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' This is to certify that the thesis entitled ON AN I‘TERATIVE METHOD TO APPROXIMATE THE NATURAL MODES 0F VIBRATIONS 0F LINEAR ELASTIC CONTINUA presented by Richard Dale Rabbitt has been accepted towards fulfillment of the requirements for Master ' 5 degree in mm Engineering 0 5 Major professor 0-7639 MS U is an Affirmative Action/Equal Opportunity Institution MSU LIBRARIES 42—— \- RETURNING MATERIALS: Place in book drop to remove this checkout from your record. FINES wiIl be charged if book is returned after the date stamped below. ON AN ITERATIVE METHOD TO APPROXIMATE THE NATURAL MODES OF VIBRATIONS OF LINEAR ELASTIC CONTINUA BY Richard Dale Rabbitt A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Mechanical Engineering 1982 ABSTRACT ON AN ITERATIVE METHOD TO APPROXIMATE THE NATURAL MODES OF VIBRATIONS OF LINEAR ELASTIC CONTINUA BY Richard Dale Rabbitt For various differential and integral operators it is necessary to employ approximate techniques to find represen- tations of the mode shapes and frequencies of vibrating domains. The technique of variation-iteration has been shown to converge to the fundamental eigenfunction for a class of linear systems. Connection of the iterative pro- cedure to the static problem is investigated along with extension of the procedure to higher modes. Several Sturm- Liouville systems are presented as illustrative examples. ON AN ITERATIVE METHOD TO APPROXIMATE THE NATURAL MODES OF VIBRATIONS OF LINEAR ELASTIC CONTINUA Submitted By Richard D. Rabbitt Date Master of Science Candidate Approved By Dr. Mathew A. Medick Date Professor Thesis Advisor Dr. John A. Brighton Date Chairperson Mechanical Engineering ii ACKNOWLEDGMENTS I would like to express my gratitude to Professor Mathew A. Medick for his time, encouragement, and comments during the preparation of this thesis. A special thanks is also extended to the Case Center for Camputer Aided Design. Finally, I wish to thank my parents, Mr. and Mrs. Robert R. Rabbitt, for their encourage- ment and support. iii LIST OF TABLE OF CONTENTS FIGURES O O O O O O O O O O O O NOMENCLATURE . . . . . . . . . . . . . Chapter 1. 2. 3. INTRODUCTION 0 O O C C O O O O O O 1.1 The Spirit of Mathematical Physics 1. 2 The Problem Statement . . . 1.3 An Introductory Description of the Work . . . . . . . . . . THE METHOD OF ASSUMED INERTIAL LOADS. . . .1 Motivation for the Method . . . .2 Fundamental Approximate Inertial Load .3 The One Dimensional Wave Equation . .4 The Two Dimensional Wave Equation on a Uniform Circular Domain . . . .5 The Two Dimensional Wave Equation on an Annular Domain . . . . . . 2.6 Eigenvalue Estimations . . . . . NNNN N AN ITERATIVE PROCEDURE TO IMPROVE APPROXIMATE EIGENFUNCT IONS O O O O C C O O O 3.1 The Variation-Iteration Procedure The Integral Approach . . . . The Differential Approach . . . Iteration of Higher Modes . . . Variation-Iteration of the One Dimensional Wave Equation . . . Variation-Iteration of the Annular Membrane . . . . . . Variation-Iteration of the Uniform Circular Membrane . . . . . . w 0) wwww O 08 WIbWN o \l iv Page vi viii Pdmtfl (fl (a HFJ I4 la H vb haw mu 31 31 33 35 43 46 48 52 Chapter Page 4. The Extension to Higher Order Modes . . . . 56 4.1 The Principle of Monotonicity. . . . 56 4.2 Modification of the Integral Operator . . . . . . . . . . 58 4.3 Modification of the Differential Form . 60 4.4 Exclusion of Lower Modes in the Taut String. . . . . . . . . . . 65 4.5 Generation of Zero Iteration Eigen- functions for the Taut String . . . 70 4.6 The Uniform Circular Membrane. . . . 73 5. CONCLUSIONS . . . . . . . . . . . . 88 APPENDICES Appendix I. Riccati Transformation . . . . . . . . 91 II. Second Order Transformations to Obtain Self-Adjoint Forms . . . . . . . . . 92 III. Inner Product Routine. . . . . . . . . 94 IV. Monotonicity Principle . . . . . . . . 96 V. Leighton's Theorem (1962) . . . . . . . 97 VI. Operators of Class 2 . . . . . . . . . 99 VII. A Finite Dimensional Example . . . . . . 100 VIII. Orthogonality of Eigenfunctions . . . . . 103 IX. Boundary Expansion Solution of the Membrane Operator . . . . . . . . . . . . 105 X. Finite Dimensional Analog to the Boundary Expanstion Technique . . . . . . . . 115 REFERENCES. 0 O O O O O O O O O O O O O 119 Figure 2.1. LIST OF FIGURES Fundamental Eigenfunction of the one Dimensional Wave Equation - Zero Iteration . Fundamental Eigenfunction of the Circular Membrane - Zero Iteration. . . . . . . Fundamental Mode of Annular Membrane for Low Values of R2/R1. . . . . . . . . Fundamental Mode of Annular Membrane for High Values of R2/Rl and Fixed Boundary Conditions. . . . . . . . . . . . Fundamental Frequency vs. Annulus Size for Fixed-Fixed Boundary Conditions. . . . . Fundamental Mode of Annular Membrane for Free-Fixed Boundary Conditions and R2/R1 = 1000 O O O O O O O O O O O O O Fundamental Frequency vs. Annulus Size for Free-Fixed Boundary Conditions . . . . . Free-Fixed, Fixed-Fixed, Frequency Comparison. Fundamental Mode of One Dimensional Wave Equation-“FirSt Iteration. o o o o o o Eigenvalue Error vs. Number of Iterations . . Fundamental Mode of Unifonm Circular Membrane "FirSt Iteration o o o o o o o o 0 Zero Iteration Second Eigenfunction Based on Approximate Lower Modes . . . . . . . Zero Iteration Third Eigenfunction Based on Approximate Lower Modes . . . . . . . vi Page 13 16 20 21 22 23 24 25 49 50 55 74 75 Figure Page 4.3. Zero Iteration Fourth Eigenfunction Based on Approximate Lower Modes . . . . . . . 76 4.4. Zero Iteration Fifth Eigenfunction Based on Approximate Lower Modes . . . . . . . 77 4.5. Estimated Frequency vs. Mode Number for One Dimensional Wave Equation . . . . . . 78 4.6. Frequency Ratio vs. Mode Number for One Dimensional Wave Equation . . . . . . 79 4.7. Zero Iteration Second Eigenfunction Based on Exact Lower Modes . . . . . . . . 80 4.8. Zero Iteration Third Eigenfunction Based on Exact Lower Modes . . . . . . . . 81 4.9. Zero Iteration Fourth Eigenfunction Based on Exact Lower Modes . . . . . . . . 82 4.10. Zero Iteration Fifth Eigenfunction Based on Exact Lower Modes . . . . . . . . 83 4.11. Zero Iteration Second Eigenfunction of the Circular Membrane . . . . . . . . . 87 vii NOMENCLATURE 1. Constants an Arbitrary bn Arbitrary cn Arbitrary e Arbitrary h Step Size k Arbitrary i Length Rn Constant Radius T Tension a Arbitrary B Arbitrary An Eigenvalue wn Eigenvalue En Convergence Parameter 2. Index Variables viii 3. Coordinate Variables r Radial Position t Time xn Spatial Position 6 Arbitrary 8 Angular Position 4. Functions f(x) Arbitrary g(xn) Inertial Load p(xn) Weight q(xn) Inhomogeneity r(xn) Coefficient Function u(xn) Eigenfunction w(xn) Admissible Function y(xn,t) System Solution 6(xn) Approximate Eigenfunction p(x) Density 1(t) Time Dependency 5. Operators A Arbitrary Spatial Operator of Class 1 Partial Differential Operator (xn,t) Kingtic Energy Operator L M Q Energy Operator T V Potential Energy Operator V Laplacian Operator ix 7. Comparison Functions Restricted Comparison Functions Arbitrary Continuous Functions Once Differentiable Functions Twice Differentiable Functions Special Symbols Summation Open Interval (Does Not Include Boundaries) Closed Interval (Includes Boundaries) Inner Product On Real Hilbert Space CHAPTER 1 INTRODUCTION §l.l The Spirit of Mathematical Physics Since the seventeenth century, physical intuition has served as a vital source for mathematical problems and methods. Recent trends and fashions have, however, weakened the connection between mathematics and physics; mathematicians, turning away from the roots of mathe- matics in intuition, have concentrated on refinement and emphasized the postulational side of mathematics, and at times have overlooked the unity of their science with physics and other fields. In many cases, physi- cists have ceased to appreciate the attitudes of mathe- maticians. This rift is unquestionably a serious threat to science as a whole; the broad stream of scientific develOpment may split into smaller and smaller rivnlets and dry out. It seems therefore important to direct our efforts toward reuniting divergent trends . . . R. Cournant, 1953 [Chl] $1.2 The Problem Statement The goal in studying vibrating distributed parameter systems is to solve forced and free initial value problems for various domains. A valid and useful technique of addressing initial value problems is through model analysis (or eigenfunction expansions). The intuitive origin of modal techniques allows for inspection of the resulting theory on physical grounds, and hence lends itself to rapid qualitative evaluation of vibrational features. In addition to the wealth of qualitative information, exact solutions to initial value problems may be found through the use of eigenfunction expansion theorems. The intent of the present work is to exploit a marriage of mathematics and physics in order to obtain modal information. Eigenfunction expansions exist for a large class of operators describing the motion of linearized continuums. Finding the eigenfunctions (or mode shapes), however, is’ often complicated by spatial boundaries that are not expli- citly expressed as coordinate lines. If the domain boundaries cannot be easily mapped into a set of coordinate lines, then spatial coupling will be present, and the operator will not admit to separation of variables [MF1, pp. 497]. Inability or difficulty in reformulating the problem such that the boundaries become coordinate lines, motivates the use and development of approximate methods to determine eigenfunctions and eigenvalues. A number of approximate methods are available to deter- mine modal information. The type of approximation obtained differs from.method to method. The boundary expansion technique, for example, yields a least squares solution whose weight is dependent on the selected basis (Appendix IX, X]. In contrast to many of the popular techniques, the iterative procedure to be presented in the subsequent chapters converges to the exact eigenfunction. In applications, however, truncation of the iterative procedure is necessary and approximations of the eigenfunctions are obtained. To solve an initial value problem the set of eigen- functions must span the solution space of the system. An eigenfunction expansion theorem may be applied only if the solution space is spanned by the set of eigenfunctions. If the eigenfunctions are approximate, or truncated, then the solution space will not be spanned and an exact solution may not be obtained. Approximate eigenfunctions, however, may be used to find least squares solutidns:or to improve dis— cretization procedures such as Rayleigh—Ritz. The following chapters discuss the derivation of a set of approximate eigenfunctions and eigenvalues based on an iterative sequence of static problems. Subsequent applica- tion of the approximate eigenfunctions is not addressed. §I.3 An Introductory Description of the Work Beyond the scope of the present work, the long term goal is to formulate an iterative procedure to determine the mode shapes and frequencies for systems having complex boundary shapes. As a starting point in addressing this problem, considerable effort has been placed on problems of simple geometry for which exact solutions are known. Illustrative examples and fundamental techniques are pre- sented in the hope that future work will extend the analysis to systems having higher order spatial domains. With this motivation, Chapter 2 discusses the physical foundation of the procedure by approximating the fundamental eigenfunction with the aid of a degenerated D'Almbert reverse inertial load. This result is refined in Chapter 3 to obtain the exact fundamental eigenfunction through an iterative sequence of static problems. In order to use the same procedure to calculate higher modes, a new system is formulated in Chapter 4 such that the nth mode of the original system becomes the fundamental mode of the new system. With this result it is possible to apply the iterative procedure to find the nth eigenvalue and eigenfunction. The analysis is based on well established variation- iteration, or Schwarz-Iteration, procedures. Several supple- ments to the classical methods are presented and include: 1. A physical argument motivating the selection of an initial trial function. 