonlinear Differential Equations Satisfied 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, op. c i t . , pp. 367-370. 28. Cunningham, W.J., op. c i t . , pp. 135-137. 29. Soudack, A.C, "Equivalence of the One-term Ritz Approxi mation and Linearization in a Chebychev Sense", American Mathematical Monthly. Vol. 72, No. 2, February, 1965. 

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