'*”Iiiéiiéfifiv‘zfitwgz_:>33‘.j:":'; ‘f A STUDY OF THE DYNAMIC RESPONSE OF AN INFINITE ELASTIC PLATE SUBJECTED TO A STEADILY MOVING TRANSVERSE FORCE Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY CLIFFORD CHI-FONG CHOU 1972 IIIIIIIIIIII , 1112.9” 01064 4700 I III/III III/III This is to certify that the thesis entitled A STUDY OF THE DYNAMIC RESPONSE OF AN INFINITE ELASTIC PLATE SUBJECTED TO A STEADILY MOVING TRANSVERSE FORCE presented by CLIFFORD CHI-FONG CHOU has been accepted towards fulfillment of the requirements for Ph. D. degreein Mechanics ”as 5/2/5451, / Major professor Date Z561 /7/1 y/‘KZ 0-7639 ? gramme By BUBK BINDERY INBu t LIBRARY amosns 1' 1 SPHINGPOII’TJICHISA!‘ at? 2 3 If"; ABSTRACT A STUDY OF THE DYNAMIC RESPONSE OF AN INFINITE ELASTIC PLATE SUBJECTED TO A STEADILY MOVING TRANSVERSE FORCE By Clifford Chi-fong Chou The dynamic response of an infinite elastic plate subjected to a steadily moving transverse force is studied. The Timoshenko plate theory incorporating the effects of shear deformation and rotatory inertia is used. The transverse force, being taken as a two-dimensional delta function, is assumed to move at a speed that is either intersonic or subsonic relative to two plate velocities. The problem is then formulated as an initial value problem and solved by a Fourier-Laplace transform method. By using the Tauberian theorem, the steady-state response solutions are completely determined without inverting the Laplace transform. The solutions are given in the forms of integrals and are inter- preted as Green's functions for a class of moving force problems governed by the Timoshenko plate theory. A STUDY OF THE DYNAMIC RESPONSE OF AN INFINITE ELASTIC PLATE SUBJECTED TO A STEADILY MOVING TRANSVERSE FORCE By Clifford Chi-fong Chou A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Metallurgy, Mechanics and Materials Science 1972 TO MY PARENTS ii ACKNOWUEDGMENTS The author wishes to express his indebtedness to Dr. David H.Y. Yen for his guidance in the initial stages of the research, for the devotion of a great deal of his time during the investigation, and for the many valuable comments offered by him. Gratitude is also due to Dr. George E. Mase, the author's academic advisor, for his constant encouragement throughout the years of the author's stay at Michigan State University. Grateful acknowledgments are also due to Drs. G.L. Cloud, C.J. Martin and M.Am Medick for their serving on the author's doctoral guidance committee. Thanks are also due to Mrs. N.L. Barnes for her superb typing of the thesis. Finally, the author wishes to extend his special thanks to his wife, Chu-hwei, for not only her understanding and encouragement, but also the many sacrifices that she has suffered during the long course of this study. This research has been supported by the National Science Foundation under the Grant No. GK-l474. iii Chapter I. II. III. IV. TABLE OF CONTENTS ACKNOWLEDGMENTS LIST OF FIGURES INTRODUCTION 1.1 Historical Review and Motivation 1.2 Organization of the Dissertation FORMULATION OF THE PROBLEM 2.1 Statement of the Problem 2.2 Mathematical Formulation METHOD OF ANALYSIS 3.1 Solution by Integral Transforms 3.2 Inversion of the Transformed Solutions 3.2.1 w-Inversion 3.2.2 g-Inversions 3.3 Evaluations of the Auxiliary Functions 3.3.1 The Subsonic Case 3.3.2 The Intersonic Case DISCUSSIONS AND CONCLUSIONS 4.1 Discussions 4.2 Concluding Remarks LIST OF REFERENCES iv Page iii 18 18 21 22 28 46 46 54 58 58 75 76 Figure 10 11 12 13 14 15 16 LIST OF FIGURES Geometric Description of the Problem Sign Conventions of the Plate-Stresses Closed Contour cm in the Complex w-Plane Branch Points and Branch Cuts in the Complex g-Plane in the Subsonic Case Branch Points and Branch Cuts in the Intersonic Case Path of Integration in the Subsonic Case Path of Integration in the Intersonic Case Contour of Integration for T > O in the Subsonic Case Contour of Integration for T < O in the Subsonic Case Path of Integration for T > O and T > ggL in the Intersonic Case 2 Path for T < O and T < 1%1' in the Intersonic Case 2 P and Equivalent Contours in the Intersonic Case 1" and Equivalent Contours in the Subsonic Case Wave Fronts in the Supersonic Case Wave Patterns in the Intersonic Case Wave Patterns in the Subsonic Case Page 10 13 23 31 32 35 36 4O 41 44 45 63 64 66 67 68 CHAPTER I INTRODUCTION 1.1 Historical Review and Motivation Among the many studies on the dynamic responses of elastic bodies, the problem of moving loads on structures has received increasing attention in recent years. In many engineering applications, structures may be thought of as being made of sec- tions of plate subjected to moving surface loads. Examples that may be easily mentioned are roads and bridges under the action of running vehicles, spacecrafts subjected to moving impact loads in ammospheric reentry, and loads on submarines. A problem of basic importance in engineering mechanics is that of an elastic plate, assumed of infinite extent, excited by a moving transverse load that has been applied for a sufficiently long duration of time. One is then led to find, mathematically, the so-called steady-state solution of the governing equations of an elastic plate subjected to the moving transverse force. For the problem mentioned above, one must first face the task of finding a suitable mathematical model to describe the plate. Certain mathematical models, of varying analytical complex- ity and physical validity, have been developed to describe the dynamic behavior of elastic plates. we shall only mention here the classical plate theory, the Timoshenko plate theory [1,2] and Mindlin's approximate theories [3]. The classical plate theory assumes that plane sections remain plane during the process of deformation and considers only the translatory inertia. This approximate theory leads to a physically defective non-wave character [l,4,5,6,7], i.e., no expected wave front separating the disturbed and undisturbed regions. It predicts that the disturbance is sensed everywhere in the plate immediately upon the application of a load. The velocity is inversely prOportional to the wave length and hence becomes infinite as the wavelength approaches zero. Because of these reasons, the classical theory is judged inadequate for treating a number of dynamic problems. As a refinement of classical plate theory and yet remain- ing a comparatively simpler analytical nature, the improved plate theory, known also as the Timoshenko plate theory, was deduced from three-dimensional elasticity by Mindlin [2]. The improved theory, incorporating the effects of the shear deformation and rotatory inertia, results in two bounded wave velocities corresponding to two modes of motion in the plate. The accuracy of the improved theory can be assessed by comparing the dispersion curves, i.e., the velocity-wavelength curves predicted by this improved plate theory,with the corresponding curves obtained from the solutions of the exact equation of elasticity theory. Better agreement on the basis of dispersion curves can also be achieved, as was done by Mindlin, by a proper choice of the constant which relates the average shear stress to shear strain. Mindlin's study shows that the improved plate theory gives a close agreement with the exact theory in so far as the lowest branch of the dispersion curves is concerned. The discrep- ancy between the two theories in the next higher branch of the dis- persion curve, however, becomes more severe. This indicates that the nmproved theory is valid only up to moderately high frequencies. Since a general disturbance may be considered as a superposition of wave-trains of all