ABSTRACT FREE PERIODIC VIBRATIONS OF CONTINUOUS SYSTEMS GOVERNED BY NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS by Paul Thomas Blotter Approximate expressions are obtained for the frequency- amplitude relations and for the nonlinear mode shapes for a general class of continuous systems governed by nonlinear partial differential equations. The formulation applies to problems in one space vari- able and one time variable, in which nonlinearities in the displace- ment and its spatial derivatives are involved. Some typical systems in this general class include strings, circular membranes, beams and circular plates on nonlinear elastic foundations or with immov- able boundary supports vibrating at large amplitudes, as well as elastic media with nonlinear constitutive equations. Two different techniques are developed and used. The first involves a modified perturbation approach. The second approach involves a linearization using ultraspherical polynomials. The general expressions obtained are applied to several dyn- amic systems. Numerical results are cataloged in the form of . graphs and tables and Compared with those obtained by other authors using different methods. The results include those for several non- linear continuous systems whose solutions are not available in the literature. FREE PERIODIC VIBRATIONS OF CONTINUOUS SYSTEMS GOVERNED BY NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS By Paul Thoma s Blotter A THESIS Submitted to Michigan State University in partial fulfillment of the requirements DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1968 «fix / . \\‘ ACKNOWLEDGMENTS To all the individuals who have helped and encouraged me during the research and writing of this dissertation, I offer my sincere thanks. Foremost appreciation is extended to my com- mittee chairman, Dr. David H. Y. Yen, who supplied the original motivation and continued to give generously of his time and valu- able counsel throughout the study. The guidance of other com- mittee members, Dr. R.W. Little, Dr. W.A. Bradley, Dr. G.E. Mass and Dr. N.J. Hills is also greatly appreciated. Grateful thanks are extended to the National Science Foundation and the Department of Metallurgy, Mechanics and Materials Science of Michigan State University for providing funds to finance my graduate program. Also, the sacrifice by my wife and three boys is acknowledged. ii TABLE OF CON TENTS ACKNOWLEDGMENTS LIST OF TABLES LIST OF FIGURES I. INTRODUCTION . II. » PERTURBATION METHOD 2 . 1 Introduction and Historical Background 2 .2 General Perturbation Method for a Class of Nonlinear Partial Differential Equations 2 . 3 Mode Configuration and Amplitude for Nonlinear Dynamic Systems III. APPLICATION OF PERTURBATION RESULTS 3. 1 Restoring Force Nonlinear in Displacement 3.1.1 Uniform String 3. 1.2 Prismatic Beams . 3.1.3 Beams with Variable Geometry 3.1.4 Thin Circular Plates 3. 2 Continua Having Immovable Supports and Large Deflections 3.2.1 Elastic Bars 3.2.2 Elastic Plates 3 . 2 . 3 Membranes . 3. 3 Nonlinear Material Properties IV. NUMERICAL RESULTS . 4. 1 Solutions for Equations of Motion with Restoring Forces Nonlinear in Displacement . 4.1.1 String . 4. 1.2 Prismatic Beams . . 4.1.3 Beams with Variable Cross Section iii Page ii ix 18 21 21 24 27 29 33 34 34 38 42 46 49 50 50 55 66 4.2 Numerical Solutions for Continua Having Immovable Supports and Large Amplitudes of Vibration . . . . . 4.2.1 Elastic Beams with Immovable Supports . . 4.2.2 Vibrations of Circular Plates at Large Amplitudes . 4. 2. 3 Vibration of Membranes at Large Amplitudes 4. 3 Numerical Solutions for Continuous Media Having Nonlinear Constitutive Equations -. V. A GENERAL SOLUTION USING ULTRASPHERICAL POLYNOMIALS 5.1 Introduction . . . . . . 5.2 Linearization of a Class of Nonlinear Equations of Motion and a Corresponding Frequency- Amplitude Relationship . . 5. 3 The Application of Ultraspherical Polynomials to Approximate Frequency -Amplitude Relationships . . . . . 5.3.1 The Restoring Force Nonlinear in Lateral Displacement . 5. 3. 2 Elastic Beams with Immovable Supports 5. 3. 3 Circular Plates Vibrating with Large Amplitudes . . . 5. 3.4 Beams Having Nonlinear Constitutive Equations . 5.4 Alternate Methods to Approximate Frequency- Amplitude Relations by Using Ultraspherical Polynomials 5.4.1 Nonlinear Vibrating String Reduced to the Mathieu Equation . . 5.4.2 Integral Transform Methods . IV. SUMMARY AND CONCLUSIONS LIST OF REFERENCES . . . APPENDIX A iv Page 81 81 89 95 100 109 109 111 118 118 120 122 123 125 125 128 134 139 143 LIS T OF TABLES Table Page 4. 1. 1-1 Frequency Numerical Values for a Uniform String Resting on a Cubic Nonlinear Elastic Foundation . . . . . . . . . . . . . . . 52 4. 1. 1-2 Nonlinear Amplitude Parameters for a Uniform String Resting on a Cubic Nonlinear Elastic Foundation 52 4. 1.2 -1 Nonlinear Amplitude Parameters for a Uniform Beam Simply Supported and Resting on a Cubic Nonlinear Elastic Foundation . . . . . . . 59 4. 1.2 -2 Nonlinear Amplitude Parameters for a Uniform Beam Clamped-Supported and Resting on a Cubic Nonlinear Elastic Foundation . . . . . 59 4. 1.2 —3 Nonlinear Amplitude Parameters for a Uniform Beam Clamped-Clamped and Resting on a Cubic Nonlinear Elastic Foundation . . . . . . . . 6O 4. 1.2 -4 Nonlinear Amplitude Parameters for a Uniform Cantilever Beam Resting on a Cubic Nonlinear Elastic Foundation . . . . . . . . . . . . . 6O 4. 1. 2 -5 Frequency Numerical Values for a Uniform Beam Simply Supported and Resting on a Cubic Nonlinear Elastic Foundation . . . . . . . . 61 4. 1.2 -6 Frequency Numerical Values for a Uniform Beam Clamped-Supported and Resting on a Cubic Nonlinear Elastic Foundation . . . . . 61 4. 1. 2 —7 Frequency Numerical Values for a Uniform Beam Clamped-Clamped and Resting on a Cubic Nonlinear Elastic Foundation . . . . . 62 V Table Page 4. 1.2 -8 Frequency Numerical Values for a Uniform Beam Cantilevered and Resting on a Cubic Nonlinear Elastic Foundation . . . . . . . . 62 4. 1.3-1 Nonlinear Amplitude Parameters for a Variable Section Cantilever Beam with a Nonlinear Displacement Dependent Forcing Function as Described in Example 1 . . . . . . . . . . . 70 4. 1. 3-2 Nonlinear Amplitude Parameters for a Variable Section Cantilever Beam with a Nonlinear Displacement Dependent Forcing Function as Described in Example 2 . . . . . . . . . . . 7O 4. 1. 3-3 Frequency Numerical Values for a Variable Section Cantilever Beam with a Nonlinear Displacement Dependent Forcing Function as Described in Example 1 . . . . . . . . . . . 71 4. 1. 3-4 Frequency Numerical Values for a Variable Section Cantilever Beam with a Nonlinear Displacement Dependent Forcing Function as Described in ExampleZ . . . . . . . . . . . 71 4. 1.3-5 Frequency Numerical Values for a Variable Section Cantilever Beam with a Nonlinear Displacement Dependent Forcing Function as Described in Example 3 . . . . . . . . . . . 72 4. 1. 3-6 Nonlinear Amplitude Parameters for a Variable Section Cantilever Beam with a Nonlinear Displacement Dependent Forcing Function as Described in Example 3 . . . . . . . . . . . 72 4.2 . 1-1 Nonlinear Amplitude Parameters for a Uniform Beam Clamped-Hinged with Immovable Supports 83 4.2. 1-2 Frequency Numerical Values for a Uniform Beam with Simply Supported Immovable Supports................. 84 4.2. 1-3 Frequency Numerical Values for a Uniform Beam with Clamped-Supported Immovable Supports................. 84 Vi Table Page 4.2. 1-4 Frequency Numerical Values for a Uniform Beam with Clamped- Clamped Immovable Supports . . . . . . . . . . . . . . . 85 4.2. 1-5 Nonlinear Amplitude Parameters for a Uniform Beam Clamped-Clamped with Immovable Supports................ 85 4.2.2 -1 Nonlinear Amplitude Parameters for a Clamped Circular Plate Vibrating at Large Amplitudes 91 4.2.2 -2 Nonlinear Amplitude Parameters of a Simply Supported Circular Plate Vibrating at Large Amplitudes . . . . . . . . .. . . . . . . 92 4.2 .2 -3 Frequency Numerical Values for a Clamped Circular Plate Vibrating at Large Amplitudes 92 4.2 .2 -4 Frequency Numerical Values for a Simply Supported Plate Vibrating at Large Amplitudes 92 4.2. 3-1 Frequency'Numerical Values for a Circular Membrane Vibrating at Large Amplitudes . . 97 4.2. 3-2 Nonlinear Amplitude Parameters for a. Circular Membrane Vibrating at Large Amplitudes . . 97 4. 3-1 Nonlinear Amplitude Parameters for a Uniform Beam Simply Supported and Having the Nonlinear Constitutive Equation as N = E (E + hta). . . 102 4‘. 3-2 Nonlinear Amplitude Parameters for a Uniform Beam Clamped-Supported and Having the Nonlinear Constitutive Equation as N = E(E+h€3) 102 .4, 3-3 Frequency Numerical Values for a Uniform Beam Simply Supported and Having the Nonlinear Constitutive Equation as N: E (e + hca) 103 4. 3-4 Frequency Numerical Values for a Uniform Beam Clamped-Supported and Having the Nonlinear Constitutive Equation as N= E (6+h63 ) 103 vii Table Page 4. 3—5 Frequency Numerical Values for a Uniform Beam Clamped-Clamped and Having the Nonlinear Constitutive Equation as N = E(€+ hea) 104 4. 3-6 Nonlinear Amplitude Parameters for a Uniform Beam Clamped-Clamped and Having the Nonlinear Constitutive Equation as N=E(€+h€3).............. 104 viii Figure 4.1.1-1 LIST OF FIGURES Page Normalized Mode Shape of a Uniform String Resting on a Duffing-Type Nonlinear Foundation 53 Frequency-Amplitude Response for a Uniform String Resting on a Duffing-Type Nonlinear Foundation . . . . . . . . . . . . . . . . 54 Normalized Mode Configurations for Beams on Nonlinear Elastic Foundations . . . . . . . 63 Frequency -Amplitude Curves for Various Beams Resting on Duffing- Type Nonlinear Elastic Foundations . . . . . . . . . . . 64 Nonlinear Frequency -Amplitude Responses for Various Beams Resting on Buffing-Type Nonlinear Elastic Foundations. Results from using Approximate Linear Mode Shapes are Compared with Results from Linear Eigenfunctions 65 Normalized Nonlinear Modes for Cantilever Beams with Variable Cross—Sections Resting on Nonlinear Elastic Foundations. Time = 1. 617‘ . 73 Frequency-Amplitude Response Curves for a Cantilever Beam with Constant Cross-Section . 74 Frequency -Amplitude Response Curves for a Cantilever Beam with Variable Cross-Section as Described in Example 1 . . . . . . . . . 75 Frequency -Amp1itude Response Curves for a Cantilever Beam with Variable Cross-Section as Described in Example 2 . . . . . . . . . 76 ix Figure Page 4. 1. 3—5 Frequency -Amplitude Response Curves for a Cantilever Beam with Variable Cross-Section as Described in Example 3 . . . . . . . . 77 4. 1. 3 -6 First Order Amplitude -Frequency Curves for Beams with Exponential Varying Cross-Sections Resting on Nonlinear Elastic Foundations . . 80 4.2. 1-1 Linear and Nonlinear Mode Shapes for a Uniform Beam with Clamped-Clamped Immovable Supports . Amplitude Corresponds to Displacement . . . 86 4.2. 1-2 Normalized Mode Shape for a Uniform Beam Clamped-Supported and Vibrating at Large Amplitudes................ 87 4.2. 1-3 Frequency -Amp1itude Curves for Beams Vibrating with Large Amplitudes and Having Various Boundary Conditions . . . . . . . . . . . . 88 4.2.2 -1 Normalized Mode Configurations for Various Plates with Large Amplitudes of Vibration. Time=0.4"7 93 4.2.2 -2 Ratio of Nonlinear to Linear Period vs. Ratio of Amplitude to Thickness for Circular Plates with various Boundary Conditions . . . . . . 94 4.2. 3-1 Normalized Mode Configurations for a Membrane Vibrating with Large Amplitudes . . . . . . 98 4.2. 3-2 Ratio of the Nonlinear to Linear Period vs. Nondimensional Displacement for Vibration of a Circular Membrane with Large Amplitudes . . 99 4. 3-1 Normalized Mode Shape for a Uniform Beam Simply Supported and Having a Nonlinear Constitutive Equation of N = E(€ + hea) 105 4. 3—2 Normalized Mode Shape for a Uniform Beam Clamped-Supported and Having a Nonlinear Constitutive Equation N = E (c + 11053) 106 X Figure 4.3—3 4.3-4 Page Normalized Mode Shape for a Uniform Beam Clamped-Clamped and Having a Nonlinear Constitutive Equation N = E (c + 1163). . . 