STUDIES ON THE NON-LINEAR VIBRATIONS‘ OF SYSTEMS WITH ONE ANT) Two TDEGREE- OF- FREEDOM Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY DAVID OWEN SWTNT 1972 "flu”-..unamw» w i ’1 ‘ , gm 4. LIBRA ’2 Y ;; - I; ‘1 . . s U -‘ ' ,0 Lla‘fizm' This is to certify that the thesis entitled Studies on the Nonlinear Vibrations of Systems with One and Two Degree of Freedom. presented by David 0. Swint has been accepted towards fulfillment of the requirements for Ph. D. degree in MEChaniCS 261 V—L 1/ 24 [7 Soda Major professor ifl‘ ‘7 2 Date .1f—v4 .. “1’ [k 1 0-7639 q’msm 139% I ABSTRACT STUDIES ON THE NONLINEAR VIBRATIONS OF SYSTEMS WITH ONE AND TWO DEGREE-OF-FREEDOM BY David Owen Swint This study presents an ultraspherical polynomial (U.P.) approximate method for representing natural vibrations for one and two degree-of- freedom spring—mass systems possessing nonlinear restoring forces. The nonlinear equations of motion are approximated by a set of linear equations over appropriate intervals of amplitude. General multilinear U.P. relations are develOped for approximating nonlinear restoring forces. The special cases of odd restoring forces, f(x) - ax + 8x3, sin x, and sinh x, are examined in detail, and used in the one degree-of-freedom system to obtain approximate expressions for period-amplitude relations. In general, for these odd functions considered, the bilinear U.P. approximate method gave improved period- amplitude relations as compared to previously published approximate period-amplitude relations obtained from linear U.P. approximations. In examining the two degree-of-freedom system a special case is treated whidh shows that where the linear U.P. method predicts a certain solution, the bilinear U.P. method does not. This special case is solved exactly by a finite difference technique and is shown to support the bilinear prediction. Finally, the finite difference method, which was developed for solving the two degree-of-freedom system exactly, is applied to other cases where the linear U.P. method predicts more than the usual two modes of vibration (superabundant modes). The finite difference method not only reveals the region over which these superabundant modes are present but also shows how these modes approach, for large amplitudes, limiting values predicted previously by other authors. STUDIES ON THE NONLINEAR VIBRATIONS OF SYSTEMS WITH ONE AND TWO DEGREE-OF-FREEDOM By David Owen Swint A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1972 (“$5 .% (£9.7QJCA¢%}M ACKNOWLEDGMENTS Sincere appreciation is extended to Professors David H. Y. Yen, R. W. Little, G. E. Mase and N. J. Hills for serving on my guidance committee. These gentlemen offered inspiration as teachers and continued encouragement during my graduate study at Michigan State University. I wish to especially thank my committee chairman, Professor Yen, who supplied the original motivation and who continued throughout this study to give generously of his time and of his valuable counsel. My graduate program was sponsored by the Air Force Institute of Technology, United States Air Force, in conjunction with a special program of the United States Air Force Academy. I also wish to thank Colonel Philip J. Erdle, Professor and Head, Department of Engineering Science, Mechanics, and Materials, United States Air Force Academy, for his assistance in this assignment and his confidence in me. Lastly, a special acknowledgment is due my wife, Sharon, for her efforts in typing this work, and more importantly, for her love and understanding during long periods of being without a husband. It is only through her patience that this work was completed. TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES I. INTRODUCTION 1.1 Background Survey . _ 1.2 Properties of the Ultraspherical Polynomials. 1.3 Applications of U.P. to Nonlinear Vibration Problems. 1.3.1 Problems of Single Degree—of-Freedom 1.3.2 Problems of Multiple Degree-of-Freedom. 1.3.3 Problems Involving Continuous Systems. 1.4 Two-Line Approximation to Nonlinear Vibration Problems. 1.5 Organization of Dissertation. II. BILINEAR ULTRASPHERICAL POLYNOMIAL APPROXIMATIONS 2.1 Ultraspherical Polynomial Approximation to a Nonlinear Function. 2.1.1 Limiting Cases. 2.1.2 Slope Parameters for the Case of a Nonlinear Odd Function Over a Symmetric Interval. 2.1.3 Slope Parameters Derived by Minimizing Methods. 2.2 Slope Parameters for Some Special Nonlinear Odd Functions. 2.2.1 f(x) - ax + 8x3. 2.2.2 f(x) - sin x. 2.2.3 f(x) - sinh x. III. SINGLE DEGREE-OF-FREEDOM SYSTEMS 3.1 Free Vibrations. 3.2 Forced Vibrations. IV. A SYMMETRIC TWO DEGREE-OF-FREEDOM 4.1 Equivalence of Approximation Techniques. 4.2 Discussion of the Regions for the Asymmetric Mode. 4.3 A Special Case of Anand's Problem. 4.3.1 Anand's Solution. 4.3.2 Bilinear U.P. Solution. 4.3.3 Exact Solution. 4.4 Superabundant Modes. V. SUMMARY AND CONCLUSIONS Page iii iv 15 21 24 25 42 42 52 54 63 63 78 89 9O 95 99 99 100 111 116 134 LIST OF REFERENCES APPENDIX A APPENDIX B APPENDIX C APPENDIX D APPENDIX E APPENDIX F 137 139 140 142 145 147 149 LIST OF TABLES Table 3.1 Dimensionless Period-Amplitude Relations for Various Nonlinear Single Degree, Free Vibration Problems. page 67 iii LIST OF FIGURES Figure Page 1.1 A Single Degree-of-Freedom System. . . . . . . . . . . . . 5 1.2 Graphical Approximation to the Non-Linear Function Sin x. O O O O O O O O I O O O O O O O O O O I O O O O O O 8 1.3 Period-Amplitude Curves for the Free Vibration.of a Single Degree-of-Freedom System With Sin x being the NOnlinear Restoring Force. . . . . . . . . . . . . . . 8 1.4 An Unsymmetric Coupled Spring-Mass System. . . . . . . . . 10 1.5 A Symmetric Coupled Spring-Mass System. . . . . . . . . . .10 1.6 Ergin's Bilinear Approximation to the Nonlinear Function f(X) . (I)! + X3. 0 o o o o o o o o o o o o o o o o o o o o .12 .2.1 Nonlinear Function of One Space Variable Approximated by Multiple Linear Approximations. . . . . . . . . . . . . 16 2.2 A Nonlinear Function C(x) Represented Between x0 and xtl by an Odd Function About z=O (xsxo). . . . . . . . . . 18 2.3 Nonlinear Function of One Space Variable Approximated by Multiple Linear Approximations. . . . . . . . . . . . . 34 2.4 Nonlinear Function Approximated by a Bilinear Approximation as xf+xm, k1+k,.k2+¢ . . . . . . . . . . . . 45 2.5 Nonlinear Function Approximated by a Bilinear Approximation as xt+0, k1+3flaxlxao, kz+k . . . . . . . . .45 2.6 Nonlinear Function Approximated by Line Segments Joining Points Along the Curve f(x)=ox + 8x3. . . . . . . .46 2.7 Bilinear U.P. Slope Parameters, k1 and k2, versus Amplitude, xm, for the Nonlinear Function f(x) - x + x3 (xt/xm = 0.2). . . . . . . . . . . . . . . . . . 49 2.8 Bilinear U.P. Slape Parameters, k1 and k2 versus Amplitude, xm, for the Nonlinear Function f(x) - x + x3 (Kt/xm = 0.5). C C C C O O C O O C C O O O O O O 50 iv 2.9 2.10 2.11 2.12 2.13 2.14 2.15 3.1 3.2 3.3 3.4 3.5 3.6 Bilinear U.P. Slope Parameters, k and k2, versus Amplitude, x , for the Nonlinear Function f(x) - x +‘x3 (xii/xm - 0.8). . . . . . . . . . . . . . . . . Bilinear U.P. Slope Parameters, k1 and k2, versus Amplitude, , for the Nonlinear Function f(x) - sin x. (xt xm - 0.2). . . . . . . . . . . . . . . . . . Bilinear U.P. Slope Parameters, k1 and k , versus Amplitude, , for the Nonlinear Function f(x) - sin 3 (x:?xm - 0.5). . . . . . . . . . . . . . . . . . Bilinear U.P. Slope Parameters, k1 and k2, versus Amplitude xm, for the Nonlinear Function f(x) - sin x (xt/xm - 0.8). . . . . . . . . . . . . . . . . . Bilinear U.P. Slope Parameters k and k2,versus Amplitude, xm, for the Nonlinear Function f(x) .Vslnh X (Xt/xm a 0.2) o o o o o o o o o o o o o o o o Bilinear U.P. Slope Parameters, R1 and k2, versus Amplitude, , for the Nonlinear Function f(x) - sinh x (x:?xm - 0.5). . . . . . . . . . . . . . . . . . Bilinear U.P. Slope Parameters, RI and k2, versus Amplitude, xm, for the Nonlinear Function f(x) - Binh x (Xt/xm ‘ 008). o I o o o o o o o o o o o o o o 0 One Degree-of-Freedom, Free Vibration.System Period- Ratio.r/r .versus Amplitude xm.for f(x) . x + x3 Xt/meUS). O O I O O O O O O O O O O O I O O 0 One Degree-of-Freedom, Free Vibration System Period Ratio T/T versus Amplitude xIn for f(x) = sin x (xt/%.059. O O O O O O O O O C O O O O I O O O C O O O 0 One Degree-of-Freedom, Free Vibration System Period Ratio T/T versus Amplitude xm for f(x) = sinh x (Ktyxmf05)o o o o o o o o o o o o o o o o o o o o o o o 0 One Degree-of-Freedom, Free Vibration System Error Plot of Period Ratio T/T versus Amplitude Ratio xt/xm 3 for f(X) - X + X (xméOo )0 o o o o o o o o o o o o o o 0 One Degree-of—Freedom, Free Vibration System Error Plot Period Ratio T/T versus Amplitude Ratio x lxm 3 0 t for f(X) ' X‘+ x (xm=100)o o o o o o o o o o o o o o o 0 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio T/T versus Amplitude Ratio x /x 3 O t m for f(X) - X + X (xmé200)o o o o o o o o o o o o o o o o 55 56 .57 60 61 62 .68 69 70 71 72 73 3.7 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio T/T versus Amplitude Ratio xt/xm for f(X) - X + 33(Xm303.0). o o o o o o o o o o o o o o o o 74 3.8 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio r/ro versus Amplitude Ratio xt/xm for f(X) - Sin X (XE? 1.0). o o o o o o o o o o o o o o o o 75 3.9 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio T/T versus Amplitude Ratio xt/xm for f(x) I sin x (me3.0). . . . . . . . . . . . . . . . . .76 3.10 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio T/To versus Amplitude Ratio xt/xm for f(X) a Sinh X (Xm=1.0). o o o o o o o o o o o o o o o o 77 3.11 One Degree-of—Freedom, Forced Vibration System Period Ratio T/To versus Amplitude for f(x) I x + x3 and Unit Step Function Excitation. . . . . . . . . . . . . . . .83 3.12 .A Nonlineax'Restgring Force f(x) MOdified to Give a New Restoring Force C(x) With a New Equilibrium Position x a x0. 0 O O O O O I O O O O O O O O O O O O O O 0 O O O O 84 4.1 Conservative Spring-Mass Two Degree-of-Freedom System. ... .90 4.2 ool Graph Showing the Six Regions for the Asymmetric Mode for a Two Degree, Symmetric, Free Vibration System. . .98 4.3 V I ff(y) dy Relation for a Nonlinear Restoring Force Approximated Bilinearly (y §_yt). . . . . . . . . . . 102 4.4 V I ff(y) dy Relation for a Nonlinear Restoring Force Approximated Bilinearly (y :_yt). . . . . . . . . . . 102 4.5 Frequency versus Amplitude Relation (a=-1, o1=0). . . . . . 114 4.6 Total Energy Curve Represented in xlsz-Space. . . . . . . .115 4.7 Variation of Frequency with Amplitude. . . . . . . . . . . .118 4.8 Frequency versus Amplitude Relationship (cz= .32). . . . . .120 4,9 Relation Between Amplitudes of Vibration of the Masses, a - .32. O C C O O C O O O C C C O ’ C O O O O C O O O O O .121 4.10 Frequency versus Amplitude Relationship (0 = 0). . . . . . .122 4.11 Relation Between Amplitudes of Vibration of the Masses, a s 0. O O O O O C O O O O O O I O O O O O O O O O O O O O .123 vi 4.12 4.13 4.14 4.15 Frequency versus Amplitude Relationship (0 = -.2). . Relation Between Amplitudes of Vibration of the Masses, a - -02. Frequency versus Amplitude Relationship (a = -.3). . . . Relation Between Amplitudes of Vibration of the Masses, a . -030 o o o 0 vii .125 O 126 .127 .128 I. INTRODUCTION 1.1 Bagkground Survey. Nonlinear vibration problems occuring in engineering are usually difficult to solve exactly and only a relatively few have been so solved. Approximate methods used to obtain solutions of nonlinear systems vary widely. Broadly speaking, such nonlinear vibration problems are grouped as those which are nearly linear (slightly nonlinear) and those which are strongly nonlinear. The bulk of work to date has been carried out on the former group primarily for two reasons. First,it is reasonable to suppose that certain phenomena known to exist in the related linear system are only slightly changed in the slightly nonlinear case; and second, the majority of physical systems falls into this group. Some approximate methods for obtaining solutions of slightly nonlinear vibration problems are the iteration method, the classical perturbation method, and the method of variation of parameters. The iteration method [2] usually consists in adopting the linear solution as a first approximation. Upon substituting this first approximation into the system of equations, a second approximation is obtained. This process can be repeated to obtain greater refinements provided that it is convergent. The classical perturbation method is based on generating solutions to nonlinear vibration problems from known linear solutions 2 which lie close to the nonlinear ones. This is accomplished by expanding the desired quantity in a power series with respect to some small parameter and converting the nonlinear problem into a set of linear problems. The method of the variation of parameters [2] consists in adopting solutions which appear to be simple harmonic solutions, but the amplitude and phase of which are assumed to be slowly varying functions of time. The amplitude and phase are the solutions of a new set of nonlinear (auxiliary) equations. These auxiliary equations are integrated approximately by utilizing the pr0perty that the quantities of interest vary only slowly with time. This research is concerned with yet another approximate technique of analysis-that of expanding the nonlinear terms of the problem in terms of ultraspherical polynomials. This expansion is made over an appropriate interval which best corresponds to the actual motion of the nonlinear system and linearization achieved by truncating the expanded series after the linear term. Denman introduced this approach first in 1959 and since then, with his co-workers, has intensively examined and extensively used this method in a number of studies [3,4,5,6,7]. The problems proposed in this dissertation are described below in Section 1.5 and are primarily motivated by the work of Denman and his co-workers. It is thus prOper at this point to review in.Sections 1.2 and 1.3 first the properties of the ultraspherical polynomials and the use made of these polynomials primarily by Denman and his co-workers. After this a two-line approximate method used by Ergin is presented in Section 1.4. In view of the work to date section 1.5 gives a brief overview of the organization of this dissertation. 