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RANDOM VIBRATIONS OF NONLINEAR ELASTIC SYSTEMS By RICHARD EDGAR HERBERT A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA April, 1964 ACKNOWLEDGMENTS The author wishes to express his sincere gratitude to Dr. William A. Nash, Chairman, Department of Engineering Science and Mechanics, for serving as chairman of his supervisory committee and for his constant advice and encouragement throughout the author's entire graduate studies program. He would also like to thank Dr. T. S. George, Professor of Electrical Engineering, Dr. A. Jahanshahi, Assistant Professor of Engineering Science and Mechanics, Dr. I. Ebcioglu, Assistant Professor of Engineering Science and Mechanics, and Dr. R. G. Blake, Associate Professor of Mathematics, for serving on his supervisory committee and for the various stimulating discussions he has held with them over the past few years. Final thanks go to the Air Force Office of Scientific Research for their sponsorship of this program. TABLE OF CONTENTS ACKTIOWLEDGMENTS . . LIST OF FIGURES . . ABSTRACT . . CHAPTER I. INTRODUCTION . . . II. THEORY OF PLATES . . . 2.1. Analysis of Deformation . . 2.2. Equations of Motion . . 2.3. Boundary Conditions . . III. THE RESPONSE OF LINEAR SYSTEMS TO RANDOM EXCITATION . 3.1. Stochastic Processes and Probability Theory . 3.2. Response of Linear Systems . . IV. THE FOKKERPLANCK EQUATION AND ITS APPLICATION TO SOME NONLINEAR LUMPED PARAMETER SYSTEMS . . 4.1. Classification of Random Processes . 4.2. The FokkerPlanck Equation . . V. APPLICATION OF THE FOKKERPLANCK EQUATION TO NONLINEAR ELASTIC SYSTEMS . . . 5.1. General Theory . . . 5.2. Some Special Cases ... . a. Simply Supported Beam . b. Simply Supported Plate . . iii Page ii v vi 1 5 5 13 22 24 24 29 35 35 40 49 49 59 59 63 . . . Page CHAPTER VI. NUMERICAL INVESTIGATION . 6.1. Simply Supported Beam 6.2. Simply Supported Plate VII. CONCLUSIONS . . LIST OF REFERENCES . . BIOGRAPHICAL SKETCH ......... . . . . . . . . . . . . . . . . . LIST OF FIGURES Figure Page 1. DEFORMATION OF A COORDINATE LINE . 6 2. DEFORMATION OF AN ELEVEN[ OF VOLUME ... .. 6 3. MEANSQUARED DEFLECTION AT CENTER OF BEAM FOR SMALL NONLINEARITIES . .... ... .. 72 4. MEANSQUARED DEFLECTION AT CENTER OF BEAM FOR LARGE NONLINEARITIES . . ... 74 5. MEANSQUARED DEFLECTION AT CENTER OF PLATE ... 79 Abstract of Dissertation Presented to the Graduate Council in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy RANDOM VIBRATIONS OF NONLINEAR ELASTIC SYSTEMS By Richard Edgar Herbert April, 1964 Chairman: Dr. William A. Nash Major Department: Engineering Science and Mechanics The principal objective of this research was to develop a method for determination of the response to random excitation of structures having geometric nonlinearities. This would account for large or finite deflections of such structures whereas work done in the past has accounted only for small deformations leading to linearized equations. The method of attack was to expand the transverse deflection of the structure in terms of the eigenfunctions of the linear problem and the longitudinal deflections in terms of orthogonal functions satisfying the boundary conditions. The equations governing the series coefficients of the expansions are then easily derived from the EulerLagrange variational equations and are shown to be nonlinearly coupled. The assumption of uncorrelated loading then permitted the identification of the phase space of the series coefficients with a Markoff process and thereby permitted derivation of the FokkerPlanck equation governing the first order probability density function of the series coefficients. A stationary solution of this equation was obtained. While the solution found is valid for several structural elements with various boundary conditions, a simply supported beam and a simply supported plate were investigated in detail. It was found that the type of nonlinearities considered had the effect of reducing the meansquared response. Furthermore, it was found that the meansqUared value of the first mode still represented a good estimate of the total meansquared deflection of the structure but in contradistinction to the linear theory the effects of the higher modes must be considered in calculating the response of the first mode. vii CHAPTER I INTRODUCTION The rocket and jet propulsion systems of modern air and space vehicles have given rise to a host of new problems in the theory of vibrations. The pressure fields generated from these systems vary randomly in both time and space over a wide range of frequencies. These random pressure fields can cause severe vibrations to the vehicles and their components. The importance of these types of vibrations is evidenced by the fact that in 1958 and again in 1963 a special summer program on random vibrations was held at the Massachusetts Institute of Technology. The two volumes of the book "Random Vibrations," edited by Stephen H. Crandall (1, 2) were outgrowths of these programs. In the first volume it was pointed out that "Some of the strictly mechanical problems are still incompletely understood and the tools for handling them are relatively crude." This statement stands in evidence as to the need for research in this important field. The first satisfactory treatment of stochastic motion was presented by Einstein who studied the random motion of a free particle. Smoluchowski generalized this theory to other types of Brownian motion and since then many important contributions have been made to the theory, notably by Fokker, Planck, Ornstein, Uhlenbeck, Chandraseklar, Kramers, and others. Numbers in parentheses refer to the List of References. 2 On the purely mathematical side, some of the outstanding contributions have been made by Wiener, Kolmogoroff, Feller, and Doob. Generally speaking, the theory of linear lumped parameter systems, i.e., systems governed by ordinary differential equations, excited by stochastic driving functions has reached a relatively high level of sophistication and the technique of spectral analysis developed by Rice (5) is generally adequate for the solution of problems of this type. For nonlinear lumped parameter systems, the theory is not quite as refined. For small nonlinearities an approximation procedure known as "equivalent linearization" has been developed (6, 7, 8). The procedure consists of replacing the nonlinear system with an equivalent linear system. The crux of the procedure is to choose the stiffness matrix of the linear system so that the mean square error of the governing equations is a minimum. When the nonlinearities are not small this method obviously fails. Another approximate procedure applicable to systems with small nonlinearities is the "perturbation method" (see reference 2). The idea in this method is to assume a series solution in powers of a parameter which represents the nonlinear component of the system. A set of linear differential equations governing the coefficients of the expansion can then be generated. These linear equations can then be handled by more established techniques. For systems with nonlinearities which are not small an alternate approach must be made. For a complete set of references see entries (3) and (4) in the List of References. Such an alternate approach exists in the identification of the trajectory in the phase space with a Markoff process. This assumption, together with that of Gaussian input permits the derivation of a partial differential equation, known as the FokkerPlanck equation, governing the probability density function of the response of the system. This concept will be more fully exploited in Chapters IV and V. We mention here only its shortcomings. These are that the assumption of a Markoff process implies white noise input which can be questioned with regard to realizability and furthermore, solutions to the FokkerPlanck equations are known only in a few special cases (9, 10, 11). Despite these shortcomings it appears to be the most fruitful approach for analyzing nonlinear systems excited by stochastic functions. The picture is worse for continuous systems, i.e., systems governed by partial differential equations. The linear theory has been attacked by Eringen (12) who used the generalized Fourier analysis developed by Wiener (13) to the problems of vibrating beams and plates. Other special linear continuous systems have been studied by many authors (1420). Chapter III contains an outline of the standard approach