TEE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING SMALL LIQUID OSCILLATIONS IN MOVING CIRCULAR AND ELLIPTIC CYLINDRICAL CONTAINERS Ernst.Co Glauser 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 1966 December, 1966 IP-763

Doctoral Committee: Professor Bruce G. Johnston, Chairman Professor Glen V. Berg Professor Samuel K. Clark Associate Professor Ivor K. McIvor Professor Victor L. Streeter ii

ACKNOWLEDGMENTS The author is deeply indebted to Professor Bruce G. Johnston, Chairman of his doctoral committee, for suggesting the topic of this dissertation and for the help and encouragement during its preparation. He is also very grateful for the suggestions and the advice given by all other members of his doctoral committee, The author wishes to express his appreciation to the Horace Ho Rackham School of Graduate Studies of The University of Michigan for granting a pre-doctoral Fellowship under which most of the research was carried out and to the Computing Center of the University of Michigan for the use of the IBM 7090 computer in calculating and plotting the numerical results. The reproduction of the thesis was undertaken by the College of Engineering Industry Program at The University of Michigan. For this valuable assistance the author is sincerely grateful, iii

TABLE OF CONTENTS Page ACKNOWLEDGENTS...................... iii LIST OF TABLES................... vii LIST OF ILLUSTRATIONS................. xiii LIST OF SYMBOLS........................ x CHAPTER I. INTRODUCTION................... 1 1.1 Initiation of Research............ 1 1.2 Review of the Research on Dynamics of Liquids in Moving Containers.......... 1.3 Investigational Procedure............ 5 II. FREE OSCILLATIONS OF AN IDEAL INCOMPRESSIBLE LIQUID IN A VESSEL AT REST................. 7 2.1 Fundamental Equations............ 7 2.2 Some Properties of the Solution......... 14 2.2.1 Solvability of the Problem........ 14 2.2.2 Orthogonality of the Eigenfunctions... 14 2.2.3 Criterion for the Comparison of the Lowest Natural Frequency in Different Containers 15 2.3 Solution for the Circular Cylindrical Container 18 2.4 Solution for the Elliptic Cylindrical Container 21 2.5 The Numerical Evaluation of Mode Frequencies.. 26 2.5.1 Circular Cylindrical Container...... 26 2.5.2 Elliptic Cylindrical Container...... 27 2.5.2.1 Characteristic Numbers and Coefficients.......... 27 iv

TABLE OF CONTENTS (Continued) Page 2.5.2.2 Series Expansions of the Mathieu Functions........ 32 2.5.2.3 Characteristic Equations and Eigenvalues........... 33 III. THE STOKES-ZHUKOVSKII PROBLEM........... 37 3.1 The Stokes-Zhukovskii Potentials........ 37 3.2 The Equivalent Inertia Tensor.......... 40 3.3 Solution for the Circular Cylindrical Container.................... 43 3.4 Solution for the Elliptic Cylindrical Container.................... 49 3.4.1 Cosine - Elliptic Case.......... 50 3.4.2 Sine - Elliptic Case........... 57 3.5 Numerical Results................ 60 IV. AN EXAMPLE FOR THE OSCILLATIONS OF A CONSERVATIVE SYSTEM WITH A LIQUID MEMBER........... 62 4.1 Dynamic System................. 62 4.2 Solvability and Nature of the Solution.... 6 4.3 The Equations of Motion............. 65 4.4 Circular and Elliptic Cylindrical Container,. 71 4.4.1 Circular Cylindrical Container..... 71 4.4.2 Elliptic Cylindrical Container...... 72 4.4.2.1 Cosine - Elliptic Case..... 73 4.4.2.2 Sine - Elliptic Case...... 74 4.4.3 Generalization of the Solutions..... 74 4.4.4 The Equations of Motion in Dimensionless Form................... 76 v

TABLE OF CONTENTS (Continued) Page 4.4.5 Summary................ 79 4.4.6 Examples for the Response of the System................ 80 V. SUMMARY AND CONCLUSIONS.............. 88 APPENDX.......................... 91 1. The equivalent Moment of Inertia of an Ideal, Incompressible Liquid Completely Enclosed in a Rigid Elliptic Cylindrical Container....... 91 2. Eigenvalues and Modal Constants for the Natural Modes of Liquid Oscillations in Elliptic Cylindrical Containers...............!92 REFERENCES.......................... 111 vi

LIST OF TABLES Table Page I The Ratios } = Mc/I for Elliptic Cylindrical Containers with Ratios a/b 5 1.... 94 II The Ratios k = M/I for Elliptic Cylindrical Containers with Ratios a/b -1.......... 95 Eigenvalues and Modal Constants for Modes of the Order (2m+l,n), for III a/b =.6667, b/a = 1.5000.. 96 IV a/b =.7143, b/a = 1,4000.......... 97 V a/b =.7692, b/a = 1.3000.......... 98 VI a/b =.8333, b/a = 1.2000............ 99 VII a/b =.9091, b/a = 1:.1000............ 101 IIx a/b = 1.0000, b/a = 1.0000............ 103 IX a/b = 1.1000, b/a =.9091............ 104 X a/b = 1.2000, b/a =.8333............ 106 XI a/b = 1.3000, b/a =.7692...... 108 XII a/b = 1.4000, b/a =.7143............ 109 XIII a/b = 1.5000, b/a =.6667............ 110 vii

LIST OF ILLUSTRATIONS Figure Page 1 Simplified Ladle System........ o. 1 2 General Container............... 7 3 Two Enveloping Containers............. 16 4 Circular Cylindrical Container......... 18 5 Elliptical Coordinates |, q........... 22 6 Characteristic Curves a m+(q) and b2m+ (q). 29 7 Motion of a Completely Filled Container.. 37 8 Closed Circular Cylindrical Container...... 43 9 Closed Elliptic Cylindrical Container...... 49 10 The Ratio X(h/a) = M /I for Liquid Filling Elliptic Cylindrical Containers with Shape Ratios a/b = 2/35 1, 3/2.... o............... 61 11 Dynamic System with Liquid Member......... 62 12 Force as a Unit-Step Function........... 81 13 Significance of the Plotted Time Dependent Variables a, a, a/a....... o o o o 82 14 The Influence of the Surface Waves. Response of the Systems No. 1, where the Surface Waves are Suppressed, and System No. 2, where they are Free to Appear o 84 15 Effect of Including Higher Modes in the Analysis. Response of the Systems No. 3, 4 and 5, where one, two and 18 Lower Modes are Considered..... 85 16 Influence of the Ellipticity of the Container. Response of Systems No, 6, 7 and 8, which Differ Only in the Ellipticity a/b of the Container. a/b = 2/3, 1 and 3/2.. o o...... o.o. 86 viii

LIST OF ILLUSTRATIONS (Continued) Figure Page 17 Response of System No. 9 over a Longer Period of Time.................. 87 18 Reference System for the Inertia Parameter X... 91 ix

LIST OF SYMBOLS A(m) i-th coefficient in the expansions of the cosine-elliptic Mathieu functions ce and Ce, Equations (91a), (92a) (in~) ~m Bi i-th coefficient in the expansions of the sine-elliptic Mathieu functions se and Se, Equations (91b), (92b) C Volume occupied by the liquid in the position of equilibrium Ca Equation (182) mn Cb Equation (183) Integrals involving modified Mathieu mn functions Cc Equation (184) Ce (t, q ) Cd Equation (185) mOn Ce Modified Mathieu function of the first kind and of order m, m one solution to Equation (74) Cf Fourier coefficient corresponding to the cosine-elliptic m,n mode type p n Equation (172) Cq Surface integral and numerator in the expression for m,n Cf, Equation (176) Cr Surface integral and a part in the denominator of the m,n expression for Cf, Equation (177) m, n F(x) Function of the independent variable x F. (0.) Functional which is minimized by the Stokes-Zhukovskii potential 0i' Equation (108) F n-th coefficient in the Fourier expansion of the auxiliary n function * in the case of a circular cylindrical container, Equation (137) G Axis of rotation in the system presented in Figures 1 and 11 I Moment of inertia of the solidified liquid with surface S with respect to the axis of rotation through its center of gravity S x

I Dimensionless I /t p a5 c c I. e (i,j)-th component of the inertia tensor of a rigid body I Moment of inertia of the container W with respect to the w axis of rotation through its center of gravity S w J Bessel function of the first kind and of order m m Kif ~ Body force per unit mass on a liquid particle [K] Resistance matrix, Equation (222) [M] Inertia matrix, Equation (222) M Equivalent moment of inertia of the liquid with rigid surface S with respect to the axis of rotation through its center of gravity S M. (i,j)-th component of the equivalent inertia tensor of an ideal, incompressible liquid completely filling a closed, rigid container, Equation (112) O Origin of the Cartesian coordinate system (x,y,z) O Origin of the Cartesian coordinate system (x,y,z) P Liquid particle or point in the system (xy,z) Qi Modal constant, Equations (248), (252), (256) and Tables III to XIII of the appendix. R. Modal constant, Equations (249), (253), (257) and Tables 1 III to XIII of the appendix S Surface or area of the liquid surface in the position of equilibrium S Center of gravity of the liquid with surface S, Figure 11 S Center of gravity of the container W Sa Equation (197) m,n Sb Equation (198) Integrals involving modified Mathieu'mn functions Scm~n Equation (199) Se (E, q ) Sd Equation (200) m, n xi

Se Modified Mthieu function of the first kind and of order m m, one solution to Equation (74) Sqc Surface integral involving the sine-elliptic mode type m, n cm, result is given in Equation (201) m, n Sr Surface integral involving the sine-elliptic mode type cp, result is given in Equation (202) m,n T Axis of rotation in the system of Figures 1 and 11 T Kinetic energy T Kinetic energy of the liquid in the: system of Figure 11, Equation (206) U. Value of a basic integral of cp. appearing in the equations 1 of motion (221), Equation (223) *' 4 U. Dimensionless U./it a 1 l V Potential energy V Potential energy of the liquid in tlhe, system of Figure 11, Equation (213) V. Value of a basic integral of pi appearing in the equations of motion (221), Equation (224) * Vi Dimensionless V.i/ a3 W Container, wetted walls or area of the wetted walls of the container W. Value of a basic integral of.i appearing in the equations of motion (221), Equation (225) W 2 i Dimensionless Wi./ a X,Y,Z Components of I in the system (x,y,z) xii

a Radius of a circular cylindrical and principal axis in x-direction of an elliptic cylindrical container a Characteristic number corresponding to the cosinem elliptic Mathieu function ce (r,q) b Principal axis in y-direction of an elliptic cylindrical container b Characteristic number corresponding to the sine-elliptic m Mathieu function se (T,q) c Length parameter in the definition of the elliptic cylindrical coordinates, Equatiorn (54) ce Ordinary Mathieu function of the first kind and of order m m, one solution to Equation (73) d Separation constant introduced in Equations(67) and (68) d Distance between axis T and surface S in the system of Figure 11 f Separation constant introduced in Equations (44) and (63) f Distance between center of gravity S and axis T in the c c system of Figure 11 * f Dimensionless length parameter f /a c c f Dimensionless length parameter f /a w w g Acceleration of gravity = 386.4 in/sec h Depth of the liquid in containers with flat bottom h. Directional derivative of the Stokes-Zhukovskii potential 0i in the direction of the unit normal v i,j Integer numbers mostly used in indices k n given in Tables III to VII and IX to XIII of the m,n m, n appendix Length of the rods connecting the axis T and G in the system of Figure 11 xiii

2 x-component of the unit normal v m Integer number mostly used in indices m y-component of the unit normal v m Mass of liquid of volume C C m Mass of the container W w n Integer number mostly used in indices n z-component of the unit normal v p Pressure in the liquid Pi Generalized coordinate corresponding to the liquid mode cp., defined in Equation (208) q Parameter in the canonical forms of Mathieu's differential equations, Equations (73) and (74) qi Generalized coordinate in Lagrange's equations, Equation (220) q nEigenvalues of q for liquid oscillations in elliptic m. n cylindrical containers, introduced in Equation (75) r Radial coordinate in the system of circular cylindrical coordinates (r,O,z) s Parameter in the canonical forms of Mathieu's differential equations s(t) Displacement function of point G in the system of Figures 1 and 11 se Ordinary Mathieu function of the first kind and of order m, m one solution to Equation (73) t Time u x-component of the liquid velocity v u Generalized velocity vector with the components ul,u2,.,u6 v Velocity of a liquid particle in P (x,y,z) v y-component of the liquid velocity v xiv

v. A2i+3/A 2i+l Equation (86a) 1^ 21+3 21+1' ^ ^i3(8ia) v Velocity of the point 0 of a rigid body v,v,v Components of v with respect to the system (x,y,z) w z-component of the liquid velocity v wl1 Je-, Equation (93) w2 -V et', Equation (94) x Independent variable x,y,z Coordinates in a Cartesian coordinate system which is fixed with the container W x,y,z; Coordinates in a Cartesian coordinate system fixed in space z Coordinate in the Cartesian, circular and elliptic cylindrical coordinate system xv

a(t), P(t) Generalized coordinates for the system in Figure 11 7Y ~ Dimensionless parameter a/b 5 ~ Derivative of the function 5(t) with respect to t 6' Derivative of the function 5(t) with respect to the dimensionless time = V t a C Dimensionless parameter h/a 5G^ Function of the free liquid surface Tq ~ Coordinate in the elliptic cylindrical coordinate system (opaz) 0 Coordinate in the circular cylindrical coordinate system (r,0,z) dt ~ Dimensionless parameter m /m c Dimensionless parameter M /I \i. j Ratio M./I.. 1i, 1J iJ u1 ~ n-th zero of the function - J (x) mn dx m v Unit outward normal to the container walls W 2 a C. Dimensionless frequency D. - of the i-th mode cpi in 1 i g the infinitely deep circular or elliptic cylindrical container. Numerical values are given in Tables III to XIII of the appendix rT ~ Dimensionless time - t a ~0 Velocity potential 0i'.2'. 06 Stokes-Zhukovskii potentials corresponding to the generalized velocities ul, u2,.,u6 c0 Stokes-Zhukovskii potential of the liquid with surface S in the system of Figure 11 with respect to the axis of rotation through the cneter of gravity S cp. i-th natural mode of 0 xvi

PiCP2'CP3 Functions of one variable only derived from cp by the separation of variable procedure X Dimensionless parameter I /Ic Auxiliary function used in the evaluation of the StokesZhukovskii potentials, Equations (126) and (161) 1'2 ^ 3 Functions of one variable only, derived from | by the separation of variable procedure ic. Frequency of the i-th natural mode ci CD. Dimensionless frequency parameter c. - i1 1~~~~~~ g XD Angular velocity of a rigid body with respect to the point 0 C),,cD Components of c with respect to the system (x,y,z) Wox" oyF Moz 0 o' J x Z)v? x xvii

I, INTRODUCTION 1.1 Initiation of the Research The author's interest in the problem of liquid oscillations in moving containers arose with his study of the dynamic behavior of hot metal ladles and their contents. Ladles are huge containers, used in the steel producing inj1G6~ ~ ^dustry to carry up to 400 Ad ----- S(t) tons of molten steel. When the ladle is carried by a li/ l[~ -moving crane, it is part | / fIof a dynamic system which, when simplified for an analytical investigation, l~ ~/ ~~/ ~will assume the configura/~~~~~,'s^tion given in Figure 1. ~~~~~~/ ~The ladle W is engaged by Figure 1 the hooks of the crane G Simplified Ladle System in the trunnion axis T in the trunnion axis To The motion of the crane along the craneway is characterized by the displacement function s(t). The following characteristics of the system and the motion will determine the analytical tools to be used for its investigation. - The molten steel in the ladle has a temperature of about -6 2 5250~C at which its kinematic viscosity is ~825 o 10 m -1 -6 2 -1 sec compared to the value of lo01 ~ 10 m sec for - 1 -

- 2 - water at a temperature of 20~C. The liquid is practically incompressible. - The ladle walls and bottom consist of heavy steel plates with welded seams, protected against the molten steel by a thick lining of fire resistant brick. -The motion always starts from rest and the acceleration, but more important, the rate of change in acceleration of the crane is always very small, thus causing only small oscillations when the system is stable. - The friction in the hinges T and G may safely be neglected. The problem may therefore be simplified to the investigation of small oscillations of a conservative system with a rigid container partially filled with an ideal, incompressible liquid. It was soon relaized in the course of the analytical work that in order to give a more general and more widely applicable solution, very fundamental problems had to be solved first. There exist hardly any attempts to solve the general liquid oscillation problem in elliptic cylindrical containers, which is the obvious generalization of the simple circular cylindrical case. The eigenvalues, eigenfunctions and inertia properties had first to be found to an extent necessary to solve the general forced oscillation problem. It was therefore decided to put the investigation into a much broader framework, obtaining results which may be applied for a wide class of similar problems.