107 Frequency -Amplitude Curves for Uniform Beams Having a Nonlinear Constitutive Equation of the Form N = E(('+ he 3) 108 xi I. INTRODUCTION The primary objective of this research is to develop approxi- mate expressions for the frequency-amplitude relations and for the nonlinear mode shapes for a general class of nonlinear continuous systems, the free periodic motions of which are governed by nonlinear partial differential equations. The formulation is sufficiently general and applies to a wide class of nonlinear elastic continua problems in one space variable and one time variable. The governing equation of motion is assumed to contain a restoring function that is nonlinear in the transverse displacement and its spatial derivatives. Some typical systems in this general class, to which the analytical results are eventually applied, include continuous structures such as strings, membranes, beams, and plates on nonlinear elastic foundations, or withimmovable boundary supports vibrating at large amplitudes, as well as elastic media with nonlinear constitutive relations. Essentially two different techniques are developed. The first is a modified perturbation approach and the second involves a linearization using ultraspherical polynomials. In Chapter H the modified perturbation method is presented. The historical development of the perturbation technique as applied to nonlinear dynamic systems and other more recent contributions are briefly discussed. The fundamental differences of the method developed in this paper are compared with those approaches used by other authors. Since a more detailed introduction to the analytical procedure forms part of Chapter H, it is only emphasized at this point that the final integral expressions for frequency and displacement are of a general nature and require only that the type nonlinearity be speci- fied and that the eigenfunctions and eigenvalues for the associated linear problem be known. Both first and second order approximations to frequency-amplitude response and transverse displacement are found. The linear case is taken as the zeroth order. In Chapter III these general expressions are applied to partic- ular continuous systems. Solutions are found for uniform strings, prismatic and variable cross-section beams, and circular plates resting on nonlinear elastic foundations. As examples of dynamic systems having a restoring function that is nonlinear in the derivatives of the transverse displacement, solutions for vibrating beams with immovable or springed end supports, circular plates and membranes vibrating at large amplitudes, and beams made of materials with nonlinear constitutive equations are also considered. The applications demonstrate the relative simplicity of the perturbation method devel- oped here as compared with other existing perturbation approaches. The particular results obtained inChapter III are further specialized in Chapter IV by specifying various boundary conditions for the above mentioned systems, whichthenlead to the linear eigen- functions and eigenvalues. Numerical results are then presented. These numerical approximations are compared with those obtained by other authors using different methods. The results also include those for several nonlinear systems whose solutions are not available in the literature. In Chapter V a method independent of the perturbation theory is presented whereby a class of nonlinear partial differential equa- tions are approximated by equivalent linear partial differential equations with variable coefficients. It is then necessary to solve the linear equations. The linearization is achieved by approximating the nonlinear restoring force over the amplitude span by the linear term of a set of ultraspherical polynomials. This method is an extension of a similar one previously used for problems of a single degree of freedom. General expressions obtained are again applied to the dynamic systems considered in Chapter 111 and the results are found to be in good agreement with those obtained by the pert- urbation methods. A brief summary of results as well as conclusions are con- tained in Chapter V1. II. PERTURBATION METHOD 2. 1. Introduction and Historical Background A classical approach for treating nonlinear partial differential equations is the method of perturbation. The method is reliable in finding an approximation to some unknown solution in the neighborhood of another solution which either is known or can be found with relative ease. The method is generally very cumbersome when-applied to nonlinear partial differential equations. Several variants of the perturbation method have recently appeared in the literature dealing with the nonlinear vibrations of continuous systems. Stoker [ll-studied the problem of a tightly stretched elastic vibrating string with fixed end points, embedded in nonlinear restoring springs distributed continuously along its length. .He developed the nonlinear infinite degree of freedom problem as an analogue of the one treated in Poincare's theory [41] with finite degree of freedom and governed by nonlinear ordinary differential equations. As a specific application, a procedure customarily followed for treating Duffing's equation was used byStoker to establish the first order frequency-amplitude relationship in the case of acontinuous string on a Buffing-type nonlinear elastic foundation. The partial differential equation of motion was modified slightly by adding frequency terms to both sides of the equation, whereby. in effect the homogeneous part was in resonance with the external periodic forcing function. This O particular -maneuver, however, was avoided in his book [2] published sometime later, where this so-called degenerate case-was solved by assuming a linear mode shape as a first approximation and following the Lindstedt [2] perturbation scheme, terms contributing to aperiodic motion were made to disappear. (It is interesting to note that higher order free periodic vibrations described by Stoker's equation of motion with the forcing function evanescent exist only if the coefficient on the cubic nonlinearity is identically zero. ) Han [3] made a somewhat different modification of the classi- cal Lindstedt method to study a simply supported beam on a nonlinear elastic foundation vibrating at amplitudes compatible with the assump- tions of the small displacement theory. First order terms were obtained through the usual concept of balancing coefficients of equal powers of a perturbation parameter. To obtain second and higher order terms would require additional series expansions involving the perturbation parameter. The approach would become unwieldy when applied to more complicated systems. Carrier [5] applied the perturbation method to study a vibrat- ing string having fixed ends and at displacements sufficiently large to induce a variable tension. Chu and Herrmann [6] solved the nonlinear coupled equations of motion of a vibrating rectangular elastic plate with hinged immovable edges. ,Eringen [7], along withV'Chobotovand Binder [8], developed an analogous procedure in order to solve the coupled equations describing the vibrationof membranes'at large amplitudes. .. . Keller and Ting [9] presented another "perturbationapproach to nonlinear problems. Rather than substituting power expansions of some small parameter directly and equating coefficients of like powers, they used Taylor's series and repeated differentiations to establish a system of inhomogeneous linear equations. Orthogonality conditions of the solution of the homogeneous equation and the inhomogeneous parts of equations that are necessary for the existence of solutions of the system of equations then yield higher order approximations for frequency-amplitude relations and mode configurations. , Evensen [10] has recently applied the approach developed by Keller and Ting to obtain first order frequency-amplitude relations for uniform beams with clamped-clamped and clamped-supported immovable boundary conditions. The perturbation scheme considered in this section is a generalization of an approach initially developed by McQueary and Clark [4], who determined first and second order approximations to the nonlinear frequency and mode shape of a continuous string and first order approximations to a membrane, both supported by nonlinear elastic foundations. The approach was later followed by Mack and McQueary [11] to obtain second order results for a membrane on a Duffing-type nonlinear elastic foundation. The set of recursive inho- mogenous linear equations, found through the usual power expansion technique or by the method of Keller and Ting, is solved by series expansions in the product space of linear spatial eigenfunctions and trigonometric time functions. First and second order frequency- amplitude relations and mode shapes for nonlinear dynamic systems are found by substituting this general series type solution into the linear recursion formulae and using the orthogonality properties of both the spatial and time functions. No restrictions are made a priori to limit th such as a sions dexw and its de systems t l dgenfunct frequencit hons are continua e Supports i to limit the procedure to a particular type of vibrating elastic continua, such as a beam, plate, membrane or string. Furthermore, the expres- sions deve10ped allow a general nonlinear function of the displacement and its derivatives and are applicable to a broad class of dynamic systems to be solved. The final results are explicit once a knowledge of the spatial eigenfunctions of the associated linear problem, along with the linear frequencies is available. Particular applications of the general solu- tions are made to vibrating elastic continua on nonlinear foundations, continua experiencing large deflections, systems with immovable supports and materials having nonlinear constitutive equations. 8 2. 2 General Perturbation Method for a Class of Nonlinear Partial Differential ELuations. [A method is developed to determine periodic solutions and fre— quency-amplitude relations for equations of motion governing free vibra- tions of nonlinear continuous systems. The dynamic system includes a general type restoring force nonlinear in the displacement function and its spatial derivatives. Consider the periodic motion governed by the following nondimen- sional equation M 2 _ where Lx is an autonomous* linear differential operator of order 2n, u is a dependent function of the spatial variable x and time t, to” is a fre- - quency parameter, 6 is a small parameter which depends upon the physi- cal constants of the system and either occurs naturally or is artificially introduced. The a - are coefficients dependent upon x and the Nj are non- J linear autonomous differential operators given by Nju = fj (u’ux'uxx """" ”B (2. 2. 2) where fj is a polynomial of finite degree in u, ux, uxx ----- ug with the notation that u E u . g 9:90,” n It will be assumed that the operator Lx is self adjoint for every t in the space of functions defined by the homogeneous boundary conditions Diu(0,.t)=0 i =1,2 ------ p (Z. Z. 3) Dju(L,t)=0 j=p+l, ----- 2n a An Operator in which the time t does not appear explicitly..,but only as a differential dt, is called autonomous. 9 where the D's are also autonomous linear differential operators of degree < 2n. For periodicity it is assumed that u-(Xi t) = Dix: t + 211’) (2. 2. 4) Furthermore, without loss of generality, the origin of the time scale will be selected so that ut (x, O) = O (2.2. 5) i. e. the system has zero initial velocity. The initial configuration will not be specified. Now to apply the perturbation method the dependent variable and the square of the nonlinear frequency parameter are expanded in infinite series in 6 as follows m u =23 £1 ui = uo + Eul. + 621123 +-- (2.2.6) 1 = O a: . . we a 2': £1 (0?: (.03 + use + 63009 .. (2,2,7) i=0 1 O 1 2 where 6 is the so called perturbation parameter. Before the above expansions are substituted into the equation (2. 2. 1), special attention must be given to the nonlinear function fj‘ Each fj may be expressed as a Taylor's series for a function of several variables about a given geometric configuration, which is taken to be the solution or mode shape of the associated linear problem defined by setting 6 equal to zero in equation (2. 2. l). The function fj is then written as N ‘1 1 a a E '1?! [‘“W’Rf‘s‘wtr f- (u, u ---u) = J x € :0 k + _________ (u€-%€]fj(u09uox ----- uo€)+Rn (2.2.8) 10 where Rn is some remainder defined in the usual sense of Taylor's theorem. The following notation has been implied in equation (2. 1. 8), namely ij (uo, uOX, ""uog) _ ij (u, ux, ----u€) (2. 2. 9) BuOX aux u _=_ 3: x‘ x u where u u 0 ’ ox , --- uog refer to the linear mode and its derivatives with respect to x. Replace the displacement function u by the perturbation expansion (2. 2. 6) and fj is written ' _ o --- o“ €u1€+ €2u2€ + ----) (2.2.10) Upon substituting the function (2.. 2. 10) into (2. 2. 8) Taylor's expansion about the linear mode 110 follows as f- = f + f [Gu + (an + —---J + f [Gu + 6211 +---) 1 2 11x 1x ax J u _j 2 + f [Cu + 6 u + ---] + ------------ uxx 1xx 2xx %“j 2 2 1"" a a + + + nu] + [c + + nu] fuul:€u1 E u2 'gfuxux 111x 6 u3x _j r 2 3 +éfu u Lcu1xx+£ uaxx+---] + --------------- xx xx -1' + f [cu + (211 + -H] [fu x + £211 + ---]+ --------- uux 1 a 1 3X with be'u 11 -j -' a +fuu [£111 + (Qua + ---_J[€u1xx + 6 uaxx+ ---]+ ---- (2.2.11) xx with the notation :3 ___ Bafj (u, ux, ---u€) ‘— uu}{ an all): z ox 11g : uog being unde rstood. Return now and consider the equation of motion in its entirety. Substitute the expansions (2. 2. 6) and (2. 2. 7) into equation (2. 2. 1), remem— bering that fj have already been expanded as given in equation (2. 2. ll). Collecting coefficients of like powers of 6 and equating these to zero lead to a system of equations as follows 6°: Lxuo + 01:intt = 0 (2.2. 12a) 1 ' 2 3 1% -5 ‘ ' Lxui + “5 u1tt ”“1 uott ' .=1 “1“ (2.2. 12b) 9 2 = a _ a E . Lxua + wouatt 402 nott (lulu1tt M 'j -j 'j n _ E a.[fu 111 + f uxulx +---- fugung (2. 2.12c) i=1 J 3. a = 2 _ 3 _ 3 E . Lxu:3 + (.00 uatt «a uott (.02 ultt €01 uatt 1:17 “j -1 + +1}? 3 + it t? 3:101ij us + f uxuax ----- uuul ‘1qu 1x+ "" -i 'j + ] +fuuxu1u1x + fuuxx u1uxx ---- e' = Lxui + w: uitt = Ni (2.2. 12d) ui Will equati. tilt. In that ti depen i0 501 btgm by tk fora 12 In the recursion formulae (2. 2. 12), all the displacement functions will be assumed to satisfy the same boundary conditions as given in “1 equations (2.2.3), the same initial condition (2. 2. 5),and (2. 2.4) for period- icity. ']he initial configurations ui (x, 0) are assumed unspecified. It is noted that the inhomogeneous part Ni appearing in the (i + 1) th equation only i ; j is1 i=1 (bserve that in arriving at the series (3.1.1) the nonlinear function has been assumed to be odd in u so that '+ a,u2J 1 > 0 (3.1.2) 3 1 s: m HM: 3' For practical purposes, the series (3.1. 