Journal of Ship Research, Vol. 36, No. 2, June 1992, pp. 154-167 Dynamic Characteristics of a Submerged, Flexible Cylinder Vibrating in Finite Water Depths A. Ergin,' W. G. Price,' R. Randall, 2 and P. T e m a r e l 3 This paper presents experimental data and theoretical predictions of the dynamic characteristics (natural and resonance frequencies, mode shapes) of a flexible cylinder vibrating in air and at fixed positions below a free surface in water of finite depth. The flat-ended, thin cylindrical shell of overall length 1284 mm, external radius 180 mm, thickness 3 mm is made of mild steel. In the experiments, the shell was tethered (i) at 0.21, 0.23, and 0.68 m depths below the free surface in water of depth 1.6 m and (ii) at 0.25, 1.5, and 3.5 m depths in 4 m of water. The resonance frequency data recorded provide measures of the influences of free surface, cylinder position, rigid boundary, water depth, etc. occurring in the fluid-structure interaction process. The theoretical predictions are derived from a three-dimensional hydroelastic mathematical model which, through the calculations of the generalized fluid Ioadings, accounts for the influence of free surface and rigid boundaries, position of submerged cylinder, neutral buoyancy or, as in the present case, with tethers and buoyancy effects. An extensive comparison of results is included. The experimental restrictions of water depth, cylinder position, etc. and the fluidstructure interactions are assessed and illustrated through the calculated resonance frequency values. 1. Introduction SITUATIONSin which flexible structures are in contact with a fluid are commonplace. The response of ships excited by an irregular seaway, aircraft in flight, water-retaining structui'es (that is, dams, storage vessels, etc.) under earthquake loading and the behavio~ of flexible pipelines in normal operation are just a few examples. Wherever the contact occurs the presence of the fluid invariably modifies the dynamic behavior of the structure; consequently, the fluidstructure interaction process has been extensively studied [see, for example, Bishop et al (1979, 1980), Hosoda et al (1989), Price et al (1988), Huang (1986), Muthuveerappan et al (1979)]. There are many factors affecting the dynamic response of a submerged flexible structure, but in cases where the specific acoustic impedance of the fluid approaches that of the structural material, the crux of the problem lies in defining the applied fluid loading. When a structure is shaken in a fluid the pressure loadings are not known a priori but depend on the structural surface motions. The equations of structural and fluid motion are inexorably linked and must be solved simultaneously unless certain assumptions are made to uncouple them. Junger (1952), Bleich and Baron (1954), and Warburton (1961) obtained results for the vibration of infinitely long cylinders surrounded by or containing a fluid by matching solutions at the interface for standing waves in the fluid with the vibrations of the cylinder. These analytical methods produce solutions at every point in the spatial continuum and are therefore necessarily limited to a few specific cases in which the wet surface of the submerged body can be described in terms of a single coordinate. When geometries become more complicated or when the fluid environment cannot be considered as an infinite do1Department of Ship Science, Southampton University, Highfield, Southampton, U.K. 2Royal Naval Engineering College, Manadon, Plymouth, Devon, U.K. 3Depa_~nent of Mechanical Engineering, Brunel University, Uxbridge, Middlesex, U.K. Revised manuscript received at SNAME headquarters Aug. 22, 1991. 154 JUNE 1992 main, alternative solutions must be sought. The so-called doubly asymptotic approximation (DAA) [Geers (1971)] which enables the radiated pressure to be specified in terms of surface motions has been extensively employed. This is an amalgam of the plane wave approximation (PWA) [Mindlin & Bleich (1953)] and the virtual mass approximation (VMA) [Chertock (1970)] which approach the exact solution at high and low frequencies, respectively. The total pressure field surrounding any point on the submerged body is divided into components associated with the incident, reflected, and radiated pressures. The PWA provides the initial high-frequency response of the interaction and the VMA adds the low-frequency response. The