wavelengths, the improved theory would not be expected to provide reliable results for those components in the high-frequency-short-wave range. Miklowitz [8] has made an intensive study on the infinite plate problem and has devised a criterion for excluding the undesirable portion of the dynamic reaponse. Mindlin's comparison and Miklowitz's study appear to confirm the adviseability of using this refined, and yet mathematical tractable model for treating dynamic plate problems. A totally different but more systematic approach to deriving approximate theory was also suggested by Mindlin [3],who proposed a general procedure for constructing approximate plate theories of various levels of approximation. By expansion of the displace- ment in an infinite power series of the thickness coordinate and integration over the thickness, the three-dimensional equations of elasticity are converted into an infinite set of two dimensional equations, which are truncated to yield the approximate equations of different orders. The zero-th order and the first order of these approximate theories give rise to the same equations of motion as in the classical and the improved plate theory respectively. Higher order approximate theories would provide further refinement over the improved theory. As they involve highly complicated mathematical equations, no attempt will be made to use them in the work here. We shall use the improved plate theory in the present study. The study of the dynamic responses of elastic beams and plates of infinite extent to time-dependent disturbances has long been a topic of basic interest and importance in applied mechanics. Numerous investigations have been made and reported. Problems of the impact-response type, concerning the propagation of disturbances caused by suddenly applied delta function type forces, have been solved for a Timoshenko beam [9]. Uflyland [10] first employed the integral transforms to solve the problem of an infinite beam subjected to a step shear load. The same problem was later re- examined theoretically by Dengler and Goland [11]. Numerical re- sults were obtained for the solutions in [11] by Goland, Wickersham and Dengler [12] and were compared with their experimental data. They found that the results compare favorably if the high- frequency portion is ignored. Boley [13] and Boley and Chao [14] derived solutions for a semi-infinite Timoshenko beam under a variety of end loadings, using a transform technique. Their results, as compared with those obtained from classical beam theory, showed again that good agreement was obtained only for the low frequency-long waves of the lowest mode of vibration of the beam. Solutions to the above mentioned problems can also be obtained by the method of characteristics. Leonard and Budiansky [15] took this latter approach to derive travelling wave solutions for various finite and infinite Timoshenko beam problems. Tang [16] also attacked the problem of compressional waves in a rod by the method of characteristics and commented on the approach he used. One common feature of the analytic solutions for a Timoshenko beam under an impulsive disturbance is that two wave trains will result. One of the wave trains prOpagates with the shear-wave veolcity while the other travels with a bending-wave velocity. The corresponding plate problem was studied by Miklowitz [8], who considered an infinite plate subjected to a suddenly applied concentrated shear load. The wave fronts emanating from the disturbance were found to consist of two outgoing concentric circles centered at the point of application of the load. The dynamic response of the plate vanishes identically beyond the outer wave front. It is also shown in the literature that these two wave fronts correspond to high-frequency distrubances [8]. The behavior across such wave fronts has been investigated by Jahsman [17]. A different type of problem concerns elastic beams and plates under travelling loads,and several authors have contributed results along this line. Florence [18] has employed the Laplace transform technique to study a semi-infinite Timoshenko beam under a concentrated load moving at a constant velocity. A similar solution to the problem obtained by using classical theory is that of Kenney [19], who studied the steady-state respdnse of the beam, with the effects of foundation stiffness and foundation damping being included. Other treatments dealing with the dynamic behavior of a Timoshenko beam on an elastic foundation have been made by Crandall [20], Achenbach and Sun [21] and Steele [22, 23, 24]. Solutions of the analogous problem of plates subjected to moving forces include that given by Morley [25], who con- sidered a plate subjected to moving step loads applied at both faces of the plate with speeds slower than the shear wave velocity. Achenbach and his co-workers [26] have employed a Timoshenko plate model to describe the dynamic response of an elastic layer resting on an elastic half-space. They considered only a moving line load travelling at a speed less than shear wave speed . Most recently, Yen [27] has advanced an extension of this type of problem. He studied the dynamic response of an elastic plate of infinite extent to a steadily moving transverse force based on the Timoshenko plate theory. The work.was re- stricted to the supersonic case of the problem, i.e., it was assumed there that the speed v of the moving force exceeds both wave Speeds v1 and v2. He took the moving force in the form of a two dimensional delta function and was concerned with the construction of a fundamental solution. Seeking the steady- state response of the plate, he treated a moving coordinate as time and formulated the problem as an initial value problem for a system of hyperbolic differential equations. The problem was then solved by a Fourier-Laplace transform method. It was found in his solutions that the wave fronts form the edges of two "Mach" sectors whose common vertices are at the instantaneous position of the moving load and that the plate will remain undisturbed beyond the outer wave front. Mention here should be made of the work of Payton [28, 29] who has treated the very same problem within the classical plate theory. In this study, the "intersonic' and the "subsonic" cases of the same problem treated by Yen, i.e., v2 < v < v and 1 1 respectively, will be studied. The present study, V Au.%.xvc 11 because the number of independent variables can usually be reduced upon introducing moving coordinates in the former problems. For our plate model we use the so-called improved or Timoshenko plate theory. According to this improved theory, there can propagate in the plate two waves with distinct wave speeds, _ /___!L___ _’ ' . . v1 — 2 and v2 — p , with v1 > v2. For an explanation 9(1-u) of the notation here, see Section 2.2. We are here interested in IQ the analytical solutions for the steady-state response of the plate for the cases v2 < v < v (intersonic) and v < v (subsonic), l 2 as the case v >Iv (supersonic) has been investigated previously 1 [27]. 2.2 Mathematical Formulation We begin by writing down the dynamic equations for the plate. Assume that D is the flexural rigidity of the plate, 6' the modified shear modulus, u the Poisson's ratio, and p the density. All these quantities are taken to be constant throughout the plate. The governing equations of motion for a homogeneous, isotropic, elastic plate may then be given in terms of the so-called "plate-displacements" w, Ax and wy as follows 2 3 a I D _ 2 a1 _ I 5!. = Eh X §i(1 u)V wx + (1+p) ax] G hmx + ax) 12 5:2 (2.1a) 3 52¢ D - 2 51 - q, 3!. = Eh_.