1) is now truncated after the first term. Thus the equation (2. 2.1) becomes 2 3 L + + -.-. Xu (1) utt 6 alu 0 (3.1.3) 21 23 u = Alka(x)cos t + £[ r1215‘(in)vncos t + 3:13“ A”) Vncos 3t ] (3.1.7) In addition to the above, the following functions as defined in Chapter II will be needed to determine the second order frequency- amplitude relations for continua on Duffing-type foundations; (3.1.8) Upon substitution of equations (3. 1. 4) and (3.-1. 3) into equation (2. 2. 28), the second order frequency-amplitude relation follows as 3Alk[q: :324J‘:1arvk3qux-t NM8 I ;:i(31q) 4 i:°‘1rvk3vq dx] (3.1.9) If the first order expressions for amplitude (3.1. 6) are substituted into equation (3. l. 9). the second order frequency-amplitude relation becomes 2 2 L 3 (I0 aerk qux) J‘Lkalrv BqV dx C”22"'13"1”‘1k[q229 (92 + (O ) - 33) E1 (3. - 9:.) (3.1.10) 24 These above expressions for continuous media vibrating on non- linear elastic foundations will be further applied to more Specific systems, such as strings, beams, plates, and membranes, in the remaining parts of this s ection. 3.1. 1. Uniform String Let us consider a taut uniform string, fixed at both ends and attached to an elastic foundation which has both linear and a cubic spring response. It is assumed that the initial tension and subsequent displacements are of such magnitude that the tension is considered constant throughout the motion. The displacements, however, may be sufficiently large relative to the supporting foundation to warrant the inclusion of a small nonlinear term in the restoring function. The equation of motion is written as , a2; a2; - —3 -'I' + o + Ku + Knu = 0 (3.1.1.1) -2 -2 Ex at where T is the constant tens ion, 0 mass per unit length, K the linear spring parameter, K 71 the nonlinear spring parameter and 11 represents the transverse displacement that depends on the spatial variable x and time t. The equation (5'. 1.1.1) may be nondimensionalized so that the string is of length “IT and the period of vibration fixed at 2 ‘n' by intro- ducing 2 u=Lu t=a)t a1=££— T IL_ L2 %_ 7T2 x=Lx a) “9—; a) €=nL T1r (3.1.1.2) 25 ' The nondimensional equation of motion takes the form 2 3 uxx + a) utt + alu + aliu — 0 (3.1.1.3) which is similar to equation (3.1. 3) if the operator Lx is defined as (3.1.1.4) To determine the first order frequency-amplitude relation one simply employs equation (3.1. 5). For the boundary conditions of the problem, the normalized linear eigenfunctions are Vk(x) = J27w sin kx (3.1.1.5) Let us consider perturbation near the first linear mode. By taking 2 uo(x,t) = A11V1(x)cos 1: so that (130 = {22: 1, the first order is 1 given by L 2 _ g 2 ,7— . 4 . For constant 011 the equation (3. l. l. 6) is easily integrated and the frequency-amplitude response is given by 2~ 2 2 2 9 w wo+£wl 1+0:l (alAll Br. (3117) which agrees with those of Stoker [l] , Keller and Ting [9] and McQueary and Clark [4] obtained previously upon replacing A IbYfir—fé A , where 1 A is the maximum amplitude. The amplitude parameters for the first order nonlinear mode are obtained through a straight-forward substitution of equation (3.1.1. 5) into equation (3.1. 6). The only nonzero terms remaining are 26 (1) _ _ 3 _ 2 __ (1) _ 3 2 21.31 — 3/16 A /(-1 + 9(00 - a1) (3.1.1.8) A33 —- 1/16 A /( 9 + 9000 al) and the mode shape is written as u=u +€u = Asinxcost+EA(1)sin3xcost o 1 13 + €A(311) sin x cos 3t + (A213) sin 3x cos 3t (3.1.1.9) The second order frequency-amplitude relation may be deter- mined in a similar manner. Recall that vk = f7’2 'n' sin kx (3.1.1.10) and 5213: k2 + a1 (3.1.1.11) Equation (3. 1.10) then yields (02: 3/256 (J2/7r A )4[ 9 (3.1.1.12) 2 11 2 (-9 4. a) .. a ) o 1 + 92 + 12 1 — - - + - (1 9wo a1) ( 9 9000 041) In order to reduce (3.1.1.12) to its previously published form 2 2 9 21 24 a) = (13° + 3'3 6A - 4096 6 A (3.1.1.13) 27 one simply makes the substitution A = J'n'72 A and sets a = 1, 11 3. l. 2. Prismatic Beams The dimensional nonlinear differential equation of motion of a vibrating prismatic beam resting upon a nonlinear elastic foundation and restricted to small displacement theory may be written as 4— .— EIB_:+x-A-a—2—-_‘21+KE -3-0 Bx g at (3.1.2.1) where E is the elastic modulus of the material, I the second moment of area, 'y the weight per unit volume, A the cross sectional area, 13 the transverse displacement as a function of 52 measured along the beam and time t, K andKfl the linear and nonlinear foundation par- ameters respectively and g the acceleration due to gravity. Performing the variable changes 4 u=Lu t=a>t al=—I'<—L_: EI'rr x =IIEX (I) =(\E145w 6 = an (3.1.2.2) 7AL4 one obtains from (3.1. 2. l) the following nondimensional equation of motion for a beam of length “IF and fixed period of vibration of 277‘ 2 3 u + + = I O O x +wutt alu aleu O (3123) which is of the general form of equation (3.1. 3) if the operator Lx is defined as (3.1.2.4) 28 The corresponding linear equation is found by setting c equal to zero in (3.1. 2. 3). u + a) u + a u = 0 (3.1.2.5) Assuming that harmonic motions of the beam exist, one finds that by separation of variables the mode shapes of the motions must satisfy the following ordinary differential equation V 2 VI - (Q - (11 where Q is the vibration frequency. ) v = 0 (3.1.2.6) It is easily shown that for nontrivial solutions V(x) of (3. l. 2. 6) to exist, 92 must exceed a To see this let us multiply both sides of l . (3.1. 2. 6) by V and integrate over the span of the beam L L jvade - ((22 - a1) jvzdx = 0 (3.1.2.7) o o A which may be rewritten as L j vade ($22 - a) = (1), (3.1.2.8) 1 2 I v dx 0 Upon integrating the numerator by parts, it follows that L L L I vade = vaII .- vIvII + I (VII)2dx (3.1.2.9) O O O O Now the first two terms on the right vanish upon applying the boundary conditions, a fact that is implied by the assumption of self-adjointness of the operator Lx‘ Hence 29 L J" (VII)2 dx = O 52—0: 1 a 0 (3.1.2.10) L 2 [CV dx Thus the numerical value of (11, if it is positive, determines a lower bound of the vibration frequencies of the system. Results on beams available in the literature obtained by perturb- ation techniques have beenlimited to first order only when boundary conditions are other than simply-supported. However, beams with other common boundary conditions such as fixed-fixed, fixed-hinged and cantilevers, even though they involve complicated eigenfunctions, can be easily programmed to the computer and higher order nonlinear approximations are readily obtained by using the integral form of Chapter II. The numerical results of several beam problems will be presented in the following sections. 3.1. 3. Beams with Variable Geometry Beginning with Kirchhoff's work in 1879 the literature contains several investigations of linear transverse vibrations of beams with variable cross sections. A synopsis of the historical development with references may be found in the work of Wang [.14], who applied hyper- geometric series to such problems. Relatively little, however, is avail- able concerning nonlinear oscillations of such beams, Variations in cross section offer no restriction to the general expressions of the previous chapter, and section 3.1, provided that knowledge of the linear problem is available . 30 The dimensional nonlinear partial differential equation of motion for a beam with a variable cross section with linear and cubic restoring forces is given by 2 2- 2- 62 E1 8:21 + yA B 121 + ku + k'nu3 = 0 (3.1.3.1) Bx Bx 9 BE Let us consider the case in which cross section varies in the following manner — y A n 31-5— : ° ° 3‘- (3.1.3.2) 9 g L x n+2 E1 = E I (—) (3.1.3.3) o o L n k = k (5) (3.1.3.4) o L where the constants are previously defined and n, which specifies “ the particular geometry, may be either an integer or noninteger. Upon making the changes of variables 4 'k L 1.1 '-"- L11 ‘1‘; ‘-"'— a) E a = _____O 1 E I 'rrz 4-2 0 1 (3.1.3.5) 'y A L IE - 2 O O _ 2 x = X (D = 6 — ”L L E I 2 O 097T the nondimensional equation becomes 2 2 2 n ”3+2- xn+2 i—g-wy wzxn B )2] + alxnu + alex u3= O Bx Bx Bt (3.1.3.6) The equation is again similar to equation (3.1. 3), after being divided through by x“, if Lx is taken to be 31 2 2 L = J;- a 2 3:“2 —§-— + 011 (3.1.3.7) x x Bx Bx2 The linear equation corresponding to (3.1. 36) merits some attention at this point, which is found by setting 6 equal to zero. After separation of variables the spatial part of the linear equation is 4 3 2 2 ii!- + 2(n+2) x {-1—} + (n+1) (n+2) d_’\21 - K4V = 0 dx dx dx (3.1.3.8) I where R4 = (QZ—al), and $2 the frequency, and V describes the mode shape of vibration. Upon introducing the operator notation l) E g; the above equation is factorable to the form (xD2 + pp + $82) (xD2 + on - Ezw = 0 (3.1.3.9) where P and Q are constants to be determined. When the factorizations are expanded, one can show that P = Q = n + l (3.1.3.10) Substitution of P, Q, and D into the factored equation (3.1. 3. 9) yields the equivalent pair of equations to be solved as 2 x d—y- + (n+1) Q + K2 v = o (3.1.3.11) 2 dx dx dzv dv ~2 x —— + (n+1) — - K v = 0 (3.1.3.12) dxz dx These expressions are forms of Bessel's equation and the solutions follow as 32 _ -n/2 1: 15 L2 V(x) — x [ ClJn(2kx ) + C2Yn (2kx ) + C3In(2kx ) . 15 + C4Kn (2kx )] (3.1.3.13) if n is zero or a positive integer and _ -n/2 8 45 1: V(x) — x [Cth(2kx ) + C2J_n(2kx ) + C31n(2kx ) + c I (21935)] (3 1 3 14) 4 _n . . . if n is neither zero nor a positive integer. Further development of the linear problem to obtain eigenfunctions and eigenvalues requires a knowledge of the boundary conditions. Several particular examples will be considered later. However, one important point to be emphasized here is that well known solutions of variable section beams without a foundation term can be readily adapted to the similar problem with a foundation only if the foundation parameter varies in the same manner as the mass or cross sectional area, the density of the material being assumed constant. In particular,'aWinkler type foundation can be easily coped with only when the beam height is constant with respect to length. With n =1 the cross section is a wedge of constant depth and it represents a Winkler type problem, in the case n = 2 the cross section is a double wedge or pyramid and a Winkler foundation is not implied. For n - 3/2 a parabolic-wedge type section again excludes the Winkler definition. The nonlinear problem is 33 by no means restricted by the preceding remarks, and several examples are worked out in the sequel, including other cross sections, such as those with an exponential varying geometry. 3.1. 4. Thin Circular Plates The partial differential equation for vibrating plates on nonlinear foundations under the classical small displacement assumptions is given by 2 _ u 2 4- B DVu+p _ 3 Bt + k5 + knu = 0 (3.1.4.1) where, in addition to the constants defined earlier, D is the conventional . E 113 . -' . 4 plate stiffness 120"“ ) , p is the mass per unit area and V stands for the biharmonic Operator. To nondimensionalize the equation the following variable changes are made - - ka4 = t = ._._. u an wt (11 D r = ar a) = 4 a) E = 17a 0 a where a is the radius of the plate. The nondimensional equation of motion of a circular plate of unit radius becomes 4 2 Vu+wu +au+a 3 tt 1 lEu - 0 (3.1.4.3) which again is similar to equation (3.1. 3) with the operator 2 2 4 B + l B B. B x V 011 r2 r Br) 2 Br) + a1 HIH (3.1.4.4) It‘ll-ll II-l ll! 0' I’ll: I‘ll 34 The linear mode shape and calculated frequencies for plates without a foundation for various boundary conditions are well known in the literature [21, 22, 26, 31]. Direct substitution of these well known results on the corresponding linear problem into equations (3. l. 5), (3. l. 6), and (3. 1.10) yields first order frequency-amplitude relations, mode shapes and second order frequency - amplitude relations for the nonlinear problem. 3. 2. Continua Having Immovable Supports and Lage Deflections 3. 2. 1. Elastic Bars In the classical theory for the transverse vibration of elastic bars, axial extensions of the bars are not considered. One end of the bar is usually considered free to move such that the effect of the changes in axial tension during motions is negligible. Woinowsky-Krieger [15] studied the transverse vibrations of hinged bars using elliptic functions and showed that the axial tension increased the frequency of vibration. Burgreen L16], Eringen [l7]and McDonald [.18] studied similar problems, for simply supported beams. Recently, Evensen 1.10] obtained first order perturbation approximations for the frequency-amplitude relations for clamped-clamped and clamped-hinged beams of uniform cross sections, as well as for hinged-hinged beams. Periodic vibrations studied in this section include systems for which the above-mentioned classical theory is not applicable. Both the initial tension and that induced by deflections are considered. The present approach leads to results that agree with those in the existing literature. In addition, second order frequency terms and nonlinear mode configura- tions are also presented. 35 When longitudinal inertia is neglected the free vibration of a uniform beam with end conditions ranging from spring-supported to immovable is [.15] . 