-31.2 Review of the Research on Dynamics of Liquids in Moving Containers With the exception of some recent and highly theoretical papers (12), investigations on dynamics of liquids in moving containers deal with an ideal, incompressible liquido The problem becomes an application of the classical theory of hydrodynamics, which has been treated by Lamb (13) in a most authoritative manner. There seem to be very few problems for which he did not give at least some hints as to possible ways of solution. Moreover, with the exception of a few studies of longitudinally excited (7) or rotating circular cylindrical containers, all theoretical studies assumed the liquid motion to be irrotational with respect to an internal frame of reference. Since the fluid is acted upon by a rotation free force field, it can be shown (16) that the motion of rigid tank boundaries cannot produce rotational flow if the flow was initially irrotational. The investigations assume, furthermore, small surface slopes, small displacements and small velocities. Despite these restrictive conditions, the solution for more general container shapes under both small translational and rotational oscillations may face tremendous numerical difficulties. This is why the first attempts to consider containers partially filled with liquid in dynamic systems replace the liquid by a mechanical analog, spring-mass systems or physical pendulums, or tried to make some intuitive assumptions regarding the motion of the liquid. This is the way civil engineers found the response of the liquid in water tanks subjected to earthquakes (8, 9)~ The same crude approach was used in the aircraft industry and later in the design of liquid propellant rockets

- 4 - to investigate the influence of fuel sloshing on flight stability. Exact solutions to the linearized hydrodynamic differential equations have only been found for a few very regular container shapes. The liquid oscillations in circular cylindrical containers under small translational and rotational oscillations seem to have been extensively investigated (2, 3). For this case, the assumptions of the linear hydrodynamic theory have frequently been experimentally justified (7, 10). Abramson (1) reports excellent correlation between theorectical and experimental results for the case of steady-state horizontal oscillations. In recent years, some excellent Russian papers have appeared, covering different aspects of the liquid dynamics in moving containers, (4, 16, 17, 18). These have treated the problem from a very general and highly mathematical point of view and have provided much inspiration and guidance for the author's researcho

-51.3 Investigational Procedure This research involves a systematic investigation of liquid oscillations in moving circular and elliptic cylindrical containers as a basis for developing a solution of a more complex dynamic system of which the container is a part. The equations of motion are based on the assumption of an ideal, incompressible liquid, potential flow and surface waves of small amplitude and small slopes. Following Poincare, the displacement of the liquid from the equilibrium position is expanded in a certain series of functions with time dependent coefficients and the problem is reduced to the solution of an infinite set of equations of second order. This is a classical method in the investigation of continuous systems and it is a very powerful tool in this case because this certain series of functions, the eigenfunctions and associated eigenvalues can be obtained by a separation of variable procedure. The solution for the circular cylindrical container is well known and its derivation is included because it is a limiting case of the solution for the elliptic cylindrical container, thus furnishing a check on the results. The author believes that the oscillation problem for systems involving elliptic cylindrical containers is solved here for the first time and is the main contribution of this dissertation. A main characteristic of this approach is the partition of the liquid motion into two parts: (1) the motion if the surface is replaced by a rigid lid, and (2) the wave motion at the surface. The first contribution can be derived from the Stokes-Zhukovskii potentials which are completely determined by the shape of the container alone. They are the basis used to compute the components of the equivalent inertia tensor for the liquid

- 6 - completely enclosed in a rigid container. The second contribution, the wave motion, is then expanded into an infinite series with time-dependent coefficients with respectto the complete orthogonal set of modes of free vibration. The research has led to the following results: Dimensionless tables and curves for the equivalent moment of inertia of an ideal, incompressible liquid completely enclosed in an elliptic cylindrical container with respect to rotations about the two principal axes through its center of gravity. - Tables of modal constants which permit the evaluation of the contribution of the n-th mode in the system of second order differential equations of motion for a certain class of dynamic systems having elliptic cylindrical containers partially filled with liquid. - As an example, response curves for the ladle system in Figure 11 are presented in a dimensionless form.

II. FREE OSCILLATIONS OF AN IDEAL INCOMPRESSIBLE LIQUID IN A VESSEL AT REST 2.1 Fundamental Equations In the following, the classical, linearized equations for the small oscillations of a liquid are derived with the help of Hamilton's principle. Consider an ideal, incompressible liquid enclosed in a fixed shell (Figure 2). It is assumed that the free surface S in the position of equilibrium coincides with Z fl~y ~ the plane determined by the x- and y-axis of the Cartesian - c —- J-^ coordinate system. C <NOI o.^designates the volume occupied by the liquid in \ /~A the position of equilibrium T~Figure 2 ~and W stands for the wetted Figure 2 General Container walls of the container. Any liquid particle P at the point (x,y,z) at the time t is described by: Velocity of the liquid particle: v = v (u, v, w) Body force per unit mass: = = K (X, Y, Z) Pressure: p Density: P (u, v, w) are the components of v and (X, Y, Z) the components of K in the coordinate system (x, y, z). - 7 -

- 8 - Since the gravity,g is the only body force acting and the liquid is assumed to be incompressible, = i (O, 0, -g) and p = const. Finally, with the equation for the free surface, z = ~ (x, y, t), the kinetic and the potential energy of the liquid are given by the formulae (Refo 13, Art. 174): T = | f 2v dC (1) C gV = 2 2dS (2) 2 S Hamilton's principle can be stated in the form t (T - V) dt = 0 (3) t] which says that the actual path followed by a dynamical process is such as to make the integral of the function (T - V) stationary. The liquid enclosed in volume C is subject to the following conditions: - Impermeability of the shell, e.go no velocity perpendicular to the wallo V ~ v = 0, for P C W (4) v is the unit outward normal to the container walls, - Continuity V v = 0, for P C (5)

-9The condition at the free surface is a consequence of Equation (5). To avoid a flow through the free surface, a liquid particle has to satisfy the condition w u + v +, for P S (6) W = u x + y + t, For the assumed small oscillations with small surface slopes, this simplifies to w for P e S (7) It is usual to impose the condition of irrotationality on the motion of the liquid. This is not necessary for this linearized problem. It has been proven in Ref. 16, pp. 242, that the vortical component of the velocity vector v remains constant if second order terms are neglected. The pressure and the shape of the free surface can be determined with the same degree of accuracy by the irrotational component alone, which can be found in this linearized problem independently of the vortical component. The condition v = V0 (8) where 0 is the velocity potential, will give a first degree accuracy of the problem, no matter whether the motion is actually rotational or irrotational. Applying Hamilton's principle to the energy expressions (1) and (2) yields

- 10 - 10 0 = J Jv dC g k dSdt tl C S 0 = 2 tovdC- g / dS dt t -C S which after introducing the velocity potential 0 becomes o = J JVOV50 dC - g J; 5o dS dt (9) Green's theorem is used to transform the volume integral of Equation (9) into an intergral over the free surface S. Using in addition the continuity condition, Equation (5), and the condition at the container walls, Equation (4), the following relation is obtained: V 5 dC = 0 dS (10) C S Together with the surface condition, Equation (7), which is used in the form Xz = X, for P C S (11) Equation (9) can be rewritten 0 = 6j /0 5 -- g C 5) dSdt (12) 1 S

- 11 - Integrating Equation (12) by parts and using the isochronism of the variations, leads to: 0 = + g t 5 dSdt (13) t S Due to the arbitrariness of the variations, Equation (13) is satisfied only when 0 = + g A, for PRe S (14) This important equation gives the relations between the surface function: and the velocity potential 0. It can be used, together with Equation (8), to give the governing differential equation, the boundary conditions and the energy relations in terms of the velocity potential 0 only. Differential equation and boundary conditions: From Equations (5) and (8): V2= = 0, for P e C (15) From Equations (4) and (8): V 0 = 0, for P e W (16) From Equations (14), (7) 2 and (8) + g = 0,for P e S (17) Energy relations: From Equations (1) and (8): T = 2 / (V 0)2 dC (18) C From Equations (2) and (14): V = pg f( ) dS (19) S

12 To study the question of natural oscillations of the liquid, it is assumed that 0(x,y,z,t) = p(x,y,z) sinmt (20) cpisthe natural mode and X the natural frequency of liquid oscillations. If Equation (20) is introduced into Equations (15) to (19), corresponding relations in terms of the natural mode (c and the natural frequency C can be derived. Differential equation and boundary conditions: V2P = 0, for P C C (21) v V p = 0, for P C W (22) 2 - ~-cg = 0, for P S (23) Extreme values for energies: T = - (Vp)2 dC (24) max 2 C 2 2 aV - oL dS (25) max 2g S Equating Equations (24) and (25) yields the Rayleigh quotient 2 (V @) dC 2) C Xt = _-C - (26) g 2cp dS S which has many advantageous properties in numerical computationso These properties are discussed extensively in the literature, e.g. Reference 22, pp. 486 ff. While the numerical procedures which are based on Rayleigh's principle may be the only approach to the general liquid

- 13 - oscillation problem, there exist a few very special cases where the solution can be obtained by rigorous integration of the boundary value problem, Equations (21) to (23).

- 14 - 2.2 Some Properties of the Solution 2.2.1 Solvability of the Problem It has been proven (Reference 16, pp. 268 ff.) that the problem which is described by Laplace's differential equation, Equation (15), and the boundary conditions, Equations (16) and (17), has the following properties: -For the motion of a liquid about the equilibrium position in a finite container there exist natural oscillations, e.g. solutions of the form of Equation (20). - The eigenvalues Co are positive, of finite multiplicity n and form a sequence increasing without bound. - The eigenfunctions cp form a sequence which is complete in S. 2.2.2 Orthogonality of the Eigenfunctions 2 Under the assumption that all eigenvalues cD are positive and disn tinct, the orthogonality of the corresponding eigenfunctions cp can be proved quite easily. Let (m and cp be two modes associated with the m n 2 2 eigenvalues XD and C. Both modes of course satisfy Equation (21) m n V 2m = 0, for P c C (27) 7V2 = 0, for P CC (28) n Green's theorem leads to: fFq) n n m dS = 0 (29) Jm z n n S

- 15 - But at the surface, both functions have to satisfy Equation (23). This leads to the two relations: U -m n cp cp dS = 0 (30) g S m n )cP 6cp — g (C2 c2) m n dS = (31) 2 3m n z m n 2 2 For m f n, D is according to the assumption different from U, m n therefore f m pn dS = 0 (32) S C aPm Sa)n __ dS = O (33) S 2.2.3 Criterion for the Comparison of the Lowest Natural Frequency in Different Containers Reference 16 gives on page 241 a criterion which has proved to be extremely useful in checking numerical results. Consider two different containers with equal liquid surfaces S but where the walls W of the first container completely envelop the walls W2 of the second container (Figure 3). Let 9 be some function which satisfies the Laplace equation in C1 and the surface condition, Equation 23, in S. Since C1>C2, the following inequality holds for cp (V )2 dC1 > (V)2 dC2 (34) C1 C2 Co

16 - Let (1)2,(l)) d(2)2 (2)) correspond to the lowest mode in the two containers. Equation (26) (1)2 f(7cl)2dC2 Figure 3 (2)2 / C c g 2 (36) Two Enveloping (2) Containers j S Since (2 is a minimum, the right hand side of Equation (36) must become bigger whenP (2) is replaced by any other admissible function. Though p(1) does not satisfy the boundary condition in W2, it is according to the general theory, Reference 6, pp. 398 and 461, an admissible function because the variational principle is of the free boundary condition type. Therefore, 2 (vl) dC2 -<2- -2 (37) g (1) as S Using the inequality (34), the right hand side of inequality (37) becomes the one of Equation (35), which means <(2) < (1) (38) The result can be formulated as follows: If two containers with the same free liquid surface are such that the walls of the first can completely envelop the walls of the second,

- 17 - then the corresponding lowest natural frequency will be greater in the container whose volume is larger.

- 18 - 2 3 Solution for the Circular Cylindrical Container This solution is described by Lamb in Reference 13, Art. 191. It is given as a basis for later developments. e/ jPY ~ Consider a cylindrical shell 7- iZof radius a, filled with an ideal, incompressible,( /^\Q \liquid up to the depth h X u |(Figure 4). The problem -i —- will be discussed in cirIj[!^ I g|cular cylindrical coordinates, C which are assumed as indicated in Figure 4. The natural modes of oscillation have to satisfy Equations (21), (22) and (23) which are given below with Figure 4 respect to circular cylindriCircular Cylindrical Container cal coordinates. 2 2 2 r2 + _P + + 2T +2 0, for P: C (39) Or2 5r 2 2 =, for P IC W (40) -? — c = 0, for P c S (41) Z~ 9 41

- 19 - Using the separation of variable procedure with cp(r, e, z) = cp(r) cp(e) cp3(z) (42) Equation (39) becomes -1. 1.... r2 1 + r 1 + + r2 = 0 (43) 1 1 C2 (3 Equation (43) requires that all the ratios (Pi/ci) are constants. In particular It CP 3 2 ( - f (44) 3 The solution to Equation (44) which satisfies condition (40) is (P3(z) = cosh f(z + h) (45) whereas condition (41) leads to the relation Xo = g ~ f ~ tanh (f * h) (46) In a similar way t! 92 2 2 - m- (47) (p2 Since Equation (47) determines the number of nodal diameters of some particular mode shape, it is obvious that m has to be a positive integer m = 0, 1, 2,... Equations (44) and (47) reduce Equation (43) to the parametric form of Bessel's differential equation 2 + (f2 -m2) = (48) r + (2 r2 2~ = 0 (48) in.4. 11r) -- mA1

20 - which in this case has a solution of the formo ep (r) = J(f o r) (49) 1 m The remaining boundary condition, Equation (40), requires dcp (r) - J (f r) = 0, for r = a (50) dr (P1 dr m (0d If x = m are the zeros in the derivative of the m-th order Bessel m,n function satisfying J (x) = 0 (51) dx m then the corresponding separation constant f is simply f = (52) mn a The indices of the eigenvalues are chosen such that m = 0, 1, 2, o o corresponds to the order of the Bessel function or the number of the nodal diameters while n = 1, 2,. o. numbers the positive zeros to Equation (51) in increasing order. If finally the separated functions are put together according to Equation (42), the expression for the (m~n)-th natural mode is P] = J ( o ) cos m 0 cosh 1 53) ~mn m m,n a mn a

- 21 - 2.4 Solution for the Elliptic Cylindrical Container This solution is presented by N. W. McLachlan as an illustration for the application of MVthieu functions, Reference 14, Arts. 16.20 ff. The method of solution is very similar to that used for the circular cylindrical container, which will be a limiting case in the procedure. The Laplace equation can be separated when elliptic cylindrical coordinates are introduced, which are related to the Cartesian system in the following way: x = c cosh 5 cos T y = c sinh 5 sin T (54) z = z The corresponding coordinate surfaces are: Right elliptic cylinders: 2 2 x + - _ 1 (55) 2 2 2 2.. c cosh c sinh Right hyperbolic cylinders: 2 2 x y (56) 2 2 22 2 c cos r c sin r Planes parallel to the x-y plane: z = z (57) Figure 5 shows the lines of intersection between the surfaces of Equation (55) and Equation (56) with the plane z = z. The fundamental

22 - system of equations, Equations (21), (22) and (23), when written with respect to elliptic cylindrical coordinates takes the form, ___________ ( 2 2 - + - + d- = 0, for P C (58) c (sinh + sin2r ), for zC = 0, for P CW (59) 2 Y~y,b - g p = 0, for P C S (60) 120~ 90~ 60o.=2o0 Fi 70e Figure 5 Elliptical Coordinates t, T If it is again assumed that a solution of the form p ( T9, z) =- p(J) cp2() cp3(z) (61) exists, then Equation (58) becomes~ tl t! lt l~ ep1 + L2 1~, —2 -- t- -+ ~ /+ =, for P e C (62) c (sinh f + sin r1),1 c2 3 The solution for cp is exactly the same as in the case of a circular