__JL §{(1 u>V Iy + (1+u) By] G (Ty + By) 12 atz (2.1b) 2 2w G'h(v w + s) + q 0. We express the transverse force in the form of a two-dimensional delta function as Q(X,Y,t) = 6(X) 6(Y‘I‘Vt) H(t) (2-3) with H(t) here being the Heaviside function. For time t < 0 the plate is assumed to be undeformed and at rest. Thus we have the initial conditions l4 w(x,y,0_) = H(an:O_) = 0 II C Vx(xay s0_) = ¢x(x)Y:O_) (204) ll 0 Iy(xsy :O_) = Iy(xs}’:0_) where the dots denote the differentiations with respect to time. We also expect for any finite t > O that the following conditions must be satisfied at infinity. lim w(x,y,t) = 0 IXI-s IYI—m lim ¢x(x,y,t) = O (2.5) Ix -*°° IYI-m lim I (x,y,t) = 0 . Ix -'°° y IYI-‘w Ultimately, we shall be concerned with the limits of the time-dependent solutions as t a m, i.e., when the steady-state solutions are reached. We introduce the moving coordinate T defined as T = y‘+ vt . (2'6) The transverse force in Equation (2.3) now takes the form q(x,r,t) = 5(x) 6(T) H(t) (2-7) by regarding x, T and t as the independent variables. Re- ferring to the x, T, t variables, we have for any function f that depends on x, y and t 15 f(x:Y:t) = f(X,T'-Vt,t) = f'(X,T,t) 0 (2.8) It then follows that a_f._,a§_'_ ax ax l afqaL (2.9) by M fdaeflae at at 8T Making use of Equations (2.9) we now rewrite Equations (2.1) as 3 D 2 hi _ I W = 22.. .a. 2 §[(1-p.)v Ix + (141,) ax] c hwx + 3;] (g: + v M) Ix (2.108) 12 D 2 32 a! fie. 21.2 + - 'h + = + 2. b §[(1-u)v I, (1+u) a,r] G [Iy 8T1 12 (at v 8T> Iy ( 10 ) G'h(V2w +.§) + 5(x) 6(T) H(t) = ph(§E-+'v 3;)2w (2.10c) aIx aI =.___ ._;l 2 i Bx + 8T ( .lOd) 2 2 2 where V = a—§'+ a—2 and the primes on the displacements have 5x 8T been dropped. The corresponding initial conditions (2.4) and the con- ditions at infinity (2.5) now become W(X,T,0_) = VI(X,T,O_) = 0 II C IX(X,T,0_) = IX(X:T:0_) (2.11) II 0 Iy O, 16 lim W(X,T,t) = O \XI-s we lim I (X,T,t) = O (2.12) x aw x ITI~°° lim I (X,T,t) = 0 . IXI—w y III-m Similarly, the plate-stresses in Equations (2.2) become BI BI = —;E ._JL Mx D O the solution w together with all its derivatives must tend to zero as Ix‘ a m and ‘T‘ a m. We thus have a completely formulated mathematical problem in terms of w only. CHAPTER III METHOD OF ANALYSIS In this chapter, the mathematical problem formulated above for the system of Equations (2.10) will be solved by the method of integral transforms. A double-Fourier transform with respect to x and T and a Laplace transform with respect to t will be performed to reduce the set of Equations (2.10) to a set of algrbraic ones, which are then solved for the transforms of the plate displacements. The inversions of the Fourier trans- forms are accomplished by the method of contour integration. How- ever, as we are only interested in the limits of the solutions as t a m, the inversions of the Laplace transforms are unnecessary. By a well-known Tauberian theroem [30], the limits of the solutions as t tends to infinite are given by the limits of the Laplace transforms as the Laplace transform parameter tends to zero. As will be discussed later, the limiting process of letting the Laplace transform parameter tend to zero offers at the same time the needed guidelines for selecting contours of integration for the Fourier inversions. Our final analytical solutions take on the form of integrals in one variable. 3.1 Solution by Integral Transforms The equations in (2.10) subjected to the conditions (2.11) and (2.12) will be solved by performing a double-Fourier transform l8 19 with respect to x and T and a Laplace transform with respect to t. The double-Fourier and Laplace transformation of a function f(x,T,t) will be denoted by :F(m,§,p), in which the tilde and bar over a variable designate, respectively, the Fourier and the Laplace transforms [32]. A combined transforma- tion and its inverse are defined by ?(w,§,p) = f; [:0 J": f(x,T,t)e-iwx ' igT ‘ ptdx dT dt (3.1a) and ~ 1 m m 1* ' + ' + t f(er,t) = T AC If-os I-oo f(w3§ rp)e1wx lgT P 8n i dw dg dp (3.1b) with w and g being real and Re p > 0. Here w and g are the Fourier transform parameters, p is the Laplace transform parameter and C denotes the Bromwich contour. Equations (2.10) are now transformed into D 2 2; ‘3'. g, g. §'[(1 ' u)(-w - é )IX + (1 + u)1we] - G'h(Ix + iww) h3v2 Z; = P————‘12 (3 + 1g) yx (3.2a) D 2 2 £1 .5- . g: .g. 2 [(1 - p)(-w - g )Yy + (l + u)1§¢] - G h(‘yy + igw) 3 2 N h v p_ , 22 = + .2 p 12 (v 1%) Iy (3 b) G'h[-(u)2 + ng: + $]‘+ %'= phv2($-+ i§)Z: (3.2c) Q = iwa +'i§Iy (3.2d) where 1:, I;, I;, and :3 are the transforms of w, Ix, Yy, and Q respectively. 20 ~ The solutions to the algebraic Equations (3.2) for 1:, N '2‘ Ix, I], and ‘3 can be easily obtained. These solutions may be expressed as :; _ l. iw wa’g’p) ’ ' D mm») (3'3) 5’- _ - I __31_§.__ wa’g’p) — D pQ(w.§,p) (3A) 5 1tum2+gmf1 “M” = ’ '13 raw-gm) ' (3'5) and 1+ D2[C£A(%+ig)2+w2+g2] g. _ l. phvz (3 6) w(w’g’p) ” D poms») ° where 2 2 2 2 QUDIEIP) = [ID + ID1(§,P)] [w + w2(§,P)] (3'7) in which (g ‘ i E) _ 2 L mien») = s2 - -—-7—1- {C(g - 15) + Am -113) + .232} (3.8a) (g " i R) _ 2 2 L wig,” = 52 _ ____.2__V_ {C(g - i, 5) - A[(§ ~15) + a 1“} (3.81)) with E = v2(L2' + 1—2-) (3.8c) v2 V1 A = v2(1_2 _ '13) (3.8d) v2 V1 _ 373 v a - Ahv (3.8s) 21 Here in (3.8a) and (3.8b), for definiteness, the branch for .22 2’5 . .. [(g - 1 v) + a ] has been taken that IS real and posrtive when g is real and p is zero. Equations (3.4) - (3.7) yield the solutions in the trans- formed plane. They will be inverted in the next section. Before proceeding, let us for convenience introduce the auxiliary functions ;(X,T,p), E(x,T,p) and ;(x,T,p) defined as . + . eiwx igT 30mm) = -1—2 I“ dw d: (3.9a) 4T1 '°° " '°° PQ(w»§.P) , 2 iwx + i§T ‘ ___ A. °° 0° (13) e 8(x,—r.p) (”T2 I-.. I-.. w(w,§’p) do as; (3.9b) . 2 iwx + i§T _ 1 m Y(X,T,p) = ZTT—z J-” fiat) (1:3 (irgrp) dw dg (3.9C) It will be seen later that all plate-stresses and plate- displacements can be expressed in terms of these auxiliary functions and their derivatives with respect to x and T. It is thus only necessary to invert these auxiliary functions E, B and §. 3.2 Inversions of the Transformed Solutions The auxiliary functions defined in Equations (3.9) will be inverted in three steps: First the Fourier transform with respect to w will be inverted by the method of residues; second by appeal to the Tauberian theorem, the paths of integra- tion in the g-plane are determined; then, third the inversion of g-integration will be performed by contour integration. The 22 processes are presented in the following sub-sections. 3.2.1 m-inversion To expound the process of inversion, we shall commence by considering the w-inversion. The m-inversion here is quite elementary and can be easily performed by using the method of residues. Note that the integrands of the inner integrals in Equations (3.9) have simple poles in the complex w-plane, located at the zeros of Q(w,§,p). The location of the poles of the integrands of Equations (3.9) can be calculated by setting Q(m.