advantage of this approach lies in its inherent simplicity and computational efficiency against which must be weighed a certain level of inaccuracy and uncertainty as to the frequency range over which that inaccuracy exists. The first-order doubly asymptotic approximation (DAA1) [Geers (1978)] generally overpredicts the fluid resistance at intermediate frequencies and thus overdamps the structural vibration. Refinements DAA2c, DAA2m [Geers & Felippa (1983)] and the Inertial Damping Collocation Approximation (IDCA) [DiMaggio et al (1978)] have been proposed and these produce improved results for problems such as the response of a submerged spherical shell. Despite the uncertainties which surround them, DAA models have received widespread use and are often utilized in conjunction with discrete-element representations of structures, introducing yet more inaccuracy and uncertainty into the overall approach describing the dynamic behavior of the fluid-structure interaction. A third alternative is the three-dimensional hydroelastic approach which will be adopted here; this has been discussed in detail elsewhere [Bishop et al (1986)]. This analysis consists of two parts: first, the dry or in-vacuo analysis in which the structure vibrates freely in vacuo in the absence of any structural damping or external force; and second, the wet analysis introduces the fluid actions which are applied as an external loading to the flexible structure. This paper presents results obtained from analytical, experimental, and numerical methods and these describe the 0022-4502/92/3602-0154500.53/0 JOURNAL OF SHIP RESEARCH d y n a m i c b e h a v i o r of a t h i n c y l i n d r i c a l shell of f i n i t e l e n g t h w h i c h is s e a l e d a t b o t h ends. T h e n u m e r i c a l m e t h o d can be a p p l i e d to a n y s h a p e of body and h a s b e e n successfully u s e d for d i f f e r e n t k i n d s of s t r u c t u r e s such as S W A T H (small wat e r p l a n e a r e a t w i n hull) a n d j a c k - u p r i g s [Fu & P r i c e (1987), F u et al (1987), Price et al (1985)] t r a v e l i n g in i r r e g u l a r seaways. It c a n a c c u r a t e l y predict t h e d y n a m i c b e h a v i o r of t h e s t r u c t u r e as w e l l as t h e i n d u c e d l o a d i n g a n d s t r e s s e s a t a n y p o s i t i o n in t h e s t r u c t u r e . T h e c y l i n d r i c a l shell u n d e r invest i g a t i o n was chosen to d e m o n s t r a t e t h e a p p l i c a b i l i t y of t h e t h e o r y to s t r u c t u r e s such as s u b m a r i n e p r e s s u r e h u l l s a n d offshore pipelines, a n d b e c a u s e of t h e a v a i l a b i l i t y of d e t a i l e d e x p e r i m e n t a l d a t a d e s c r i b i n g its v i b r a t i o n c h a r a c t e r i s t i c s in air and in water. The t h i n cylindrical shell is of overall l e n g t h L = 1284 m m , e x t e r n a l r a d i u s R = 180 m m , shell t h i c k n e s s T = 3 m m a n d is m a d e of m i l d steel. T h e a s p e c t r a t i o s L / R a n d T / R w e r e selected to be r e p r e s e n t a t i v e of a part i c u l a r class of s u b m a r i n e hull, t h o u g h in no w a y does t h i s shell t r u l y m o d e l a r e a l s u b m a r i n e s t r u c t u r e . E x p e r i m e n t s w e r e performed in t o w i n g t a n k s of fixed w a t e r d e p t h s and t h e c y l i n d e r was p o s i t i o n e d h o r i z o n t a l l y a t diff e r e n t d i s t a n c e s below t h e free surface. T h i s a l l o w e d t h e inf l u e n c e s of c y l i n d e r ' s position, w a t e r depth, free s u r f a c e a n d b o t t o m b o u n d a r i e s on t h e d y n a m i c c h a r a c t e r i s t i c s to be assessed. In all tests, t h e c y l i n d e r was k e p t s t a t i o n a r y , t h a t is, w i t h zero f o r w a r d speed, a n d forced into v i b r a t i o n by a mec h a n i c a l exciter. I n t h e dry a n a l y s i s p h a s e of t h e h y d r o e l a s t i c a p p r o a c h t h e s t r u c t u r e was idealized by a f i n i t e - e l e m e n t m o d e l a n d t h e s e r e s u l t s ( n a t u r a l frequency, m o d e shape, etc.) w e r e c o m p a r e d w i t h t h e c o r r e s p o n d i n g e x p e r i m e n t a l data. A t t e m p t s w e r e m a d e to m o d e l m a t h e m a t i c a l l y t h e r e a l i t y of t h e e x p e r i m e n t by a c c o u n t i n g for t h e effects of f i n i t e w a t e r