345 326 325 E I _4 - (To + T) _2 + p _2 = 0 (3.2.1.1) B2: Bic B‘t .m- where T0 is the initial axial tension and Tthe induced tension, which is approximated ~- L — 2 _ T =‘E'L-Aj (L3 ) dx (3.2.1.2) 2 L -_ B x with 0‘53 A "’ = ,. — 3.2.1.3 A A 1 L K ( ) where K is the spring constant of the supports relative to the axial displacement. Other symbols are defined a priori. Note that for immovable supports K is infinite and A = A. Upon introducing the following variable changes and definitions - _ TOL2 u = L u t = a) t (3 = 2 EI‘H‘ 4 2 (3.2.1.4) _ 1L - 2 _ LIL ‘2 = L x .- L X a) - 4) a) E 21TI I'rr ' and the dimensionless equation of motion becomes 7’ 2 2 u -Bu -€J(u)dxu +cnu =0(3.2.1.5) xxxx xx 0 x xx tt which is in the form of equation (2. 2. l) Lu+w2u +60£Nu=0 x tt ' 1 1 h w ere B4 B2 L = — - f3 _ (3.2.1.6) x 4 2 B): B3: = j Nlu f1 (ux,u X 7" 2 xx) = Jo(ux) dx ux (3.2.1.7) I cllliw/ I! II Ill ‘- I'll lull! 36 and a = -1 (3.2.1.8) To determine first order frequency-amplitude results use is now made of the general form given by equation (2. 2. 23). With the weighting function taken as unity one obtains 2 1 ' 21)“ 7T -j , a) = I I a f (u ,ux (x) cos t dxdt j=1 1 7r A11 0 O 1 x x) v1 (3.2.1.9) Note that the linear mode vibration is written as u = A V (x) cos t (3.2.1.10) o 111 i. e. perturbations in the neighborhood of the first linear mode are considered. The nonlinear function fjbecomes -j 3 3 7T 2 f = (ux,u x) Allvl,xx cos t I (V1,x) dx (3.2.1.11) i=1 After direct substitution and integration with respect to time, (3. 2.1. 9) reducesto 2 _ 7T 2 a) _ Z 111le: V11V xxdxj (V1.23 dx (3.2.1.12) 1 which can be shown to agree with those of Evensen. The complete set of nonzero amplitude parameters is obtained from equations (2. 2. 24) as q > 1 (1) i-Afl 7f W A 1 - 2 7-f‘2 v v1 dx I (v1 ) dx q (w - w ) 0 q ' O ' q o (3.2.1.13) q > 0 l 3 A 7r 7r (1) 4. 1 2 A = L__ v v dx (v ) dx q3 (0)2 _ 9(02) [0 q 1.xx )0 1.x (3.2.1.14) Thus, the first nonlinear mode of vibration is G O (1) (13) = + + u A11V1.cos t EAql Vq cos t ZAq Vq cos 3t q=2 q=1 (3.2.1.15) To compute second order terms derivatives of the nonlinear restoring function fj are needed. These are . w ‘J _ = I t 2 .= f — f1(ux'uxx) u u uoxx O ox dx 3 l ox' oxx (3.2.1.16) _. B f1 2 f3 = = I 2 u dx 3:1 ux B u ox oxx oxx 0 ox (3.2.1.17) . B f V -J _ 1 = I 2 = fu B u ox oxx (nox) dx 3 1 xx 0 (3.2.1.18) with all other derivatives of fj being zero. The second order frequency- amplitude relations now follows from equation (2'. 2. 28) with the weight- ing function again being equal to unity as W “’22 = “finigio‘fl, 2::de lexZAi111)Vq,xV1dx 38m +-:-j:1vxdxj:v1quzlAq(13)vq'xvldx + ablw J:(v,21x)dxi;2:(qll)vq,xxvldx 'TT )OWLX) 6‘" i: 31’" A213” q,xx Vldx] (3.2.1.19) + Ali- Numerical results corresponding to several particular boundary con- ditions will be discussed later. 3. 2. 2. Elastic Plates The dynamic analogue of the von Karman large deflection plate theory of equilibrium was proposed by Herrmann [l9] . In a later paper [6] Herrmann applied these earlier results to study free vibrations of rectangular plates with hinged immovable supports.. T he coupled non- linear equations of motion were solved by a perturbation method and frequency-amplitude relations for moderately large deflections were obtained. However, the recursion formulae involved in the perturbation scheme remained coupled and the iterative process became somewhat involved. Berger [20] decoupled the static nonlinear deflection equations such that one of them assumes a quasilinear form and is integrable by assuming that the strain energy due to the second invariant of the middle surface strains may be neglected. Wah [21] extended the 39 Berger formulation to large amplitude vibrations of rectangular and circular plates. A modified Galerkin approach was preposed whereby the first of a system of nonlinear equations was solved in terms of elliptic functions to approximate some salientparameters of the non- linear system, such as the frequency of vibration. Gajendar [22] followed the same method of Wah for large amplitude vibrations of plates on elastic foundations. In the following, the Wah decoupled nonlinear equations of motion describing axisymmetric plate vibrations are taken to be 2 .— B t where N 12 a at": 2 - - D 2 2 dr a h 0 which further reduces to a ;i = LL2— u(V2u) rdr (3.2.2.3) D 2 2 a h for simply supported or clamped edge conditions. The independent spatial variable is now r instead of x while other symbols remain as previously defined. The following variable changes and dimensionless constants 2 - -- 1 u=au t= t €=‘——2: - -2 D 2 h (3.2.2.4) r=ar (D = (1) 4 pa are introduced so that the nondimensional equation of motion is 40 4 2 V u + w utt - 6 f1 — 0 (3.2.2.5) where 1 2 2 f1 = I (u ) rdr V u (3.2.2.6) O r The above is of the form of equation (2. 2. l) with a1 = -l M = 1 2 2 4 B 1 B B Lx = v =(_—2+;B—;) (——2 + '1‘ 53-1) (3.2.2.7) B.r B:r r With the nonlinear function defined as f(uu)=(u +"]"'u)j‘1()2d 32 8) 1 r' rr rr r r 0 ur r r ( ' '2' the first order frequency-amplitude relations follows immediately from equation (2. 2. 23) with the weighting function now equal to r. After performing integration with respect to time the result is 1 2 3 2 _ - n—A (1)1 — 4 1] [r(Vl V1, + Vlv], ) dr I 1 )Zdrj (r V1 r 0 (3.2.2.9) The amplitude parameters follow from the expression (2. 2. 24) a ‘ 1 2w 1 A1) _ 1 2 " 2 I (V ) rdr (r V V (JP V(q2m2_ a) ) 0 1,]? IO Jo q l,rr 0 q + v v )dr cos pt cos3t dt (3.2.2.10) q 1,1: 41 and after integration with respect to time one has q > 1 3- a 1 A(1)- AL 1‘ I (V )2 rdrj‘ (rVV + VV )dr ql ((1)2- (132) 0 1,r q 1,rr q 1,r ° q ’ (3.2.2.11) l q > 0 (1) _—Z___—a 1. = +7 Aq3 (9w 2 ):J‘ (v1r1rdrio (r qul,rr qu1,r) dr 0 . (3.2.2.12) The first order mode shape is of the form of equation (2. 2. 25) with the constants defined as above. For the second order approximation, the following nonzero derivatives of the function f. are needed "a'i-=u IZu rdr+luJ2u rdr+1ju2rdr Bur r r r r r r 'r 1 (3.2.2.13) B f =J u 2 rdr (3.2.2.14) Bu t rr 0 Upon substitution into equation (2. 2. 28) and factoring out A31 one finds (”221111141p10715391):izvm:d:io,(rvirr1vl,r) 2 Ag; vq,r V1 dr + j:(V11)2 rdr 1:22Ap(1q)vq,rv1 dr q P ,1 1 °° °° * + J V1 20rer ZZAPq (1) Vq rr Vl rdr] (3.2.2.15) 0 Oq p ' Obviously, the normalized eigenfunctions Vn are the familar Bessel functions for circular plates. Detailed computations are programmed on a digital computer and the results will be presented in the next chapter. 42 3 . 2 . 3 . Membranes The question naturally arises as to whether the relative magnitude of the strain energy introduced by the second strain invariant is negligible in large amplitude vibrations of membranes as was found to be the case for vibrating plates. Since the formulation of the membrane problem excludes any contribution due to bending, the reliability of results obtained in such a manner is speculative from a theoretical point of view. The fact that the Berger - Wah development of the plate problem found justification upon comparison with known results suggests that the validity of the membrane analogy be studied in a similar manner. So Eringen [7] and Chobotov [8] have studied the membrane problems using the coupled nonlinear equations. Their results can be used for comparison purposes. Timoshenko [23] gives the strain energy due to stretching of amembrane as (b = —Ell—§ I I [e2 - 2(1-1))e2 ] dxdy (3.2.3.1) 2(1-1) ) where e = the first invariant of strain = 6 + 6 in rectangular coordinates X = (r + (a in cylindrical coordinates and e2 2 the second invariant of strain e g g .. — 7 in rectangular coordinates 2 x y 4 xy : £ 69 in cylindrical coordinates with circular symmetry. r Upon neglecting the second strain invariant and including the work of some external load q (3.2. 3.1) reduces to 43 ¢= jinn—271162 dxdy = I J qwdxdy (3.2.3.2) The first strain invariant expressed in rectangular coordinates is 2 2 Bu Bv 1 Bw 1 Bw =— ——+-_ _ where u and v are displacements in the plane of the plate and w the transverse deflection. By principles of the calculus of variations and integ ration by pa rt 3 _ Eh _ Be_ Be_ B Bw 61-1432 iiiéuax évay 5wa %Bx) _6w_a§_;(e_:_%)]dxdy-Ij1q6wdxdy=0 (3.2.3.4) Since 51.], 5V and 5ware arbitrary, the coefficients must vanish in- dependently. Therefore B_e_ _ 22.- Bx _ o a _ 0 (3.2.3.5) "< 2 _5_ Q3311) + .5— (e in) + 9153—)— = 0 (3.2.3.6) 44 From the first set of equations one deduces that e is constant with respect to x and y . Defining q as the inertia term one then has the following equation of motion 2 2 2 - Eh at2 It is now convenient to consider cylindrical coordinates and write Hooke's law as 1 E = "" (N - uN ) hE r r 6 (302.308) 4L. 66 - hE (N6 UNr) where N designates the stress and c the strain. The total strain e can be written as e = 6 + 69 = 6 + 69 + 60 = constant (3.2.3.9) where 60 is initial strain and the primed quantities refer to the induced stress and strain. Since the sum is constant it follows that 1 1 (11;) 1 1 - hE (Nr + N9 ) (3.2.3.10) which may again be written as 2 2 _(1_-g_)_ 1 1 Bu 1 Bw u le l Bw hE (r N6) r12 Br +r+EB +2?) B9 (3.2.3.11) Equation (3. 2.3.11) is now multiplied by rdrde and integrated over the area. Assuming circular symmetry and that on the boundary v is continuous and u evanescent, one obtains 45 ' 2 c 1 + £61 = 12 I (%-¥*) Ed; (3.2.3.12) r 27ra A By equation (3.2. 3. 9) and (3.2. 3.12) the equation of motion including the initial stress is written as 2" 2 - - a -No vzw- Eh 2 I (:—¥-) rdr V2w+o-—g‘=0 (1-u)27ra A at (3.2.3.13) where No is the initial tension in the membrane. After the following variable changes and definitions are introduced w = an t = 56?: E = ar 6 = Eh (3.2.3.14) (1-u)No the nondimensional equation of motion takes the form -V2u+w2u +0: 6f =0 (3.2.3.15) tt 1 l where 1 2 f =1. 5 u rdr V2 u (3.2.3.16) 1 a r O a = 1 and u is now the transverse displacement. The remaining manipulations to obtain the nonlinear dynamic results are identical with those in the plate problem . 46 The results will involve the expressions V1. and (Dn, the linear membrane eigenfunctions and eigenvalues respectively. 3. 3. Nonlinear Material Properties Free vibrations of continuous structures with nonlinear material prOperties represent another class of problems that can be treated by the general theory presented in Chapter II . In this section the motion of a homogeneous slender beam of uniform cross section and having a longitudinal plane of symmetry passing through the centroid will be considered as an example. The usual small deflection theory is also assumed. Obviously, the method could be extended to other elastic structures, and continua with combinations of large deflections, elastic foundations etc. The equation of motion for the free vibrations of a beam with nonlinear viscoelastic material properties as presented by Sethna [24] is modified such that material time dependency is excluded. For a material described by N=E(€+h£3 ) (3.3.1) where N is stress, E strain in the x direction, E and h constants, the equation of motion is written as |___.| 4- 2’- — _ - .. ‘ EI 5__.l1__+QAau+EIh|:3§_2_u_ fi+682u/83g_2 l -4 -2 - - - _ ax 9 at 2 3x2 8x4 ‘3x3\3x3 = 0 (3.3.2) The following variable changes and definitions are introduced 4 - - ._ I h37r u = Lu 1: = wt 5 = ‘2'?- 4. 11L (3.3.3) It _ gEIl'IT 8 x = L x w "-4— w L 47 where 11 and I2 are the second and so called fourth moments of inertia and other notation has already been defined. The nondimensional equation of motion now becomes uxxxx + wzutt + {(uxx)2 uxxxx + 2 uxx(uxxx)2] = 0 (3.3.4) which is again similar to the general equation (2. 2.1) with 64 LX = 5x4 011 = 1 M = 1 (3.3.5) f = (u ) u + 2 u (u .) (3.3.6) With a knowledge of the nonlinear restoring function and the linear mode, the first order frequency-amplitude relations follows directly from equation (2. 2. 23) as 2__3_ 2 ”’T 2 2 ml — 4 A11 Jo [v1(v1,xx) V1.xxxx + 2 V1V1,xx(vl,xxx) ]dx (3.3.7) The nonzero amplitude parameters for the nonlinear mode shape are easily determined to be 3 3 "A A(1)_4 11 (1)]. ql — 7T 2 «02-a32) Jo [Vq(vl,xx) V1.xxxx + 2 qu1.xx 0 q >21 dx V ( l I -XXX .1 48 1A 3 (3.3.8) q>0 A( 13) = (:2 A112 )I: [V( V,1 xx) V1 xxxx 2 + 2 vq Vl, xx(v 1, xxx) 1 dx After differentiating the function f with respect to u, ux, uxx --- and evaluating the results at points along the linear mode shape as before, the second order frequency-amplitude relations follow from equation (2. 2. 28) directly as 7r U322: All Jo [2 Vl,xxvl.xxxx + 2(V1 xxx’q2121A A2913 vq.xxv1 + [ 4v1,xxvl,xxx] Mtg]; q,xxx 1 q=1 +[(V,)lxx2]zA(qpl)vq,xxxxV1dx(;6 +£6) q"1 4 1p 4 3p (3.3.9) It is important to note that a third power of the amplitude is implicitly contained in the amplitude parameters, consequently the first order frequency depends on the square of the amplitude, the second order on the quadruple and so forth. IV NUMERICA L RESULTS Numerical results obtained through using the expressions derived in the previous chapters are presented in this chapter. Whenever possible, the results are compared with those existing in the literature. Higher order approximations to the nonlinear frequency~amplitude relations and to the nonlinear mode shapes are given which complement the existing solutions. In addition, numerical results are also presented for a number of nonlinear cpntinuous systems for which nothing has been published in the literature. A CDC 3600 digital computer and a Newton-Cotes numerical integration technique were used in obtaining the results. 