THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING DYNAMIC RESPONSE OF A THIN-WALLED CYLINDRICAL TUBE UNDER INTERNAL MOVING PRESSURE Sing-chih Tang A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the University of Michigan Department of Civil Engineering 1962 January, 1963 IP-602

Doctoral Committee: Professor Bruce G. Johnston, Chairman Professor Glen V. Berg Professor Samuel K. Clark Professor John H. Enns Professor Lawrence C. Maugh

ACKNOWLEDGMENTS The author is deeply indebted to Professor Bruce G. Johnston, chairman of his doctoral committee, for help and encouragement during preparation of this thesis. He is also grateful for suggestions given by all other members of his doctoral committee. The author wishes to express his appreciation to the Air Force Special Weapons Center for sponsoring a research project from which the author initiated this work; to the Institute of Science and Technology of The University of Michigan for granting a pre-doctoral Fellowship under which this thesis was completed; and, to the Computing Center of The University of Michigan for the use of the IBM 709 computer in calculating the numerical examples. The final copy of the thesis was typed and reproduced by the College of Engineering Industry Program at The University of Michigan. For this help in typing, drawing, and printing of the thesis, the author is sincerely grateful. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENTS............................................ 9, ii LIST OF FIGURES................................................... vi NOMENCLATURE............... Z......Zo..........o...... ix ABSTRACT...................................................... xiii INTRODUCTION...................................................... 1 I EQUATIONS OF MOTION OF A THIN-WALLED CYLINDRICAL TUBE.................................................... 6 A. Assumptions and Approximations....................... 6 B. Basic Equations of Motion for Free Vibration in Cylindrical Coordinates............... 6 C. Approximate Equations of Motion for Free Vibration.................................... 8 1. More Exact Theory Corresponding to Timoshenko Theory in Beam Vibration........... 8 2. Elementary Theory Corresponding to Euler-Bernoulli Theory in Beam Vibration........ 14 D. Approximate Equations of Motion for Forced Vibration.................................. 14 1. More Exact Theory.............................. 14 2. Elementary Theory............................... 15 E. Approximate Equations of Motion in Dimensionless Form................................. 15 1. Elementary Theory.............................. 16 2. More Exact Theory............................. 16 II STEADY STATE WAVE PROPAGATION IN A THIN-WALLED CYLINDRICAL TUBE........................................ 18 A. Elementary Theory.................................. 18 1. Equation of Motion for Free Vibration........... 18 2. Frequency Spectrum............................. 18 3. Velocity Spectrum.............................. 19 iii

TABLE OF CONTENTS (CONT'D) Page B. More Exact Theory.................................. 36 1. Equations of Motion for Free Vibration.......... 36 2. Frequehcy Spectrum.......................... 36 3. Velocity Spectrum.............................. 39 C. Comparison of Results from Both Theories..4....... 41 1. Frequency Spectrum............................. 41 2. Velocity Spectrum............................. 42 III STEADY STATE RESPONSE OF A THIN-WALLED CYLINDRICAL TUBE WITH INFINITE LENGTH UNDER INTERNAL MOVING PRESSURE........................................... 43 A. Solution of the Equation from Elementary Theory...., 43 1. Case for V < Vco o.............................. 47 2. Case for V > Vco.............o o...o...... 50 B. Solution of the Equation from More Exact Theory,.... 56 1. Case for V < V^co.6.0.....G.......... 58 2. Case for V > Vco............................... 59 2. Case for V > Vco..9 C. Discussion.................................... 60 1. Soundness of the Sinusoidal Wave Solution for the Case V > Vo0........................... 60 2. Comparison of Results from Both Theories........ 61 IV TRANSIENT RESPONSE OF A THIN-WALLED CYLINDRICAL TUBE WITH SEMI-INFINITE LENGTH UNDER INTERNAL MOVING PRESSURE........................... 62 A. Solution of the Equation from Elementary Theory..... 62 1. Formulation of the Problem..................... 62 2. Method of Solution.....................,.o.. 63 3. Interpretation of the Solution................. 66 B. Solution of the Equation from More Exact Theory..... 68 1. Case for V < Vco o..o..o.......o................ 69 2. Case for V > Vco................................ 74 iv

TABLE OF CONTENTS (CONT'D) Page C. Numerical Examples.................................. 93 1. Fourier Integral Solution for the Velocity Range V < Vco........................ 93 2. Numerical Solution for the Velocity Range V > Vco........................... 97 D. Discussion of the Numerical Results................. 108 1. Case for V < Vco............................... 108 2. Case for V > Vco............................... 109 V EXTENSIONS OF THE NUMERICAL SOLUTION................ 110 A. Forcing Function Due to Non-Uniform Pressure or Pressure Front Moving with Non-Uniform Velocity. 110 B. Arbitrary Boundary Conditions at X = 0......... 110 C. Tube with Finite Length............................. 110 D. Consideration of the Inertia Force in Axial Direction.............................. 111 CONCLUSION........................................................ 113 REFERENCES........................................................ 115 APPENDIX.......................................................... 117 v

LIST OF FIGURES Figure Page 1.1 Resultant Forces and Moment Acting on an Element......... 11 1.2 Cross Section of an Element........................... 11 2.1a Frequency Spectrum for h/R = 0.1 (Complex Arm)............ 20 2.1b Frequency Spectrum for h/R = 0.1 (Real and Imaginary Arms). 21 2.1c Frequency Spectrum for h/R = 0.1 (Real Arm Near 2 = Q* from More Exact Theory).................................... 22 2.2a Frequency Spectrum for h/R = o.o6 (Complex Arm)............ 23 2.2b Frequency Spectrum for h/R = 0.06 (Real and Imaginary Arms) 24 2.3a Frequency Spectrum for h/R = 0.03 (Complex Arm)............ 25 2.3b Frequency Spectrum for h/R = 0.03 (Real and Imaginary Arms) 26 2.4a Velocity Spectrum for h/R = 0.1 (Complex Arm).............. 27 2.4b Velocity Spectrum for h/R = 0.1 (Real Arm)................. 28 2.4c Velocity Spectrum for h/R = 0.1 (Imaginary Arm)............ 29 2.5a Velocity Spectrum for h/R = 0.06 (Complex Arm)............ 30 2.5b Velocity Spectrum for h/R = 0.06 (Real Arm)......,..... 31 2.5c Velocity Spectrum for h/R = 0.06 (Imaginary Arm)........... 32 2.6a Velocity Spectrum for h/R = 0.03 (Complex Arm)............. 33 2.6b Velocity Spectrum for h/R = 0.03 (Real Arm)................. 34 2.6c Velocity Spectrum for h/R = 0.03 (Imaginary Arm)........... 3 3.1 Moving Pressure in a Tube with Infinite Length............ 44 3.2 Inversion Contour for (X - VT) >....................... 48 353 Inversion Contour for (X — VT) 0..................... 48 3.4 Positions of Poles when V > Vco Without Damping.......... 51 3.5 Positions of Poles when V > Vo With Damping.............. vi

LIST OF FIGURES (CONTD) Figure Page 4.la Pressure Front Moving to the Right...................... 67 4.lb Pressure Front Moving to the Left..................... 67 4.1c Static Pressure...................................... 67 4.2 Four Characteristic Lines from the Source.............. 78 4.3a Typical Elemen t 81 4.na Typical Elem ent...................................... 81 4.3b Element at Boundary................................ 81 4.4 Triangular Region Used in Computation.................. 85 4.5 Pressure Front in X - T Plane When Vl < V < Vc2....... 91 4.6 Pressure Front in X - T Plane When V > Vc2............. 94 4.7 Deflection at X = 20 for V = 0.1811 (V < Vco) from Elementary Theory....o................. 95 4.8 Deflection at X = 40 for V = 0.1811 (V < Vco) from Elementary Theory................................ 96 4.9 Total Deflection at X = 20 for V = 0.1811 (V < Vco) from More Exact Theory................................ 98 4.10 Total Deflection at X = 40 for V = 0.1811 (V < Vco) from More Exact Theory............................... 99 4.11 Deflection at X = 20 Due to Shear for V = 0o1811 (V < Vco) from More Exact Theoryo.............. 100 4.12 Deflection at X = 40 Due to Shear for V = 0.1811 (V < Vco) from More Exact Theoryo...................... 101 4.13 Deflection at X = 20 for V = 0o3561 (Vco<V < Vcl) from More Exact Theory.........................o 102 4.14 Deflection at X = 40 for V = 0.3561 (Vco < V < Vcl) from More Exact Theory........................ 103 4.15 Deflection at X = 20 for V = 0o7754 (Vcl < V < Vc2) from More Exact Theory............................o 104 vii

LIST OF FIGURES (CONT'D) Figure Page 4.16 Deflection at X = 40 for V = 0.7754 (Vco < V <cl) from More Exact Theory................................ 105 4.17 Deflection at X = 20 for V = 1.600 (V > Vc2) from More Exact Theory................................. 106 4.18 Deflection at X = 40 for V = 1.600 (V > Vc2) from More Exact Theory...................................... 107 5.1 Reflection of Characteristic Lines Between Two Boundaries.................... 112 viii

NOMENCLATURE A,A1,A2 dimensionless amplitudes, (amplitude)/h B1,B2 constants of integration C dimensionless constant proportional to the coefficient of the viscous damping C1!C2 paths of integration Eh3 D flexural modulus of a plate, 12(1-v2) FF1iF2 functions G shear modulus Hi(i=l,..,8) functions I IbI, g I simproper integrals I2, Ibl Ib2 IslIs2 improper integrals L dimensionless length of the finite tube, 1 1-v2 M dimensionless bending moment, hh M Mxx transverse bending moment per unit length of undeformed mean circumference N dimensionless wave number, (wave number) x h/vE, or parameter in the Fourier transform Nxx resultant normal force per unit length of undeformed mean circumference on the plane perpendicular to the axis of the tube NGQ resultant normal force per unit length of the undeformed tube on the radial plane P P dimensionless internal pressure, 12 G Qx Q dimensionless resultant shearing force, -7h-G Qx resultant shearing force per unit length of undeformed mean circumference on the plane perpendicular to the axis of the tube in the radial direction R mean radius of the tube ix

S dimensionless distance from the pressure front (distance from the pressure front) x \/2/h T,To dimensionless time coordinates, h /Ji vdAt AT,AT' dimensionless time increments, h V dimensionless phase velocity, or dimensionless velocity of the pressure front, v/vd Vco dimensionless first critical velocity, VcO = 26g, from elementary theory, VcO defined on p. 40 from more exact theory Vcl dimensionless modified shear wave velocity in C plate, 6 = vs/vd, or dimensionless second critical velocity Vc2 dimensionless dilatational wave velocity in a plate, 1, or dimensionless third critical velocity W total dimensionless deflection, w/h Wb dimensionless deflection due to bending, wb/h Ws dimensionless deflection due to shear, ws/h ~WQo~ dimensionless maximum radial static deflection for P=l, 1/g2 WTo dimensionless radial deflection at To, (radial - deflection)/h W,WsWb functions transformed from W, WS, Wb, respectively X dimensionless coordinate along the axis of a tube, h Y symbol for unknown value a radius of the circular contour in contour integration on the complex plan eon strain component of B-plane in a-direction co f (i=l,...,6) functions g dimensionless parameter depending on the ratio of the thickness to the mean radius of a tube, g2 E (h)2 x

h thickness of the tube wall dT k slope of the characteristic lines, d 1 length of the finite tube m imaginary part of the dimensionless wave number, N, or imaginary wave number (spatial attenuation number) n real part of the dimensionless wave number N, or real wave number p internal pressure per unit area r radial coordinate t time coordinate u displacement of a point on the middle surface of a cylinder in x-direction ux displacement in x-direction ur displacement in radial direction UG displacement in G-direction uz displacement in z-direction v velocity of the moving pressure front vI velocity in radial direction _ — v1 v dimensionless velocity in radial direction, d2 Vd vd dilatational wave velocity in a plate, p(l-V2) vs modified shear wave velocity in a plate, JKP WqWb.Ws total deflection, deflection due to bending, and deflection due to shear in radial direction along the tube respectively x coordinate along the axis of a tube yy1 positive constants z local Cartesian coordinate in radial direction xi

7foga ~ shearing strain on D-plane in a-direction in engineering notation 6 dimensionless parameter, ratio of the modified shear wave velocity to the dilatational wave velocity in a plate, 2 (-2V2)G E ~ angular coordinate K shear correction factor ~~~X ~dimensionless wave number due to static pressure, VI v Poisson's ratio p mass of the tube per unit volume Tag stress on P-plane in a-direction Toa(cag) average sharing stress on P-plane in a-direction fx angular rotation of the radial line Qn^ ~ dimensionless frequency (frequency) x h/\12 vd, or parameter in the Fourier-transform QX2sc ~ maximum dimensionless frequency with complex wave number CD angular velocity of the radial line 0cu~) ~ dimensionless angular velocity of the. radial line, h 12 vd xi xii

DYNAMIC RESPONSE OF A THIN-WALLED CYLINDRICAL TUBE UNDER INTERNAL MOVING PRESSURE Sing-chih Tang ABSTRACT The dynamic response of a thin-walled cylindrical tube under internal moving pressure is analysed. Two kinds of approximate equations of motion are used, corresponding to the elementary Euler-Bernoulli theory and the more exact Timoshenko theory of beam vibration, The work contains three major parts which are as follows: 1. Steady state wave motions in the tube wallo Frequency spectra (frequency as a function of wave number) and velocity spectra (phase velocity as a function of wave number) are plotted based upon two kinds of approximate equations of motion. The frequency spectra are useful for studying the transient response of the problem; and the velocity spectra are useful for the steady state response, since in steady state response the velocity of the moving pressure front is identical to the phase velocity of the wave propagation in the tube wall. 2. A study of the steady state response for a tube with infinite length under moving pressureo This is analysed by means of the Fourier transform. In the case of the pressure front moving with velocity greater than critical, the solution is obtained by introducing a viscous damping term in the equation of motion and setting it equal to zero in the limito