to these problems. As in lumped parameter systems, the method of "equivalent linearization" has recently been used to study continuous systems governed by equations containing small nonlinearities (21). Until now there has been no analysis presented for the problem of continuous systems which are governed by equations containing large nonlinearities. It is the purpose of this paper to show how, with the assumption of a Markoff process and Gaussian input, the FokkerPlanck equation can be 4 used to determine the response of nonlinear elastic plates excited by uncorrelated stochastic loadings. The nonlinearities of the plate which are considered are those arising from geometrical considerations. We still make the usual assumptions of small strains, shears, and extensions compared to unity so that Hooke's law is valid, but we allow the rotations to be moderately large, i.e., small compared to unity but large compared to the shears and extensions. This will be the result when the deflections of a plate are not small relative to its thickness, a situation very common in structures subjected to vibrations produced from jet and rocket engines. CHAPTER II THEORY OF PLATES The theory of the large forced vibrations of elastic plates is presented in this chapter. The equations governing the large vibrations of beams are obtained by properly simplifying the expression for the kinetic potential. 2.1. Analysis of Deformation We start our study of plates with the geometry of deformation. Let the position of a point in an undeformed body at time to be designated by x, y, z referred to some rectangular coordinate system X, Y, Z. After deformation, at time t, the point has moved to a new locationF 7 1 referred to the same X, Y, Z coordinate system. Then we can write = Z+ (2.1) The functions JL I ,UT represent the projections onto the X, Y, Z axes of the displacement at time t of a point which was at the position x, y, z at time to. If in equations (2.1) we set x = xo = constant, y = yo = constant, we obtain the equations (2.2) These are the equations of a line which at time to was parallel to the Z axis. At time t it is some curvilinear line as indicated in Figure 1. Thus, while the x, y, z form a rectangular coordinate system in the undeformed body, they form a curvilinear coordinate system in the deformed body. Therefore, if we speak of a stress 6jL we mean the stress on a Z Fi4. 1. Deformation of a coordinate line. X 1! C b Undeformed Deformed Fig. 2. Deformation of an element of volume. a. Lagrangian coordinate system. b. Eulerian coordinate system. surface which was originally perpendicular to the Z axis and acting in a direction normal to this surface (see Figure 2). On the other hand a stress 5, will be the stress on a surface which in the deformed body is perpendicular to the Z axis and acting in the Z direction (see Figure 2). Having dispensed with these preliminary considerations of our coordinate system, we may now proceed to our analysis of strain. If we introduce the following parameters e =e . y3 + (2.3) I L +4 w = e ( I) W, = CC then the components of strain can be defined as (22) 6^ = e t[e1 +(e(+_w)1 +(e W^ S= ez, = e,+ey e,)' +(I eI, I t = e+tle^1 +j e2+ w+ci( e, L] ee ) + equations (2.4) can be written as (22)(2.4) e, e xzz+ W,) z = + = e .L + ze  We) + e, ez+L4) 4c2 x W, )( e,,+ y) E.x z ei.4 e,,(x ez wzd, ) ^x ,x +( I e, ^ e7y + ,( We now make the assumption that the strains and rotations of a volume element are small compared to unity. The assumption of small strains permits us to use Hooke's Law which is generally not valid for large strains. When we add the assumption of small rotations, equations (2.4) can be written as (22) 4 e L + + e= +2L + C ) (2.5) A further consequence of our assumption is that the quantities (W , L, ~ T can be interpreted as the components of the rotation vector of a volume element. The retention of their squares in equations (2.5) is due to the fact that while these quantities may be small compared to unity they may be large compared to the strains so that their squares may be of the same order of magnitude as the strains. For flexible bodies this is quite often the case. We are now ready to start our analysis of thin plates. We consider that the plate is of constant thickness h and that the XY plane forms For an excellent treatment of these points see reference (23), p. 47. 10 the middle surface of the plate before deformation. Thus, the Z axis is normal to the plate. Therefore, after deformation the xy surface will form the middle surface and the z axis will be normal to that surface. As is always the case in a "Strength of Materials" type of analysis we must assume a form for the displacement field. For a thin plate with moderately large rotations we take, M%^ =r ) JU(YLt,) rL^ a___Ur (2.6) VCrCX, UZ, 't)=T CJCL U)fk As was pointed out by Biot (24) equations (2.6) are those of the von Karman plate theory. They are tantamount to assuming that a straight line originally normal to the middle surface of the plate remains straight and normal to the middle surface after deformation. This is the assumption of classical plate theory. Furthermore equations (2.6) permit a stretching of the middle surface of the plate which in classical theory is omitted. The use of equations (2.6) yields the following results for the parameters given by equations (2.3) 11 p ^1 3 a^2 ev E 3T C) 19 2. e~y = : z Jy aa~ ia a  kL +F2 zj. a^ a4w (2.7) C z = a u? aur z  The rotations in the plane of the plate can be considered fairly small so that in using equations (2.7) to calculate the strain components we neglect U . Thus, making use of (2.7) the strain components become 12 a 2Ur r 4 CIVI 5CY = ,L (in, EIL^ Z 'a I*4 (2.8) where &y La ~% +2 (2.9) The quantities E. 1, E y, * are obviously the components of the strain of the middle surface. + I_(J.. L f 214/ Fu = __L L E =0 I +,la"~ b(itI _ t 4!X au a>< + AA_ __ a )a Y 13 That E.X2 = Ez = 0 points up the fact that we are neglecting the deflections due to the transverse shear stresses, an approximation that is valid in most cases. 2.2. Equations of Motion To derive the equations of motion governing the forced vibrations of plates we employ Hamilton's principle in the form s Kdt =(2.10) where K is the kinetic potential of the system given by K TV (2.11) Here T is the kinetic energy of the system and V the potential. Equation (2.10) states that the displacement field assumed by a system is such that the kinetic potential is an extremum. For an elastic structure V consists of two parts, the internal strain energy Vs and the potential energy of the external loads Ve. Thus, \4 J [.(2 +.12) V (2.12) cs~l~~b~B, +5~ 14 and e ff[ j + ( +i as (2.13) SL where the first expression is integrated over the volume of the body and the second over the entire surface of the body. The quantities E ,L + are the components of the boundary traction in the X, Y, Z directions, respectively. Since in our analysis we have assumed small strains, no distinction has been made between integration over the deformed and undeformed body nor between the stresses and the pseudo stresses d which actually should be used in equation (2.12). For a plate with no shear stresses on top or bottom, equation (2.13) reduces to h h/l. Ve [ff u s A h (2.14) + J + ur) d.td or Ve= (2.15) kit a.4 15 where c%(x.c k H2 (2.16) and the line integral is taken around the entire cylindrical boundary of the plate. We consider only the case when the forces on the cylindrical boundary are the forces of constraint so that the second integral of (2.15) will vanish and we have Ve jy^^ )t) c ) Ms (2.17) The retention of the second integral in (2.15) would lead to the boundary conditions but since these have been established in other papers (25, 26) we will not derive them here. They will be briefly discussed at the end of the chapter. Returning to equation (2.12), the strain energy, with the help of equations (2.8), can be written as s ly r z 2 Z(2.18) +i; ][( +() +dtcs 16 Integration over z yields \V + N I + N, E+ (2.19) S V2Nr 9ur Ur M )0. I ZM ^s where NO = dtLz. M,%. =fz gCZ hlz h12. N,,= az M 7 z I7 (2.20) hli N1= f dz M% = fza6 dz h/2. and we have assumed k/l S z =0 h/i The kinetic energy of an elastic plate is given by T f( + + d (2.21) V 17 or after substitution of (2.6) and integration over z T= rh r+ ] + S(2.22) where p is the mass density of the plate and the dot indicates differentiation with respect to time. The second contribution to the integral is due to rotatary inertia. Since this only effects the higher modes of vibration we will disregard it. Thus we write T f/( A. Z Z) (2.23) S Combining (2.17), (2.19) and (2.23) with (2.10) gives 4 S ++ N M,, (2.24) M MI+2. ctwl1dS o Application of the techniques of the calculus of variations to (2.24) would yield the equations of motion in terms of the stresses and 18 displacements. The stressdisplacement relations would then yield the equations strictly in terms of the displacements (see reference (26)). For reasons which will become apparent later we will, with the use of the stressdisplacement relations, obtain the kinetic potential in terms of displacement only. For an isotropic, elastic, material the stressstrain relations are =J S+U)E (2.26) E +V)E .