- 23 - cylindrical container. This means T3 f2 (63) p3 cp3(z) = cosh f(z + h) (64) and 2 g f. tanh (f - h) (65) With the separation constant f, Equation (62) takes the form P 1 2P 2? g- + fc sinh t + + fcsin 2 = (66) which requires that _~ + 2c2.2 2 + f2c2 sin2 r = d (67) cP2 (])it --- + fc2 sinh2 ~ = -d (68) (P1 where d is some constant. Equations (67) and (68) are equivalent to (22 22,P, + f - d - f2C cos 2 jCP 2 = ~ (69) 1cp - d - f2c- cosh 2)cp = 0 (70) 2 2 f - d == (71) 2 2 f *c q (72) ~1 - =q

- 24 - the canonical forms of Mathieu's differential equations are obtained. p + (s - 2q cos 211) 2 = 0 (73) p - (s - 2q cosh 20) c = 0 (74) Equation (73) is simply called "Mathieu's Differential Equation" while Equation (74) is usually known as the "Modified Mathieu Differential Equation". Both equations, however, are very closely related due to the fact that Equation (73) takes the form of Equation (74) simply by putting in for r. There are two conditions from which the parameters s and q can be determined. The first is the very obvious requirement that cP2(r) must have a period of 2i. This leads to a functional dependency s = s(q) which is well known in the form of the Characteristic Numbers (Refo 14, Art. 3.25). The other condition is derived from Equation (59), requiring a zero derivative of cp1() at the boundary E = B of the container. In fact, there exists an unbounded set of distinct, positive eigenvalues q and corresponding frequencies m, n 2 2g 2h 2D =~ - q tanh -Vq m,n c tn h V (75) which satisfy these conditions. The properties of the solutions to Equations (73) and (74) are extensively discussed in Reference 14 upon which the subsequent work will draw heavily. It is sufficient for this investigation to select from the various

- 25 - possible solutions to Mathieu's differential equations only the following two cases: Cosine-Elliptic Case: CP2m+1n 2m1 n= 2m m+l, n 2em+l (T)q2m+1,n cosh c 7n(z+h) (76) Sine-Elliptic Case: CPm = Se 2m+ q (,q se ll m cosh (2 2m+l n (z+h) 2m+ln 2m+2m2m+llnn 2m+l m+l,n c (77) Equation (76) represents the class of natural modes which are symmetric with respect to the (x-z) plane and antisymmetric with respect to the (y-z) plane. Similarly Equation (77) represents the modes symmetric to the (y-z) and antisymmetric to the (x-z) plane. These functions are fundamental for liquid oscillation problems where the nature of the system dictates a motion with the same symmetry properties. The most basic problem is now the evaluation of the eigenvalues q21 +. The selected numerical procedure for the solution of this problemwill be presented in the next section blem will be presented in the next section.

- 26 - 2.5 The Numerical Evaluation of Mode Frequencies 2.5.1 Circular Cylindrical Container The evaluation of mode frequencies for this case involves the solution of the transcendental equation d Jm(X) = O (78) Solutions pn are tabulated to some extent in the literature, e.g. mn Reference 11. If additional values are needed, a rapidly convergent Newton-Raphson procedure can easily be developed with the help of the following relations: m+(x) = 2m (x) J (x) (79) (x 1 x m m-1 d j ((X) (x) - j.(: (80) dx m 2 m-l - m+l (80) This means that derivatives of Bessel functions can be expressed in terms of Bessel functions of the same kind and higher order Bessel functions can be computed from lower order functions. The mode frequencies can be computed from p. with the relation mn 2 = 1 g tanh (1 mm,n - m,n a m,n a (81) The applications in Chapters III and IV will only require solutions to Equation: (78) for m = 1. The lowest 20 values of 1.,n are given for completeness in column SIGA in Table IIX of the Appendix completeness in column SIGMA in Table IIX of the Appendix.

- 27 - 2.5,2 Elliptic Cylindrical Container The determination of eigenvalues for the two cases given in Equations (76) and (77) is discussed in this section. Since these two cases will lead to relations which are very similar, each will be given the same number with the letter a attached when the equation corresponds to the cosine-elliptic case and the letter b when it corresponds to the sine-elliptic caseo 2.5.2.1 Characteristic Numbers and Coefficients As indicated in Section 2.4 the solution to Equation (73) must be periodic, with period 2,o An expansion of the solution into a Fourier series will impose this condition. oo ce (Q,q) = A2i+1 cos (2i+l)ri (82a) 00 co se (,q) B2i+l sin (2i+l)r1 (82b) If these series expansions are made to satisfy Mathieu's differential equation, Equation (73), identically, the following recurrence relations are obtained, They are given in the form in which they were used in numerical computations.

- 28 - A 2 5 -s -_ 1 A5 s - 32 1 A q A 3 3 A1 A 2 A A2m+3 s - (2m+1)2.1 2m+l 2m+1 (83a) A2m-1 s -(A2m+ —--!(83)a - q A2m+ 3 2i+1 q A A 2 (2m +3 - -2 2m+ A2m+ 3 2i+1 s - (2i+) - 2i+3 q "2i+1 ______ ~_~_~_ ________ m = O, 1, 2,. o o The corresponding relations between the coefficients B2i+l of the sineelliptic case are the same except for the first equation B3 S B3 = s- 1 + 1 (83b) B1 q _ + 1 (83b) B q These recurrence relations are satisifed for every value of q by the characteristic numbers, which are continuous functions in qo s = a2m+l (q) (84a)

- 29 - s = 2m+ (q) (84b) These functions are qualitatively plotted for m=O, 1, 2 in Figure 60 a5 The integer m=O 1, 2, ^^^'^" ~^~~^.^ ~ designates the order of the Mathieu function and its 9 t~\ (3 significance can be visualized by considering the -_ \,\_ xN __ \' __ solution of Equation (73) \\ \X \ for q = 0o Then Figure 6 Characteristic Curves a2m+l(q) and b2m+ (q) ce2m+l(B,o) = cos (2m+l)T, a2ml(O) = (2m+l)2 (85a) se2m+l(,o) = sin (2m+l)i, b2m+l(0) = (2ml)2 (85b) The roots of Equations(83) can be found byatrial and error procedure. In order to explain the algorithm, Equations(83a)are rewritten with

- 30 - A2i+3 v. = Ai (86a) 2i+l and under the assumption that sl is a good approximation for a2m+l s - 1 v = - - 1 o q Sl 2 1 v = _ -__ 1 q v 0 (1,-) s1 - (2m+1)2 1 v =1-.,. _. m qC vM1 (87a) (1,2) i m sl - (2m+3) )2 Evml v _ some v. equal to zero. If a second pair of values v (2,1) and v 2,2) are 1 mm obtained from another approximation s2 = sl + As, then a better approximation s can be expected from a simple linear interpolation. V(1,1) - V(1,2) 3 =ss (1,1) (1,2) (2,1) (2, (2) (88) m m m m -m+l 2m+l The iteration procedure, which was put together on the basis of this equation,converged to the desired characteristic number a2m+l (b2 +l) except for some small regions of mainly higher values of q, where it

- 31 - converged to the next higher or the next lower characteristic number, no matter how good the original approximation was. This problem was solved by arbitrarily decreasing m in Equations (87) by one in the first case and increasing by one in the second case. When the recurrence relations, Equations (83), are satisfied, they determine at the same time the coefficients of the trigonometric expansions, Equations (82), up to some constant factor, which has to be determined from a normalization convention. It has become common practice to normalize according to the condition 21T r1 2 0 2t 1 J se2 (Tj,q)dT = 1 (89b) js 2 m+ 1 0 which by using the expansions of Equations (82) leads to i [A(2+l 2 = 1 (90a) 00 (o2 i iB(2m+li = 1 (90b) This normalization avoids the possibility that coefficients may become infinite if one coefficient happens to become zero. The characteristic numbers and the coefficients are tabulated extensively in Reference 19. But to avoid interpolation in tabular

- 32 - values and to insure that the characteristic numbers and the coefficients were available to an extent required by the following applications, the outlined procedure was programmed for automatic computation. 2.5.2.2 Series Expansions of the Mathieu Functions The expansions of two solutions for the ordinary Mathieu differential equation, Equation (73), have already been given. They are, if the order of the functions is included in the notation: cem (Iq) = E (2re+l) cos (2i+l)q (91a) i= i+l se21(rq) = B sin (2i+l)r (9lb) i=O Due to the indicated close relation between the two Mathieu differential equations, the corresponding solution to the modified equation can immediately be giveno 00 (2m+l) Ce (2 q) = A(2mi ) cosh(2i+l)S (92a) 2m+l' 2i+l i=O 00 (2m+l) Se2m+(q) = B2i+l sinh(2i+l)~ (92b) 2m+l = 2i+1 It is proven in Reference 14, Art. 3,22 and 3~23, that all four series and the p-th derivatives obtained by term by term differentiation are absolutely and uniformly convergent in any closed interval of the real argument. Nevertheless only Equations (91) can be used in the given form for numerical computationso With the accuracy of an ordinary electronic computer, series (92) will become unstable for arguments q which are

- 33 - still of great practical interest. The literature gives, however, series expansions which have far better convergence properties. In this investigation, expansions into Bessel function products were selected. With w1 = e te (93) w2 = et (94) Equations (92) are written in the form: Ce (Sq) Ce2m+(i q) Ce2m+l0 2+'2)ce {2' ~ i) (2m+l) [ Ji, )+Ji ( )J1 ( = ~ ----- ~ 2 — E (- 1) 2i+l Ji(wl)Ji+l(w2)Ji+l(wl)J(w2 V(A (2m+l)) i0 L J 21 (95a) Sem+l(O,q)sen2m+l(2, i (2m+l) i 0, fo( ) E ( —) B2+l e) (96bw2 ]) Vq - 1(2m+l) 2 L J (95b) 2.5.2.3 Characteristic Equations and Eigenvalues The condition at the boundary B = yB, Equation (59), leads immediately to the characteristic equation for this problem, T 1Ce l (g) = 0 for (=B (96a) d Se- (,q), for = (96b) d~~~ 2m~~~~~l ( 9b)

- 34 - To each of these modified Mathieu functions of order (2m+l) corresponds an unbounded set of distinct, positive eigenvalues q, which satisfy the characteristic equation (96). They are arranged in increasing order and characterized by the number n = 1, 2, 3, To find the roots of the characteristic equations, the Mathieu functions are expanded according to Equations (95). Considering the fact that the proportionality factor of these expansions never vanishes for q>0 (Reference 14, Art. 3.30, Reference 15, pp. 134), which is quite obvious from physical considerations, the characteristic equations which are suitable for numerical computations take the form o 0 (- )iA(2m+ Jl) )Ji+l(w2) + W Ji(W )J (w2) i=0 2i L. i 0 (2m+1)[ (97a) + o (-l)iA(2m [-lJi (W)Ji(w) + WJ.i+(w)J(w ) 97a) i 0 2i+l i+11i 2i+l 1 i 2 l)iB(2m+1) [ Ji(w)j (w) + wJ (w J (9b) - Z (-1)i B2+l),i (w )Ji(w) + w J (w )J!w) i 2i+l L lil 1 + 2 2Ji+l 1 i 2 where w1 and w2 are of course taken at the boundary E = ~Bo Though these equations are incomparably more difficult than the characteristic equation for the circular cylindrical container, similar methods may be used to find the roots. Equations (79) and (80) are again fundamental to reduce higher order Bessel functions and derivatives of Bessel functions. Once the right sides of Equations (97) can be calculated with reasonable speed and accuracy, the Method of False Position (Regula Falsi) may be employed to find the roots.

- 35 - The actual numerical evaluation of the roots to the characteristic equations (97) is fundamental in the discussion of liquid oscillations in elliptic cylindrical containers. It turned out that the solution of this problem was at the same time the most difficult and the most timeconsuming part of the whole investigation. To simply evaluate the right hand sides of Equations (97) for an arbitrary value of q, the expansion coefficients have first to be determined. They result as a by-product in the iteration for the characteristic valueso In order to insure that the characteristic value is obtained which belongs to a certain order of the Mathieu function, this iteration has to be started with a very good approximation. This approximation might be the characteristic value belonging to a closely adjacent parameter q. This dictates that the search for the roots has to proceed in small steps starting from q = 0, for which the characteristic value can be immediately given, to higher values of qo In each step, the characteristic value, the expansion coefficients and the sum of the infinite series in Equations (97) has to be determined. Only when a change of sign in the right hand side occurs within one step, the Regula Falsi procedure may be applied to find the zero. The iteration was performed until the root k2m+l =qm has been obtained with a relative precision of at least 105 This procedure became more and more difficult for hiher values of q and higher order (2m+l) of the functions. In every case it. was carried out until the accuracy requirement could not be satisfied any furthero

- 36 - The roots k + are tabulated in column K of the Tables III 2m+l,n to VII for the sine-elliptic case and in Tables IX to XIII for the cosine-elliptic case. The frequencies which correspond to the eigenvalue or root of the characteristic equation of order (2m+l,n) can now be obtained from the following relation i1-twog- 1_ (98)_n oh A i CD"~29 =- qcosh tanh - q cosh (98) 2m+-l n a V mlnB a -mtlnB /

III. THE STOKES - ZHUKOVSKII PROBLEM It has already been shown by Stokes and Zhukovskii that a closed container completely filled with an ideal and incompressible liquid, whose absolute motion is irrotational, is dynamically equivalent to some rigid body. This means that its dynamic properties can be completely described by its inertia tensor. 3.1 The Stokes - Zhukovskii Potentials Consider a closed container of arbitrary shape, completely filled with an ideal, incompressible liquid (Figure 7). The container with the volume C and the walls W is moving with respect to the system (x, y, z) with the velok.z ATT city v and the angular velocity C which have in the body coordinate system - (x, y, z) the components _7 \ (\'l ~v 0 (vx v v, v ) and(ox, |\ X ~ C~ l,oy' o ). r is a posi< - \ /V I y~oy oz tion vector in the system 0 \ \Al (x, y, z) and v ( m, m, n) Ai is again the unit outward Figure 7 normal to the container Motion of a Completely walls W. walls W. Filled Container The velocity potential 0 has of course to satisfy the Laplace equation. V20 = 0, for P e C (99) - 37 -

- 38 - and the boundary condition = v'v; + (a( x rV, for P e W (100) V o 0 Since Equation (100) can be trivially rewritten in the form =v + v m +v n + w (yn-zm) + o (za-xn) + c (xm-y5) for P W ov ox oy oz ox oy oz ) (101) it is obvious that the assumption = vo0 + voy02 + v 0 + +ox04 + Doy05 + ooz06 (102)' oxil oyx2 oz3 ox4-6 introduces functions 0i which depend on the shape of the container only. According to an accepted terminology, they are called the StokesZhukovskii potentials. They are harmonic and satisfy the boundary conditions --- = -- = m, = n (103) for pure translations and 604 -S = yn - zm 5 = z~ - xn (104) av -;= xm - y~ for pure rotations.

- 39 - The Stokes-Zhukovskii potentials for pure translation can immediately be given. It is e.g. - = 1 -- + m -- + n -- (105) dV as by oz which identically satisfies the boundary conditions (103) when a0 a0 a0ax = 1, -o 0o (106) Using a similar reasoning for 02 and 03 gives 01 = x, 02 = Y 03 = z (107) The determination of the Stokes-Zhukovskii potentials for the case of rotations is far less trivial and may even become an extremely difficult problem in more general cases. But here, the Rayleigh-Ritz procedure may again effectively be applied since the potentials 0 minimize the functional Fi(0i) = J(V i)2 dC - 2 jihidW (108) C W a0i where hi is the value a in W, Equations (103) and (104). In fact i Fi (0i)= 2 Jv0iV 0i dC - 2 fb ih dW C W which with the help of Green's theorem leads to bF.i(0) = - 2 1J0V20 dC + 2 jb0 dW + 2 f60i I dW C W W Fi(0.i) vanishes identically for arbitrary variations only when 0.i is harmonic and satisfies the boundary conditions of the problem.