2. An extension of the analysis to a direct differ- ential form for general Sturm-Liouville systems. 3. An extension to higher modes through the formula- tion of a new problem in a differential form with- out the use of an integral kernel. These results simplify application of the procedure to a given problem by eliminating the need to find an associated integral kernel. CHAPTER 2 THE METHOD OF ASSUMED INERTIAL LOADS §2.1 Motivation for the Method The method of assumed inertial loads is based on the premise that static behavior of the system under study is well understood. By well understood it is meant that an intuitive feel or qualitative description is widespread and that standard techniques exist to expedite the calculation of static deflections under various loads. The class of pro- blems for which this premise is valid is quite large and is increasing rapidly in size with the aid of digital computers. It is precisely the availability of this mass of knowledge that motivates the use of static results to approximate dynamic behavior. An important link between static equilbrium and dynamic behavior was first introduced by Jean D'Alembert (1743) and is now known as D'Alembert's Principle [Tl pp. 300, BDl]. Based on Newtonian mechanics, D'Alembert reasoned that since the sum of the forces on a particle results in an accelera- tion mx, then the application of a force equal to -m; would produce a condition of "dynamic equilibrium." This principle is the basis of the inertial load concept. If D'Alembert's Principle is extended from particle dynamics to the realm of a continuously distributed group of particles then a reverse inertial load may be defined that would produce a condition of dynamic equilibrium in the distributed media. The required inertial load will be in general a function of space and time. For systems that admit to synchronous motion the inertial load may be written as a product of a function of space along and a function of time alone. The function in space represents the spatial distribution of the inertial load. In general the spatial distribution is not known until the problem is solved completely. If this un- known function causes difficulty in obtaining dynamic equilibrium solution(s) then it may be advantageous to replace the unknown spatial distribution with an approximate function. The selection of an appropriate approximate function is not immediately obvious, however insight may be gained by considering the static equilibrium problem. §2.2 Fundamental Approximate Inertial Load Suppose we have a partial differential equation describing the motion of a linearized continuum having the following form. (2.2.1) Mly] = Lly] + pytt = 0 where L is of class 1 as defined in Appendix VI subject to natural boundary conditions and y = y (x1, x2, . . . x t) n! and the following standard notation has been adopted. a 32y 322 :1 = _ Yt at ' yxt axat ' Ytt at2 Restricting the discussion to systems that undergo synchron- ous motion it is assumed that (2.2.2) y(xl, x , x , t) = u(xl, x 2 n xn) T (t) 2’ .0. For (2.2.2) to be valid it must be able to satisfy the differential operator M and its associated boundary con- ditions. Many differential operators are separable in space and time, and admit to synchronous motion. Their associated boundary conditions, however, are not separable unless they represent coordinate lines [MF1,;KL 497]. At face value this restriction requires the boundary conditions to be independent of time in order for a separable solution to exist. In the general case, however, it may be possible to find a invertable map such that the domain boundaries are mapped into coordinate lines. If this map exists then synchronous motion exists. With this assumption (2.2.2) is substituted into (2.2.1) to obtain the following result. (2.2.3) L[u(xlx xn)'r(t)] = -pu(x1x (5) 00.x T n) 2 2 tt L[u(,>5)T(t)l = -pu<§)rt (t) t Investigating the simplest case, it is assumed that the time dependence is harmonic and that L is strictly a spatial operator subject to natural boundary conditions. With these assumptions (2.2.3) becomes (2.2.4) L[u] = lpu where p = p(xlxz... xn) u = u(xlx2 ... xn) A = w2 i T(t) = Ae wt Equation (2.2.4) is in standard eigenvalue problem form with eigenfunctions denoted by the set {un}. The right hand side of (2.2.4) represents an inertial load, and the left hand side represents various restoring forces. If the inertial spatial distribution is approxi- mated, (2.2.4) may be written (2.2.5) L[¢S] = Apg where LQ ll g(x1x2, ..., xn) = approximate inertial load static deflection 9 II Equation (2.2.5) is simply a description of the static deflection ¢s when subjected to a load lpg. Since it is assumed that static behavior is well understood the solution of (2.2.5) will cause little difficulty. Given the static solution the problem is reduced to selecting an inertial load 9 that represents the actual inertial load in the system. If a static deflection ¢s for a particular load 9 can be found by equation (2.2.5), then that deflection will be unique. For operators of class t, however, equation (2.2.4) has many solutions; these solutions define the set of eigenfunctions Inn}. The eigenfunctions are con- tained within the set of all possible inertial loads {9}; hence, selection of an arbitrary inertial load may approximate any, or none, of the eigenfunctions. It is therefore necessary to select an inertial load to approxi- mate a particular vibrational mode out of the sequence. In studying vibrations of a particular domain, it is often possible to make sharp, qualitative, statements concern- ing the shape of the fundamental mode. Similar statements are not as easily obtained for higher modes. Availability of this knowledge motivates approximation of the funda- mental mode rather than a higher mode. Use of the inertial load concept for higher vibrational modes is discussed in Chapter 4. If the fundamental mode of a given system has no interior zeros, nodal lines, or nodal surfaces, then the inertial distribution within the domain will be of one sign. Existence of a fundamental mode of this type may be shown for a large class of problems based on comparison theorems. Leighton's comparison theorem for example (Appendix V) may be used to prove existence in the general 10 Strum-Liouville case. Assuming the fundamental mode has no interior zeros (surfaces, lines or points), it is possible to select an inertial load of one sign that also has no interior zeros. Under this load the static deflection as given by equation (2.2.5) will be defined as an approxima- tion to the fundamental eigenfunction. Insight concerning the qualitative shape of the fundamental mode may lend a hand in determining a spatial distribution of the inertial load. In the event that no qualitative information is available--as would be the case in a numerical scheme--it is useful to approximate the inertial load with an unbiased constant over the domain. Substituting a constant into the static problem, the following equation is obtained. (2.2.6) L[¢{°)] = ep where e = arbitrary constant ¢{O) = ¢{°)(xl,x2 ... xn) = approximate eigen- function P = P(xl,x2, ... xn) = weighting function (0) The superscript (0) denotes that ¢ is a zero iteration approximation, and the subscript 1 denotes that ¢1 is an approximation to the first eigenfunction. If the approximate eigenfunction ¢l(°)offers a better qualitative description of the actual eigenfunction than the arbitrary constant e then progress has been made. Improve— ment of the spatial representation suggests an iterative 11 process that would use the (n-1)th order approximation as an inertial load when calculating the nth order approximation. This hypothesis is investigated and applied in Chapter 3. Prior to investigating more accurate iterative pro- cedures it is desirable to acquire some insight and confidence in the zero order approximation. To this end attention is turned to specific simple example problems for which exact solutions are known. §2.3 The One Dimensional Wave Equation Consider as an example the one dimensional wave equation. (2.3.1) VtZO, 05x31 C yxx = ytt Suppose the displacement u(x,t) is constrained to zero at x = 0 and x = 1. y(0,t) O VtZO (2.3.2) y(1,t) = 0 VtZO Assuming a separable solution in the form of (2.2.2), the spatial equation is (2.3.3) -c2uXX = Au Considering the zero order approximation ¢{O) (2.3.3) becomes (2.3.4) ¢{°) — e 0 xx ‘Where eo is an arbitrarly constant. Equation (2.3.4) re- presents static deflection of the domain under a constant 12 uniform load. To obtain the static solution (2.3.4) may be integrated directly. (0) _ xn $1 - f f eodgdn + elx + e2 2 (o) _ eox (2.3.5) ¢l — ——E— + elx + e2 The constants e0, el and e3 may be related by the boundary {0)(0) = ¢{°)(1) = 0.0 the approxi- mate fundamental eigenfunction becomes conditions. Requiring ¢ (2.3.6) ¢{°) = bl(x2 - lx) where bl = eo/2 and is arbitrarily. (0) The shape of ¢l is compared to the actual eigenfunction in Figure 2.1. Using the approximate shape the eigenvalue is bounded by Rayleigh's Quotient. (see section 2.6) du 2 )2 < deEE) dx (2.3.7) ( " idlufldx = Wu” (HE The approximate eigenvalue associated with ¢{0) is defined by Rayleigh's Quotient as (n) 2 m _ (n) (2.3.8) ( C ) - R(¢m ) (o) 2 w _ (0) ( C ) " R(¢l ) coflumumuH ouom t coauooom m>m3 HmcoflmcoEHo moo one mo cofluocsMcomHm HoucoEmocom 13 A\x mUCMUmHQ omuflaoEHoz F r _ _ coflpmaflxoummfi coauououH OHoN coausaom uomxm H .m 8ng AxVD mesuaamea 14 It is a common result that Rayleigh's Quotient is stationary near an eigenvalue. This indicates that the approximate eigenvalue may be better than the approximate eigenfunction. ('better' has not been defined) For the shape given by (2.3.6) it is found that w(0) _ /I6c . 