§,P) = 0 (3-10) from which it is found that they are located at u) = i iw1(§,P) (3-118) and w = :_iw2(§,p) (3.11b) The w-inversion may be carried out by closing the contour in the upper half-plane for positive x and in the lower half- plane for negative x. For x > O, we consider now the closed contour Cw comprising the straight line F: -R < Re w < R; and the infinite semicircle C1: ‘w‘ = R a m in the upper w-plane as depicted in Figure 3. The integrands of Equations (3.9) are analytic functions 0f the complex variable w everywhere in the upper m-plane except at isolated simple poles. Let F(w) denote such a typical 23 w-plane ’ —- — — ~ ‘ z’ ’ \\\ " ‘L / \ 1’ . ‘\ / 1“’1 \ l / / I 1032 \u)‘ = R -* 0° \ ’ \ I I I l L r I -R I“ o R Figure 3 Closed Contour C m in the Complex w-Plane 24 integrand. Then, since the contour Cw encloses two simple poles in the upper half plane, by Cauchy's integral theorem and the calculus of residues we have (IF + IC1)F(w)dw = 2ni Z (Residues of F enclosed by cm) (3.12) As R a m, the integral along the infinite semi-circle C1 can be shown to vanish, i.e., AC F(w)dm = 0 provided x > O 1 It follows from Equation (3.12) that l -—'[ F(w)dw = i 2 (Residues of F enclosed by C ) (3.13) 2n T w Upon using (3.13), the auxiliary functions can be written as — 1 co elmX'I'ig'r a(x,T,p) = §;-I_miz [Residues of EEYETEjg) in the upper half of the w-plane]d§ (3.14s) , 2 iwx+i§T "‘ 1 co . (IF) 8 . , , ='—- ’ R d f 1 the r 8(x T p) 2n I-a32 [ e81 ues o pQ(w,§,p) 1n uppe half of the w-plane]d§ (3.14b) . 2 ' ' 1 (1w) eiwx-I-igT y(x,T,p) = a; I-miz [Residues of pQ(w,§,p) in the upper half of the w-plane]d§ (3.14c) As illustrations, only the details of the calculations of residues for 3(x,T,p) will be presented below. We have 25 iwx + igT Residues of pQ(w.§.p) iwx+i§T . e = Resrdues of 2 2 2 2— P(u) +w1) (u) +w2 iwx-I-ig'r iwa-igT = Residues of - e + e (2 2)(2+w2) (2 2)(2-Iw2) pu’i‘wz ‘” 1 pwl-wZ u’ 2 J eiwx+i§T = lim ~(u) - in) ) + , ( 2 2 2+w2 LII—+1031 p col-Luz) (u) 1) iwx+i§T lim (u) - in) ) e . 2 ( 2 2 ( 2+w2) III—+1032 p wl-wz) u) 2 w1x+i§T -m2x+i§'r z: -e + e 2(im1)(mi~w:)p 2(iw2)(wi-w:)P It is to be recalled that only those roots which lie in the upper w-plane in Equations (3.11) are used in the above calculation. Thus, from Equation (3.14s), it is found that -w1x+i§T -m2x+i§T — 1 co e e 0(xaTIP) = Z; f-m [E 2 2 + 2 2 —] d8 pu’1(‘”1"“’2) puL’2("L’1“”2) which may be rewritten as -w X+i§T -w x+i§T ‘ l 00 e 2 1 on e 1 F a(x.«r,p> =;;j_m 2 2 as -;j_w 2 2 d. Pw2(w1-w2) pw1(w1-w2) for x > O (3.15) For negative x, the contour must be closed in the lower w-plane and the calculations proceed in exactly the same manner. __<::_ET__ PQQDIEIP) , It may also be observed that as the expressions 26 . 2 igw . 2 igT SiSl—§———- and Slgl—£———- in —' - a d _ res ectivel are pQ(w,§,P) PQ(w.§.P) 0’: B n Y P y even functions of w, it follows that a, B and y are even functions of x. Consequently, from Equation (3.15), we may infer that igT-(n2\x‘ ig'r-w1\x‘ 0° e co e (1(x3T9p) — 411 I-oo 2 dg - 41-1 [.09 2 2 dg w2(w1-w2)p w1(w1-w2)p for all x Similar expressions may be obtained for a and Q, but the details will be omitted here. The results after the w- inversions will be expressed, for convenience, as 3(X.'r.p) = 31(X.T.p) + 5209mm §(x,'r,p) = EIGHTH?) + §2(X.'r.p) §(x,w.p) = ?l + §2(x.np) where -m2‘x‘+i§1’ _ 1 m a1(x,'r,P) = my“, e 2 2 dé w2(w1-m2)P -u) ‘X‘+i§7 "' 1 on e 1 a2(xn.p) = - EL,” 2 2 d5 w1(w1-w2)P 2 -w2‘x\+i§'r _ 1 m . 31(X,T:P) = mI-m (I: e dg 2 2 u"2(““1"“’2)p 2 ~m1‘x‘+i§7 _. 1 ’ (3 (3 (3 (3. (3 (3 (3. .16a) .16b) .l6c) 17a) .17b) .17c) 17d) 27 wZ‘x‘i-ig'r —' 1 m u’28 Y1(X:T:P) = Z1: j-oo 2 2 dg (30179) (wl’wz)P w1\x\+ig¢ _ 1 co (.016 Y2(X.T.p) = - EL“, 5": (3'1”) 2 2 Gnl-w2)P It is observed that all the expressions given in Equa- tions (3.16) are of the form mm?) = 2.17:1: Elei§"d§ + 21:11]: 32(x,§,p)ei§“'d§ = 11:10:00 E(X:§2P)ei§Td§ (3.188) in which E(X3§3p) = 31(X:§ap) + 32(x,§,p) (3.18b) where 5(X,T,p) is a generic auxiliary function. An inspection of expressions for m1(§,p) and m2(§,p) given in Equations (3.8b) and (3.8c), respectively, reveals the complex conjugacies that * u)1(X,'§,P) w2(x,§,p) (3018C) and * w2(X,'§,P) = w1(x:§:P) (3018(1) where * denotes complex conjugation. With the aid of (3.18c) and (3.18d), it follows that the relations :- ark G1(X:"§ap) = 62(X,§,P) (30189) 28 and E,(x.-r~;.p) = El (3.180 hold for real g and Re p > 0. Thus 1E(x,-§,p) in the integrand given in Equation (3.18a) is the complex conjugate of Ekx,§,p) whenever p is real and positive. We shall make use of this fact in carrying out the g-inversions. 3.2.2 g-inversions As was pointed out earlier, we are primarily interested in obtaining the steady-state solutions as t tends to infinity. It thus suffices to regard p as a small positive parameter. The integrands in the expressions given in (3.17) have branch points in the complex §-plane coinciding with those of w1(§), w2(§) or wi(§) - w:(§). From the expressions for wi(§) and w§(§) given in (3.88) and (3.8b), it is easily deduced that the integrands have branch points at g = g1, g = g2 where g1 and g2 are given by i(§-+ a) (3.19a) an H III and III 52 1(5 - a) (3.19b) 2 2 2 k which are those of w1(§) and m§(§) due to [(§ - i 5) + a 12. Furthermore, on setting wi(§) and w:(§) equal to zero 2 (g ' i 5) -— 2 2.2 2 a g -—'2———{C(§-iv)-_I-_A[(§-iv) +a]}=0 29 and solving for 5, one finds that 2 £3 ; i 5+ C(23) (3.20a) V 2 xi ‘34 E 1 v - 0( 2) (3.20b) are the branch points for both w1(§) and w2(g); and, in addition, when v > v > v 1 2 v4 1 ‘ 2 2 Vlv2 2 g5 a b - i 5 2 2 + C(93) (3.20c) (1 - 12—) <1 - 12-) V1 V2 and v4 1 ' 2 2 Vlv2 2 56 = -b - 15 2 2 + 0(95) (3.20d) (1 - "—2) <1 - 1’3) " V1 V2 where b = 3a 2 (3.20e) v v 5 2m - —2—)<1 - 7)]2 V1 V2 are the branch points for w1(§) and m2(§); and, when v > v2 > v, g5 is a branch point for w1(§) and 56 is a l branch point for w2(§)- From the eXpressions given in (3.19) and (3.20), it is expected that the character of the disposition of the branch points will be basically different depending on the magnitude of v relative to the plate speeds v and v . The locations of the 1 2 branch points for the two cases v1 > v2 > v and v1 > v > v2, which have been referred to as the subsonic case and the 3O intersonic case respectively, will now be discussed. In the subsonic case, §1 lies along the positive imaginary axis, and 52 on the negative imaginary axis of the g-plane; g3 and g4 both lie on the positive imaginary axis; g6 is located in the third quadrant and g5 is in the fourth quadrant of the g-plane. These branch points are shown in Figure 4. The path of integration, i.e., the real §-axis, now separates the branch points g1, g3, g4 from g2, g5, 56. Branch cuts must be chosen and they should not cross the real g-axis. This is achieved by introducing branch cuts from g1, g3 and g4 to + im, and from g2, 55, g6 to - im as shown in Figure 4. For small positive p, the integrands can then be made analytic in g and single-valued in the slit g-plane, with ml, wz and [(g - i E)2 + a2]% being taken to be the branches with positive real parts when g is real. In the intersonic case, b is imaginary and all the branch points are now located on the imaginary g-axis, with g1, g3, g4 and :5 being above the origin and g2 and :6 below the origin as shown in Figure 5. To render the integrands single-valued analytic functions of g, branch cuts are introduced from §1, g3, g4 and g5 to + im and from g2 and g6 to - im as shown in Figure 5. Again, in the slit g-plane we shall take the branches of wl’ m2 and [(g - i 5)2 + a2]% to be those with positive real parts when g is real. It should be noted that the branch cuts in both the sub- sonic and intersonic cases are symmetrically located with reSpect 1mg *Re g Figure 4 Branch Points and Branch Cuts in Complex g-Plane in the Subsonic Case 1mg ’Re L‘L¥d#fihhfififf¥rs v7r un un 0‘ N [J [11141 (77Ifll’ Figure 5 Branch Points and Branch Cuts in the Intersonic Case 33 to the imaginary axis. On account of the fact that the integrands in Equations (3.16) become complex conjugates to the original ones if g is replaced by -§ and the symmetry property of the branch cuts, the integration from -m to '+w is equal to twice the real part of the corresponding integration from 0 to +m. The expressions for a, E and §' in Equations (3.17) can thus be