ABSTRACT The dynamic response of a thin-walled cylindrical tube under internal moving pressure is analysedo Two kinds of approximate equations of motion are used, corresponding to the elementary Euler-Bernoulli theory and the more exact Timoshenko theory of beam vibration. The work contains three major parts which are as follows: 1. Steady state wave motions in the tube wallo Frequency spectra (frequency as a function of wave number) and velocity spectra (phase velocity as a function of wave number) are plotted based upon two kinds of approximate equations of motion. The frequency spectra are useful for studying the transient response of the problem; and the velocity spectra are useful for the steady" state response, since in steady state response the velocity of the moving pressure front is identical to the phase velocity of the wave propagation in the tube wall. 2. A study of the steady state response for a tube with infinite length under moving pressureo This is analysed by means of the Fourier transform. In the case of the pressure front moving with velocity greater than critical, the solution is obtained by introducing a viscous damping term in the equation of motion and setting it equal to zero in the limito xiii

3. A study of the transient response for a tube with semi-infinite length under moving pressure. This is analysed by means of the Fourier sine transform in the case of the pressure front moving with velocity less than critical, For over critical velocity, the transient response based upon the equations of motion from the more exact theory is analysed numerically by means of the method of characteristics. Numerical results for different velocities of the moving pressure front are computed. xiv

INTRODUCTION It is known that the dynamic effect on the stresses in the tube wall is very large when a pressure front with high velocity moves down a tube. Static analysis of the stresses in the tube wall is valid only when the pressure front moves with low velocity. If the velocity is supersonic, dynamic analysis of stresses has to be employed, For instance, when a shock wave is transmitted in a tube, static analysis of stresses in the tube wall will be inaccurate. Shock tubes (i.e., tubes in which shock waves are generated) have been widely used in testing models. A method for analysis of the dynamic stresses in the tube wall will be presented. Exact solutions based upon actual properties of the pressure front and complete three-dimensional theory of elasticity are too complicated to admit analytical treatment. Idealized forcing function due to pressure front and approximate equations of motion must be employed. These simplifications are adequate for the purpose of calculating the stresses in the tube wall. The pressure front is assumed to be moving with constant velocity parallel to the axis of the tube and the intensity of the pressure is assumed to be uniform. Approximate equations for axially symmetrical motion of a thin-walled cylindrical tube due to Lin and Morgan,(l) and Herrmann and Mirsky(2) are valid for a wide range of frequencies, particularly for higher frequencies. These equations involve terms due to rotatory inertia and shear deformation that are significant for higher frequencies. For low frequencies, neglect of these terms will not cause any serious error. Equations omitting these terms are due to Love.(3) -1

-2The difference between equations containing these terms and equations omitting these terms is analogous to the difference between equations of transverse vibration of a beam by Timoshenko theory(4) and by Euler-Bernoulli theory. For higher frequencies, the equation based on the Timoshenko theory should be used, In this work, the two approximate equations are employed to analyse the stresses in the tube wallo Further simplifications are introduced, so that additional assumptions have to be made, ioe., the inertia force parallel to the axis of the tube is neglected and the resultant longitudinal stress across the thickness of the wall on the plane perpendicular to the axis of the tube vanishes. These simplifications make sense if the strain energy due to radial motion is large compared with that due to axial motion.This work presents the case in which radial motion predominates. After the simplifications mentioned above, these two kinds of approximate equations become identical to the equations of Euler-Bernoulli beam and. Timoshenko beam both on an elastic foundation. Euler-Bernoullie beams on an elastic foundation under a concentrated force moving with constant horizontal velocity have been investigated by many authors. In the investigation of dynamic stress in rails under the wheel of a locomotive, Timoshenko(5 6) formulated this problem. Using Fourier series to solve the problem of a beam with finite length, he found that the dynamic effects was insignificant because the horizontal velocity of the wheel was small compared.with so-called critical velocity which depended on the flexure rigidity of the beam as

-3well as the foundation stiffness. Ludwig(7) solved a similar problem, but the beam was infinite in length and the force moved with velocity either less, equal, or greater than the critical. His interest was in the steady state response, so he assumed that the moving force had already acted on the beam for a long time. Mathews(8) used the Fourier transform to solve the rail problem for the steady state response, but his moving force was such that the magnitude of the force varied sinusoidally in time. Dorr(9) formulated a problem -- a semi-infinite Euler-Bernoulli beam on an elastic foundation with one end simply supported, under a concentrated force moving from that end with constant horizontal velocity. He used the Laplace transform to solve this problem. He calculated the inverse transform by the asymptotic method, so the duration of the moving force on the semi-infinite beam had to be infinite and this then was a steady state solution too, This steady state solution with the velocity of the moving force greater than the critical is physically meaningful, He also derived a formula in terms of Fourier integrals for a moving force with velocity less than the critical suddenly applied at the middle of an infinite beam, For a moving force with velocity greater than the critical, he used a power series to solve the transient problem, but no numerical results were given. The effect of viscous damping on this problem was first studied by Kenney.(1O) In the case of velocity greater than the critical and with no damping, he took as the physically meaningful solution that which was approached in the limit by a system whose damping approached zero, Crandall(ll) used this idea to solve a Timoshenko beam on an elastic

-4foundation under a concentrated force moving with constant horizontal velocity. In a paper -- Transmission of Shock Waves in Thin-Walled Cylindrical Tubes -- Niordson,(l) using the equation identical to that of Euler-Bernoulli beam on an elastic foundation, found the radial deflection of the wall for the steady state response under a moving pressure front. Supplementing Niordson's contribution, the present work gives a solution based on equations taking into consideration the rotatory inertia and shear deformation under the same forcing function as Niordson's. The Fourier transform is used. Both the forcing function and the method of solution are different from those of Crandall. In this work the transient response of a semi-infinite tube under an internal pressure front moving with constant velocity parallel to the axis of the tube is investigated here for the first time. The result obtained is the main contribution of this thesis. The standard method of solution is to use the Laplace transform, but the inverse integral involves too many branch points to be dealt with. The Fourier sine transform is used to solve the transient problem when the velocity of the pressure front is less than the critical, A numerical method to solve finite difference equations is used when the velocity is greater than the critical, since the Fourier sine transform fails in that case. For investigation of vibrations, first of all, the wave propagation in the tube wall has to be understood, The frequency spectrum: wave number (complex, real, and imaginary) versus real frequency of the radial vibration of the wall, and the velocity spectrum: wave number

-5versus real phase velocity of the wave propagation are plotted. A minimum phase velocity under wnich a wave with a real wave number can be propagated along the wall is foundo This phase velocity is identically the same as the critical velocity defined by Timoshenko, Mathews, Ludwig, Dorr and Kenney for Euler-Bernoulli beam and by Crandall for Timoshenko beam.

I. EQUATIONS OF MOTION OF A THIN-WALLED CYLINDRICAL TUBE A. Assumptions and Approximations 1. All assumptions in the linear theory of elasticity are employed. 2. Thickness h of the tube wall is small compared with the mean radius R of the tube. h is also small compared with the wave length of the disturbance. 3. Motion of the tube wall is axially symmetric so that no displacement occurs along the circumference. Displacements, strains, and stresses in the wall are independent of the angular coordinate. 4. Radial lines remain straight after deformation. 5. Displacement of the tube wall in the radial direction is uniform through the thickness of the wall. 6. Inertia force in axial direction is neglected, (2) and the resultant longitudinal stress across the thickness of the wall on the plane perpendicular to the axis of the tube is set equal to zero (13) 7. In the elementary theory, terms due to shear deformation and rotatory inertia are neglected. B. Basic Equations of Motion for Free Vibration in Cylindrical Coordinates Let (r, 9, x) be the cylindrical coordinates of a point within the tube wall, ur, u, ux be the corresponding displacement components. -6

-7Let the stress tensor be ct-r rre Crx,~ Co( =3^, ( &xr ZN< SX. where T^ = T Tr x = Txr and Tqx TxG Equations of motion(l1') without body force in r, G, and x directions are e~rr D t Ircl~rXt v / s8 bLl dr r+I a +- fx &r P z (1.2) ar Cr + +LrxlI ___ a u (1.4) r +a+r Crx z (1.4) From the assumption, TrG Tx = 0, Q- = 0, and u0 = 0, Equation (1.5) is automatically satisfied. Equations (1.2) and (1,4) can be simplified to ~ ~err e3x V/Ur ___ r r (1.5) Ir+ ^ -^-Z - (1.6) + xr r

-8C. Approximate Equations of Motion for Free Vibration 1. More Exact Theory Corresponding to Timoshenko Theory in"Beam Vibration (Shear and rotatory inertia terms included) Let r = R + z and Trr Tzz, rx = Tzx Ur = Uzj then the equations of motion become eaz, + t'ears _ E C US a i -ax_ aR - f Ft (1.7) at a+ ~ =y z (1.8) Following Herrmann and Mirsky,(2) we assume that t((x/,,()= (Xt) - ^(04 ) (1.9) (xi,3 t)-'(CX ) (1.10) From Equations (1.9) and (1.10) strain components can be computed as e~ —< = -u ax (l.o) ev= r,- t r =- R+Z (1.12) -; j.. j+j Ur (1.13) and all other strain components are zero. With Lloyd,(l5) let w be separated into two parts k(xA') = 4b~x,) + X,&) (l.lt)

-9where wb is due to bending and ws is due to shear. Then Equation (1o13) becomes put x- t x Xax ) = aO or yL^(X't) 9a t (1.15) Non-zero strain components can be written as l = a - (1.16) <@ =; >b__+ ArS(1,17) /r^={>~~~~~~~~ 7-^h~ (1,18) By Hooke's law, the non-zero stress components can be computed as 6= [r z[ - e- + Y.C... =a L7_Z a + (1.19) ^ =!j- L eOe + re xx =, E r ~z [+ EC r' - -, -, ~ (1 20)

-10^x= - (1.21) where TZ is the average shearing stress through the thickness of the tube wall, K is the shear correction factor which depends upon Poisson's ratio(2) and is approximately equal to that of the rectangular beam. Let Nxx be the resultant normal force per unit length of undeformed mean circumference in the.x direction, Nig be the resultant normal force per unit length of the undeformed tube in the 9 direction, Qx the resultant shearing force per unit length of undeformed mean circumference in the radial direction, and Mxx is the transverse bending momentper unit length of undeformed mean circumference. Moment and resultant forces are shown in Figures 1.1 and 1.2. They can be expressed in terms of displacement components and their derivatives as follows I ^ Z, ( ro + )d2 k_ _ From the assumption NX = O, it follows that x _r (1.22) Nee= J 1 dt +_ vzj (1.,2) fz

-11/ /... Xxx z___:_/!oQx'R N969 Rd Figure 1.1 Resultant Forces and Moment Acting on an Element. N Sean ee Figure 1.2 Cross Section of an Element.

-12Using Equation (1.22), N$9 can be expressed in terms of wb and ws as follows: Nee = -R (C^.Ib <s 5) (1.24) ~=1 h T7x (I0 t -)ds -x 6 (1.25) L Mx- f_ h x(l+ i )?d? I M = — D ~>(z (1.26).Eh3 where D = Eh3 12(1-v2) Since for thin-walled tube 1 >> I where h hK - R 2 2

-13By Equations (1024), (lo25) and (1o26). the approximate equations of motion can be expressed in terms of wb, ws, and their derivativeso Multiplying both sides of Equation (lo7) by (R + z) and integrating with respect to z over the range -h/2 to h/2, gives the equation of translation in the radial direction: xt~JeQ~~~h~Jb~~~~t~~) (1o27) ax R h at (1027) Multiplying both sides of Equation (lo8) by z(z + R) and integrating with respect to z over the range -h/2 to h/2, gives the equation of rotation: aMr Q =- - th (1028) D MXX rz axa3,\rbtL~ (1. 28) In terms of ws and wb only, Equations (1.27) and (1.28) become t&3-, _.... h.t6 ) or P G-C(fi 7 ^ r's)= d3(s E 9(t(() (1.29) -Z Dc6 3 - hR a3-b D x orx or "a + rt t- (1.30)

-142. Elementary Theory Corresponding to Euler-Bernoulli Theory in Beam Vibration If the rotatory inertia is neglected, Equation (1.28) becomes or = x x (1.31) Combining Equations (1.27) and (1.31), and setting ws = 0, then a2Mxx _ t _. ree b -5- R — h z or PD 3 - + a' 2TJ +5 g = o (1.52a) Putting wb = w, then D 0 4y ir + Shba- 0 (1.32b) which is identical to Niordson's.(12) D. Approximate Equations of Motion for Forced Vibration Let p(x,t) be the internal pressure per unit area, and p(x,t) acts outward along the radius of the tube, 1. More Exact Theory A forcing function p(x,t) must be added to the equation of motion in the radial direction, while the equation due to rotation remains