~ Solving the first two of these equations for O. and 0 gives CILJL C^+4 V ) + V (2.26) 19 Equations (2.26) and the fourth equation of (2.25) can be substituted into equations (2.20) and with the aid of (2.8) the required integration can be performed. The result is the forcedisplacement relations N 2. M~~~ LI EK1'^j N +' l M =a Y() 2 (2.27) Here, J iz0 1) E + IX I+ 1) 2 + U) d ^) M C )  20 We have again employed the assumption o _lNz and further h/z jzadz = 0 h/z. We can now combine (2.11), (2.17), (2.19), (2.23), and (2.27) to finally obtain the kinetic potential strictly in terms of displacements. Thus, K Arr itL +V w+ ] (2.28) E ( +I +) Here, the first term represents the kinetic energy, the second the potential energy of the lateral load, the third the membrane energy and the fourth the bending energy. 21 Having the kinetic potential we could employ Hamilton's principle and obtain the three equations of motion governing u, v, w. In the ensuing analysis, we will be more interested in equation (2.28) than in the equations of motion so that the latter will not be derived here. The simplification of equation (2.28) to the beam equations can be made by setting V / ,a/c equal to zero so that we have for the kinetic potential of a beam undergoing large deformations the expression F =i+t (2.29) L l This equation can be simplified further by disregarding the longitudinal inertia and by writing s, L w so that we have ac,wr ~Lcjtd g f 2 (2.30) K f mI for the kinetic potential of a beam undergoing moderately large vibrations. 22 To obtain the linearized equations of motion of a plate we disregard the membrane stresses and the longitudinal inertias so that the kinetic potential becomes S K0+rPtLf+ ['I L. &T. JS (2.31) Application of Hamilton's principle yields (27) (2.32) for the equation governing the small vibrations of an elastic plate. A linear viscous damping term 13 has been added to account for the damping phenomena. 2.3. Boundary Conditions For equation (2.32), governing the small vibrations of elastic plates, two conditions are needed on w along the entire boundary. For example, the boundary conditions at a clamped edge are w = 0 and l/lIn = 0 where n is the direction normal to the cylindrical boundary in the xy plane. A simply supported edge has the conditions w = 0 and ew/ di = 0 along the boundary. Other boundary conditions are given in (27). 23 When we consider the large deflections of plates as governed by equation (2.24), additional boundary conditions must be prescribed. In addition to the two conditions on w we need one condition on u and one condition on v along the entire boundary. The case of a clamped plate is the simplest for then we have u = v = 0. For a simply supported edge we have un = 0, where un is the longitudinal displacement normal to the cylindrical boundary in the xy plane. The other boundary condition is obtained from the condition that the shear stress along the boundary 4Cn vanishes. From (2.25), (2.8), and (2.9) we have _+ _n au_ar Z u ) (2.33) Since un = w = 0 along s then S( )l(2.34) along s and the condition on us at the boundary is dos/M = 0. Other types of boundary conditions can be considered. For example see (25). CHAPTER III THE RESPONSE OF LINEAR SYSTEMS TO RANDOM EXCITATION The basic aim in this chapter is to review a method of solution of equations of the type (3.1) L(u+) mw = q+rw< where L is a linear, spatial, differential operator, the dot indicates differentiation with respect to time t, and q(x,t) is a function which is not completely deterministic. We first must specify what we mean by solving the equations. Since the load q is not deterministic, i.e., only certain statistical properties of it are known, then quite naturally all we can expect to know of w are certain statistical properties. The process or experiment for which w is the result is said to be a stochastic process and w itself is called a random variable. Before attempting to solve equation (3.1) it seems natural to first give a brief introduction to the mathematical descriptions of stochastic processes. More complete and rigorous descriptions of such processes can be found in any number of books (28, 29, 30). 3.1. Stochastic Processes and Probability Theory From the mathematical point of view a stochastic or random process is a collection of functions y (t), y (t), yN(t) for which there 24 25 exists a probability measure. Each yi(t) is called a sample function or record and the entire collection is called an ensemble. We relate this mathematical model to the following physical model. Consider a certain experiment which can be repeated under similar conditions a large number of times, e.g., the thermal noise arising across a set of identical resistors. The outcome of each experiment is a different function y (t) so that we say the function y(t) which we are measuring is a random variable. On any given trial we cannot predict the outcome so that only certain statistical information concerning the process y(t) can be determined. The basic functions that define a random process are the following set of probability density functions: W1(Yl,tl) dyl = probability of finding y in the range (yl, y1 + dyl) at time tl. W2(Y1,tl; Y2,t2) dy1 dy2 = joint probability of finding y in the range (yl, Yl + dyl) at time t1 and in the range (Y2, Y2 + dY2) at time t2. W3(Yltl; Y2,t2; Y3,t3) dy1 dy2 dy3 = joint probability of finding y in the range (yl, yl + dyl) at time tl, in the range (Y2, Y2 + dy2) at time t2 and in the range (y3, Y3 + dy3) at time t3. We continue on in this way indefinitely. These set of functions must fulfill the following conditions. 1. VV >, i. The use of the word variable for these types of functions is traditional. 26 3. 0 o W C. ( ,t; ...^,w =J*m)) f *Wn(y..i; ..** m, ...^8*v3cd,* since each Wn implies all previous W,. It may happen that under a shift of the t axis the functions are unaffected. Such a process is called stationary and we have W1(y1) dyl = probability of finding y in the range (yl, y1 + dyl). W2(Y1,y2; t2,t1) dyI dy2 = joint probability of finding a pair of values separated by a time t2 t1 in the range (yl, y1 + dyl) and (y2, y2 + dy2). And so on. For experimental work the condition of stationarity is almost a necessity. It is tantamount to assuming a steady state condition, i.e., all transients of the system have disappeared. Quite often it is necessary to deal with more than one random process. That is, we may be concerned with several random variables Yl(t), Y2(t), YN(t). Defining the process we then have the following probability density functions: W1(y11, Y21, YN1, tl) dyll dy21 dN1 = probability that yl falls in the range (y11, yll + dyll),' 2 in the range (Y21' Y21 + d21)' S. YN in the range (YNI, YN1 + dyN1) at time tl. W2 (Yl, Y21i YN1, tl; Y12' Y22' YN2' t2) dyll dYN1 dY12 N2 = joint probability that yl falls in the range (yll, 11 + dyll) S. YN in the range (YNI, YN1 + dYN1) all at time t, and that yl falls in the range (YNl, YN1 + dN1), yN in the range (yN2' yN2 + dyN2) all at time t2. And so on again. For simplicity we can use vectorial notation and treat the N variables as components of an N dimensional 27 vector. Then in place of the above we may write Wl(yl,t), W2(j,tl; Y2,t2), etc., with m (Ylm, Y2m' YNm) One of the commonest and most useful density functions is the normal or Gaussian function. For a random variable y(t) the Gaussian probability density function is defined as (31), W t )C C ... = where rf\^ = A< n > = n) and where is the covariance matrix of elements 3C= TC,, )tl wYnmt The symbol < > indicates ensemble average and R(tn,t m) is the correlation function defined by If, = < i n n > If the process is stationary then R Ut*,t 1 = R C *,) = (3.3) =R(.C) 28 Tn= fL ='M and the process is