49 50 4. 1 Solutions for Equations of Motion with Restoring Forces Nonlinear in Displacement. 4.1.1. String Perhaps the simplest example to which the methods developed in the previous chapter may be applied is that of a vibrating string. In section 3. l. l the first order frequency-amplitude relation is found from equation (3.1.1.7) to be w2 = wz +0: 35.42 (4.1.1.1) where (11 is a dimensionless quantity and A is the amplitude of vibration defined such that A = Alllvl \max (2.3.4) It is to be noted that the amplitude A represents the maximum displace- ment of the linear mode and since 6 and A51 occur only through the 2 11 , we may for convenience set€ = l and regardAlz1 as being productcA small. The first order nonlinear mode shape as written in equation (3. 7) with 0Ll g 1 has nonzero amplitude parameters as contained in Table 4.1-1. The normalized linear and nonlinear mode shapes are plotted in. figure 4. 1-1, where the amplitude is not taken as A but defined such that CA11 : “577- EA (4.1.1.2) 51 The determination of the second order correction term to the nonlinear frequency requires additional eigenfunctions and eigenvalues from the linear problem. However, in the case of a string, spatial eigenfunctions beyond n = 3 contribute nothing as a consequence of the special nonlinearity assumed. With the information from table 4.1-2, the second order frequency -amplitude response is Z _ 2 .2. 2,21. 2 4 4.1.1.3 w ‘ “’0 +160‘1A 4096"“1A ( ) Figure 4. 1-1 compares the first and second order approximations for small amplitudes. As mentioned earlier, the first order result given as (4.1.1.1) agrees with those obtained by other authors. 52 Table 4, l, 1-1,Frequency numerical values for a uniform string resting on a cubic nonlinear elastic foundation. LINEAR FREQ LIE’INEAR a a FIRSTDRDER SECOND QRDER _. (RIDE k"0 = n “f ‘ A3 A: .4410"3 A?110'3 1 2 0.5625 0.3581 2 5 6.59179 2.67155 3 10 -5.12695 -Z.07787 4 17 -5.12695 -Z.07787 5 26 -5.12695 -2.07787 6 1 Table 4. 1. l-2.Nonlinear;amplitude parameters for a uniform string resting on a cubicE nonlinear elastic foundation. I 1 n = spatial function 3 AA”), = amnA113 _ m = t1me function i A143,; : dmnA3 aOrnn n m l 2 3 4 5 6 7 1.4920 1 . 0.0 0.0 0.0 0.0 0 0 10_2 7.4603 4.9735 0 0 0 0 0 0 0 0 3 10.3 0.0 10_3 . . . . dmn 2.3437 1 0.0 -2 0.0 0.0 0.0‘ 0.0 10 1.1718 -7.8125 3 10—2 0.0 0 0 0.0 0.0 0.0 £33655.“ umooflcoc 29$ wcflmsfl a co mcflmou memento. 8.33:5 a mo 253m 358 woufimaflhoz .HA Aw ouswmh A1: xv 009330 53 0.0 w.o Nd ed m.o v.0 m.o N.o ~.o , // / / 11 N m \ - / , / / \ \ \ \ 1. m w / I IIIII \\ \ m / / \ m. / / cw. onofiiH. \ 4v m. / / \ W... / / N. I QEdH \ \ [Tm V / / h. 0 I \ \ w / \ \ ..al / /. n... / / .o .o u QEMH . \ a- o w... / \ e / \ w... / // I \\ \ I: N. IV / ll II \ .18 w. Jot i \\ 0 $52 Amocficoz II I I 1- o £002 .190ch Dimensionle ss Amplitude (6A1- - / 54 Second Order Response —. Lmear Response/ First Order Response % J. i ‘ + : 1+ 1 1 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 Frequency Ratio w/wo Figure4.1. 1-2. Frequency-amplitude response for a uniform string resting on a Duffing type nonlinear foundation. 55 4.1.2. Prismatic Beams Uniform beams on nonlinear Winkler type foundations as described in section 3.1.2 are considered here for simply supported, clamped-clamped, clamped-hinged and cantilevered boundary condi— tions. The first order nonlinear frequency-amplitude relation with r(x) = l was found to be a)? = 231-11211 fralle‘ldx ° (3. l. 5) The dimensionless parameter a1 may be taken as unity and the maxi- mum amplitude of vibration A is related to All by A=)>.H|vllmax (2.3.4) Wylie [25] , along with several other authors gives the exact linear eigenfunctions for the above -mentioned boundary conditions. Substitution of these eigenfunctions into equation (3. 1. 5) gives the 2 correction term ml to be added to the linear frequency in order to obtain the nonlinear response. After substituting these linear eigen- functions into (3. l. 5) and performing the integrations the first order frequency—amplitude relations for beams with the various boundary conditions are, simply supported wz = (02 +.35812€A2 0 11 (4.1.2.1) 56 clamped -suppor ted (02 = (0(2) + .40324 (A121 (4.1.2.2) c lampe d -c lampe d 2 2 Z w = w + .44210 (A 0 ll ( (4.1.2.3) and for a cantilever beam (02 = (0% + .56070 (A121 (4.1.2.4) Linear eigenfunctions and frequencies beyond the fundamental mode necessary for nonlinear mode shapes and second order nonlinear frequency terms are taken from reference (26]. To determine the amplitude parameters for the nonlinear configuration it is recalled that 11' 3 (1) _ i A f 3 > n - 4 11001V1Vndx n 1 2 2 3.1.6 17 3 (1 __'i 3 V V dx A3n-4A11J‘00211n2 n>0 (wn - 9w0) with .002 , (.02 andV known, the constants are determined as 0'1 n 0 n shown in tables 4.1.2-1, 4.1.2-2, 4.1.2-3, and 4.1.2-4. A continuous graph of these nonlinear mode shapes appears in figure 4. 1.2 -l. The second order frequency-amplitude results are obtained from equation (3. l. 10) by direct substitution of the linear information and then 57 followed by integration. Since the integration involves a series, a suf- ficient number of linear eigenfunctions must be taken to insure conver- gence is relatively good using only a limited number of linear functions. Nonlinear frequency -amplitude relations that include the second order approximation are taken to be a)?” = (.03 + 0. 35812€A121 + .002063624111 (4.1.2.5) for the simply-supported, w2 = 01% + 0.403246A121+ .001376c2Af1 (4.1.2.6) for clamped-supported, (.02 = (0(2) + 0. 44210€A121+ . 000743€2A11l (4.1.2.7) for clamped-clamped, and w2 : 0102+ 0.560706A121- .0199562A111 (4.1.2.8) for a cantilever beam. These results are plotted as continuous curves for small amplitudes in figure 4. l. 2 -2. Den Hartog [271 , by applying Rayleigh's energy method, has approximated the linear frequencies of vibration by assuming some spatial function which does not satisfy the differential equation of motion, but satisfies the boundary conditions. For the case of a cantilever beam without a foundation term, a quarter cosine wave approximation yields an approximate linear frequency only 4% above the exact value. A full 58 cosine wave approximates the linear frequency of a clamped-clamped beam within 1. 3%. These results can be extended to linear beams on linear elastic foundations by including an additive constant, i. e. the 2 2 linear frequency p now becomes p — 011 , and close agreement between the approximate and the exact is again obtained. For the case of beams on nonlinear elastic foundations, approxi- mate spatial functions also yield results similar to those obtained from exact eigenfunctions. The nonlinear analogues of the above -mentioned linear systems result in the frequency-amplitude relations as 2 _ 2 2 w _ (00 + 0.60756A11 (4.1.2.9) for the cantilever with a quarter cosine wave approximation and 2 2 2 for the clamped-clamped beam with a full cosine approximation. These results are plotted in figure 4.2 -3. 59 Table 4. 1.2 -1. Nanlinear amplitude parameters for a uniform beam simply supported and re stingon a cubic nonlinear elastic foundation. 1 ' 1 g n = spatial function Aiéz = am nAll3 5 m time function "drnnA , 1 1 a'rnn ) 1 f 1 inin' .1 2 3 4 3 5 6 l 7 j 1 1.4920 g g 1 0 0 10.3 0 0 0.0 0 0 0 0 5 1 3 7‘469: 0.0 6'2169 0.0 ‘ 0.0 0 0 0 0 ' g 10' , i an... '1 1 2.3437 E 3 1 0 o 0 0 0.0 0 0 o 0 g 3 1.1718 9,7656 3 L .10‘2' 0.0 10-4 0.0 0.0 0.0 0.0 E Table 4. 1.2 -2. Nonlinear amplitude parameters for a uniform beam clamped-supported and resting on a cubic nonlinear elastic foundation. ) n : spatial function A3912 = amnAu3 i m = time function 161-“) 3 ‘ v ' Amn " dmnA ‘“ —1 amn n T ‘ n, 1 2 3 1 4 5 l 6 i 7 1.1718 1.1952 -1.2506 -5.1047‘ 3 1 10'3 10‘3 10'4. 10‘6 f 3 4.8836 -2.0894 5.3277 —4.5560 -l.7657 10‘3 10‘3 10'4 10'5 10'6 1 dmn 1 ) -1 8260 -8.0384 3.6351 6.4116 . 2 10-6 10-8 10-10 10"13 2 t 3 é-1.7125: 3.2558 -3.5832 1.3242 .2.2178 g 3 10'4J 10'6 10‘8 10‘10 10713, E 3 60 Table 4. 1.2 -3. ,Nonlinear amplitude parameters for a uniform beam clamped-clamped and resting on a cubic nonlinear elastic foundation. (l) _ 3 a n = spatial function Amn - amnAn ) ‘ m = time function A1921 : dmnA3 i a'rnn n m 1 2 3 4 l 5 6 7 ~ ' 1.0695 -2.9363 -3.9313 1 0.0 0.0 0.0 _ . 10"3 10'5 , 10 7 3 3.0007 5.3925 0 0 -1.0346) 0 0 —1.3311 3 10‘3 0'0 10‘4 ' 10'5 ' 10'7 1 dun, , 2.8144 -1.4429 -3.6174 A 1 0.0 10-3 0.0 10-12 0.0 10.17 4.0827 1.4189 0 0 -5 0842 0 0 -l.2248 0 0 . _ _ 3 .10'5 10'8 10 13 10 17 Table 4. 1.2-4. Nonlinear amplitude parameters for a uniform canti- lever beam resting on a cubic nonlinear elastic foundation. l 1 3 ' n 2 spatial function Ariiri = amnAll 1 m = time function 1 . Ariiri = drnnA3 amn n ‘ i “L 1 2 3 4 5 1 6 g 7 1 4.9370 -2.1920 1.9292 -3.4120 i 1072 10‘3 10'4 10'5 2.0731 -1.9225 -9.5072 6.8424 -1.1629) ; , 1 3 10'2 10‘2 10‘4 10‘5 10'5' 1 1 dm n __= R_~"_..i_..w. 1., “.1, ...._.._...._....__..). 1 , )-3.6140 6.6730 -2.5422i1.9427 ; i 10'4 10'7 10’9 10"11 j 1 g 3 3.1.9673§1.4073 2.8942 -9.0163 6.6214 3 i ' i _ ‘ _ - _ ' - 1 7 10 31 10 4 10 7 1010 1012. . 1 61 Table 4.1.2 -5. Frequency numerical values for a uniform beam simply supported and resting on a cubic nonlinear elastic foundation. -..W - ”LINEARLINEAR FREQ.‘ FIRST ORDER w, SECOND-ORDER mg “ M21333 ____f”2_:n4+f" 5 A2 A21: 114(10)‘3 Aifl(10)’3 1 '2 0.56250 0.35812 2 17 . 6.59179 2.67155 -211- 82 ___ .2193933__ 2.06303 --,_ 4 257 -1... ___ 5.09033 2.06303 5 626 ' 5.09033 2.06303 Table 4.1.2 -6. Frequency numerical values for a uniform beam clamped - supported and resting on a cubic nonlinear elastic foundation. -L114NOE§1ERL1NE£°1R4FREQ. jIZRST ORDERZ wf spoon-51) 01111213219. 11%;; n i (00m +0: A All $18—00) A1100).~ ' $1“ ; 2 . 0.55635 0.40324 ; 2 17 . 7 3.67310 1.93057 __» 3 i 82 i = 2.65111 1.39341 4 e 257 » f 2.61869 1.37637 5 ‘ 626 3 2.61856 1.37630 “—n...“ 0 . 62 Table 4.1.2 -7. Frequency numerical values for a uniform beam clamped-clamped and resting on a cubic nonlinear elastic foundation. LINEAR LINEAR FREQ; FIéiST ORDER ‘0? : SECOND ORDER «)3 if?” 2 A .4121 f A4(10)"3 1».ff(10)'3 1 ; 2 :0.55067 0.44210) 2 Q ' 17 1 § 2.05826 1.32667 3 5 82 1 1 1.15694 .745720. 4 3 257 ; 5 1.15694 .745720- 5 ? 626 . 1.15286 .743089 9 ' 6562 : 1.15286 .743088: Table 4. 1.2 -8. Frequency numerical values for a uniform beam cantilevered and resting on a cubic nonlinear elastic foundation. .9 LINEAR) LINEAR FREQ. FIRST ORDER w? ESECOND ORDER 093 .MMODE l 1 A2 A2 i A4(10)'3 Af‘luor3 1.7””1 g 2 )0.44037 0.56070; .fl_ 2 . 17 3 A -1.18956 -1.92844 ‘*_ 3 . ' 82 - .1.22920 -1.99271 4 L 257 -1.23035 -l.99458 5 626 ~l.23045 —l.99474 fi‘w« m ' ‘- r~m~mh l“.~- W 2:033:53 umocficoc co mason. no.“ mCOfidquCcoo opoE poufldfiuoz .TN A .v oudmfm 04 mod w.o P b q 2 63 / £6on umocfieoZI I I 3602 amoewd .o 1) AA}; 00:3me 0.0 m.o «4.0 m.o Nd H6 h I b b . . 4 . a d pounommdm 398mm pomEflU u poagmao I 1 pomemmapomcuflo .mo 1H 0 -mN .0 mad .0 V 3) epniuduxv S S 911101 suauiiq IT. 8 §( T l E ) 2 1 Dimens ionle s s Amplitude ((A1 64 _ Clamped- Clamped / , Clamped-Hinged / / / / / / A)- / / / " / / .. / Simply Supported / / .- . Cantilever / / / / ., / / / / " / First Order Response Second Order 1" Response 1 1 1 I & ' ' I I r 1.0 l. l 1.2 1.3 1.4 Frequency Ratio 0.3/0.)o Figure 4. 1. 2-2. Frequency-amplitude curves for various beams resting on Duffing type nonlinear elastic foundations. 1 .1. 2 2 11 Dimensionle s s Amplitude (6A 65 Clamped- Clamped Beam Cantileve r Beam _ _ _ Exact Linear Mode Approx. Linear Mode 1 J A J r 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Frequency Ratio w/wo Figure 4. 1. 2-3. Nonlinear frequency-amplitude responses for various beams resting on Diffing type nonlinear elastic founda- tions. Results from using approximate linear mode shapes are compared with results from linear eigenfunctions. 66 4.1. 3: Beams with Variable Cross Section The following nonlinear results are examples of the theory and application of section (3. 1.3). Beams and restoring functions as described by taking particular values of n in equations (3. l. 3.2), (3. 1.3.3), and (3.1.3.4) are considered. Linear transverse vibra- tions of beams of variable section without any restoring force are treated by Cranch and Adler [28] using simple beam thory and the solutions involve Bessel functions. These linear results are ex- tended to include a linear restoring function and thereby supply the necessary linear eigenfunctions and eigenvalues for computing the results. Let us consider a cantilever beam with a coordinate system so chosen that the distance x measured along the beam originates at the free end. The boundary conditions at the clamped end give the fr equency e quation 1 l 1 l " '2‘ § ‘2' _ Jr1 (ZkLa) In+l(2kL )+ Jn+1(2kL )In(2kL )_ 0 (4.1.3.1) and the linear mode configuration follows as n 1 1 1 _ '- s ‘2’ _ s %] Vn(x) — Cx 2 [In(2kL )Jn(2kx ) Jn(2kL )In(2kx ) (4.1.3.2) where k4 = (02 - Oi , L :17 , x is the dimensionless length variable, Jn and In are the Bessel and Modified Bessel functions of the first kind. Example 1. We now consider a cantilever beam with rectangular cross 67 section and set 11 = 1. If 011 and E in equations (3.1. 3.2), (3. 1.3.3) and (3. 1. 