3 1.2 Properties of the Ultraspherical Polynomials. The ultraspherical polynomials (U.P.) [20] are sets of polynomials orthogonal on the interval (-1,1) with respect to the weight function l-l/Z (l-xz) , each set corresponding to a value of A>-l/2. They may be obtained from (A) (A) -A+1/2 d ‘1 Hn-l/Z Pn (x) I An (l-xz) (—) (l-xz) dx where n (X) (-1) 1(A+1/2) I‘(n+2k) An - A 4 o 21:1 n! 1121) I‘(n+A+l/2) (A) {-1)n 2n n! I A a: 0 An (2“) ! (A) Here A3 is a normalizing factor, n is the degree of the U.P., and A is an index, i.e., a parameter identifying a particular subset of polynomials. A function, f(x), defined on the interval [-A,A] may be expanded in terms of these polynomials as Z (A) (1) f(x) - n-o an Pn (t) (A) where x I At and -A:x§A, ~15tgl, with coefficients an given as 1 (A) A-l/Z f f(At) Pn (t) (l-tz) dt (1) "1 an 1 (1) 1-1/2 I [Pn (t)]2 (1-t2) dt -1 Some frequently used subsets of the ultraspherical polynomials are: 1'0, Chebyshev polynomials of the first kind; XII/2, Legendre poly- nomials; AI1,Chebyshev polynomials of the second kind; and A+w, the powers of x. Appendix A contains additional properties of the ultraspherical polynomials. 1.3 Applications of U.P. to Nonlinear Vibration Problems. The following is a brief summary of some of the problems in which the U.P. method has been applied. 1.3.1 Problems of Single Degree-of-Freedom. The governing equation of motion for the free oscillation of the single degree-of-freedom system in Figure 1.1 is §+E(x)-o where f(x) I w: sin x and mo = Vmg/i Exact solution. The solution for the period can be expressed exactly in terms of elliptic integrals [4] as 4 n/2 d¢ 4 'r I— f = —- K(k,Tr/2) ° 1/2 w (l-k2 sinzd) w 0 where K(k,fl/2) denotes the complete elliptic integral of the first kind and k I sin (A/2), with A being the amplitude of motion. Figure 1.1 A Single Degree-of—Freedom System 6 The above can be rewritten nondimensionally as 2 III I- K(k,1T/2) 0 n where 2n 1' B ..— o w One-line apppoximation. Denman and his co-workers [3,4,5,6,7] introduced U.P. in solving single degree-of—freedom, free and forced, vibrating systems. The nonlinear function was expanded in terms of U.P. and truncated after the linear term. The vibration period was amplitude dependent, a fact not present when the same function is expanded by the Taylor series and truncated after the linear term. This is illustrated by the examples that follow. 1) pgaylor approximation. Replace sin x by the linear term of the Taylor series expansion as sin x 2 x and the equation of motion becomes x + mg x I 0. The approximate frequency relation is given by w* = m 0 or r*/to I (Zn/wo)/ (Zn/mo) = 1, where * indicates an approximation. 11) U.P. approximation. Replace sin x by the linear term of the ultraspherical polynomial expansion as (1+1) (A) (A) x 2 sin x = a1 P1 (-;)= (T) I‘()\+2) JA+1 (A) X where F is the Gamma function and Jn is an ordinary Bessel function of order n. The equation of motion becomes .. 1+1 x + mg (13‘) MHZ) Jxfl (A) x - o The approximate frequency-amplitude relation is given by 1 1+1 “2 i (E) T Thus T*(A9A) zit/(0* mo 2 A+l "1/2 a = "_ 3 (—) I‘(A+2) JA-l-l (A) 1:0 2U/wo (0* A or, r*(0.A) (2 -1/2 f” A = 0’ —— = —) J (A) To A 1 -1/2 T* (1/2.A) 3/n/2 J3/2 (A) for A a 1/2, __ = To (A)3/2 iii) Graphical approximation. A simplified graphical approximation as suggested by Denman and Liu [6] is now summarized here. This graphical approximation has been shown to be related to the one-line U.P. method. This procedure is illustrated below for the function f(x) I sin x as: 1. Plot the nonlinear function f(x) of the system as f(x) vs x. 2. Select an amplitude x = A and draw a straight line from the origin to an ordinate at A, such that the maximum error due to this straight-line approximation to f(x) in (O,A) is minimized as shown in Figure 1.2 for A = O and A = 3fl/4. f(A) = sin A (Radians) Amplitude, A Figure 1.2 Graphical Approximation to the Non—linear Function Sin x #4 h‘ P‘ O O O a- a~ a: Period Ratio, T/T H N ...: O O 0 (D p D b D p Amplitude, A Figure 1.3 Period-Amplitude Curves for the Free Vibration of a Single Degree-of~Freedom System With Sin x being the Nonlinear Restoring Force 9 3. The slope of this straight line is an approximate equivalent "spring constant" corresponding to this value of A. From this the approximate period ratio T*/To can be obtained as: .. “’0 g ,4: a: ff (slope @ A = 0) x + kx I 0 where w = VE- o w* = JE'a 70.467 (slope @ A 3fl/4) Therefore, 2n/ {0.467 1 T*/T = I = 1.46 2n/1 0.689 This value is then plotted as a point on the T/To versus A graph in Figure 1.3. 4. The process is repeated for various values of the amplitude A, and the graph of T*/To versus A is obtained. The graph T/To versus A in Figure 1.3 contains, in addition to the graphical approximation C, the exact result E, the linear Chebyshev approximation C, the linear Legendre approximation L, and linear Taylor approximation T; to the function sin x representing a "soft" restoring force. The graphical method is quite simple and can be used whether f(x) is expressed in terms of simple functions or numerically as a load-displacement plot. 1.3.2 Problems of Multiple Degree-of-Freedom. In addition to the use of U.P. in the study of single degree-of- freedom prOblems, Liu [7] examined a two degree-of—freedom, free and forced (with damping), vibrating systems represented in Figure 1.4. The coupling spring between the two masses was of the cubic type. This nonlinear spring was then linearized (one-line approximation) in 10 FT" x1 x2 linear cubic spring spring -1 —/vv\q .2 rYIIIV III/l III/IT Figure 1.4 An Unsymmetric Coupled Spring-Mass System x1 x2 cubic 6 cubic cubic spring spring spring m1 V V m2 O _ /7//77 777/7 ////// Figure 1-5 .A Symmetric Coupled Spring-Mass System 11 terms of ultraspherical polynomials. Frequency-amplitude expressions were obtained, tabulated, and compared with numerical results. More recently, Anand [18] examined a symmetric two-mass system using a method which will be shown in this research to be essentially a one-line U.P. approximate method (Figure 1.5). Aside from the efforts of Liu and Anand, however, very little has been done in the application of the U.P. approximate method to multiple degree- of-freedom systems. 1.3.3 Problems Involving Continuous Systems. Blotter [8] has applied the ultraspherical polynomial (one-line) method to systems governed by nonlinear partial differential equations in one space variable and one time variable. He assumed an autonomous system with the nonlinear term being the nonlinear forcing functions. A linear mode of deflection was assumed and this allowed him to linearize the nonlinear forces. This method was then applied to typical systems of strings, bars, circular membranes and plates on nonlinear foundations and with immovable end supports vibrating at large amplitudes. He found that if the Chebyshev polynomials (AIO) are used, the frequency-amplitude relationship agrees exactly with the first order perturbation solutions. 1.4 Two-Line Approximation to Nonlinear Vibration Problems. Ergin [9] introduced a two-line segment (bilinear) approximation for the nonlinear restoring force to obtain solutions to a number of single degree-of-freedom, transient—load problems. Ergin approximated the nonlinear function by two straight-line segments, each of which was determined by its slape and a point through which it passes. The problem reduced then to solving the same number of linear equations 12 f(x) = ax + x3 f(x) I 82(X) Amplitude, x Figure 1.6 Ergin's Bilinear Approximation to the Nonlinear Function f(x) I ax + x3 13 as there were line segments with the prOper matching of displacements and velocities at the transition points. For simplicity Ergin chose the slope k1 of the first line segment as the slope of the function at the origin, i.e., linear Taylor approximation, (Figure 1.6). The remaining task was then to establish the location of the transition point xt between the two line segments and the slope k of the second 2 line segment by minimizing the mean square error. Ergin's two- line approach has provided the motivation in this dissertation to construct, as a natural extension, a two-line ultraspherical polynomial approximate method. 1.5 Iggganization of Dissertation. In Chapter II a general development of the U.P. approximation to a nonlinear restoring force is presented, giving two bilinear approximations over some apprOpriate interval containing the equilibrium point. The bilinear U.P. approximation is then shown to degenerate into a one-line (linear) U.P. approximation as the transition amplitude point xt connecting the two lines either approaches zero or the maximum amplitude xm. Next, a mean square error minimizing method is used to generate a similar bilinear approximation. This is shown to agree with the bilinear U.P. approximation when certain conditions are satisfied. Three examples of odd, nonlinear restoring forces f(x) = ax + 3x3, sin x, and sinh x are then approximated by this bilinear U.P. approximation. In Chapter III one degree-of—freedom free, undamped, nonlinear vibration problems are solved for the three nonlinear restoring forces whose bilinear approximations were derived in Chapter II. Also included 14 in Chapter III is the one degree-of—freedom forced, undamped, nonlinear vibration problem involving a step function excitation with the restoring force being f(x) I x + x3. Chapter IV treats a two degree-of-freedom free, undamped, symmetric nonlinear vibration problem with cubic nonlinear restoring forces. An approximate technique which was developed by Anand and which yielded more than two modes of vibration (superabundant modes) is shown to be essentially the one-line U.P. approximation. A special case in which Anand's development [18] predicts these superabundant modes is then solved analytically using a bilinear U.P. analysis. The bilinear U.P. method shows no evidence that such a superabundant mode exists. An exact solution by.a finite difference method of this special case is then shown to agree with the bilinear U.P. prediction. Also included in Chapter IV are additional exact solutions of the two degree-of-freedom problem considered. The results then enable us to argue that the superabundant modes for Anand's problem do approach, for large amplitudes, Rosenberg's [15] straight-line superabundant modes. A brief summary of results as well as conclusions are contained in Chapter V. II. BILINEAR ULTRASPHERICAL POLYNOMIAL APPROXIMATIONS In this chapter we consider nonlinear functions of one space variable represented by two bilinear approximations over some appropriate interval which includes the equilibrium point. Next, the simplified case of the U.P. bilinear approximation of an odd function is considered and is compared against another bilinear approximation obtained by a mean square error minimizing method. Finally, three odd functions ax + 8x3, sin x, and sinh x serve to illustrate the procedure leading to their U.P. bilinear approximation. 2.1 Ultraspherical Polynomial Approximation to a Nonlinear Function. In this section we consider the nonlinear function G(x) shown in Figure 2.1. One may think of G(x) as representing some restoring force in the study of vibrations of physical systems. By equating G(x) to zero and solving for the real roots, one equilibrium point, xo say, can be found. On either side of the equilibrium point a particle will in general experience a different spring force. For the case above, the spring force is approximated bilinearly on each side of the equilibrium point over the total interval from a to B as follows. In the interest of simplifying the algebra, the origin is shifted horizontally to x0. Next, the two slope parameters k1 and k3 are computed for the intervals [0, Izmlll and [-Izm3l, 0] respectively, where Izmll I Ixtl-xol, Izm3| = [XO-xtzl and xt1 and xtz are suitably chosen transition points. To obtain the line k2, 15 16 G(x) 60(3) ' ‘ G(X)\ , k G (2) o 4 1 0 k3 ’ : 9 :- : 8m 4Fzm3 ‘V k Cow) T“ 112 h 1 l -y ....‘fi x0 ———L— zml k 2 1 xtl ym E. 3 __ Figure 2.1 NOnlinear Function of One Space Variable Approximated by Multiple Linear Approximations 17 the origin is shifted horizontally and vertically to xtl. The line k2 is then computed for the interval [0,ym] where Iyml I IB-xtll. This is followed by yet another shift in the origin horizontally and vertically to xtz, where the line k4 is computed for the interval [- lsml, 0], Ian] I Ixtz-al. In this way four linear functions are constructed, each computed by the one-line method. In each case, the shifted nonlinear restoring function is expanded in the ultraspherical polynomials {Pn(l)(t) , where each Pix) (t) is of degree n in t, and the polynomials are orthogonal on the appropriate symmetric interval. This is motivated by an approach by Howard [10] which concerns the construction of an odd function over the appropriate half interval. As an illustration, the shifted function on [0, Izmll] is 60(2) I G(z + :0) where x I z + x0 (Figure 2.2). In order to expand Go(z) on [0,Izm1] in polynomials orthogonal on the symmetric interval [-Izmll, Izmlll, we construct an odd function which coincides with Go(z) on [0, Izmlll. A function G(x) continuous on {-5, E] has the ultraspherical poly- nomial expansion m (A) (A) x m (A) (A) (A) (A) G(x) I 2 an Pu (...). 