depth, b o u n d a r y , t e t h e r s , etc., b u t not viscosity, in t h e e v a l u a t i o n of t h e fluid l o a d i n g s in t h e w e t analysis. T h i s p r o v i d e s i n s i g h t into t h e m a g n i t u d e s a n d r e l e v a n c e of t h e v a r i o u s effects i n f l u e n c i n g t h e p r e d i c t i o n s a n d allows a r e a l i s t i c c o m p a r i s o n b e t w e e n experimental measurements and theoretical estimates. Nomenclature a = generalized mass matrix a~ = an element of generalized mass matrix, a A = generalized added mass matrix A(ao) = generalized added mass matrix evaluated at infinite frequency A(¢o) = frequency dependent generalized added-mass matrix A(¢o) = A(¢o) + A(~) A~. = an element of generalized added-mass matrix, A A~8(:¢) = an element of generalized added-mass matrix evaluated at infinite frequency, A(~) Arc(Co) = an element of frequency-dependent generalized added-mass matrix, A(¢o) A . , A . A , , , = amplitudes of component vibrations b = generalized damping coefficients matirx B = generalized hydrodynamic damping coefficients matrix B~ = structural damping matrix B(oo) = generalized hydrodynamic damping coefficients matrix evaluated at infinite frequency B(¢o) = frequency-dependent generalized hydrodynamic damping coefficients matrix B(o~) = B(¢o) + B(~) B~, = an element of generalized hydrodynamic damping coefficient matrix, B B,,(~) = an element of generalized hydrodynamic damping coefficients matrix eval- JUNE 1992 Br,(¢o) = c = C = D det D (det D)* F h(v) = = = = = hra(~) = h*(~) = h~*(,) = H = HA~) = = H~(~) = K = L = L/R,T/R = m = M = n = N = uated at infinite frequency, B(~) an element of frequency-dependent generalized hydrodynamic damping coefficients matix, B(¢o) generalized stiffness matrix generalized fluid stiffness matrix characteristic matrix determinant of matrix D complex conjugate of (det D) forcing amplitude vector generalized velocity impulse response functions matrix an element of generalized velocity impulse response functions matrix, h(v) modified generalized velocity impulse response functions matrix an element of modified generalized velocity impulse response functions matrix, h*(~) distance between body and wall a component of Fourier transform real part of component of Fourier transform imaginary part of component of Fourier transform structural stiffness matrix overall length of cylindrical shell aspect ratios number of axial half-waves structural mass matrix number of circumferential waves number of principal coordinates p(t) = principal coordinates vector ps(t) = sth principal coordinate P = external forces vector Q(t) = generalized fluid-structure interaction and all other external forces vector R ~ ( t ) = rth generalized external force due to sth reponse of structure r = integer index R = external radius s = integer index t = time T = shell thickness u = mode vector u = displacement component in axial direction U,(/,U = structural displacements, velocities, and accelerations vectors respectively v = displacement component in tangential direction w = displacement component in radial direction x = axial coordinate distance xo,Yo,Zo = defines position of mechanical forcing oscillator ~r = rth structural damping factor p = density of fluid (b = angular coordinate -~(t) = generalized excitation force vector -~o = amplitude of generalized excitation force ¢o = circular frequency cot = natural frequency of dry structure (¢Or)~ = resonance frequency in air or natural frequency in vacuo (fOr)we,= resonant frequency in water = time variable JOURNAL OF SHIP RESEARCH 155 2. Experimental study A detailed experiment was devised to collect data describing the vibration characteristics of the thin cylindrical shell in air and water. These data are used for comparisons with theoretical predictions. It was appreciated that geometrical irregularities in the cylinder, such as large variations in the circularity of the cross section or nonuniform wall thickness, would produce poor definition in the frequency response data at or near resonance; therefore the cylinder was manufactured to exacting tolerances and its dimensions were only limited by the manufacturing process and the need to place transducers inside the shell. Fifteen measurement sites were selected at positions where the vibration patterns of a cylindrical shell consisting of standing waves in both the axial and circumferential directions