3.4) are both constant, then the cross section varies in such a manner that the beam depth is constant, the height varies linearly and the foundation parameter varies as the area. Upon substitution of the linear eigenfunction into equation (3. l. 5) with n = l and arbitrarily setting a1 : 1 , the first order nonlinear frequency approximation is found as 2 _ 2 2 w — wo + 0.7665 (All (4.1.3.3) To compute the nonlinear mode shapes, the first five linear eigenfunctions are taken. The amplitude parameters for the nonlinear mode shape are given in table 4.1. 3-1 . Furthermore, the normalized nonlinear mode shape is graphed in figure 4. 1. 3-1. Inspection of the results given in table 4. 1. 3-3 indicates good convergence of the second order frequency (3.1. 10) including second order corrections is given as 4 11 (4.1.3.4) 2 (2) + 0.76656A121- .086162A The results are plotted in figure 4.1. 3-3. Example 2. By setting 11 = 2 both the width and depth are a linearly tapered if a rectangular cross section is assumed and the beam represents a pyra- mid. Also, by the change of variables H II M: H o a: 0 r = rO(X/L) A = "r20 (4. 1.3.5) 68 the results with n = 2 represent a circular cross section with a uniform taper. Both 0L1 and E are again assumed constant. The first order nonlinear frequency-amplitude response is 2_ 2 2 w -w0+0.8257(An (4.1.3.6) The nonlinear mode shape is described by equation (3. 1.7) with the constants defined in table 4. 1. 3-2. These mode shapes are plotted in figure 4.1. 3-1. The second order frequency-amplitude relation, again computed by truncating after the fifth linear eigenfunction, is found from table 4.1.3-4 as 2_ 2 2 2 4 w — “’0 + 0.8257(A11 - .08257£ A11 (4,3,7) A frequency graph appears in figure 4. 1. 3-4. Example 3. If one sets 11 = 3/2, the beam has a parabolic width and linearly tapered height with 'yl and E constant. Again the restoring parameter varies as the area. Upon substitution into equation (3.1. 5) withOl E 1 , the first order approximation of frequency is 2 _ 2 2 a.) — “’0 +.’7961£A11 (4.1.3.8) The nonlinear mode shape has constants according to table 4. 1. 3-6 and the nonlinear frequency-amplitude relation including the 69 second order perturbation from table 4.1.3-5 is 2 2 2 4 0 + .7961 (All 2 ..08766 All (4.1.3.9) These results are graphed in figure 4.1. 3-5. It'll'yii'l‘l‘l'llll '1'! cl"! ll.“ fIJI 70 Table 4. 1. 3-1. Nonlinear amplitude parameters for a variable section cantilever beam with a nonlinear displacement dependent forcing func- tion as described in Examjle 1. A“) = a 3 1 n = spatial function mn mn ll . 1 i m = time function ASL} = dmn 3 i 1 amn E 1n,“ 1 2 3 4 3 5 6 2 7 E 3 -3.8703 -7.1743 -2.0219 -6.9666 g 5 g 1 10’2 10'3 10"3 10'4 g 1 8.2756 2.5774 -3.6437 -7.7010 -2.4566 1 1 E 3 10'3 10‘2 10"3 10‘4 10'4 3 i 3 d 1" i 1 mn . -4.6487 -4.0444 -5.6202-JL1320’ , 1 - - - - '. . 10 5 10 7 10 9 10 11 1 l 3 1. 9388 3. 0958 -2. 0540 -2.1405 -2. 8676 -4 -5 -7 -9 -11 10 10 10 10 10 Table 4. 1. 3-2. Nonlinear amplitude parameters for a variable sec- tion cantilever beam with a nonlinear displacement dependent forcing function as described in Example 2. n = spatial function A2111 = amnAfl : m = time function A“) :-d A3 1 W mn mn a mn n I 3 nn 1 2 3 4 5 g 6 1 7 ; -2.4515 -6.0515 -2.0445 -8.2047§ g 1 42 -3 —3 -4 1 : 10 10 10 10 , - 1 3.9439 9.5634 -3.9678 -8.4l43 -3.0024( < 3 10"3 10‘3 10'3 10‘4 10'4 1 2 {1” __fi_ mn 1 =__4W-M--W1 1 1-5.2391 -4.2723 -5.0183 -7.0747 g .10” 10'8 10'1010'12 3 3.1824 2.0437 -2.8012 -2.0652 -2.5889 10'5 10"6 10‘8 10'10 10'12 1 Table 4.1. 3-3. function as described in Example 1. 71 Frequency numerical values for a variable section cantilever beam with a nonlinear displacement dependent forcing ‘LthSEfi‘ER gig-NEAR RREQ.__§1ZRST ORDER 71f: Sfcogm CREEK-(2); -9- 1.9- ”a 3 4. A11 .A 10. A1110 1 71 =4.61 0.4035 0.7665 _ q 2 12 = 7.80 =’ -1.81787 -6. 56106 1.. 3 Y3 = 11.0 -2.26738 —8. 18345 4 1'4 = 14.1 -2.36215 ~8.52547 5 ”1’5 = 17. 3 -2. 38759 -8. 61730 Table 4.1. 3-4. function as described in Example 2. Frequency numerical values for a variable section cantilever beam with a nonlinear displacement dependent forcing ' éLfingfiAERigljF BBS-310.: FZIRST ORDERZEf r SEC-qND OREER-Zgi 1 .n 1° _'_jr? 1 A All ‘A 10 A1110 1 1 71 = 5.91 1 0.3988 0.8257 ‘ 2 72 =9.20 “ 4.30134 -5.57869 3 ‘93 = 12.4 -1.74476 -7.47958 4 74 = 15.6 41.86696 -8.00342 5 75 = 18.8 34.90836 -8.18092 72 Table 4. 1. 3-5. Frequency numerical values for a variable section cant1lever beam With a nonlinear displacement dependent forcing function as described in Example 3. ' LINEAR 1LINEAR FREQ? FIRST ORDER of" 'é'sEEOND ORDER E ; MiDE «fighfgfla l A2 , A2 1144104 AfilOT: 1 . 71= 5.27 30.4010 0.7961 i 2 l 7’2 z; 8'51 ' -1.58874 -6.25986 .7 3 j 73 = 11.7 206616 -8. 14098 g 4 1 74 = 14.84 -2.18589 -8.61271 1; r: 5 75 = 18.01 1 222290 -8. 75856 f L ‘ ’1 j Table 4. 1. 3-6. Nonlinear amplitude parameters for a variable section cantilever beam with a nonlinear displacement dependent forcing function as described in Example 3. n = spatial function A”) = a A3 7 _ , f _ (fin mm 31 ' m - t1me unction A = d A { mn mu 3 . 1 mn _‘ 1 m“ 1 2 3 4 5 6 7 1.. -3.0800 -6.8065-2.1441 -8.1026 1 10“2 l 10‘3 10'3 10'4 5.6373 1.5512 -3.8700—8.4659 -2.9078 3; 10‘3 10'2 10'3 10'4 10' d mu 5 -1.4963 -1.3062 -1.7203 -2.5772 11 10'5 10'7 10'9 10‘11 3 7.5420 7.5360 -7.4272-6.7924 -9.2493 g 1015 10'6 10‘8 10'10 10‘12 l 73 .ho .H u 08:“. .mcowudeDOm udoczcoc co mcmumon mcofloom 1906.3 03.62.»; £33 mason. uo>ofiucmo no.“ movoc: nmocflcoc ponfimfiuoz A125 0053me o; «6.0 m6 >6 pd m.o v.0 m6 _ _ m a m 0:00 won cash. no 38.6.“th 286m 6626? 2:... £23 6:32.41 236m 23.680 666 £23 66306.3 cofioom m mono «sauna—OD .7» 4 .v 2&5 T m.o V3) 91311411di ssawoisuamtq II 8 %( 1111.1!lll‘lll||lllllllbl"tl In‘il' ll.‘ ll 1 2' ) 2 1 Dimensionless Amplitude (6A1 74 — Linear / / Response / / Second Order Response First Order Response % L ; : ; .L 1 J 1 5 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 Dimensionless Frequency 00 Figure4. 1.3-2. Frequency-amplitude response curves for cantilever beam with constant cross section. I!Ilill.‘|".dll“¥ll(ll )5 2 11 Dimensionle s s Amplitude (6A __; ! 75 Linear Response Second Order Response ' First Order Response l l l l L l A 1 r T v .9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 Dimensionless Frequency (0 Figure 4. 1. 3-3. Frequency-amplitude response curves for cantilever beam with variable cross section as des- cribed in Example 1. ,5 2 1 Dimensionle s s Amplitude (6A1 76 \ Second Order \/ Response Linear /-Response /—— First Order Response 1r 1 l 1 l l l l W— v— 1 f 3.0 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Dimensionle s 3 Frequency 0) Figure 4. 1. 3-4. Frequency-amplitude response curves for cantilever beam with variable cross section as described in Example 2. 1 § ) 2 11 Dimensionless Amplitude (6A —’1,fi 2.4 77 Linear Response Second Order Response First Order Response >1... 1 L 4 J J j l I f V f 2.5 2.6 2.7 2.8 2.9 3.0 3.1 3.2 Dimensionless Frequency (.03 <1- Figure 4. 1. 3-5. Frequency-amplitude response curves for a cantilever beam with variable cross section as described in Example 3. 78 In order to include examples Of variable section beams involv- ing trigonometric solutions rather than the Bessel functions, let us consider the nonlinear vibration Of beams with cross sectional areas, moments of inertia and displacement dependent restoring functions that vary in an exponential manner . The nonlinear partial differential equation of motion is taken as equation (3. l. 3. 1). By making the vari- able changes —— 4 'uzLE t’wt 01=K0L Aon‘BX/L 4 E117 O I---'I.0ex/L 4 2 2 _gEI17w x={—; w — 0 4 ean2 K=K0ex/L yAOL The dimensionless equation Of motion follows as 2 2 a x/L azu Zx/L 5u+ x/L+ (x/L3=o 1.216 v. We 3:2" “1"“ ““16 .. The linear frequencies and mode shapes as given by Suppiger [29] are extended to include foundation terms. They are 2 _ 3.12 x V1(x) = Ge. 77- [cos h 3'17 X - 1.115 sin h 31'117 x - cos 31,109 x + 7.43 sin 3&09 x] 79 for a simply supported beam, and w: = a +£4.1773)4 -_x. V1(x) = Ce 2" [cos h 4,175 x - 0. 98 sin h 4;,75 x + 0. 99 sin 45711: 4. 71 ] -cos x for a clamped-clamped beam. 7' The first order nonlinear frequency for the simply supported system follows from equation (3. l. 5) upon integration as 2 _ 2 2 w — “’0 + 0.3578(An and for the clamped-clamped beam 2 _ 2 2 w — (.00 + 0. 4325 (All where the amplitude A11 is again defined in the normalized sense._ These results are plotted with015 l in figure 4.1. 3-6. 1 '5 ) 2 11 Nondimen s ional Amplitude (EA 8O Clamped- Clamped _.._. Simply Supported 1! A I L l l I v I v v y I v 1.0 1.1 1.2 1.3 1.4 Frequency Ratio (51 ) 0 11 Figure 4. 1. 3-6. First order amplitude-frequency curves for beams with exponential varying cross sections resting on non- linear elastic foundations. 81 4.2. Numerical Solutions for Continua 1'1an Immovable Supports and Large Amplitudes of Vibration. 4.2. 1. Elastic Beams with Immovable Supports Numerical results for the nonlinear problem of a uniform beam with immovable end supports are given in this section. The system is described in section 3.2. l and the dimensionless equation of motion is taken as equation (3.2. l. 5). Linear frequencies and eigenfunctions are again taken from Wylie [ 25] for simply supported, clamped—supported and clamped-clamped boundary conditions . The first order nonlinear frequency-amplitude relation is given by equation (3.2.1.12) as 2 3 2 " ’7 2 “’1 : 'ZAIIIO VlVl,xxdx‘fo (V1.11) d" (3.2.1.1.12) Upon substituting the linear eigenfunctions and performing the integrad tions on the computer one obtains 2 2 2 w - coo + 1.17809 (A (4.2.1.1) for the simply supported, (.02 = w?) + 1.407486A2 (4.2.1.2) 82 for the clamped-supported, and 2 012 = «1(2) + 1.45168 (A (4.2.1.3) for the clamped-clamped end conditions. The amplitudes are defined as before, where A = All ‘Vl ‘max (2. 3.4) i. e. A is the maximum dimensionless deflection and A11 is associated with the normalized eigenfunction. It is to be noted that the product 6 A2 is equivalent to 31517 multiplying the dimensionless ratio of the deflection to the radius of gyration of the beam cross section. These first order results are identical with those‘of Evensen [10]. The nonlinear mode shapes are determined upon-substituting higher modes and corresponding frequencies of the linear problem into equation (3. 2. l. 15). The amplitude parameters are given in tables 4. 2. 1-1, and .4. 2.1-2. The mode shapes are graphed in figures 4. 2. l-2 and 4. 2. l -2. Second order frequency-amplitude results follow directly from equation (3.2. 1.19) and tables 4. 2. 143, 4. 2. 1-4 and 4. 2. 1-5 as . 402 = 403 + 1.17809 6A2 - .0192765 €2A4 (4.2.1.4) for simply supported, 2 2 2 + 1. 40748 (A. 4 - .0119564 (ZA (4.2.1.5) 83 for clamped - supported, and w3 = 4003 + 1.45168eA2 - .00602962 (314‘ (4.2.1.6) for clamped-clamped end conditions. These results are plotted in figure 4.2. 1-3. There are no second order approximations available for comparison. Table 4. 2. 1-1. Nonlinear amplitude parameters for a uniform beam clamped-hinged with immovable supports. n = spatial function Ar(nlr1 i amnAll3 m = time function Ag: = dmnA3 amn 1 Z 3 4 5 6 7 l. 6373 3. 0874 9. 0027 3. 3543 l. 4718 7. 2679 l 10'2 10’3 10'4 10'4 10"4 10‘5 1.7423 3.4531 1.2534 3. 1934 1.1477 4.9697 2.4399 3 10'2 10'2 10'3 10'4 10'4 10'4 10'5 -2.5514 -2.0765 -2.6l68 -4.2139I-7.9930 -l.7074 10 10 '11 10'13 10’14 -6.lO97 -5.3810 -8.4298 -9.2823 -l.4418 -2.6989 -5.77319 '5 10'8 10'10 10 10'13 10'15 Table 4.2.1-2. 84 Frequency numerical values for a uniform beam with simply supported immovable supports . "LEI/INgfiiER LINEAQR fREQ. FIRST QZRDER «3‘1 SECONP OZRDER 03% 1.1.1 wozn A A 10- 1 2 1.17809 2 16 1.92765 3 81 1.92765 4 256 1- 1392765 5 625 1.92765 6 1296 1.92765 7 2508 1.92765 Table 4.2.1-3. Frequency numerical values for a uniform beam with clamped-supported immovable supports. LINEAR LINEAR FREQ, FIRST ORDER (.0? 'SECOND ORDER—63: MSDE «134%? A7- 4410'2 1 91: 3.9266023 1.40748 2 72: 7.0685827 - .886755 3 13:10.2101761 -1.111530 4 1g=13.3517688 -1.167424 5_#_ 3g=16.4933614 -1.185525 6 3%;19.6349541 -1.192560 7 97:22.7765468 -1.195649 85 Table 4. 2. l-4. Frequency numerical values for a uniform beam with clamped-clamped immovable supports. LINEAR INEAR FREQ. FIRST ORDER of SECOND ORDERwa MOIDE “62: (Hr—)4 A2 A4110” 8—1 1 91: 4. 7300408 1. 45168 2 )5: 7. 8532046 5. 695165 3 )3: 10. 9956078 -4. 837814 4 34:14.1371655 -4. 837814 5 yg=17.2787596 -5.839504 6 36:20 4203525 -5.839504 7 77:23. 561945 -6. 029620 Table 4. 2.1-5. clamped-clamped with immovable supports. Nonlinear amplitude parameters for a uniform beam 1 3 n = spatial function A1112, = amnAn m = time function A9121: dmnA3 ‘ amn '1 6 7 m 1 2 3 4 5 —3.6299 6.3668 -2.3333 7.9605 1.1252 1.84453 1 10'7 10'3 10‘8 10'4 10' 10 9.4495 5.6967 2.9627 -8.6565 2.7790 3.82991 6.23%3 3 10"3 10‘7 10‘3 10'9 10‘ 10’ 10 dmn -2..2138 1.6753 -2.6538 3.9139 2.386% 1.70:): 1 10'10 10'7 10‘14 10'11 10' 10- 1.2856 3.4743 7.7960 -9.8456 1.3663 8.1422 5.7428 - - - ‘ -18 -15 3 10'4 10‘10 10 8 10 15 ‘ 10 11 10 10 86 .«coSOumammU O» mucoamouuou ovum—“38¢. .muuommsm. “639588“ bongo? uvomaflo 5?» Emma Euoflcn a ROM moans? oboe unvoicoc van .3254 A; .N .w 0.3th Adfinv 0093me DJ 0.0 w.o 5.0 0.0 m.o v.0 m.o ~.o H6 P _ di- “F — 4 « :v .o OEMH km .0 METH I hm OFFH. l ovoz AOOCZCOZ l .l I «.602 ummcmd a( 3V3) apnindmv SS aluotsuamiq 87 .muvsfiaam Owufl um mcmumun? was Oeuuommnpmuvonrhflo ammo, 5.83:5 m “Om Omanm ovOE UONMHOEHOZ A1: xv 00:3me .N; .N .4 253m OH 0.0 mo 0.0 m.o «4.0 Md «.0 H6 \ \\ \ \ \ / \\ \ / \\ \ // \\\ \ / \\ \ \\ II \ th 06TH \\ / \ hear“... oboe/H 235302 I I I 352 .53ch V3) 91313111di S S 91uoisusmtq II E. 8 T ( a)? Dimensionle s s Amplitude (EA 2. 