completely defined by the correlation function R(T) and the mean m. Furthermore, when ensemble averages may be replaced by time averages (known as the ergodicity property) then Tur)= e^ T/ ^(3.4) T The Gaussian density function can be extended to cover two or more stochastic processes. A discussion of this situation can be found in (31). When a constant parameter linear system is driven by a Gaussian random process then the output of the system is also a Gaussian random process. For such a situation, knowledge of the mean and correlation function of the output permits us to write, down any multivariate density function for the output process. If the input is nonGaussian then, in general, the output is nonGaussian and the mean and correlation function no longer completely define the process. For such a situation no general method exists for finding the probability density function of the output. One other useful function in stochastic processes is the power spectral density. It can be defined as the Fourier transform of the correlation function. Thus, o (3.5) F Wo)PRt dO FjWJ~~c~ex?01 29 The quantity F(to) dr3is the amount of power in the frequency range (W, u + dw) and hence the name power spectral density. The inverse transform is R f r F (3.6) d0 Note that o (O) fFCo ) = J _," 'R(0)F ,I VC)L + (3.7) 0C = total power of the process. 3.2. Response of Linear Systems Since the correlation function for linear Gaussian systems is so important we will outline here a method of obtaining this function. This will later permit comparison between linear and nonlinear theories for mean squared values of displacement. We start with the equation governing the system in the form of (3.1). We formally seek a solution in terms of normal modes. That is, we consider L Cu += o (3.8) and seek solutions satisfying the boundary conditions. Let these be Sw.,.t (3.9) ur C < eC~) 30 whereUon is the natural frequency of the nth mode. Substituting (3.9) into (3.8) yields 2 C (o)= 0 Mo C O( (3.10) We formally seek a solution to (3.1) in the form U(X ,t)=o C,C(r ) (3.11) n=t Substitution of this into (3.1) gives L LC+P o< m or = q Li) M(3.12) Invoking (3.10) reduces this to Mm+. + el + YLr 2= 0 )L (3.13) A basic property of normal modes is that of orthogonality, i LO( (TI)O<( C = ^^ < ~(3.14) where Cmn is the Kronecker delta. Thus, multiplying (3.13) by O(m(x) and integrating over all x gives the result 31 ^+, a +GLto M=^J^.^c<=f ) +(3.15) The solution to (3.15) may be written in the form ,(*. jf(^*.t)dt (3.16) where GJ (3.17) the inverse Fourier transform of [(AX: .eFsV The function hn(T) is the impulse response function of equation (3.15). From (3.11), (3.15), and (3.16), we have, as a formal solution for each sample response w(x,t), the result O( )= c )JO,,(x' }, a7 (3.18) where the lower limit on hn(!) has been changed to cX since this function is zero for 2' < 0. 32 The cross correlation function is the statistical average of w(x,t) and w(x',t +21) which from (3.18) is x (L,) dc, 8t,J) (1) LL The function 4(' ,e, t z.)= ) < \ ($ ^,t7 (3.20) is the cross correlation function of the load q and it completely determines the cross correlation of the response IU via equation (3.20). An important special case of 4 is (L. o,, I = SY^Lc) s(I rj (3.21) where S is the Dirac delta function. This permits no correlation in space nor time and is obviously not physically realizeable. However, for lightly damped systems it represents a fair approximation to reality. 33 Substitution of (3.21) into (3.19) gives For the important case of lightly damped systems we have for all n so that (3.17) becomes T4 .. aon = 0 Z Substitution of (3.23) into (3.22) would yield the cross correlation function of the response. Upon setting x = x', Z" = 0, we obtain the mean squared response at the point x L i,<< M J (3.24) ct3 = nzi (3.22) (3.23) R~iz4 em I 34 For a simply supported beam, 2. 2. so that (3.25) For systems with more than one space variable, e.g., plates or shells, the method of attack is essentially the same. It hinges on the method of normal modes. For complicated structures these functions are sometimes difficult to obtain and approximate techniques must be employed. CHAPTER IV THE FOKKERPLANCK EQUATION AND ITS APPLICATION TO SOME NONLINEAR LUMPED PARAMETER SYSTEMS The classification of random processes is discussed in this chapter. It is shown how the assumption of a Markoff process permits derivation of the FokkerPlanck equation governing the conditional probability density function of a set of stochastic variables. It is then shown how this equation has been used to obtain the stationary firstorder probability density function governing some nonlinear lumped parameter systems. 4.1. Classification of Random Processes In order to discuss the classification of random processes it is necessary to introduce the concept of conditional probability density functions. We define these functions in the following manner (see reference 31): P2 (Y1,tl 72,t2) dy2 = probability that, if y has the value y at time t1 then y will have values in the interval (y2, y2 + d2) at time t2 (t2_0 tl). Pn (Y',tl; Y2,t2; n1 tn 1 n tn) dn = probability that if y has the values Yl, Y2, n1 at the respective times tl, t2, . tn1 then will have values in the interval (y, yn + dy') at time tn (tn tn1 tl). 35 36 With these definitions we will then have W( ,(,t, ;,,ih43 = W (= .,N z(j).I . (4.1) and so on. The Pn must fulfill the following conditions (4.2) which follow from definition. Some further important properties of the conditional probability density function are: = /P ,% (4.3) + t1wo ff  . _ ca0 4P0 00 i0 >v E3 / 0 1 37 so that t:z (?,t ,^,^) = ,( ^.) C^ (4.4) since it is certain that y2 = yl at t2 = ti. Here (Y2 Yl) is the Dirac delta function. We also have te (4.5) +t.a so that t ^C.,i z ^,t,,t^~ ,(4.6) since there is no statistical dependence between values of y observed at times sufficiently separate. We are now ready to start classification of random processes. The simplest type of process is the purely random process for which we have (4.7) so that from (4.1) we have (4.8) *** .C^ ^n*) 38 This last equation tells us that the purely random process is one in which any ym and n for tm # tn are statistically independent. The next more complicated process is known as a Markoff process. It is the situation in which all the information is contained in the secondorder probability density function W2 (y',tl; 72,t2). For the definition of the Markoff process we have PRrld ,SC^ji~.,t, nn,(4. 9) This equation tells us that the probability that y has values in the range (n, Yn + dyn) at time tn given that it takes on the values yl1 Y2, Yn1 at times tl, t2, tnl respectively depends only on the value of' at the previous time tn_. Substitution of (4.9) into (4.1) gives .C0 (4. 0) so that the process is completely specified by P2 (yn1 tnl I n, tn) since W1 (Y1,tl) is found from the relation (4.11) We can continue on in this way for more complex problems. Thus P3 (Yn2, tn2; Ynl, tn1 0n, tn) defines the next more complicated 39 process, P4 ( 3' n3; Yn2' t2; 7nl, tn Yn>, tn) the next, etc. However, in this analysis it is the Markoff process with which we are concerned. Therefore the higherorder processes will not be discussed. It might seem that equations (4.2) are the only restrictions on P2 (71, tl 1 Y2, t2). However, for a Markoff process this is not the case. P2 (Yi, tl 2, t2) must satisfy the Smoluchowski equation, which we will now establish (see reference 31). We start from the equation, W3(2 jVk ^ (4.12) integrate over yo and employ (4.9) so that we have 7. '.O (4.13) But, (4.14) so that upon using this in (4.13) we arrive at Smoluchowski's equation in the form TL_ OQt t 7= (4,.15^ ^ *) 40 This is the basic equation of a Markoff process. In the next section we show how, with the proper assumptions, this equation can be used to derive the FokkerPlanck diffusion equation. 