could be adequately defined. Eight measurement sites were located at 45-deg intervals around the circumference at a cross section where the lower mode shapes display significant amplitudes of motion, that is, at a distance 552 mm from one end. The remaining seven measurement sites were located at regular intervals in line along the length of the cylinder. Each site was fitted with an accelerometer, mounted on a stud inside the shell, recording radial motion. Attempts were made to record the motion of the flat end plates. This was more difficult due to the extra thickness of the sealing plates and glands and the limited energy which is delivered to the ends when shaking the structure near the center. A single-point excitation was applied in a vertical direction in the plane of the eight accelerometers using an electromagnetic shaker driven through a flexible push rod into a piezoelectric force gage. Two series of tests were performed in two separate water tanks at the Royal Naval Engineering College, Manadon. In the first tank with dimensions 7.5 m long, 1.7 m wide and containing water to a depth of 1.6 m, the cylinder was tested at various depths of immersion. The experimental results of the dry analysis and the wet analysis for 0.21, 0.23 and 0.68 m distances below the free surface in the first water tank are tabulated in Table 1. In a later series of tests the problem of measuring the response of the ends was addressed. A shaker was installed inside the cylinder so that measurements could be obtained for greater submerged depths and to provide a means of applying the force directly to the ends. The disadvantages of this experimental procedure lie in the difficulty to isolate the response of the shaker mounting system from the cylinder's response, and the total force input was not measured by the force gage. This was satisfactorily resolved by providing a very stiff shaker support structure manufactured from laminated carbon fiber sheets and attaching it to the cylinder near the junction of the flat ends to the curved cen- tral section--this being a position of minimum movement. By measuring the response of the shaker support structure before and after attachment and comparing the results with the previous series of tests it was possible to establish the spurious results due to shaker interaction and to eliminate them. These tests were performed in the deep end of the towing tank in a section measuring 4 m long by 4 m wide by 4 m deep. The results from the receptance measurements for a position near the original driving point are shown in Table 2 for 0.25, 1.5 and 3.5 m depths of immersion from the free surface. The signal-to-noise ratio for this point is less than in the previous tests due to the modes of vibration not being directly excited. Although the resonance frequencies shown in Tables 1 and 2 are of similar order of magnitude, their sensitivity to tank depth and immersion depth as well as to experimental procedure is notable. Generally, the effect of submerging the cylinder in water was to reduce the appropriate resonance frequency by a factor of approximately 0.5; however, this decrease is not uniform for all mode shapes. Modal properties (that is, resonance frequency, damping factor, and modal constant) were extracted from the frequency response data using a straightline technique [Dobson (1987)] which is particularly accurate at separating closely spaced modes. Variations in resonance frequency confirm the anticipated trends of increasing resonance frequency as the depth of immersion decreases. Although there is evidence of twin peaks at resonance caused by the circularity of the tube being less than ideal, the modes are well defined with low damping loss factors (in the order of 10-4). The test model was not designed specifically to monitor the response of the ends. There was a hatch plate at each end and the transducer connection cables fed in through a central gland. The hatches were screwed on to the ends and then glued with a rubber sealing compound to prevent the ingress of water. Consequently, the ends were not uniform although they were manufactured from the same gage steel as used for the curved portion. The aft-end plate did not contain any gland openings and generally produced better receptance data. The end modes are characterized by large loss factors (an order of 10 -2 ) probably due to the rubber around the sealing hatch. In the experiments, the cylindrical shell was not neutrally buoyant and therefore was held in position in the water by four light tethers attached from fixed positions on the floor of the tank to diametrically opposite positions on the circumference at the ends. Thus, the tethering system created a port-starboard symmetry with the vertical plane through the cylinder's axis lying in the plane of symmetry. 