- 40 - 3.2 The Equivalent Inertia Tensor To be able to write the following relations in a more compact way, Equation (102) is rewritten in the form 6 0 =i^r i (109) ui can be considered as components of the generalized velocity vector u. The expression for the kinetic energy is T = 2 (V0) dC (110) C which is transformed by Green's theorem into T = f0 dW (111) W Introducing Equation (109) into the energy expressions yields a relation of the following form 6 6 L L UU. U. M.. 2 i=l j=l 1 M.. is a component of the equivalent inertia tensor and given by the two equivalent expressions M.. = p (VW )(V ) dc (113) C M.. = p -0 - dW (114) w This inertia tensor with its 36 components can be simplified very considerably. Equation (113) establishes immediately the symmetry of the tensor. M.. = M.. (115) 1J Ji Since the gradients of the Stokes-Zhukovskii potentials for translatory

- 41 - motion, Equation (107), are mutually orthogonal unit vectors, it can be concluded from Equation (113) that M22 = M3 =pC (116) M12 = M23 = ~ (117) Consider now as an example the component 14 = p jx(yn - zm) dW W which can be written as L4 = P y x dW - z dW W W Both terms in this equation are zero according to Green's theorem. This means M14 = 0 and from similar considerations 4 = = M6 = (118) Consider next M15 P p x(z - xn) dW W which again is rewritten as M5 - ^xz^ $ dW - pjx 7 dW W W Green's theorem transforms both terms into p xz - dW = p z dC W C W C Hence M =- - p fz dC C

- 42 - Similar calculations lead to M2 = pfzdC C The results can be summarized as follows: = -M15 = pfzdC C C M6 -M31 = PydC C All components given by Equations (ll9) vanish if the origin 0 of the coordinate system (x, y, z) coincides with the centre of gravity of the liquid. With this result, the analogy with a rigid body as originally stated by Stokes and Zhukovskii is complete. The expression for the kinetic energy, Equation (112), reduces now to the well-known quadratic form 3 6 6 2 i 2 T = tC^ ^i + 2 4'4i j ij ( i=l i=4 j=41 For comparison purposes it is appropriate to introduce X.. which stands for the ratio between the component M.i of the equivalent inertia tensor and I.. of the rigid body inertia tensor.

- 43 - 3.3 Solution for the Circular Cylindrical Container This solution was given by De Veubeke in Reference 21. Since his work proved to be of very fundamental value for the derivations in the next section, it is briefly presented here. Consider the closed circular cylindrical container of P/ |.^ Xheight h and radius a in the reference system (x,y,z) as [ shown in Figure 8. Because 0\ 0. / ~K of the symmetry of revolution y\ / | with respect to axis z, 06=0. Therefore 46 = 6 = 66 = ~ (121) Due to the symmetry of and antisymmetry of V 05 with 0| | f X respect to the (y-z) plane, I s: Cl l tX~M = 0 (122) The only components which thereFigure 8 fore remain to be determined are Closed Circular Closed Circular M,and M which are obviously Cylindrical Container 44 55which are obviously equal. The first problem is the determination of the Stokes-Zhukovskii potential 0. It has to satisfy the following conditions:

- 44 - - in the liquid C r2 2o + r 5 + 2 2 _ = 0 (123) 2 or 2 -- - + r 2 rr ae - on the cylindrical surface r = a, where 1 = cos 0, m = sin 0, n = 0 )05 -8 = z cos 0 (124) - on the flat ends z =+ 2 where 1 = m =, n = + 1 0-2 5 = -r cos (125) The boundary conditions suggest the following assumption for 05: 5 = [z r + T (r,z)]cos 0 (126) If this assumption is introduced into the Laplace equation and the boundary conditions, it can be found that the auxiliary function t(r,z has to satisfy the following conditions: r + r + (127 0. for r = a (128) = -2r, for z + (129) Using again the separation of variable approach with V(r,z) = l(r) 2(z) (130) Equation (127) becomes 2 ~1'K 2'2 r + r- + r -- 1 = 0 (131)'K K'V2

- 45 - This leads to the two separated differential equations t - = 0 (132) 2 2 r 21 + r + (r - 1) = 0 ~(133) where f is the separation constant. The appropriate solution to Equation (133) is *l(r) = Jl(fr) (134) The boundary condition (128) requires't(a) = JX(fa) = 0 (135) which yields the eigenvalues f = j1 /a. The solution to Equation l,n l,n (132) which corresponds to fln and which satisfies the symmetry requirement indicated by condition (129) is 2 (z) = sinh (ln Z) (136) The general solution to Equation (127) which satisfies condition (128) and is antisymmetric with respect to the (x,y) - plane is therefore 00oo n -nFn J1 (ln a) sinh (,n a) (137) The expansion coefficients F have to be determined with the remaining n condition 6* = _ 2r, for z = (138) Hence -2r = a nVl,n cosh __nf h_ -2r = a ZFnln cosh 2 a J1 (1 a ) (139) The coefficient F can now be determined according to the general theory n of Fourier - Bessel series, Reference 20, ppo 221. If for simplicity x r is substituted for -, it is obtained: a

- 46 - F 2 x J (1 nx) dx F.=.- - 1 l.n d (140) n (,n h ~1 cosh 2 _ (,1 x) dx.,n 2a o, The Lommel integrals appearing in Equation (140) can be expressed in terms of Bessel functions of the same kind and ordero It is according to Reference 21, pp. 29, 12 1 x J (Il n) dx = 2 1 ( ln) (141) 1 l 2 l1 (l,n 0o l,n and according to Reference 20, pp. 218, o (,x)dx = 1 ln (142) After substituting Equation (141) and (142) into Equation (140), the expansion coefficient F is obtained in the form n 4a2 n ln (,n 1) cosh J1 (h) (143) which finally completely determines the Stokes-Zhukovskii potential 05. ~~~~[ 00% 1~~~~~~~~~~ ~ 5 coZ ]] cos81(144 05 = zr + Z nJ1 (1,n a) sinh (IJ. )cs 0 (144)'5 L n= n 1 1=n a ln a Equation (114) is now used to determine 55 of the equivalent inertia tensor. 55 = P 5 (zl - xn) dW (145) W M5 is built up of two parts, the contribution of the cylindrical surface (l5 )1, and the contribution of the flat ends (M55)2.

- 47 - - Cylindrical surface r = a, where 1 = cos 0 and n = 0 2 t? 2 2 o~ 4 a3sinh ([i *) 1 551 +. Lza2I 2 l, n a 2 (M = p J h z a - -----— L z cos hdedz 0 -~ (L i — -i) cosh ( l -)J n ln 2a n (1[ h F 3 51 hln -tanh ln (w) = -A p a () -8Z 2 aln/ (146) M51 L T a n=l 1 (3 2 1,,n (l,n - Flat ends z = + - where 1 = 0 and n = + 1. Symmetry in the integrand of Equation (145) with respect to the (x,y) - plane allows simply to double the contribution of the end z = + - (M5)r = - 2p f r2 cos-a r n - 1 (1, a dOdr'- 2 o-.' 2 2- ) O 0~ n~l~l,n lJn 1 ln, The evaluation of this integral involves again the application of Equation (140). The result is: (MX55) = P a la)+8Z, h 1 2a (147) (55 2 3'"[ 14,. * oo tanh -22n~l,nn -1 ) Consider now again Equation (139) which after substitution of F n by the expression in Equation (143) becomes 00 4a r (1_44 2r = E - 4 (n a) (1 ) n=l (42 n ln l,n 1 ln Multiplying both sides by r dr and integrating between 0 and a immediately furnishes the sum of the following infinite series:

- 48 - <oo 8 —------ = 1 (149) n=l 2 2 - 1ln,n Adding both contributions to 55 considering Equation (149) leads to the final result tanh ln 155 = (a) 4 - pa) 16 ( -) ] (150) It is of interest to note that the corresponding term in the inertia It is of interest to note that the corresponding term in the inertia tensor of the rigid body with the same geometrical dimensions would be I 5 P a (l) + 4 (b ) (151) 55 12 a[ 4

354 Solution for the Elliptic Cylindrical Container Consider the closed elliptic cylindrical container of height h, major axis a and minor axis b in the reference system (x,y,z) as shown in Figure 9. It can be immediately concluded from the symmetries of +y the container that M 5=M6=M6 O (152) 45 = M46 = M56 = 0 (152) O r V c^ 0J The remaining three components, however, do not vanish and 1Z they can be determined. It has to be pointed out here Ai<~11~~~~ I I that in view of the application in Chapter IV where Co will be oz U -- ------ o0|| X zero, only M4 and M45 will be 01 44 5 evaluated below. The solution | ^~~y~ C |to this problem will therefore, - l ______not be complete. Figure 9 Closed Elliptic Cylindrical Container The problem will be discussed in an elliptic cylindrical coordinate system, which was defined by Equation (54). Since the coordinates 5 and r do not have an immediate geometrical meaning, the needed geometrical relations have first to be expressed in terms of E and i. The following relations can be immediately derived from Equations(54). - Arc length on the confocal hyperbola r dv =c /cosh 2~ - cos 2r d (153) ~2

- 50 - - Components of the unit outward normal v to the confocal ellipse | 1 = 12 sinh ~ cos_ (154) \/cosh 2 - cos 2r m = 2 - osh 1 sin _ (155) \/cosh 2t - cos 2T - Value of ~ at the boundary of the container a a + 1 R= 2 log a (156) ~B = 2 a b It can be seen from Equation (156) that i is defined and finite only for shape ratios >>1. But this will not cause more than formal difficulties since M44() =M 1 (157) It suffices therefore to evaluate both M4 and M5 for shape ratios a/b>l. Since the two solutions involve two different types of Mathieu functions, it is more appropriate to characterize them by the type of functions rather than by the expected result. 354.1 Cosine - Elliptic Case The study of this case will lead to 05 and M55 for a/b>lo 05 has to satisfy the following conditions in terms of elliptic cylindrical coordinates - in the liquid C 2 2 \ h:l5 + c + —- ) o (158) c (cosh 2e - cos 2 ) atg an az

- 51 - - on the cylindrical surface E = gB, where i is given by Equation (154), m by Equation (155) and n - 0. a0 = c z sinh E cos T (159) - on the flat ends z = + h, where m 0, n =+ 1 a0 -z = - coh cosh cos (160) It is now assumed that a solution of the following form exists 05(E, r, z) = c z cosh E cos q + *(|, r, z) (161) If this assumption is introduced into Equations (158), (159) and (160), a similar but simpler set of conditions is obtained for the auxiliary function 4. 2 2.2 _ 2+ = 02 for P e C (162) c (cosh 2t - cos 2T) a2 a az H = 0, for = B (165) z =- 2c cosh cos T, for z + - (164) The separation of variable approach is used again with (,sr,, Z) = - ( 2) 2(ri) 33(z) (165) This transforms Equation (162) into a set of three separated differential equations for the functions'l', $' and r.

- 52 -. - _ J = 0 (166) 3 2 3 c 2 + (s - 2q cos 2n) t2 = 0 (167) $1 - (s - 2q cosh 2) 1 = 0 (168) s and q are the separation constants and the parameters in Mathieu's differential equations (167) and (168). It is the boundary condition (164) which suggests the solution in the class of cosine-elliptic Mathieu functions of integer order. The solution to Equation (167) and the corresponding solution to Equation (168) are therefore 2(T) = Ce2m+l(',q) (169) $1(~) = Ce2m+l(Eq) (170) V has to satisfy boundary condition (163). This yields the eigenvalue problem d Cem+l(' ) = 0, for e = Ad (171) which has already been discussed in section 2.5.2.3. If finally the solution to Equation (166) is taken such as to satisfy the symmetry in the problem as indicated by boundary condition (164), the general solution to Equation (162) which satisfies condition (163) is: 00 00 *(ETz2) = c nice2(h, ) 2mln ~ = c V;20mc+2re1l, n 2m+ 2m+l, n = c 1 n~Z2m+1ns nh 2m+l 2m+l,n (172) Cf has now to be determined using the only remaining condition

- 55 - = - 2c cosh 5 cos q, for z = + (173) which yields - 2c cosh t cos v 2 2.....2h =c Lf00 2 cosh 2mln m=0 n=l n c c 2m ( q2m+ln ) Ce 2m+l 2m+ln Ce2m-c+lq2mnln) Ce2m^l(5,2mcl,n) (174) It is justified according to Reference l4, Art. 16.20, to consider Equation (174) as a double Fourier series of the absolutely integrable function (-2c cosh | cos r) with respect to the complete orthogonal system Ce2m+l(,q2m+l,n) Ce2m+l(t'q2m+ln) in the domain (05-t5< B)(0 -<-T 2A). This means that Cf2m can be given by the expression 1 C2m+l,n 2m+ln h / —--- Cr q2m+l,n csh i,n 2m+ln (175) where 1 B r2i Cq2m+ln =- -J 2 cosh t cos r ce22m+l(Bq2m+l,n)Ce2m l('2m+ln 0 0 (cosh 2t - cos 2T)dtdrT (176) and 1 2 (C2 lCr = _ ce (v^,) )2 (c q)(cosh 2t - cos 2T) 2m+l,n = Ce2m+l ( 2m+l,n )C 2m+l t 2m+l,n 0 0 (177) The evaluation of Cq2m+l, and Cr n will lead to two different types of definite integrals, one involving ordinary and the other modified

- 54 - Mathieu functions. Using the series expansion in Equation (91), the solution to the first type of integrals can immediately be giveno 27C 2 71 2ce2m+1(Q2m+1,)dT iZO 2i+1 ] (178) 1 o 2 [A(2m+l, n)1 A(2m+1,n) (2m1+ln) 1 2m+ 1 n 0n1 i=1 2i+3 (179) 1 fce (I'i )cos qld = A(2m+ln) (180) 0 2m+l nl A2+ln) (180) 0 1 ce (ml(q )cos Tj cos 2TdT, = 1 [A(2m+l,n)o+ A(2m+l,n) 2m+l 2m+l,n 2 1 3 (181) It should be remembered that due to the adopted normalization convention, Equation (90a), the right side of Equation (178) will be unity. The second type of integrals is: r B Ca = Ce (2 q ) d5 (182) J2mln 2m+l 2m+,n18 2+ - 1 Ce2m+( Cem++ ( Cn) cosh 2dd= (185) B Cmc 1n = Ce ( lq ) cosh 2 cosh 2id (184) 2m+l,,n 2m+l 2m+l,n 0 EB If Ce is represented by the expansion given in Equation (92a), a series representation of these integrals cod be 2m+derived Suc a Ca series representation of these integralsn) coshuld be derived Such a(18)

- 55 - series would not be suitable at all for numerical computations. McLachlan (Reference 14, Art. 14.22) expressed the integral in Equation (183) in terms of certain derivatives with respect to the parameter q, which however, can only be evaluated by a numerical process. Since it does not seem possible to put these integrals into a rigorous mathematical form, which is at the same time suitable for numerical computations, it was finally decided to apply Simpson's rule for numerical integration. Ce2+ (,q) is a continuous and smooth function in the interval 0!~i~<^ B and can be evaluated by a series with good convergence properties, Equation (95a). It can therefore be expected that the numerical integration will yield a result which is sufficiently precise for all practical applications, provided the interval /A is taken sufficiently small. Consequently, the expressions for the modal constans Cq and Cr become: (186) Cq A 2rl(2m+ln) A(2mlnC) + 2m(2m+ln) 21 m+1,n + 2 1 3 Cr Cb - Ca *&1 (2m+l,n)2 ] E (2m+l,n) A(2m+l, n) Cr2m+l n b2m+1 n 2Cam+l, n 2 LA 2i+l 2i+3 i=O (187) If all the results are now put together according to Equations(175), (172) and (161), the following expression for the Stokes - Zhukovskii potential 05 is obtained. 2 "" 0> sinh2 V p _g___ z Cq 5 = cz cosh cos n + c2 EO sinh2 V +ln 2mln m=On-1 g, cosh|- q21, 2m+l,n Ce2m+l(rq2m+l,n) Ce2ml(,'q2m+ln) (188)

- 56 - 05 is the basis for the evaluation of the tensor component M55, 55 = p 5 (zl-xn) dW (189) W which consists of the parts (55)1 corresponding to the cylindrical surface and ( 5)2 corresponding to the flat ends. Cylindrical surface B = yB, where n=O and ldW = c sinh tB cos n dq dz (M55)1= p c sinh B J J z 05 ((BY,'z) cos T dr dz o (190) 5 )1 1h 3 - C 1 Cq (2m+l,n) -(-5 = 1(hL) sinh 2 B + sinh B Z2m+n Cr2m+l,n itP,c5 T cB B' 2q2m+l, Cr A1 pm=0 n=l 2mCe1( B9 4I n ch_ 1 tanh hq ------ 2m+l( B'q2m+ln cm+ n'n2mll,n h - Flate ends z = +, where 1 = 0 and n = + 1 Symmetry allows again to double the contribution of the end h (M55 ) = -p c3 J J cosh 5 cos r1 0 (tnr9)(cosh 2e-cos 2T)d2dq o 0t (M55)2 1 h 1 tanh1'/I. = c5 - 16 h ~Vsinh 4B~ + sinh 2 jB)+ 4 i 2 Cr2-1 (191) In order to simplify these relations slightly, consider again Equation (174). Introducing Equation (175) leads to