2 an Pn (t) = a0 Po (t) nIO n=0 (A) (A) (A) (A) (A) (A) . . . + a1 P1 (t) + a2 P2 (t) + 33 P3 (t) + where -1§_t :_1 and (-€§_le§) with the coefficients an(A) written as 18 G(x) T T Figure 2.2 A Nonlinear Function G(x) Represented Between x0 and xt by an Odd Function About z=0 (x=xo) 1 19 _(A) [1 G(Ec) P (t) m(A,t) dt (A) -1 n a = n (A) f1 [P (t)]2 w(A,t) dt -1 n 1-1/2 and where the weight function w(A,t) is defined as w(A,t) = (l—tz) and A>-1/20 Parameter k1(l,zml): On the interval [-|zm1|, Izmll] -[G(-z + x )] -|z I < z<0 G (2) - O ’ m1 _ O G(z + x ) O < z ('2 I o ’ -' m1 Expanding Go(z) over this symmetric interval in terms of these polynomials and truncating after the linear term, we obtain (A) 2 G (2) - 81 "' [k 0‘92 )1 z o Izml 1 m1 where, a(A) [10 ('2 lt) P(A)(t) ( ) d 7 k1()\,zm ) a -— . -— -1 () e z z A l mll l mll {1 [P1 (t)]2 w(A,t) dt .. -1 J o (A) 1 I -[G(-]zm1It + x0)] P1 (t) w(k,t) dt Izmll (A) l 2 2! [P1 (t)] w(A,t) dt 0 (A) + g 1 G(Izmllt + x0) P1 (t) w(A,t) dt (2,1) 20 .(A) (A) By recognizing that P1 (-t) I —P1 (t) and w(A,t) is the positive definite weight function, equation (2.1) is rewritten as, (A) f1G(|zm It + x0) P1 (t) m(A,t) dt 1 o 1 k (A z ) I (2.2) 1 9 m1 . Izmll f1 [p (A) 2 1 t)] w(A,t) dt 0 where Izmll= Ixt1 —xo| In a similar manner the parameters k3, k2 and k4 become (A) 1 _ _ 1 g [G( lzm3|t + x0)] P1 (t) m(A,t) dt k (1,2 ) I 3 "'3 ‘zml (A) (2.3) 3 f1[P1 (t)]2 w(A,t) dt 0 where Izm3| = lxo ’ xtzl , k2(}‘9kls xtl’ym) " (A) '7 fl [G(Iym]t + xtl) - k1(xt1 -xo)] P1 (t) m(A,t) dt := 1 o lyml (2.1.) 1 (A) 2 h, f [P1 (t)] w(A,t) dt .1 0 where Ile I IB-xtll, and 21 k4(}u k3, xtz’ 8m) (A) 1 £1-[G(-Ism]t + xtz) -k3(xt2-xo)] P1 (t) w(A,t) dt lsml (A) f1 [Pl (t)]2 m(A,t) dt 0 (2.5) where lsml I Ixtz-ol 2.1.1 Limiting Cases. If we allow the transition points, x t1 and xtz, to simultaneously approach the extremes in amplitude, B and a , the four slope parameters k1, k2, k3 and k4 will degenerate into two slope parameters identical (for AIO) to that obtained by Howard [10]. Applying these limiting processes to each of the four slope parameters (2.2), (2.3), (2.4) and (2.5) respectively we obtain (A) Ilc(|zm1|t + x0) p1 (t) m(A,t) dt 1 o Lim [k1(1,zm1)] I Lim I I xtA+8 x +8 2 (A) 1 ‘1 “1 f1 [pl (0]2 m(A.t) at AIO i=0 0 £1G(IB-x°|t+xo) T1(t) w(0,t) dt . (2.6) lB-xol f1[T1(t)]2 w(0,t) dt 0 22 and Lim “‘30“... )] I Lim 1 . art-2* a 3 xt; a Izm3| A- o A= o - 1 (A) " i -[G(-Izm3I t + x0)] p1 (c) w(1,t) dt (A) f1 [p1 (t)]2 w(1,t) dt 0 ..J F n [1 -[G(-|x6-o| t + x0)] T1(t) w«3.t) dt 3 1 O Ira-0| (2.7) f1 [T1(t)]2 w(0,t) dt 5 O ...J and Lim [k2,(A,k1,xt1,ym)] . Lim _1__ "t‘f 5 851* B lyml A ' 0 A 3 0 (2.8) (A) ‘ £1 [9(Iyml t + xt1)-k1(xt1-xo)] P1 (t) w(1,t) (it (A) f1 [P1 (t)]2 w(A,c) dt 0 23 and Lin [k (1,1: ,x ,s )] =Lim 1 . 4 3 t m act-go 2 xt-2>o. Ian] A - o A = 0 j_’ (2.9) 1 (A) 7 g '[G('l8ml t + xt2)-k3(xt2-xo)] P1 (t) w(A,t) dt 1 (A) 2 +00 1' [P1 02)] (00¢) dt J O Upon examining equation (2.6), we find P _1/2 a f1 G (IBI t) t (1-t2) dt Lim [k1(1,zm)] _ 1 o ° ‘6’ B 1 _l—B|- -1/2 1 f1 t2 (1-t2) dt J A - o " o (2.10) a ...—L. .ll f1 '1/2 |B| n o GO(IBI t) t (l-tz) dt where Go(lBlt) . G(IBIt + x0), T1(t)-:/;, IBI = IB-xol. I: [T1(t)]2 (1-t2) dt = n/2 Similarly, equation (2.7) yields, -1/2 Lim. [k3(A,zm )1 - _;;,__5_ [1 ~Go(—|Alt) c (1-t2) dt (2 11) xt+ a 3 |A| 1T 0 . 2 AIO 24 where Go(-IAIt) I G(-IA|t + x0) and [AI = Ixo-al The limiting cases of k1 (2.10) and of k3 (2.11) represent the two slope parameters (Chebyshev polynomials, i=0) that approximate G(x) over the interval [o,8]. These agree exactly with Howard's result. The limiting cases of kg (2.8) and of k4 (2.9) are meaning- less since the respective amplitudes ym and 8m approach zero. 2.1.2 Slope Parameters for the Case of a Nonlinear Odd Function Over a Symmetric Interval. Restricting our consideration to nonlinear odd functions over a symmetric interval I-xm, xm], we take the origin as the equilibrium point and equate the absolute values of the transition points about the equilibrium point. In Figure 2.1 we take G(x) as an odd function with onO to establish the equilibrium.point as the origin and then observe that Izmll I xt and Izm3| I -xt specify equal transition amplitude points. Substituting these relations into (2.2) and (2.3) we find that k1 and k3 yield the equivalent slopes f1 wit dt 1 o k =1, .._ (2.12) 1 3 xt flw t2 dt 0 where A-l/Z . w = w(A,t) = (l-tz) x a xtl ' xtz 'E.I G(xtt) 9 x I x t 25 Similarly, from (2.4) and (2.5) we find that k2 and k4 yield the equivalent slopes f1 mugI t dt-klxt [1 u) t dt l o k -k a— (2013) 2 4 y f1 w t2 dt m. o where ym - xmfxt g I G(ymt + xt) x I ymt + xt 2.1.3 Slope Parameters Derived by MinimizingTMethods. In this section we shall first discuss a number of techniques for minimizing the error between the approximating function and the nonlinear function. We then apply a technique modeled after an approach by Ergin [9] to derive slope parameters for the linear and bilinear approximations which correspond to equations (2.12) and (2.13). A variety of methods has been proposed to minimize the error between an approximating function and the actual nonlinear function [21]. Ergin investigated three methods: 1) That the work done per cycle by the bilinear and the nonlinear spring forces is the same; 2) that the mean square error between the bilinear and the nonlinear spring forces is minimized; and 3) that the spring forces are equal at the maximum displacement point. Ergin's development centers around the second approach. In this research we will investigate a fourth method which can be regarded as a generalization of Ergin's approach. 26 Linear Approximation. The approximation of a nonlinear function fo) over a symmetric interval by a polynomial g(x) of any degree can be achieved by minimizing the integrated square of the error, subject to some appropriate weight function. We impose at this point one condition on these polynomials: pn(t) is a polynomial of nth degree in t. (2.14) Thus, we express the square of the error as, x EZI lf-g n I f m w(x) [f(x) - g(x)]2 dx * ”x m or, xm E2 = f w(x) [f(X) - aOPOOt) “319109 ----- -«=1kpk(2t)]2 dx "X “I By a change of variable, x I xmt, m(x) and pn(x) are represented as w(t) and p(t) respectively since they are arbitrary at this point. Now, E2 becomes k E2 - 1'1 x1, Mt) [f(xmt) - 2 anpm(t)]2 dt -1 n=0 and by squaring the integrand, we have k 122 -- f1 xm ...(c) [f(zqnt)]2 dt + 2 0 an2 I: xm (0(t) [pn(t)] 2 dt n: — * Snyder [11] defines this relation as the least square norm of the difference of two functions over an interval. 27 k -2 )2 an flights“) f (xmt) ph(t) dt nI0 -l k k +2 2 z anal I1 Xmm(t) pn(t) p,(t) dt (2-15) nIO 2-0 -1 nit At this point we impose the orthonormal relation f: w(t) pn(t) p,(c) - an, (2.16) where 6“, is the Kronecker delta. Now equation (2415) can be simplified by using (2.16), k 22 - x1 1m w(t) [f(xmt)]2 dt + A: an2 x r1 o.)(t) [pn(t)]2 dt -1 n=0 m -1 k -2 2 anxm [1 p(t) f(xmt) pn(t) dt n=0 -1 The partial derivative of E2 with respect to each an must vanish in order to make E2 minimum. Thus 322 -;—- - o - zanxIn i: w(t) [pn(t)]2 dt-me i: w(t) f(xmt) pn(t) dt an Hence, 1 {1 Mt) f(xmt) pn(t) dc an . (2.17) I1 w(t) [pnm]2 dt -1 28 The set of polynomials corresponding to a particular weight function that satisfy the conditions (2.14) and (2.16) may be determined step by step.* Since po(t) is of zero order, we write po(t) = a and determine this constant from the normalizing condition [1 w(t) azdt - 1 (2.18) —1 Since Schelkunoff [12] has shown that a weight function of one, th) I 1, yields the Legendre polynomials, a natural extension would 1-1/2 then be to use the weight function, w(l,t) I (l-tz) , associated with the ultraspherical polynomials. The Legendre polynomial is a special case of the ultraspherical polynomial when A- 1/2. Thus, 1-1/2 with.m(1,t) I (1-t2) equation (2.17) reduces to -1/2 1/2 (t) a - [[1 w(A,E) dt] . r(A+1)/[/?'T(A+1/2)] po -1 Next, we determine p1(t) I b +»ct from the orthonormal conditions, I1 w(A,t) po(t) p1(t) dt = f1 w(A,t) [a] [b + ct] dt -1 -l 0 and, f1 m(A.t) [p (t)]2 dt = 1 -1 1 * The process is known as the Gram-Schmidt process [22]. 29 Solving for b and c, we find 1/2 p1(t) - [2r(a+2)/[/h r(x+1/2)]] t Proceeding to the next polynomial and next, and next, we find a definite pattern which reduces to the form, -1/2 212 J? r(A+1/2) P(2l+n) (A) (A) pnCI) . ' Pn (t) = cnpn (t) (n+x) r(n+1) P(l+1) r(2x+1) (2,19) After some algebraic manipulations Cn can be reduced to the form (see Appendix B for details), 1-21 -1/2 2 n F(2l+n) (x) pn(t) - Pn (t) (n+1) (r(x))2 P(n+l) (A) where the P11 (t) polynomials are called the ultraspherical polynomials. These ultraspherical polynomials are orthogonal but not normalized. The constant multiplier, C is the normalizing factor. n, Upon substituting equation (2.19) into equation (2.16) we obtain, (1) (A) f1 w(A,t) CnPn (t) c, P£ (t) dt = 6n -1 2 or, (A) (A) -1 11 m(l,t) Pn (t) P2 (t) dt = 5n2 [cncg] -1 30 roan*£ II A l—ZA -2 2 n PCZXHn) n=£ K (Cu) .5 (n+1) Ir(x)12 r(n+1) A¢o Having shown that the ultraspherical polynomials can be generated from minimizing the integral of the squared error relationship, we now show the connection between the al coefficient of the approximating polynomial and the slope parameter k. Now, consider approximating a nonlinear odd function f(x) over a symmetric interval by a linear approximation--that is, we truncate the approximating polynomial, g(x), after the linear term. k f(X) = g(x) = 2 an pn (x) nIO letting k I l, g(X) . aopo(x) + 8191(X) or, by change of variable, x = xmt (A) (A) g(xmt) I aop°(t) + a1p1(t) I aoCoPo (t) + alclPl (t) 31 Since f(x) is an odd function, only the linear term survives, there- fore BCth) I aICIIZAt] I nalt 01‘ g(X) = [nallxh] x - kx (2.20) where n I 2ACl and k = naI/xm From equation (2.20) (A) Ilmcx,t) f(xmt) C1P1 (t) dt 11 w(l,t) f(xmt) t dt a1 -1 =31_ 0 == (A) n f1 w(l,t) [0121 (c)]2 dt f1 w(l,t) t2 dt -1 0 Thus, f1 w(l,t) f(xmt) t dt 1 0 8(3) ‘3 JJ- "’ x ‘h n f1 0 w(A,t) t2 dt f1 w(l,t) f(xmt) t dt g(x) I 'i;' o x = [k] x f1 w(l,t) t2 dt 0 (2.21) 32 Denman [5] obtained this general relationship for k by another method, namely, by expanding the nonlinear odd function in terms of ultraspherical polynomials and truncating after the linear term. Based on these observations, the mean square error method was applied in generating polynomial approximations to smaller segments of a larger interval. Qualitatively, this provides a means of obtaining a multilinear polynomial approximation over the total interval and in particular, a bilinear ultraspherical polynomial approximation. Bilinear Approximation. If we consider a general nonlinear function f(x) derivable from the potential function V(x) I ff(x) dx and assume that the origin has been shifted to the local minimum potential point (dVCx)/dx I O), we can create two odd functions about this minimum potential point (one function that coincides with the non- linear function for negative arguments and one that coincides for positive arguments). Having done this, we can compute the slope parameters RI and k3. Similarly, by two more shifts of the origin the slope parameters, k2 and k4 can be found. Now, we proceed to show under what conditions the mean square error method generates the ultraspherical polynomial approximating function. With no loss of generality we can simplify the algebra by choosing a nonlinear odd function with local minimum point at the origin. The integral expression to be minimized is 2 h 2 XI“ 2 z - "f-g" - f w(x) [f(x)-g(x)] dx ... 2r w(x) [f(x)-g(x)] dx 33 X . 2 fxt tub!) [Hid-300]?- dx + f mob!) [f(X)-s(X)]2 dx 0 Kt The first integral can be simplified by a change of variable, x = xmt. The function in the second interval is shifted so that its origin is the transition point xt. As shown in the Figure 2.3 below an odd function is created, followed by a change of variable to represent the interval as [-1,l]. Thus, k 32 = xt i: th) [f(xtt) - i=0 anpn(t)]2 dt k + ym f1 w(t) [fo(t)- z bnpn(t)]2 dt ‘1 n=0 where fo(t) is the odd function defined as fo(t) I -[f(-ymt + xt) - a1n], -l §_t < 0 N 2.22 [f(Ymt + It) '3101» 0 < t j.1 ( ) and .J Cl I normalizing factor n I 2101 ym ' xm-xt Squaring the integrands, we have 34 f(x) f (y) 'xm Amplitude, x Figure 2L3 Nonlinear Function of One Space Variable Approximated by Multiple Linear Approximations 35 k 22 - 11 x w(t)[f(xtt)]2 dt + 2 a3 xt rlwcc>lpn(t)]2 dt -1 t nIO -l k ‘ 1 ( ) ( ) ( ) d -2 2 It I t f xtt pn t t nIO an -1 k k 1 2 +2 2 Z anazxt f1 w(t) Pn(t) P£(t)dt + Ym {1 ”(t)[fo(t)] dt nIO 2-0 -1 nfz k 2 1 2 k 1 +-2 b y f w(t)[p (t)] dt —2 z b y f w(t)f (t)p (t) dt n” n m-l n n-onm-l O n k k 1 ( 23) + 2 ’3 73 b b y f w(t)p (t) p (t) dt 2. nIO EIO n z m -1 n 2 nil At this point we impose the orthonormal relation I1 w(t) pn(t) p£(t) dt - an, (2.24) -1 The error relation can now be simplified by substituting equations (2.22) and (2.24) into (2.23) to obtain k 22 - x, f1m(t) [f(xtfiflz a: + z anzxt 11 w(t) [pn(t)]2 dt -1 nIO -1 k -2 z anxt 11 w(t) f(xtt) pn(t) dt + ym 11 w(t)[f0(t)]2 dt nIO -l -1 k k + 2 b3 ym I1 wtpn12 dt -2 z bnym r1w(t)[fo(c>1pn(c) dc nIO -1 nIO -l (2.25) 36 The partial derivatives with respect to an’bn and xt must vanish in order to minimize E2. Taking first 3(E2)/3an I 0 for n i l and recognizing that afo(t)/3an I 0 for n # l we obtain, 3132 -;:- - 2 anxt {1 m(t)[pn(t)]2 dt - 2xt f: w(t) f(xtt) pn(t) d, . o n Thus, fl w(t) f(xtt) pn(t) dt -1 an 8 . 11 w(t) f(xtt) pn(t) dt x1 w [pn‘_1/2 for A) -.5. So technically, the A I -.5 case cannot be considered in an ultraspherical polynomial expansion. Taking the limit of (2.40) and (2.41) as A + -.5 however, we obtain Lim [k (1,x )1 = Lim. [a + 38x 2/[2(x+2)]]= a + SK 2 (2.44) 1+’-.5 1 t 1+—.5 t t and Li k A = + B 2 + + 2 2.45 A+E.5 [ 2( ’ym)] a (Km xm¥t xt ) ( ) 48 Taking values from Figure 2.6 we have H -—l- I slope of first line segment I f(x)/xt x XIx t t a 3 (l/xt) [axt + th) (2.46) I o + 2 Ext H2 xm'xt I slope of the second line segment Ill/(xm-xtn {£00 I -f (x) I } X'Jfin x-xt I [l/(xm-xt)] [axm + me3—axt-th3] :- 2 2 a + 8(xm + xmxt + x ) (2.47) t Comparing equation (2.44) with (2.46) and (2.45) with (2.47) we see that the A I -.5 bilinear limiting case is equivalent to a linear interpolation of the nonlinear function. Dependence of RI and k on Amplitude. For the nonlinear function 2 f(x) I ax + 8x3 with a I l and B I 1, the dependence of RI and k2 upon the transition amplitude xt and the maximum amplitude xmis shown in Figure 2.7, 2.8 and 2.9. By comparing these figures we observe that the k2 curves approach k, the one-line U.P. approximation, for small xt (Figure 2.7) as previously predicted by (2.43). Similarly, we observe that the k curves approach k for large xt (Figure 2.9), as 1 previously predicted by (2.42). Slope Parameters, k1 and k2 35 3O 25 20 15 10 49 D ’ d I k2 (A=-05) A I . _ , /* k2 (A=0) / / . / L k (AIO), linear U.P. I slope parameter .’ / /' y /' u?’ . / /o/’ k1 (A80) 1’ / ’ . l/za/‘O’ ’ ’::L;::.-‘.::r21¥55£=:85£ZI- 0 1 2 3 4 5 Amplitude, xm Figure 2.7 Bilinear U.P. Slape Parameters, k1 and k2, versus Amplitude, , for the Nonlinear Function f(x) I x + x , (xt/xm I 0.2) Slope Parameters, RI and k2 35 30 25 20 15 10 50 k2 (A=-05) ’ k2 (AIO) k2 (AI6) II ‘ ’/ k (AIO), linear / U.P . slope / parameter Amplitude, xm Figure 2.8 Bilinear U.P. Slope Parameters, RI and k2 versus Amplitude, xm, for the Nonlinear Function f(x) I x + x3, (xt/xm I 0.5) Slope Parameters, RI and k2 35 30 25 20 15 10 51 w r j ’r 1 . , / I / I I k2 (4:0) . ‘7» k2 (AI-.5) I I ' I (b I, I q l I I l I ‘ k (AIO), linear . 5 U.P. slope . l' I, parameter I I, ’ k1 (A=-.5) * . // k1 (AIO) . I 1 I I I o I I //H5 I / . / . / ’/ ’ /’ , K ’ / //A " l/ , /‘r / _ / // / A k1 (1=6) . ’ I A” ff / ,g"" --av”::;:>”1r I l a, . /’M—"‘M‘ 0 l 2 3 4 5 Amplitude, xm Figure 2.9 Bilinear U.P. Slope Parameters, RI and k2, versus Amplitude, , for the Nonlinear Function f(x) I x + x , (xt/xIn I 0.8) 2.2.2 f(x)9I‘§in x.- 52 we again substitute the nonlinear function f(x) I sin x into equation (2.12) and (2.13) to obtain k1(A,xt) ' .1; xt substituted back in (2.48) to yield 1‘10“.) - [rennet/2) Also k2(A,xm-Xt) ' " 1-1/2 ) f(l-tz) [sin (xtt)] t dt 0 A_1/2 (2.48) f1(l-t2) t2 dt 0 J \- The integrals in (2&43) are evaluated in Appendix E and are X+l " 1-1/2 1-1/2 1 f1 (1-t2) [sin(xtt)] t dt - klxt f1(1-t2) t dt 0 o XII/2 xm—xt 11 (1-c2) :2 dt O (2.50) The integrals in (2.50) are evaluated easily following similar steps to those in Appendix B so that k2(A,mext) _:3E1;EE_P(A+2) l1? (xm-Ixt) pom/2) .f 53 1‘ 0+2) + cos xt JA+1 C IX ) 1+1 x“ t < ) 2 2n xm-x 21203.2) co (--1)n (—-% + —— 2: 2 sin x mext nIO P(n+l/2) F(n+A+3/2) (2.51) For comparison we note that the one-line ultraspherical polynomial approximation is [P(A+2)/(xm/2)A+1] JA+1 (xm) (2.52) _Limiting Cases: In the limit as xt-+xm we take the limit of (2.49) and obtain Lim [k1(A,xt)] - Lim [r(1+2)/(xt/2)“1] xt+xm xt-rxm A+1 F(A+2)/[xm/2] JHl (xm) a k (2.53) Similarly, taking the limit of (2.51) we obtain -2k1 xt F(A+2) Lim [k2(A,x -xt)] I Lim xt+xm m xt+xm I;(xm-xt) F(A+3/2) A l + 1‘(A+2)/[(xm"xt)/2] + [ °°S xt JA+l (Km—x9 J x -xt 2n °° n (-—“5 ) + .ZESliEL sin x E (-1) m ’51:”: [ t nIO I‘(n+ 172) 1"(n+A+ 3/27] '* (2.54) 54 we consider the second limiting case x£+0. The bilinear U.P. approximation again degenerates into the linear U.P. approximation. To see this we take the limit of (2.49) and (2.51) as x£+0 and obtain A Lim [k1(A,xt)] -‘an [r(1+2)/(xt/2) +11 [51+1 (xt)] = o (2.55) xt+0 xt+0 and ‘A+l 21:0 Ik2(A.mext)J I (F(A+2)/(xm/2) ] Jx+l (XE) I k (2.56) t As shown previously for the cubic nonlinear case, the bilinear ultraspherical polynomial approximations degenerate into the linear ultraspherical polynomial approximation as seen upon comparing (2.53) and (2.56) with (2.52). Dependence of k1 and k2 on Amplitude. For the nonlinear function f(x) I sin x the dependence of RI and k2 on both the transition amplitude xt and maximum amplitude xm is shown in Figures 2.10, 2.11, and 2.12. Again, comparing these figures we observe that the k 2 curves approach k, the one-line U.P. approximation, for small xt (Figure 2.10) as predicted by (2.56). Similarly, the k curves approach 1 k for large xt (Figure 2.12) as predicted by (2.53). 2.2.3 f(x) : sinh x. Into the equations (2.12) and (2.13) we substitute the nonlinear function f(x) I sinh x to obtain ' 1 A-l/Z ‘ f (1-c2) [sinh (xtt)] t dt k1(A,xt) -.1_ o (2.57) A-1/2 f1 (l-t2) t2 dt L o J Slope Parameters, RI and k2 0 U1 -05 55 \ ~‘\ . ‘ \\ \ a P \\ \ ‘x p \ ,///r ‘\ \ \K ‘\ P k2 (A30) \ V k (AIO), linear U.P. slope parameter ' Amplitude, xm L J j I Figure 2.10 Bilinear U.P. Slope Parameters, R1 and k2, Amplitude, xm, for the Nonlinear Function f(x) I sin x, (xt/xIn I 0.2) versus c U! Slope Parameters, R1 and k2 C) “'05 56 '\ . \. ' IA , \ . ‘\ D \ d \\5. . \ k (AIO), linear ‘ \ U.P. slope \ parameter . F \ . ~ ‘1 . X \Y/kz 0:0) 5 3 a \ v 1 2 ‘\ 3 D \\ I! Amplitude, xm \.\ k (Ag-.5) \ 2 \ J P \ . k2 0:6) \ ' (h \ ¥{ \ h a A a l \\ A Figure 2.11 Bilinear U.P. Slope Parameters, RI and k2, versus Amplitude, xm, for the Nonlinear Function f(x) I sin x, (xt/xm I 0.5) Slope Parameters, k1 and k2 l. 0 U -05 57 P I k1(AI6) . ‘V‘x ~~ ":'_—_—__‘h_'-77“‘-—o-._ \ \ ‘ b \ ~ . " \ \ K ‘ ‘1 r- \ \\\ “\ \ \k1(AI-.5) .. I ( \ . \ \ k1(x'0) , ‘\ ‘I \‘ 4 \ A. \ \ ' ‘4 \ I. - ‘\A I ‘ \\ k(AI0), linear ‘\\ d ' U.P. slope \ parameter J T \ \ ‘q q _ \ b \ \ d \ \* s \51 X : ¢ 1 I \ 2 - Amplitude, x \ . \ .- \ k2(}\=6) \ ‘ \ k2(A=0) \ q . \Y“////,I1: ' 4 \ \ k2(A"-.5) ' \ \ \\ In \ q 1 L 1 L X Figure 2.12 Bilinear U.P. Slope Parameters, k1 and k2, versus Amplitude xm, for the Nonlinear Function f(x) I sin x (xt / xm I 0-3) 58 The integrals in (2.57) can be evaluated easily following similar steps to those in Appendix E, and it follows that 0 ) 0+2) ix --—————___ k1 t 2+1 Cxt/Z) ”2+1 (xt)] (2.58) where Ian) I (Ii)n J (ix) is a modified Bessel function. n Also I2k1xt F(A+2) + Ir(x+2)/(xm~x,/2)A+11- IF'me-xt)P(A+3/2) k2(laxmfxt) - coshxt IA+1 (xm-xt) 2r(>.+2) .. [(xm-xt) /2]211 + sinh xt Z (2.59) xm-xt nIO I(n+1/2)P(n+A+3/2) we note also that the one-line ultraspherical polynomial linear approximation is k - [maven/29“] 1,11 (xm) (2.60) Limiting Cases. In the limit as xtIxm we take the limit of (2.58) and obtain Lim [1.10.59] uLim [r(1+2)/(xt/2)‘+1] IA+1(xt) x *1 Xt+xm t m 11 - Uneven/2)“ 1m (25.) = k (2.61) 59 Similarly, taking the limit of (2.59) we obtain Lim IKZQ aim-9%)] I °° X t‘VXm we consider the second limiting case xtIO. The bilinear U.P. approximation again degenerates into the linear U.P. approximation. To see this we take the limit of (2.58) and (2.59) as xt+0 and obtain Lim [k (71.2%)] = Lim [I‘(A+2)/(xt/2)>‘+l = o xt+0 l xt+0 and Lim Ik2(A,xm-xt)] = [r(x+2)/(xm/2)“1] I,“ (xm) = k (2.62) xt+0 Dependence of k1 and k on Amplitude. For the nonlinear function 2 f(x) I sinh x the dependence of k and k2 on both the transition amplitude 1 xt and maximum amplitude xm is shown on Figures 2.13, 2.14, and 2.15. Again, comparing these figures we observe that the k2 curves approach k, the one-line U.P. approximation, for small xt(Figure 2.13) as predicted by (2.62). Similarly, the RI curves approach k for large xt (Figure 2.15) as predicted by (2.61). Slope Parameters, k1 and k2 / 25 . I. , I k(A=0), linear ' / U.P. slope / 20 1 parameter / / / I 15 . ’ // . / / 10 . I ./ J I / 60 k2(A-.5) { k1(A=-.S) 5. k1(A=0) ’/ .- . I / .. ’4’...— .. k1(x=6) ‘ ‘ __-... 1.1-.- .--.~«r:"’-..' _ -_,_1- - . _ - . _ IW:¥-EJ o a L a L L 0 l 2 3 4 5 6 Amplitude, xm Figure 2.13 Bilinear U.P. Slope Parameters k1 and k2, versus Amplitude, , for the Nonlinear Function f(x) I sinh x) (xt/xm I 0.2) Slope Parameters, k1 and k2 35 3O 25 20 15 10 61 I l I I I k2(l=0) I k (A=O), linear U.P. 310pe parameter _L Amplitude, xm Figure 2.14 Bilinear U.P. Slape Parameters, R1 and k2, versus Amplitude, xm, for the Nonlinear Function f(x) = sinh x (xt/xm = 0.5) Slope Parameters, RI and k2 62 35 r ‘ T I 1 r T I I I I I 30 .- I p—k2(A“o5) ‘ II k2(A= —0)\ 25 b q I k2(}\= —6) [A 20 _ ’ I k(A=O), linear ' U. P. slope parameter 15' ‘ I, / / / / I / 10 P I f/ I, T / I / ‘/ /,/’ I- / ’ ‘ S ’ / R104” I / ’A / 6 x / ( , ’ / .../«zfi' ’1 f 1 \ 4 4“, ‘ £=_:__::—_o-———-———— ‘0 - ‘ 0 l l I l L O 1 2 3 4 5 6 Amplitude, xm Figure 2.15 Bilinear U.P. Slope Parameters, RI and k2, versus Amplitude, xm, for the Nonlinear Function f(x) - sinh x (xt/xm = 0.8) III. SINGLE DEGREE-OF—FREEDOM SYSTEMS In this chapter, both free and forced vibrations of nonlinear undamped, single degree-of-freedom systems are examined. The non- linearity occurs in the restoring forces. The objective here is to show how improved period-amplitude relations are possible with bilinear U.P. approximation method. 3.1 Free Vibrations. The governing equation is of the form '3 + f(x) - 0, complete with initial conditions x(0) = xm and i (O)= 0 When the nonlinear force f(x) is approximated bilinearly, the equation of motion reduces to the two linear differential equations ‘31 + klxl = 0, leixt (3.1) and 382 + kzxz + xt(k1-k2) = 0, xtilxl (3.2) where xt is the transition point as discussed in Chapter II. The conditions, at time t - O, are then x2(0) = xm and x2(0) = O (3.3) and x1 and x2 are matched at xt, say at t = tt. x1(tt) = XZ(tt) and x1(tt) = xZ(tt) (3.4) Equation (3.2) is multiplied by x2 and integrated once to yield 63 64 an expression for £2 . 2 1/2 x2 ' [ Cz-lxzxt(k1'k2) - k2 x2 ] (3.5) where C2 8 (k1‘k2) Ithxm] + kzxg‘ and the initial conditions (3.3) have been used. Similarily, when equation (3.1) is multiplied by i1 and is integrated. and the conditions (3.4) are used, one finds ‘ 2 1/2 x1 - [Cl-klxl] (3.6) where a .. _ 2 2 Cl (k1 k2) [thxm xt ] + kzxm The period of motion, T, may be computed by using equations (3.5) and (3.6) as xt 2 -l/2 Km 2 -1/2 4 [ g [Cl-klx1 ] dxl + £ [CZ—2x2xt(kl-k2)—k2x2] dxz] t (3.7) The right hand side above can be integrated out explicitly and the results expressed in closed form. Assuming k1>0, then for k2<0, k2 - O and k2>0 respectively, we have, after some manipulations, 65 For k2 < O: r 2 1/2 I 1 2 "1 k b xt - ( :2) .... a 1 +_ ..t - - 2 ... - _ To 7;; n an fit. W) 23—5?- is? 1 L. VI—k-l _1< t (“1(2) ) + cosh b (3.8) V "k2 Km- ('kg) Fork -0: 2 r '5 1/2 -1 1/2 .1... In % 1 + g [2(xm-xt) J tan [2(xm-xt) ] TO 1 W xt Kt c .J (3.9) For k2 > 03 b 1/2 _ xt + _. J— = 1 1+2 —tan [k— (xm+b)1_ k2 1 7E]- 1r 1 X: k2 o b x + —- m k2 1 x -+ b 1 " t +%ms< a) 2 b Km + {2 J (3.10) where -l/2 v 8 n $9 H l w N v U H II N :3 \ 8 II N :1 A 3‘ p... V O 66 For purposes of comparison other period-amplitude relations are for the same system also given in Table 3.1. The one-line U.P. approximate period, T, is calculated using values for k previously given in Chapter II. Comparison of Results. The dimensionless, one-line U.P. and exact period- amplitude relations given Table 3.1 are now compared against the bilinear U.P. approximate method in Figures 3.1, 3.2, and 3.3 for the nonlinear functions x + x3, sin x, and sinh x, respectively. Relations derived previously in Chapter II for k and k2 1 are substituted into the appropriate T/To relation derived by the bilinear U.P. method- equation (3.8), (3.9) or (3.10)—- to yield the corresponding period-amplitude relationship. An amplitude ratio xt/xm = 0.5 was assumed. The symbols UPl and UP2 refer to the linear and the bilinear U.P. approximate methods, respectively. The parameter is varied between. A=-.5 and A = 6 to show qualitatively how the results are effected. These figures show how insensitive the bilinear U.P. results are to changes in A , while for the linear U.P. results the contrary is true. A closer look at Figures 3.1, 3.2,and 3.3 reveals further insight into the magnitude of the error by the linear and the bilinear U.P. methods. By looking at one maximum amplitude value xm from any of these figures the error between the exact and linear U.P. is fixed. However the bilinear U.P. method is also dependent upon the amplitude ratio xt/xm. Quantitatively, Figures 3.4 through 3.10 give for a particular maximum amplitude xm a measure of the error between the 67 Nonlinear Dimensionless Period-Amplitude Relations Restoring Force To Exact Linear One Line U.P. f(x) Te/To Tg/To TlTo === x + x3 2n 2K(k1) 1 3 2 1/2 —-— 1/2 1 + $72 A. n(l+x2m) where 2 1/2 klgsj-nell In 2(1+xm2) A+1 1/2 1 s n x 211 '12?K(k) l [(L31‘0-1-2) JA+1(xm)] xm where k = 81 :nm) 2 A+1 sinh x 2n 1 % sech(_xl_n. KW) 2 where y=tanh(};¥) [(i.) P___ I J UP2 (A'O) - I ' I UPl'linear U.P. solution I A UP2-bilinear U.P. solution 2.0 b / I 1.5 ’ L ll Li 47 —1l 0 1 3 Amplitude, xm N Figure 3.2 One Degree-of—Freedom, Free Vibration System Period Ratio T/To versus Amplitude xm for f(x) = sin x (xt/xm=.5) 70 UPl (A-6) .4 ~ UP2 (A-6) U21 (x--.5) upz (x--.5) U91 (A=0) .3 ' UP2 (A-O) UPl = linear U.P. solution UP2 = bilinear U.P. solutiOL my. . . .‘\‘ 0 2 3 4 5 6 Amplitude, xm Figure 3.3 One Degree-of—Freedom, Free Vibration System Period Ratio T/To versus Amplitude xm for f(x) = sinh x (xt/xm=.5) 5L- 4 4- - 03? -I __ __ __ __._..__ _lflflPiifiil) _________ «z r Percent Error in Period Ratio T/T /’ N / ‘ \r: UP2(A=1) .01 .1 .2 .3 .4 .5 .6 .7 .8 .9 .99 Amplitude Ratio xt/xm Figure 3.4 One Degree-of-Freedom, Free Vibration System Error Plot of Period Ratio T/To versus Amplitude Ratio Xt/xm for f(x) = x + x3 (xm=0.5) Percent Error in Period Ratio T/To 5 - \(UP2(A=1) \ \ \ 4 L \ \. \ ___ X (RIM—---“--- 3 \ \ \ “ UP2(A=.5) \ \ 2. b \\ \ \ \ \ \ \ 1 I" s \ \ J, -‘ ”<7..- .\. \ (”ElixiiEL- /_,'___ "\.~-~ \\ \K /7 fl UP2(A=.125)./‘\~~.~_- ‘ \ O .‘ 4_ ‘ UP1(A=0) _ . ‘ w upz /"< \ ‘1 ' ’ UP2(A=-.125) " ,z"’ ' _ _ _ _ UP1_Q=-_.125) __ H E i 1 L _L l J L L J .01 .1 .2 .3 .4 .5 .6 .7 .8 .9 .99 Amplitude Ratio xt/xm Figure 3.5 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio T/To versus Amplitude Ratio xt/xm for f(x) = x + x3 (xm=1.0) Percent Error in Period Ratio T/To 73 T \ i 5 _ \VIUP2(A=1) I q \\ I I ‘.\ l 4 P \ I a L_____- \ .1: UP1(A=.5) _L \ \ / 1- \ /;J s.‘ \ ‘\ fUPlflATlZS) A. \ Z s“¥\ /, . ‘v .... ‘\ UP2(X=.125) o n._¥ r v . r ...—R. l s- N__/ \ I W0)\ A AV -1 W \ UP2(A-0) ’///,d' UP2(A=-.125) ‘ L 1 L .d 1 1 1 J.