rewritten as 'LDZIXI+1§T 516mm) 92—: we 2 2 dg (3.21a) w2(w1-w2)P wlIxI+i§T - R a2(xn.p) = - fifg e 2 2 dg (3.21b) m1(w1-w2)p 2 wz‘x‘+i§T EIGHTH» = - 15—: j: g e 2 2 dg (3.21C) w2(w1-w2)p 2 m1|x|+i§¢ §2(x.'r,p) = 1% j"; g e 2 2 dg (3.21d) w1(w1-w2)p ‘u’zIxI+i§T _ Re co wze Y1(X,T,P) = 2; IO 2 2 dg (3.219) (wl-w2)p w e-ml‘x\+1§T —(x'r)=-R—ej“"1 dg (321f) Y2 ’ ,p 2“ 0 (wZ-wz) 0 l 2 In order to obtain the limits of the solutions as t tends to infinity, we apply the following Tauberian theorem [30, 31] lim C(t) = 1im+ p 5(1)) t—m p—O 34 where G(t) stands for a function of t and 6(p) its Laplace transform. When G is taken to be any of the auxiliary functions in Equations (3.21), a steady-state solution is reached in the limit as t a m. Thus G(x,T) = lim G(x,'r,t) = 1im+ p 5(x,«r,p) (3.22) t-m p—O Therefore, when only the steady-state solutions are sought, it is not necessary to invert the Laplace transforms. We next investigate the limits of the auxiliary functions as p a 0+. In the subsonic case, as p approaches zero, we have gl 4 ia, g2 a -ia, €3,54 a O, g5 a +b and g6 a -b. The path of integration in Equations (3.21) must be deformed to F1 which lies along the entire positive real g-axis except that it circumvents above the branch cut from O to +b as depicted in Figure 6. In the intersonic case, we have g1 a ia, g a -ia, g3,§4 a 0, £5 « ib and £6 a -ib as p approaches zero. The path of integration is deformed to F2 which runs along the entire positive real g-axis except circumventing below the branch point at g = 0 as shown in Figure 7. Upon applying the Tauberian theorem in Equation (3.22) to Equations (3.21), we obtain the following expressions for the auxiliary functions a, B and y as t tends to infinity: szxI-I-ig'r al(x,T) = %E If, e 2 2 dg (3.23a) m1Ix|+ig¢ 02(Xfl') = -%§jr e 2 2 dz; (3.23b) j w1(m1-w2) 35 kkk sin-Re § a V V ‘ x f N -L A L in K V 0 6 F: 2 g \\\\\ {LKILhKNKKKLNKL\ AI 11 V 1 ~ I‘I‘I‘I Figure 6 Path of Integration in the Subsonic Case 36 1mg Se g 37 I 50 (D 9—3 .1 If“ m 81(X9T) " ' 'Z—T; 2 2 dg (3.23C) 82(xur) = g dg (3.23a) ( ._ lie “’28 d (3 23 ) Y1 X2T) - 2" fr. 2 2 g ° C J (wl-w2) -w1IxI+i§T Re wle 220“) = - ZIP. 2 2 as; (3.2315) _] (ml-102) Here ml and w2 are the limits of the expressions given + in (3.8a) and (3.8b), respectively, as p ~ 0 and assume the following values 1v u"1 = /2 [cg " “52 + 32) 2115 (3-23g) (92 = [2 [cg + A(§2 + az)€]!5 (3.2311) where C = 2 - E 2 v2 =2 _y.2.-._2 (3.231) V1 V2 The contour T j = 1, or 2, is determined by the speed j, v in accordance with whether it is subsonic or intersonic. In terms of the axuiliary functions in Equations (3.23), the steady- state solutions for the plate-diSplacements are w(X,T) = 1% {a=-%[mmo-wwmo] 02%) nfifl)=-%%cfin) 61%) gon)=-%§aen) 02%) Likewise, the plate-stresses can be expressed as MX(X,T) = ’[Y(X:T) + LLB(X,'T)] (3°253) My(x3'r) = {8(X9T) + LL'Y(X,'T)] (3.251)) 2 Myx(x,T) = -(1 + n) i;;; a(x,¢) (3.25c) QX(X.T) = g; E- 925 a(x,w) - v(x,T)] (3.25d) chx.¢> = g; [- 925-e] (3.25e> where a(x,'r) = a1(x,'r) + a2(X.'r) (3-268) 8(x3T) = 61(xaT) + 82(X,T) (3.26b) w(xn) = y1(x,'r) + Y2(X.T) (3.260) It now remains to evaluate the integrals given in (3.23), the appropriate contours F and F2 having been shown in 1 Figures 6 and 7. We shall now consider the subsonic and inter- sonic cases separately. i) The Subsonic Case: In the subsonic case, the contour F in Figure 6 may be 1 further deformed for more explicit evaluations of the integrals. In view of the integrands of the auxiliary functions, it is 39 observed that on a circular arc CR with radius Ig‘ « m and center at the origin we have w1(§) = I; ’9? = 5*; (3.27a) _ Efl=L (02¢) —§/ 2 v1 (3.271» where v 2 l = ___ = 3.2 v1 I A 71:73! W ( 7c) l v 2 2 = ___ = .2 V2 C-A F7 (3 7d) 2 v Hence, as IEI a m x . -w1\xI+i§T - V2 + lgT e a e (3.28a) §|xI , - +1 -w2IXI+1§T v1 5" (3 .28b) e a e We choose CR in the first quadrant for T > O and in the fourth quadrant for T < O. The closed contours for T > 0 and T < 0 are shown in Figure 8 and Figure 9 respectively. More specifically, for T > O, the closed contour com- prises the following paths: (1) the path of integration F1 from 0 to R along positive real g-axis; (2) CR: a circular arc Ig‘ = R from g = R to g = iR, R a w; and (3) -C11: from g = iR to g = i0 along the positive imaginary g-axis, circumventing the branch points at g = ia 4O Figure 8 Contour of Integration for Subsonic Case T > O in the 41 Figure 9 Contour of Integration for Subsonic Case T < O / / 1’ CR in the 42 and g = 0. Since there are no singularities of the integrands inside this closed contour, by Cauchy's integral theorem, we have, for a typical integrand g(§), (§)d§ +’ (§)d§ ‘ (§)d§ = 0 J”I‘lg ICRg [C 11g Since an inspection of the integrands shows that the integral along CR vanishes, it immediately follows that 8(§)d§ = g(§)d§ frl fcll which implies that the contour F1 may be deformed into the path C11 in Figure 8. Similarly, for T < O, we follow the same arguments to deform the contour F into the path C 12 consisting of 1 L1 + L2 + L3 +'L4 as shown in Figure 9 where (1) £1 + L2: from g = + i0 along the positive g-axis to g = +b and then return to g = -i0 by circumventing the branch cut around the branch point g = +b; and (2) L3 + L4: from -iO to -im by circumventing the branch point at g = -ia. ii) The Intersonic Case: In this case, it is observed that as \§\ a m ,E;é.= 13 w1(§) d g 2 v2 (3.29a) Skbé =‘§_ m2(§) d E‘/ 2 v1 (3.29b) where 43 _ 2 _ v1 V1" C+A- 2 (3.29c) V-V V 2 _ 2 v2 _ A_C — (3.29d) V'Vz Hence, as \g\ a m -ml‘x\+i§T ei§(T - 13%) e a (3.30a) -w2\x‘+i§T - Sill-KL + igT e d e 1 (3.30b) Thus, for 01, BI and v1, we close the contour in the first quadrant in the complex g-plane for T > O and in the fourth quadrant for T < 0. Similarly, for 02, 82 and Y2’ we may introduce a closed contour in the first quadrant for x . V , as shown in 2 2 Figure 10 and Figure 11, respectively. Following the same T > 1%13 and in the fourth quadrant for T < arguments as in the subsonic case, we may deform the path F2 as follows: x . For T >"L---L in a V 2 a1, 81 and v1, F2 is deformed into the new path C21 which 2, 82 and v2 and for T > 0 in traces the positive imaginary g-axis exceptthat it goes around the branch point 5 = O, E = ia and g = ib as shown in Figure 10. X For T < 1—1' in a , B and y and for T < 0 in V2 2 2 2 01, 81 and v1, F2 is deformed into C as depicted in 22 Figure 11 which runs from g = -i0 to g = -ia, g = -ia to g = -ib and g = -ib to g = -im, connected by two small 44 Figure 10 Path of Integration for in the Intersonic Case T>O and T>llL V2 45 ' m3 / Figure 11 Path for T < O and T < lgl' in the 2 Intersonic Case 46 semi-circles at the branch points at g = -ia and g = -ib. More detailed evaluations of the above integrals along the newly deformed path in each case will be presented in the next section. 3.3 Evaluations of the Auxiliary Functions We now evaluate the integrals for the auxiliary functions 01(Xfl), 012(Xfl), SIGHT). 320cm). Y1(X,T) and Y2(X,’T) given in (3.23), following the deformed contours as shown in Figures 8, 9, 10 and 11 above. 3.3.1 The Subsonic Case The contours C11 and C12 shown in Figures 8 and 9 will be employed in the subsonic case. Thus, for T > O: w2\x‘+i§7 _ Re e al(x,T) - E Io 2 2 dg (3.31a) 11 w2(w1-w2 -w ‘x‘+i§T - 5‘?- e 1 d? 3 31b 02(X,T) - " 211 J‘Cll (2 2) g (. ) u’1 U"1“”2 E28 '0.) 2x‘ ‘+1§T 610m) = - —jC11d§ (3.3lc) M0» wz) “2 -w1‘x \+i§T B (xn) = dg (3.31d) 2 13%ch w1(w1‘wz) -wz‘x‘+i§T Re wze v1(x,T) = - fife 2 2 dg (3.31e) 1 ml - wz) 47 v (x,¢) = — d§ 2 2n fell (mi - mi) and for T < 0: fiDZ‘X‘+i§T _ 32. e d al(x,T) - 2n Iclz ( 2 2 E wz m l-mz) m \x‘+i§T Re e 1 02(X,T) = ' ET—IJ‘C 2 2 dg 12 w1(w1 - wz) 2 w2\x‘+i§¢ _ 132 § 61(X,T) " ' 211’ jC e 2 2 (1% 12 w2(w1 - wz) “2 -m1‘x‘+i§T a,(x.