- 57 - 00 00 Cq2 cosh, cos i 2 n ce (vq Ce ( ^q O0 — Cr 22m+ l 2m+l,n 2m+l q2m+1,n (192) Multiplying both sides by cosh; cos T (cosh 2| - cos 2T) d5 dr and integrating over the ellipse with axis a, b leads immediately to o 2ooc X 00 Cq2m+ 1 8 E Z. = - sinh 4 B + sinh 2 (19) m=OI-0Cr 2m+l,n M5 can now be written in the final form p cS5 24= c sinh 2 B + sinh 5B E Cr2+ A 24 p C~LBi~.- Cr 1 it p q5 -mOn= 24m+lcn 2m+l,n C) 00 C2 Ce +1 B'2m+ l,n){ - tanh( + E f (194) 1 2 m ~ l m o n.l 2 m + l1 n lm n2Vm+l n tanhc h q/m l c 2h Cem5 - l()sinh 2 + sinh 4.B+ sinh 2 mIn (195) ac5 24I B 16 2 B 3.4.2 Sine - Elliptic Case This case yields 04 and M44 for a/b >. Since the procedure is completely analogous to the one developed in the previous section, only the results will be given. If the eigenvalues q2m+l are the roots of the characteristic equation,

- 58 d Se1 (Eq) = 09, ~ _= (196) then with f B 2 Sa2m+1n J Se 2m+1(, q 2m+, (1:97) Sm+bln Se2 l ( q2m ) cosh 2E d (198) o rtB SC2m+n Se (2m+( q2m+n) sinh E cosh 2E dE (199) 2m+11, n 2mil 2m+1,n 0 rB 2+Sd = J Se2m+ ( l, q sinh E dE (200) 2rdm+ 1,n 2ml 2m+ 1,n o and S B(2~m<l,n)SC 1 (B2m+l,n) -(2nml,n) q2m+ln 1 2m+l,n 2 1 Sd2m+ln (201) Sr =,Sb - Sa2m+ F B1 (2m+l,n)1 2 (2m+l,n) B(2m+l,n) 2m+l,n 2m+l,n 2m+l,n 2 1 21+i 2i+ J 1=0 (202) the expression for the Stokes - Zhukovskii potential 04 is: - 2 sinh TV^ Sq, p4 = - c z sinh E sin T + c 2 | c 7 ) m _-n rn=O n=l q2m+l,n coshlh V'rmln r2rm+ln se2m+l ( 2m+, n ) Se2m+1 (q'2m+1 n) and finally:

- 59 - 5 1h 211 ) 02m+l, B(2m+l,n) sinh 2 SB - cosh B E B P c 5 24 m=rB2q 2m+l,n 2m+l,n tanh ~q2m+ln C00 Sq2mtl 2n Se tanh l __,r h a+zS -B' c2m+1 n C 2 fm+1nZ Srn21 tanh c I2mn - (20) l V2m+l, n The relation which corresponds to Equation (193) is oo oo Sq2 1 8 ZE E 1 = i s4 s i4 - sinh 2 (204) m=On=l 2m+l,n The corresponding inertia component for the rigid body is 55C5 = I() sinh 2 B + - t sinh 4 - sinh 2 E (205) r p c

- 6.0 - 3-5 Numerical Results On the basis of known eigenvalues ln and q l which have been,n 2m+l, n determined for different ratios a/b in the previous chapter, the evaluation of the corresponding equivalent moments of inertia is a straight forward numerical problem. The only question is, whether the limited number of modes on which the computation has to be based due to numerical difficulties in the evaluation of higher eigenvalues, are enough to produce a result with reasonable accuracyo It is beyond the scope of this investigation to give a rigorous answer to this question by discussing the remainder in the infinite series of Equations (194) and (203). It was, however, clearly apparent from the performed numerical calculations that in no case did the higher half of the given modes influence the required 4 digit accuracy of the resulto The results are summarized in Tables I and II of the Appendixo Values are given to four significant digits for parameters a/b and h/a which seem to be of most practical interesto As an illustration, the functions X(h/a) for a/b = 2/3, 1 and 3/2 are plotted in Figure 10. The influence of the ellipticity (a/b on X is surprisingly smallo There is, however, a small domain of h/a in which consideration of the deviations from the curve for the circular cylindrical container appears warranted and this is exactly the domain of most practical applicationso

1.0.8 with Shape \ati as a/b = 2/3, 1, 3/2 0 cr 0 _'_____________ ___ __ 0 1 2 3 4 5 6 7 8 SHAPE RATIO 6= h Figure 10. The Ratio %(h/a) = M/Ic for Liquid Filling Elliptic Cylindrical Containers with Shape Ratios a/b = 2/3, 1, 3/2

IV AN EXAMPLE FOR THE OSCILLATIONS OF A CONSERVATIVE SYSTEM WITH A LIQUID MEMBER 4.1 Dynamic System to o6R /(,,/ I, o Figure 11 Dynamic System with Liquid Member - 62 -

- 63 - Consider the system in Figure 11. A container W. symmetric with respect to the (xz) and(y,z) plane, is filled with an ideal, incompressible liquid, which has the free surface S in the position of equilibrium, The container is carried by the crane with axis G which moves parallel to the coordinate axis x according to some displacement function s(t). The crane and the container are connected by two weightless rods which are hinged to the rotation axis T of the container and the axis G of the crane. Besides the parameters which are explained in Figure 11, the following system properties will enter the final equations. m:Mass of the container W w I Moment of inertia of the container with w respect to the axis of rotation through its centre of gravity S. w m Mass of the liquid Co M: Equivalent moment of inertia of the liquid with surface S with respect to the axis of rotation through its centre of gravity S I Moment of inertia of the solidified liquid with surface S with respect to the axis of rotation through S 0: Stokes- Zhukovskii potential corresponding to M c C: m-th mode of free oscillation in the m container W at rest. 4L2 Solvability and Nature of the Solution Two general theorems about the solvability and the nature of the solution for a conservative system with a finite number of members and a finite number of cavities partially filled with liquid are given in Reference 16, pp. 268 ff.

- 64 - Theorem 1l If a system consists of a finite number of conservative members and contains a finite number of cavities partially filled with a liquid and if the potential energy of the system has a minimum in the equilibrium position (eogo is statically stable), then there exist natural modes in the motion of this system about the equilibrium position, the frequencies of these oscillations are positive numbers and form a denumerable sequence such that lim c = oo, n n - oo a finite number of natural modes correspond to each eigenvalue, the system of natural modes is complete, which means that any free motion of the system may be presented by a superposition of oscillationso Theorem 2: If the potential energy is not a minimum in the equilibrium position, then there is at least one negative quantity among the CD, which means that the motion is unstableo

- 65 - 4.3 The Equations of Motion The main problem here is the derivation of expressions for the kinetic and potential energy of the liquid. In order to take full advantage of what has been done in Chapter III, the liquid motion is thought as being composed of two parts. The first part is the motion which would occur if the surface had been replaced by a solid cover while the second part considers the influence of the surface waves. On this basis, the expression for the kinetic energy of the liquid may be written in the form: c= piJi + )Vo + (s - a-f( + ))V7x +V0]2 dC c 2 Jc c C (206) In this expression 0 is the potential of wave motion in the container. Expanding the integrand and integrating wherever immediately possible gives T = 2 (a + )2 M + J(V0)2 dC c 2 c 2 c 2 C + - - f(Cz + )fVxVdC + p( )( f )) x + )JV dC C C + p(s - a- f(a + x))/ V x dC (207) C If now the potential of wave motion is expanded with respect to the natural modes of free oscillations c, 0(x,y,z,t) t) = (t)c(Xy,) (208) an assumption which is clearly justified by the completeness of the system of eigenfunctions (Section 2.2.1), the integrals encountered in Equation (207) take the following forms:

- 66 - V)2dc = ZP 2W f (p2ds (209) C m s C m S J x V 0dc = iPm j x (cPmdS (211) C m S,7 V0 dC = E CP dS (211) C m S dX c = (212) placed - b by- co cp and finally because of the orthogonality of the g m m eigenfunctions, Equation (209) became the normal form of the homogeneous quadratic in p. The expression in Equation (212) simply vanishes because the centre of gravity of the liquid with surface S was chosen as the origin. To obtain an expression for the potential energy of the liquid, the same decomposition of the liquid motion may be considered. If S(x,y,t) is again the free surface function, then c c= g(a + f - cos a - f cos(ac+))+ 2d - p g/(a+ / xdS S S (215) The unlknown surface function G can be eliminated with the help of the surface condition given in Equation (7). a Zm a ) (217) m z0

- 6,7 - which becomes, when it is assumed that the motion starts from rest -(x,y,t) = Pm( a ) (215) m z=O0 Now the integrals encountered in Equation (213) can be given in the form dS = P dS (216) S g m fxddS = g Epw s cp dS (217) T = [ - - fw(m+)] + -d (&1+ ) m r,2 M + [s - f- c(a+)] + c (cx p)2 2g LPm(P m mmd mS using again (Eua ) ) to replace dS Adding finally the energies of the container, which is a simple rigid body, the total energies of the system can be given by the expressions:.2m I + Q[s - a - f )] Pn f xcds (218) 2 ~m 2 MS m S

- 66 - V = m g[e + f - cosa - f cos(o+)] + me g[ + fc- I cosa - fCCos(a+)] + L E p2m4 f2 ad 2g'm mM m S S Since these energies are expressed in terms of the generalized coordinates q = a,,, p, p,..., Lagrange's equations can be used to derive the equations of motion of the system. Since in this case = 0 and = 0, they may be used in the simplified version d aTi + 5v = 0 ( = 1,2,'') (220) Application of Equation (220) to the energy expressions (218) and (219) leads to an infinite system of coupled linear inhomogeneous differential equations.

- 69 - | | Pm m=l,2,... 2 m m f +m) c f (221) + Mc | + mc c c c -- + c g(S+fc + I 2 r +v I... cV ~ m f2 - p f J dS +w fw c m w Sm f2 _2 c c / c _ + Mm c+ m f____ 2 2 S_ _ m=2 a/2 r w P-~'~m m=l,2,... +m g(e+f) +m g f /~r+J+ dS W +m~ g(~ef) +f S +me+ g f) f fpdS +m f w w m w w S +m g f +m f _. m 2 _..__m f +P, JCC dS -P fJxcpas

- 70 - The equations of motion may be written in matrix notation [M]{q} + [K]{q} = {s} (222) where both matrices M and K are square, symmetric and infinite. The coefficients of these matrices are given in Equations (221), omitting the symmetric part of the offdiagonal elements. These infinite matrices must be made finite in numerical computations by neglecting the contributions of higher modes. Since the cut-off point depends on the required accuracy and on the nature of the forcing function, it has to be determined in every case. But keeping in mind the restrictive assumptions of the linear theory, it is in most cases sufficient to consider only a few lower modes. After the system has been truncated, the motion can be found by using any of the well known numerical procedures for the solution of equations of the type (222). The functions p (t) do not have an immediate physical meaning, but they may be used as a basis to compute the complete motion in the liquid at any time t. In most applications one will only be interested in the behavior of the liquid surface. Equation (215) gives the function for the free surface at any instant t. In Equation (221), the properties of the liquid appear in the mode frequency wu and three different surface integrals. They have to be evaluated considering a specific container W. evaluated considering a specific container W.

- 71 - 4.4 Circular and Elliptic Cylindrical Container The problem of free oscillations of liquid in these two container shapes has already been treated in Chapter II. To be able to give all coefficients in Equation (221), the three following integrals have to be determined Ui = cTidS (223) V. = Jx cdS (224) S W = Ji dS (225) S In order to give tabular values, it is furthermore assumed in this section that only the surface motion in the point P (a0,0,) is of practical interest. 4.4.1 Circular Cylindrical Container The natural modes in this container are according to the derivations in section 2.3 CP =J (1 r-cos mG cosh({> - mn m m,n a co mcn a m = 0,1,2,... (226) n = 1,2,3,.. The Stokes - Zhukovskii potential 0 at the surface S can be derived from Equation (144). c =r a2tanh X * 2 r lZjl - 2 ) (~ J a ] (227) Assuming now that the natural mode cp is introduced into Equations m, n

- 72 - (223), (224) and (225), then it is immediately apparent that U and m, n V vanish except for m = 1, while the corresponding W vanishes m~n mn as a consequence of the equations of motion. This means that only modes of the type l- = J( r)cos & coshLn [ ) (228) 1^n = J a l,n a,n a will contribute to the motion of the system. Using the orthogonality relation a J r l lm ) aJ1 n a)dr = O, if m f n 0 1 ~,m a, n a and the integral relations given in Equations (141) and (142), the following relations are obtained after some obvious calculations: Un = a 4cosh(ln ) 2 [1() 2 tanh( X h ln 1 n 2 a ln W1 = i a cosh 1,n a 2 2 (231) nn n=l 4.4.2 Elliptic Cylindrical Container Here it is again essential to distinguish between the cosine-elliptic and the sine-elliptiche first case will furnish the solution for shape ratios a/b>l and the second case for ratios a/b<l.

- 75 - 4.4o2,1 Cosine-Elliptic Case The expression for the natural mode is given in Equation (76). C2m+1,n 2 m+ 21( m+ n)ce 2m+l 2m+11, n c o2s (zh) m = 0,1,2..., n = 1,2,3... (233) and the epxression for the Stokes-Zhukovskii potential 0c at the surface can be derived from Equation (188) h 2 oo oo tanh c cosh ~ cos + c.. ce2 +l ) m= 0 n=l Crm+n 2m+l,n n Ce2m+l ('q2m+l,n) (234) Using the orthogonality relation B 2 ce2m+l( q2m+l,n Ce2m+l q2m+l,n 2i+l 2i+lj) Ce2i+l 2i+lj 0 0 (cosh 2t - cos 2q) dE dr = 0, if m ~ i and n j j and the definitions for Cq2m+,n and Cr2+ given in Equations (176) and (177), the following expressions can be obtained after some obvious calculations: c4,'2h - F1-h\ tanh( V +m1lJ (235) c21n cosh / (236) 2mnl,n 2 2m+11n chr V2 c236)

- 74 - c 2 2h W =Jt Cr cosh (237) 2m+l,n 2 2m+l, n c 2m+,n The motion of the surface in P (a, 0,0) is (5B' 0t) c I P 2m+, n (t) 2m+1n sinh( 2h + m=O nl ln Ce2m+l(SB q2m+ln )Ce2m+l (O q2m+l n) (238) 4.4.2.2 Sine-Elliptic Case An exactly analogous procedure leads to the following results: 4 2h +l,n Sq2m+ln osh c t vanh L V2m+ln J (239) c3 2h V = t i c (240)o V2m+l1 l2 2m+l,n ) (240)n 0 2 2h W =t= - Srm cosh (241) 2rn2m+ 1. n 2 m+l,n 00 00 (6B' 2 Z ) c -m ( 2m+1,n sinh 2h ) m=On=l 2m+ln 2mln 2me l ~ B'~2m~l ))se (242) S2+l(B' 2m+l,n) e2m+l (2' 2m+l,n (2 4.4.3. Generaliziation of the Solutions The three solutions to U., V. and W. show a very close similarity. It should therefore be possible by simply redefining some constant terms to obtain only one expression which then would be applicable to all three cases. If again qpi is the i-th natural mode and a. the corresponding eigenvalue, then these unified expressions are:

- 7'5 - U, = ga~cosh()i a )[2 atan2 (243) V. = 4 a cosh( 2 t i h)Q (244) od1=Ic aii W. = g a c osh a R W = It a2cosh2 i h)R (245) 1i a i 00 ((x=a, y=O, t) E= -P(t) sinh.i h) S (246) The modal constants take different forms in each of the three cases: circular case 01,n = 9n (247) Qln 2=5^, (248) Qn,n 2 ) R = - n) (24-9) l,n 2 2 JL ln Sln l( 1,n) (250) - cosine-elliptic case for a/b>l 1 a/b + 1 B = 2 log a/b - 1 2ml,n = 2 V2m+ln cosh B (251) ~2m+ln C2 n 2m+l,n 2cosh3( Cr 2m+ln R p —(253) 2m+ln 2", 2m+ln = Cem+ (B cl, 2m+l (0m+l,n) (254)