“ 1 J\ .9 .01 .1 .2 .3 .4 .5 .6 .7 .8 Amplitude Ratio xt/xm .99 Figure 3.6 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio T/To versus Amplitude Ratio xt/xm for f(x) = x + x3 (xm=2.0) Percent Error in Period Ratio T/To UPl(A-.125) o I L~~\3\ \UP2(A-.125) l J l J l .01 .1 .2 .3 .4 .5 .6 Amplitude Ratio xt/xm k:// mums-.125) . g \l a: lo 8 Figure 3.7 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio T/TO versus Amplitude Ratio xt/xm for f(x) - x + x3 (xm=3.0) Percent Error in Period Ratio T/To I ..- 75 r I I I l I T T I 1 A j! - 1 b «I UP2(A=-.125) UPl(A=-*—.125)~~a:!E -a’\'~- ‘ - i, “__7 ‘ -o_____.”. . ‘ 3" ;‘:_ ' ___.-__..-..——:= ,, / UP2(A=0) \ _ / UP2(A=.125) \ 1 ' ,c [mu-.5) \. I 1" A/ UP1(_A_=_.§)\ '. /( \ . //’\upz(A=1) ‘. UPl(A=l) {______________>x_ __.h r 1 L 1 1 lL l L J l l L {U .01 .1 .2 .3 .4 .5 .6 .7 .8 9 .99 Amplitude Ratio xt/xm Figure 3.8 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio t/ro versus Amplitude Ratio xt/xm for f(x) - sin x (xm=l.0) Percent Error in Period Ratio r/ro 76 UP2(A=—.125) 2 ’ . -2 . I// A: m 1) UP2(A=.125) ’ , UP2(A=.5) I , / (.44 will 4. .01 .l .2 .3 .4 .5 .6 .7 .8 Amplitude Ratio xt/xm Figure 3.9 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio T/To versus Amplitude Ratio xt/xm for f(x) - sin x (xm=2. O) Percent Error in Period Ratio r/ro I fir U 1 V F T U f j 5 r . 4 t . 3 . . 2 r UP1(A=1)-_§_\_ _____ i \ 1' ‘(urzosn I, \ ¥ fauna. 5) ‘ K I . \ \ UP2(A=.5) / / mug-.139 _‘\.\ m2Q=._125 ._\--: ~‘\:.*-§‘ -..- I, I \ \IIP1().-- .125) upz(A=-.125) -1 L UP2(A= 0) l 1_i _l_ j A L l l j .01 .1 .2 .3 .4 .5 .6 .7 .8 .9 .09 Amplitude Ratio xt/xm Figure 3.10 One Degree-of-Freedom, Free Vibration System Error Plot Period Ratio T/To versus Amplitude Ratio xt/xm for f(x) = sinh x (xm= 1.0) 78 exact, linear U.P. and bilinear U.P. methods for a range of A parameters and of‘xt/xm values (ijt/xmfl) which includes xt/xIn = .5. Figures 3.4, 3.5, 3.6 and 3.7 ShOW'thiS dependence for function xI+ x 3; Figures 3.8 and 3.9 for sin x; and Figure 3.10 for sinh x. The A = O and xt/xm = .5 values over the maximum amplitudes considered, in general, give the better bilinear U.P. approximate results for the three nonlinear functions investigated. 3.2 Forced Vibrations. In this section an undamped, forced vibrating system is examined. This system is solved first as a forced vibrating problem by approximating the restoring force bilinearly about its own equilibrium point (which differs from the point about which oscillation occurs). Next, this same system is solved as a free vibrating problem by approximating the equivalent nonlinear restoring force by four lines about a point called the minimum potential point where oscillation occurs. Only the case of a step function excitation and of a cubic restoring force is investigated to illustrate the procedure. One illustrative problem is solved by the former approach, yielding results comparable to the one line U.P. solution. The governing equation is of the form K + f(x) = F(t) = Fou(t) complete with the initial conditions x(0) - O and 11(0) = 0 where u(t) is the unit step-function (u=0 for tO) and F; is the amplitude of the applied excitation. 79 Solution as a Fbrcgdeibration Problem. Like for the free vibration system the equation of motion can be approximated bilinearly by two linear differential equations 1 1 1 F(t) Fou(t), lxl :_xt, (3.11) and 3% + k _ _ 2 2X2 + XtCRI k2) - 1701103), lxl : xt (3.12) where x is the transition point as discussed in Chapter IT. I: The initial conditions are then x1(0) - o and i1(0) a o, (3.13) and x1 and x2 are matched at xt, say at t = tt x1(tt) - x2(tt) and i1(ct) = i2(ct) (3.14) Following a procedure similar to that of the free vibrating problem in Section 3.1 we obtain the period of motion, T, as X Xm 2A dx T s 2 [ f t -——- -+ f 2 J 0 i1 Xt i2 Nondimensionally, the period of motion becomes ' T 1 xt -l/2 -1/2 2A -1/2 __' a "‘ f (#1) [ZQX-XZ] dx + f [e + fx + gxz] dx TO N O xt (3.15) where Q = Fo/kl 8 = “k2 -1/2 e = (kl-k2) xt2 To = 21r(k1) 80 The right hand side above can be integrated out explicitly and the results expressed in closed form. Assuming k1>O, then for k <0, k = O and 2 2 k2>O respectively, we have, after some manipulations, For kzi): T l -l/2 —l xt—d 11 1 1/2 — 3 — (k1) sin " + — J + — log[ (e+f2A+g4A2) To 11’ Idl 2 [g— /_ f J 1 [ l/2 +2Ag+-—— -—log (e+fx +gx 2) z/E fg‘ " t f + X if; +.__. I (3.16) t 2]; For k2 = O ‘1' 1 -1/2 -1 xt—d 11 4A xt -— --- [ (kl) [ sin ( ) +—] + (1--—) (3.17) 1’ 1r Idl 2 5 x 2A 0 1 For k2>__0_: T 1 ‘1/2 -1 x -d 11 1 1 4 - - A —f _ -*[(k1) [ Sin ( t ~)+—] 4‘ sin < g To 1r I‘ll 2 I“; V’fr-4eg 81 The dimensionless, bilinear U.P. period-amplitude relations derived above are now used in solving the special case f(x) - x + x3. The slope parameters RI and k2 derived in Subsection 2.2.1 for f(x) - x + xzaare now substituted into one of the above equations and the results are compared in Figure 3.11 against the linear U.P. and the exact solution, obtained by a quadrature method. As Figure 3.11 shows both the bilinear U.P. and the linear U.P. method closely approximate the exact solution. For this function f(x) = x + x3, the bilinear U.P. method does show improved results over the linear U.P. method for the larger amplitudes. However, this slight improve— ment does not warrant the additional effort required compared to the linear U.P. method except for highly nonlinear functions. Solution as a Free Vibration Problem. By a change of variable the governing equation is reformulated as a free vibration problem as 5; + f(x)-Fou(t) = 0 or where 60;) = f (x)-Fou(t) Introducting the change of variable x = z + x0 we obtain a formulation where z = 0 becomes the local minimum potential point as previously defined in Chapter II. Thus we have that '7: + E(z+xo) = o (3.19) 82 with the initial conditions 2(0) - --x0 and 2(0) = 0 where x0 is the local minimum potential point obtained by setting 6(x) - O and solving for x, (see Figure 3.12). Derivation of the Period vs Amplitude Relationship. The equation of motion (3.19) is approximated by the four linear differential equations 24 4- kaz4 + (xtz-xo) (k3-k4) = O -xo§a§f|zm3| and Z3 '0' R323 = O -|Zm3|.:?:p and 21 + klzl 3 0 0:2: Izmll and I C) 22 -+ kzz2 +(xtl-xo)(k1-k2) ‘ IzmlI£?£.A The initial conditions are then 24(0) - -XO and 54(0) ' 0 and 23 and 24 are matched at, say t = t2 23(t2) - 24(t2) and 53(t2) ‘ é4(t2) and 21 and 23 are matched at, say t - to 21(to) - 23(to) - O and él(to) = é3(to) (3.20) (3.21) (3.22) (3.23) (3.24) (3.25) 83 a 1 r I i \ UP1 0-5) 0 )____ b “ UP2 (2"6) . .1 \‘ UP1().--.5) A __-— \ UP1 (A=0) D )___ \: UP1=linear U.P. solution V UP2-bilinear U.P. solution \ r \ \ \\ L _ 1 \ y r ‘ ‘ o ;\ Exact ‘\\ \\ ' \ 7 : § \\§\ \ \_ Q ~- ""7 P L 1 0 l 2 Amplitude, xm )- )- W b Figure 3.11 One Degree-of—Freedom, Forced Vibration System Period Ratio T/To versus Amplitude xm for f(x) = x + x3 and Unit Step Function Excitation 84 f(x) f(x) (f(x) 6(x) -E(z + x0) xt 2 :,¢fl ‘ z x xo xt; B = x0 + A Figure 3.12 A Nonlinear Restoring Force f(x) Modified to Give a New Restoring Force G(x) With a New Equilibrium Position x - x o 85 and 21 and 22 are matched at, say t = t1 22(t1) = zl(t1) and é2(c1) = él(tl) (3.26) Obtaining the first integral of each of the four linear differential equations and solving for the constant of integration, we find the period-amplitude relation as A dz -lzm I dz 0 dz3 T = 2 f .T_-=2 f 3 '———§ + f . + —xo 2 -x0 24 -lzm3| 23 (3.27) 2 dz A dz + | “11' -—! f 2 O 21 Izml 22 where 1/2 24 :[C4-V4] with 2 C4 - kéxo - 2(xt2-xo) (k3-k4) x0 and, < II k2+2 - .. 4 424 (xt2 x0) (k3 k4) 24 1/2 23 = [C3—V3] with and, with with Cz and 86 2 C4 - (xtz-xo) (k3—k4) 2 k323 1/2 [Cl-V1] l 1 1/2 [CZ-V2] _ 2 _ C1 + (xt1 x0) (k1 k2) 2 kzz2 + 2(xt-xo) (kl-k2) 22 87 The slope parameter k1, k2, k3, and R4 are given by equations (2.2), (2.4), (2.3) and (2.5) respectively, with G(x) replaced by ECx), that is, (A) 1 {315 ([2!n | t + x0) P1 (t) w(>\,t) dt .(3 23) k1(_A,zm1) ' | 70 ' zml [1 [P1 (t)1.2 w(x9t) dt 0 where Izmll = Ixt1_xol 1 __ (A) 1 (f) [G(Iyml t + xtl>-k11 P1 (t) mom) dc k20nklaxt9Ym) = —-"|' (A) lym f1 [P1 (t)]2 m(A,t) dt 0 u (3.29) where lyml = IB-xtll = szo-xtll 1 _ (A) 1 g -[G(-Izm3|t + x0)] P1 (t) w(A,t) dt k (492 ) = (A) 3 “’3 I2 I 11 [P1 (017—1110.: dc m3 0 where (3-30)u I2, I = Ixo-xtzl 88 (X) 1- — - - _ 1 f0 [G( lsm't + xtz) k3(xt2 xo)] p1 (t) w(A,t) dt k4(A.k3,xt2,8m) a rw— (x) lsml ‘ ’1 [P1 (12)]2 w(A,t) dt 0 (3.31) where lsml a lth- “I: lxtzl While the special case of f(x) = x + x3 was not solved by the four-line U.P. method, (3.27) should give a good approximation for those cases where the restoring force is highly nonlinear. | Ill ill I [I ‘II III I II." II 1.1l lJnllllllul. 1| III] I n IV. A SYMMETRIC TWO DEGREE—OF-FREEDOM NONLINEAR SYSTEM The symmetric two degree-of-freedom problem shown in Figure 4.1 has been investigated by a number of authors [13, 14, 15, 16, 18]. Only two of these authors, however, have investigated the questions of existence of more than two "normal modes". Here the term "normal modes" in nonlinear systems is understood in the sense of Rosenberg [17] (see Appendix F). In [15] Rosenberg commented on the existence of four distinct normal modes for the two degree-of-freedom symmetric system in which the nonlinear springs are of homogeneous degree. The details are given in Appendix F. More recently, Anand [18} uncovered this multiplicity of normal modes for the symmetric two degree-of-freedom system.with cubic nonlinearities. In Section 4.1 Anand's approximation will be shown to be equivalent to a linear (one-line) ultraspherical polynomial approximation. A discussion of the six regions of the different modal patterns predicted by Anand which depend on the values of two parameters of nonlinearity is then reviewed in Section 4.2. In Section 4.3 a special case of the symmetric problem is examined which, according to Anand's approach, indicated the presence of an additional normal mode but which, according to the bilinear (two-line) ultraspherical polynomial approach as well as the exact solution, denies the existence of this additional mode. Section 4.4 contains additional calculations and discussions on this multiplicity of normal modes (superabundant modes). 89 90 4.1 Equivalence ovapproximation Techniques. Anand considered the symmetric'problem shown in Figure 4.1 where 3 3 f(x) klxl + m a1 x1 W f(x ) = k x + m a x 3 2 l 2 1 2 (4.1) The equations of motion are mail + f(xl) + f(xl-Xz) = 0 mx2 + f(xz) - f(xl-xz) = 0 Anand defines two new parameters, w 2 = (k + k1)/m and m12 = k/m. 2 In these parameters the equations of motion become -- 2 3_ 2 _ 3= ’ x1 + m2 x1 + 01x1 ml x2 + a(x1 x2) 0 (4.2) £2 + mzzxz + a1x23 -w12x1-a(x1-x2)3 = 0 (4.3) x1 X2 f(x ) f(x -x ) f(x ) l 2 “‘ AKA: “‘ ///// iii/7 //7// Figure 4.1 Conservative Spring-Mass Two Degree-of-Freedom System. 91 into which.solutions of the form x1 = A cos wt and x2 = B cos (mt + B) are substituted. Since the system is conservative there is no need for assuming the phase angle 8. Without loss of generality we may let 8 = O at this point. After some algebraic manipulation, terms involving cos wt, sin wt, cos 3wt, and sin 3wt are obtained. Upon disregarding the superharmonic terms and equating the coefficients of the harmonic terms to zero one obtains 3 9 9 3 — (ad-<11) A3 -- QAZB +—01AB2 -—a B3 + (wzz-wzfll -w123 = 0 (4.4) 4 4 4 4 3 9 9 3 --aA3 +-—-aAZB ----aAB2 +~—-(a +o1)B3 -w12A + (wzz-w2)B = 0 (4.5) 4 4 4 4 Equations (4.4) and (4.5) can be obtained by yet another approximate method. If the nonlinear terms in the equations of motion, (4.2) and (4.3), are linearized in terms of ultraspherical polynomials (one-line approximations) with respect to appropriate amplitudes: and if a normal mode solution is assumed (as was assumed by Anand), then equations (4.4) and (4.5) result provided the Chebyshev polynomial (i=0) is used. The details are presented below. In terms of ultraspherical polynomials, the nonlinear terms of equations (4.2) and (4.3) are linearized as follows: 92 (i) linearize x1, with respect to A; (ii) linearize x2, with respect to B; and (iii) linearize xl-xz, with respect to C, where C = A—B The resulting equations of motion are '° 3A2 3C2 (x -x ) = 0 2 .______ _ 2 a 1 "1 + “’2 x1 + “1 [2(1+2)]x1 “’1 x2 + [2(A+2)] 2 332 .. 2 [ ] 2 3c2 x2 + w x + a1 x2 -ml x -a (x -x ) = O 2 2 2(x+2) 1 2(A+2) 1 2 x A 1 Substituting a solution of the form{ }-{ }cos wt into these x B 2 equations yields 3A2 -Aw2 + w22A + a _— 2(A+2) 3c2 ] A «.1123 + a[ ] (A-B) = o (4.6) 2(A+2) 332 3c2 ] B -w12A - a[ ](A-B) = O (4.7) -Bw2 + m22A + 011 [ 2(A+2) 2(A+2) Substituting A-B for C in (4.6) and (4.7) yields respectively 93 (a+a1) A3 - --2- aAZB + 9 aA32 - u 3 a 83 2(A+2) 204-2) 2(A+2) 2(A+2) + ((1122-1112) A -w123 = o (4.8) and - 3 (1A3 + -—--9——0LAZB - —2—— CLAB2 + —-—-3-———- (OH-011) B3 2(A+2) 2(A+2) 2(A+2) 2(l+2) -w12A + (mzz-wz) B = 0 (4-9) Finally when A is set equal to zero (the Chebyshev polynomial), equations (4.8) and (4.9) reduce to (4.4) and (4.5) as mentioned before, Thus we have shown that Anand's solution method is equivalent to the ultraspherical polynomial approximation with A = 0. Now, subtracting (4.9) from (4.8), or equivalently, (4.5) from (4.4) with A = O, we obtain (3/4 011 + 3/2 a) (A3-B3) - (9/2 aAB + 1.12 -m22 -w12) (A-B)= o (4.10) and, adding (4.9) and (4.8) we Obtain 3/40.1 (A3 + B3) — (1.2-(.22 + (1112) (A + B) = o (4.11) 94 As Anand pointed out in his study, equations (4.10) and (4.11) yield three pairs of solutions for A and B which constitute the possible modes of vibration. The possible modes are summarized below. Symmetric Case (A -B): The relation A = B represents the in— phase or symmetric mode. Dividing equation (4.11) by A + B and sub- stituting A = B into this resulting relation yields 3/4a1 A2 - (m2 -w22 + ml?) = o (4.12) This is the frequency-amplitude relation for the symmetric mode. Antisymmetric Case (A = -B): The relation A = -B represents the out-of-phase or antisymmetric mode. Dividing equation (4.10) by A—B and substituting A = -B into this resulting relation yields (3/4a1 + 6a)A2 - (wZ-wzz-wlz) = 0 (4.13) This is the frequency amplitudeerelation for the antisymmetric mode. Asymmetric Caseg(A # B and A # -B): The relations A # B and A {-B represent Anand's third mode which he terms the asymmetric mode. Dividing equations (4.10) and (4.11) by A-B and A+B, respectively, we obtain (3/401 + 3/2a) (A2 + AB + B2) — (9/2a AB + mz-wzz-wlz) = O (4.14) 3/4a1 (A2 -AB +32) - (412-0122 + ml?) = 0 (4.15) Solutions of equations (4-14) and (4.15) will be discussed below. 