¢> = Re I 2 as C 12 :1(w2 - wz) -u)2‘x\+i§T R “’ze Y1(X:T) = Z: IC 2 2 dg 12 u’1 ' U”2 ”hex-“’1‘ Xl+1§T Y2(X’T) = ' hie 2 d5 (.01 " (D2 It is easily seen, as an inspection of Equations (3.31) (3.31f) (3.323) (3.32b) (3.32c) (3.32d) (3.32e) (3.32f) and (3.32) will reveal, that the following remarks hold true for the integrations along the paths 011 and C12: (1) Integrations along the small semi-circles whose radii are designated as r1 and r2 contribute nothing as r1 and r2 tend to zero, (2) Integration along the small semi-circle whose radius is designated as r0 contributes nothing as r tends to zero 0 whenever the integrands of the auxiliary functions in Equations 48 (3.31) and (3.32) are of the order 0(lg), where s < l, at the § branch point g = 0. With these remarks, it then easily follows that, for T > 0, the contributions to the integrals 01,2, 81,2 and Y1,2 come from the linear paths L1 and L2 in Figure 8, namely, 0‘1,2 For 31,2 : [C11 = le +ij2 (T > 0) (3.33) V1,2 Similarly, for T < O, the contributions to 01 2, 81 2 and 3 3 come from the linear paths L1, L2, L3 and L in Y1,2 4 Figure 9, that is, 0’1,2 For 51 2 : $012 = [£1 + [L +»f£ + [L4 (T < 0) (3.34) 2 3 For the integrals al and a given in (3.32a) and 2 (3.32b) for T > 0 and T < O, the order s of the branch point at g = O exceeds unity. Consequently, the contributions from integrating around the small semi—circle do not vanish as rO tends to zero. We thus invoke the convolution theorem. We first express a1 and a2 as, for T > 0 Es iéT 2n C1161(§)GZ(E,X)e d5 (3.358) (11(Xfl') a2(x,T) — - gfi-jc Gl(§)G3(§,x)eigTd§ (3.35b) 11 that is, as the inverses of the products 61(§)Gz(§) and 49 Gl(§)G3(§) respectively. Then the convolution theorem states that such inverses can be expressed as convolution integrals as follows: a1(x,'r) = 15:; J”; g1(T - T')g2(T')dT' (3.36a) and R a2(X,T) = - EE-fg g1(T - T')g3(T')dT' (3.36b) where G1 and 31’ G2 and g2 and 63 and g3 are transform . . 1 pairs. In the present case, for T > O, G 13 taken as '- 1 1g and G and G as the remaining factors of the integrals. 2 3 Since the inverse g(T - T') is the Heaviside unit step function, we thus obtain, for T > O a1(x,T) =‘fg a:(x,T')dTI (3.37a) a2(x,T) = f; a:(X,T')dT' (3.37b) where * . -w2‘x\+i§T a1(x,T) = %§'fL +L 1§e 2 2 dg (3.38a) 1 2 w2(w1 - wz) -w1‘x‘+i§T * . a2(x,T) = - %9-fL +L 1§e 2 2 dg (3.38b) ” 1 2 w1(w1 - «02) After the above indicated integrations are carried out, we obtain, after omitting the details, the following expressions for the auxiliary functions. 50 For T < 0: 01(x,T) = I; Q:(X,T')dT' * ' b ' al(x,T ) =/%nA f0 féélqh cos(gT )cos(§2€‘ (p3\x‘)d!; QT l - Mfg/911% [($99 Coshcza ch‘XDcosCé; (p10\x\) + (p10 sinhféC (pglxh -sin(€c‘ cplo‘x‘)]dC J" 1 co - 2/211A «Ya f—QCQZCPS cos(§% cp5\X\)dg 1 U + z/znA I: fawn. “Mfg W‘X‘m (3'398) _ 1 b ,1; . 81(X,T) - - [ZTTA IO $1ch Sin(gT)cos(§% cp3\x\)dg QT l - m‘ra [Le— [(pgcosh(éL (pg‘xhcoscgg‘ cplolx‘) O (96R + (plosinhCég‘ cpg|x\)sin(‘§g cplolx‘)]d(; l m‘zgeg'r .. 2/2nAJ8 CPZCPS COS(§%(P5‘X‘) QT l z/znAELéjgj c05<§§w4|xhd§ (3.3%) .4. + 51 Qp3 . Z/ZnA.yO\/C@1 Sln(QT)COSC£% m3\x X‘)dg _1_ a e57 411A f0 foP6 v1(x,T) = - .022; \We .1014) - cplosinhcgg wg‘x‘)sin(§£ QplO‘X‘)]dg 1 58‘;T + 4/2ncha,/Q@2 COSQ§g @5‘X\)dg eQT “he Z/ZTTA j: fCfiPZ COS%©4‘X‘)dQ (3.39C) a2(x,T )= [0% *(X T')dT * . 1 b l , 02(X,T ) = fznA f0 fQCPICP3 cos(§T )Cos(‘§% (p3‘x\)dg QT 1 ([L (LC: - 31:; I; 78 COSh 2 (pg‘X‘)COS 2 cplolxl) C(96R + .plosmhg ¢9\x\>sin<§—9 cp10\X\)]dC, l egT firmew QT 1 co +mfa7§eggfcos<fiwmdé (3'40” _£L_ sin(QT)cos(‘[; (p3\X\)dQ 91¢ 3 /2 23;___ 10 Wm“ WM) 2.22.122) 82(X’ T) = /2_TT_A [O + @1OSinhogg w9‘x‘)3i"0§; w1olx‘)]dg :2e QT -Z/ZT‘IA f: (pwa cosC§§ CPS‘X‘) 2:21.213. 52 1 b ‘93 . y2(x,T) = 2/211A [0 [$91 s1n(QT)cos(‘§% cp3‘x\)dg QT 1 (\[L .. 4—“: I; f;% [mgcoshcgg‘ cp9\x\)cos 2 LPIO‘X‘) - cplosinl‘xczfl cplo‘x‘)sin(‘2LC‘ cp10\x‘)]dg QT 1 m‘Pse + 4/2nA [a 7E5; “38% ¢5\X‘)dg <9 eh 4 fQQPZ - 2er j: cosQ/gz @4‘x\)dQ (3.4%) where 2 21/ (P1: (Q +a)2 2 21’ W2 = (C 'a)2 % 0 T * U I 011(Xfl) = I0 01(X,T )dT * e‘TC 01(X,T)= 1A21’1'J‘Oe‘7II—II—ngc‘p6[R[q>9C081”‘(LCL (pg‘x X‘)COS(& cplo‘x‘) + cplosinhczfi cpg‘xhsinfél cplo‘xbjdg +7—211A J": 7—CCP2:5 :}{OS(‘%QPS\ deg Z/ZTTA I: [ET—2:} cosffggpa ‘XDdQ (3.428) "QT 1 e 51(x,T) = " 2—11A7 j: [LR—g; [cpgcoshcgg (p9\x\)cos(‘§; CPIOH‘) + (PIOSinhéLg (pg‘xhsinCé; (p10\x‘)]dg 1 + Z/ZnA fig“— (9;: :OSC/Lg‘ W4\X\)dt; -2/2nAf%—cp:1r\coscf% cps\ x‘)dQ (3.42b) “TC Y1(X,T)= --4__nA [02'— [“36 [cpgc:sh(‘fl gp9\x x‘)cos(‘a; CPIO‘X‘) - (plosinhCéL cpg‘x‘)sin(€g cplo\x‘)]dg U1 1 - M21118 Eva chp: :OSCérz> (P4 ‘XDdC 1 + 4/2nA E5 fzgpz “330% €95l>dg (3.142(2) S4 = T * 1 t a2(X.T) f0 a2(x,T )dT 'QT * ' 1 02(X,T ) = my; [2961‘ [c.pgcoshfgg gpg\x‘)cos(‘§L cplo‘xp + ‘PloS’inhcg; 0, and in evaluating a and 2’ 82 Y2 for T > 131-. The path C in Figure 11 will be used in 2 evaluating a1, 81 and Y1 for T < O, and in evaluating a 22 2, 62, Y2 for T < L51 . V2 As it was in the subsonic case, the integrations around the semi-circular paths about the branch points in Figure 10 and Figure 11 vanish as the radii tend to zero. It follows that only the linear paths L1, L5 and L5 or Li, L5 and L' along the 3 branch cuts contribute to the integrals. After some lengthy manipulations, we obtain the following results: For T < O * 01(X,T) = I; 01(X,T')dT' * . 1 m 01(X,T )= /TnA ja‘/::ZTQC cos€§%‘ QA X‘)dg (3.44a) QT 81(X,T) = _l;__j“’¢23i__ Q§g wa‘x‘)dg (3.44b) /2TTA a (p :p C OS 2 4 _ 1 m ¢le Y1(x,T) - - Z/QWA fa y2;;_'COSQ§% Qq‘x‘)d§ (3.44c) and for T < 161 2 _ T * I t 02(X2T) - I‘X‘lvz 02(X,T )dT 56 “:(X’T') " 7—211”: a/€:2‘Ps °°S°§%‘Ps"“)d9 32(x,T)= -./2nA I: ¢::5 dcosQég ¢5\x‘)dg 1 Y2(X’T)=Z/'21TA ra,/Z:: TCOS€§%‘ $5‘ x‘)dg In the same manner: for T > 0 01(x,T) = I; a:(X,T')dT' , . _ 1 m e-QT' (11(Xfl') — -/21TA jam C03C§§ cph‘xbdg 1 81(X3T) = szTA jalL— (P:CP4 “2.080% @4‘X\)dg 1 Q/e-QT Y1(X’T) = ' Z/Zrm J: fécpz “35% CPIJXDdQ and T >145-L v2 02(X,T) = "rx‘lvz Q:(x97')dT' a:(x, T') = -—'jO‘/;:T 6R [mgcoshClg Q9|X X‘)coscég @101X\) + (glosinhcgQ (p9\x\)sin(‘§Q ¢1O\x\)]dg e-cT 1 b + m I. m; 608% «aw 82(X,T) = iIO Re $6 ‘££‘--:-g—T'[cpgcoshC‘l'Q @9‘X x‘)cosC£g Q10‘x‘) + (plosinhcgge :pZ‘xDsinCZ-C’ (plo‘x‘ndg Mscfi— /2nA J‘lia— (p 82.95 f2Cp 5\X\ ) d'; (3.453) (3.45b) (3.45c) (3.46a) (3.46b) (3.46c) (3.47a) (3.47b) 57 1 w”) = - 2.7137;ch °°Sh<€§cp91xh°°s<§ mm - $1091.}.ch .9\x\>ssmc§9 cplo\x\>1dc se-QT +Z/ZnA Ia [2.92 ”3% CP5‘x1)d£ (3.47c) where $2, 94, $5, $9, $10 and R are defined as in (3.41) with the exception that b = A8 I in the 2 (1-'—)(1- ) V V 1 2 present case. CHAPTER IV DISCUSSIONS AND CONCLUSIONS In the preceding chapter, the inversions for the auxiliary functions a(x,T), B(x,T) and y(x,T) have been carried out and the results presented in the form of integrals. The integral representations given in Equations (3.39) through (3.43) and Equations (3.44) through (3.47) constitute the complete solutions for the subsonic and intersonic cases, respectively, of the mathematical problem formulated in Chapter II. These steady- state solutions depend on the distance from the load but not on the distance from the initial point of application. Some important features of the solutions will now be discussed. This chapter will then be concluded with some further remarks. 