-15unchanged ItG7a - 4h(t ^) -C r= -h +x<)) or axz) -; = - |- p(xt) (1.33) 2, Elementary Theory -X4 + FRh tR S t = (<^t) (1.34) E. Approximate Equations of Motion in Dimensionless Form For convenience of numerical computation, the following dimensionless variables are introduced for x, w, Wb, ws, t, and p respectively --- h T -h — w='G?- __ 2 E where vd = -2 is the square of the dilatational wave velocity in a plateo

-161. Elementary Theory Replacing x, w, t, and p by X, W, T, and P respectively in Equation (1.34), the equation of motion in dimensionless form is obtained e + foin + i = ( mters p (1.36) Let the following dimensionless parameters be defined (1.37) r5_ (,I- IC6 I2 where v2 is the square of the modified shear wave velocity in a plate, s = |- - dz (i)z (1.38) Equation (1.36) becompes a+t 0, w It Z C - P (1.39) 2. More Exact Theory Replacing x, wb, ws, t, and p by X, Wb, Ws, T, and P respectively and introducing dimensionless parameters g2, 82 into Equations (1.5533) and (1,30), the following dimensionless equations are

-17obtained a atb w) - ta 3Wbw = p- (l.40a) t 6- t.s;a TZ = (1o40b)

IIo STEADY STATE WAVE PROPAGATION IN A THIN-WALLED.CYLINDRICAL TUBE A, Elementary Theory (corresponding to Euler-Bernoulli theory in beam vibration) 1. Equation of Motion for Free Vibration For steady state wave propagation in the wall of a tube with infinite length under free vibration, the dimensionless equation of motion from this theory is 3 C + w -+- aMv = ~ (2.1) where dimensionless variables W, X, and T as well as dimensionless parameters 5 and g are defined in Chapter Io 2, Frequency Spectrum —-Wave Number (real, imaginary, or complex) versus Real Frequency of Vibration Assume the following solution for Equation (2.1) w- = A e(2.2) then N and S must satisfy the following algebraic equation N+ t l^3-, 2- O (2.3) or \N+ = _2 - _ (204) Since Q is always real, the spectrum has three arms, i.e., when Q2 > 62g2 N is either real or pure imaginary, and this corresponds to the real or -18

-19imaginary arm; when Q2 < g2 62 N is complex and this corresponds to the complex arm. Let N = n for the real wave number, N = im for the imaginary, and N = n + im for the complex in the frequency spectrum. The frequency spectrum is plotted only for the first quadrant in Figure 2.1 through 235, since the spectrum is symmetrical with respect to all coordinate axes. In those figures, v is taken to be 0.3, K to be 0.833 and h/R to be 01o, 0O06, and 0003. 30 Velocity Spectrum —-Wave Number (real, imaginary, or complex) versus Real Phase Velocity Assume the solution of Equation (2.1) has the form A = A e x-T (2.5) where N is the wave number of the traveling wave, N may be real, imaginary, or complex; V is the phase velocity of the traveling wave, V is always real.o If Equation (2.5) is a solution of Equation (2.1), V and N have the relation N4- "VN " --. =0' o (2.6a) or ~^=^~[^f-^9~~~~~~' Z (2)(2.6b) When V4 - 2g2 < 0, N is complexo Let Vco be \J2bg which is the minimum phase velocity (critical velocity) with which a sinusoidal wave can be propagated. In this case V is less than Vco. When V4 - 48s2g2> or V > Vco, N is real and N has two real values for a given Vo N cannot be imaginary if V is realo The velocity spectrum is plotted only for the first quadrant in Figures 2.4 through 2,6, since the spectrum is symmetrical with respect

.L =0.1 R 0.030 0~. 030,N =n+im 8 —g MORE EXACT THEORY ~ e,.^-..___^ ~~.... —- ELEMENTARY THEORY,~/! YY/~~ 0.020 - // \\r / \\ I// |PROJECTION OF 11//I~~~~~~~ \ ~cCOMPLEX ARM ON PROJECTION OF COMPLEX ARM l-n PLANE\ // ON 41 - im PLANE \ / /(FREQUENCY VERSUS SPATIAL\ ~, ~ ATTENUATION NUMBER) I./ o0.010 I I Im 0.10 0.05 0 0.05 0.10 n Figure 2.la Frequency Spectrum for h/R O.1(Complex Arm).

h 0.1 0.7 I N =im N =n /^ ~~~~~/ ^-REAL ARM 0.6,\SECOND REAL / ^r -ARM / ARM /FIRST REAL ^^ "S ~/ / ARM a~~~~~~~~~~~~~~~~ 0.5 - IMAGINERY ARM / 0.4 < / / / ^-/ H ~~\ ^IMAGINERY ARVNb.3 / ~\ (FREQUENCY VERSUS / ~\ SPATIAL ATTENUA- / \ TION NUMBER) / / x^ ~ ~ 0.2- / / — MORE EXACT THEORY \ —— ELEMENTARY THEORY \ ~" as~~~~~~~~~~~~~~~~~~~0 ^SEE FIG 2.1c ----------------------— ____________________________________ 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 L4 im n Figure 2.lb Frequency Spectrum for h/R =0.1 Real and Imaginary Arms.

h 0.1 N=n 0.02753800 - =g 0.02753700 0.02753600 / 0.02753500 0 0.010.020 0.30 n Figure 2.lc Frequency Spectrum for h/R a 0.1 (Real Arm Near n = Q* from More Exact Theory).

N=n+ im |t —----- ---- MORE EXACT THEORY h 0.06. ---— ELEMENTARY THEORY 0.015 f/ v- PROJECTION OF / COMPLEX ARM ON n-im PLANE.( FREQUENCY VERSUS SPATIAL ATTENUATION NUMBER PROECTON OF COMPLEX ARM ON,- - n PLANE 0.010 < 0.005 - I t I I I I OJO. 0.05 0 0.05 0.10 im n Figure 2.2a Frequency Spectrum for h/R - 0.06 (Complex Arm).

0.7 h 0.06 SECOND REAL ARM REAL ARM N =im 0.6 / N=n / _ $,, / N = / At xFIRST REAL / ~~~I / ARM IMAGINARY ARM / (FREQUENCY VERSUS 0.5/ / SPATIAL ATTENUATION // NUMBER) / 0.4 / 04/ / I / O~~~/A_/ IMAGINARY // / \I // 0.3 / \/ ||/ /MORE EXACT THEORY \ /......ELEMENTARY THEORY \ \ ~-2 "/ \ \0.2 / 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 im n Figure 2.2b Frequency Spectrum for h/R = 0.06 (Real and Imaginary Arms).

1h 1X0.03 N=n+im R = --- | MORE EXACT THEORY.. —. ELEMENTARY THEORY 0.0.10 8g -PROJECTION OF COMPLEX ARM ___ —-___ s S ON nl- n PLANE PROJECTION OF COMPLEX ARM ON a- -im PLANE (FREQUENCY VERSUS 0.005 SPATIAL ATTENUATION NUMBER) 0.10 0.05 O 0.05 0.10 im n Figure 2.3a Frequency Spectrum for h/R - 0.03 (Complex Arm).

h00 =0.03 if R SECOND REAL.6 -ARM 0.6 / N =im I REAL ARM a~~~~~~~~~~~~~~~~~~ N n ~^' ~~~N=n I.5 / / vFIRST REAL ARM I~~~~~~~~~~~~~~I.~4 /.3 / ~~~~\ ~~~~~~/ ----- MORE EXACT THEORY \ ~~~~~~~~~~~~~~~~/ M\ AG AR —-- AM 02/ELEMENTARY THEORY \ -,_-IMAGINARY ARM / v. / FREQUENCY VERSUS 02 (SPATIAL ATTENU- / \ ATION NUMBER) / "x *./~~~~~~~~~~~~~~~~~~ I ~\ I 0.6 0.4 0.2 0 0.2 0.4 0.6 0.8 1.0 1.2.4 im n Figure 2.3b Frequency Spectrum for h/R = 0.05 (Real and Imaginary Arms).

h =0. I RR 030MORE EXACT THEORY 0.30 ----- ELEMENTARY THEORY N =n + im 0.25 y Cq FROM ELEMNARY THEO. -.> VCOFROM MORE EXACT THEO. 77 / 0.20 // PROJECTION OF COMPLEX / ARM ON V -im PLANE / / (PHASE VELOCITY VERSUS / SPATIAL ATTENUATION / NUMBER) / / PROJECTION OF rO / 0.15- / COMPLEX ARM ON 7 / I V-n PLANE f 010 -~Q1O p1~~~ ~ 0.05 0.20 0.15 0.I0* 0.05 0 0.05 0.10 0.15 0.20 im n Figure 2.4a Velocity Spectrum for h/R = 0.1 (Complex Arm).

V h = 0. I MORE EXACT THEORY R ---- ELEMENTARY THEORY 2.0 N=n I.5 REAL ARM Vc2 = I SECOND REAL ARM 1.0 \ VCI =S — FIRST REAL ARM 0.5" ^ VCOFROM ELEMENTARY THEORY -VCO FROM MORE EXACT THEORY 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 n Figure 2.4b Velocity Spectrum for h/R S 0.1 (Real Arm).

R0 v _ _ _ MORE EXACT THEORY (NO IMAGINARY ARM FOR N =im ELEMENTARY THEORY) VC2= I 1.00 0.90 I) 0.80 s 0.70 Vc= 1 0.60 -IMAGINARY ARM 0.50 (PHASE VELOCITY VERSUS SPATIAL ATTENUATION NUMBER) _IJ' i I' I, I,', I, l' A 1.80 I. 70 1.60 1.50 1.40 L30 1.20 LIO 1.00' im Figure 2.4c Velocity Spectrum for h/R - 0.1 (Imaginary Arm).

h =0.06 V 0.06 V_ MORE EXACT THEORY - - ELEMENTARY THEORY N =n+im 0.20 VCO FROM ELENTARY THEQ PROJECTION OF COMPLEX ARM.V. _ FRMMREATT. ON V -im PLANE FROM MORE EXACT THEO.ON OF COMPLEX PROJECTION OF COMPLEX (PHASE VELOCITY VERSUS // ARM ON V-n PLANE SPATIAL ATTENUATION PL NUMBER) 0.15 / 10 0.05 0.15 0.10 Q05 0 0.05 QIO 0.15 im n Figure 2.5a Velocity Spectrum for h/R - 0.06 (Complex Arm).

v h =0.06 MORE EXACT THEORY N =n -- ELEMENTARY THEORY 2.0 1.5 REAL ARM SECOND REAL ARM Vc2: I 1.0 v,, ~s/ Vci = -8 - 0.5 05.FROM ELEM. - ITHEORY E FIRST REAL ARM VOFROM MORE EXACT THEO. 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 n Figure 2.5b Velocity Spectrum for h/R.= 0.o6 (Real Arm).

h =0.06 - MORE EXACT THEORY V (NO IMAGINARY ARM FOR N =im ELEMENTARY THEORY) ____Vc2 I 1.00 0.90 IMAGINARY ARM (PHASE VELOCITY VERSUS 0.80 SPATIAL ATTENUATION NUMBER) 080 0.70 V.60 0.50 1.90.80 1.70 1.60 1.50 1.40 1.30.20 1.10 1.00 0 im figure 2.5c Velocity Spectrum for h/R o0.06 (Imaginary Arm).

V h 0.03 R Vco FROM ELEMENTARY THEORY "/ ~ VCO FROM MORE EXACT THEORY MORE EXACT THEORY // 0.12 --— ELEMENTARY THEORY,// /A |N =n+im 0.10 PROJECTION OF COMPLEX ARM ON V-im PLANE (PHASE VELOCITY VERSUS SPATIAL ATTENUATIONo08 PROJECTION OF COMPLEX NUMBER) ARM ON V- n PLANE0.06 / - Q04- I[~~~~ ~~0.02 0.10 0.08 0.06 Q04 0.02 0 0.02 0.04 0.06 0.08 0.10 im n Figure 2.6a Velocity Spectrum for h/R = 0.03 (Complex Arm).

v R =0.03 MORE EXACT THEORY N-... —- ELEMENTARY THEORY N=n 2.0 1.5 - |-,0 -. 0 REAL ARM SECOND REAL ARM VC2 1.0 VcI = 8,/ \ O VoFROM ELEM. S' — FIRST REAL ARM o 0 FROM MORE EXACT THEO. 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 L8 2.0 n Figure 2.6b Velocity Spectrum for h/R O.03 (Real Arm).

h MORE EXACT THEORY V R *0.03 (NO IMAGINERY ARM FOR N =i m ELEMENTARY THEORY) Vc2= I 1.00 0.90 IMAGINERY ARM (PHASE VELOCITY VERSUS SPATIAL ATTENUATION NUMBER) 08 0.70 0.50 I I I I I I I I I'~, 1.80 1.70 1.60 1.50 1.40 1.30 1.20 1. 10 1.00 0 im Figure 2.6c Velocity Spectrum for h/R = 0.03 (Imaginary Arm).