95 4.2 Discussion of the Regions for the Asymmetric Mode. Anand discusses the effect the nonlinear parameters on the existence of this third mode. Eliminating w2 from equations (4.14) and (4.15) we obtain A2 + 132 - Us (20; -a1)AB + 4/3a w12= 0 (4.16) Solving this equation for B, we get a]. a1 a 4 2 B =[1-— ]A + [—[—-l -l A2 - -— (1)12 1/ 2a a 4a 3a (4.17) where A has been arbitrarily assumed to be larger than B. For a real, physical solution we require that al > 4a (4.18) which places the following restriction on A 160 wlz A2 > (4.19) 3a1 (a1 - 4a) By considering three cases in the (cal) plane we can identify the regions where this asymmetric mode is present. Case I. For a > O and (i) 0 < al This subcase represents a system with all springs hard? Equations (4.18) and (4.19) place a restriction on amplitude A; A cannot be too small. * A nonlinear spring force is considered hard if its first derivative increases with increasing displacement, and soft if its first de- rivative decreases with increasing displacement. 96 (11) 0 _<_ 61 (iii) 0 > a §_4a Asymmetric solution does’not exist. This subcase represents a system with the coupling spring being hard and the outboard spring being soft. Equations (4.18) and (4.19) restrict the amplitude in that A cannot be too small. 1 Case II. For a = O and (1) a1 > 0 or a1< 0. Taking the limit of (4.17) as a + O we obtain -al A a a 4 B= + -—1 -—1A2-—wi ”2 2a a 4a 3a Expanding the second term of (4.17) by the binomial expansion we find ..a a A 8 m 2 a I B 2 - A+ 1 1--J-—_+ .0... 2a 2a 3 1120112 As a tends to zero, .01A 01A 4 (D21 Lim B 2 Lim + _, + o o o a + O a + 0 12 97 (ii) For a = 0 and 01 = O, Asymmetric solution does not exist. Case III. For a < O and (1) a1 < 0. If .21 <1, a1 > 4a and the amplitude, A, in re- (11) 01 = 0 (iii) 40 stricted in that A cannot be too large, i.e. 16a w z A2 < 1 3n1 (a1 -4a) If 31": 1, al.3’4a; asymmetric solution exists 4a regardless of the amplitude A. Asymmetric solution exists with no restriction on amplitude, A. a1 > O Asymmetric solution exists with no restriction on amplitude, A. In addition to the conditions given by equations (4.18) and (4.19), which place restrictions on the parameters a and a1, and the amplitude A, the frequency w must also be real in order to guarantee real asymmetric solutions. This additional requirement w2 > 0 must there- fore also be satisfied. Cases I, II, and III are displayed below in Figure 4.2, 98 Amplitude, A, restricted (cannot be too small) No asymmetric mode \ \ \\ ‘ \ A restricted , \ (can't be too small! test" \/ / “I not Amplitude, A, / not may Figure 4-2n a1 Graph Showing the Six Regions for the Asymmetric Mode for a Two Degree,Symmetric, Free Vibration System 99 4.3 A Special Case of Anandjs Problem. A8 a test of Anand‘s approximate approach, a special case is examined so that the bilinear U.P. approximation method as well as the exact solution can be compared against it. To avoid undue algebraic difficulties inherent in solving the symmetric problem by the bilinear method, the coupling spring is the only nonlinear term allowed and is of the soft cubic type. In particular, we can illustrate this case in Figure 4.1 with the coefficients m = 1, k1 = 1, k = 1, a1 - O and a - -l substituted into (4.1). The restoring forces given by (4.1) then reduce to f(xl) - x1 f€xz) - x2 (4.20) f(xl-xz) = (xl-xz) - (xl-xz)3 4.3.1 Anand's Solution. This special case falls within Case III mentioned previously, which indicated that for the asymmetric mode no restriction is placed on the amplitude of XI, A. However, a check still is required on the frequency, w. Substituting the above coefficients into equation (4.15) and simplifying we find that the frequency is real, 100 In addition, substituting the parameters from this special case give for the in-phase mode the frequency relation which simplifies to the same expression as for the asymmetric mode. For the out-of-phase mode the frequency relation simplifies to k1 + 2k w2 = wzz + wlz + 60 A2 = ---- + 6a A2 = 3(1-2A2) 'm These values are plotted in (Figure 4.5) for comparison with bilinear and exact frequency relations, yet to be determined. Thus, we may conclude that for this special case, Anand's approximate method does predict an asymmetric mode. We will now apply the bilinear ultraspherical polynomial approximation to this same special case and show that the bilinear solution can be expressed in analytic form. 4.3.2 Bilinear U.P. Solution. This special case of the symmetric two degree-of—freedom (free vibration) problem is formulated in this subsection by the two-line ultraspherical polynomial approximation. This problem is shown schematically by Figure 4.1 and the coupled equations of motion are 3x1 111352 - an (4.22) 31:2 x 2 x 2 (xl-x )2 (x —x )1+ with -u -= -U(xl,x2) =—1- +—-2-+——L - —1——2— (4.23) 2 2 2 4 These equations are approximated bilinearly by equations (4.21) and (4.22) but with U(x1,x2) approximated by 101 2 2 X1 X2 2 2 and V(x1-x2) = k3(x1-x2)2 + 1/2 (k4-k3) (xl-xz-yt)2 2 = ‘3. Y2 + “NM-R3) (y-yt)2. y _>, yt (4.26) 2 Simplified, the bilinear set of approximating equations becomes \ I C) mil + klxl + k3(xl-x2) - ) y g y, (4.27) | C> m§2 + klxz - k3(x1-x2) - .. \ mx1 + klxl + k3(x1-x2) + (k4-k3) (xl-xZ—yt) = O )y 3 yt (4.28) figz + klxz -k3(X1-X2) -(k4-k3) (xl-xz-yt) = 0 J The slope parameters R3 and R4 are the linear approximations to the nonlinear function over the appropriate intervals. 102 The term V(x1-x2) is obtained by summing the areas under the force-displacement plot for the approximating slope parameters. For example, Figures 4.3 and 4.4 represent a function approximated bilinearly for the y §.Yt and y :_yt intervals, respectively, f(y) 0 y Yt Ym Amplitude, y Figure 4.3 ‘V a [f(y) dy Relation for a Nonlinear Restoring Force Approximated Bilinearly (y §_vt) f(y) V(y) = A1 + A3 - A2 2 - 2 _ 2 A2 A = kay + k4(y yt) _ k3(y yt) «“x 3 2 2 2 $ Al ’: ‘J:>"‘§i"‘ -__. (where k4<0 in this case) \\\ y :,vt 0 y, y ym Amplitude, y Figure 4-4 V =.ff(y) dy Relation for a Nonlinear Restoring Force Approximated Bilinearly (y Z.Yt) 103 mil + klxl +'k3 (xl-xz) - 0 (4.29) y §.Yt m§2 + klxz - k3(x1-x2) = 0 (4.30) mil + k1x1+ k3(x1-x2) + (k4 - k3) (xl-xz—yt) = O (4.31) yzyt miz + klxz - k3(xl-x2) - (RA-k3) (xl-XZ'YC) = 0 (4.32) These equations can be rewritten in a simplified form by substituting y . xl-xz and, for the y.: yt region subtracting equation (4.30) from equation (4.29) to yield the following equations \ ms: + (k1 + 2k3)y = 0 (4.33) > y i y. mil + klxl + k3y -- 0 J (4.34) Similarly, for the Y.Z.Yt region we can simplify these equations by subtracting equation (4.32) from equation (4.31) and by the change of variables 2 = y-2F and u = xl-F, where F = yt(k4—k3)/(k1 + 2k4) to obtain m2 + (2k4 + k1) z = 0 (4.35) Y Z_Yt mfi + klu + k42 = 0 (4.36) This simplification uncouples equations (4.33) and (4.35) above. 104 The solutions to these four simplified equations have eight constants of integrations which are evaluated in terms of four initial conditions and four matching conditions at y = y The initial conditions are t. x1(0) a A., i1(0) = 0 x2(0) = B . i2(0) = O or, equivalently , x1(0) = A, 21(0) = O y(0) = xl(0)~x2(0) = AeB a c, and § = i1(0)-i2(0) = o The matching conditions involve equating displacements and velocities between the y §_yt and y3__yt regions at the transition amplitude point. Recall that yt, the transition amplitude, is already known since the slope parameters, k3 amd R4, are functions of yt' In this special C case yt "" has been assumed. 2 y _3 yt Region: Using these initial conditions, oscillatory motion is started in the y_>__yt region and governed by the equations of motion (4.35) and (4.36). By substituting a solution of the form {3 - {3 a»: into (4.35) and (4.36), the resulting roots of the frequency equation become 1/2 1/2 w3 = (kl/m) and w4 = [(k1 + 2R4)/m] 105 The corresponding amplitude ratios are T k1 -w32 m 81 n— 2.- 8 0 3 R4 8 -1 a - kl-w42 m = 2 2 The above solutions are uniquely determined by normalizing the eigen- vectora. The general solution is then represented as u x1 - F v11 v21 - - a1 cos w3t + a2 C08 w4t (4.37) z y - 2F v12 V22 where 1/2 1/2 w3 = (kl/m) and w4 = [(k1 + 2R4)/m] and 2 -1/2 -l/2 -1/2 v11 = (81 + 1) = 1 v21 = (822 + 1) = (5) 2 -l/2 -1/2 —1/2 v12 = 81 (81 + 1) = o v22 = 82 (822+ 1) = 2(5) Using the initial conditions x1(o> = A. 340) = c 106 the constants a1 and a2 are obtained from (4.37) and (4.38) X1(0) -F } A-F V11 V21 { '{ }”1{ } { } y(0) -2F C-2F V22 v12 Since V12 = 0, 1/2 a2 = (C-2F)/V22 = (5) (C-2F)/2 and A—F- 82V21 a1 =-——~ a .A - c/2 V11 At t = tt = t1 and from equation (4.37) Yt 'ZF = alvlz cos w3t1 + a2V22 cos w4 t1 but V12 8 0, thus yt -2F 1 _1 yt -zp cos w4t1 = or, t1 =-—— cos ——————- (4.38) C “'ZF (1)4 C -2F Having completely determined the solution for the y 3.Yt region, we are in a position to evaluate 31: 31, Y: and 2 at the transition amplitude, yt. Thus, at t = t = t1 t X1(tt) It V1 V21 F = a1 COS (1)3111 + 82 C08 m4t1 + Y (tr) Yt V12 V 2F 22 (4.39) 107 . ‘ . ‘ —w3 a1 sin w3tl - w4a2 sin w4t1 tht) yt V12 V22 (4.40) These values for xt, yt, it and yt will now serve as the initial conditions for the solution in the y §_yt region. x G 1 : Y _<_ yt Region: By substituting a solution of the form{ } = )eimt y H into equation (4.33) and (4.34), the resulting roots of the frequency equation become _ 1 2 _ 1/2 ms — (kl/m) / and w6 - [(k1 + 2k3)/m] The corresponding amplitude ratios are 2 ‘H kldm w52 H k1 dm w6 - 2 3 G k1'+ k3 -mm5 C k1 + k3 ~mw6 The above solutions are uniquely determined by normalizing the eigenvectors. The general solution for this region is then represented X “11 { 1}_{ } (bI cos wst + b3 sin ms t) Y w12 w21 +{ } (b2 cos w6t + b4 sin 0161:) W22 by (4.41) 108 where 1/2 1/2 ms = (kl/m) and “6 - [(k1 + 2k3)/m] and -1/2 -1/2 -1/2 w11 3 (Y32 +1) 8 1 2 W21 = (Y42 + 1) = (5) ' -1/2 -1/2 -1/2 W12 ' v3 (732 + 1) = 0. wzz = v4 (v42 + 1) = 2(5) Using as initial conditions the xt,)q:, it, and §t previously obtained, the constants of integration b1, b2, and b3 and b4 can be evaluated. At t = 0, we have yt w12 "22 it w 1421 { . } 3 11 (1)5133 + (1)6134 yt w12 w22 Solving for the constants we obtain 1/2 b2 I (5) yt/Z (4.42) 131 xt - yt/Z (4.43) 109 1/2 . b4 ' (5) Yt/(Zwe> (4.44) 1. [ , §t ] ' b3 -—- xt -—— (4.45) ms 2‘ Having completely determined the solution for the y‘iyt region, we can now establish the relationship between the initial amplitudes A and B, or equivalently, A and C. To do this we make use of the fact that for normal mode solutions the masses must simultaneously pass through the equilibrium position. Thus, at t - t2, x1(t2) - 0 and x2(t2) = 0 or equivalently, x1(t2) = 0 and y1(t2) = x1(t2) - x2(t2) = 0 Upon substituting these conditions into (4.41) we obtain (t ) 0 w {XI 2 }-{ }-{ 11} (b1 cos w5t2+ b3 sin 035 132) Y (t2) 0 w12 WZI (b 03 t + b sin w t ) W 2 ° “6 2 4 6 2 (4.46) 22 Since W12 = 0, we have b2 cos w6t2 + b4 sin ”6 t2 = 0 or, tan (06:2 "— = " 6 and t2 =— tan [ J 110 and, 0 . W11 (b1 cos wstz + b3 sin ms t2) + “21 (b2 cos w6t2 + b4 sin w6t2) (4.49) However, the coefficient of W21 is zero by equation (4.47). Substituting xt, it, and yt from equations(4.39) and (4.40) respectively into equations (4.43) and (4.44) yields b a x --—— a --— cos w 1 t 2 2 3 1 and l y m C C 133 - __ it ._ ...—E ] I: .- _.l [A — —]Sin w3t1 = “[A- "" ]Sin L03t1 ms 2 (D3 2 2 Thus equation (4.49) becomes C 0 s [A-- ][cos m3t1 cos w3t¢ — sin w3t1 sin w3t2] 2 or, C 0 = (A-g) [cos (.3 (t1 + 122)] (4.50) Thus, either A - C/2 = 0 or, cos w3 (t1 + t2) = 0 If A _ C/Z n 0, then A = -B and represents the out-of—phase mode. The frequency-amplitude relation is then m = Zn/T = 21r/[4(t1 + t2)] 111 where t1 and t2 are given by (4.38) and (4.48), respectively. If cos (113(t1 + t2) - 0, then w3(t1 + t2) - w/Z and the period is T a 4(t1 + t2) - 4[n/(2w3)] = 2n/w3 1/2 where w3 = (kl/m) , so this represents the in—phase mode. These frequency-amplitude relationships are plotted in Figure 4.5 along with Anand's solution obtained previously. This analytic solution shows that only the in-phase and out—of-phase modes are possible. The exact solution to this special case is next presented to resolve this dilemma between the conflicting predictions by Anand's equivalent one-line U.P. approach and the bilinear U.P. approach. 