2.6— 2.4 2. l. l. l. l. l. O. O. 2 8 6 0.4 O. 2 88 .- Simply Supported _ \_ Clamped-Supported — Clamped- Clamped First Order Perturbation Results Identical with Evensen's I- — Second Order Perturbation Results W l l I l A l T F I I 1.'0 1.1 1.2 1.3 1.4 1.5 Frequency Ratio (w/ 04)) Figure 4. 2. 1-3. Frequency-amplitude curves for beams vibrating with large amplitudes and having various boundary conditions. 89 4. 2.2 Vibration of Circular Plates at Lag Amplitudes The numerical solutions contained in this section pertain to axisymmetric clamped and simply supported plates vibrating at ~ large amplitudes as described in section 3.2.2, where the dimen- sionless equation Of motion is given as equation (3. 2.2. 5). Information on the corresponding linear problem is taken from reference [26]. For a clamped axisymmetric plate with zero initial radial tension the frequency equation is Jn+1 (Y) + In+1(Y) = 0 Ju M 1:104 (4.2.2.1) 4, and for the simply supported plate the frequency equation is taken as Jn+1(7) + In+1(7) 2 My) Inm 1'” <4-2-2-2> The corresponding linear eigenfunction for both boundary conditions follows in the form Vn(r) = C [1.1mm (vnr) - Jowna) Iownr) ] (4. 2 . 2. 3) Notice that Poisson's ratio appears explicitly in the frequency equation for a simply supported plate, but not in the linear frequency equation for a clamped plate. Consequently, clamped plate vibrational response is independent 0f 1) .but this is not true for the simply supported plate. As pointed out by Berger [20] in the case Of the static analogue, 90 neglecting the strain energy due to the second strain invariant can be interpreted as neglecting part of the variation of the deflection caused by a change in U . Cognizant Of the linear results for the problem, the first order approximation to the nonlinear frequency follows from equation (3. Z. 2. 9) as 2 (2) + 3.28138 6A2 (4.2.2.4) for the clamped plate, and a? = 003 + 3.59788 6A2 (4.2.2.5) for the simply supported plate. The amplitude parameters are computed according to equations (3. 2. 2.11) and (2. 2. Z. 12) and appear in table 4. Z. 2-1 and 4. 2. 2-2. The mode shapes for both boundary conditions are plotted in figure 4. 2. 2-1. Substitutions of the linear frequencies and eigenfunctions into equation (3. 2. 2. 15), along with the already computed amplitude para- meters yields the second order approximation to the nonlinear frequency as 4 002 = 013 + 3.28138 6A2 - .0325862A (4.2.2.6) in the case of a clamped plate, and (.02 = «1(2) + 3.597886A2 + .0257ZEZA4 (4.2.2.7) for a simply—supported plate. In order to compare these results with those Obtained by Wah using a modified Calerkin approach, the dimen— sionless amplitude is plotted against the ratio of the nonlinear period 91 to the linear period in figure 4. 2. 2-2. Tables 4. 2. 2—3 and 4. 2. 2-4 con— tain the nonlinear results corresponding to the order Of the linear eigen- function used in the series of equation (3. 2. 2.15). Table 4. 2. 2-1. Nonlinear amplitude parameters for a clamped circular plate vibrating at large amplitudes. (1) _ 3 n = spatial function Am“ - amnAll : ' ' (1) _ 3 m t1me function n _ dmnA E q =— amn n 1 Z 3 4 5 6 7 m -l.2213 1.9440 -5.2062 1 10'2 10'3 10'4 1.4371 ~9.3599 7.2529 -l.7955 3 10' 10'3 I 10'4 10'4 Table 4 . 2.2-2. 92 ported circular plate vibrating at large amplitudes. Nonlinear amplitude parameters Of a simply sup- (1) 3 n = spatial function Amn = amnAll .. . - l m - time function AInII _._ dmnA3 amn E nl m l 2 3 4 5 6 7 2.9392 -3.2279 1 10'3 10‘4 4.3281 1.2666 -l.1155 3 10‘2 10'3 10‘4 Table 4.2.2 -3. Frequency numerical values for a clamped circular plate vibrating at large amplitudes. LINEAR LINEAR FREQ FIRST ORDER 031’; SECOND ORDER??— “‘3.“ 93 = 711 As 4:. A‘ A: 1 71 = 3.1961 3.281.38 35.7989 2 72 = 6.3064 -.O36912 -4.35053 3 Y3 = 9.4395 -.031483 -3.70157 4 74 = 12.577 -.O32580 -3.84009 Table 4.2.2-4. Frequency numerical values for a simply supported plate vibrating at large amplitudes. LINEAR LINEAR FRE FIRST ORDER 0:? SECOND ORDERw§ 14 11E 2.. 4 (r). “b “ 7n A1; A5 A“ A1: 1 ‘n .=2.22 3.59788 25.2322 2 7% = 5.45 .026509 1.30368 3 73 = 8. 61 .025724 1.26519 93 54 .o n 083 comumnnfiw mo movzufimgm smug n3“? mega 35:9, MOM mcomumusgmcoo OUOE @0335qu .HIN .N .v 9",:th 3602 umocficoz 0632 Hamid .o h. 41- 63 .5 HOHEOO Eouh 00:3me 06 m.o v.0 m.o Nd ad 4? i- n n . _ 1 . / S .o u 3 // 3.3% UOqummflm >Hm8mmI I / // 3 mo 6.63; :3 / 33m UOQENHO / .. mod III. 04 iIZI/IJVB) apniudmv ssatuoisuamiq I NONLINEAR PERIOD LINEAR PERIOD 94 l. 0 ‘ Clamped Plate (All Values Of V) \ \ \ \\ 0. 9 T \\ \ \ \ \\ \ Simply Supported Plate 0. 8 v (v = 0. 3) \ \ \ \ O. 7 " \ \ \ __ _ Wah's Galerkin Solution \\ First Order Perturbation \ .. . _ _ _ __ Second Order Perturbation \\ 0. 6 \ 0T2 0'.4 0.6 018 1'.0 AMPLITUDE THICKNESS Figure 4. 2. 2-2. Ratio nonlinear-linear period vs ratio amplitude-thickness for circular plates with various boundary conditions. 95 4. 2. 3. Vibration Of Membranes at Large Amplitudes The following numerical results are Obtained through the expres- sions deve10ped in section 3. 2. 3 describing large amplitude vibrations Of circular membranes. Using the linear theory from reference [31] and substituting into equation (3. 2. 2. 9), one Obtains the first order fre- quency - amplitude relation (.02 = 00(2) + 3.375 6A2 (4.2.3.1) The nonlinear modal constants are found from equations (3. 2. 2.11) and (3. 2. 2.12). Numerical values are given in table 4. 2. 3-1 and the mode configuration at different times are plotted in figure 4. 2. 3-1. By equation (3. 2. 2. 15) and after some manipulation the second order frequency-amplitude relation is L02 2 403 + 3.3756A2 + .060962A4 (4.2.3.2) The relationship between the amplitudes is again A = All \vl [max (2.3.4) where A is the maximum deflection of the linear mode, )Vl I'max is the maximum value of the normalized eigenfunction, E the perturbation para- meter and All the amplitude associated with the normalized linear mode. In order to compare the results with those of Chobotov and Binder [8] the ratio of the nonlinear to linear period is plotted against a dimension- less amplitude parameter defined as 2 A2 _ 6WO Eh 4—2 a “’0 PII'”) (4.2.3.3) 96 where in the notation of Chobotov, W0 is the maximum central displace— ment Of the nonlinear mode. Figure 4. 2.3-2 indicates exceptionally good agreement even at large amplitudes. 97 Table 4.2. 3-1. Frequency numerical values for a circular membrane vibrating at large amplitudes. LINEAR ngEAR FREQ FIRST ORDER w? SECOND ORDERwa MODE w : Y9 2 a 4 4 n 9 n A A11 A All 1 ‘n .22.404 3.3756 25.0323 2 72 =:5.520 .06219 3.42018 3 73 ==8 654 .06142 3.37775 4 74 = 11.792 .0609? 3.35286 Table 4.2. 3-2. Nonlinear amplitude parameters for a circular mem- brane vibrating at large amplitudes. n : spatial function A1912, r: amn ll m : time function Ag; : dmnA3 amn m n 1 2 3 4 5 6 7 1.9470 -2.5605 -l.7831 l 10‘4 10’4 10‘4 1.8046 -7.4388 «2.5783 -9. 1010 3 10“1 10"5 10"4 10‘5 .movsuZQEN omnmfi no“? $53.3? wcauagme m MOM mcomfimuamflcoo owed: wouflmauoz .Tm .N 2v 9%:th AA\ NV heucmu Eouh 0053me o .H o .o w .o h .o o .o m..o v .o m .o N .o a .o I I. I II“ " # m .r # fl Jr / I I I I I I I I / I I I I I I I I / I I II / / G / w. / 10.." 9 / 4 u / Bio 68E. m. / m / m. / S / S / . m8” 0 N V mm // o o :H m x .m. // w. / ) / 11 3 IIII o m IV 352 hmchcoZ II I I I J ribs 3. GU02 umocmd T NON LINEAR PERIOD LINEAR PERIOD 1.0 0.8‘ 0.6 0.4 0.2 3 99 — - — Chobotov and Hnder's First Order Perturbation of Couples Equations Equations _ __ _..Second Order Perturbation of Uncoupled Equations 2. 0 4. 0 6. 0 8. 0 10. Dimensionless Amplitude X Figure 4. 2. 3-2. Ratio of the nonlinear to linear period vs nondimensional displacement for vibration of a circular membrane with large amplitudes. First Order Perturbation of Uncoupled 100 4. 3 Numerical Solutions for Continuous Media Havirg Nonlinear Constitutive Equations. Several different types of dynamic systems including plates, membranes and beams with nonlinear elastic materials are solvable by means of the general theory presented in Chapter II. However, in order to illustrate the numerical procedure, only application to beams as developed in section 3. 3 is considered. The motion is as- sumed to be defined by the dimensionless equation of motion (3. 3. 4). The necessary results on the linear problem for beams with various boundary conditions are contained in Wylie [25]. Substituting these results into equation (3. 3. 7) and performing the integrations on the computer, one obtains the first order frequency-amplitude relations as I 012 = mg + .18750 (A2 (4.3.1) for the simply supported beam, 0.12 = on?) + 1.1042 (A2 (4.3.2) for the clamped-supported, and 2 (.0 = w 2 (2) + 4.8474 (A (4.3.3) for the clamped-clamped beam. The amplitude A has been previously defined as the maximum deflection of the linear mode. After truncating the series in equation (3. 3. 8) to include the sixth linear eigenfunction, the amplitude parameters follow as in tables 4. 3-1, 4. 3-2, and 4. 3-6, where A11 is associated with the normalized eigenfunction as before. 101 Second order nonlinear frequency-amplitude relations follow from equation (3. 3. 9) as 2 2 4 O) = to: + .18750 (A - .01186562A (4.3.4) for the simply supported case, (.02 = (.03 + 1.1042 (A2 - .383176213.4 (4.3.5) for the clamped-supported, and 0.12 = 402 + 4. 8474 G A2 - 4.12345€2A4 (4. 3. 6) 0 for the clamped—clamped beam. Kauderer [32] has approximated the case of the simply supported beam by assuming a particular solution of higher harmonics and equating coefficients. The first order frequency relation obtained here is identical with his approximation. Figure 4. 3-4 shows the frequency response for all three boundary conditions. Figures 4. 3-1, 4. 3-2, and 4. 3—3 are graphs of the nonlinear mode shapes compared with the corresponding linear configuration at different times. It is interesting to note how the nonlinear contribution changes the algebraic sign during a period of vibra- tion and the maximum deflection of the clamped—supported beam changes position along the span at different times. 102 Table 4. 3-1. Nonlinear amplitude parameters for a uniform beam simply supported and having the nonlinear constitutive equation as N = E (6 + bee) _. - . (1) __ 3 ’ n - spatial function Amn - amnAll I. m = time function A“) = d 6A3 3-- mn mn F"” a ’ L mu 1, I S n a ; 3&1 1 2 1 3 4 1 5 I 6 7 g 0 0 4.4g62 I 3 1 . 10- 0.0 0.0 I 0.0 5 3 4.9735; 0 0 1.6578 0 0 0 0 f 0 0 1 I - O O O i 10-3 I ' 10 3 3 I I 3 JP '" I __ mn i 1 7.03uL’ ‘ I I 0 0 10.3 0.0 0 0 0 0 1 g I 7.8125 2.6041 ; '3 -3 0-0 -3 0.0 0 0 0 0 1 I 10 10 1 I Table 4. 3-2. Nonlinear amplitude parameters for a uniform beam clamped-supported and having the nonlinear constitutive equation as N = E (6+ hi3) n = spatial function A”) = a A3 mn mn 11 1 m = time function A”) _ d . 3 I - _ mn mn j a. a 3 mn n,“ 1 2 3: 4 I 5 I 6 I 7 -2.1208 -5.1031 -1.0420 -3.03433-1.2659f - - I I 7 1 10 10‘.3 10’2 . 10'3 i 10‘3 a 1.3670 -4.4729 p2.0716I-3.6963'-1.0382;-4.2744i 3 -2 -2 43 3 -3 é -4 E 10 10 I 10: 10 10 3 10 s ' 8 mn _‘ _........_. , _ 11 I 3 3048‘ 3.432? w3.0288.'3.8111I 6 8748‘ 1 ' E 10 10 I 10 . ‘-4.7938( 6.9701 1.3933 1.07441 1.3039; 2.3213 ‘ . 3 I -4 , -5 -7 -8 ~10 . -12 I 1 . 10 1 10 10 10 10 1 10 , é Table 4. 3-3. 103 Frequency numerical values for a uniform beam simply supported and having the nonlinear constitutive equation as N = E (E + his) .- leNEAR LINEAR FREQ. FIRST ORDER Aw? * SECOND ORDER (0% ' MgDE w§=n‘ : A2 A11 .4410'2 21111110"3 1 1 g 0.18750 0.1194 Q 2 16 I 3-1.46484 .593678I 3 81 i F -1.18652 -4.80879“ 4 256 -1.18652 -4.80879 5 625 , -1.18652 -4.80879 6 1296 ‘ -1.18652 -4.80879 Table 4.3-4. Frequency numerical values for a uniform beam clamped-supported and having the nonlinear constitutive equag tion as N = E(€+h€3) ELINEARILINEAR FREQ. FIRST ORDER «if % SECOND ORDER mgr“; I MEDE I ”53%“? I A2 A121 A410"I 1471*110‘I - I 1 ‘ 71 — 3.9266023I1.1042 0.8006? 2 172 = 7.0685827; } -.805985 -.423622. 3 I73 :10.2101761I ; -.990191 -.520440‘ 4 £74 :13.3517688I 1 -3.23279 -l.69914 5 575 =16.4933614I '- -3.67639 -1.93229' 6 "376 =19.6349541§ i -3.83l78 -2.01396§ 104 Table 4. 3-5. Frequency numerical values for a uniform beam clamped- clamped and having the nonlinear constitutive equation as N = E (6+hé3 ) LINEAR LINEAR FREQ. FIRST ORDER 915 SECOND ORDER"~!a2 MODE 2_ ‘Y 4 fl n “’6 ‘ (1,9) A2 A3 A4 A‘ 11 11 1 71: 4.7300408 4.8474 3.8953 2 72 = 7.8532046 .19074 .12294 3 74 =10.9956078 -1.26516 -.81547 4 9; 14.1371655 -1.26516 -.81547 5 3g = 17.2787596 -3.54851 -2.28723 6 7; 20.4203525 -4.12345 -2.65782 Table 4. 3 -6. Nonlinear amplitude parameters for a uniform beam Clamped-clam ed and having the nonlinear constitutive equation as N = E (e + he 3)) . . (1) 3 n : spatial function Amn = amnAn m = time function Arlilrl -._-. dmnA3 a’mn “ 1 2 3 4 5 6 7 m 2.7386 -4.3225 8.5521 -2.1976 -7.8798 1 10‘6 10'2 10‘7 10“2 10‘3 3.1573 -4.2980 —2.0114 3.1728 4.67%9 —2.68837 3 10‘2 10‘6 10‘2 10‘7 10’ 10' drnn 1.6702 -l.1374 9.7250 -1.0799 -7.2610 1 10"9 10‘6 10’13 10'9 10'13 4.2956 --2.6212 -5.2927 3.6079 -3.7700 ~2.4775 3 10'4 10‘9 10’7 10'13 10‘10 10'13 105 _Time : O. 0 Linear Mode _ _ _ Nonlinear Mode .1162 \ z 0. \ f5 \ I \ E \ ”a 0.4 I Time=0,21T \ E < \ \ (D . \ \ x g \ 5, 0 3 ~ Time=0.411\ \ E .5 “ \ Q - - ~ \ \ \ \ \\ \ 0 2 r \\\ \ \ \ 0.1“ 0.5 0.6 0.7 0.8 0.9 1.0 Distance (x/L) Figure 4. 