-36to all coordinate axes. In those figures, v is taken to be 0.3, K to be 0.833 and h/R to be 0,1, 0o06, and 0.03. B. More Exact Theory (Corresponding to Timoshenko Theory in Beam Vibration) 1. Equations of Motion for Free Vibration For steady state wave propagation in the wall of a tube with infinite length under free vibration, the dimensionless equations of motion from this theory are {a -fWW —W,)-S- I= 0 (2.7a) el 9$s _ 8Z( Wb at Ws)0 L g2 i + - T - = (2.7b) where dimensionless variables Wb, Ws, X, and T as well as dimensionless parameters 5 and g are defined in Chapter I. 2. Frequency Spectrum Assume the following solution for Equation (2.7) rW=& _ A, eI, L -(NX - T) (2.8a) l5 AZ - AT) (2.8b) where Q is always real, then Al and A2 must satisfy (a-x ) A1 _ (-, t- ) A~ = o (209a) (W2-) AI + A z = (2.9b) For a non-trivial solution of Al and A2, the determinant of the coefficients must be zero and this then is the frequency equation

-37or 7Z4_r[?(t - sMt~) SC(a-=. z _ a(2.10) +Z = Z t ^ (I ~ 5) 11- ^ ~ [(Z0;Z^ 2 ))'f jJy- ),X (2.11) If N is complex, then For 2 to be a maximum, the above expression should be equal to zero, namely, _ }- 7 -2t -,1 L,, t —_- -,- )= ~.= ~ Let Q~be the root of the above equation, then j(2 = 5 [_, _ J) + _ 26 ( _, iL)az])tz Taking the negative sign 2 _ S-,_ __, -'(_- )) ] since az=' ();2 <' I ~: -'~~ G ~z _.z 0 there is no real rooto Taking the positive sign

-38" L T F Lf ^ ) +t E ({ z-E- 2 _l_ (2.12) --'- Z' ) ~ i (52 [ - (higher order terms > 0)] so that 2 < 82g2 and only one real positive root exists. Taking the positive sign before the radical in Equation (2.11) N- +) - _ - ~ )] If 0 < 02 < K*, N is complex and this is the complex arm in the frequency spectrum. If * < KQ, N is real and this is a part of the first real arm in the frequency spectrum. Taking the negative sign before the radical in Equation (2.11) N'= L 1 i ) 1 +s ) -c u1( t a- )(I-) (p If.2* > Q > 0, N is complex, and N is conjugate to that in the previous part. If only first quadrant is used, this region need not be plotted due to symmetry of the diagram. If 2* < K2 < 5g, N is real and this is another part of the first real arm in the frequency spectrum. If bg < I < 8, N is imaginary and this is the imaginary arm in the frequency spectrum. If O > 5, N is real again and this is the second real arm in the frequency spectrum, The frequency spectrum is plotted only for the first quadrant in Figure 2.1 through 2.3, since the spectrum is symmetrical with respect to all coordinate axes. In these figures, v is taken to be o.3, K to be 0.833 and h/R to be 0.1, 0.06, and 0.03.

-393o Velocity Spectrum Assume the solution of Equation (2~7) has the form -F^b = 84 e L-(X —^T) (2.13a) WTS = A.e i (X -v(21b) where V is always real. Substitute into Equation (2)7), relations of A1 and A2 are obtained (M'-'t ) 41 [j)-Az + _( )- 2jAZ 0 (2.14a) Nz(V-| ) A -t g' Az -0 (2.14b) For non-trivial solution of Al and A2, the following determinant must vanish |D (. - ) ((2:15) or (V -I)( -l) N4- (l+ -] h (2.16) Nz tt)^v((22[ v (^-......).-( )(..-1)- (2.(17)

-40If the positive sign is taken before the radical in Equation (2.17), N becomes infinite at V = 1 or V =. V = Vcl=5 is the dimensionless modified shear wave velocity in a plate and V = Vc2=l is the dimensionless dilatational wave velocity in a plate. From Equation (2.17), we see that N becomes complex if the argument under the radical is less than zero and N becomes either real or pure imaginary if the argument is greater than zero. Thus we can define V = Vco to be that which makes the argument zero [v It* ) -;-r1 4-(. - a ) (c, -I ) or f1- s^)*v4Z t Zf2l+2a'-A) V^-^ f(a-+S^)-O v = ^J^ \ - _2( I +2 -Z ) ~ I 4 (I ^-rf.-f (I —.. Since 452 > g and V is real, a positive sign is taken before the radical, then e W = 0Z-lr H-zWR- Vf^I) + (l+2Z- 3)2-(CI -- 2- f (2.18) There are three critical velocities Vco, Vcl, and Vc2, which divide the spectrum into four regions. These regions are discussed separately. Based upon Equation (2.17), the velocity spectrum V versus N can be plotted. There are four regions:

-41i. 0 <V < VC N is complex, and this is the complex arm in the velocity spectrum. ii. VCo < V < Vc1, where Vcl 8 N is real for both positive and negative signs before the radical in Equation (2.17). This is a part of the first real arm in the velocity spectrum. iiio Vcl < V < Vc2, where Vc2 = 1 N is real for the negative sign before the radical in Equation (2.17) and this is another part of the first real arm in the velocity spectrum. N is imaginary for the positive sign, and this is the imaginary arm. iv. V > V2 N is real for both signs before the radical in Equation (2.17). These are still another part of the first real arm and the entire second real arm. The velocity spectrum is plotted for the first quadrant in Figure 2.4 through 206, since the spectrum is symmetrical with respect to all coordinate axesO In these figures,- v is taken to be 0.3, K to be 0.833 and h/R to be 0.1o, 0o06 and 0.03. Co Comparision of Results from Both Theories 1. Frequency Spectrum When Q < 6g, spectra from both theories are with complex wave numbers, and they almost coincideo The reason is that the frequency is very lowo When Q > 6g, the first real arm of the spectrum by the more exact theory has the same shape as the real arm of the spectrum by the elementary theoryo For frequencies a little bit over 8g, they are

-42close. The imaginary arms of the spectra from both theories are quite different except these for low frequencies. There is no second real arm for the elementary theory. 2. Velocity Spectrum When V < Vco, the velocity spectra with complex wave numbers for both theories almost coincide. When V > Vco, the velocity spectrum from elementary theory exists only with real wave numbers, These with smaller real wave numbers lie almost on the corresponding part by the more exact theory. The rest are far from the more exact theory.

III. STEADY STATE RESPONSE OF A THIN-WALLED CYLINDRICAL TUBE WITH INFINITE LENGTH UNDER INTERNAL MOVING PRESSURE Ao Solution of the Equation from Elementary Theory Let p be the intensity of the pressure whose front moves with velocity v as shown in Figure 3o.o In the ideal case, both p and v are constanto For the steady state response the pressure is assumed to have acted for a long time. Introduce the dimensionless variables P and V for p and v respectively P = g (3.la) 12 icG V = -- (3.ob) From Equation (1.39) the equation of motion is -h52 -- ^1-=? -~<x< -~<-r< (2 7;:= +;)iJ t X -T o = 3 Pa (3.2) where r P when X < VT P (XT).= 0 when X > VT Since this is to find the steady state responses W(xT) is required to be bounded everywhere instead of assigning any boundary and initial conditions for Wo The partial differential equation given by Equation (3.2) can be transformed into an ordinary one by means of the Fourier transform(l^ then it can be solved. Take the Fourier transform with respect to T and the transformed function is w = J W(X,5T)eTdT (3T3) -43

- \ r 7 \o P~~xv X = VT Figure 5.1 Moving Pressure in a Tube with Infinite Length.

-45Because of the convergence of the transformed forcing function P, imaginary part of 2 must be greater than zeroo The inversion formula is given by W(XJT)= 2T (X-2,)e-'& Td2 (3.4) where the constant y > 0, To justify this transform, the integrals given by Equations (3.3) and (3.4) have to be proved convergento It is assumed this is the case, and when W(X,T) is found one can verify that this W is a solution simply by substituting into Equation (3.2)o After the Fourier transform, Equation (3.2) becomes nS) ='Z _ e TAT (3~5 cr4 X4R-2) = ^ e (3.5) with JW(X,O) < co for all positive* value of X. Since we do not need the boundary conditions, the particular solution of Equation (3.5) is enougho Assume it is w (X,,. )- Ae vX To satisfy Equation (3o5), the coefficient must be A L= * For negative values of X, the non-homogeneous term in Equation (3~5) will be changed, but the final solution for W is the same. For convenience, use positive values of X to find W

-46The particular solution is VPZP e ~x w (X,, ) 4- (3.6) Finally, inversion by Equation (3.4) yields -T) tvLP I <X <~T (3 7) Put N = - into Equation (3.7) and it becomes (Xz 1 N (3.(X T8) TT) = P ^o+L ( N(4- JZ 2 ) *N.8) where - ylj =- Y and is greater than zeroo From Equation (3o8), it is obvious that W(XT) depends on (X-VT), the distance from the pressure front, because this is a steady response. To compute the integral given by Equation (3~8), The residue theorem(17) is to be used. Before applying the residue theorem, the poles of the integrand must be investigatedo The poles are the roots of the equation N ( 4 _ z NZ + jZ ) o or N-=O and N4 - V2 N + 2= The latter is identical to Equation (206) which expresses the relation between the phase velocity and the wave numbero From Figure 2o4 through

-472.6, N's are complex when V is less the critical velocity Vco ( = XJ& ) and real when V is greater than Vcoo These two cases are discussed separately. 1. Case for V < Vc In this case, N's are complex and let them be N =-~ n~ (3 9) a. Deflectionfor Positions Before the Pressure Front Arriving, i.eo, (X-VT) > 0 Take the contour as shown in Figure 3.2, then, tJ J eIsX-^ _)a JCI N4- V ^) I. =residues at N1 and N2 As "a" approaches infinity, by Jordan's lemma i VT(X-)T.,. N (, 4- V='+'-) so that [a~~, e^^'N(' VT) iL' NCN______ _ _ _ N V= residues at N1 and N2 LZ;r ~L"J (N4- -Vzz 1vz - z e e CX - ( VT) LZ(f;L)covsfn(~X.-.VT) -*,2 )zsin n(X-VT)1 (3.10) Since (n2 + m2)2 2g29 the deflection becomes WX,7T) = -^z e- X - VT) cn - T) (X)-n - (3 1 X c, 0 ( VT) — 2n sin n(-'VT) (3.11) 2 WI n~~~~~~~3o1

-48im N - PLANE / I / N2 NI Y, 0 N3 N4 Figure 3.2 Inversion Contour for (X - VT) > 0 N- PLANE im N2 Ni y, \ on n N3 N/ Figure 3.3 Inversion Contour for (X -VT) < O

-49b. Deflection for Positions After Pressure Front Having Arrived, i.e.,, (X-VT) < 0 Take the contour as shown in Figure 3.3, then -] j -'. e NX-T) r N (2VT) Vzl,N4- z)'Cz iC$- NW)d M = residues at O, N3, and N4 As "a" approaches infinity, by Jordan's lemma M (CX-VT) J' e L J so that'.+t, 9( ( _J(X- V7T) 0^ 0 t%- (ZJ-Z 2.+S ) diN = residues at 0, N3, and N4 - -I - e 7 i. ) __ _rt _ _ COs n ( X- vT) t 4`4rn)iLSin ( VC-vT)] (3.12) The deflection becomes W(XT) - er (X-vT7). 3%Z Z~7[co5n(X-VT)+ z s 5 in n (X-VT)] (3o13) c. Conclusion W(X,T) given by Equations (3.11) and (3o13) satisfies the equation of motion given by Equation (3.2); it is bounded for all values of X and T; and it depends only on the distance from the pressure front. Hence it is a solution of Equation (3.2).

-50For simplicity, the radial deflection can be expressed in terms of the distance from the pressure front S = X - VT (3.1.4) Introduce a function of S F(S) = e-' IS [cos n' —Z s n n 1 I (3.15) The deflection can be expressed in terms of F(S), ie.e. w(s) = - () for S < 0 (3.16a) W(S) = z- F(S) for S > 0 (3.l6b) 2o Case for V > V ~________co In this case, N's are real and let them be N = 0, ~nl, and ~ n2 (3o17) All poles of the integrand in the integral given by Equation (308) are on the real axis as shown in Figure 3o4~ By the residue theorem, the deflection can be found as follows W(XT) = 0 for (X-VT) > 0 (3o18a) since there is no pole above the real axiso W(X,T) - ~ C ( nIa T ) cO5 fl n(X -vT)1 I1 nZ- n~ Y1,z n2 for (X-VT) < 0 (3 18b) where n1 and n2 are positive and n1 > n2,

-51N- PLANE iim.... -4 ": - v- n -n1l -n2 0 n2 nI Figure 3.4 Positions of Poles When V > Vco Without Damping

-52It is obvious that the following term can satisfy the homogeneous differential equation given by Equation (3~2)o WL(X,T)= - *-a I *. Cos n1 (X=VT) (3 19) If this solution is super-imposed on the solution given by Equation (3ol8), the resultant solution W(X,T)= n ^= C 053r(X-VT) for (X-VT) > 0 (3.20a) W(X,T) = Lz-' — COSnz(X-VT) for (X-VT) < 0 (3 20b) still satisfies Equation (3o2) and is bounded every-whereo It is difficult bo distinguish which of the solutions given by Equations (3.18) and (3o20) is the true one. By means of Kenney-s concept of vanishingly small damping, a physically sound solution can be found and it is the true solution, For taking into consideration the viscous damping, a viscous damping force term is added to Equation (3~2), then it is X4W, + a+C +P C (3.21) P when X < VT where P (XT)= 0 wh.en X > VT | W(X,T) I< o everywhere) and C is a dimensionless constant proportional to the coefficient of the viscous damping in the tube wallO By Equation (3o3), Equation (3o21) after transform becomes T + -a( sZ _ -cC W = e1 (3.22) d ~ u J2 ~