4.2. The FokkerPlanck Equation Before actually deriving the FokkerPlanck equation, let us look at the differential equations with which we will ultimately be dealing. Consider an N degree of freedom system governed by the differential equation Fi^ ty (4.16) where y = (y1, 2, YN, Y, y2, yN)' Ym are the N variables with which we are concerned and ym are their derivatives with respect to t. Further F is a deterministic vector valued function. This equation would produce a deterministic trajectory in the 2Ndimensional phase space and would be completely determined by equation (4.16) and the specification of y at some time to. b Let us add to equation (4.16) a stochastic forcing function f(t) so that we now have a set of equations governing each sample of y which have the form SF(tIV)+ 1) (4.17) dj; The trajectory of the phase space is now a stochastic process. Clearly, the position y(t2) at the end of any infinitesimal interval of time depends only on the value" (tl) at the beginning of the interval (t2,tl) 41 and on the stochastic forcing function f acting during this interval. We now make the assumptions that the forcing function f is Gaussian with zero mean. We further assume that the forcing functions acting on the system at any two small consecutive time intervals are statistically independent. These assumptions make it necessary for us to take < ^0 (4.18) < +(t) ({tC t) > =!?z (4.19) where fm is the mth component of f and Rmn is some function of m and n. Furthermore, since the position at the end of an infinitesimal time interval depends only on the value y(tl) at the beginning of the interval, and on the forcing function acting during the interval, which according to our assumption is independent of the forcing function acting outside the interval, then the trajectory of the phase space is a Markoff process. It is completely defined by the conditional probability density function P2 (yl,t \ Y 2,t2), which must satisfy the Smoluchowski equation (4.15). With the proper assumptions equation (4.15) can be used to derive the FokkerPlanck equation governing P2 (y1,t1  2,t2). The derivation can be found in many places (4, 10, 31). Here we follow the derivation given in (10). To start the derivation of the FokkerPlanck equation we consider the first and second moments of the displacement of the phase point in 42 an infinitesimal time. These are J ^C~, k^ 'J ^'taf(4.20) We assume that these are of order A t and that all higher moments are of higher order of A t. The first assumption insures the existence of the following limits. 1 L(4.21) CL M Jatro, t ,,%y% J It has been pointed out in (4, 10) that these assumptions are tantamount to the assumption of a Gaussian process for the disturbances. Having made these preliminary assumptions we consider an arbitrary scalar function R(y) which vanishes sufficiently fast to zero at infinity. Multiplying the Smoluchowski equation (4.15) by this function and integrating over the entire phase space gives (4.22) and we have interchanged the order of integration. We now develop R(y) in a Taylor series in (y x), (4.23) atid aid 43 Upon using (4.23), (4.21) and the assumptions concerning the higher moments, the righthand side of (4.22) becomes JFC^)O^ C4)P ^ (4.24) + o A0 , Integrating by parts, writing y for x and putting the result in (4.22) gives IR p.( .U,t\ ?4 0c^ 1 (4.25) ,ZN ZN ,Z la ~ i n Y\?t +01~ b I V.0 tV ^Ml 2.. 0 oA ~ x ane +t r ^ 44 Taking limits as A t 0 yields I 2(4. 26) Since R(y) is arbitrary, the bracketed expression must vanish, leaving us with the FokkerPlanck equation. n + Y(4.27) This is a parabolic diffusion equation. The required solution is the positive one with (4.28) If all transients of the system have died out and a steady state condition has been reached, then P(Yo0ol ',t) W1(Y,t) so that the FokkerPlanck equation becomes ^ ^T^^^ d~m~E 2T ^ f(4.29) It still remains to be shown how equation (4.29) can be used to determine W1 governing the variables of equation (4.17). The connection, of course, is through the moments dm and dmn. Indeed we had, Ct)P M^ 1S'LC_ %kAi)4;. (4.30) 4L40 4t vv ')T2 45 Now at the beginning of the time interval xm had the value ym and at the end the value Ym + AYm so that dxm = d(Aym) and we have SL Aiw YLUC' (4.31) Similarly, it can be demonstrated that d, L" t' % > (4.32) &t .0o 0 *'&* These moments can now be computed from the differential equation (4.17) and thus the corresponding FokkerPlanck equation is fully derived. The method is best illustrated by examples. Therefore, to close this chapter we consider two lumped parameter systems which have been analyzed by this method. The first system is a onedegree of freedom, nonlinear oscillator. The equations of motion are Also (4.33) Also 46 Writing yl = y, Y2 = equation (4.33) is equivalent to the two equations (4.35)The coefficients of the FokkerPlnck equation re therefore The coefficients of the FokkerPlanck equation are therefore 'A d At 0 .... < I A = 0 (4.36) Substitution of these coefficients into equation (4.29) yields Substitution of these coefficients into equation (4.29) yields Ho ,,  :a aw, The solution to this equation as given in (9) is T Lil (4.37) (4.38) at;o Ak d' ato at*' o~t ei ILd~ t r3 Ir rsu.trZC~~I1VJ.~ =O b~L LI~TP IL ~d ~ ~ 47 where C is a normalizing constant. If k(y) is linear, then this reduces to the Gaussian density function as it should. The second illustration of the application of the FokkerPlanck equation to lumped parameter systems consists of a loaded nonlinear string as analyzed by Ariaratnam (11). The N equations governing the system are of the form (4.39) S[ALlw~, +* LL 2.LJ + m (Ai=o To simplify these equations, the deriving functions and the response are expanded in terms of the eigenfunctions of the linear problem. Thus, (4.40) Substitution of (4.40) into (4.39) yields (A) (4.41) A,+ Wt,, +_% L at o <4241% This equation may be used to derive the moments appearing in the Fokker Planck equation. The solution of the resulting equation, as obtained by Ariaratnam is 48 W, C=e~B crsA+ +~fS1\N+1A I m=  (4.42) where C is a normalizing constant and This expression can be used to obtain the mean squared displacement of the various masses on the strong. The FokkerPlanck equation has been used in the past to solve several nonlinear lumped parameter systems. In the next chapter the responses of some nonlinear continuous structures such as beams and plates are investigated by this method. CHAPTER V APPLICATION OF THE FOKKERPLANCK EQUATION TO NONLINEAR ELASTIC SYSTEMS In this chapter it is shown how the FokkerPlanck equation can be used to investigate the finite responses of plates which have been subjected to white noise excitation. A general solution to the Fokker Planck equation is given which is applicable to plates with any boundary conditions. Detailed solutions are presented for a simply supported beam and a simply supported plate. 5.1. General Theory In Chapter II it was shown that the forced vibrations of an elastic plate are governed by the following equation where K f f(& ++r 4' ^ifWfc (5.2) S and Vs is the strain energy. Instead of applying variational techniques to equations (5.1) and (5.2) and thus obtaining the three equations of motion governing u, v, w we proceed as follows: We expand each sample function of w and q in a 49 50 series of the eigenfunctions of the linear problem. This is valid as long as each sample of w and q has continuous derivitives up to fourth order (reference 32, p. 370), a condition we now assume. Thus, we write NW The infinite series has been terminated at some N, which is later to be specified. Of course this invalidates the equality sign of equations (5.3) and (5.4). However, as long as the infinite series representing q and w converge, the finite sum can be made as accurate as desired by properly selecting N. Similar to equation (3.10) the eigenfunctions of the plate must satisfy the equation ( v > ,, W = o (5.5) where the \rm are the eigenvalues determined from the frequency equation. In addition the Wmn must satisfy the appropriate boundary conditions. Each sample function of u and v is also expanded in an infinite set of functions. We choose this set to be orthogonal, to satisfy the boundary conditions and to be such that term by term differentiation 51 of the infinite series is possible. Thus, (5.6) These infinite series have also been terminated with the previous argument concerning convergence still applying. For convenience the same N has been chosen for all series. An example of a proper expansion would be kL Ot) I MnnIL AAM Tlt (5.7) for a clamped rectangular plate of sides a and b and with the origin of the coordinate system at a corner of the plate. This would be a double mt x nlvy mV x nITy Fourier series with terms such as sin a cos cos  sn b cos m x os n y omitted. These terms can be omitted if we consider a b the extension of u onto the intervals (a,0) and (b,0) to be an odd function. Now since (5.7) is a Fourier sine series, which is zero at x = 0, x = a, y = 0, y = b, then it is continuous throughout the entire xy plane and may be differentiated term by term (see Theorem I on page 137 of reference 32). Therefore, for a clamped rectangular plate equation (5.7) is a series expansion of u which satisfies the boundary conditions and the conditions for at least one differentiation provided each sample u is integrable in the Lebesgue sense, a condition we now assume. 