4.1 Discussions The analysis presented in the preceding chapters was carried out in order to obtain the steady-state dynamic reSponse of an infinite plate subjected to a steadily moving force. The so-called improved or Timoshenko plate theory was employed. The speed v of the moving force was assumed to be intersonic and subsonic relative to the two plate Speeds v1 and v2, i.e., v1 >'v > v2 and v1 > v2 > v, respectively. Based on the physical evidence that the plate no longer remains quiescent ahead 58 59 of the moving force in either of the cases here, the problem was formulated mathematically as an initial value problem as in Chapter II. A combined integral transform, namely, a double Fourier transform with respect to spatial coordinates and a Laplace transform with respect to time was performed to solve this initial value problem. It is noted here that a direct application of a double Fourier transform to the equations governing the steady- state plate responses would result in the difficulty of defining the correct contour of integration properly for the Fourier integral to yield the unique physical solution. This difficulty was circumvented here by the present formulation and by the performance of Laplace transform in the present study. The use of the Laplace transform provides a parameter p, which is considered as a small positive quantity but different from zero, corresponding to which the behaviors of singularities arising in the transformed plane are better understood. In the limit as p tends to zero, the path for the contour integral was determined definitely and the steady- state solution as the large time limit of the corresponding initial value problem was obtained without the necessity of inverting the Laplace transform. In mathematical terms, our solution given in Equations (3.3?) through (3.47) can be interpreted as a fundamental solution, known as the Green's function, of the differential equations of the improved plate theory given in Equations (2.1). Being defined through the use of a two-dimensional Dirac delta function, the Green's function was constructed with the aid of a combined Fourier and Laplace transform method. 60 The Green's function represents mathematically the response of a Timoshenko plate to an idealized point force with infinite intensity. It thus possesses a singularity under the load. In a given physical problem, however, the load is actually distributed over some finite area with finite intensities. An integration of the Green's function over the distributed forces will lead to a solution that is continuous and finite under the load. With the aid of the above interpretation for a Green's function, the responses caused by a wide variety of moving loads such as line loads or spatially distributed loads can be obtained by simply superimposing the solutions corresponding to a point moving load on the plate. This leads to performing an integration along the line or over the area where the load is acting, with Green's function serving as the kernel of integration. The Timoshenko plate model was used in the present study. One should realize that this model has the defects of not being able to predict the high-frequency-short-wave portion of the dynamic response. Using a Fourier analysis, a general point force can be considered as a superposition of infinite wave trains of all wave lengths. Thus, the portion of the present analysis of the dynamic response corresponding to those waves in the high- frequency range should be regarded as questionable. The reliability of the results when a problem 1H; analyzed by the improved plate theory can be assessed by using Miklowitz's criterion [8]. 61 It is also worth mentioning here that an alternative approach can be taken in choosing an appropriate contour of integration without performing a Laplace transform. Introduction of a small damping parameter c, which accounts for an energy loss, will lead to a consideration of behavior of solutions in the transformed plane because of the presence of this parameter and will allow us to define the path of integration so that the radiation condition BY is satisfied. This is accomplished by adding the terms 3 525-, 3‘! e SEX. and e 2%. to the right-hand side of Equations (2.1a), (2.1b) and (2.1c) respectively and considering the original problem as the limiting case as e approaches zero. In this manner, the expressions for mi and w2 2 involving the parameter 6 take the following form mi = g-[cg - A0;2 + a2 - 1:2)k] (4.1a) w: = g1 [cg + A(§2 + a2 - i—gifi] (4.1b) where A, C and a are defined in (3.8d), (3.23i) and (3.8e) respectively and d is a positive constant. It can be shown from Equations (4.1a) and (4.1b) that an addition of this small damping constant will yield the same branch points in the transformed plane as those resulting from a small Laplace parameter in the present analysis. In the evaluation of auxiliary functions a(x,T), B(x,T) and y(x,T), it is noted that the integrands have multiple branch points as discussed in Section 3.2. Let the path of integration from -m to +m be denoted by F. A selection of contours 62 equivalent to F is given in Figures 12 and 13 for the intersonic and the subsonic cases respectively. The contours shown in Figures 12-a, 12-b and Figures l3-a, 13-b, l3-c are all suitable for the evaluations of the auxiliary functions. In Figure lZ-a there are barriers each being parallel to the real axis, while in Figure lZ-b, the barriers are on the imaginary axis. The paths of integration can all be enclosed by adding the paths along the branch cuts as well as large circular arcs at infinity in the lower or the upper slit g-plane depending on whether T’< O or T > 0. The equivalence of each path with F can easily be seen by using Cauchy's integral theorem [See 33], for none of the contours completed via the broken lines encloses a singularity. This is because the contributions from the dotted arcs approach zero as the radius becomes infinite and those from the small circles around the branch points approach zero as the radii shrink to zero. So the contribution of integrals along the path P is equivalent to those along the linear paths along the branch cuts introduced. As the integration along Figure lZ-a yields a more slowly convergent solution involving a sine or cosine term in T, the path shown in Figure lZ-b is thus adopted for the evaluations of auxiliary functions as it gives rise to more rapidly convergent integrals. Similar arguments and reasonings also apply in the subsonic case. Thus the path shown in Figure 13-c is used to obtain the solutions for the subsonic case instead of those shown in Figure l3-a and Figure l3-b. Some important features of the dynamic response of an elastic plate subjected to a moving force can be described with _. Im‘é / I \ \ z I \ z \ / \ fin \ ’ \ , i: \ I \ s 1" 1 ’ <——Re g I ‘ 0 A7 I ____________“p x \L / a"? 