3.1623 c (2.3.9) 1 -_-1_ ~ 1 The exact value is _'19 3 3.14159 c (2.3.10) ml - 1 1 A relatively reasonable agreement. §2.4 The Two Dimensional Wave Equation on a Uniform Circular Domain Consider as a second example the two dimensional wave Operator governing the motion of a uniform circular membrane having clamped boundary conditions. The problem is written (2.4.1) szzy = ytt Vt_>_0, v0, vo_<_r_<_R y(R, e, t) = 0 ve, VtZO (2.4.2) yr(0. 9. t) = 0 ve, Vt_>_0 where Y = Y(rr 6! t) 15 Once again it is assumed that a separable solution of the form (2.2.2) exists such that the problem is reduced to an eigenvalue problem in space. (2.4.3) y = u(r, 6) T(t) (2.4.4) c V u = -Au Considering a zero order approximation the following static problem is formulated. (2.4.5) v2¢(°) = e Taking advantage of the symmetry of ac, (2,4.5) reduces to (2.4.6) ¢(°) + l eiO) = e l r rr 0 This may be integrated to obtain the zero order approxima- tion. (0) .1. §— (ra¢l ) = e r Br 3r 0 (0) _ 2 (2.4.7) o1 (r.e) — eor + ellog(r) + e2 Constraining this to meet the boundary conditions specifies el and e2. (2.4.8) ¢{°) = b1(r2 - R2) Figure 2.2 compares this result to the exact solution. Application of Wirtinger's inequality in the form of Rayleigh's Quotient yields 16 .coflpmumuH cums I ocmnnfioz unasouflu gnu mo coauocsmcomflm Hmucofioocom m\H mocmpmflo Hafiomm GONHHMEMOZ coaumaflxoummd coflumumuH ouom cofluoaom uomxm IIIIIIIIII .m.~ eusmflm Ages mesunamea l7 (o) 2 2 Bo (2.4.9) wi0) fd r ( a: ) dr ()— C Id r (¢{°’)2 dr =2. R2 Hence (o)_,r6-C: ml -———- (2.4.10) — R 2.449 wk) The exact eigenvalue is given to four significant digits by [M1] (2.4.11) wl = 2.405 wk: An agreement of two significant digits without iteration. §2.5 The Two Dimensional Wave Equation on an Annular Domain Suppose we extend the results of section 2.4 to a con- centric annular domain 2 as shown below. f a ‘2? 18 Consider two sets of boundary conditions. The first boundary condition is a clamped-clamped condition and is given by II C u(Rl, e, t) ”'5'“ ve, VtZO u(Rzl 8: t) = 0 The second condition is clamped along the outside boundary and free along the interior boundary. ll 0 ur(Rl, 6! t) (2.5.2) V6, VtZO u(RZ, 6, t) = 0 Case I The form of the first approximate eigenfunction is given by (2.4.7). (2.5.4) ¢ = r e + e 1 o 1Log‘r) + e 2 Constrain (2.5.4) by applying (2.5.1). Evaluating the arbitrary constraints, it is found that 2 _ 2 (2 5 5) e = (R2 R1) 6° ' ‘ 1 Log (Rl/RZ) Ri Log (RZ/RI) + R: Log (R1) (2.5.6) e2 = ( Log (Rl/RZ) ) eO Once again Wirtinger's inequality may be used to obtain the lapproximate eigenvalues. 19 2 2 Id reg—‘3) dr (2.5.7) (— - — fd r(u)2 dr Normalizing the system such that R1 = 1.0, the estimated frequency mic) may be found for various values of R2. For this same normalization constant the estimated eigenfunctions may be calculated. These results are shown in Figures 2.3, 2.4, and 2.5. Case II Again the form of the approximate eigenfunction is given by (2.4.7). (2.5.8) ¢{°) = rzeo + elLog(r) + e2 Constraining this form to meet the fixed-free boundary con- ditions specified by (2.5.2), the constraints el and e2 may be specified. (2.5.9) el = -2R e 2 2 (2.5.10) = ((-R2) + (2R1Log R2))eO e2 The approximate mode shapes and frequencies are shown in Figures 2.5 and 2.7. The fundamental frequency of the free-clamped annular membrane is compared to the fundamental frequency of the clamped-clamped annular membrane in Figure 2.8. As expected, the lower curve corresponds to the free-clamped membrane. 20 .Hm\mm we modao> 30g new mcmunfimz umHsccé m0 @602 Hapcofimocsm .m.~ madman OoNHH OoHHH (L TS: menuflamsa 21 can A .mcoHuHGCOU muooasom ooxflm m\mm mo mmsHo> swam u0w mcmunsoz uoascc¢ mo moo: HopcoEoocsm .v.m ousmflm moeebmno fiancee o A5: mesuflamee OOH n m\Nm oooa n m\ m 22 .mcoflpflocou mumocsom ooxHMIomme How oNHm moaocc< .m> accosooum Hmucmamocsm H Ao.H n m Ou CONHHMEHOZV mm Numocsom ooflmuoo am A.___ A 2 . [IL b It .l p T j 1 1 1' 1 1 i 1 1| j .m.m 8.3on 93 com . Nucofiwwum 3m 8* sum 23 .o.oa n Hm\~m use mcofluflocou wumocoom omxflmaooum How mcmunsoz Hoaocc¢ mo moo: HmucmEmocsm .m.m ouomflm ca mocmumflo Hafiomm an m II a: Auvs mesbflamem 24 .mcoHuHUcOU xumocsom ooxflmiooum new muflm moaoccd .m> hocosooum HmucoEmocsm .n.m ousmflm .o.H n Hm on emanamsuozc mm wumocsom oofimuso Hm e A 2 b ,It PI h _ b. p p S 1' r mm“ 1. . H 1 o a m .. 5mm 1 e. I "K T H u N s l mocmnwmum mm. 1 1 as“ j T Tl 1.81 T 1 9mg .COmHummEoo >ocooqmnm .omxflhlomxflm .omxflmlmmum .m.m ousmflm Ao.H u Hm on eouflamsuoz. m m mumocoom moflmuso Hm w 1 r /// 11 IINT // 25 o.H n m I am“ r Lame u 2 3 T R N A I Nocmowoum T j and T I . 203350 33:58 emeueme. 1 mm.— L 1 mcoflu. :00 wumocoom omxflmnoonmrlg W T 26 §2.6 Eigenvalue Estimations Up to this point eigenvalues have been approximated using Wirtinger's inequality or Rayleigh's Quotient. The intent of this section is twofold; first to show that Wiritinger's inequality is valid for a given class of systems, and second to expand the inequality to include higher order modes. The expanded inequality will be neces- sary in order to apply the inertial load technique to higher modes as presented in Chapter 3. Within the large class of problems that admit to eigenvalue estimation by Rayleigh's Quotient, the following analysis will restrict itself to problems that may be reduced to the following Sturm-Liouville form. (2.6.1) %;(r(x)%§u(x))+-p(x)u(x) = 0 or (2.6.2) L[u] = %;(r(x)§-§u(x)) = Ap*(x)u(x) where p (X) = -Ap*(X) By the Prfifer transformation it is known that an eigenfunction expansion exists for the general boundary con- ditions given below. [A1, pp.222-229] -(2.6.3) o1 u(a) + azu'(a) = 0 l O 81 u(b) + 82u'(b) 27 where the notation u' = g% has been adopted In order to simplify the following analysis several assumptions are placed on the above system. First of all, it is assumed that r(x)>0 on the open interval (a,b) and that r(x) and p(x) are continuous on (a,b). It is also assumed that the boundary conditions are not of the mixed type. These constraints are written r(x) > 0 on [a,b] (2.6.4) r(X). p(X) ECIa.b] a 91 2 = 8182 = 0 Denote the set of comparison functions on [a,b] by 0 b (2.6.5) o = {W8D2[a,b], fr dw 2 a (3;) dx < °°I w(a)W'(a) = w(b)'w'(b) = 0} Wirtinger's inequality states: Given u(x) ¢ 0 on (a,b) is a solution to (2.6.1) and (2.6.3) with constraints (2.6.4), then for every ch b 2 (2.6.6) fr(%¥) dx 3 fbpwzdx a a = only if w = ku where k is an arbitrary scalar. The intent of this analysis is to show that the following 'corollary must hold: 28 Corollary: Denote a new set of comparison functions orthogonal to the eigenfunction ul(x) on [a,b] such that (2.6.7) 8: = {w=D2[a,b]; w(a)w' (a) = w(b)w' (b) = o, f:(gy)2dx < w < w(x) u (x) > - 0} a dx ’ ’ l — Given u2 # 0 on (c,d), u(c)u'(c) = u(d)u'(d) = 0 and a jgc < d i b so (c,d) c (a,b), then for every wcfli we have (2.6.8) fdrw'zdx 3 fdpwzdx : = if w = ku2(x) c c Proof: Let co, we0* a B 2' 22 = f (rw' -2ww'rv + rv w )dx a . Integrate the center term by parts. 8 B = f (rw'2+rv2w2)dx - f rvdw2 a a B B B = f (rw'2+rv2w2)dx - [rvw2]| + f w2(rv)'dx a a a 8 (2.6.10) f [rw' , a B a 2 + w2((rv)' + rv2)]dx - [rvwzll 29 Equation (2.6.10) must hold for all choices of v 8 c(a,b). Clearly one possible choice for v is the function < u 5L3: (2.6.11) N u2(X) ae 0 on (c,d) Where u2(x) is a solution of (2.6.1) subject to the constraint = 0, and u2(x) # O on (c,d). With this choice of v it is possible to apply the Riccati Transformation as presented in Appendix I. In doing so equation (2.6.1) becomes (2.6.12) (rv)" + rv2 + p = 0 Substituting this into (2.6.10), the following inequality must hold. 8 B 2 2 ru'w2 (2.6.13) f (rw' - pw )dx - [——Ir—JI 3_0 o a We are interested in the limit of this expression as o+c and 8+d. Consider the limit of the second term. (2.6.14) lim [51—17;]! c,d u c Noting that r(x) is bounded; if the boundary condition is on the slope then clearly (2.6.14) is zero when evaluated at that boundary. If the boundary condition is on displace- ment then by uniqueness the slope will not be zero and it is seen that 30 (2.6.14) lim [ru“w2] lim [$3] c,d u c,d u Since w(x) and u(x) approach zero if the boundary condition is on displacement, we may use L'Hospital's rule to evaluate the limit. (2.6.15) lim [£3] = lim [2ww'] c,d u c,d u' Again by uniqueness w' ¢ 0 and u' # O on the boundary [c,d], so the limit must in fact be zero. Applying this equation, (2.6.13) reduces to d 2 2 (2.6.16) f (r(w') - pw )dx 3 0 c = if w(x) = ku2(x) where k is an arbitrary scalar. The form (2.6.16) is nice in that any two points may be selected such that u(c)u'(c) = u(d)u'(d) = 0. These two points define the domain of integration and must be only within the interval [a,b]. Since the set of comparison functions is restricted to the orthogonal complement of the fundamental mode, then (2.6.16) will bound the second eigenfunction from above. This analysis may be extended to the nth mode by including (n—l) orthogonality conditions in the set of functions 0*. CHAPTER 3 AN ITERATIVE PROCEDURE TO IMPROVE APPROXIMATE EIGENFUNCTIONS §3.1 The Variation-Iteration Procedure Chapter 2 demonstrated that for a particular class of differential operators it is possible to obtain an approxi- mation of the fundamental eigenfunction by degenerating the inertial term within the operator. By doing so, relatively accurate eigenvalues, and to a lesser degree eigenfunctions, were generated for some simple examples. More complicated differential operators, less accurate inertial load approxi- mations, and a fundamental need for "exact" modal informa- tion necessitates improvement of the zero iteration approxi- mation. As indicated in section 2.2, the spatial distribution of the inertial load is represented exactly by the eigen- functions. Recall equation (2.2.4). (3.1.1) L[u] = Apu The function u on the right hand side comes directly from the spatial distribution of the inertial load. If the 31 32 inertial load is approximated as being evenly distributed over the domain, the zero order approximate eigenfunction is obtained from equation (2.2.6). (3.1.2) L[¢{°)] = ep where e = constant of arbitrary magnitude p = p(x1 .. x2) = weight function ¢{°) = ¢IO)(X1 ... xn) = zero order approximate eigenfunction. If the function 0:0) represents the fundamental eigenfunction better than the assumed inertial load (in this case a con- stant), then it would be reasonable to use the function ¢{°’ approximate eigenfunction. With this motivation the first as an approximate inertial load in calculating a second order (first iterated) approximate eigenfunction may be defined as follows. (3.1.3) L[¢{1’1 = p(¢{°’) If representation of the eigenfunction continues to improve, it may be valuable to define the nth order approximate eigenfunction. (n-l) (3.1.4) L[¢(n)] = p (41 ) l The interest is in determining if this sequence of problems will converge to the exact eigenfunction. 