3-1. Normalized mode shape for uniform beam simply supported and having a nonlinear constitutive equation of N = E (c + hez). A «w: + 3 H M Z “53.250 ozgfiumcoo Hdocficoc .m muffins was pouuommsmupomemao Econ 8.83“; m new ommnm 0608 pmnfimdfluoz .Num .v 0&9th AA\ 5 mucoumwfl 106 0.0 wd v.0 ed m.o ¢.o md Nd To 0 0 4 n . ,\ \ // \ \\ / \ \ / \ / \ \ / \ / \ \ / \ / \ \ / / \ / // x. \ x / \ / \ \ / 54 u 085, \ / \ / k n cash. \ / . \ / \ / \ / \ / \ / £52 Hmocficoz I..l.ll / \ Illl: / \ mpoz nmwsmq // \\ , V3) 9pn411dwv s s aluoi s uamiq II 8 g T 107 A mu 3 + 3 H M Z ~53.»an oEfiSEmcoo nmocflso: m magma, pad pomEdHoupvamHu Ewen 8.8.3”; N no.“ 093m upon» ©9338qu .Mum .v vufiwmrm A]: 5 00:3me .H omo m5 5o 9o mno Yo m5 Nd To b b b L h p b s A _ A _ _ q kw .o 08TH. In RN .o 08TH o .0 08TH. “,5on umocflcoz I I I .352 .3354 III: / \ l. V3) epnzntduxv s s awoysuswiq II 8 §( 1: D imen s ionle s s Amplitude .0J .24. 108 Clamped- Clamped _ Clamped-Supported Simply-Supported First Order Identical with Kauderer's Solution First Order Response _. — — Second Order Response -’\/~ + 1r : s 1.0 1.1 1.2 1.3 1.4 Frequency Ratio (JD/0L)o Figure 4. 3-4. Frequency-amplitude curves for uniform beams having a nonlinear constitutive equation of the form N = E (c + h (3). 109 V. A GENERAL SOLUTION USING ULTRASPHERICAL POLYNOMIALS 5. 1. Introduction In this chapter a method is presented for determining the amplitude- frequency relations for a class of nonlinear continuous systems under- going periodic motions. The method applies to systems governed by nonlinear partial differential equations in one space variable and one time variable, in which the nonlinear terms are assumed due to non- linear forcing functions which depend on the displacement and its spatial derivatives, but do not depend on time explicitly. Recently Denman et a1 [33, 34] developed a method for treating nonlinear vibration problems with one degree of freedom. By lineariz- ing the nonlinear spring forces using a set of ultraspherical polynomials over the interval over which the motion takes place, they were able to obtain approximate amplitude-frequency relations with fair accuracy. Some attempt has also been made to extend the method to systems with two degrees of freedom [36] . In this present chapter, the method of ultraspherical polyno- mials is extended to nonlinear continuous systems of the type described above. An obvious difficulty immediately arises because the maximum displacement varies from point to point and these maximum displace- ments are not known in advance. To overcome this difficulty and to achieve the linearization of the nonlinear forces, one must initially assume some appropriate “mode of deflection. " In cases where the 110 linear mode of small vibration is known, this linear mode is taken to be the mode of deflection. Otherwise some suitable approximation to the linear mode has to be used. Next, an amplitude parameter is introduced so that the maximum displacements are determined by the product of the amplitude parameter and the normalized linear mode. Using ultraspherical polynomials, the nonlinear force at each point is replaced by a force that is linear in the displacement. This results in a linear partial differential equation. Together with the initial and boundary conditions, one is thus led to solve a linear eigenvalue problem and it is the determination of the eigen- value that leads to the desired amplitude-frequency relation. In order to illustrate the procedure, the general form of the nonlinear equation of motion is linearized and a frequency- amplitude relationship is established in section 5. 2. The developed expressions are then applied in section 5. 3 to typical systems of strings, bars, circular membranes and plates on nonlinear founda- tions, vibrating with large amplitudes, vibrating with immovable end supports, or consisting of nonlinear elastic materials as described in the previous sections. It is found that for these cases, if the Tchebycheff polynomials of the first kind (a special case of the ultraspherical polynomials found by setting A = O) are used, the frequency results agree exactly with those of the first order pert- urbation solutions. In addition to the general solution of the linearized equation of motion by eigenfunction expansions, a uniform string on a Duffing type foundation is solved by reducing the linearized equation to the form 111 of the well known Mathieu equation, and a circular membrane supported by the same type foundation is solved by the application of Hankel transforms. The latter two procedures simply suggest that once the nonlinear equation of motion has been linearized, several approaches could be used to obtain the solution to the linearized problem, de- pending on the type of physical dynamic system. 5. 2. Linearization of a Class of Nonlinear Equations of Motion and a CorrespondingFregency-Amplitude Relationship Consider the following dimensionless form of a nonlinear partial differential equation of motion 2 L u + 00 + = x utt qu 0 (5'2°1) where Lx denotes some linear differential operator of degree Zn in the spatial variable x. Nx is an Operator such that qu denotes a nonlinear restoring function with the pr0perty that > uNXu _ 0 (5.2.2) for all u, and is assumed representable in the form on - 2n+l qu — 2012!?”- u (5.2.3) n=1 where ‘1an is either constant or at most a functional of the associated linear spatial eigenfunctions or derivatives thereof. qu must be mathematically well defined at all points along the continua. .WE . i ii '1- t. I ‘ at ‘ 112 Eventually periodic solutions u(x, t) = u(x, t+27r) are sought that satisfy 2n time-independent homogeneous boundary conditions of the form ll O H II Diu(0,t) 1,2,... p (5.2.4) Diu(L,t) II 0 H II ptl, ....2n where Di are linear differential operators of order \,A) Pl (AVi(x)) (5.2.9) where P1A is the linear ultrashperical polynomial of degree A , , and g(x,>\.A) is some function yet to be determined. In order to determine g(x,A,A), equation (5. Z. 9) is multiplied u 2 X45 X u [1 - (9Vi(x)) ] P1 (5Vi(x)) (5°2°10) and integrated with respect to u from -AVi(x) to AVi(x), x being treated as a parameter. It then follows that 114 . A- I.AVJ'b‘lil u 1 _(_____u ) 2] 15P A _____u ) du -AVi(x)x AViIX) 1 AViIx’ ghh’hA) = +AV. (x) - 2 X 45 I-.. .. ,[1- (r: {—7)} [P (.————. (.01 .. (5.2.11) which can be reduced to 1 2 A -% IONXSAVi [l - s] P: (s ) ds g(x.A.A) = 1 (5.2.12) I, [1 — .2 M53912 .. upon setting 3 = u/AVi(x) and using the fact that both qu and P1(u/AVi(x)) are odd functions of u. When the terms in the series expansion for qu given in equation (5. 2. 3) along with the normalized P}; (s) * written as A Pl (3) = 2A3 (5.2.13) are substituted into equation (5. 2.12) one obtains Appendix A contains a more complete discussion of ultraspherical polynomials and normalization. 115 1 A-k - 2n+1 2n+2 2 a2n+1(Avi) IO 5' [1 - S 1‘ d8 92n+1(x.A.A) = ZA [1 [l - 32] X-% 32 ds 0 (5.2.14) for nonnegative integer n. After making the variable change 3 g ylV/Z and some manipulation, the integrated result follows as - 2n+1 9211+]. I I " 2A 0 o X) - 1/2 where the constant H2n+l are given in terms of Gamma functions as = P(n+§(2) L§A+2) H2n+1 r(3/2) r.+2) (5.4.1.5) 127 Equation (5. 4.1. 4) will now be recognized as the Mathieu equation, and the solution which vanishes at x = 0 and x = 75 and reduces to sin x for small q (or A) occurs when .. ___.1_2_1.-_3 p — b1(q) _ 1 q 8 q + 64 q ... (5.4.1.6) The corresponding solution of X(x) is given by X(x) = se1(x,q) = sin x - % sin 3x (5.4.1.7) q2 [Si___l‘_1___ 5x sin 3x _ sin x + 192 64 128 Both functions b1(q) and se1(x, q) are plotted in [37] . Combining (5. 4.1. 6) and (5. 4.1. 5) one obtains the following amplitude-frequency relation A2 2 2 2 .2 9 “1A 9 (51A ) 1 + 8 (1+2) 512 ($1277— + "’ (5'4'1'8) where Q is the fundamental frequency for the linear problem 1 2 91 _ 1 + a1 (5.4.1.9) Stoker [l]has given the first order perturbation solution for this same problem. Except for some differences in the notations he obtained 128 2 2 _9_ 2 a) — $21 + 16 alA (4.1.1.3) which coincides with the first two terms of the results: given in (5.4.1.8) if one sets A = 0, i. e., if the Tchebycheff polynomials of the first kind are used. ' P”! 5. 4. 2. Integral Transform Methods The frequency-amplitude relations for nonlinear systems can also be obtained by applying integral transform techniques to the linearized equation of motion. For example, the nonlinear vibrating string of the previous section, or likewise a simply supported uniform beam, could be investigated by using the sine integral transform. In order to illustrate the general procedure, the Hankel transform is used in this section to predict frequency-amplitude relations for a nonlinear vibrating membrane. The dimensionless dynamic equation governing the axisymmetric motion of a vibrating circular membrane attached to a cubic founda- tion is taken to be 2 2 ‘ a -(au+l§'l)+wzau+au+au3=o rar o 2 r B t2 3 (5.4.2.1) Equation (5. 4. 2. l) is of the form (5. 2.1) if 2 .5— .l. .5. '5 r 0 ‘ and N u = a 11 (5.4.2.3) It is observed that for 013: 0, the first linear mode of vibration is Jo(k1r), where Jo is the Bessel functiOn of the first kind of order zero and k1 = 2. 40483. . . is the first zero of Jo. Introducing the amplitude parameter A one may represent the maximum Hisplacement at r by AJo(klr). The nonlinear force given in (5. 4. 2. 3) is then A _11..__ AJO (kl r) in the same manner as was done in Section 5.1. Thus by equation approximated by the linear ultraspherical polynomial Pl (5. 2. 19) with V1(x) = Jo(k1r) one has BZu +1 6 _w282u _[a + 30‘3 + 6 r2 r B r 6 t2 0 20. 2) '(5.4.2.4) 2 2 A Jo (klr)] u — 0 Upon separating variables by writing sin t u(r,t) — R(r) cos t (5.4.2.5) one finds that R(r) satifies 2 Q—B + $93 + [p + q J2 (k r)] R = 0 (5.4.2.6) 2 1' dr 0 1 dr 2 3a A where p = (112-010 and q =- 2—(3352) '. In addition R(r) satisfies the boundary conditions R(O) = Finite R(l) = 0 (5.4.2.7) 130 It is noted that (5. 4. 2. 6) plays the same role as the Mathieu equation (5. 4.1. 4) does in the string problem. To solve the eigenvalue problem posed by (5. 4. 2. 6) and (5. 4. 2. 7) the method of Hankel transforms is employed. Let 1 Rn = JOrR(r)Jo(knr)dr (5.4.2.8) where kn denotes the nth zero of Jo. Equation (5. 4. 2. 6) is now multiplied by rJo(knr) and integrated with respect to r from 0 to 1. After integrating the first two terms by parts one obtains 2 - 2 - + = (p kn )Rn q I r R(r)JO (klr)Jo(knr)dr 0 (5.4.2.9) To simplify the integral on the left it is noted that R(r) has the following eigenfunction expansion in terms of Jo(knr) R J (k r) R(r) =2“;1 m 2° m (5.4.2.10) (k) Substituting (5. 4. 2.10) into (5. 4. 2. 9) and interchanging the order of summation and integration, one obtains Q (p-k2)R+2an §=O (54211) n n mn m "' m=1 where 1 1 a =————'J rJo (km r)Jo2 (k 1C)r)J (kn r)dr run 2 J1(km) 0 (5.4.2.12) 131 The integrals that appear above have been computed and tabulated by McQueary and Mack in [38] . In order to obtain the desired amplitude-frequency relation it is necessary to determine the relations between p and q in (5. 4. 2.11) under which nontrivial solutions for Rn occur. To achieve this the series in (5. 4. 2.11) is now truncated after m = N. Equation (5. 4. 2. 11) then becomes a set of N homogeneous linear algebraic euqations in Rn, n = l, 2, ...... , N. By setting the determinant of the coefficients equal to zero N general relations between p and q result. The desired relation is the one for which all the nontrivial Rn except R1 should tend to zero as q(proportional to A2) tends to zero. The relations may be improved by taking a larger N. Let us instead describe here an alternative iterative pro- cedure by which the desired relation between p and q may be developed as a power series in q for p. It is observed that by setting N = 1 one obtains from (5. 4. 2.11) and (5. 4. 2.12) the following solution. it = 0 n > 1 , ii 9! 0 (5.4.2.13) n l and p = 1.12 (5.4.2.14) Equation (5. 4. 2. l4) simply states that for linear vibrations in the first mode the frequency a) = 03 = k + a (5.4.2.14) is independent of the amplitude A. Now taking N = 2 one obtains from (5. 4. 2.11) the following second order determinantal equation 132 [(p - 1.12).» 2qall] [(p - 1.22) + 2:21.122] 2 - 4q 412.521 = 0 (5.4.2.15) which may be rewritten as PM” 2 i an (p - k ) p - k2 = -2qa - #224 1 + o14“ the expansion corresponds to a Taylor's series of an analytic function about the origin. 143 144 A function f(x) expandable over the interval [-A,A] in these polynomials may be written (33). f(x) = 2 A: P: (x/A) (A.2) n=O where the coefficients are obtained in the usual way by multiplication of the weighting function and integration over the span as +1 I fiAx)PnA(x/A)(1 - x2)x-;5 dx )1 -1 An = +1 A 2 A‘gi (AO3) j [Pn (x/A):I (1 - x) dx -1 It is important to observe that 50nP3(x/A) is unchanged if Pn(x/A) is multiplied by some constant. Consequently, the normaliz- ation factor Aifrom equation (A. 2) is not unique and any convenient normalization constant is permissible. In this treatise Anxis taken as n -1 + + - Ank ___ (n ) I‘LL 31) l"(n 2;) (AA) 2 n! I"(2>.)1"(n+>(+;5) for all subsets except i=0 . Sincevr(0) is undefined, for the case of the Tchebycheff polynomials for the first kind the constant is taken to be 145 n n I An — (2n)! (A'S) As already mentioned, the approximating function is not altered by redefining An. ‘I1111111111111‘5