-55The particular solution for the above equation is -? TP. J12 W(X -2) = - - eL- vX (3.23)'V4Q4 Al- L) + The inversion yields W(X,T) =5 j ILd 277 oaQj (l —JZ —- d _F2Z) VT oo^,,+, N(X-VT) = tT P [ J1 ___ ^_______jW (3.24) 27T J- hj4- z-,_ +, (. z_, V +- 7) where Yl > 0. The poles of the integrand in the integral given by Equation (3.24) are the roots of the equation N( N4 - V Z- iCV~ + VN Z) - o or N= 0 and N - -'zh - icvN - zz - 0 (3.25) Assume J = -- d rn,, z + i z (3~26) where n1, n2, mi, and m2 are real, ni > n2; values of m1 and m2 are small compared with those of nL and n2 and they approach zero as C becomes vanishingly small. Insert N —= n + im into Equation (3o25); the following equations are established n4 —I zrn z -w,4 _ Vz(lz — wn)-+dVr -+ I''2 =- ~ (3o27a) 4 (nZ- w^Z)- - - 0-V=O (3.27b) Since m is small compared with n, the above equation can be reduced approximately to r4 — VZ n2z Z $ = o (3.28)

-54and 4)nz — zm z- Cv _- 0 so that m =(329) Z(2n2 z ) (3 29) Solutions for Equation (3028) are nl — 2 (3.30a) n2- =4 2 (3o30b) 2 2 Since V > Vco (Vc = 425g) n n2 are real and positive. The corresponding m's are _mv, > 0 (3 31a) 1 2v4_4s- z Cmv - - < 0 (3031b) - - V-4Zy The poles of the integrand in the integral given by Equation (3024) are shown in Figure 3 5 a. Case for (X-VT) > 0 By the residue theorem, as C approaches zero, the deflection is'WIAX,7) =n nl;(X- VT) or w()= hnz -) cOS 2C-VT) (3032a) I I Zz, cs ~ CX-vr

-55N- PLANE im miam2 APPROACH ZERO AS COEFFICIENT OF DAMPING VANISHES 0 0 nl+im n, +im, 0 - n2 +m. i n2+m2 i Figure 3.5 Positions of Poles When V > Vco with Damping

-56b. Case for (X-VT) < 0 By the same procedure, the deflecetion is W(X,T) =^ Cos n (X - T) or w(s) = - - Co 05nz (3-32b) B, Solution of the Equation from More Exact Theory Introduce the same dimensionless variables P and V for p and v respectively as in Equation (3o1), the equations of motion from Equation (1.40) are I -f2WW +Wt~s ui-I'~A.... L -?-) (3.33a) Wax+, - -o 0 (3.33b) P when X < VT where P(X,T) = 0 when X > VT instead of assigning any boundary conditions, Wb(XT) and Ws(X,T) require bounded values everywhere. Use Equation (363) to transform Equation (3-33) into ordinary ones - Z ) (bz L?) (3034a) jfz Z +'Ws + _ Wa m 0~ (3.34b) with IWb(X,Q)I< co and I|W(X,Q) 1 < oo for all positive* values of X. * See note on Page 45

-57The particular solution for Equation (3.34) is FWb(X,J2) =,e xr (3.35a) Ls ( CRZ) = aze t (3 35b) where A1 -4 -a -. -.) A = 2 = v....^- IV(4 +.^) ] } The inversion from Equation (304) yields Wb (xT) T= z Al e e d - " JplJ (3 36a) - J-P +L AV9-O -j-t z- (3 36b) LTT e- oo { D- ^f)^^ ^} C) 0 where N = V aand yl > O0 Poles of the integrand in the integrals given by Equation (3~36) are the roots of the equation N {(Vz-1)- -I) )N4- Z(1)- 2] +- 2 =0 or N= 0 and ~vr -,)(~ _-).4- tvZ(, +)-t 1 -t +,^f^ o

58The latter is identical to Equation (2,16)o From Figure 2.4 through. 2,6, N's are complex when v < V real when V < V < V two real and co co cl two imaginary when Vcl < V < Vc2 and real when V > Vc2o These four cases are discussed separatelyo lo Case for V < V co In this case, N's for the latter equation are complex, N = ~ n ~ im (3~37) By the residue theorem, the deflections due to bending and shear are'W ) = 7 - e-a,oi-T e( — sn (X-V)-T) (v -_) 4-.' 5(TXT)= (w P-' 41n Sin n(X- VT) for (X - VT) > 0; T(XT)=r — l e^C-VT) VCosn (X-VT) + Z M Sin(2-T)j j Z Z12 2 O'vnb( ea. e rn(X-v7) Ws (T) =v —) 4 — ~ Sn n ((X-VT) for ( X- VT) < 0. Again introduce S = X - VT, and define a function of S RFS) =e-' [GcosnS- ^ srnn I then Wt$) = -- - for S < 0 (3o38a) bv(W) -= t ]F(C) for S > 0 (3o38b) -WVsS)= - - Sin ng

-59for all values of S 2. Case for V > V In this case, the solution for Equation (3~33) is not uniqueo In order to find a physically sound solution, a viscous damping term which vanishes in the limit is also introduced into Equation (3~33) _W I? vz (WW t Ws )- a)2 -z ^)a-(x,T) (3.40a) -b73C +5Z ws - b o (3o40b) P when X < VT where P(X,T) = 0 when X > VT By means of the same technique as used in the elementary theory, the following solutions are found in the limit as C vanishes: a. Vco clV V - 4 P Cos ni2S TWbS () =- ( L- ( )(-^ * nrl;- ) for S < 0 (3.41a) _ 54p cos nS w^ =- 43. Cos n,2 ~for S > 0 (3.41b) Wb~'=,-,)tv~_lC 6L h-)' ) n?,j nz) Swv S () -= ( 2 P Cos nl) for S < 0 (3.42a) WS (S) =- (Vz-") I-n,- z) for S > 0 (3.42b) where S = X - VT, n1 and n2 are the positive real roots of Equation (2.16) and nl > n2 is assumedo b. Vcl <V < V2 z + (^)(L )) co + Z^] for S < 0 (3.43a)

-60Wb(S):-(vL2I)(V2L_) ZOZ(^tZ'rZ) for S > 0 (3.43b) WJS( ) = v(c Ms — — osS) ( for S < (3.44a) ws (S)= ^^ ^ * ^z+r) 9for S > 0 (3.44b) Where ~n are the real roots of Equation (2.16) and,imi are the imaginary roots of Equation (2.16)o c. V >Vc2 Kbt ( — B L )+(lw^, sn v-,) VIn for <0 (3.45a) rb(S) = 0 for S > 0 (3.45b) -ws(2) = P: 5 ( Wn -1 -os nZSf for S < 0 (3046a) wusts) =0 for S > 0 (3046b) where nl and n2 are the positive real roots of Equation (2ol6) and nl > n2 is assumed. The reason why Wb and Ws are zero before the pressure front arrives lies on the fact that there is no disturbance, since the dilatational wave velocity Vc2 is less than that of the pressure front in this case. Co Discussion 1. Soundness of the Sinusoidal Wave Solution for the Case V > Vc Since the deflection curve of the intersection of the middle surface and the radial plane is a sinusoidal wave extended to infinite length, the energy in this system becomes infinite' It is doubtful whether this result might violate energy principle~ Actually it does not as was pointed out by D~rr(9) and Kenney(lO)1

-612. Comparison of Results from Both Theories a. V<V co The formula given by Equation (3.16) for W has the identical form as that given by Equation (3~38) for Wbo From the velocity spectra in Figure 2.4 through 2.6, it is obvious that they are almost identical, i.e., n's or m's are nearly the same for both theories. The real part of the wave number is small, so that the frequency of vibration is low. In this case, the total deflection in the elementary theory is nearly equal to that due to the bending in the more exact theoryo In calculating the bending stress, the deflection formula given by Equation (3.16) may be used without any serious error. b. V > Vc From the velocity spectra in Figure 2,4 through 2.6, they are quite different for both theories except the part for small wave numbers. In this case, the high frequency of vibration is caused, so the deflection formula given by Equation (3016) from the elementary theory may not be used. Instead, the formula given by Equation (3-38) from the more exact theory should be used in calculating the bending stress,

IV. TRANSIENT RESPONSE OF A THIN-WALLED CYLINDRICAL TUBE WITH SEMI-INFINITE LENGTH UNDER INTERNAL MOVING PRESSURE In this chapter, a cylindrical tube with a semi-infinite length is dealt with. The boundary condition at one end of the tube is that the periphery is simply supported in the radial direction. The pressure front starts to move at this end, The assumptions as to the intensity and velocity of the pressure front are the same as those in Chapter IIIo Ao Solution of the Equation from Elementary Theory In this section, only the case for the velocity of the pressure front less than the critical, i.e., V < Vco is investigated When V > Vco, the elementary theory will involve serious error. 1, Formulation of the Problem From Equation (139), the equation of motion which is valid in the quarter plane defined by X and T is 4 + W; 7 t +- - = OP(X,T), x>o, T>O (4.1) P when X < VT where P(X,T) = { O when X > VT At the near end, the radial deflection and its second partial derivative with respect to X are assumed to vanish, since the periphery is simply supported in the radial direction. At the far end, no disturbance is assumed, so that the radial deflection as well as all its partial derivatives with respect to X vanish, The boundary conditions are -62

-63W(O,T) = 0, a (X = (4.2a) l m'tg T)_ } XzT) = 0) X-0o X —Ou a /,. Y ~wayj)' T) (4.2b) IJ W 1T) o 2O a=x1=T) =,.VOU a'wcx' No initial displacement and velocity are assumed radially, so the initial conditions are W(X',0)= 0, w(X,o = 0 (4.5) 2. Method of Solution The partial differential equation given by Equation (4.1) can be transformed into an ordinary one by means of the Fourier sine transform.(l8) Take the Fourier sine transform with respect to X, then the transformed function is W(N,T) = SJ W(X,T) SinNX dX (4.4) where N is real. The inversion formula is given by W(X,T) = f J W(N,T) SinNX dN (4.5) To justify this transform, the integrals in Equations (4~4) and (4.5) have to be proved convergent. It is assumed this is the case, and when W(X,T) is found one can verify that this W is a solution simply by substituting into Equation (4.1). Due to the boundary conditions given by Equation (4.2), Equation (4.1) after transform becomes

-64d'w + ( N4t ) W = ( (1- Cos VTN) (4.6) The initial conditions after transform are W(N,0) =,')dT 0 (4,7) By the convolution theorem, the solution for Equation (4.6) with initial conditions given by Equation (4 7) is FW(m,T)=- t' - io (I- coSVUN)Sin C(T- P')dZ -= aI - -Co'5-aS ) 3 CT- svS 7 (4.r8) where Rn = _+ -h 62Z (4.9) is identical to the frequency Equation (253). The inversion yields W(X,T) = -I W (N,T) SinXN dN -any'P 2 frgsinX N i.rCoSVT nxw jd -a Jorrt j2 d -J - dN - rf: 1 &- Jt S] -in2 XCco2 Td- (4.10) The first two integrals on the right hand side of Equation (4.10) can be evaluated in closed form by the residue theorem, and they are

-65IT " )-R ] N4 V^a ^z) N2 Z N Tr 2-(M^+ y ) Jos t (M4- 1 "nC2 n.). z e-'(X- VT)[cosn( VT) -,sn n(X+vT)] - CoS A X for (X - VT) < 0 (4.11a) e-(-. [ COS Y (X- 7T)- z (N4 —~z- -- C o n,..m are both real and positive and A= - - (4.12)

-66Let the last integral on the right hand side of Equation (4.10) be L t_ r [ 2 _ __ s NTX ciJ2T dJ or - T o (N.4) - V Mrz c+ 4ta Td a (4.13) It is very difficult to evaluate I2 by means of the residue theorem, since the integrand involves branch pointso Numerical integration will be used to compute 1I o The final result is W(2C,7) = 1, TX,T) -ft 1 (X, T) (414) 3~ Interpretation of the Solution a, Contribution Due to I. The term involving (X-VT) in Equation (4,11) is identical to Equation (3,16) which is the steady state solution with the pressure front moving to the right as shown in Figure 4ola; the term involving (X+VT) is the steady state solution with the pressure front moving to the left as shown in Figure 4olb; and the term involving XX is the static solution due to the pressure, as shown in Figure 4l1co The solution superimposed by these three parts can satisfy Equation (4,1) and boundary conditions in Equation (4~2), but the initial conditions in Equation (4.3) cannot be satisfied, In order to satisfy the initial conditions, 12 must be super-imposed,

-677-ACTING OUTWARD /00 +- I -' F x Figure 4.la Pressure Front Moving to the Right -ACTING INWARD / V'< * Figure 4.lb Pressure Front Moving to the Left ACTING INWARD iACTING OUTWARD ( p -co +cb Figure 4.l Static Pressure

-68b. Contribution Due to 12 The solution due to 12 is to correct the initial conditions given by the solution due to I1. This can be interpreted as the transient term and it will disappear as T approaches infinity. This is true from the Riemann-Lebesgue lemma. B, Solution of the Equation from More Exact Theory From Equation (140), the equation of motion which is valid in the quarter plane defined by X and T is - - ( Wb + w/ K TS) — 7 Tbt ) _ -p?) (4.15a) i~ t- H SZS - zw = 0 (4.15b) >o, T >o fP when X < VT where P(X, T) = 10 when X > VT The boundary condition at the near end is simply suported along the periphery of the tube and no disturbance is assumed to exist at the far end; they are WJb (OT) + WT (O,T) = 0 (4.16a) A)co/n _ 0 i Wb( T).m Wlb,XJ,T) -o, Ii A; O (4.16b) X —o x —o X