33 The question of convergence may be confronted in one of two ways. The first method would be to define an inverse operator and formulate an integral process that may be evaluated for convergence. The second method would be to work directly with differential inequalities. The following two sections address convergence of the integral and differ- ential forms respectively. §3.2 The Integral Approach It can be shown that solving the homogeneous differential equation given by (3.1.1) is equivalent to solving the following corresponding homogeneous integral equation ICHl, pp. 358]. (3.2.1) u(x) = Af K(x,€)p(§)u(€)d€ d Once again restricting discussion to self-adjoint operators L, the kernel K(x,€) will be symmetric and is related to the eigenfunctions by the following relationship [CH1, pp. 360]. un(x) un(§) A (3.2.2) K(x,€) = Z n 1 Expression (3.2.2) is called the Bilinear Relation and will be accessed in a later section. Given this integral formulation, the expansion theorem allows any solution u of (3.2.1) to be written as an infinite ,series of eigenfunctions [81, PP- 218]- 34 00 (3.2.3) u(x) = nil cn un(x) Define the first trial function in the series as ¢(°) o O I o o n I and write ¢( ) as an 1nf1n1te series in un CD (3.2.4) ¢(o) = )3 c u (x) n=l n n In order to generate the first order approximation suppose 0(0) is substituted into the following integral. (3.2.5) ¢(1) éK(x.€)p(€)¢(o)(€)d€ (1) = E CD I K(X,€)p(€)un(€)d€ n D 1 But since un(x) is an eigenfunction equation (3.1.5) must hold such that , (1) 0° unIX) (3.2.6) ¢ = Z c n=l n An Substitute this into (3.2.5) °° u (E) (3.2.7) ¢(2) = f K(x.8)p(€){ c n } d€ D _ n A n—l n °° C ¢‘2’ = 2 T2 f K(X.€)p(€) un(€) 86 n=1"n D °° C (3.2.8) em = ): izun n=l In ‘In general the iterative process will yield (3.2.9) ¢(i) = 2 Given a system whose eigenvalues are characterized by (3.2.10) Al<12 (3.3.2) 11 = V89 (V v) 0 = {veD2[a,b] : chZ, veC[a,b]} Considering all functions ch, the one function correspond- ing to a minimum value of the above inner product ratio will be the fundamental eigenfunction. In using such a charac- terization it is often difficult to determine where to search within the set 0 and usually impossible to search the entire set of admissible functions. Since the aforementioned sequence of approximating functions converge to the funda- mental eigenfunction, a viable method of proceeding through the set 0 is through the approximating sequence. If the first approximate eigenfunction is substituted into the supremal quotient (3.3.2) we find 37 1(1) _ l _ (u(n)’ u(n)> We will have uniform, monotonic, convergence if (3.3.5) 11‘“) 3 11(n+1’ Vn (3°3'6) An - vc0* 9* = {ch; = o : Vk = 1, 2, ..., (n-1)} 38 Where the functions uk are the eigenfunctions associated with each 1k. Having existence, convergence of the approximating sequence is shown for partial differential operators that reduce to the following Strum-Liouville form. (3.3.7) 5% (r(x) 3% (u(x))) + p(X) u(x) = o u(a) 3‘}, (u(a)) = 0 (3.3.8) d u(b) a; (u(bll = 0 Where the boundary conditions have been limited in order to apply Wirtinger's inequality. Note that (3.3.7) may be written in standard eigen- value form. (3.3.9) L[u] = Ap*(x) u(x) where (3.3.10) p(x) = -Ap*(x) Consider functions r(x)>0; r(x), p(x) eC(a,b). Recall that L is self-adjoint and gives rise to the following weighted inner product. b (3.3.11) = f pfg dx a :For all admissible functions in the set a, Wirtinger's inequality states 39 b (3.3.12) f (rw'2 = pwz) dx 3 o 3 (3.3.13) 8 = {weD2[a,b], w(a)w'(a) = w(b)w'(b) = 0. fb rw'2 a dx < w, w(x)¢0 on [a,b]} Using this inequality, define for any approximate eigen- (n) function u a parameter €(n) by the following equation. 2 2 (3.3.14) e‘n’ = 1b (r(x)(3§u‘n’(x)) - p(x)(u‘“’(x)) )dx a The constraint w(x)#0 on (a,b) must apply to all approxi- mate eigenfunctions uIn) such that u(n)#0 on (a,b). This constraint requires all approximate eigenfunctions to be of one sign on (a,b). If the sequence of approximate eigen- functions converges to the fundamental eigenfunction then the fundamental eigenfunction must also be of one sign. This is reasonable for many systems, and in fact characterizes the fundamental mode of Sturm-Liouville systems. Leighton's theorem, as given in Appendix V, may be used to prove existence of a single sign fundamental eigenfunction for Sturm-Liouville systems. Assuming the first approximate eigenfunction in the approximating sequence is an element of a, monotonic con- (n) (n+1) vergence will be shown if e > e . Consider the (following proof by contradiction. 40 Suppose that (3.3.15) €(n) 5 c(n+l) '2 2 b +1 '2 +1)2 fb (ruIn) - pu(n) )dx 5 f (ru(n ) - pu(n )dx 8. a rewrite as (3.3.16) 2 2 u u 2 2 fb(r(x)(u(n) - u(n+l) ))dx 5 fb p(X)(u(n) - u(n+l) )dx l— a -.....__._____.___l )3 ..-. . J If I I LFT RT First consider the left hand side LFT = fb r(x)(u(n)' - u(n+l)')(u(n)' + u(n+l) )dx a i fbr(x)(u(n) _ u(n+1) )d(u(n) + u(n+1)) a Integrate by parts to obtain the following expression 1: I b LFT = r(x)(u(n) - u(n+l) )(u‘n) + u(n+l))l a (3.3.17) -fb(u(n) + u(n+l))d{r(x)(u(n) - u(n+l) )} a 41 Since the approximate eigenfunctions are comparison functions they will meet the boundary conditions and the first term in the above equation will vanish. Recall equation (3.3.1) (n)) (n-l) 5%(r(xlagu = -p(X)u and substitute into (3.3.17) -fb(u(n) + u(n+l) ) {-p(X)u(n-l) + p(X)u(n) }dX a LFT (3.3.18) fbp(x)(u(n) + u(n+l’)(u‘“'1) - u(n))dx a We could have integrated (3.3.16) in another way as follows. LFT = Ib (n). + u(n+l)') d(u(n) — u(n+l)) a r(x)(u Again integrate by parts. (3.3.19) LFT = fb p(x)(u(n) - u(n+l))(u‘n‘1) + u(n))dx a Equate (3.3.18) and (3.3.19) to obtain the following recursive relationships. 2 (3.3.20) LFT = % fb p(x){u(n) - u(n+l)u(n-l)}dx a (3.3.21) LFT = % rb p(x){u(n)(u(n-l) - u(n+l))}dx a ' Substitute (3.3.20) into (3.3.16) 42 2 I - fb p(x)u(n+l) dx 5% fb p(x){u(n) + u(n+l)u(n 1)}dx 3. a 1 b (n)'2 (n+1) (n-l) i|§ f {r(x)u + p(x)u u )dx 3. (3.3.22) = % 1b p(x){u(n’1)(u(n+l’ - u(n))}dx a and similarly with (3.3.21) 2 (3.3.23) fb p(x)u(n+l) dx i% fb p(x)un(u(n-l) - u(n+l))dx a a add (3.3.22) and (3.3.23) 2 (3.3.24) fbp(x)u(n+l) dx 5 % fb p(x){u(n+1’(u(n"1’ - u(n’)}dx a a = % fb p(x)u(n+l)(u(n-l) - u(n))dx 3 (3.3.25) = '% fb u(n+l) 5%{r(x)(u(n)' - u(n+l)')}dx a Integrate by parts such that (n+l)'(u(n)' _ u(n+l)' (3.3.26) = % fb r(x)u )dx a write as b (n+1)2 b (n+1)'2 b (n+1)' (n)' 4f p(x)u dx -f r(x)u de: -f r(x)u 11 dx a a a From Writinger's inequality :2 2 fb r(x)u(n+l) dx 3 fb p(x)u(n+l) dx a a 43 hence (n+l)' 2 I (3.3.27) 3fb p(x)u(n+l) dx i (-)fb r(x)u u(n) dx a a Integrate the left hand side by parts once again 2 2 (3.3.28) 3fb p(x)u(n’l’ dx 5 (“)f p(x)u(n) dx a or b (n+1)2 (n)2 (3.3.29) f p(x){3u -+11 }dx i 0 a In general Sturm-Liouville theory it is common to require b 2 (3.3.30) f p(x) (u(x)) dx > 0 a in order to satisfy existence and uniqueness criteria [A1, pp. 205]. With this requirement on p(x), (3.3.29) is clearly a contradiction and the thesis is supported. (3.3.31) €(n) > €(n+l) Equation (3.3.31) shows that convergence is uniform and monatonic. §3.4 Iteration of Higher Modes The iteration procedures given in sections 3.2 and 3.3 will converge to the fundamental eigenfunction providing a particular iterated eigenfunction ¢(n) is not exactly equal to an exact eigenfunction [81, pp. 231]. If the interest is to apply the same procedure to the second mode then it is necessary to remove the first mode from the problem. This 44 may be done by formulating a new differential operator that excludes only the first mode from its solution space, or by preventing components of the first mode from entering into the procedure. Consider first the formulation of a new integral opera- tor. Recall the original problem in its integral form. (3.4.1) u(x) = If K(x.€)p(€)u(E)d€ d As shown in section 4.2 the Bilinear Relation allows the writing of a new kernel whose solution will be identical to the original problem less the fundamental mode. If the original kernel K(x,€) is known then the new kernel R(x,g) may be found such that the iteration applied to R will yield the second eigenvalue. Often, however, the original kernel K is not known until the problem has been solved, and hence this technique is of little use in obtaining new information. Rather than an integral form, it is more common to have a differential form. Modification of the differential opera- tor to remove the fundamental mode is not as straight for- ward as modification of the integral operator. If a modified differential operator is obtained (through techniques as discussed in Chapter 4) then the new operator will yield a fundamental frequency and mode shape identical to the second mode of the original differential operator. In the event that a new differential or integral opera- tor is difficult to obtain, it is possible to restrict the set of approximate eigenfunctions such that the fundamental 45 eigenvalue is not introduced. Consider an eigenfunction expansion of the first trial function. (3.4.2) ¢(O) = Z cnun Suppose components of 6(0) in the un direction are sub- (0) tracted from ¢ . (0) (3.4.3) ¢new E cnun n-2 (0) o u (o) _ (0) __ < ’ D In general, at each iteration it is required that components of the fundamental eigenfunction be subtracted from the generated function. (3.4.5) 6) With this formulation it is clear that as i+w the second eigenfunction will dominate and ¢(1) = (3.4.6) 11m new 11 i+oo 2 The ability to iterate a static deflection to a higher mode will be considered in Chapter 4. The remainder of this chapter will be devoted to presenting several examples illustrative of the iterative procedure. 46 §3.5 Variation-Iteration of the One Dimensional Wave Equation Consider once again the one dimensional wave equation, as presented in section 2.3, with clamped boundaries. 2 _ (3.5.1) c yxx - ytt Vtzo, Ofixfil Y(0.t) -- 0 (3.5.2) V830 y(1.t) = 0 After separating time from space we obtain the following zero order approximation. (See section 2.3) (o) _ 2_ (3.5.2) 01 - bl(x x) The first order approximation is given by (l) _ 2 (3.5.4) ¢lxx — bl(x - X) By direct integration it is found that (3.5.5) 61(1) = b1(T%x4 - lx3) + e X+ 6 e 3 3 Application of the boundary conditions (3.5.2) yields the approximation. (3.5.6) 6 (l) = b (x4 - 2x3 + x) l 1 01(1) is compared to the exact eigenfunction in Figure 3.1. After only one iteration, the curves are nearly impossible to distinguish without extensive magnification. In this case Wirtinger's inequality yields 47 (3.5.7) wl(1) E 3.14181 C Recall that the exact value is given by _ nc z 3.14159 c The second iteration is given by (2) = (1) (3.5.9) ¢l ¢l XX = b1 (x - 2x3 + x4) hence (3.5.10) ¢1(2) = b (x - %x3 + x5 - %x6) Successive iterations yield the following set of approximate eigenfunctions ¢l(o) = x - x2 ¢1(1) = x - 2x + x4 (3.55.11) ¢1(2) = x (1.66667)x3 + (1.000)x5 - (0.33333)x6 ¢l(3) = x (1.66471)x3 + (O.82353)x5 - (0.23529)x7 + (0.05882)x8 ¢1‘4’ = x - (1.66450)x3 + (0.82246)x5 - (0.19582)x7 + (0.03254)x9 - (0.00653)xlo 61‘s) = x - (1.647124)x3 + (0.817511)x5 - (0.190803)x7 l 2 + (0.026389)x9 - (0.00287O)x1 + (O.OOO48)X1 48 The exact series representation of the fundamental mode is given by. 2 3 4 5 — .. ELL I_X_ .. (305.12) 1.11 — X 3! + 5! 