3 / \ / \ / \ / \ \ / / (a) 1m§ / / I ‘ \ \ \ /l/ \ / \ ’ \ // ‘ \ / , \ J l ~‘—= R \ 0 T ' e g \ I \ / / \ \ / \ / / \ / \ \ I / (b) Figure 12 F and Equivalent Contours in the Intersonic Case Figure 13 F and Equivalent Contours in the Subsonic Case 65 the aid of the figures as sketched in Figures 14, 15 and 16. In these sketches, the origin of the moving coordinates and the black dot denote the instantaneous and initial positions of the load, respectively. The arrivals of the waves generated by the load at the time of application are represented by two concentric circles centered at the black dot. The lines drawn from the position of the force tangent to these circles are known as Mach lines. The dynamic response of the plate in the SUperSOHiC case depicted in Figure 14 consists of two families of traveling waves whose envelopes form two Mach sectors trailing behind the load [See 27]. Consequently, the plate remains undisturbed beyond the outer sector. Figure 15 shows the features of the response of the plate when the load moves intersonically. As expected physically in this case, the outer Mach sector does not exist and only the inner Mach sector corresponding to the slower wave front prevails. The wave pattern in the subsonic case as indicated in Figure 16 displays, consequently,no wave fronts at all, as t tends to infinity. Since the solutions in integral expressions for a(x,T), B(X,T) and y(x,T) in Equations (3.44) through (3.47) are analytic everywhere away from the Mach lines, it is, thus, of interest to investigate the behavior of the solution across the wave front. The behavior of the solutions across the wave front in the intersonic case can be determined immediately from the inverse Fourier integrals given in Equations (3.17) by applying Tauberian theorem. This procedure is accomplished by first expressing the 66 sumo UHCOmnmasm wsu aw mucoum m>w3 «H shaman 67 .r memo ochmuoucH mzu cw msumuuwm m>m3 ma muswwm - . y \ t sumo cacomnsm ozu CH mcumuumm m>m3 0H ouswwm 69 i§(T-‘x‘/V2) integrands in appropriate form as G(§)e near the wave front, then, by expanding G(g) in descending power series in g and inverting the integral term by term. In general, such a series expansion of G(g) can be truncated after terms of order %; This is due to the fact that terms of order 15- or higher yield terms that are continuous across the wave front. On the other hand, the constant term in the series corresponds to Dirac . . l delta function type behav1or across the wave front and the term of‘- 5 gives the desired finite jump or step function discontinuity across the Mach line T = |x\/v2. Imposing |§‘ dim on the afore-mentioned inverse Fourier integrals and upon inversions of the leading terms (both constant term and term of order é), the following jumps result: Across x = V2T [a] = 0 [B] = V2/2A [y] = l/ZVZA [w] = 4V2/G'h 2 (l +'V ) = - ______JL_ [<9] 2DAV 2 [Yx] = O (4.2) Y = 0 [-y] 2 1 + ”V2 [Mx] =- ZAV2 u, + V: [My] = ' 7E— 70 Q ~ - V26(x-V2T) x 2 2 V 6(x-V T) Q N 2 2 y 2 where [ ] denotes the jump of the quantity from T > $-' to T O. In this limiting case, the finite jump across T = 3;. remains unaltered. The behavior acrossing the degenerate Mach line are obtained as follows: II C [a] - LAT/3i 6(T) 0 m 1 [Y] [w] = 0 "1161' ‘9 WI) 0 (4.3) r"1 ué l_.J ll _ h Hy] ' W D M~b_5_(‘_|')_ y 4/3A' 2A 7l [Qx] = 0 32,] = 0 in which the substitutions C = A = [(vi-v:)/v:]% and a = 4/3/Ah have been made and [ ] denotes jump from T > 0 to T < 0. These quantities agree with those obtained by Yen [27] in the study of the supersonic case. The explicit expressions given in Equations (3.39) through (3.43) and Equations (3.44) through (3.47) represent the fundamental solutions of the steady-state responses of a plate for the subsonic and intersonic cases, respectively. It is noted that the integrands of these integrals have singularities at either or both of the limits of integration which are integrable in the sense of Riemann integrations. In principle, these solutions can be readily evaluated numerically. We wish here to investigate the integrals asymptotically at large distance from the load. To obtain asymptotic approximations for a(x,T), 3(x,T) and y(x,T) at a large distance from the load, the integral representations for the solutions can be put into the form of an integral of a real variable g as ‘_ X1 -Ecose[g-h1(g)tane-ig1(g)tane] F1(R,e) = f61£1(g)e dz; (4.4) with T = E cos 9 M E2 = x2 +'T2 '1? sin e (4.5) 72 where i. is real positive parameter. For large E, it is evident that the usual asymptotic methods, i.e., the method of stationary phase and the method of the steepest descent are no longer practical. We thus, alternatively, will study the asymptotic behavior of the solutions for large T along the path of the load. In this particular case, an inspection of the solutions reveals that the solutions contain i) integrals of the form x, Th2(€) F2(T) = $52 £2(g)e dg for x = o (4.6) where h2(g) is a real-valued function of the real variable g, f2(g) is real-valued continuous function in the real interval (62,x2), where 62 >>O, k2 > 0; and ii) Fourier integrals of the form x3 iTk(g) F3(T) =.f53 f3(g)e dg (4.7) where (63,x3) is again a real interval, k(§) is a real-valued function and f3(g) is an integrable function so that F3(T) as defined exists for all real T. Various mathematical methods are available for finding the asymptotic approximations for the integrals defined in (4.6) and (4.7). The so-called Laplace's method is applicable to the integral, given by (4.6), while the method of stationary phase can be applied to (4.7). More details on the mathematical justifications of these methods can be found in the work of Erdelyi [34]. In what follows only the leading terms in the 73 asymptotic representations of auxiliary functions for the subsonic and intersonic cases respectively are summarized. Subsonic case: The asymptotic expressions obtained for x = O are: for T > 0 * l {(5) e .l a —75 x. + a” 110“) 3 new (no) l :Q/Z) -aT l 3 ~ +e 0(-) (4.8) “(4403’2 3” T _ _1.___ m... e-aT 001;) 211048)}; ./T T and for T < 0 *,g2.11“ 11-1 em 1 7f$%-cos(bT + 4) a an «(11.1)15 (be)35 “ (Assam)Li M 1 I‘Q/Z) + fl @r cos n .. _ (b + —) (4.9) B 1101.03” M372 "(M/WT T ‘* -aT A Y ~e 0(T) Intersonic Case: Similarly, the asymptotic evaluations will lead to the following results in this case for x = 0: for T > 0 74 * 1 11(5) e'a" 1 A 1 1 Q ~ + '— _+ 0(— n“”372 /T w(Aa)(ac)%‘/28 C T T2 1 113/2) e-aT 1 1 B .. - - —+ 0(—-) (4.10) "(Aa)3/2 T3/2 /2n(ac)%/aC T T2 % ~,_.1__t1§1-e'"“’“) 1 11.1. 1 Y 2n(Ac)% ‘ffi 2" A8 /23 C T + 0(TE) and for T < O 8T * e 1 A 1 l Q ~ - --+ O(-— momma);i (2“ C T T2 81’ 8 ~ - e .5 l+0(1—2-) (4.11) /2n(aC) faC T T aT % e (8C) 1 A l _1__ Y"- ZnAa /2aCT+O(2) T From Equations (4.8) and (4.9), it is seen that whereas the leading terms of the auxiliary functions decay as T- for both T < 0 and T > 0. They contain oscillating factors only for T < 0. Equations (4.10) and (4.11) show that the solutions .31- consist of terms decaying as ‘-;—- for both T > 0 and T < O as well as terms decaying as T- for T > 0. The amplitude of the leading terms behaving as T-% dominates the response pattern of the plate behind the load. It also follows from the above equations that a, and hence the deflection w, increases as T15 for large T > O, which is consistent with-Payton [28]. This implies that the points behind the load in the neighborhood of the T-axis are much stronger affected than ahead of the load. 75 4.2 Concluding Remarks In this thesis, the study of the steady-state responses of an infinite plate to a concentrated, normal load in uniform motion over the plate surface was made within the realm of the improved plate theory. The closed form solutions are obtained in terms of both definite and indefinite integrals. These results are valid for the intersonic and subsonic cases of the moving load problem, where the speed of the force lies in between v1 and v2 and smaller than either of these two, respectively. The solutions may be referred to as the fundamental solutions or the Green's functions and, in conjunction with superposition principles, they can be used to obtain the dynamic response of an elastic plate to an arbitrary disturbance Spatially distributed over the plate surface. The present study represents an extension of earlier work by Yen [27], in which the Fourier-Laplace method was applied to treat the supersonic case of the moving force problem. The present effort of analyzing the intersonic and subsonic cases requires a different transform technique as well as a different mathematical formulation. The present cases were studied by performing a double Fourier and a Laplace transform to an initial value problem. 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