-69For convenience in later use, the boundary conditions given by Equation (4.16a) are changed into the form WTb (ojT) =0 v W 7(T) = o (4.17) These two forms of boundary conditions at the near end are consistant, b2w since b will vanish in the limit of X = 0 if Ws(O,T) = 0 and;x2 Wb(O,T) = 0 are inserted into Equation (4.15b). The initial displacement and velocity due to both bending and shear are assumed zero, therefore the initial conditions are:(3o) = 0 W ~', o (4.18) 35 (X,o)=o, OVSo) o 1. Case for V < Vc The Fourier sine transform is also used to reduce Equation (4.15) to ordinary ones. By Equation (4.4), Equation (4.15) with boundary conditions given by Equation (4.17) becomes *-N v': -a (fb t+ s) - a d (a ^ tf ) = _ (1- Cos SVT) (4.09a) l- N ~ Wb +- W s - -( 0 (4.19b) From Equation (4.19b), Ws can be expressed in terms of Wb and its

-70second derivative ws =. T' -t-^ dTZ (4.20) Eliminating Ws and daWs in Equations (4,19a) and (4Q20), a fourth order s dT2 ordinary differential equation is obtained (d4^R ^Z; 2 +[52. 2. 21 dT ar4.(t c - + t NZ ~ )j * z d * (: + 3 T) t~4:] Wb =-? (-1 Cos MVT) (4.21) The transformed initial conditions are f b(,o) =o! d~)=o IdT j (4.22) dT W'(,)o.. dT o Solutions for Equations (4.21) and (4.20) with initial conditions given by Equation (4.22) are + PN - - F COS JvT] (4.23b) where F1 = N+9 ~;NZ t Z (4.24a)

-71Fz = (V'-)( VrJ- ) N4 — [ I'VZ(lt* ) -,5'Y] N 4 (424b) = ^ ~ NZti~)3 f^(I+~^) ] y [ N^) - [ Z(MA+r)4t,]s (4.25a) f -z- CH-,J(+)- -) + / [ (-)I. -^Z)']tWz ['DN(.,) -.i 2 ~'z. (4.25b) z ~ h = 1' ) i "; -N —Z)- (4.26a) 32 \( ^ 4 i- F 1 VZ l 7(4.26b) In Equation (4.23), the last term on the right hand side is for the particular solution and the first two terms are solutions for the homogeneous equation. Since the particular solution cannot satisfy the initial condition in Equation (4.22), two additional solutions for the homogeneous equation are super-imposed in order to match the homogeneous initial conditions. Equation (4.25) is identical to the frequency Equation (2.10), so that nl is the frequency in the first real arm and n2 the frequency in the second real arm in the frequency spectrum. All are with real wave number N. By the inversion formula given by Equation (4.5), Wb(X,T)

-72and Ws(X,T) can be found Wb( -'Tr ) B f F. I _ SF_____I + orjo a r\ "n 7^ nmASd + — f Jf 3z os - S2zT Sn NdM = Ib + I b + 1b (4027a) N t- I W P ET)'?j[S FV C65MVT1SnXM) dXl + r 5zI B, ( 4'-Q;) Cos5Q,T 5 n X J d + — f00 BzC r r- -2^ ) ~o~ 2zT Son X M d cM _ A 151 + I (4.27b) In the above expressions, Ib and Is are due to forced vibration; Ibl and Is are due to free vibration corresponding to the first real arm; Ib2 and I 2 are due to free vibration corresponding to the second real arm. It will be shown later that in the case V < Vo, the contribution due to the second real arm of free vibration is small, so that it can be neglected.

-73The improper integrals in Ib and Is can be computed without any difficulty be means of the residue theorem, but these integrands in Ibl, Ib2, Isl and Is2 involve too many branch points to be computed by analytic methods. Numerical integration will be used to compute them, By the residue theorem, Ib and Is are computed as follows: z 0 co fI' o 5V' wvAO- — )Tr) x IbtXT) = lr- s'T> lf ( _ - sT ) s?-nX _d -= -- - - r -' tr)coSn (X-V) +'Sn (x —T)] -- 3 &-o (6on Ylo - A s,-o n n X) for (X - VT) < 0 (4.28a) I (X T) = r (X - VT) [5os n C(- "T) - Y'- - 5s r) (X-VT7) - - e- T)[ (os,5 (2C-r ) - SArn x:v)O) for (X - VT) > 0 (4.28b) where N = n + im is the root of the Equation (2.16), i.e., (VV-l) (VS-S ) tj4 - ['V( | t)- 5 1 az t - 2 = O

-74and n, m are both real and positive; No - no + imo is the root of the equation N + ~ NZ t Ba - = o and no, mO are both real and positiveo I( LT) Sn noXo ~2-p e-v nX-VTI t (4), 4 n srn n (XC- -T) >,Q-(X+-VTT) - ~. ~ e- si n (X+' T) (4.29) The solutions due to Ib and Is have the similar form as that due to Ii in the elementary theory, so they have the same physical interpretation. The solutions due to Ibl and Ib2 or Isl and Is2 are employed to match the homogeneous initial conditions. Ibl or Isl corresponds to 12 in the elementary theory, but there is an additional term, Ib2 or Is2, in the more exact theory, since there is a second real arm in the frequency spectrum. 2. Case for V > Vco In this case, the Fourier sine transfrom cannot be used, since the transform formula given by Equation (4.4) diverges, The standard method of solution is touse-the Laplace transform with respect to time variable T, to reduce the partial differential equation into an ordinary one, then the transformed function is to be determinedby solving the ordinary differential equation. The difficulty lies in the fact that the

-75integrand of the inversion integral involves too many branch points to be dealt with. The method used in this work consists of two parts, First, the particular solution which satisfies the non-homogeneous Equation (4.15) only is found as that in Chapter III; then a solution of the homogeneous equation (set P(X,T) = 0 in Equation (4015)) is super-imposed to the first one in order to match the boundary conditions given by Equation (4,16) and the initial conditions given by Equation (4,18). The solution of the homogeneous equation with specified boundary conditions as well as initial conditions is to be determined numerically by the method of characteristics. Since the forcing function P in Equation (4,15) is a stepped one which has a discontinuity at X = VT, the method of characteristics is difficult to apply directly in Equation (4o15)o This is the reason why the solution has to be calculated in two parts, The homogeneous equation is obtained by putting P = 0 in Equation (4.15), then b- c Xf (Wb, )- aZ btt =0 (4.30a) +lXX 3^5- a^ = o (4030b) X >o T > o The boundary conditions are specified at X 0 as T-Vb(To,) -+ Ws,(OT) = -,(T) (4.31a),T X = = (T) (4.51b)

-76The initial conditions are specified at T = 0 as |w, tx>o)= C3a().' EWagV qt5)2Cx) (4.32a) W5 (X,o~) = X) ^ = ( (4.32b) Let k be the slope of the characteristic lines of the system in Equation (4.30), k is to be determind by the following Equation7l9)! h_ -Z ) _ 1 =0 o (k'- I) or (l Z- ) (J 2- I) = 0 There are four real k's, i.e,, so that there are four sets of real characteristic lines for Equation (4530) and the system can be reduced to a fourth order hyperbolic differential equation. The characteristic lines are dTJ7* ~ =K= s (4.33) Since 6 is the dimensionless modified shear wave velocity in a plate and "1" is the dimensionless dilatational wave velocity in a plate,

-77shear waves are propagated along the characteristic lines with slope +, and dilatational waves along the characteristic lines with slope - s + 1 o There are only two kinds of waves in this elastic system(l5) Each can be propagated with positive or negative velocity, so there are four sets of characteristic lines along which they are propagated from the source as shown in Figure 4.2o For convenience in writing out the finite difference equations, defined along characteristic lines, four first order simultaneous equations(29) are used instead of two second order simultaneous equations given by Equation (4.30). Five new dimensionless variables are introduced. They are dimensionless total deflection in the radial direction: V7 = b - W s (4.34a) dimensionless velocity along the radial direction: - = ~ (4.34b) where r, - (LAb b+ Ss) at dimensionless angular velocity due to bending: I~ - d Z (4.34c) where atax

-78- CHARACTERISTIC LINE ALONG WHICH THE DILATATIONAL WAVE IS PROPAGATED -. — CHARACTERISTIC LINE ALONG WHICH THE MODIFIED SHEAR WAVE IS r- jPROPAGATED i \ I SOURCE X Figure 4.2 Four Characteristic Lines from the Source.

-79dimensionless resultant shearing -or ce: G- (4.34d) and dimensionless bending moment: M =.h'Z M Xx (4.34e) The four simultaneous equations are CX -a f-P I aT (4.35a) a- = <r- Go (4355b) a, (4.35c) =T - X -(4.35d) Since W = w0-i- T T To where WTo is the radial displacement at T = ToQ there are only four unknowns, M, T, Q, and v, to be determined by the above four equations with boundary conditions specified at X = 0 and initial conditions specified at T = 0 o Along the characteristic lines, these equations are simplified as follows: along dT = 1 dX dM — 2Q ld t -d =O (4.36a)

-80along dT -1: LdX dM'+3z-Q T - d - = o (4.36b) dT 1 along = dQ - wldciT - c J7 + J)dT =~ (4.36c) along d =.: dX & 2Q +7'cT aZt* - IT =7 - 0 (4.56d) All derivations in this section are shown in detail in the Appendix, For solving for M,, Q and 7, Equation (4036) is expressed by four finite difference equations for corner points of an element DAPB bounded by two sets of characteristic lines of the dilatational wave family as shown in Figure 4.3a. Equations (4.36c) and (4.36d) are defined on characteristic lines of the shear wave family, so that their finite difference equations are in terms of values at P, A', and B'. If all values are assumed varied linearly between A and D or B and D (21) () then values at A' and B' can be expressed by YA' = YD+ - (A (4537a) YBwe= YD=M T, (Y (43o7b) where Y=MM, Q c., v, or W.

-81T P \ T A AT' A /' B D X Figure 4.3a Typical Element. T P A AT' /B' D Fgr X Figure 4.3b Element at Boundary.

-82AT' can also be expressed in terms of S aT' 7= I (4q38) The following difference equations are established along PA: (Mp- MA)- — (Qp+ QA) t (p - ~^A) = 0 (4o39a) along PB: (p — HM) t -T (Qp-t QS) - ()p- - 8)0) O (4039b) along PA', (Q-QA') -'(Wp + W) — -( -T A)-A+ q(ep+% ))= O (4o59c) along PB': (up- Q8.) i a+ (tWp v;C W S () z - Tp- "l P +) = ~ (4o39d) where W = WD + (vp + D) AT (4.40) If M, w, Q, and v are known at A, D, and B, they can be found at P by solving Equation (4o39). Equation (4.39) is written explicitly in the following table:

-83TABLE 4o1 Mp Qp WP VP Given Values 1 - _ 1 0 MA +QA -AT,+ 22 ~g2 A' gATAT' 2 gAT' 2AT' 2 2 MB QB _ AT AT 2 2 i ^-. 6g2A ATV1 gT 2a 6 2,L bg TLT I+ 1 - VD 2~, (wD + WB, f, _ z~~,+ 2' aFor finding those values at boundary points, the four equations can be reduced to two, since any two of the four unknown values are specified there. If M and v are specified, the other values can be found if all the values at B and D are knowno The two equations are shown in the following table: TABLE 4,2 Qp |p Given Values 62' -1 -(Mp- MB) 62TB B 22 B 2B AT' 6.g2~, (Wp + wB, -AT 2 (_ + B) - (WP. VB,,')I %', where the triangle PBD is shown in Figure 4o3b o

-84If M, C, Q and v are specified on the line X = T and M and v are specified on X = 0 as shown in Figure 4.4, X and Q at All can be found by expression in Table 4.2. Now M, C, Q and v are known at A11, A01 and A02, those values can be found at A12 by the expression in Table 4.1o By successive use of the expression in Table 4.1, they can be determined at the points through Aln o By repeating the previous procedures, values within the triangular region OAonAnn can be found, The expressions for boundary conditions on X = 0 and X = T are different for the three velocity regions, so that the three cases are discussed separately. a, Velocity V of the Pressure Front Such that Vco < V < Vcl From Equations (3o41) and (3.42), the particular solution for Equation (4,15) is Wb (XT) - J2 -J p cgo5n,(X-VT) for (X-VT) < XWb( Xj) z4 c& n' (X-VT) for (X-VT) > 0 7n~b(IT)aP c$ (X - VT ) WS(XT) = cO' (-T- for (X-VT) < 0 6"I Cos o nX-vT) W (X, T) =, CEw) ^ ) for (X-VT) > 0 where n1 and n2 are real positive roots of the equation

-85T Ann A2n Ain 2T 1A2 A03 A XAo02 2 T Aoi - x 0 Figure 4.4 Triangular Region Used in Computation

-867S- ( - 0 N4 - V2(I+ ) -3 \Z +j- O and n1 > Y Fz Since nl is the root of the above algebraic equation, the expression = T)6 XP coTnY(X2-r b,' ^o(v^-l _sL~a) y,2(,Z-n,2z) for all values of (X - VT) iW5 JX,7 ) (Vt 6I) (^_ L_ a 7) is the solution of the Equation (4.30) which is the homogeneous form of Equation (4.15). If these two are added together, a second particular solution of Equation (4.15) is obtained XTb (T)) - 1 E - _ _____ co An,(-VT) c.5 nz(X —VT) - Q_ T()^-lX -(V-XNz "-T n- for (X - VT) <'.b(X7T)= 0 for (X - VT) > 0 (4.41a) tW5( X;T) = -:)2- n) CO n ( X-VT)- Co 52z X-vT' for (X - VT). <,0 W /( X,T) 0 for (X - VT) > 0 4.41b)