0000 = 3 5 7 ul _ x - (1.64493)x + (0.811742)x - (0.190752)x ... Convergence of the coefficients in the approximating series to the exact coefficients seems evident from this example. This result is illustrated in Figure 3.2. §3.6 Variation-Iteration of the Annular Membrane As a second example the clamped-clamped annular membrane as introduced in section 2.5 will be considered. Recall the problem statement. 2 2 (3.6.1) C V y = ytt VtZO, V9, V0_<_r_<_R y = Y(rlelt) y(Rl,6,t) = 0 V0,Vt30 (3.6.2) y(RZIeIt) = 0 The approximating sequence is defined by (2.4.6) (n+1) 1 (n+1) (11) (3.6.3) o + — ¢ = ¢ lrr r 1r 1 The zero order approximation is given by. (3.6.4) ¢1(0) = eor2 + elLog(r) + e2 49 .coaumuouH umHHMIIcoHumoom o>mz HmcoflmcoEHo mco mo moo: Hmucofimocom A\x mocmumwo omNHHmEHoz _ - » h :oflumumuH umuwm :ofluoaom uomxm IIIIIIIIII .H.m museum Axes mosufiemem 50 b R)? .wGOHumumuH mo HoQEoz .m> uouum moam>comflm .N.m whooflm mcoflumuoucH mo Monaoz pomxms alllllmvooa Houum ucoonom 51 Where the constants en are defined in (2.5.5) and (2.5.6). The first order approximation may be calculated directly by integration. (l) 1.31 1L1 _ (o)_ 2 (3.6.5) r 3r (r 3r ) - ¢1 — eor + ellog(r) + e2 8 a (1) 8r (r 3r ) eor + elrlog(r) + ezr 391(1) eor4 r210g(r) r2 ezrz r 8r = 4 + 61‘ 2 ' Z ) + 2 + e3 ‘ (l) eor4 elr2 e2r2 (3.6.6) o1 = —I0— + -—z—(log(r)-1) + __4— + e3log(r) + e4 Recall (R 2-R 2)e (3.6.7) el = 18 (Rl/R ? 9 1 2 (R 2(log(R /R 2) + R 21og(R ))e e = 1 2 l 2 1 o 2 log(R1/R2) The boundary constraints may be used to evaluate the remain- ing constants. T I - 4 2 '1 e3 [ 1 -1 ] e081 R1 } _ 4--——4e (log(R )-1 + e (3.6.8) = log(R2) 109(R1) 16 4 1 1 2 e4 1°9(R1/R2) eoR24 R22 L J :76" "' T{e1(1°9(R2)'1) + 92L 52 It is noted that the approximate eigenfunction contains the fundamental components of Bessel and Weber functions that make up the exact eigenfunction. For comparison the following expansions are written. 2 4 6 (3.6.9) j (r) = 1-———£——— +-———E——---——JE———-+ ... ° 22(1))2 24(2))2 2°(31)2 . °° (’1)m(§)2/m 1 1 (3.6.10) yo(r) = jo(r)1og(r) - ' Z [a +_m:I + ... + l] m=n (m! ) 2 §3.7 Variation-Iteration of the Uniform Circular Membrane Recall the problem statement of the fully clamped uniform circular membrane. 2 2 (3.7.1) c V y = V530, V6, OjriR Ytt (3.7.2) y(R,8,t) = 0 VCZO, V6 Yr(orert) = 0 In the spatial domain equation (3.1.4) defines the approxi- mating sequence and is written (n+1) l. + _ 4’1 (n) (3.7.3) 41 r (n+1) rr r =¢1 Section 2.4 gives the zero order approximation as (3.7.4) 61(0) = eo(r2-R2) 53 Calculating the first iteration it is found that (1) 39 l .2. .__l _ (o) = e0(r2 - R2) solving for ¢1(1) (1) 4 2 2 391 = 5 _ R r r 3r eo(r 2 ) + el 4 2 2 (3.7.6) ¢l(l) = eoIIE -.E_%_) + elln(r) + e2 apply boundary conditions ¢1r(l) (o) = 0 + el = 0 ¢ (1) (R) = 0 + e =-§Bfi e l 2 16 0 Hence the approximate eigenfunction is given as 4 2 2 4 (1) _ ‘£_ _ R r 3R Figure 3.3 illustrates the shape of this approximate eigen- function. Referring to section 2.6, Wirtinger's inequality may be applied to estimate the eigenvalue. 2 (1) du (01 ) 2 _ fD. r(Ef) dr C _ f (3.7.8) ( 2 D r(u) dr 54 (0.02866807) ” R2(O.00494817) hence (1) ~ 9 (3.7.9) ml ~ 2.4070 R Recall the zero order approximation and the exact value (3.7.10) (O) 2.449 wk) (3.7.11) w 2.405 wk) Comparison of Figures 3.3 and 2.2 illustrates improve- ment of the eigenfunction with one iteration. 55 .cofiumuouH umuflhllmcmuofioz HmHsoHHU Showed: mo moo: Houcofioocom .m.m ousmflm m\H mocmumfio ooNflHmEuoz L p p - cofluwumpH umuflm coauoaom uomxm Axon 333mg CHAPTER 4 THE EXTENSION TO HIGHER ORDER MODES §4.l The Principle of Monotonicity Chapters 1 and 2 demonstrate the ability to use approxi- mate inertial loads in obtaining a relatively accurate repre- sentation of the fundamental eigenfunction. It was shown that a sequence of functions could be formulated beginning with an unbiased, single sign, inertial load that would con- verge monatonicly to the fundamental eigenfunction. To begin the sequence qualitative aspects of the fundamental mode were exploited. Similar qualitative information, however, seems to be absent in the study of higher order modes. In order to utilize the information and techniques available for the fundamental mode in the study of higher modes it is often possible to formulate a new problem such that the second mode of the original problem becomes the fundamental mode of the new problem. Once this new problem is formulated it may be treated in the manner described in Chapters 1, 2, and 3. This treatment will yield the original system's second mode shape and frequency. In formulating a new problem whose solution space excludes the first eigenfunction of the original problem it 56 57 must be shown that statements evident for the first eigen- function may be carried over to the nth eigenfunction. Assuming the form (4.1.1) L[u] = Apu where L is of class K as defined in Appendix VI. Recalling the supremal characterization of eigenvalues and eigen- vectors as given in section 2.3, it is natural to persue dynamic equilibrium of the (n+1) mode as described by equation (3.1.2). _ inf (4.1.2) In — ve0* 0* = {ve0; = 0 Vk = l, 2, ..., (n-1)} This characterization indicates that any eigenvalue and associated eigenfunction may be represented as the "funda- mental" if the set of admissible functions is restricted to the orthogonal complement of the lower (n-l) modes. Modification of the set of admissible functions may be in- convenient or bulky and it may be desirable to achieve this goal by an alternative formulation of the differential operator itself. Without addressing how to formulate such an operator Weinstein and Stenger were able to show that fundamental mode techniques would apply to the new problem [WSl, pp. 21]. The principle of monotonicity insures this ability. 58 §4.2 Modification of the Integralggperator If it is possible to find the integral operator associ- ated with L and its boundary conditions then the bilinear relation allows simple removal of the fundamental mode from the integral operator. Consider the operator (4.2.1) u(x) = If k(x,€)p(€)u(€)d€ D Where k(x,€) is the symmetric kernel associated with the self-adjoint operator L. For L of class K the bilinear relation is [CH1, pp. 360] 0° un(x)un(€) (4.2.2) k(x,€) = X 1 n=1 n With this relationship it is possible to define a new kernel R(x,€) such that its integral form will include all modes of the original operator excluding the fundamental. Suppose we let R be equal to k minus all components of k in the 111 direction. u1(x)u1(5) (4.2.3) Ema) = k(x,€) - 1 1 From this it is clear that (4.2.4) R(x,€) = )3 in integral form (4.2.5) u(2,(x) = 1;; k(X.€)p(€)u(2)(€)d€ 59 Where the new eigenfunctions are identical to the original eigenfunctions with the exclusion of the original fundamental mode. The new fundamental is exactly the original second mode. This may be written in terms of k(x,€) ul(X)ul(€) (4.2.6) u(2)(x) = Ag{k(X.€) - 11 }p(€)u(2)(€)d€ In general to formulate a problem having the original nth mode as its fundamental, the integral becomes n-l uk(X)uk(€) (4.2.7) u (x) = Af{k(x,£) - Z In) D k=l A k }p(t)u(n) (ma: If the kernel is known then each.mode shape and eigenvalue may be calculated as if it is the fundamental. This formula- tion allows each successive mode to be calculated by variation-iteration. With this result a method of determining all eigen- functions by variation-iteration combined with initial estimate inertial loads has been established. The integral formulation just presented is quite nice mathematically and lends itself to simple proof. In applications however, the problem is often stated in differential form and its associ- ated integral kernel is not known. If finding the kernel presents a substantial task then it may be advantageous to abandon the integral procedure and consider a direct differ- ential approach. 60 54.3 Modification of the Differential Form The mathematical modeling of a system by Lagrangian or Newtonian mechanics ultimately gives rise to a differential form. It is the availability of the differential form that motivates presenting results similar to the previous section from the standpoint of differential operators. The standard differential form is given by equation (2.2.4), and is reproduced here. (4.3.1) L[u] = lpu P (4.3.2) T Associated Natural I Boundary Conditions ( M Combination of (4.3.1) and (4.3.2) define the differ- ential operator in a functional format. In the interest of distinguishing the equation from the constraints, however, L will be referred to as the differential operator and the boundary conditions will be referred to simply as the boundary conditions. Modification of the integral operator resulted in subtracting components of the kernel represented by the fundamental eigenfunction from the kernel itself. u1(xn)u1(§) X1 (4.3.3) R(xn,§) = k(xn,§) - u1(xn) is the fundamental eigenfunction and l, is its associated eigenvalue. Since the kernel and the eigen- function are characteristic of the differential operator 61 and its boundary conditions, equation (4.3.3) suggests that a similar subtraction in the context of the differential form may induce changes in the operator, the boundary conditions or both. Independent of the form of the modification, questions of existence and uniqueness must be addressed. If it is possible to show existence, uniqueness, and that the resulting operator is of class Z, then extension to the differential form would be direct and complete. Such a form, however, has not been derived. As an alternative to direct modification of the operator the following analysis con- siders a variational formulation that introduces an ortho- gonality condition into the differential equation. Placing the appropriate orthogonality relationship in the variational form results in an inhomogeneity that is introduced into the new differential equation. This in- homogeneity is not a direct modification of the operator; however, has the ability to exclude lower modes from the system. Consider a system that can be expressed as an energy integral in the form of Hamilton's Variational Principle. t (4.3.4) 61’ 1 (L + Oldt = o where 6 = variational operator L = Lagrangian O = additional energy terms 62 Suppose the Lagrangian may be written in an integral form as follows (4.3.5) L = I (T[y(xn.t)] - V[y(xn.t)])dxn D where T is the kinetic energy operator and V is the potential energy operator. Also suppose any additional energy terms may be written as integrals of the form. (4.3.6) Q = f Q[y(xn.t)]dxn D Omitting the arguments and substituting into (4.3.4) yields. t (4.3.8) 6 I 2 I (T-V+Q)dxndt = o tlD This is referred to as the variational energy form. If Green's formula or integration by parts is applied then the original partial differential equation will be formulated. Assuming this partial differential operator is of the form defined by (2.2.1) in section 2.2 then it is possible to separate time from space in the following way Mly(xn.t)] = o Lly(xn.t)l + pytt(xn,t) = 0 L[u(xn)I(t)] + p(xn)u(xn)ttt(t) = 0 Dropping arguments we find (4.3.4) L[u] = Apu 63 Since L is of class 1, the eigenfunctions {ul,u2 ...