52 If we'now substitute equations (5.3), (5.4) and (5.6) into equation (5.2) and perform the indicated integration, we obtain (5.8) where V is the strain energy Vs after integration and bw c)n= j(C U)iS (5.9) S Here the orthogonality property of the eigenfunctions has been used. To satisfy equation (5.1) we consider the umn, Vmn, Wmn as generalized coordinates and apply the EulerLagrange variational equations, which are of the form d a( .. (5.10) where the 9m are the generalized coordinates. Application of (5.10) to (5.8) yields the following set of differential equations 53 h"" I ^ w ~ ~ p~_ _ V (5.11) xJ ph /3~n t~ I (yk where we have introduced the same linear viscous damping term of the linear problem (see equations (2.41) and (3.15)). We now write (5.11) as a set of firstorder equations by setting 'r T (5.12) Equations (5.11) then become, 4Lrmtr = ",a" Vf A m M ~ hQa.r 5U~m C*) (5.13) av 0 & VIVT%_ ~Tm ~ 54 k b I I These equations constitute a set of stochastic differential equations. They are of the same form as (4.17). The vector y has the components umn, umn vmn, Vmn, Wmn, mn for all values of m and n. This is a total of 6N2 components. The stochastic driving functions fm are equivalent to the qmn. Therefore, if we assume that the qmn belong to a Gaussian random process and that < % > =0 (5.14) then the arguments of Chapter IV hold and the trajectory in the phase space of equations (5.13) constitute a Markoff process. As was demonstrated in Chapter IV, this implies that the stationary probability density function W1(y) must satisfy the following FokkerPlanck equation a ..I WI) m1f 55 where, Al*0 A (5.16) mn A~bl0 o X Equations (5.14) imply that 'e' ,i^^c;,y with The moments dm and dmn of equation (5.16) can be calculated from the differential equations (5.13). For bookkeeping purposes the following notation is introduced: cl = Aff.  L A U,, > C9 AA) VlA X f.1 < > (5.19) 56 ALr tn )I~ 4tk At o At c _* A Cm rs Atco at AtO 1f LX lff % 'A LmL AV T > ( A ,,kTA1VS > /'A U);' a Ur "> Then, with the aid of equations (5.13) these moments can be calculated to be Ai^o A* Alt0 At 1 k o  I,, a 10 ,.,V." M (5.20) J M c^ Tftn. 57 dUr  wn. 4 Ip,'_ CLvt7L r1,5 J 1 m^nvn'~s 4tht tat ;I v ( J ^ (^>^ W d:LLA > all other dmnrs = 0. Substitution of these expressions into equation (5.15) yields the following FokkerPlanck equation: in~ m L i ~~~l ) (5.21) ^m5r.M a f S ,), , a^J^1Q~~'b11^ I' V V V. 'V% bC 9e n W + a 6wr^v% At p0  12"'eS & S ( 3V (~'h~L plyvr r FL V% + ~3, av,, (,M V,.)  58 This is indeed a complex equation for which there exists no standard technique for obtaining the solution. By a process of trial and error the author was fortunate in obtaining a solution for the case in which S= ZN (5.22) This imposes the restriction that the load is completely uncorrelated in time and space, i.e., <^G^Z=tN' ^:x)t') t) (5.23) The solution to (5.21) with the restriction of (5.22) is S^Jv4 Z < ^z ^(5.24) Z where C'' is a constant to be obtained from the normality condition. The general uniqueness theorem to the FokkerPlanck equation given in (33) tells us that this is the only solution. It is to be noted that the distribution of the velocity variables is Gaussian. If we integrate over these variables from 0 to +00 we 59 obtain the probability density function of the umn, mn and wmn. Thus, W,,( ,....,...3,... = C'^fL ] (5.25) The constant C' is obtained from the normality condition. We are ultimately interested in the probability density function of Wmn since from it we can determine quantities such as mean squared displacement, mean squared stress, etc. To obtain this we integrate equation (5.25) over all umn and vmn so that v^ (3 .. =r .pf l TeLJl^ cku. (5.26) co 00 1NL FOL3 This is as far as can be gone in such generality. To proceed further we need to compute V. Several important cases are presented in the next section. It should be noted in passing that when the load has some spatial correlations it would be advisable to employ some approximate technique to solve equation (5.21) governing the probability density function of the modal amplitudes. 5.2. Some Special Cases a. Simply Supported Beam The kinetic potential of a beam with moderately large vibrations is given by equation (2.29) so that the strain energy is O . T. V^ dW EP (5.27) 60 The eigenfunctions of the linear problem are sin mrx so that we L write UC7^,t = an XJ (5.21 n=r V The first four equations of (5.20) are satisfied identically since u = v = 0 and since the strain energy has been expressed independently of u and v. The probability density function of the wm is then given by W,CC, 94 fc'^ L,1 (5.2( with V given by (5.27) after substitution of (5.28). Thus where .. [ i (5.3 where E NoL T' (5.31) N, N=L L. 8) 9) 0) 61 Here 6G is the mean square deflection of the first mode of the linear so problem obtained by setting =0. We have replaced N'/L by No, which is interpreted as the power spectral density of the average load acting on the beam, i.e., Go L L ao 0 CO (5.32) N'L 9&t) =N' where q is now to be interpreted as force per unit length and we have used the fact that The static deterministic counterpart of No for a uniformly loaded beam 2 is, of course, q i.e., the square of the constant load per unit length. It is seen from equation (5.30) that the nonlinearity of the beam causes the probability density functions of the modal amplitudes to become nonGaussian. Furthermore these variables are no longer statistically independent. The mean squared response of the beam is given by NN A (5.34) < r 2,= = Y < QA inr M Aw AM I. 15 V%' V I k L. 62 where cp oo oD 0o Because equation (5.30) is an even function of the wn we will have, < bx W. >= 0 r*V (5.36) so that the modal amplitudes are linearly independent. The mean squared stress in the beam depends upon < ( which is given by 2  In the linear problem, 4 wm 2 is of the order i/m4 so that the infinite series does not converge. This was discovered by Eringen (12) and attributed to two factors: (a) The S functions appearing in equation (5.23) and (b) The inadequacy of BernoulliEuler beam theory. To obtain expressions for meansquared stresses, Samuels and Eringen (34) investigated a Timoshenko beam and Crandall and Yildiz (35) later investigated many different beam models with different damping mechanisms. It was found that these more refined theories produced finite mean squared stresses. 63 2 Since we have only an integral representation for wm2 we cannot rigorously investigate equation (5.37). However, there is really no reason to believe that the introduction of the membrane stresses would cause (5.37) to converge. To investigate the mean squared stresses in the nonlinear problem, it would be most desirable to consider a more refined beam model. 2 An approximate expression for wm > can be obtained by substituting the expression [. R \..... L~j (5.38) into (5.35) and performing the required integration. The result, after some juggling, valid only for "moderately large deflections" is 150, r ^ (5.39) b. Simply Supported Plate We consider a rectangular plate simply supported on all four sides. For the linear problem the eigenfunctions are Av w 2L rt .(5.40) 64 where a and b are the lengths of the sides of the plate in the x and y directions respectively. Earlier in the chapter it was shown how, for a clamped plate, a double Fourier series could be constructed for the functions u and v, which satisfied the boundary conditions and the conditions for differentiability. For a simply supported plate we define the extension of u on the interval x = (a,O) to be odd and on the interval y = (b,0) to be even. Then we can write So=o 0. b Now by Theorem I of reference (32) it is evident that this series can be differentiated with respect to x or y at least once. The summing of the series with n = 0 is necessary since a cosine expansion must include a constant term. For convenience we have also started the summation at m = 0. For the function v we can proceed along the same lines. We may therefore write for the three displacements ,'o Co a. " 65 For uniformity we have started all sums at m = n = 0 and terminated them at m = n = N1 so that the total number of terms of um is still N2 The kinetic potential of a plate undergoing large vibrations is given by equation (2.28). The strain energy is therefore given by (5.43) where the Ex' Ey' xy are given by (2.9). If we now substitute (5.42) into (5.43), perform the required integration and substitute the result into (5.25) we obtain, after some rearranging \l CAL ^ 1 o,' n == n~h V vr A& %A. % 'V * 66 vM~ (E t^ fftE 2ff.rr m a. (\ vb) b . x a? 0> + ltp HEN' E' As) ~1 ev s A tErrEEt ^.ur^^ K^^.1) x +1.L ULn I +___ +ErZZ AE ts vLl  "Ysv k, + I.L , w p r 2 Wi. A W r ' OL~t ftf ":,b .