3. Analytical approximations Arnold and Warburton (1953) investigated the vibrations of finite length, thin cylindrical shells with freely supported Table 1 Experimental results of dry analysis and wet analysis for various depths of immersion from free surface in 1.6 m deep tank (*see Fig. 1) Dry Expr. (co,),#y (Hz.) 194.0 198.0 241.4 300.6 336.6 387.0 403.0 156 JUNE 1992 0.21 m, (¢o,),~t (Hz.) 101.6 113.4 122.4 153.6 203.0 220.5 243.6 0.23 m. (co,),,~, (Hz.) 98.4 109.4 120.6 149.5 201.5 218.6 242.0 0.68 m. ((o,)w,, (Hz.) 97.5 108.7 118.3 149.1 200.9 217.0 241.3 Modes m-n 1-2 1-3 End 1" End 2" 1-4 2-3 2-4 Table 2 Experimental results of a wet analysis for various depths of immersion from free surface in 4.0 m deep tank 0.25 m. 1.50 m. 3.50 m. Modes (o~,)~,, (o~,)~,~ (0~,),,,~l m-n (Hz.) (Hz.) (Hz.) 99.4 109.9 122.1 141.1 199.4 214.1 240.1 97.1 107.9 118.9 141.0 198.9 215.4 238.9 96.4 106.7 114.4 128.6 195.9 216.3 239.2 1-2 1-3 End 1 End 2 1-4 2-3 2-4 JOURNAL OF SHIP RESEARCH and fixed-end conditions in vacuo. Assumptions were made in order to restrict the problem to the linear domain (that is, the displacements are small in comparison to the thickness of the shell). Furthermore, the Kirchhoff hypothesis was assumed. Three displacements components in the axial u, tangential v, and radial directions w are assumed to vary harmonically for the freely-supported end conditions and are expressed in the form: u cos ,cos v=Avsin(m~)sin(n6)cos(~t) Mathematical + model BdU + KU = P Analytical In Water (co,),,,,, 100.0 110.0 203.5 219.0 243.0 Numerical In Vacuo (co,)a,r 197.2 203.5 349.6 391.2 412.2 Numerical In Water (¢o,),.~t 99.3 114.7 217.8 225.6 261.0 Modes m-n 1-2 1-3 1-4 2-3 2-4 MLI + KU = O (3) The dynamic characteristics (that is, mode shapes, natural frequencies, etc.) of the structure are determined from this equation. In the hydroelasticity theory proposed by Bishop and Price (1979), equation (2) may be written in terms of principal coordinates p(t). The (N x 1) principal coordinates matrix, p(t), is a solution of the equation ap(t) + bp(t) + cp(t) = Q(t) (4) where a, b and c denote (N x N) generalized mass, damping, and stiffness matrices, respectively. The (N x 1) generalized force vector Q(t) describes the fluid-structure interactions and all other external forces (that is, wave forces, etc.) and N indicates the number of principal coordinates required to describe the responses of the structure to an arbitrary generalized external force. It has been shown that this generalized external force matrix, Q(t), may be expressed as Q(t) = -(Ap(t) + Bib(t) + Cp(t)) + ~(t) (5) where A, B and C are (N x N) generalized added mass, generalized hydrodynamic damping, and generalized fluid stiffness matrices, respectively, and .~(t) denotes a (N x 1) generalized external force vector caused by waves, mechanical excitation, etc. It is well known that for surface vessels oscillating with forced frequency co in calm water or excited by sinusoidal waves of frequency co, the hydrodynamic coefficients contained in the elements of A and B vary with frequency (see, for example, [Gerritsma & Beukelman (1964)]). Therefore, a more general expression to describe the generalized external force was proposed by Bishop, Parkinson, and Price (1977) in the form F Q ( t ) = R ( t ) +~.(t) = - / +m h(T) p ( t - x ) d r +~(t) (6) 3- where h(x) denotes the (N x N) generalized velocity impulse response function matrix [see also Cummins (1962)]. Here, an element Rrs(t) represents the rth component of the generalized force due to the sth response of the structure and is given by +~ Rrs(t) = - f_~ hr~(7)[gs(t - ~)d~ (2) where M, Ba, K denote the mass, structural damping and stiffness matrices, respectively. The vectors U, U and [J represent the structural displacements, velocities, and accelerations, respectively, and the column vector P denotes the external forces. In an in-vacuo analysis, the structure is assumed to vibrate freely in the absence of any structural damping and external forces