} are orthogonal with respect to weight p and span the solution space of (4.3.9). The orthogonality condition is written (4.3.10) = 6nm n, m = l, 2, 3 ... where 6nm is the Kroniker delta In the very special case of constant weighting Appendix VIII shows that the following orthogonality relationship will also hold. dum (4.3.11) (un"Td§ > = anm n, m = l, 2, 3 ... Consider for now, however, systems with functional weighting such that equation (4.3.10) is the most restrictive form of orthogonality. Writting the inner product in integral form (4.3.12) é punumdxn = 6nm In order to place (4.3.12) into the variational energy formulation it is necessary to include time dependence in the following way. (4.3.13) fbuu'r 2 D nnmdenT6 T b nmm n Certainly integration and variation over the interval tl to t2 must also be zero. t t 2 2 _ 6! f bnpunume dxn T 5f nm t1 D t 64 This integral is of a form that may be substituted directly into the variational energy equation (4.3.8). t2 t2 2 (4.3.14) 6/ f ("t-v-+Q+bnpunum'rm )dxdt = (Sf 6 t1 D tl anzzbndt Suppose the fundamental mode is known (un = ul) and an attempt is made to exclude 111 from the new system. If we simply let um represent any of the higher modes but 111 then (4.3.14) will yield t (4.3.15) (Sf 2 I (T-V+Q+bnpu1ut2)dxdt = 0 t1_ D This is in the form of Hamilton's Principle where the orthogonality has been included. If u = 111 equation (4.3.15) will not be an identity and hence mode 1 is excluded. If u is equal to any eigenfunction other than the fundamental then (4.3.15) is an identity, and the higher modes will be included. In general the first n modes may be removed by the following variational formulation. tl n (4.3.16) 6f f (T-V+Q+pyT 2 biu.)dxdt = 0 t1 D 1 1 1 In the case of constant weight the orthogonality may be expressed in terms of the first derivative and (4.3.16) has the alternative form 7 t2 n d (4.3.17) 6! f (T-V+Q+pyt_2 a; biui)dxdt = 0 t1 D 1=l 65 Note that the arbitrary amplitude bi may be absorbed into the eigenfunction ui. It must be stressed that (4.3.17) is valid only if p(xn) is a constant and not a function or space or time. In general p(xn) will be a function of space and (4.3.16) must be used. Through Green's formula or integration by parts, equation (4.3.16) will yield a new differential form whose eigenfunctions are identical to the original problem but exclude the fundamental. With this result modification of the differential form is complete. To gain some insight: several examples are considered in the following sections. §4.4 Exclusion of Lower Modes in the Taut String The taut string is governed by the one dimensional wave equation and is given by 2 (4.4.1) c yXX = ytt Vt_>_0 , 05x52 Consider for the sake of example fixed boundary conditions I O Y(Ort) (4.4.2) Vt_>_0 y(l,t) I 0 Note that the orthogonality relationship has unit weight and hence equation (4.3.17) may be used. t2 (4.4.3) 5! f (T-V+Q+pYT t1 D 1 l IIMU FL ui)dxdt = 0 D.- X 66 First consider exclusion of only the fundamental mode so (4.4.3) reduces to t1 (4.4.4) 6f f (T-V+Q+py'tui )dxdt = 0 t1 D x To derive (4.4.1) the energy T-V+Q may be written t (4.4.5) 51 2 f (DYt 6(yt) - Tyx 6(yx) t1 D - 5(ucpi T))dth = 0 x Where $1 has been substituted for ui . This assumes that x x ¢i has been iterated to a reasonably accurate shape. x To evaluate (4.4.5), define the following terms. Term (1) = f 2 I“ (oyt 6(yt))dxdt t1 0 t2 1 Term (ii) = f f (TyX 6(yx))dxdt t o 1 t2 1 Term (iii) = f f 6(y¢1 T)dxdt t1 0 x Consider one term at a time. .i) Term (1) 2 t2 3 f Dyt Gyt dt = f Pyt EE- (6y)dt t t 1 l t t =py 5Y|2'IZL(QY) 6ydt t t at t 1 t1 67 But we have no variation in y at t1 and t2 so the first term is zero and t2 t2 (4.4.6) I pyt éyt dt = -f pytt 6y dt t t l 1 ii) Term (ii) lg Ty ny dx = f£ Tyx g; (6y) dx 0 0 £ = Tyx 5y I - Ii 3:; (Tyx) 5y dx 0 o Impose natural boundary conditions such that l Tyx 6y I = O o and term (ii) becomes 2 l (4.4.7) (J; Tyx 6yx dx =2; Tyxx 6y dx iii) Term (iii) 2 2 f 6(y¢1 T)dx = f (cpl x o T6y+y16¢l +y¢i 6T)dx o x X X The function ¢i is specified, so the variation must be x zero. 68 2 = f (¢l 16y+y¢1 5T)dx O X X Expand the second term 2 R = f ¢l T5de+T5T f ¢1 u dx 0 x o x By orthogonality we require (for systems having con- stant weighting) Hence, term (iii) reduces to l 1 (4.4.9) I 6(y¢l T)dX = f $1 Tdydx o x o x Combine equations (4.4.4)-(4.4.9) to obtain 2 . _ _ _ (4.4.10) f f (TyXX Dytt ¢le)6ydxdt — 0 Variation of y is arbitrary so the integrand must vanish for all x and t in the domain for all time. (4.4.11) TyXX = pytt - ¢le = 0 Vtzo, vogxgz ‘ The original boundary conditions remain natural. (4.4.2) y(0,t) = y(2,t) = o vtgo 69 Following the procedure outlined in Chapter 1, time may be separated from space and the inertial term may be re- placed with a constant eo as follows. (4.4.12) C ¢2 = e + The associated boundary conditions are (4.4.13) ¢2‘°’(o) = ¢2(°)(2) = 0 Equation (4.4.12) may be integrated directly in order to obtain the zero iteration approximation of the second eigen- function. x2 (44 14) ¢‘°’(x)=—fof”¢ (F,)d€dn +e° +ex+e ° ° 2 2 l 2 1 2 c p x From the first boundary condition we have ¢2(°)(o) = 0 = c2. From the second boundary condition -1 eo£2 (4.4.15) C1 = 7 [7 £ +-—%— f C D n f $1 (€)d€dn] X Recall that we must also satisfy the orthogonality condition 2 (4.4.16) f ¢1 (x) ¢2 (x) dx = 0 o This condition will specify the magnitude of the constant e0. We now have a method of applying the inertial loading technique to the second mode. Once the second eigenfunction is determined we can formulate a third differential equation 70 that will yield the third approximate eigenfunction. The differential equation used to find the third approximate eigenfunction is given by (4.4.17) Ty xx = tht + (¢1 (x)+¢2 (x))r (t) X x with boundary conditions (4.4.18) y(0,t) = y(£,t) = 0 In general the result is (4.4.19) Ty ¢ 1 px Htfifl = pytt + T P XX Such that orthogonality is maintained. (4.4.20) <¢n, ¢m> = 6nm n = l, 2, 3 ... Where 5nm is the Kroniker delta. Recall that variation-iteration allows the approximate eigenfunction to be calculated to a high degree of accuracy. At any point in the sequence of fundamental mode problems. §4.5 Generation of Zero Iteration Eigenfunctions for the Taut String Determination of the nth eigenfunction and eigenvalue by the variational energy formulation is highly dependent on the accuracy of the previously calculated (n-l) eigen- functions. From the derivation given in the previous section it is seen that the nth mode will in general be less accurate than the (n-l) mode. This building up of errors motivates some numerical study. 71 The zero iteration approximate eigenfunction is the function that is generated when a non-biased constant inertial load is placed on the system as described in section 2.2. To investigate errors induced by the worst case, a sequence of zero iteration eigenfunctions are calculated based on knowing only rough zero order approxi— mations of the previous modes. The first five eigenfunctions are shown in Figures 4.1-4.5. The figures illustrate increased errors in nodal locations and weighting with each successive mode. The degree to which each mode is affected is illustrated in eigenvalue estimates given in Figures 4.5 and 4.6. It is seen from Figure 4.6 that the frequency ratio increases in near linear manner with mode number. This observation indicates that the procedure is relatively insensitive to the mode number if the previous modes are known exactly. To investigate this premise, suppose the first n modes have been determined to a high degree of accuracy and the (n+1) mode is determined based on the "exact" lower eigen- functions. Disregarding computational errors, the varia- tional derivation in section 4.3 appears to be insensitive to the value of n (mode number). If this is true then calculation of higher modes based on exact lower modes should yield approximate zero iteration eigenfunctions and eigenvalues having an error independent of n. Consider the exact eigenfunctions. 72 _ . knx (4.5.1) uk — bk51n (—§—) Hence the zero iteration (n+1) eigenfunction is defined by n d (4.5.2) ¢ = -2 bk 3; (uk) + bn+1 (n+1) _ xx -1 integrating twice (4.5.3) ¢(n+l) = kil bk f uk(€)d€ + bn+1x +e1x+e2 Substituting the exact eigenfunctions into (4.5.3) yields. n knx 2+kil bkCOS (T) 2 (4.5.4) = b x +e1x+e ¢(n+l) n+1 Since ¢ is a comparison function it must meet the (n+1) prescribed boundary constraints such that n k (4.5.5) e1 = -b - z bk (1+(-1) ) (4.5.6) in general n _ 2 Efl§., .. k - (4.5.7) ¢(n+l) - bn+1(x -x) + 1:1 bk(cos( 2,) (1+(l) )X 1) where the ratios bo/bk are obtained from the orthogonality constraints.. (4.5.8) <¢ Figures 4.6-4.10 illustrate an improved ability to match the exact zeros. The approximate eigenfunctions continue to include larger errors at the higher frequencies. Figures 4.5 and 4.6, however, show a marked improvement over approxi- mate eigenfunctions based on zero iteration lower modes. §4.6 The Uniform Circular Membrane Recall the differential form governing motion of the circular membrane (4.6.1) c2V2y = ytt VtZO, ve, V0_<_r§R y(1,e,t) - 0 V6, Vt_>_0 (4.6.2) yr(0,e,t) — 0 V6, Vtgo If the first n modes are to be removed from the system then we require (4.6.3) = 0 V m = l, 2, ... n where (4.6.3) is a weighted inner product (4.6.4) ( = f rfgdr d With this constraint the variational energy form yields the following new equation. n 2 2 _ 1 Y - ytt + p T 2 u r (4.6.5) c V 74 H\x monopmwo omNflHmEHoz . L h ' ’ "'- cofiuwumpH oumN coHusHom uomxm IIIIIIIIII .mmooz H6309 mumEonumm¢ so pmmmm cofluocSHQmmflm pcoomm coflumumpH oumN .H.s musmflm Axvs mnsuflamsm 75 .mmooz Hozoq ouweflxoummfl co Ummmm cofluocsmcmmflm CHH£B coflumumnH onom H\x mocmumwo pmNflHmEHoz - p b P GOHubmHmHH OHGN :oHHSHom uomxm IIIIIIIIII a) .m.¢ musmflm Axvs waspflamsm 76 .mmooz umBoq mmeonumm< co Ummmm coauocsmcmmflm nuusom coaumume oumm b h 4. H\x mocmumflo omuflamauoz L b cofluwumuH oumN coflusaom pomxm .m.v musmflm Axvs a mnsuflamaa 77 .mmooz um3oq mumEonummfl co pmmmm coauocsmcomflm cuwflm coflumumuH oumN .v.