4.3.3 Exact Solution. To set up this special case so that it can be solved exactly, the method suggested by Rosenberg is applied. The definition of normal modes as applied to nonlinear free vibration conservative systems may be found in Appendix F. Because the system of governing equations are conservative, the sum of the kinetic and potential energy is equal to the total energy for the system and is a constant, no. At the equilibrium position, all of the energy is transferred into kinetic energy. At the maximum amplitudes where the velocity of each mass reverses, all of the energy is transferred into potential energy. Rosenberg has shown that if we start the motion by initially displacing the masses, the trajectory of the masses in the (x1, x2) plane traces out a path toward the equilibrium position. If, for a particular set of initial maximum amplitudes for the masses, the path traced out passes through the equilibrium position with all the displacements 112 simultaneously equal to zero, then this curved path represents a modal relation and is one solution curve for the system considered. Equivalently, this same problem can be formulated in reverse by starting the motion with initial velocities specified at the equilibrium position and moving toward the maximum amplitude position. If, by assuming some initial velocity ratio between the masses, the path traced out by a finite difference method using small time steps intersects the U0 = -U curve orthogonally, then the relation defining this path is called a modal relation. As already mentioned this modal relation is, in general, a curved (not straight) path. A finite difference method is formulated after this latter approach to arrive at the exact solution. This method is applied to the special case problem solved previously by the two approximating methods in Sections 4.3.1 and 4.3.2. The finite difference method is programmed on the digital computer using small time steps, At - .01. Taking advantage of the symmetry of the potential function, the entire xlsz- space can be represented by considering just the region between 9 - ~450and 45°. The first step toward finding all possible normal modes for the xlsz- space is to sweep through angles between -450 and 450 in the (x1,x2) plane for preset total energy valves. Specifying the total energy is equivalent to specifying the maximum amplitudes A and B for x1 and x2 respectively. This sweeping procedure isolates where possible modal relations exist. This is done by comparing two slope terms. One slope term is the local ratio, Ax2/Ax1, which when parallel to the gradient of the energy curve yields a modal relation. The other slope term is the slope at the corresponding point tangent to the total energy ellipse and is used as a check on the orthogonality between sz/Axl and the tangent to 113 energy curve. When these slope parameters are orthogonal, the corresponding x1 and x2 values represent one point on the modal curve. The procedure leading to a normal mode solution is to select initial velocities (V10 and V20) at the equilibrium position (origin in Figure 4.6) in the x1 and x2 directions where x10 = 0 and x20 = 0. Then Axl and sz are calculated using these initial velocities and the time step as X1 = V10(At) and X2 = V20 (At) At point 1, a new x1 and a new x2 are calculated as x1 = x10 + Axl and x2 = x20 + sz The corresponding velocity changes are l 3U(x1,x2) AVl = 5"]. (At) =— __ } (At) m 3x1 and H l aU(xl,x2) sz = x2(At) = —[____ ](At) m 3x2 Thus, the new velocities at point 1 become V1 3 V10 +‘AV1 V2 = V20 + AVZ Frequency, w 2.5 1.5 ...: O 114 i I u r a =-l ol = 0 k = l Out-of-phase mode 1=1 = 1 Bilinear ‘K\\‘-In-phase mode Exact, Bilinear, and Anand's _ i Exact l L l 0 .2 .4 .6 .8 1.0 Amplitude, x1 Figure 4.5 Frequency versus Amplitude Relation (a=-1, al=0) 115 Total Energy Curve, "0 = -U(XI,XZ) x2 x2 xz,...3.‘.2,.... 9 x1 I” Figure 4.6 Total Energy Curve Represented in x1x2U—Space 116 At this point a check is made to determine whether V1 has changed sign. If not, use the values at point 1 and repeat the above steps and proceed point by point until the velocities of the masses change sign. When the velocities do change sign another check is made to see whether the two slope terms are orthogonal. This procedure is repeated for the entire xlxzu- space, identifying all possible modes that satisfy the orthogonality requirement between the two slope terms mentioned previously. The exact frequency-amplitude results for this special case are given in Figure 4.5. 4.4 Superabundant Modes Rosenberg shows in 115] that more than two normal modes exist for a two degree-of—freedom conservative, symmetric, homogeneous system. As explained in Appendix F a homogeneous system here is defined as one in which all of the nonlinear restoring forces have the same degree of nonlinearity. For example, such a system shown in Figure F.l (Appendix F) with degree three, where 81 = 83x13 S2 = A3(x1-x2 3 S3 = a3x23 Anand [18] has similarly shown with his approximate method the existence of the asymmetric mode. However, while Anand's problem is symmetric he allows the linear terms in the restoring forces. Figure F.l thus also represents Anand's system, except that the restoring forces now include linear terms in addition to cubic terms. Therefore, for Anand's system, 117 3 S = x + ma x 1 k1. 1 l l 32 a kCXl-sxz) + MCxl'X2)3 = 3 S3 klx2 + malxz In this section exact modal relations of the frequency—amplitude dependence are formulated for various coefficients using Rosenberg's approach as outlined in Section 4.3.3. In addition, a discussion of the limiting case of Anand's problem for large values of U0 is presented. To recreate the asymmetric modes predicted by Anand, coefficients of the restoring forces used by him are also used in the exact formulation. It will be shown that these exact modal relations will start from values predicted by Anand for small amplitudes where the linear term is predominant and approach limiting values predicted by Rosenberg as the amplitudes increase, where the cubic term is predominant. Anand'scoefficients of the restoring forces will also serve as a point of departure to obtain frequency-amplitude curves by the exact method. Figure 4.7 below represents one case where Anand discovered the asymmetric mode (Figure 4.7 appears as Figure 2 in [18]). Also, from the 091 graph (Figure 4.2) this case falls within the regions where the asymmetric mode is predicted. Thus, the values a =0.32, a1=1.6, k1=.896, k=.1536, and m=l are used in the exact formulation of Case I below. Besides Case I three additional cases are considered. For these cases only the a term is varied and the other terms a k k, and m are held fixed. 1’ 1’ 118 Asymmetric mode H O U! Frequency, w ...: .5 D (1 = .32 a1 = .6 0 . L 0 S l. 1.5 Amplitude, x1 2°5' Out-of- phase mode ...: 0 UI In-phase mode Frequency, w t" .5 a = .32 01 = .64 0 1 j J 0 .5 l. 1.5 Amplitude, x1 Figure 447 Variation of Frequency with Amplitude. 119 Case Ig(g§,32). This case represents a springemass system with the outboard and the inboard springs being the hard cubic type. Figure 4.8 represents the exact frequency—amplitude curves for these coefficients. In addition, the corresponding nonlinear relation between the amplitudes of vibration of the masses is given in Figure 4.9. As shown in Figure 4.9, the in-phase mode occurs along the 6 = 450 line and the out—of-phase mode occurs along the e = -450 line. It is of interest to note in this case that for absolute values of the amplitudes less than 0.8, only two normal modes are possible. However, for absolute values of the amplitudes greater than 0.8, the asymmetric mode branches off the out—of-phase mode. As Figure 4.2 shows, Anand does predict correctly that the amplitude could not be too small for the asymmetric mode to occur. Also, as the amplitude increases the cubic term of the restoring force is predominant and the asymmetric mode approaches a limiting value, 6 = —20.9°, which is found by substituting the ms for a3 and m for A in equation (F.8). 1 3 Case II(a=0x This case represents a spring-mass system with the outboard springs being the hard cubic type and the inboard spring being linear. The exact frequency-amplitude curves are given in Figure 4.10 and the relationship between amplitudes in Figure 4.11. For absolute values of the amplitudes less than 0.36 only two normal modes exist, namely, the in—phase mode and the out-of—phase mode. However, for absolute values of the amplitudes greater than 0.36 the asymmetric mode branches off the out-of—phase mode. As Figure 4.2 Shows, Anand does not correctly predict the exact result. As the absolute values of the amplitudes increase the asymmetric mode approaches the limiting value, e=0°, again found by substituting the above coefficients into equation (F.8). Frequencv, w 120 Out-of—phase mode Asymmetric mode In-phase mode a = .32 q a1 '- 1.6 k = .1536 k1 = .896 m = 1 ‘ l l 1 l 0 1 2 3 4 Amplitude, x1 Figure 4.8 Frequency versus Amplitude Relationship (a = .32) p... Amplitude, x2 -.8 121 T 1 l In-phase mode a = .32 (11 = 106 )— k = .1536 " m = 1. l 2 3 4 5 ‘ Amplitude, x1 on...“ =-2o.9°, *“"""" Limiting value line (calc. _ from Rosenberg's approach). \ Q.\ P Asymmetric mode - Out-of-phase mode L I l Figure 4.9 Relation Between Amplitudes of Vibration of the Masses, a: .32 Frequency, w 122 a = 0. cl = 1.6 In-phase mode k = .1536 kl = .896 t m = l. ’l” Out-of-phase mode Asymmetric mode 1 1 1 1 0 1 2 3 4 Amplitude, x1 Figure 4.10 Frequency versus Amplitude Relationship (a = 0) 123 3 I 1 l P n 2 In-phase mode a = 0 . 1 F 3 3 k = .1536 s u 2 k1 = .896 5“ 6 = 0°, limiting value line m = 1- ’///’ (calc. from Rosenberg's approach) 0 % ._‘P__.L _____ 45E: % ¢--*‘ 2 3 4 5 —'36 "" . Amplitude, x1 Asymmetric mode -1 P _ ‘2 - Out-of-phase mode _ -3 J L ! of the Masses, a: 0 Figure 4.11 Relation Between Amplitudes of Vibration 124 Case III (9=-.2). This case represents a springdmass system with hard, cubic outboard springs and soft, cubic inboard spring. The exact frequency amplitude curves are represented in Figure 4,12 and the relation between amplitudes in Figure 4.13. For absolute values of the amplitudes less than 0.29 only two normal modes are present-—again, the in-phase and out—of-phase modes. For absolute values of the amplitudes greater than 0.29 the asymmetric mode branches again off the out-of-phase. Again, as Figure 4.2 shows, Anand predicts that the amplitude is not restricted, which, of course, does not agree with the exact result. As the absolute values of the amplitudes increase the asymmetric mode approaches limiting value, 6 = 5.770. before the amplitude becomes unbounded. Case IV (a=-L3). This case represents a springdmass system with hard, cubic outboard springs and soft, cubic inboard spring. The exact frequency-amplitude curves are represented in Figure (4.14) and the relationship between amplitudes in Figure (4.15) For absolute values of the amplitudes less than 0.27 only two normal modes are present-- again, the in-phase and out—of-phase modes. For absolute values of the amplitudes greater than 0.27 the asymmetric mode branches off the limiting value, 6 = 7.90, before the amplitude becomes unbounded. Again, as Figure 4.2 shows, Anand's prediction does not agree with the exact result. Limiting Cases. In this subsection we confine our discussion to Anand's system shown in Figure 4.1, with the restoring forces defined by [4.1]. we show that Rosenberg's homogeneous system with degree three does serve as a limiting case of Anand's system for large amplitudes. Frequencv, w 125 r I I I a --.2 a1 = 1.6 k k = .1536 k1 = .896 m - l. ‘ In-phase mode ’25 - /’~\ Asvmmetric mode / / / / / / x / J’////”,,. Out-of—phase mode . ° ‘ ‘° J: :@ 4% _:_~ ’7 L, 1 l .4 Figure 4.12 1 2 3 b Amplitude, 31 Frequency versus Amplitude Relationship (0 =—.2) H Amplitude, x2 126 I I l In-phase mode b m a =-.2 (11:1.6 k = .1536 _ 4 k1 = .896 6 = 5.770, limiting value line m = l. (calc. from Rosenberg's app::::hz——fl________-_.—————"”' -TM 1 L4 2 3 4 5 Amplitude, x1 Asymmetric mode Out-of-phase mode r d 1 1 1 Figure 4.13 Relation Between Amplitudes of Vibration of the Masses,(,=-.2 Frequency, w 127 In-phase mode Asymmetric a =-.3 k = .1536 . . m = l. ’ : ' O . ' Out-of—phase mode j l l 0 2 3 4 Figure 4.14 Amplitude, x1 Frequency versus Amplitude Relationship (a =-.3) 128 3 T I T a =-o3 In-phase mode 2*- al-lo6 ‘1 k = .1536 k1 = .896 m = l. KN . 1L 1 '§ 6 = 7.90, limiting value line ‘: (calc. from Rosenberg's approach) id”’¢#,,.—~— F1 ..«vrr ‘g‘ ",,,~#*"””’.-— __4/ o ”54’ .L = 1‘ . N 2 3 4 5 -.27 --- r Asymmetric mode Amplitude, x1 -1_. _ -2.. 4 Out-of—phase mode -3 .1 l J Figure 4.15 Relation Between Amplitudes of Vibration of the Masses, a=-.3 129 To show this Rosenberg's polar coordinate formulation of the homogeneous system is summarized in Appendix F. Likewise, Anand's system is re- formulated in terms of polar coordinates, is analyzed for large amplitudes, and is compared with Rosenberg's homogeneous system. Polar Coordinates-eAnand's System. The equations of motion for this system are given by (4.2) and (4.3) or, equivalently by (F.l) and (F.2) where l 2 and al 2 a3 4 4 A1 2 .— 3 .. I = — 2 —— __ .. U U(xl,x2) (xl +x2 ) + (x1 +x2 ) + (x1 x2) 2 4 2 A3 +-—- (x -x )1+ (4.51) 4 l 2 with 81 = k1, a3 = ma By introducing the polar coordinates x1 = r cos 6 and x = r sin 6 2 into (4.51) the potential function -U becomes alr2 a3r4 -U = 'U(r:6) = + (cos”6 + sinue) 2 4 A r2 A r'+ +-—l—— (cos 6-sin 6)2 + 3 (COS O-sin 0)“ (4.52) 2 4 130 Since the total energy Uo for this conservative system is constant, the potential energy —U equals Uo at the maximum amplitude position; and the kinetic energy equals Uo at the equilibrium position. Thus, the derivative of (4.52) at the maximum amplitude yields 3U 8U dU --—dr+— de=0 0 Br 36 or, dr 3U 3U ._ .-_ /_ d6 86 Sr In this case the modal relations are, in general, not straight but rather curved. However, as Rosenberg shows (Appendix F) the vanishing of the derivative of (4.52) at the maximum amplitude still defines a normal mode provided r = r(6) intersects -U = U o orthogonally. Hence, dr/de = 0 implies that au/ae = 0. Thus taking the derivative (4.52) with respect to 6 yields aU/BB = [ a3[cos36 sin 6-sin36 cos 6] + A3[(sin 6 + cos 6)(cos 0 -sin 6 )3] + Allr2 [(cos a -sin 6)(sin 6 + cos 6)] 1 r“ (4.53) After some algebraic and trigonometric manipulations (4.52) reduces to EU A1 -—- = 0 = A3rl+ cos 26[(a3/(2A3) —1] sin 26 + l + ] 39 431:2 (4.54) 131 Equation (4.54) reduces to (F.7) when either A1 = 0 (the linear coefficient of the coupling spring) or, the Al/(A3r2) term is small in comparison to the [[a3/(2A3) -1] sin 26 + 1] term. In this latter case, taking the limit of the bracketed term in (4.53) we obtain a A a 3 lim [(-1 -) sin 26 + l +-1- ]=[(-— -1) sin 26 + 1 J r+ w 2 A r2 2 3 Hence, for a positive definite potential function (a3/A3 >0 or, a3/A.3 - 0 and A3 >0) and for large amplitudes we observe that Anand's result (4.54) reduces to Rosenberg's result (F.8). While the above discussion does not constitute a rigorous proof, we will now arrive at the same conclusion by another method that is suggested by Rosenberg. Rosenberg shows in [14] that the solution x2 = x2(x1) of the system defined by (F.l), (F.2) and (4.51) is completely equivalent to finding the geodesics in the potential energy surface. The potential energy surface is a surface in the xlsz—space defined by the function U. The geodesics are solutions to the differential equation 2("0 + U)n" + [1 +(n')2][n' Ug 41”] = o (4.55) where 1/2 1/2 n = (m) x2 F; = (m) x1 Un = 8U/3n U; = 311/35 and, where U0 is the total energy and U given by (4.51). 