-87The particular solution given by Equation (4.41) does not satisfy the boundary condition at X = 0, if the following calculation is performed W(O T) = -Jb(OT) -+ Ws (cT) -t t - z (-_~^) ln'~-" n.T - -I 1 C0Snzv T} H(,CT) t 0 (4.42a)'= vL)cnf;-n. v (lv_ z-I] Is'n, —7 sm-]n n:aS = C-AIT) E 0 (4.42b) H(oT) -- - cp;4 ^C1 T,vZ l)Cvzr'-)' ~n, Z) XC0 -77 n}. -5 2 V T' Hz(T) o 0 (4.42c) The particular solution given by Equation (4o41) does satisfy the homogeneous initial conditions, since Wb and Ws are identically zero for X > VT (where'0 < X < oo and V < ) o For this reason, the values of M, Q,, are zero on the line X = T in Figure 4.40 If v = - Hi(T) (W =- H1(T)) and M = H2(T) are specified on X = 0 and no disturbance is assumed for X > T, a solution can be

-88found numerically by the previously stated method of characteristics. A final solution which satisfies both the homogeneous boundary and initial conditions is gotten by adding the said solution to the solution given by Equation (4.41). b. Vcl < V < Vc2 From Equations (3.43) and (3.44), the particular solution for Equation (4.15) in this case is 7W (X T) — r) 2-'> ~M' T)2 for (X - VT) <.0 ~AW~b (MT) V e,_X-or T-bL^,T)' LI(Tr-I) )' 2L(y)l -hvAt^) for (X - VT) >0 (4.43a) I.)=- cn(x- VT) - eV-'r)] for (X - VT) < 0'~~s L(X; T) -~Z. -x-v T) WVsP a) (X) Tt)X) for (X - VT) > 0 (4.43b) where n and mi are the roots of the equation (and n, m are real and positive+.)Z + and n, m are real and pos it ive

-89This solution satisfies neither the homogeneous boundary conditions nor the homogeneous initial conditions, if the following check is made W(o T) = ~ + T -) ) o fl TT-~ [l ZLZ]i = H3(T) 0 (4.44a) \ t(, T) - 1W(OJT) B 7 - ^ ( I? T: ^hz( -r-) -Kins nvT;- l; ^ lle-VT = HcT).0 (4.44b) M (0,_ _) _ _ _ _ __eT) M^oT'r)= — T>2C *-z (-i)()-(,-tn~-l) (co" nrT - z ) = H4(T).O 0 (4.44c) 3sp e-0X T, (.Xo) - - (,)lV- )' 2(n-+) T o T~ (2CXo) _ _ p' 0e- X l 7 C- s^) zic(-t "-) 0 o 7 (7rIV~k. Z72.) 2 (y,7 YM5

-90Specifying the boundary conditions at X = 0 and the initial conditions at T = 0, a solution can be obtained numerically. A true solution is gotten by subtracting the said solution from the solution given by Equations (4.43). To save one half of the computing time, M, q, Q and v may be specified at the line X = T in Figure 4.5 instead of the initial conditions at T = 0. It is well known that there is no disturbance before the dilatational wave arrives for this velocity range V < Vc2 M, c, Q and 7 should be zero on the line X = T. From the particular solution, M, ~, Q and v can be computed along the line X = T and they are ~-2-TLw P 4 oe-0-T) -V ^ IK-T 5 p54 _(4.45a) a XzT! —--'C-'X 2(~V)T - l ] - 7 —-. -.-..e. ~IT) (4.45b) 12C - T ^ 61 -*(Y, + 7I C -:- = H )T) (4.45d)

-91-.~T~~ ~~ ~~LINE ALONG WHICH ------ THE PRESSURE FRONT / X = o T MOVES / X=VT // /X=T / / r 4 /P/. o NO DISTURBANCE Figure 4.5 Pressure Front in X - T Plane When Vc < V < Vc2

-92A solution can be obtained numerically by the method of characteristics, if v = - H(T) (W = - H5(T)) and M = - H4(T) are specified on X = 0 and M =- H5(T), = - H6(T), Q - - (T) H(T) - HT) (W = - H8(T)) specified on X = T. The final solution which satisfies both the homogeneous boundary and initial conditions is gotten by adding this solution to the solution given by Equation (4.43), c. V > Vc2 From Equations (3.45) and (3.46), the particular solution for Equation (4.15) in this case is CT I Cos J?f; or (C X - VTQ) <_ 0 Tht, -r) = - -- 0 for (X - VT) < 0 v-W'IT) =- (z I ^ S [co~ n (X-vT) — Co nz (X-VrT)] for (X - VT) < 0 W's (CT) =0 for (X - VT) > 0 These equations are identical to Equation (4.41). The same method as was used for the velocity range Vco < V < Vcl, can be applied hereo

-93In this case, there is a disturbance before the dilatational wave arrives, because the velocity of the pressure front exceeds the velocity of the dilatational wave. Preceeding the arrival of the dilatational wave, the disturbance is caused by the pressure directly. Figure 4.6 shows clearly. C. Numerical Examples The constants used in the numerical examples are E = 30 x 106 lb/sq.in G = 12 x 106 lb/sqoin v = 0.3 K = o.8533 and h/R = 1/10 is used in all examples. 1. Fourier Integral Solution for the Velocity Range V < Vco a. Elementary Theory The radial deflections at sections X = 20 and IX = 40 with T = 0 through T = 460 are computed based upon Equation (4.14), The velocity of the pressure front V = 0.1811 is used. In computing I2(X,T) in Equation (4.14), numerical integrations by means of Simpson's rule are employed. Due to the fact that there is N5 in the denominator, the improper integrals converge rather rapidly. W/Wo at X = 20 and X = 40 versus T is plotted in Figures 4.7 and 4.8 respectively, where Wo = 1/g2 is the maximum radial static deflection of a tube with infinite length under uniformly distributed internal pressure P = 1. The results are plotted in Figures 4.7 and 4.8.

-94- -- LINE ALONG WHICH THE *T PRESSURE FRONT MOVES /X=8T X/// X=T / / / /DISTURBANCE / / / X=VT / I -'/ /'' NO DISTURBANCE Figure 4.6 Pressure Front in X - T Plane When V > V Figure 4.6 Pressure Front in X - T Plane When V > V

1.6 1.4 1.2 1.0 0 3 0.8 Os 0.6 I 0.4 0.2 0 O100 200 300 400 500 T -0.2 Figure 4.7 Deflection at X * 20 for V. 0.1811 (V < VCo) from Elementary Theory

1.4 i.2 1.0 0.8 0 0.6 ON 0.4 0.2 0100 loo200 300 400 500 -0.2, vm Figure 4.8 Deflection at X - 40 for V = O.1811 (V < Vco) from Elementary Theory

-97b. More Exact Theory The radial deflections both for bending and shear at sections X = 20 and X = 40 with T = 0 through T = 363 are computed based upon Equation (4.27). The velocity of the pressure front V = 0.1811 is used. In computing Ibl, Ib2, Isly and Is2 in Equation (4.27), numerical integrations by means of Simpson's rule are employed, Integrals Ibl, Ib2 converge rapidly due to N5 in the denominators, but integrals Is1, Is2 converge slowly. Total deflection ratio W/Wo versus T is plotted in Figures 4.9 and 4,10 respectively, Shear deflection ratio Ws/Wo versus T is plotted in Figure 4.11 and Figure 4.12 respectively. 2. Numerical Solution for the Velocity Range V > Vco Three different velocities are used, i.e., Vo < V < Vcl, Vc< < V < Vc2, and V < Vc2 They are V = 0.3561, 0.7754, and 1.600 in the examples. The radial deflections consist of two parts, one is due to the particular solution, another due to the correction of the boundary conditions. The former is based upon Equation (4.41) and (4.43) as well as Equations (3.45) and (3.46). The latter is based upon the difference equations in Tables 4,1 and 4,2 with bT = 0.5. The radial deflections at X = 20 and 40 with T = 0 through 400 are plotted in Figure 4,13 through Figure 4.18. Each figure consists of three curves, namely, one due to the particular solution, another due to correction of boundary conditions, and still another being the resultant of the preceeding two. The deflection is expressed in terms of the ratio to Wo = 1/g2 which is the maximum radial static deflection of a tube with infinite length under uniformly distributed internal pressure P = 1

1.61.4 1.2 1.0 0 0.8 0.6 0.4 0.2 1Fiue.9TtlDfctoaX /0foV=0181VctT -02 Figure 4.9 Total Deflection at X = 20 for V - 0.1811 (V < Vco) from More Exact Theory

1.4 1.2 1.0 0.8 0o 3 0.6 0.4 0.2 Io00 2300 400 T -0.2-0.4 Figure 4.10 Total Deflection at X = 40 for V = 0.1811 (V < Veo) from More Exact Theory.

3.0 2.0 0 Figure 200 Shear for V ( < ) from More Exact Theory400 -1.0 -2.0 -3.0 Figure 4.11 Deflection at X = 20 Due to Shear for V = 0.1811 (V < Vco) from More Exact Theory

3.0 2.0 1.0 0 I 100 200 300 400 0~~~~~~~~~~~~~~~~~~~~~ -2.0 -3.0 -4.0 Figure 4.12 Deflection at X - 40 Due to Shear for V - 0.1811 (V < Vco) from More Exact Theory

---— TOTAL DEFLECTION -- -- SOLUTION CORRECTING BOUNDARY CONDITION 2.5 - --- PARTICULAR SOLUTION 2.0 ~ \ y/ 1.5 / 0~~~~~~~~~~~~~~~~0 N.. / / // H / 0 01~~~~~~~~~~~~~~~~~~ I') / 100J 200,' 300 400 T Figure 4.15 Deflection at X 20 for V 0.5561 (v V V) from More Exact Theory..Figure 4.1.3 Deflection at X 20X for V n 0.3561 (Vco < v < Vc1) from More Exact Theory..

TOTAL DEFLECTION |~ - ~/^~- --- SOLUTION CORRECTING 2.0 / BOUNDARY CONDITION J/j X\ --- -PARTICULAR SOLUTION / \_ /1. - I.0 Figure 4.14 Deflection at X " 40 for V " 0.3561 (Vco < V < Vcl) from More Exact Theory.

2.0O 7|~/ /I/~ 1~ / —-./ -TOTAL DEFLECTION I1.0O/ \\ / -- SOLUTION CORRECTING 1.0 /,/ \\.' BOUNDARY CONDITION | // \ / / —-- PARTICULAR SOLUTION I~//\ /I' I Xj i\ I// O 200 _- 300 400 -1.0 Figure 4.15 Deflection at X = 20 for V = 0.7754 (Vcl < V < Vc2) from More Exact Theory.

2.0 / 0 O~~~~/ \/ // 1.0 /a^~ ~L~ |~~~ y~ —-- / --- TOTAL DEFLECTION -— SOLUTION CORRECTING BOUNDARY CONDITION H,^~/~~ \ g~/~~ ~~ — -- PARTICULAR SOLUTION / 1.0 - Figure 4.16 Deflection at X a 40 for V - 0.7754 (Vco < V < Vcl) from More Exact Theory

0 2.0 7 1 ^ Q _ f ^\ // ------- ~~~~~~~~~~TOTAL DEFLECTION 1.0 / \I \ // ~~~~~~~~SOLUTION CORRECTING I \\ /~/ --- BOUNDARY CONDITION / ^// ~ ~ —---- PARTICULAR SOLUTION 0~~~~~~~~~~~~~~~~~~~~~~ 100 -300 9.0 400T -1.0 Figure 4.17 Deflection at X - 20 for V - 1.600 (V > Vc2) from More Exact Theory.

2.0 I~~~~~~~~~~~\ 0 I,/ / ---— ~ TOTAL DEFLECTION /$~ I \\ / ~~~~~~ —-— SOLUTION CORRECTING 1.0 BOUNDARY CONDITION / ^ -- PARTICULAR SOLUTION 0~~~~~~~~~~~ \\~~/ 100 200 300- 400 T -1.0 Figure 4.18 Deflection at X - 40 for V 1 i.6oo (v > Vc2) from More Exact Theory

-108Do Discussion of the Numerical Results 1. Case for V < VCO In both elementary and more exact theories, the velocity V = 01811 of the moving pressure front is taken for numerical computation. Figures 4~8 and 4o10 or Figures 4~9 and 411l show that responses at X = 20 or X = 40 for both theories are almost the same. Before reasoning why, the formulas have to be investigated firsto Equation (4,14) expresses the deflection based upon the elementary theory; Equation (4.27a) expresses the deflection due to bending and Equation (4,27b) expresses the deflection due to shear based upon the more exact theoryo From the numerical results shown in Figures 4.11 and 41o2, the deflection due to the contribution of shear in the more exact theory is very small, so that it has no significant effect on the total deflection and only the comparison between the total deflection W in the elementary theory and the bending deflection Wb in the more exact theory is giveno In this velocity range, namely V < Vco, the velocity spectra shown in Figure 2,4 through 2.6 are nearly alike for both theories, For a given velocity V, the corresponding complex wave numbers N = n + im are approximately equal for both theories. Formulas for Ib and I1 have the identical form. That is why I1' Ib o I2 in Equation (4.14) is due to the contribution of the real arm of the frequency spectrum in the elementary theory I-bl in Equation (4.27a) is due to the contribution of the first real arm of the frequency spectrum in the more exact theory0 Those two real arms are very close for small wave numbers. The integrands both in 12 and Ibl have an order of N5, where N is the real wave

-109number, so that there are no significant contributions due to large wave numbers N or high frequencies. For this reason 12 % Ibl There is an additional term in Equation (4.27a), namely Ib2 which is due to the second real arm of the frequency spectrum. From the numerical results, it has a magnitude of 10- of the maximum total deflection. In conclusion, the reason why the total deflections are approximately equal lies in the fact that the high frequency modes have no significant contributions in this velocity range, 2, Case for V > Vco For the numerical solution part, M, X, Q and v (W) can be found at the same time in the process of calculation, but only the total radial deflection W is plotted in Figure 4o13 through 4o18. In comparison with the particular solution, the deflection due to the particular solution part is relatively important both at X = 20 and X = 40 * Boundary effects are not significant for a section even with moderate distance from the end.