¢ musmflm a H\x mocmumfla omNHHmEHoz s r b h b d! .xvs mosuflamaa coaumnmuH cums soapsaom uomxm IIIIIIIIII 78 .cofiumsqm m>63 HmcoflmcoEHo moo MOM Hmnasz moo: .m> mocmovmum Umumaflumm mGSHm> COH¥MM®UH OHmwN I G modam> uomxm I o HmnEDZ moo: n w PPp-irPhpfihlnbp pr-p bP-P . I. WUjI 11 1 I 1 j‘T' 1111 .m.v musmflm 3 mm"... mocosqmum 79 .coflpmsqm m>63 HmcowmcmEflo mco How Hwnfisz moo: .m> oaumm wocmsqwum .o.v musmflm w-prrwpprpjppL-flblP mmpoz Hm3oq uomxm co commm I < mmpoz Hmsoq coaumnwuH OHmN co Ummmm I 0 “69852 woos 1: T m k. r d.“ wumEHumms Umxm u. 3 rwflummnwocmsmmum NJ 80 .mmpoz Hmzoq uomxm so Ummmm cowuocsmcmmflm pcoomm coaumuwuH oumu .h.v musmflm . H\x mocmumflo ommfiamfiuoz fl Axvs mosuflflmea coflumumuH oumN :oHusHom uomxm IIIIIIIIII 81 .mmpoz umzon uomxm co pmmmm :oauocsmcmmflm phase coflumumuH oumN .m.v musmflm . H\x mocmumwo omNHHmEHoz a b P b b “I Axvs 665335 coaumuwuH oumu coHuSHom uomxm IIIIIIIIII .mmpoz umsoq uowxm co pmmmm coflpocswcmmwm apnsom :ofipMHmpH onmm .m.w musmflh H\x mocwumfla cmNHHmEHoz i 82 p (u . . «I Axe: mcsuflamem coflumumoH Ohms coflusaom uomxm IIIIIIIIII 83 .mmpoz Hmsoq pomxm co Ummmm cofluocsmcwmflm Sawflm coflumumUH . H\x mocmumfla omNflHmEuoz oumN 1 a p r r p 6 . \\\ III \\\ III \ \ I s s I T . . x s \ o o n J s p o s a s a s a . p . a . v \ v \ 0 \ I. a u ’ ~ ’ s a \ v s a s O s ’ 5 fl \ a x a s a s f a \ a \ pr \ III\\ Il|\\ II\ coaumumUH oumN coaunaom pomxm IIIIIII .oa.¢ mucous a... Axvs monogamsa 84 Where up is the pth eigenfunction. The fundamental mode of (4.6.5), as approximated by a constant inertial load, will be the same as the (n+1) mode of the original system. Consider solutions symmetric in theta and remove the time dependence to obtain (466) u +lu=—u+——zur ' ' r 2 Using the constant inertial load introduced in Chapter 1 we find (0) n (4.6.7) %--g—r-(r3?%fl)=eo+—%— 2 ur c p p=l p First consider the second mode based on the approximate fundamental mode given in section 3.7. (0) NZ _3_ (4.6.8) g; (r T) =eo+e1 (%-%+16) th Integrating twice it is found that e r 6 4 2 (4.6.9) ¢2(°) = ° + e (EL—- U1 \1 CW” .b ox .b e2 = 0 (4.6.10) 85 hence O‘ 2 4 2 (o) _ £;__ 1 r _ £_ 3r _ 19 (4'6°11’ ¢2 “ eo( 4 4’ + el(_76 6 + 64 "76) The constants eo and e1 may be related by the following orthogonality condition. (0) (l)> <¢2 r (bl = 0 (4.6.12) 3 r ¢2(O)(r) ¢l(l)(r) dr = 0 Recall that (4.6.13) 41(1)(r) = a (r4 = 4:2 + 3) The inner product yields (4.6.14) e1 5 (8.3720951) e0 Therefore, the zero iteration second eigenfunction is given by (4.6.15) ¢2(°) = (0.0145349)r6 - (0.130814)r4 + (0.642442)r2 - 0.5261629 The eigenvalue is given by Wirtinger's inequality 2 2 (O) f u “’2 _ n “7’ O” C _ f r(u)2 dr D (4.6.16) ( hence 86 (4.6.17) (0) _ g (1)2 - 5.592 R The exact value is given by — E Figure 4.11 shows the approximate in this section. eigenfunction obtained 87 .mcmunEmz HmHDUMHU map wo cofluocsmcmmfim pcoomm coflumumuH oumN .Ha.v musmflm . m\n mocmumflo pmNHHmEMoz i Axes mnsuflamsa coapmumuH Cums uHSmom pomxm IIIIIIIIII CHAPTER 5 CONCLUSIONS Under the premise that static behavior is well under- stood, the present work shows that eigenfunctions may be obtained through an iteration of static deflections. The ability to iterate the static problem to the dynamic case was argued on physical grounds for a large class of opera- tors, however was proven only for a very limited group of operators. Physics suggests that the iterative procedure may be extended to higher order problems having more compli- cated spatial domains or operators. Since the iterative procedure converges to the funda- mental mode, a technique was developed to formulate a new problem such that the fundamental mode of the new problem is equivalent to the nth move of the original problem. If the new problem is based on approximate lower modes, then orthogonality must be enforced at each step in the iterative procedure. If orthogonality is not maintained then the iterated solution will gravitate to the lowest mode. In some cases, the approximate eigenfunction may be close 88 89 enough to the actual eigenfunction such that a finite number of iterations may be applied without enforcing orthogonality. Several simple examples were presented to illustrate the behavior of the procedure. Rapid convergence is apparent for eigenfunctions having distinct, separated eigenvalues. The zero iteration approximation appears to be insensitive to the mode number, indicating similar convergence for high frequency modes. In addition to quantitative results, the ability to visualize static deflections lends the procedure to qualita- tive analysis. Upon formulating a new problem for the nth mode, the nature of the new inertial load may be inspected in order to gain insight to vibration in the mode. The technique, as presented is complete for a small class of problems, and may be implemented on a computer in order to inspect various vibrational problems. The class of problems for which the procedure has been proven is restricted to those problems that reduce to Sturm-Liouville systems. Clearly this class must be enlarged in order for the procedure to compete with other methods. When expanding the procedure to more complicated Operators and domains several additional questions must be considered. 1. The effect of repeated or complex eigenvalues. 2. Existence of mixed derivatives within the operator. 90 Ability to characterize qualitative features of the fundamental mode to efficiently begin the procedure. Computational efficiency of the procedure compared to alternative techniques. Advantages when used in conjunction with a descreti- zation procedure. APPENDICES APPENDIX I RICCATI TRANSFORMATION APPENDIX I RICCATI TRANSFORMATION Let r(x) and p(x) e c2(a,b) and let u be a solution of d d _ (l) E)? (r(x) a u(X)) +p(x) u(X) - 0 such that u(x) # 0 on [a,b]. If we let u'(x) (2) V(X) = u(x) then v(x) will satisfy the Riccati equation (3) d—: (r(x) v(x)) +r(x) v2(x) + p(x) = 0 Conversely if v is any solution of (3) on [a,b] then there exists a solution of (1) = u(x) # 0 on [a,b]. 91 APPENDIX II SECOND ORDER TRANSFORMATIONS TO OBTAIN SELF-ADJOINT FORMS APPENDIX II SECOND ORDER TRANSFORMATIONS TO OBTAIN SELF-ADJOINT FORMS (l) a2(x)y" + al(x)y' + ao(x)y = 0 Consider the following transformations. A. Transform the independent Variable by _ x a1(s) (2(x) — exp; a2(s) ds then (1) becomes (2)* 3% (r(x) gxx) + p(X)y = 0 . r(X) > 0 B. Transform the Dependent Variable by _ x ds t - i E757 , r(x) > 0 0 then (2) becomes (3)* y" + Q(t)y = 0 C. A second Dependent Variable Transformation if we have (4) . y" + p(x)y' + q(x)y = 0 92 93 let y = uv and v = exp{ - % fx P(€)dg} then (4) becomes (5)* 11"+(q‘711'92‘%""“=° *Self—Adjoint forms. APPENDIX III INNER PRODUCT ROUTINE APPENDIX III INNER PRODUCT ROUTINE Consider Simpsons l/3 rule In -11. : f(x)dx — 3(fo+411+2f2+...+2fn_2+4fn_l+fn) n = 2%2 , n = even Program SUM = 0.0 A = 0 LIMITS B = 1 N = 2000 H = (B-A)/N COUNT = 0 DO 20 X = A, B, H IF (X.NE.(A.OR.B))GOTO l M=l 1 IF (COUNT.EQ.0) GO TO 5 M = 2.0 COUNT = 0 GO TO 10 5 COUNT = 1.0 M = 4.0 10 CONTINUE WEIGHT = [WT FUNCTION] 94 20 95 FX = [F(X)] GX = [9(X)] SUM = SUM + M*FX*GX*WEIGHT CONTINUE COUNT = H*SUM/3.0 WRITE "THE INTEGRAL IS =" COUNT END APPENDIX IV MONOTONICITY PRINCIPLE APPENDIX IV MONOTONICITY PRINCIPLE Let A* be an operator of class 2 and A be an operator of class 2 such that (1) D(A*) < D(A) and (2) i VueD(A') Then A is dominated by A* and we write A.§ A*. From Rayleigh's Principle we clearly have 41.5 11*. The following theorem shows that every eigenvalue of A is not greater than the corresponding eigenvalue of A*. Theorem If A* and A satisfy (1) and (2) then the eigenvalues of A* and A satisfy the inequalities A. 5 A.* (i = l, 2, 3 ...) The proof follows from the Minimum-Maximum Principle and is given by Weinstein and Stenger [W81]. 96 APPENDIX V LEIGHTON'S THEOREM (1962) APPENDIX V LEIGHTON'S THEOREM (1962) Let r, r1 6 c(a,b) such that r(x) > 0 on (a,b) and let p, p1 e c[a,b]. d dw _ 6; (r1 6;) + P1“ - 0 (l) w(a) = w(b) = 0 has a nontrival solution w such that 2 . b d 2 (2) 3(w) = g ((r-r1)(a§) + (pl-pr )dx 5 0 then every solution a of d du _ (3) fi(rfi)+pu—O TE must have a zero on (a,b) unless u u(x) \ w(x) 0 /' V\\ \b u(x) m 97 98 Two fundamental results are 1. If the mass is decreased we have increased frequencies 2. If the spring is increased we have increase frequencies Proof: Multiply (l) by w and integrate by parts 0 = fb w((riw')' + plw]dx = - fb r w” dx + fb p wzdx 1 l a a a (4) 0 = fb plwzdx - fb rlw'zdx a a Suppose the contrary conclusion and let u(x) > 0 on (a,b) then: wen so Wirtinger's inequality states (5) Ib rw'zdx > Ib pwzdx a a Combine (4) with (5) to obtain the opposite of (2). This proves the theorem. The following corollarys come directly from Leighton's result. Corollary (Stunm 1836) Let r(x) = r1(x) and 0 < p1(x) _<_ p(x) then u has a zero in (a,b) Corollary (Picone 1909) Let r(x) > 0 and r1(x) 3 r(x) p(x) 3 p1(x) then u has a zero in (a,b) APPENDIX VI OPERATORS OF CLASS 1 APPENDIX VI OPERATORS OF CLASS 2 Let H be a real or complex Hilbert space with inner product and norm Ilfllz = Then L is of class 1 if i) L is a self-adjoint linear operator on subspace M of H. ii) L is bounded below. iii) The lower spectrum of L consists of isolated eigen- values 99 APPENDIX VII A FINITE DIMENSIONAL EXAMPLE APPENDIX VII A FINITE DIMENSIONAL EXAMPLE Placement of Orthogonality Condition Into The Differential Operator Consider a simple 2D.O.F. system shown below. ——{k=1>———— m=l ~——€<=1}—--- I m=l -—Q=1}—-3 Q C O 0 )dx]. L’ X2 (1) [1032+ 2-11§=g L o 1 j 1 2_f The modal matrix is determined to be 1 l l (2) 2:]; 2t: 1 -1 let x = u y such that (v 12+42=9 A= ° — 3 100 101 If we simply eliminate the first mode from the uncoupled set of equations and perform the inverse transformation, the resulting system may be non-physical. For example consider the following equation 00 ['00] (4) 17+ (g=9 0 l " ‘ ’ let (5) 2t§=x such that 0 0 u 0 0 (6) x + x = 0 l —1 ~ 3 -3 ~ " or simply (7) §-3z=o where (8) S = x1 - x2 and is incorporated to remove the rigid body mode. This technique will always reduce the order of the system by one and may not be applicable to infinite dimensional sys- tems. If we maintain the order of the system and write equation (6) as (9) M*x +-§*x = 0 102 It is seen that the coefficient functions in the operator have been changed, but the general form of the operator remains in tact. Recall that the original problem was symmetric (self-adjoint); the new problem defined by (9), however is not self-adjoint. This indicates that modification of the differential opera— tor to exclude previous modes may disrupt our ability to solve the problem; in particular the modification may yield a non-self—adjoint operator. APPENDIX VIII ORTHOGONALITY OF EIGENFUNCTIONS APPENDIX VIII ORTHOGONALITY OF EIGENFUNCTIONS Suppose we have (1) L[um] AmpuIn (2) L[un] lnpun where un and um are eigenfunctions. For L e 2 the ortho- gonality condition is derived as follows. (3) (un'Lum) = (unplmPum) (4) (Lun'Wn) = (inpunmm) Subtract (3) from (4) and utilize the fact that L is self- adjoint. 