- 76 - -sine-elliptic case for a/b-<l 1 log b/a + 1 2nmln = 2 V ln sinh B (255) Qq~nrcln -(256) Q2m+l,n (256 2sinh3B 2m+l,n sin2 22 n 2sinh2 B (257) 2m+ln =Seml (iB' 2m+l,n) se2nl(C' ml, (258) o2 Qm+l.nR2m+l ln 2m+l 2 2l,n Values for a 2l n l2m+ln' 2-m+l,n' S2m+ln are given in Tables III to XIII of the appendix for different ratios a/b. 4.4.4 The Equations of Motion in Dimensionless Form Investigations of dynamic systems can be made much more general if they are carried out in a dimensionless manner. It turns out that the character of the motion of this system with a circular or elliptic cylindrical container is completely determined if the system is specified by the following parameters: - length parameters f f ^ 1, ^w f. c h a 1*'f* = -1 f* = ~. - y2 a w a c a a b(259) - inertia parameters m I wX w (260) m I c c

- 77 - All other parameters which will enter the equations of motion can be derived. They are I M ~~c 5' c I C I cp a c U V W * * V * W U = — Vm = m Wm- _ m (261) m 43 m 2 t a it a 1 a *D a m D (1) = a m m g If finally the time is transformed into the dimensionless form kT ='g t (262) and if the derivative with respect to T is denoted by a prime, the dimensionless equations of motion are, if written similarly to Equation (221):

- 78 - &| l Pm | c f. +fw|; | |+. *f*()i*+ *) m (*+f* )2 +de f*(*+f*) | p l 7 w -^*+y7 w(263) 2p~ *2 c m m 1+ ^w~r + -f 7 c y c c + 2U* 7 C Y C * +w2 U* + I* (K + X) + Im (+ ) m m c C * r2 + de f *:+ w w0 c ~*2 c m m + f V 7 c y c -xp*2 * + I (X + 7C) m m + Wo W m m! Pm s i a 7 C e -+ f 2 * C * 4- (e + f de 4' f (X +f vje ve w77W w *2 7 w wm mV * inf c *f 7 * c 7 c 7 c +~ w **2 * w m m c * + - f + _f 7 c 7 c *x-4 * *2 * + c W -co V m1m 1 1 1m m

- 79 - 4.4.5 Summary To facilitate the application of the dimensionless equations of motion (263) to compute the response of the system to an arbitrary forcing function s"(T)/a, a summary of all the relations needed to evaulate the coefficients in Equationr (263)ame given The length parameters *' f = w * c h a f f - e = a w a ca ab and the inertia parameters m I W W at - -, = I m I c c depend on the specific arrangement and properties of the system. They are the basis to determine the dependent parameters 1 1+f B = log 1-7, for < 1 (264) B = log +1, for > 1 * + - sinh 2( -sn I = 3 B 1 6 c 24. 2 16 4 sinh 42B - sinh 2B ) for 7< 1 sn inh sinh * 15 1 c =12 c + e for y = 1 (265) * 1 e3 sinh 2B 1 I - + 2 1- h4 2 sinh 4B + sinh 2tB for 7>1 c 24 c 2sh2 6 4B cos h B' cosh B Values for f are given in Table I and II of the appendix for different values of e and y.

- 80 - U cosh (ai.) Qi - e - - tanh21 c) (266) i2 r. 2 V. = cosh (ai c) i (267 *2 W. = cosh2 ( ) R. (268) 2 = a. tanh (ai c) (269) 1 1 1 These relations determine all the coefficients in Equatiors (265). c(t)* p.(t) p(t) and 1 can now be computed using any suitable numerical procedure. 2 a The motion of the surface point P (x=a,y=0) can now be obtained from the relation (=ay=,t) ai sinh (ai c) S (270) a. ^- i i i i a The modal constants a.i Q.i R. and S. are given for different values of y in Tables III to XIII of the appendix. 4.4.6 Examples.for the Response of the System In order to give some idea of the character of the response,which might be expected in a specific case, Equations (265)were solved by.:the Runge-Kutta numerical procedure for a few cases which were designIjd such as to show the following effects: - Influence of the surface waves, Figure 14. - Effect of including higher liquid modes in the analysis/ Figure 15. - Influence of the ellipticity of the container, Figure iy. - Behavior of the system over a longer period of time, Figure 17. The system was subjected to the force function - in form of a unit step g function as given in Figure 12.

- 81 -.0 9 aQ Figure 12 Force as a Unit Step Function The following system properties are the same in all computed examples: = /a f =f/ f f/a c = h/a o = m /m = I /I 2,0.4.2 1.8.5.8 All the other properties are varied so as to show the intended effects. Number of Lower Response System a Liquid Modes Duration T of the Plotted No. b Included Investigation in Fig. 1 1.3 0 20.0 14 2 1,3 18 20.0 14 3 1.4 1 20.0 15 4 1.4 2 20.0 15 5 1.4 18 20.0 15 6.6667 18 20.0 16 7 1.0 7 20.0 16 8! 1.5 18 20.0 16 9 1.4 18 80.0 17

- 82 - e,- e. The?wePonse&-sof the time de^7 _'__c J = -1 pendent variables a, p and g/a in Figure 13 are plotted for |IJ~: the 9 given dynamic systems IJ0~ -~with respect to the dimensionJI:~~ ^(less time axis T =a;,t in | / Figures 14 to 17. The curves ^r /^^# Tgive rise to the following conclusions, which however'I / r'4'-' f may not be generalized to -.,, / X/~ ~ arbitrary dynamic systems. ^'-^^ / ~ They are valid for the investigated dynamic systems and for the very simple-naturd -~Ore'ing function. It may be assumed Figure 13 that some of the observations Significance of the Plotted Time Dependent are of quite general nature. Variables a,,,/a Figure 14. compares the response of system 1, in which the development of the surface waves is suppressed, and system 2, where this arbitrary restraint is removed. It shows clearly that the character of the response is very drastically changed by the wave motion, which is able to absorb a considerable amount of energy. Figure 15, however, demonstrates that this drastic change in the rigid body motion may only be attributed to the lowest mode. This figure compares the response of systems 3, 4 and 5, in which an increasing number of liquid modes are free to appear. It allows furthermore a visualization of the convergence of the analytical process in that

- 83 - it demonstrates that the influence of the second mode is about the same as the contribution of the following 16 modes together. It can therefore be concluded that the lowest mode determines the rigid body motion of the system with high accuracy, but that a sufficient number of liquid modes has to be considered, whenever the behavior of the free surface is the purpose of the investigation. Figure 16 again confirms an observation which has already been made while computing the mode frequencies and equivalent moments of inertia of a liquid in elliptic cylindrical containers. It compares the response of the systems 6, 7 and 8, which only differ in the ellipticity 7 of the container. Only the wave motion shows some appreciable differences and even these are very small. The- way in which the equations of motion were made dimensionless permitted a comparison of the influence of the different flow patterns in the three containers. The system properties illustrate clearly how one might consider this effect to simplify practical computations. Figure 17 shows finally the periodic energy exchange which takes place between the motion of the liquid and the motion of the rigid body.

w a FOR SYSTI-M NO. I W 3. _a FOR SYST-M NO.2 ____ __ C') a. CLL _ __\ C) z <[ 0 I. — FOR S STEM NO.1 ~z 1 FOR S STEM NO. c'N __2...._\ rr)'_____ -2- FOR YSTEM. ___ -3 w Z 0 2.5 5 7.5 10 12.5 15 17.5 20 m0 DIMENSIONLESS TIME T = tj.. Figure 14. The Influence of the Surface Waves. Response of the Systems No. 1, where the Surface Waves are Suppressed, and System No. 2, where they are Free to Appear

-o Z 4 JhJ W3, FOR SYSTEMS NI.3,4 AND e l| a FOr SYSTEMS NO.3,4 ND 5 IU) 0 ~ 03 ~ aOR SYSTEM NO.3- 3 \ ill 2 W) 4 4 Z 0 2.5 5 7.5 10 12.5 15 17.5 20 DIMENSIONLESS TIME T = t/T Figure 15. Effect of Including Higher Modes in the Analysis. Response of the Systems No., 4 and 5 where one, two and 8 Lower Modes are Considered w -3 NO. 5 I -4 Z 0 2.5 5 7.5 10 15 17.5 20 DIMENSIONLESS TIME r = tVT' Figure 15. Effect of Including Higher Modes in the Analysis. Response of the Systems No. 3, 4 and 5, where one, two and 18 Lower Modes are Considered

Iz w...... _ _ _ ____ _ _ _ _ _ 4 w,B FOR SYSTEM NO. 8 6 z 3 --- -- --- ---— NO. — 7.. 3 c) a FOR SYSTEMS NO. 6,7 AND 8 C 2 LIZ z cQo 0 -2 o!2 0 S 3________-r FOR SYS EMS NO. __________ N NO.?7 <1: NO. 8 W 4 z w 0 2.5 5 7.5 10 12.5 15 17.5 20 DIMENSIONLESS TIME F s tIf Figure 16. Influence of the Ellipticity of the Container. Response of Systems No. 6, 7 and 8, which Differ Only in the Ellipticity a/b of the Container. a/b = 2/5, 1 and 3/2

-5 ----------------------------------------- -5 0 5 10 15 20 25 30 35 80 DIMENSIONLESS TIME r=tJI 5; 55 40 45i 50 55 60 5 70 75j 8 DIMENSIONLESS TIME r t~r Figure 17. Response of System No. 9 over a Longer Period of Time

V SUMMARY AND CONCLUSIONS This dissertation presents a solution of the problem of small oscillations of an ideal, incompressible liquid in moving circular and elliptic cylindrical containers. The investigation is directed toward the solution of the liquid oscillation problem in hot metal ladles but all the numerical results are nevertheless given in a completely general manner so as to be immediately applicable to similar problems. The solution involves a classical approach in the dynamics of continuous media. The motion of the liquid is expanded in a series with time dependent coefficients with respect to the natural modes of free oscillations in the container, which represent a complete orthogonal set of coordinate functions for this problem. A clear picture of the liquid motion was obtained by separating it into two parts; (1) the motion if the free surface S is replaced by a solid lid and (2) the surface wave motion. This suggested that the solution to the forced oscillation problem could be obtained in three principal steps. (1) The coordinate functions and corresponding eigenvalues were obtained from the solution of the free oscillation problem of liquid in containers at rest. The solution for the circular cylindrical container is very old. In the case of an elliptic cylindrical container, the problem reduces to an eigenvalue problem with two parameters. A rather involved and time-consuming numerical procedure was developed in this case to yield enough eigenvalues and eigenfunctions to handle all subsequent problems. - 88 -

- 89 - (2) A closed rigid container completely filled with an ideal, incompressible liquid is dynamically equivalent to some rigid body with a mass equal to the mass of the system and some moment of inertia. The motion of the liquid inside the container can be derived from the Stokes - Zhukovskii potentials, which depend on the shape of the container only. A suitable assumption reduces the Stokes - Zhukovskii problem to the same eigenvalue problem which has already been solved in the first step when only the boundary conditions at the cylindrical walls are considered. The Stokes - Zhukovskii potential is now expanded in a Fourier series with respect to the eigenfunctions and the unknown coefficients are determined from the remaining boundary conditions. The Stokes - Zhukovskii potentials provide the basis for computing the components in the equivalent inertia tensor. These operations were performed to obtain the equivalent moments of inertia of the liquid in elliptic cylindrical containers with respect to rotations about the two principal axes through their center of gravity. The results supplement the known values for the circular cylindrical container and have important technical applications. (3) The.small oscillations of a conservative system having containers partially filled with an ideal, incompressible liquid can be described by an infinite set of linear,

- 90 - second order differential equations. It is demonstrated that for the ladle system with a circular or elliptic cylindrical container, the evaluation of the coefficients in these differential equations requires merely a rearrangement of some modal constants which have already been computed in the first two steps. They are therefore tabulated and may be utilized when in any dynamic system these modes are excited. The investigation closes with the presentation of some characteristic responses of the ladle system, disturbed by a simple unit-step function. The responses show the following important characteristics, which however may not necessarily be generalized to other systems and forcing functions. - The wave motion of the liquid changes drastically the response of the system. - The rigid body motion is accurately determined when only the lowest liquid mode is considered, while the surface waves may be considerably affected by higher modes. - The influence of the container ellipticity alone on the response of the system is small and it can be concluded that at least for preliminary investigations the elliptic cylindrical container may be replaced by the simpler circular cylindrical container of the same dynamic properties.

APPENDIX 1. The Equivalent Moment of Inertia of An Ideal, Incompressible Liquid Completely Enclosed in a Rigid Elliptic Cylindrical Container Table I and II give the dimensionless y inertia parameter? = M /I. for an ideal, incompressible liquid completely enclosed in the rigid, elliptic cylindrical container of _ - a DFigure 18. k is the ratio between Z _________ the equivalent moment of inertia of the liquid M and the moment of c N#^~~~ t~inertia of the solidified liquid Ic, __- I D_ _ both with respect to rotations about 51 X C i the y-axis through the centre of gravity of the liquid S. Values ________ ______ of \ are given for the shape para= a h D |meters Y =b and =, which are of most practical interest. The parameter X relates Me with I, - c cc Figure 18 which is given by the following Reference System for the Inertia. Pa rexpressions: Inertia Parameter \ - 9.1 -

- 92 - I 1 sinh 2B c5- = 2 3 2 1 6 + - sinh 4B - sinh 2tBfor Yl it P a sinh 16 sinh B B B c i 3 1 1= -- 3 +, for = 1 (271) i p a I 1 3sinh 2t ____ 13 B 1 ( h2 24c = 21 3 - 2+ +-h 4t 2 sinh B + sinh 2B or >l t p a cosh cosh where 1 1 +y = - log for 7<1 B 2 1 -y (272) t = 1 log +,for > 1 = -!' 2, Eigenvalues and Modal Constants for the Natural Modes of Liquid Oscillations in Elliptic Cylindrical Containers Tables III to XIII give a summary of constants which belong to the mode of order (2m+l,n), which proved to be of fundamental importance in problems dealing with liquid oscillations in moving elliptic cylindrical containers Column K contains the eigenvalues k2m+l, = V/qth 2m+ln 20lm+ln of the corresponding parameter in Mathieu's differential equations. This column is omitted in the case y = a/b = 1, where it does not have a meaning. The column SIGMA gives the dimensionless frequency parameters a2m+ln' which are the basis to compute the mode frequencies c2m+ln from the relation "2 = 2m n tanh (m n ) (273);2m+ln 2m+ln g 2m+ln a

Columns Q and R give the constant Q2m+l,n and R2m+l, which are the basis for the evaluation of the contribution of the corresponding mode in the system of second order differential equations of motion. Column S finally gives the constants S2m+ln which then are used to find the response of one particular point in the free surface. The application of these modal constants to the investigation of a specific dynamic system is summarized in Section 4.4.5.