Vo EXTENSIONS OF THE NUMERICAL SOLUTION A. Forcing Function Due to Non-Uniform Pressure or Pressure Front Moving with Non-Uniform Velocity If P is an arbitrary function of X and T, the numerical solution is still valid, so long as the particular solution for the equation of forced vibration can be foundo In this particular solution it is not necessary to satisfy boundary conditions or initial conditions, so it can be easily determined by the integral transform method used in the case of uniform pressure. Once the particular solution has been determined, solutions to the homogeneous equation can be super-imposed on the particular solution to satisfy the prescribed boundary conditions at X = 0 and X = To A homogeneous equation with specified boundary conditions can be solved numerically by the method of characteristics. B. Arbitrary Boundary Conditions at X 0 There is no difficulty in the application of numerical solutions to the boundary conditions at X = 0 other than simply supported. For fixed boundary at X = 0, v = 0 (W 0) and c = 0, the other two values, M and Q can be found by two simultaneous difference equations defined along the characteristic lines PB and PB' shown in Figure 4.3b, if all values at B and P are known, C. Tube with Finite Length* The particular solution for the tube with infinite length is still valid, but one additional boundary condition must be corrected * Fourier series can be applied in the solution of tubes with finite lengths. This solution is based on orthogonal mode super-position, The frequency spectrum in Fourier series is discrete instead of continuous as in Fourier transform for the infinite length tube. -110

-111at X = L, where L is the dimensionless length of the tube, In this case the characteristic lines are reflected back and forth between two boundaries as shown in Figure 5,1, and this is equivalent to the dilatational waves as well as the shear waves in the tube wall being reflected back and forth along the characteristic lineso D. Consideration of the Inertia Force in Axial Direction If the inertia force in axial direction is included, the numerical method can still be applied. There are two more first order equations due to the longitudinal translation in addition to four previously established, to determine six unknowns. One additional boundary condition, say, longitudinal force or displacement must be specified. There are still four sets of characteristic lines with the same slopes, i.e., +1 and +. The only work left is to find the particular solution due to the moving pressure. Once the particular solution is found, the part for the numerical solution will be the same except two more difference equations established along the characteristic lines with slopes +1o For any boundary or initial value problem with homogeneous equations of motion, the numerical method can be applied directly, because there is no forcing function in the equations. Problems in which longitudinal translation predominates such as the problem in which longitudinal displacement is specified at one end or the problem with longitudinal impact, are the examples.

-112T 2L;_L I X Figure 5.1 Reflection of Characteristic Lines Between Two Boundaries

CONCLUSION The velocity spectra show that the elementary theory is quite different from the more exact theory for high phase velocities of wave propagation. When the velocity of the moving pressure front is greater than the first critical velocity, the more exact theory must be used in any case. In the elementary theory for the steady state response, the solution is not unique for the case when the velocity of the moving pressure front is greater than the critical. In the more exact theory, the solution is not unique when the velocity is between the first and second critical velocities or greater than the third critical velocity. The reason for the non-uniqueness of the solution is that no specific boundary conditions are assumed and the only condition is that the radial deflection is bounded everywhere, If the boundary condition is specified at infinity, there will be no solution for the over critical case unless the damping effect is included. Fortunately damping must exist in every physical system. The zero damping response can be obtained by assuming that.the damping coefficient approaches zero in the limit. For the transient response in the semi-infinite tube, the Fourier sine transform is used in the solution of the under critical case. Only the formula for the radial deflection W is derived. If the longitudinal fiber stress in the tube wall is wanted, the bending moment has to be known. It can be obtained by differentiating W twice with respect to X in the elementary theory and Wb in the more exact theory. Both W and Wb contain improper integrals which converge -113

-114uniformly with respect to X, so it is valid to differentiate twice with respect to Xo For simplicity, if the velocity of the moving pressure front is under critical, the simple formula from the elementary theory is allowedo In this velocity range9 the response due to modes with large wave numbers or high frequencies is small. The advantage of the numerical method is that M,, Q, and v(W) can be found at the same time. It can be applied to any boundary conditions at the near end of a semi-infinite tubeo If the period of the input forcing function is very long, the longer duration, has to be adopted in the calculation, If Tmax is the duration to be taken in calculation and AT is the interval, N(N + 1)/2 (where N = TmaxjAT) stations have to be calculated. It will take a long time even for the IBM 7090 computer. If the velocity of the moving pressure front approaches the first critical, the particular solution is unstable and the amplitude is increasing as time increases. The input forcing function in the second part which corrects the boundary conditions due to the particular solution is also unstable and is increasing as time increaseso When the velocity approaches the second or third critical, one of the wave number approaches infinity. In this critical range, the validity of the approximate equations of motion as used herein is doubtful, In the neighborhood of the second or third critical velocity, the frequency of the vibration is extremely high. In these regions, the exact equations of motion from three dimensional theory of elasticity should be usedo

REFERENCES 1. Lin, T. C. and Morgan, G, W, "Vibrations of Cylindrical Shells with Rotatory Inertia and Shearo" Journal of Applied Mechanics, Trans. ASME, 78, (1956) 255-261. 2. Herrmann, G. and Mirsky, I. "Three-Dimensional and Shell-Theory Analysis of Axially Symmetric Motion of Cylinders." Journal of Applied Mechanics, Trans. ASME, 78, (1956) 563-568. 3. Love, A. E. H. A Treatise on the Mathematical Theory of Elasticity. fourth edition, Dover Publications, New York, (1944) Chapter 24. 4. Timoshenko, S. P. "On the Correction for Shear of the Differential Equation for Transverse Vibrations of Prismatic Bars." Philosophical Magazine, 41, Series 6, (1921) 744-746. 5. Timoshenko, S. P. Vibration Problems in Engineeringo third edition, D. Van Nostrand Company, Inc., New York, (1955) Chapter 5. 6. Timoshenko, S. P. Method of Analysis of Statical and Dynamical Stresses in Rail. Proceedings of the Second International Congress for Applied Mechanics, Zurich, Switzerland, (1926) 407-418. 7. Ludwig, K. D. Die Verformung eines beiderseits unbegrenzten elastisch gebetteten Geleises durch Lasten mit konstanter Horizontalgeschwindigkeit Proceedings of the Fifth International Congress of Applied Mechanics, Cambridge, Massachusetts, (1938) 650-655. 8. Mathews, P. M. "Vibrations of a Beam on Elastic Foundation." Zeitschrift fur Angewandte Mathematik und Mechanik, 38, (1958) 105-115o 9. Dorr, J. "Der unendliche, federnd gebettete Balken unter dem Einfluss einer gleichformig bewegten Lasto" Ing.-Archo, 14, (1943) 167-192. 10. Kenney, J. T., Jr. "Steady State Vibrations of Beams on Elastic Foundation for Moving Load." Journal of Applied Mechanics, Transo ASME, 76, (1954) 359-364~ 11. Crandall, S. H. The Timoshenko Beam on an Elastic Foundationo Proceedings of the Third Midwestern Conference on Solid Mechanics, Ann Arbor, Michigan, (1957) 146-159, 12. Niordson, Frithiof Io N. Transmission of Shock Waves in Thin-Walled Cylindrical Tubeso Trans. of the Royal Institute of Technology, Stockholm, Sweden, Nr 57, 1952o -115

-11613. Timoshenko, S. and Woinowsky-Krieger, So Theory of Plates and Shells. Second edition, McGraw-Hill Book Company, Inc., New York, (1959) Chapter 15, 140 Sokolnikoff, Io S. Mathematical Theory of Elasticity, McGraw-Hill Book Company, Inc., New York, (1956) 177-185. 15o Lloyd, James Ro Wave Propagation in an Elastic Plate Resting on an Elastic Foundation. Doctoral Thesis in California Institute of Technology, 1962. 16. Titchmarsh, Eo Co Introduction to Theory of Fourier Integralso Second edition, Clarendon Press, Oxford, (1948) Chapter I. 17. Churchill, R, Vo Complex Variables and Applicationso Second edition, McGraw-Hill Book Company, Inc., New York, 1960. 18. Churchill, Ro Vo Operational Mathematics. Second edition, McGraw-Hill Book Company, Inc., New York, (1958) Chapter 10. 19. Petrovsky, Io Go Lectures on Partial Differential Equationso First English Edition Translated by Ao Shenitzer, Interscience Publishers, New York, (1954) Chapter I. 20. Leonard, Ro Wo and Budiansky, Bo On Traveling Waves in Beams. National Advisory Committee for Aeronautics, Technical Notes 2874, 19535 21. Plass, H. J., Jr. "Some Solutions of the Timoshenko Beam Equation for Short Pulse-Type Loading " Journal of Applied Mechanics, Trans. ASME, 80, (1958) 379-385.

APPENDIX DERIVATION OF FOUR SIMULTANEOUS FIRST ORDER EQUATIONS USED IN THE METHOD OF CHARACTERISTICS Ao Basic Equations From Equation (1.27), the equation of translation along the radial direction is a x -"~ ((btS)) - fh a)z If w = wb + Ws and vi = t are introduced, the equation becomes aX — j z= a (A. la) From Equation (1.25), a relation between resultant shearing force per unit length and the deflection due to shear is obtained iQX-h C 6 or a- hkG( ax -- L) 2 wb If =- is introduced, the equation after being differentiated - xat with respect to t becomes - = hkG (' - ) (A.lb) From Equation (1o28), the equation of rotation is aM Q_ k3 a3 ax - x- 1 3Xtoz or a~ x = it (Aolc) -117

-118From Equation (1.26), the relation between moment and the deflection due to bending is H-'= -D xz Differentiating both sides with respect to t, it becomes 3a l ax (A. ld) B. Basic Equation in Dimensionless Form As mentioned previously, the following dimensionless variables are introduced il. J T = _ _ _ W= oY,h Izv='i f 2 \ j IZ =M =I - M/ Dimensionless parameters are g2 = __ (_b_ ) 82 = )-Z.z 6 15

-119Equation (A.1) becomes WCax I r ^ JaT (A.2a) iT = ax (AO2b) ____ _T ____(Ao2c) aM -- __ tT - - (A.2d) Co Determination of Characteristic Lines and. Differential Equations along the Characteristic Lines The total differential of M along a certain line on X-T plane can be expressed in terms of dX and dT as following: edM. M dX + JdT aT Similar relation is hold for Wc, d0- idX + adIT Four simultaneous equations for four unknown partial derivatives, ioe., 3-, 3M, 3, T, can be obtained by combination of the above two equations with Equations (Ao2c) and (Ao2d) DMJx+ i M JT JM ^X+ a dT =dM (A.3a) ax3 d:X t T T = J: (Ao.b) ax a7T

-120-'a X ~ ^ ~+ T = Q(A.5c) at + BX = 0 (A.3d) aM 6M bWu Let the determinant of the coefficients of - T X and be zero dT dX dT o o o o dc<z dT = 0 (A.4) 0 0 o I I o A relation between dT and dX can be gotten as follows: 4 =T (A.5) These are the differential equations of the characteristic lines along 6M 6M a~ aw which dilatational waves are propagated. If ~X T-T T and are dT definite along the characteristic line - 1l the following determinant dX must vanish dM dT 0 0 Id o dX dT = 0 (A.6) L-Q o o I o I I o

-121Since = 1 along the characteristic line, a differential equation is dX established from Equation (Ao6) as follows: dM —' dT t d =0 which is Equation (4.36a)o By the same procedure, the differential equation can be estabdT lished along the characteristic line d. -.1 as follows: dX dM-'QdT-c d = o which is Equation (4o36b). Other four simultaneous equations for four unknown partial derivatives, i.e., -, a and can be obtained by combination u X, T o, andT of the total differentials of Q and v along a certain line with Equations (A.2a) and (A.2b) | dJX + a- dT = cQ (A.7a) a -^I = T (A7C) a - ax = (A.7d) Let the determinant of the coefficients of A and av be zero aT

-122dX dT o o o o dX JT o (A.8) O O.L' 0 0 - o I -I O A relation between dT and. dX can be gotten axZ - ~ ~I (A.9) These are the characteristic lines along which the modified shear waves Q Q. v Q? are propagated. If.-, 7, - and are definite along the dT 1 characteristic line _- = the following determinant must vanish XA 5 G. J dT O O d o SdT dT cIT0 (A.10) - 0 I -I 0 From Equation (A.10), the following differential equation is established dT 1 along the characteristic line - = dX 5 dQ -hThWdT - Sui + dJT =o which is Equation (4.36c).

-123By the same procedure, the differential equation can be dT 1 established along the characteristic line dX - as follows: wdh iAX + EatTn _-o3d)d o which is Equation (4.36d).