%Cp~r 'D r V 'of lp 1 Ar C P ai61 ' ^P " (5.45) L ,pY S= Imp rE, pF, + ImvF4,E' 2 tfsQe "rus flL 4L I(O + per oE .,s F "p L~ lz I to ^.C\u ) + where irfvwfYpr .Wv [f,[k r 2*l , V%, 15 Ft E I K nItpos p F= mpt t TV 67 0. E = fcA  b F =. 1 = CMA'r LQ. A l cra mn Y Ampri&t. M7  0Oo P" 21 rn rIT d"34 aC.UL a.r b 4s < AAML Q'w Y r,, AIA, rnlt. Awn nA OL c C0a4 I L OL o (5.46) b 0 Qb b Va \^ ,, 104 I CsAu ac a. C"pr^ C^^CC^^. AAfti rA Ah2Sly ~Y=l 09e~OL CL CL~~~: AR~ ~nty ab es Ar. = C4IE:* AAMI b b a S .KSI co& w 68 To obtain the probability density function of the w alone we need to integrate out the umn and the vmn. If we make use of the relation (5.47) which is given on page 64 of reference (36), then W ,c(v ,,(A. = "i ,,u ,...., ..1 ,... becomes, after some manipulation becomes, after some manipulation (5.48) cf1 AI F. i x" C K [ 4 L b (5.49) *PJM^1 iu + L 0 9) b VVL AV.Z to 20. a6 .b LVW2E.n"2EEE WpWr.( Kj, Lw pys) )I 4N* #A. s r % p %  ~~(riLD) 1v. 0. ~ nnpClo~e\l ;L L*  ZEE & S 7  00 ' *ae brldr: 69 If we integrate out the vmn, again making use of (5.47), we finally obtain W, ic ( t3 E tTO I ,C )L && U 0 N;O w~.:o +CEL EEEE r Tpi o v (C & j p n. o f% (5.50) O. z2. ____ ( Tj)L_____________ l ,,' (V) 6 Q,. 4 V a z.b where as before No is the power spectral density of the average load acting on the plate, i.e., / s oO N 00 N0 = N'/ab jl0 i^O 70 Again, the effect of the nonlinearity is to cause the modal amplitudes to become statistically dependent and nonGaussian. The meansquared displacement of the plate is given by 2.QA/y S A^Tl^AA MT_ Xv^ (5.51) %V b b where Having reduced the meansquared displacement to quadratures, we must again stop and seek some approximation to finish the job. This is left to the next chapter. As was the case in the beam, the meansquared stresses for the linear plate do not converge and so will not be investigated here. The investigation of the large random vibrations of a simply supported beam and simply supported plate have been reduced to quadratures in this chapter. We leave to the next chapter the numerical investigation of specific cases. CHAPTER VI NUMERICAL INVESTIGATION In the previous chapter the probability density function of the modal amplitudes of a plate undergoing large deflections was derived. Detailed calculations were presented for a simply supported beam and a simply supported plate. The meansquared deflection of these structures was reduced to quadratures. For the beam an approximate formula was developed. In this chapter we investigate more closely some of the results of the previous chapter. For the beam, the linear, approximate nonlinear and integral representation of the meansquared displacement are compared numerically for a range of the parameters. For the plate the meansquared displacement of the first mode, which is the predominant term, is numerically investigated for different aspect ratios b/a. 6.1. Simply Supported Beam In Figure 3, the meansquared deflection at the center of the beam .SZ as determined by the linear theory, the approximate formula (5.39), and numerical integration of equation (5.35) is plotted against rB with R = 1/2. For the range of Id considered it was found, as in the linear theory, that sufficient accuracy is obtained with N = 1. It was also found that the approximate formula is valid over a small 71 72 41 0 1I 14. 0C a4 ,0 o4 4 ow 0 o 0 44 0 (3 cun 44> P14 , 73 range of 6o As long as the difference between linear and nonlinear theories is not greater than 10 per cent, this formula gives an excellent estimate of the true meansquared deflection. This is to be expected in view of the approximation made in deriving formula (5.39). Also plotted in Figure 3 is the meansquared deflection determined by numerical integration with N = 1 and R = 1/4 and 1/8. These curves indicate that for lower values of the radius of gyration the curve of the nonlinear theory begins to deviate sooner from the straight line of the linear theory. This is to be expected since with diminishing R the role of bending dimishes. Furthermore, if the beam were rectangular then h2 = 12R2 so that Z { (6.1) Now for R = 1/2, the nonlinear theory begins to appreciably deviate from the linear theory at B2 = 0.3 or at = 0.1. For R = 1/4 we have appreciable deviation at Qg2 = 0.009 or S= 0.11, and for R = 1/8 at dB2 = 0.004 or P = 0.14. Since, as is well known, the linear theory of beams is valid only for w/h reasonable. In Figure 4 the meansquared deflection as determined from the linear theory and from numerical integration of equation (5.35) is plotted against larger values of Bo2 for R = 1/2. Of the three curves, the uppermost represents numerical integration with N = 1. The lowest curve represents 4 wl2 > as determined by numerical integration with N = 3. The middle curve represents the total deflection, i.e., 74  c;o CO *r4 o, 0 41 0 1 0 rFl 0 0 0 0 4J :J3 a0 O a i" 75 4 l2 + these curves. Firstly, consideration of the beam as a onedegree of freedom system (i.e., N = 1) is no longer valid for such large values 2 2 of Bo2. Secondly, as in the linear theory, < wl2 > gives a fairly close estimate (a few per cent) of the total meansquared 2 deflection. However, in computing ( wl > it is now necessary to consider the effects of the second and third modes. That is to say, the nonlinear coupling is so strong, for the range of parameters considered in Figure 4, that the second and third modal coefficients have a significant effect on the meansquared value of the first modal coefficient. It should be pointed out that for a rectangular beam with R = 1/2 we have h2 = 3, so that with B = 1.0, the highest value in the graph of Figure 3, we have q2. C~~ II z ir ,6 (6.2) This value is not in excess of the applicability of the nonlinear theory considered nor of the values of practical interest. 6.2. Simply Supported Plate The first mode is the predominant one in the calculation of the meansquared displacement of the plate. Therefore, if in equation (5.50) we consider only the first mode we have 76 + U . L  IL __ ( xyWn%%i 0 +i. t Pw),brn  l +2..  b b14 CL" 1 L + I n  C (*>**'' ^ T1 Z. o.b +1 2. 6't a b~ 9a.b 6c4b* b"Cb'' L =.n  CeL +b 8b 801 j  aaO (6.4) C. I 3tZIn h g + + 8b..> a ^ 0 'g.',Q ( 1X 2. 0. Z.b (6.3) Now, +~v)i + "L , %w ... . z. b   + & W1 C,) = CRILf EhrrY hL _gEhrr' I Y8 2NoC\ ri 11 77 and all other Lnllll = Kmnllll = 0 so that (6.3) reduces to \N, C =C PA r  PO I[c+l1 1,g Wit E + ~lY8X Ia b o,. I=> I..+ 2 , Sb L + L_ CL (6.5) where L 0O I of = b/a This is of the form ^^} ^^C (6.6) PELe +, la^Ho ^i)" O^ where ) and o( are easily calculable once V, a and b are known. The meansquared deflection of the plate is CO (6.7) 4 vt% tII ~ (I, Cy\~L o Lt, 78 Unfortunately this integral is not tabulated and we must resort to numerical methods. In Figure 5 the meansquared deflection at the center of the plate 2 T as determined from numerical integration of (6.7) is plotted 2 against "2 for 1) = 1/3, h = 1, and two difference aspect ratios Po of = 1 and 2 = 2. For larger aspect ratios the linear and nonlinear theories give higher meansquared deflections. This is what is to be expected since increasing the aspect ratio is equivalent to moving apart one pair of supports of the plate. Also evident from Figure 5 is that the nonlinearity of the plate causes a reduction in the meansquared deflection. Furthermore, for the smaller aspect ratio the percentage deviation of the nonlinear theory is slightly greater than for the higher aspect ratio. In this chapter the meansquared deflection of the beam and plate has been investigated by numerical integration. The results indicate reduction of these quantities and a significant coupling of the modes for sufficiently large deflections. 79 J 0 Co o o c 0 0 o 6 6 l 4, 4I 0 4 cd r.l 0. 