0 = -(un,Ampum) + (Anpun,um) (5) 0 = (An-Am*) where (6) = Wuw‘h’ For L Of class 2 An ¢ Am* so we have 103 104 (7) = 0 V n# m In a similar way, consider the first derivative of (2). I _ I l (8) L[unj - Anpun + Inpun and I _ I I. (9) (Lun 1%) — (Anpun 1‘55“) + (Anpun I%) combine with equation (3) (lo) (An-An*) + ()2p un.um) = 0 Hence, if p is a constant then I. _ ' 0 d un (11) dx ' gm) < ll 0 If p, however is not a constant then (11) will not in gen- eral be valid. APPENDIX IX BOUNDARY EXPANSION SOLUTION OF THE MEMBRANE OPERATOR APPENDIX IX BOUNDARY EXPANSION SOLUTION OF THE MEMBRANE OPERATOR Suppose we have a non-simply connected domain governed by a simple membrane operator as shown below. 0) XV” . 34/ Motion of the membrane is transverse and the restoring force Fi ure I is considered tension alone. With this in mind the equation of motion is (l) utt = c V u a. c = sound speed = Jak; b. u = transerve deflection c. V2 = Laplacian operator 105 106 In polar coordinates the boundary s1 is a coordinate line and is easily defined as the circle of constant radius r. If the same coordinates are used to describe the second boundary 82 then it is possible to define a function that is satisfied along 52' With this in mind the boundary con- ditions for clamped edges are: (2) a. u(sl) = 0 u(r,0) = 0 b. u(sz) = 0 11(g(6),6) = 0 In order to impose the spatial boundary conditions equation (1) is separated in time and space. Assuming two spatial coordinates n and 6 we find. u(nlgrt) = 1P(n1€)T(t) where we select positive real w2 in order to match the vibratory nature of the solution. (3) T" (t) + (02T(t) = o 2 w 2 (4) v (101.6) + ('5’ 401.6) = o For w2 > 0 solutions to equation (3) have the form. (5) T(t) = Asin(wt +a) 107 The general solution to (1) written as an eigenfunction expansion would have the form. u(n,€,t) = n "M8 0 Anwn(n,E)sin(wnt + an) The problem is to find the eigenfunctions wn(n,g). The mode shapes are defined by solutions to equation (4) that fit the boundary conditions defined by (2). If we let the coordinates (n,§) represent standard rectangular polar, eliptical, or hyperbolic coordinates, and separate spatial variables, we find that the separated equations cannot satisfy the boundary conditions directly. Hence the eign- functions wn(n,g) are not separable in the trial coordinate systems and we have no proof that they are separable in any coordinate system. The difficulty in finding the eigen- functions wn(n,§) leads us to an approximate solution. Suppose we separate variables in polar coordinates such that W(n)€) = W(r.9). 2 w 2 v (Mme) + (E) 46.6) = o where N N 3 l a _+___.. 3r r2 <1 ll (6) 3:!” + Hue 0 N Assume that (7) ¢(r,9) R(r)¢(6) 108 Substitute into the spatial equation so 2 2 (8) (53—123) +%%§)¢ +_12?.d_§+ (%)2R = 0 dr r d6 Divide by 4R so 2 62R 1 dR 624 r (7+2? "‘7 dr = d0 # r2(9)2 R 0 c 2 2 2 i II I _ II r 2_ + r (R + r R ) = .2— = m2 = const. c2 R 0 Since w(r,6) must be single valued we must have m = interger and m2 > 0 so the separated equations become. (9) 4"(e) + m24(e) = o (m = 1, 2, 3 ...) 10 R"()+lR' )+ “’2 m)R -0 ( ) r r (r (:2- --r—2 (r) - For each value of m (m = l, 2, 3 ...) equation (10) is Bessel's equation of order m. One method of solving (10) would be to use a power series. For each value of m.we have a solution to the spatial equation of the form. (11) wm(r,0) = A1m Jm(Br)sin me + A Im(8r)sin m6 2m + A.3m Jm(Br)cosrne+-A4m'Ym(8r)cosrn0 Defining the positive x axis as the origin of the polar theta 6 coordinate we can write the symmetric and antisymmet- ric portions of (11) as follows. 109 Symmetric (12) wm(r,0) = [A1m.Jm(Br) + A2m>Im(Br)]sinm6 Antisymmetric (13) wm(r,0) = [A3m Jm(Br) + A4m ¥m(Br)]cosm6 The parameter 8 is defined as 2 2 w (14) B = — C2 Consider the symmetric case. §ymmetric Vibrations Consider the set of all functions such that (15) w = {wm(r,6), m = 1, 2, 3 ..., OfirfiR, 050521)} Suppose this infinite set forms a complete basis in the spatial domain such that we can represent an arbitrary, square integrable, function f(r,e) as an infinite sum of functions in the set W.* We write the series (16) f(r,0) = 2 Bm wm(r,0) m 1 We can drop the constant Hm since wm(r,6) already contains an arbitrary multipucitive constant. *Y is complete over the spatial domain for all f(r,6) 8 L2, = 6i°l v(rle)€D} 3 Expanding each term in (23) as an infinite series in v we find (25) 0 = A 8:48 :tfl8 1m an Pn(e) 112 The coefficients in equation (25) are given by the inner product (26) Cmn = qm(9) = [Jm(Bg(9)) + dm¥m(Bg(0))]sinm0 Since the functions un(6) are linearly independent each coefficient equation of un(6) must vanish independently. Considering this we have the matrix equation. C66 C10 "° 10 C01 C11 ... All (27) C02 .. . = O I. .J _ ' .. g A = 0 The characteristic values of c are found directly from (28) DET [9| = 0 Since the system is infinite dimensional equation (27) cannot be evluated exactly unless it is diagonal or simply diagonable. Equation (27), however, will be diagonal only if the set v consists of the eigenfunctions of the problem. Since the eigenfunctions are not known when selecting v, the matrix system will be coupled. There are basically two 113 techniques of solving the characteristic equation in the form of (27). The first would be to diagonalize the system by successive row and columns operations. If a recursive relationship, however, does not emerge it would be necessary to truncate the operation at some point. The second method of finding the characteristic values would be to truncate the 9 matrix and evaluate the determinant directly. Regard- less of the solution technique equation (27) in its infinite dimensional form defines the exact eigenvalues. Suppose we choose to truncate (27) and solve directly. Clearly the convergence is dependent on the set v. If v consists of the eigenfunctions then the truncated set of equations will yield the exact eigenvalues. Other choices for the set v will force the dimension of the truncation to increase in order to maintain the accuracy of the estimated eigenvalues. It is also interesting to note that if we modify the definition of v and select the functions qm(6) as elements of v then the double series expansion will yield a least squares solution. Selections other than v = {qm(0)} we find a least squares solution along non-orthogonal vec- tors.* Due to the complicated nature of qm(6), and lack of orthogonality, it is more convenient to allow v to be one of the standard infinite dimensional bases. The spatial variable f(r,6) must be periodic in 0 to be single valued, and in the symmetric mode f(r,6) is an even valued function. *See "finite dimensional analog to boundary expansion technique." 114 This suggestes the use of cos(ne) as the basis set v. The coefficients for the cosine basis are given by -2. (29) cmn — fl This gives us a complete straightforward technique of finding the eigenvalues. We now turn to determining the associated eigenfunctions. If we truncate (27) to a k x k finite dimensional form it is possible to find an eigenvector associated with each unique eigenvalue. Use of the adjoint matrix gives a simple method of finding the eigenvectors. (30) ep = colIADJ 2(Bp)] T [elp e2p ... ekp] The vector ep defines the mode shape associated with eigen- value and frequency Bp' Transforming this finite dimen- sional vector into a mode shape using equation (21) k (31) Ep(r,6) = mic emp(qm(8pr) + amYm(Bpr)]sinm0 The functions Ep(r,0) are approximations of the exact eigen- functions wo(r,e). Now that the eigenvalues and eigenfunctions have been approximated it is possible to solve an initial value pro— blem in the symmetric domain. A complete initial value problem.must also include the antisymmetric modes. APPENDIX X FINITE DIMENSIONAL ANALOG TO THE BOUNDARY EXPANSION TECHNIQUE APPENDIX X FINITE DIMENSIONAL ANALOG TO THE BOUNDARY EXPANSION TECHNIQUE Consider the overdetermined system of equations (1) ‘Az = b As an example let £5-‘-2x3=[‘31°1= 1 1 X 21x2 = y ~lx3 — U‘ I Define an orthonormal basis in R3 as (2) B = {Blr Bzr B3} where Bs are column vectors. We can write the columns of‘A in terms of the basis B, and the vector b in terms of the basis B. 115 116 91 = B1 + (01'82>82 + B3 (3) 92 = Bl + 82 + <02.B3>B3 y 6l + 82 + 63 Replace the matrix A and vector b in equation (1) with the equivalent representation from equation (3). Note the linear independence of the B vectors and write the system of coefficient equations in matrix form as follows. r' .Lf01183> (c2'83i_ Lfb'83>_ To approximate a vector 5 that will solve this system of equations in R2 we can truncate the system of equations to two equations in two variables. We note that the resulting solution will not satisfy equation (4) exactly but will satisfy (4) in terms of projections on the base vectors. Write the truncated set as I- . . F 1 (C1181) (C2181) [X (b181> (5) (01:82) (CZ’BZIJ Y _sb182>‘ L- 117 If we select c1=Bl and c2=82 then equation (5) yields the least error solution in teams of least squares. This least squares solution is equivalent to projecting b into the column space of A and solving for the vector 30' This solution is written -1 50 = (ATA) ATb where we note ATA = E and ATb = f It is not necessary, however, to select the columns of‘A as the basis vectors to be used in the expansion. We may select any basis as defined by set B. If we select vectors 8 other than the columns of A, the projections will be "skewed." p1 P1 9 p 0 th «1 t: "Skewed" r ogona < > < > Pro'ection Projection p’B' p,B, 3 If the projection is skewed the vector p1 will not be minimized and a least square's solution will not be obtained. Equations (4) and (5) can be expaned to the infinite dimen- sional space using a well defined inner product over the given domain. 118 This analogy indicates that expanding the approximate eigen— functions in terms of the same eigenfunctions would yield the least error solutions; hence, the fourier expansion used by Nagaya [N1, N2] may be improved by expanding in terms of bessel functions rather than sinusoids. REFERENCES REFERENCES Atkinson, F. V. A1. Discrete and Continuous Boundary Problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York (1964). Boyce, W. E. and Diprima, R. C. BDl. Introduction to Differential Equations, John Wiley & Sons, Inc., New York (1970). Courant, R. and Hilbert D. CH1. Methods of Mathematical Physics, Vol. 1, Interscience Publishers, Inc., New York (1937). CH2. Methods of Mathematical Phycis, Vol. 2, Inter- science Publishers, Inc., New York (1962). Mirovitch, L. Ml. 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