132 dzn When n" =-——-= 0, (4.55) reduces to (1&2 n " Un/UE - constant Thus, as Rosenberg says, every straight line which intersects all lines of constant potential energy orthogonally is a modal relation since it satisfies (4.55) as well as the definition of normal mode by Rosenberg (Appendix F). Rosenberg shows in [14] that homogeneous systems of degree one (U quadratic) and of degree three (U quartic) yield straight-line modal relations. Now, we intend to show how the potential [4.51] approaches a straight modal relation for large U0. Thus, we split U into the quadratic and the quartic terms to obtain U = Ul + U2 (4.56) where U1 a quadratic terms and U2 = quartic terms Dividing (4.55) by U0 and collecting terms using (4.56) we obtain 2.91 -391 8U2 _ 3U2 U1 DZ 1 2 n' 5 3“ n' 3; 5n— 2(1+— +—— n"+[1+(n)] + =0 U0 U0 U0 no (4.57) For large U0, U155U2 irregardless of the n' value. Then, (4.57) can be approximated by 0' 9.92 _ .2322 U2 85 8n 2(1 +— )n" + [1 + (n.')2] = 0 (4.58) U U 0 133 But (4.58) is just (4.55) with U = U2. Thus, for large Uo we see that this case is homogeneous and of degree three. Similar arguments hold for small U U >>U . 0’ 1 2 V. SUMMARY AND CONCLUSIONS Vibrations of discrete systems outside of the classical linear domain are no longer independent of amplitude. Since exact solutions to such nonlinear systems are, in general, difficult to obtain, approximate methods gain favor. An approximate method using ultra- spherical polynomials (U.P.) was developed in this research and used to obtain approximate solutions to certain one and two degree- of-freedom vibration problems. The nonlinearity was assumed in the form of a restoring force. The nonlinear restoring force was approximated by expanding it in terms of ultraspherical polynomials orthogonal over the interval (-l,l) with respect to the appropriate weight function, and truncating it after the linear term. The general development of the U.P. approximation presented in this research involved calculating two bilinear approximations over some appropriate interval containing the equilibrium point. The mean square error minimization method was also used to generate bilinear approximations. Relations were obtained which showed that A I -.5 was one of two conditions necessary for the two methods to agree. However, this condition was shown to be merely a linear interpolation between points along the nonlinear function. In the remainder of the research the polynomial expansion method was used because of its greater flexibility in specifying the A parameter. This method was then applied to three odd functions to illustrate the procedure leading to their U.P. bilinear approximation. 134 135 The bilinear U.P. method was used to approximate the motion of the one degree—of-freedom, undamped, free and forced, nonlinear vibrating systems. For the free vibration problem the previously obtained bilinear approximations to certain nonlinear restoring forces were used. Period-amplitude relations were graphed and revealed,in general, improved accuracy over the linear U.P. approximation. A A parameter of 130 (Chebyshev polynomial) and an amplitude ratio of xt/xm-.5 were found to consistently yield the better results. For the forced vibration problem only one case was investigated. This problem included the function f(x) = x + x3 as the restoring force and the unit step-function as the exciting force. Little improvement over the linear U.P. method was obtained by using the bilinear U.P. method. The bilinear U.P. method was then applied to the two degree- of-freedom free, undamped, symmetric nonlinear vibrating system. Anand [18] recently investigated this system using cubic restoring forces and found more than the usual two normal modes of vibration present. Normal modes for nonlinear systems were defined in the sense of Rosenberg [17]. In this research Anand's approach has been shown to be essentially the linear U.P. approximate approach. Having found this, a special case was solved by the bilinear U.P. method and its frequency-amplitude results were compared with Anand's result. Where Anand's formulation predicted the existence of this additional mode (asymmetric mode) the bilinear U.P. method denied its existence. To resolve this dilemma an exact solution was devised using a finite difference method modeled after an approach by Rosenberg [14]. The exact solution showed the bilinear U.P. prediction to be correct. 136 The finite difference method was applied to other cases so that the exact result could be compared to Anand's predictions. In some cases Anand's criterion did correctly predict the existence of the asymmetric mode. However, in other cases the predictions were incorrect. For the above system Rosenberg [14] has provided a more predictable method for determining these asymmetric modes. The finite difference method developed in this research has provided a means of linking together the works of Anand and of Rosenberg, as applied to the existence of asymmetric modes. Several avenues of research suggested by this study are as follows. The bilinear ultraspherical polynomial method could be applied to other forced vibrating systems possibly with damping present. For the symmetric two degree-of-freedom system considered here, an extension would be to generate analog computer solution solutions to compare with the digital computer results. Uhsymmetric systems could also be investigated and might yield very fruitful results in understanding other unusual nonlinear phenomenon. 10. 11. 12. 13. 14. LIST OF REFERENCES Rosenberg, R.M.,"Nonlinear Oscillations", AMR, 14, pp. 837-841, 196] . Stoker, J.J., "Nonlinear Vibrations", Interscience Publishers, 1966 . Denman, H.H., "Amplitude-Dependence of Frequency in a Linear Approximation to the Simple Pendulum Equation", Am. J. Physics, 27’ pp. 524-25, 1959 . Denman H.H. and Howard J.E., "Application of Ultraspherical Polynomials to Nonlinear Oscillations I. Free Oscillation of the Pendulum," Q. Appl. M8th., 21’ pp. 325-30, 1964 . Denman and Liu, Y.K., "Application of Ultraspherical Polynomials to Nonlinear Oscillation II, Free Oscillation", Q. Appl. Math., Denman H.H. and Liu Y.K., "A Graphical Procedure for the Approximation of the Period of Nonlinear Free Oscillations", Jour. of Appl. Mech., 31, pp. 718-19, Dec. 1964 . Liu, L.K., "Application of Ultraspherical Polynomial Approximation to Nonlinear Systems with Two Degrees of Freedom", Ph.D. dissertation, wayne State University, 1965, Detroit, Michigan. Blotter, P.T., "Free Periodic Vibrations of Continuous Systems Governed by Nonlinear Partial Differential Equations", Ph.D. dissertation Chap.V, Michigan State University, 1968, East Lansing, Michigan. Ergin, E.I., "Transient Response of a Nonlinear System by a Bilinear Approximation Method," J. of Appl. Mech., 23, pp. 635-641, 1956 . Howard, J.E., "Asymmetric Oscillations", S.I.A.M. J. Appl. Math., 16, No. 4, pp. 747-755, July, 1968. Snyder, M.A., Chebyshev Methods in Numerical Approximations, Prentice- Hall, Inc., Englewood Cliffs, N.J., 1966 Schelkunoff, S.A., Applied Mathematics for Engineers and Scientists, 2nd edition, Van Nostrand, Princeton, N.J., 1965. Rosenberg, R.M., and Atkinson, C.P., "On the Natural Modes and their stability in Nonlinear Two-Degree of Freedom Systems," ASME Trans. 81 E (J.A.M., vol. 26), 377-385, 1959. Rosenberg, R.M., "Normal Modes of Nonlinear Dual Mode Systems," ASME Trans. 82 E (J.A.M.) 2, 263-268, June 1960. 137 15. 16. 17. 18. 19. 20. 21. 22. Rosenberg, R.M., "0n Normal Vibrations of a General Class of Nonlinear Dual Mode Systems", ASME Trans. 83 E (J.A.M.) 2, 275-283, June 1961. Rosenberg, R.M., "The Normal Modes of Nonlinear n-Degree of Freedom Systems", ASME Trans. 84 E (J.A.M.) 1, 7-14, Mar. 1962. Rosenberg, R.M., "On the Existence of Normal Mode Vibrations of Nonlinear Systems with Two Degrees of Freedom", Quart. Appl. Math, 22, 3, 217-234, OCt. 1964. Anand, G.V., "Natural Modes of a Coupled Non-Linear System", Int. J. Non-Linear Mechanics, Vol. 7, pp. 81-91, February, 1972. Gray, A. and Mathews, G.B., A Treatise on Bessel Functions and Their Applications to Physics, 2nd edition, Dover Publications, Inc., New York, 1966, p. 45. Abramowitz, M., and Stegun, I.A., Handbook of Mathematical Functions, Dover Publications, Inc., New York, 1965. Ergin, E.I., "Transient Response of Non-Linear Spring-Mass Systems", Ph.D. dissertation, California Institute of Technology, 1954, Pasadena, California. Courant, R., and Hilbert, D., Methods of Mathematical Physics, lst Eng. Ed., Interscience Publishers, Inc., Vol. 1, New York, N.Y. 138 M umaanfi A324 83:1. Lu u— «4 HI I lllnsna HI .flufil‘ . 2n 4 . 3 2 3 .. 2:. $333.5... :. Amv a. .. ofiv .. 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J 0 let y = x2, _ -l B(m,n) = Zfl x2111 1 (1-x2)n dx 0 + let 8 = 2m-1flbmi-E-i-l t = n-l => n=t+1=A+1/2 t = A-l/z therefore, 3+1 1* (““2 ) run/2) - 3 2 A 1’2 = £1 x (1-x ) dx 2 F(s+2x+2) 2 145 A-l/Z f1 xs(l-x2) o I‘[S_;i] I‘(A+l/2) 2 IIWJ dx = Jfi P(A+l/2) 2 F(A+1) P(A+l/2) 2 r(A+3/2) 2A+1 /% P(A+1/2) 4 F(A+2) P(A+1/2) 2 P(A+5/2) 2 (2A+1) (2A+3) 3¢% r(A+1/2) 8 P(A+3) P(A+1/2) r(A+7/2) 8 (2A+l) (2A+3) (2A+5) 15¢% r(A+1/2) l6 P(A+4) 146 APPENDIX E EVALUATING k1 FOR THE NONLINEAR FUNCTION, f(x) = sin x [1 (l-t2)>\_1/2[sin(xtt)] t dt I (3.1) O -1/2 fl (l-tz)A t2 dt 0 .J b By changing the variable, t = c036) and by using Appendix D to evaluate the denominator, k1 becomes ffllz sin2A (6) [sin(xt cos 6)] cos 6 d 6 0 k1 = L (E.2) xt _gi_ rgx+ 1/2) _ 4 P(A+2) J Substituting the power series expansion for sin(xt cos 6) using 2 +1 x n {-1)n sin 1: =2 , and using the beta function relation n=0 (2n+1) ! 147 am 2 __1 _ B [—,-J = 2f"/ 81% (x) cosm 1(x) dx 2 2 0 we obtain after simplifying, P 2A+1 2n+3‘ + —— _— m (-1)n xi“ 1 B[ 2 , 2 ] k1 " _]_.__ Z 4 P()\+2) xt J‘n’ r(x+1/2) (13.3) n=0 (2n+l)! 2 ~— J Evaluating the beta function using Appendix D and representing the gamma functions in terms of factorials we find, m (-l)n xt2n+1 P(A+2) (2n+l) (n-1/2)! k1 =.l_ Z (E.4) xt n=0 (2n+l)! VF?' (n+k+1)! The term (2n+1)! can be simplified using a relation in Gray and Mathews [ l9], (2n+1)! = (2n+l) (2n)! = 22n(n-1/2)! n! (E.5) f‘n‘ Substituting (E.5) into (E.4) yields k1 =-—--x;i Z 2 = F(A+2) Jx+1(xt) xt n=0 (n+A+1) ! n! H1 (_ (xii/2) 2 148 APPENDIX F NORMAL MODE VIBRATIONS FOR NONLINEAR SYSTEMS Rosenberg [17] defined normal mode vibrations for the nonlinear system in terms of a vibrations-in-unison of the physical system. A system is said to vibrate in unison if the motion satisfies all of the following conditions: (i) all masses execute equi-periodic motion, or xi(t) = x10: + T), (1 = 1, "‘n) where T is a constant; (ii) there exists a time t = to when all masses pass through the equilibrium position, or xi(to) = 0, (i = l, ----n) (iii) there exists a time t = t1, # to when all velocities vanish,or xi(t1) = 0, (i = 1,""n) (iv) the position of every mass at any instant of time t is uniquely determined by that of anyone of them at the same instant, or H=xflqun. u=2,~~m are all single-valued functions of XI. The governing equations of motion for the nonlinear system shown in figure F.l are ml 1 = 9}; (F.l) 3x1 1112322 " fl 3X2 (F.2) 149 X1(t) x2(t) J; In1 W m2 8 d O O '771/ 7/// //7/ Figure F.l Coupled Spring-Mass System where U = U(x1,x2) is a positive definite potential function. Consistent with above conditions a normal mode of the system in figure F.l is defined as a function x2 = x2(x1) called the modal relation, which is satisfied for all time by periodic solutions x1 = x1(t) = x1(t + T) and x2 - x2(t) = x2(t + T) and where x2(x1) is a single-valued function of x1, in the closed domain -U(x1,x2) = Uo of the(x1,x2)-p1ane subject to the boundary condition x2(0) = 0 and which intersects the line -U(x1,x2) = Uo orthogonally. Rosenberg has shown that modal relations, x2 = x2(x1), for these nonlinear systems are, in general, not constant. However, he has shown that two classes of systems exist for which the ratios of the displacements of the two masses are identically equal to constants (i.e., x2 = c x1) for all time when the system vibrates in normal modes. One class is the homogenous case of degree "k" which may be entirely unsymmetric with respect to the masses as well as the anchor springs (outboard 150 springs). The homogeneous case refers to those systems having all springs with terms of the same degree k. For example, if k=3, then all springs would have only the cubic term present. The other class is the symmetric case in which the two masses are equal and the outboard springs are equal. Rosenberg discusses one feature of the nonlinear system having two degrees-of-freedom which is not found in the linear system. This feature is that there may exist more than two normal modes. This he illustrates by choosing a symmetric homogeneous system with the potential function U(x1,x2) =—«E§ (xl”+ x3)--fig (x]_--x2)‘+ 4 4 (F.3) By introducing the polar coordinates x1 = r cos 9 and x2 = r sin 6 (F.4) the potential function becomes U(r,9) = £:.[-a3(cos”6 + sinue) -A3(cos 9 -sin 9)q] (F.5) 4 The necessary and sufficient condition for the existence of straight modal relations for the positive definite potential function U(r,6) is 311 = 91 (9)02 (r,8) (F.6) 89 Equating, (F.6) to zero implies that 01 (6) is zero and its roots are the modal relations. For this illustrated case the modal relations are 151 ([a3/2A3-l] sin 29 + 1) cos 26 - 0 (F.7) Setting cos 26 = 0, yields 6 = :n/4 and setting [a3/2A3 -1] sin 26 + 1 = 0, yields the additional roots -1 e a -l/2 sin {2/[(a3/A3) —2]] ' for a3/A3 3_4 (F.8) 152 IES ”'Wfiifimgfuijlizu‘xggax@Lnijflxfiiflflfliflh