4i o 0) go 41 0 U X4 0 0U 1.a , Fr a ' 3 04 CO C gl 0 0 o o CHAPTER VII CONCLUSIONS This paper has been devoted to the study of the random vibrations of some nonlinear elastic systems. The equations of motion of thin elastic bodies with large deflections have been presented in variational form. General truncated series expansions of the middle surface displacements have been performed and the EulerLagrange variational equations have been used to obtain the nonlinearly coupled differential equations governing the series' coefficients.. The assumption of uncorrelated, Gaussian loading permitted the phasespace of the series' coefficients to be identified with a Markoff process. This, in turn, permitted the derivation of the FokkerPlanck equation governing the probability density function of the series' coefficients. The moments for this equation were obtained from the governing set of differential equations. A solution to the FokkerPlanck equation for the case of spatially uncorrelated white noise has been obtained. Several special cases have been worked out in detail. In particular, the probability density functions for a simply supported beam and a simply supported plate have been computed. The meansquared displacements of these systems were reduced to quadratures. For the case of the beam, an approximate formula was developed. 80 81 Numerical integration of the exact expression for the meansquared displacement of the beam has shown a reduction of this quantity as compared to the linear theory. Calculations have also shown a limited range of applicability of the approximate formula. Furthermore, it was shown that the first mode still gives a good estimate of the meansquared response but the coupling effect of the modes is so important for sufficiently large deflections, that the effect of the higher modes must be taken into account when computing the meansquared value of the first mode. Numerical integration also showed a reduction of the meansquared deflection of the plate. While this paper has presented a method of attacking the random vibrations of elastic systems with large deflections, it must be pointed out that the solutions presented are valid only for completely uncorrelated loadings. This is, of course, physically unrealizeable. Nevertheless, the results of this analysis should give some insight into the problems of the random vibrations of nonlinear elastic systems. LIST OF REFERENCES 1. Crandall, S. H., ed. Random Vibration. Vol. 1, the M.I.T. Press, Cambridge, Massachusetts, 1948. 2. Crandall, S. H., ed. Random Vibration. Vol. 2, the M.I.T. Press, Cambridge, Massachusetts, 1963. 3. Unlenbeck, G. E., and Ornstein, L. S. "On the Theory of Brownian Motion," Phys. Rev., Vol. 36, 1930, pp. 823841. 4. Wang, M. C., and Unlenbeck, G. E. "On the Theory of Brownian Motion II," Rev. Mod. Phys., Vol. 17, 1945, pp. 323342. 5. Rice, S. 0. "Mathematical Analysis of Random Noise," Bell System Technical Journal, Vol. 23, 1944, pp. 282332; Vol. 24, 1945, pp. 46156. 6. Booton, R. C. "The Analysis of Nonlinear Control Systems with Random Inputs," Proc. Symposium on Nonlinear Circuit Analysis, Vol. 2, 1953, pp. 369391. 7. Caughey, T. K. "Response of Van Der Pol's Oscillator to Random Excitation," J. Appl. Mech., Vol. 26, 1959, pp. 345348. 8. Caughey, T. K. "Random Excitation of a Loaded Nonlinear String," J. Appl. Mech., Vol. 27, 1960, pp. 575578. 9. Chuang, K., and Kazda, L. F. "A Study of Nonlinear Systems with Random Inputs," Trans. Am. Inst. Elec. Engrs., Part II, Applications and Industry, Vol. 78, 1959, pp. 100105. 10. Ariaratnam, S. T. "Random Vibrations of Nonlinear Suspensions," J. Mech. Engrg. Sci., Vol. 2, 1960, pp. 195201. 11. Ariaratnam, S. T. "Response of a Loaded Nonlinear String to Random Excitation," J. Appl. Mech., Vol. 29, 1962, pp. 483485. 12. Eringen, A. C. "Response of Beams and Plates to Random Loads," J. Appl. Mech., Vol. 24, 1957, pp. 4652. 13. Wiener, N. "Generalized Harmonic Analysis," Acta Mathematica, Bd. 55, 1930, pp. 117258. 14. Ornstein, L. S. "Zur Theorie der Brownschen Bewegung fur Systeme, worin mehre Temperaturen vorkonmen," Zeltschrift fur Physik, Bd. 41, 1927, pp. 848856. 82 83 15. Van Lear, G. A., and Unlenbeck, G. E. "Brownian Motion of Strings and Elastic Rods," Phys. Rev., Vol. 38, 1931, pp. 15831598. 16. Press, H., and Houboldt, J. C. "Some Applications of Generalized Harmonic Analysis to Gust Loads on Airplanes," J. Aeronautical Sci., Vol. 22, 1955, pp. 1726. 17. Thomson, W. T., and Barton, M. V. "Response of Mechanical Systems to Random Excitation," J. Appl. Mech., Vol. 24, 1957, pp. 4652. 18. Lyon, R. H. "Response of Strings to Random Noise Fields," J. Acoust. Soc. Am., Vol. 28, 1956, pp. 391398. 19. Nash, W. A. "Response of an Elastic Plate to a Distributed Random Pressure Characterized by a Separable Cross Correlation," Tech. Note No. 1, Contract No. AF 49(638)328, Engrg. and Industrial Exp. Sta., Univ. of Fla., Gainesville, 1961. 20. Bogdanoff, J. L., and Goldberg, J. E. "On the BernoulliEuler Beam Theory with Random Excitation," J. AeroSpace Sci., Vol. 27, 1960, pp. 371376. 21. Caughey, T. K. "Response of a Nonlinear String to Random Loading," J. Appl. Mech., Vol. 26, 1959, pp. 341344. 22. Novozhilov, V. V. Theory of Elasticity. English translation, Israel Program for Scientific Translations, Jerusalem, 1961. 23. Novozhilov, V. V. Foundations of the Nonlinear Theory of Elasticity. English translation, Graylock Press, Rochester, N. Y., 1953. 24. Biot, M. A. "Elastizitatstheorie zweiter Ordung mit Anwendungen," Z. a. M. M., Bd. 20, 1940, pp. 8999. 25. Wang, C. "Nonlinear Large Deflection Boundary Valve Problems of Rectangular Plates," N.A.C.A. TN1425, 1948. 26. Herrman, G. "Influence of Large Amplitudes on Flexural Motion of Elastic Plates," N.A.C.A. TN3578, 1956. 27. Timoshenko, S., and WoinowskyKreiger, S. Theory of Plates and Shells. 2nd edition, McGraw Hill, New York, N. Y., 1959. 28. Doob, J. L. Stochastic Processes. John Wiley and Sons, New York, N. Y., 1953. 29. Kolmogorov, A. N. Foundations of the Theory of Probability. 2nd edition of English translation, Chelsea, N. Y., N. Y., 1956. 84 30. Feller, W. Probability Theory and Its Applications. John Wiley and Sons, New York, N. Y., 1950. 31. Middleteon, D. An Introduction to Statistical Communication Theory. McGraw Hill, New York, N. Y., 1960. 32. Tolstov, G. P. Fourier Series. English translation, Prentice Hall, Englewood Cliffs, N. J., 1962. 33. Caughey, T. K. "Derivation and Application of the FokkerPlanck Equation to Discrete Nonlinear Dynamic Systems Subjected to White Random Excitation," J. Acoust. Soc. Am., Vol. 35, pp. 16831692. 34. Samuels, J. C., and Eringen, A. C. "Response of a Simply Supported Timoshenko Beam to a Purely Random Gaussian Process," J. Appl. Mech., Vol. 25, 1958, pp. 496500. 35. Crandall, S. H., and Yildiz, A. "Random Vibrations of Beams," J. Appl. Mech., Vol. 29, 1962, pp. 267275. 36. Grobner, W., and Hofreiter, N. Integraltafel. Zweiter Teil, Bestimmte Integrale, SpringerVerlog, Wien, 1958. BIOGRAPHICAL SKETCH Richard Edgar Herbert was born November 11, 1938 at Mount Vernon, New York. In June 1956 he was graduated from A. B. Davis High School. He attended the Cooper Union School of Engineering, a privately endowed, tuitionfree college. Upon receiving the degree Bachelor of Civil Engineering in June 1960, he enrolled as a halftime graduate student in the Department of Engineering Mechanics at the University of Florida. He was graduated with the degree Master of Science in Engineering in February 1962. In September 1962 he embarked upon doctoral graduate work. The author was a halftime Instructor from September 1960 to February 1962 in the Department of Engineering Mechanics except for the summer of 1961 when he was an Assistant in Research. From February 1962 to September 1962 he was a fulltime Research Associate in the same department. When he started his doctoral studies he was granted an Engineering College Fellowship and subsequently a Ford Foundation Fellowship. Richard Edgar Herbert is a member of Sigma Xi and Omega Delta Phi. 85 This dissertation was prepared under the direction of the chairman of the candidate's supervisory committee and has been approved by all members of that committee. It was submitted to the Dean of the College of Engineering and to the Graduate Council and was approved as partial fulfillment of the requirements for the degree of Doctor of Philosophy. April 18, 1964 Dean, College of Engineering Dean, College of Engineering Dean, Graduate School SUPERVISORY COMMIITTEE Chairman 4t / 7, .  . a,.sJ_ .[,  ,, ' . !^.^.C UNIVERSITY OF FLORIDA 3 1262 08553 79091111111111111 I 3 1262 08553 7909 