- 94 - TABLE I THE RATIOS LAMBDA = C/IC FOR ELLIPTIC CYLINDRICAL CONTAINERS WITH RATIOS A/B LESS OR EQUAL TO 1 A/B *6667 *7143,7692 *8333. 9091 1.0000 A/B B/A 1,5000 1.4000 1.3000 1.2000 1*1000 1,0000 B/A H/A H/A 0 1. 000 1, 00000 10000 1 0000 1.0000 0.2 *9511 *9511,9510,9509.9509 *9508,2 04.8264.82641 8257 *8254.8250.8245.4,6 *6640 *6632 *6623.6614.6604 *6593.6 *8 *4999,4986,4972.4956.4939 *4921 *8 lo10 3601.3583.3563 *3543.3520.3495 1.0 1.1.3036.3016 *2995.2973.2948 *2922 1.1 1.2.2570 *2549.2527 *2503.2477.2449 1.2 1.3 *2202.2180 *2157.2132.2105 2077 1.3 1.4.1926 *1904 *1880 *1855,1828.1800 1.4 1.5 *1736.1714 *1690 *1665,1639.1611 1.5 1.6.1622,1600,1577 *1553 *1527 *1499 1.6 1.7.1575.1554.1531 *1508 *1482,1456 1.7 1,8.1585 *1564 *1543.1520,1496 *1471 1,8 1.9.1643.1623.1603.1581.1558 *1534 1.9 2.0.1740 *1721 *1702 *1681 *1659 *1637 2.0 2.1.1868 *1851.1833 o1813.1793 *1771 2.1 2.2.2022 e2006 1988.1970.1951.1931 2.2 2.3.2195.2180.2163.2146.2128.2110 2.3 2.4.2382.2368.2353.2337 *2320 *2302 2.4 2.5 *2579.2566 *2552.2537 *2521 *2505 2.5 2.6.2783.2770 *2757 *2743 *2729 *2714 2.6 2.7.2990,2979.2966.2953.2940.2926 2.7 2.8.3199.3188.3177.3165 *3152 *3139 2.8 2.9.3407.3397.3386.3375.3363 *3351 2,9 3.0.3613.3604 *3594 *3583.3573.3561 3.0 3.5.4581.4574.4567 *4560 *4553.4545 3,5 4.0.5409.5404.5399.5394.5388.5382 4.0 4.5.6092.6089.6085.6081.6077 66073 4.5 5.0 6652.6649.6646 *6643.6640.6637 5.0 5.5.7109.7107.7104.7102.7100 *7098 5.5 6 O.7484 *7482.7481.7479.7477,7475 6.0 6,5.7794.7793 *7792.7790.7789 *7787 6.5 7,0.8053 *8052.8051.8050.8049.8047 7.0 7.5.8270.8269 o8268.8267,8266.8265 7.5 8.0.8454.8453 *8452,8452,8451.8450 8,0 8.5.8611.8610.8609.8609 8608.8607 8.5 9.0 874587 45 8745 8744,8744,8743.8742 9.0 9.5.8861 *8861.8860.8860.8860.8859 9.5 10.0.8962.8962.8962.8961.8961.8960 10.0 10.5.9051.9050 *9050.9050.9049 ~9049 10.5 11.0.9129.9128.9128.9128.9127.9127 11.0

- 95 - TABLE II THE RATIOS LAMBDA = MC/IC FOR ELLIPTIC CYLINDRICAL CONTAINERS WITH RATIOS A/B GREATER OR EQUAL TO 1 A/R 1.0000 1.1000 1*2000 1.3000 1.4000 1.5000 A/B /A 1.0000 *9091 *8333 *7692.7143.6667 B/A H/A H/A 0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 0.2.9508.9507.9507.9506,9506.9505.2.4 *8245.8241.8237.8233 *8230 8227.4.-6.6593.6582.6572.6563.6555.6548.6.8.4921.4903.4887.4872,4860.4848.8 1.0l 3495- 3472.3451.3433.3418 *3404 1.0 1.1.2922.2896 *2874.2855.2838.2824 1.1 1.2.2449 *2423 *2400 *2380.2363 *2348 1.2 1.3.2077.2050.2027.2007.1990.1975 1.3 1.4.1800.1773.1750.1730.1713.1698 1.4 1.5 *1611 *1584.1562.1542.1525.1511 1.5 1.6.1499 *1474.1452.1433.1417 *1403 1.6 1.7.1456.1431 *1410 *1392.1377 *1364 1.7 1.8.1471 *1447.1427.1410.1396.1383 1.8 1.9.1534.1511 *1493.1476.1463.1451 1.9 2.0 *1637 *1616 *1598.1583.1570 e1559 2.0 2.1.1771.1752 *1735 *1721.1709 e1699 2X1 2.2.1931 *1913 *1897 *1884.1873 @1863 2 2 2.3.2110 *2092 *2078 *2066.2055 *2046 2@3 2.4.2302.2286.2273.2262.2252 *2244 2.4 2.5.2505.2490.2478.2467.2458 *2450 2.5 2.6.2714 *2700 *2688 *2678.2670 *2663 2*6 2.7.2926.2913 *2902 2893,2885 @2879 2.7 2.8- 3139.3127 *3117.3109.3101 *3095 2*8 2.9.3351.3340.3331..3323.3317 *3311 2.9 3.0.3561.3551.3542.3535,3529.3524 3.0 3.5.4545.4538.4532.4526.4522.4519 3.5 4.0.5382.5377 5373.5370.5366.5364 4.0 4. 5.6073.6070 e6066 *6064.6062 *6060 4 5 5,0.6637.6634.6632 *6630.6628 *6627 5.0 5,5.7098.7095 *7094.7092.7091 *7090 5.5 6.0.7475.7474 *7472.7471.7470.7469 6.0 6.5.7787.7786 *7785.7784.7783 *7783 6e5 7.0.8047.8046.8045.8045,8044 *8043 7.0 7.5.8265.8265.8264.8263.8263.8262 7.5 8.O.8450.8449.8449.8448.8448.8447 8.0 8.5.8607.8607.8606.8606.8605 @8605 8.5 9.0.8742.8742 @8741 *8741.8741 *8740 9e0 9.5.8859.8859.8858.8858.8858.8857 9.5 10.0 *8960.8960 *8960 *8959.8959.8959 10.0 10.5.9049 ~9049.9048 ~9048,9048 ~9048 10.5 11.0.9127.9127 *9126 *9126.9126 @9126 11.0

- 96 - TABLE II I EIGENVALUES AND MODAL CONSTANTS FOR MODES OF THE ORDER 2*M+1.N A/B = *6667, B/A = 1.5000 M N K SIGMA Q R S 0 1 1.*0078 1.8028 *17721E 00,85813E-01 *42862E 00 0 2 2.,8239 5.0515 -.65435E-03 *10689E-03 - 17471E-01 0 3 4.5861 8.2039 *75963E-05 *12290E-06 *66656E-03 0 4 6*3448 11.3500 -.11963E-06 *12943E-09 - 23426E-04 0 5 8.1024 14*4939 *22031E-08 *13094E-12 *79145E-06 0 6 9.8594 17.6370 -.38533E-10 *96536E-16 -.22559E-07 1 1 1.8846 3.3713 *57502E-01 *92220E 00 -.13769E 01 1 2 3.7059 6*6293 -.12241E-02 *37973E-02 *78811E-01 1 3 5.4446 9.7396.32099E-04 *11777E-04 -.48475E-02 1 4 7*1920 12*8654 -.80969E-06.24144E-07 *23519E-03 1 5 8,9430 15.9977.19871E-07 *37156E-10 -.97308E-05 2 1 2.8259 5.0552.25878E-01 *54989E 01.36254E 01 2 2 4.6556 8.3282 -.14284E-02.31818E-01 -.20520E 00 2 3 6.3530 11.3646.75585E-04.22411E-03.18858E-01 2 4 8.0772 14.4489 -.29488E-05 *84817E-06 -.12413E-02 2 5 9.8143 17.5564.10231E-06.22551E-08.67304E-04 2 6 11.5578 20.6753 -.33582E-08.48244E-11 -.32427E-05 2 7 13*3050 23*8007 *11733E-09 *10752E-13.15848E-06 3 1 3.7747 6.7524 *14372E-01 *30726E 02 -.92788E 01 3 2 5.6455 10.0990 -.13881E-02.15757E 00.43390E 00 3 3 7.3020 13*0622.13081E-03.20234E-02 -.52428E-01 3 4 8.9953 16.0913 -.76884E-05,13477E-04 *45999E-02 3 5 10.7130 19.1641 *35711E-06 *55819E-07 -.31145E-03 4 1 4.7178 8.4395.90270E-02 *17568E 03.23793E 02 4 2 6*6.544 11*9038 -.12567E-02.63872E 00 -.86684E 00 4 3 8.2836 14*8182.18493E-03.11590E-01.11781E 00 4 4 9.9419 17*7847 -.16048E-04 *12905E-03 -.13443E-01 5 1 5.6540 10.1142 *61448E-02 *10418E 04 -.61'+35E 02 5 2 7.6679 13.7168 -.11077E-02,24715E 01 *17462E 01

- 97 TABLE IV EIGENVALUES AND MODAL CONSTANTS FOR MODES OF THE OR.DPR 2*-M+1,N A/B = *71439 R/4 =1.4-'0n M N KSIGMA Q R 0 1 8869 18104 *20433 003.21 85E 0'0 *52232E 00 2- 2.489.1 5.0809 -.11217E- 02 *33!35EE-03 - 730758E-01' 3 40342 8.2348 *203 56E-04 o91175E-06 *18168E-02 e 4 5.5757 11.3814 -4 972" E-0C *22cE- 08-.98658E-04 5 71161 1t4.5257 *14144E-07.55043 E-11 *51374E-05 6 8.6560 17*669; -*45379E-09 *13619E-13 -26(828E-06 0 7 1 O *10. 1957 2 0.81 1 19675E -10 *5405E- 16 1749 8 E -07 t1 1. 7144 3,4994 *59358E-0 1 *15225E 01 -,18530E 01 1 2 33341. 638057 -17943E-02.1-53 2E-01 *13046E 00 1 3 48565 9.9134 *76157E-04 *77124E-04 -.12347E-01 1 4 6. 3867 13.0368 -30364E-0 5 37822E-06 *92893E-03 1 5 7.9205 16*1677 *12078E-06 *14942E-08 -*61653E-04 1 6 9.4561 19.3022 -.41053E-08 *37129E-11 *32123E-05 2 1 2925890 5.2848 26049E-01 *11814E 02 *t6457E 01 2 2 4.2398 8.6544 -.18622E-02 *8 922E-01 -33735E 00 2 3 5.7279 11.6919 ol5914 -07 1? 09~EE-02 44983E-01 2 4 7.2353 14*7689 -*10023E-0' *1,1872E-04 -*46094E-02 2 5 8.7553 17.8718 *54917E-06*754?2E-O7.38753E-03 2 6 10.2818 2090876 -.28268E-07.38670E-09 -28952E-04 2 7 11.8118 24.e1108. 14072E-0Q,17197E-11 *20008E-05 2 8 13.3441 27.2386 -.66388E-10 64642E-14 -.12651E-06 3 1 3.4621 7*0671 1 4322F-01 *90532E 02 -*17020E 02 3 2 5.1745 1C05623 -*16825E-02 *46592E 00.76526E 00 3 3 6.6377 13.5491.24340E-03 10707E-01 -.11809E 00 3 4 8*1160 16*5668 -.2348E-04- *16 98E-03 *16045E-01 3 5 9.6171 19.6308 e17572E-05 *16756E-05 -.16907E-02 3 6 11.1305 22.7201 -*11411E-06.12338E-07 *15145E-03 4 1 4.3268 8.8321 2 89377F-02 *72524E 03 *51691E 02 4 2 6.1164 12.4851 -1 4571E-02 *21C06E 01 -.17438E 01 4 3 7*5758 15.464';.3C459E-03.57876E-01 25854E 00 4 4 9.0245 18.4213 -*44309E-04 *14479E-02 -*43952E-01 4 5 10*5027 21*4386 *44369E-05 *21928E-04 *57605E-02 5 1 5.1843 10.5823 *60536E-02 *60786E 04 -e15891E 03 5 2 7.0535 14.3979 -.12414E-02.11821E 02.41278E 01 5 3 8e5309 17*4137.33603E-03.25449E 00 -.52732E 00

- 98 - TABLE V EIGENVALUES AND MODAL CONSTANTS FOR MODES OF THE ORDER 2*M+1- N A/B =.7692, B/A = 13000 M N K SIGMA Q RS 0 1 *7551 1*8181 *24110E 00,18217E 00 *65465E 00 0 2 2*1255 5.1177 -.19760E-02 *10941E-02 -.55507E-01 0 3 3.4362 8.2733.58258E-04 *77153E-05 *52531E-02 0 4 4.7433 11.4206 -.22910E-05 *49641E-07 -.45679E-03 0 5 6.0494 14.5653 *10459E-06 *30535E-09 *38071E-04 0 6 7*3551 17e7089 -.52357E-08 *18328E-11 -.30971E-05 0 7 8*6604 20*8518.26981E-09.10136E-13 *23992E-06 0 8 9*9656 23*9944 -.11111E-10 *32421E-16 -.14053E-07 1 1 1.5139 3.6451 *62588E-01 *29668E 01 -*27311E 01 1 2 2.9161 7.0212 -.26186E-02 *31620E-01 *22389E 00 1 3 4.2073 10*1300.18641E-03.56458E-03 -*32878E-01 1 4 5.5033 13*2503 -.12314E-04 *71301E-05 *39865E-02 1 5 6.8029 16*3794.79940E-06 *72615E-07 -*42576E-03 1 6 8.1043 19.5128 -.52260E-07.65465E-09.42307E-04 1 7 9*4067 22*6486.27804E-08,35340E-11 -.32294E-05 2 1 2.2995 5*5364.26875E-01 *34429E 02 *10321E 02 2 2 3.7543 9*0393 -.24168E-02 *27755E 00 -.61575E 00 2 3 5.0246 12.0978.33093E-03 *84208E-02 *11105E 00 2 4 6.2995 15.1674 -.35666E-04.19510E-03 -*18289E-01 2 5 7.5858 18.2645.32421E-05.31968E-05 *24829E-02 2 6 8.8783 21*3764 -.27223E-06.420C9E-07 -*29787E-03 2 7 10.1743 24.4969 *21913E-07 *47685E-09 *32949E-04 2 8 11*4725 27,6226 -.17166E-08.48644E-11 -.34365E-05 2 9 12.7722 30.7519 *13087E-09.45064E-13 o34022E-06 3 1 3.0762 7*4066.14629E-01 *41594E 03 -.39185E 02 3 2 4.6055 11.0887 -*20522F-02 *19506E 01,16573E 01 3 3 5.8744 14.1440.43038E-03.64321E-01 -.28307E 00 3 4 7.1265 17*1585 -.72677E-04 *24088E-02.58575E-01 3 5 8.3946 20*2118.92214E-05.62063E-04 -.10037E-01 3 6 9.6745 23*2936 -.99142E-06.11793E-05.14524E-02 3 7 10.9615 26.3921.97185E-07 *18240E-07 -*18775E-03 4 1 3.8435 9.2541.90675E-02 *53297E 04 *15041E 03 4 2 5*4510 13*1245 -*17015E-02.14254E 02 -.47188E 01 4 3 6.7416 16.2318.47257E-03.37531E 00.66548E 00 4 4 7*9789 19.2109 -.11494E-03 *18036E-01 -.14876E 00 4 5 9.2264 22*2147.20391E-04.70265E-03,31447E-01 4 6 10.4909 25.2591 -*28125E-05.19128E-04 -.54803E-02 4 7 11.7664 28.3303.33493E-06.40016E-06.82637E-03 5 1 4.6041 11.0854,58723E-02.71997E 05 -.58265E 03 5 2 6.2849 15.1323 -.13963E-02,.11434E 03.14114E 02 5 3 7.6114 18.3260.47452E-03.21252E 01 -.16163E 01

- 99 - TABLE VI EIGENVALUES AND MODAL CONSTANTS FOR tMODES OF THE ORDER 2*M+1,N A/B.8333 B/A =- 12000 M N K SIGMlA R S O 1.6056 1.8259 *29865E 00 o30210E 00 *86656E 00 0 2 1.7133 5 1657 -o36475EE-02 40233E-02 -o10440E 00 0 3 2.7608 8.3242.18560E-03.80783E-04.16659E-01 0 4 3 8049 11 4722 - 1251 E-04 * 14956E-0.5 -o 24-605E-02 0 5 4.8480 14.6174 o7143E-0o,26406E-07 o34766E-03 0 6 5.8907 17.7612 -oa82455E-07 4 5.3 E- 09 -4.7856E-04 0 7 6.9332 20 904- e 74 319- 0 76574.E-11 64800E-05 0 8 7o9755 2400471 -,70075E- O 1 Z. 2'CE-12 -*86952E-06 0 9. 0178 27o 1 86 733-82 E-i0.2'/-78 aE-14 ol2]459E-06 1 1 1.26388 368107 691 - (0 O C062 E C1 -o47891 E 01 1 2 2.4174 702887 -o8752E-02 o.1542E 00 42582E 00 1 3 3o4535 10o4126.47222E- 0 a,24922E- ~ -o93628E-01 1 4 4.487 5 5 1 5 C 2 -, 54 L 4 7 1'. 8E- 1 1 5 5.5244 16. o568.3 6 0 5E o 5E-05 - e35328E-02 1 6 6o5631 19o7886 -o71 432E- 06 o 7!:7CE- 06 o59572E-03 1 6 6 5 6 a63 1 7 c, 7 5 f-, o 7' 4 2. 0 (' 7 3 2 Ti 7 E- 5 o 5 5 I E ~ 0 3 1 7 7.6028 229233 <80241FE-07 3 25E-1 -.5278E-04 1 83 8.6431 26o0598 -o90151E-0 0 o7208E 10.14653E-04 1 9 9o6837 29o1974 o 5296' -09 < 0 6E - 1 2 1444E-05 2 1 1.926.9 5. 098 6, 291 79E-1...79E- 03L. 25413E 02 2 2 3.147" 9. 4885 -o.32319E-0 2'4123E 01 -.14642E'01 2 3 4o1839 12.61 49 o66543E-03 67040E-01 *30370E.00 2 4 5 2033 1506884 -o13085E-03 e49-02 E-02 -a7?1 2E-01 2 5 602282 1 87786 21706E-04- 1?' 9E-'03 1].8516E-01 2 6 7 2585 21 8853 - 32199E-05 74-062 E-05 - 38. 82E-02 2 7 8e2923 25o0021 o44979E-06 5 24332E-06 *72137E-03 2 8 9.3281 28e1252 -o60584E-07 Q 71664E-08 -o12818E-03 2 9 10,3652 3102523,79612E-08 019475E09 *21775E-04 2 10 11.40 34 34.3824 o 10177E- 0.48956E- 1 - 3 5 46E-05 3 1 2o5774 7.7713 15728E-01 o4, 6 0 4E 04 -,3 5 89E 03 3 2 3.8709 11.6712 -o25870E-02 1!7729E 02 54;989E 01 3 3 4.9309 14867 o72 7 72791E-3O e61040EF O -~89155E t 3 4 5.9448 17.9242 -*20748E-03 o42407E-01.23317E 00 3 5 6.9558 20.9726 o499883E-04 *3C329E-02 - o661 I-4E- 1 3 6 7.9746 24.0444 -o99003E-05.17115E-03 o16662E-01 3 7 8.9998 27.1354 17237E-05 *77774E-05 -,37194E-02 3 8 10.0291 30.2389 -.27728E-06.3C0206E-06 76082E-03 3 9 11.0612 33.3508 o42257E-07 o10440E-07 -14-597E-03 3 10 12.0953 36'4686 -.61778E-08 332941E-09 o26.649E-04 3 11 13.1307 39.5906.87086E-09 o96531E-11 -e46744E-05 3 12 14.1672 42.7158 -ol2193E-09.28086E-12 *81514E-06 4 1 3.2195 9.7071.96864E-02.11333E 06 o74709E 03

- 100 - M N K SIGMAQ R S 4 2 4.5806 13*8112 -*20404E-02 *25646E 03 -.22243E 02 4 3 5.6756 17.1126 *71172E-03 *57335E 01 *28394E 01 4 4 6.7004 20.2025 -.25862E-03 o32879E 00 -.62735E 00 4 5 7.7037 23.2274 *84924E-04 *27866E-01 *18493E 00 4 6 8.7096 26.2605 -.22743E-04.22132E-02 -.55172E-01 4 7 9.7240 29.3191 *50091E-05.13915E-03 *14588E-01 4 8 10. 7451 32.3977 -.96754E-6 7 129 E- 19-05 -o34435E-02 5 1 3.8!959 11.6259 *49745E-02.30901E 07 -.41180E 04 5 2 5.2778 15.9133 -.16052E-02 *41921E 04,94477E 02 5 3 6.4090 19.3240 *65171E-03.62263E 02 -*98654E 01 5 4 7.4564 22.4820 -.28167E-03 *25720E 01.17979E 01 5 5 8.4644 2.55210.11495E-03.19557E 00 -,47098E 00

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- 104 - TABLE IX EIGENVALUES AND MODAL CONSTANTS FOR MODES OF THE ORDER 2*M+1IN A/B = 1.,1000. B/A *9091 M N K SIGMA Q R 0 1.3849 1.8479.42675E 00 *81080E 00'15729E 01 0 2 1.1402 5,4737 -.12071E-0 71373E-01 773E- -.32691E 00 O 3 1.8439 8.8522,21354'E-02.19640E-01 *94679E-01 0 4 2.5573 12.2769 -.33379E-03 *2'8697E-02 -. 12366E-01 0 5 3.2753 15.7240.50352E-04 *27094E-03 *10907E-02 0 6 3.9943 19.1758 -.81207E-05 *21966E-04 -*85202E-04 0 7 4*7136 22.6292 1385.8E-05 *16518E-05 *62607E-05 0 8 5.4331 26.0832 -.24593E-06 11830E-0-6 -.44135E-06 0 9 6.1527 29.5377 *44795E-07 *81877E-08 *30194E-07 0 10 6.8723 32.9925 -*82792E-08 *55206E-09 -*20181E-08 0 11 7,5920 36*4475 *15453E-08.36986E-10.13431E-09 0 12 8.3117 39.9026 -*30345E-09 *27556E-11 -*99583E-11 0 13 9.0314 43.3578.41096E-10 *10253E-12 *36958E-12 0 14 9.7511 46*8131 -.12202E-10 *209c1E-13 -*75760E-13 0 15 10*4709 50.2684 -*26326E-11 *32529E-14 *11844E-13 0 16 11*1906 53*7240 *16106E-12.19732E-15 -.73387E-15 1 1 *9149 4.3921 *74068E-01.41806E 02.14425E 02 1 2 1.7354 8.3314 -*81451E-02 *77277E 00 -*16136E 01 1 3 2*4412 11.7199 *22896E-02 o10677E 00 *52529E 00 1 4 3.1335 15.0435 -.85235E-03 *34.289E-01 -.20304E 00 1 5 3.8349 18.4105.25726E-03 *88036E-02.46621E-01 1 6 4.5456 21.8227 -.59240E-04 *13080E-02 -.61900E-02 1 7 52602 25*2532.12779E-04 o14861E-03 *65723E-03 1 8 5.97-64 28.6915 -.27309E-05 *14806E-04 -*62728E-04 1 9 6*6935 32.1342 o58036E-06,13560E-05.55758E-05 1 10 7*4112 35.5797 -.12215E-06 1 1680E-06 -.46981E-06 1 11 8.1293 39.0273 *25239E-07 *95911E-08.37935E-07 1 12 88478 42*4763 -.50533E-08.75933E-09 -*29659E-08 1 13 9.5664 45*9264.95116E-09 *57946E-10 *22451E-09 1 14 10.2852 49.3773 -.15862E-09 *44045E-11 -.17038E-10 1 15 11.-004.2 52.8290.16394E-10 28303E-12.11048E-11 2 1 1.3985 6.7137.29217E-01 *32538E 04.14058E 03 2 2 2.2891 10*9894 -.52637E-02.23550E 02 -.92206E 01 2 3 3.0344 14*5673 *20894E-02.12493E 01.21891E 01 2 4 3.7388 17.9494 -.93098E-03.17147E 00 -.77654E 00 2 5 4.4280 21.2578 *44598E-03.50653E-01 o32828E 00 2 6 5.1212 24*5858 -.19170E-03.17340E-O1 -.10760E 00 2 7 5.8249 27.9642 *59136E-04.36825E-02.19984E-01 2 8 6.5348 31.3724 -.15453E-04.54373E-03 -*26869E-02 2 9 7.2476 34.7944.38197E-055 67152E-04.31266E-03 2 10- 7.9620 38.2241 -.91019E-06.74402E-05 -.33252E-04 2 11 8.6775 4-1.6589.20818E-06.76181E-06.33046E-05 2 12 9.3937 45.0974 -.44305E-07.73287E-07 -.31099E-06

- 105 - M N K SIGMA Q R S 2 13 10.1106 48*5389 *88875E-08 *66871E-08.27984E-07 2 14 10*8279 51o9823 -*15880E-08 *58410E-09 -o24373E-08 3 1 1.8701 8*9779.15326E-01 *28546E 06 o14207E 04 3 2 2.8164 13.52-08 -.34840E-02 *10278E 04 -a61657E 02 3 3 3.5948 17.2580,15713E-02 *30476E 02.10549E 02 3 4 4.3284 20*7799 -*87757E-03 *21884E' 01 -29871E 01 3 5 5.0348 24*1711 *49772E-03.29290E 00,11124E 01 3 6 -57243 27.4814 -.27123E-03 *74159E-01 -*48040E 00 3 7 6.4125 30.7850 *14174E-03 o27855E-01.19441E 00 3 8 7.1097 34.1322 -.55251E-04 *79581E-02 -,48958E-01 3 9 7*8150 37.5181 *16675E-04 14854E-0281320E3 10 8*5244 40.9239 -*45085E-05 o21962E-03 -.11143E-02 3 11 9*2361 44.3408,12513E-05 *28446E-04 13724E-03 3 12 9.9494 47.7649 -.24890E-06.33540E-05 -.15622E-04 4 1 2.3354 11o2118.94216E-02 o26781E 08.14631E 05 4 2 3.3302 15*9877 -*24602E-02.53871E 05 -o45708E 03 4 3 4.1363 19.8577.11918E-02 *10091E 04 o60050E 02 4 4 4.8919 23.4852 -.6971.7E-03.47173E 02 -o13254E 02 4 5 5,6207 2 69839 *455i7E-03.39424E 01.40633E 01 4 6 6.3293 30.o3859 -o29924E-03 52139E 00 -o15588E 01 4 7 7.0212 33.7075.17925E-03,11349E 00 *67688E 00 4 8 7*7070 36.9995 -.10271E-03.40624E-01 -*30342E 00 4 9 8.3985 40.3195.47807E-04 1.1l408E-01.99477E-01 4 10 9.0991 43.6827 -o15084E-04 *33764E-02 -.20437E-01 5 1 2.7967 13.4264.11246E-01 *26326E 10 o15255E 06 5 2 3.8350 18,4110 -o39448E-03 *31896E 07 -.36078E 04 5 3 4.6663 22.402C.10144E-02 o40160E 05 o38170E 03 5 4 5*4409 26.1205 -o56379E-03 o13209E 04 -,68830E 02 5 5 6.1861 29.6981.37215E-03.78421E 02 o17232E 02 5 6 6.9122 33.1838 -*26307E-03 o71915E 01 -.55185E 01 5 7 7.6226 36.5944.18763E-03.94894E 00,21472E 01 5 8 8.3178 39.9318 -.12103E-03.18397E 00 -.93886E 00 5 9 9.0033 43.22.3.82998E-04.57931E-01 43626E 00

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- 107 - M N K SIGMAQ R S 4 4 6.6710 24.1366 -.47982E-03 *28978E 00 — 12317E 01 4 5 7.6230 27.5810 o23537E-03 *57782E-01 o40523E 00 4 6 8*5899 31o0794 -o83961E-04 *12616E-01 -o80213E-01 4 7 9.5875 34o6888.18166E-04 e12763E-02 o69344E-02 5 1 3*8559 13.9511 *59558E-02.21394E 07.46695E 04 5 2 5.2778 19o0959 -o12840E-02 o28968E 04 -o12069E 03 5 3 6o4086 23*1871 *68329E-03 *43369E 02 *14565E 02

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- 109 - TABLE XII EIGENVALUES AND MODAL CONSTANTS FOR MODES OF THE ORDER 2*-M+1,N A/B = 1.4000, B/A = *7143 M N K SIGMA Q R S 0 1 *6513 18613 *2.1753E 00 *26791E 00 o99716E 00 0 2 2.1435 6.1257 -.30371E-02 o12796E-01 -.59863E-01 0 3 3.6702 10.4885 *54778E-04 *91118E-04 o36395E-03 0 4 5.2072 14.8809 -.12690E-05 *35900E-06 -e13520E-05 0 5 6o7452 19'*2759 *34943E-07.11699E-08 o42793E-08 0 6 8.2835 23.6720 -11154E- ) 8.3772 E-7 1 1 - 1 358E- 10 1 1 1.*6557 4.7315 31.697E-01.11061E 01 o28017E 01 1 2 3.0942 8.8424 -28609F-02 o525 C7E -01 -o30827E 00 1 3 4.5342 12 9574 *18899 E-03 *1819cE-02 o89006E-02 1 4 -60o453 17 o27 59 -74036E-05 o15711E-O04 -*67966E-04 1 5 7.5699 21.6327.29028E-06.89553E-07 *36z450E-06 1 6 9.0998 26.0048 -.11568E-07 4C0989E-09 -.16076E-08 1 7 10o6326 30.3851 o45835E-09.15685E-11 59988E-11 2 1 2.5781 7.3675.12387E-01 o64708E 01 o77497E 01 2 2 4.1007 11.7186 -,20359E-02.1241 9E I O -o71568E 00 2 3 5o4545 15.5 875 o,36604E-03 l 1324E-01.73027E-01 2 4 6.9225 1c %782.6 -24-033E-O 2387 zE- 03 - 11768E-02 2 5 8o426C 24.C792 ol2960E-05 22E'!E-05 o10291E-04 2 6 909424 28.4128 -.65982E-07 <15795E-07 -.67208E-07 3 1 3.4605 9.8892 o63704E-02 o46715E 02 o22309E 02 3 2 5.1105 14.6044 -ol5464E-02 352C8E CO0 -ol4859E 01 3 3 6.4265 1. 3654.4 6790 3 oiE-03 oV8E- 01.27441E 00 3 4 7.8334 22.3858 -.o57274E-O04'.4566E-02 < -o10963E-01 3 5 9.3095 26.6042 e41508E-05 o3102E-4 n15358E-03 4 1 4.3266 12 3642.38241E-02 *37043.E 03.6256E 02 4 2 6o.0948 17.4173 -.11876E-02,1334r-8 01 -o31419E 01 4 3 7.4353 21.2483 o45637E-03 o 8036E'-01.61497E 00 4 4 8.7766 2500811 -o10783E-03.97436E-02 -.63475E-01 5 1 5.1842 14.8152 *25444E-02 o30740E 04.20000E 03 5 2 7.0480 20.1412 -.88287E-03.623C7E 01 -o68661E 01 5 3 8.4543 24.1603 42059E-03 o20942E 00.11785E 01 5 4 9.7538 27.8739 -.15636E-03.30473E-01 -.22738E 00

- 110 - TABLE XIII EIGENVALUES AND MODAL CONSTANTS FOR MODES OF THE ORDER 2*,,+ LtN A/B = 15000 B/A * 6667 M N K SIGMA Q R S O 1.6948 1.8643.19403E 00 *22838E 00.94810E 00 0 2 2*3834 6.3952 -.17779E-02 *64457E-02 -*30574E-01 0 3 4*1314 110O858.18973E-04 *17691E-04 *72185E-04 0 4 5*8858 15.7933 -.28134E-06.29047E-07 -.11241E-06 0 5 7.6409 20.9027 *49249E-08 *38643E-10 14563E-09 0 6 9.3964 25.2132 -*76276E-10 *29471E-13 -*10930E-12 0 7 1101508 29*9207 -*13979E-10 *31903E-14.11708E-13 1 1 1*7848 4*7892.27123E-01 *70657E 00 *23060E 01 1 2 3.3675 9.0359 -*21009E-02 *33366E-01 -*20149E 00 1 3 5.0311 13.4998 *72198E-04 o40498E-03 *19756E-02 1 4 6.7603 18*1397 -o17832E-05,14309E-05 -*62760E-05 1 5 8*5020 22.8132 *44796E-07.34094E-08 *14162E-07 1 6 10.2490 27.5011 -e11573E-08 *66398E-11 -*26668E-10 2 1 2.7998 7.5127 a10619E-01 e28453E 01 *53971E 01 2 2 4.4263 11,8771 -o16272E-02 *76800E-01 -.50576E 00 2 3 5.9849 16.0593.15926E-03 *33521E-02 *19249E-01 2 4 7.6744 20.5927 -62312E-05 o2.4398E-04 -*11986E-03 2 5 9.3949 25.2091 *21478E-06 *97311E-07 *44180E-06 3 1 3.7691 10.1136.54657E-02.14137E 02 *12983E 02 3 2 555099 14.7846 -.12259E-02.16205E 00 -*96376E 00 3 3 6.9865 18,7467 o25530E-03,14700E-01.98438E-01 3 4 8.6224 23.1363 -o15952E-04 *22696E-03 -*12421E-02 4 1 4.7167 12.6563 o32667E-02 o78602E 02.32250E 02 4 2 6-5877 170.6766 -.97264E-03.40879E 00 -*17830E 01 4 3 8*0320 21.5521.30833E-03 o38995E-01.29389E 00 4 4 906000 25*7594 -*33371E-04 *13856E-02 -*84608E-02 4 5 11.2617 30.2183.20087E-05.14365E-04.76772E-04 4 6 12.9565 34.7659 -.10214E-06 o88790E-07 -*43934E-06 5 1 5.6538 15.1707.21502E-02.46216E 03.81718E 02 5 2 7.6418 20.5051 -.76749E-03 *12621E 01 -.33426E 01 5 3 9.1097 24*4438.30522E-03.77710E-01.59942E 00

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