reducing equation (2) to the form JUNE 1992 Analytical In Vacuo (co,)~ 198.6 203.5 349.5 392.6 412.0 (1) The equation of motion describing the response of a flexible body to external excitation is given by [Bishop et al (1986)] MU Analytical and numerical calculated frequencies (Hz; 9.0 m below free surface in water of infinite depth) . where A,, A~ and A~ are amplitudes of the component vibrations; m and n define the nodal arrangements alorig the cylindrical shell and around the circumference, respectively. The parameters x, 00, and co represent the axial coordinate distance, angular coordinate, and circular frequency, respectively, and t denotes time. In this analytical approach with the freely supported end conditions, it is assumed that the ends of the cylindrical shell remain circular and this approximates, in some measure, the conditions applying at the ends of the experimental cylindrical shell. That is to say, for this thin finite length structure with both ends sealed, the end plates force both ends of the shell to remain circular. This causes the natural frequency values to increase in comparison with those derived for a cylindrical shell having perfectly free ends. In Warburton's (1961) model, the displacements of the shell were expressed in cylindrical coordinates. It is assumed that the fluid is inviscid and the motion is irrotational, the fluid fills an infinite acoustic domain, and the fluid flow is described by a velocity potential function. Therefore, in this idealization, the pressure on the cylinder acts in the radial direction. One of the assumptions introduced into this theoretical study is that the shell is infinitely long, hence the effects of the end plates are not included in the calculations. In addition, when comparing these results with experimental data it should be remembered that here the cylinder is neutrally buoyant, with no tethers attached and no boundaries (that is, free surface, tank wall, or bottom) present. The natural frequencies in vacuo and resonant frequencies in water corresponding to each mode shape were calculated by solving the analytical equations [Warburton (1961), Arnold & Warburton (1953)1. These results are given in Table 3. 4. Table 3 (7) where hr,(T)=0 if,<0and hrs(~)Ps(t - ~) = O i f ~ > t For a sinusoidal response, ps(t) = [gse i~t, equation (7) may be written in the form [Price & Wu (1989)] JOURNAL OF SHIP RESEARCH 157 R,~(t ) = - known [see, for example, Janardhanan, Price, and Wu (1992)], the principal coordinate response p(t) may be determined. As shown by these authors, the form and characteristic of h*(x) vary considerably from one type of structure to another, reflecting their different dynamic behaviors. hr~(T)e-i~dT p s e i~t = ps(t)Hrs((o) = ps(t){Hrn,(¢o) + i ¢oH~(~o)} = ps(t)HR(o)) + ps(t)H~s(to). (8) The Fourier transform f - I h,£r) cos to~d~, j_ ¢oH1~s(tO) = hrs('r) sin tordr (9) where HR(to) and H~rs(to) denote frequency dependent components and respresent the generalized damping coefficients (that is, in phase with velocity) and the generalized addedmass coefficients, respectively, [A~(¢o) =- - H~,(¢o) and Br~(to) =- -HrRs(o~)]. It follows that equation (7) may be rewritten so that R~(t) = -Ars(°°)[gs(t) - - B~(~)p~(t) - C~p~(t) h*8($)p~(t - T)dr (10) where h*(T) represents a component of the modified generalized velocity impulse response function matrix. The elements Ars(oo) and B,~(oo) represent components of the generalized added mass and the generalized damping coefficients matrices evaluated at infinite frequency. These terms are extracted from the frequency dependent data to ensure that the impulse response functions have Fourier transforms. Thus, equation (4) may be rewritten in the form [a + A(oo)] p(t) + [b + B(oo)] p(t) + [c + C] p(t) + J - i h*(r)p(t - r)dr = E(t) (11) For a generalized excitation, =~(t) = ~0 ei~t, we seek a solution in the form of p(t) = p o ei°'t and after a slight rearrangement this satisfies the equation [a + .~(¢o)] p(t) + [b + 13(to)] p(t) + [e + C] p(t) = -=(t) (12) 5. Numerical analysis Dry analysis In the dry analysis the structure is assumed to vibrate freely in vacuo in the absence of any structural damping or external force and this allows the dynamic characteristics (that is, natural frequencies, mode shapes, etc.) of the flexible structure to be calculated. The finite-element method [Nastran (1985)] was adopted for this purpose. The structure was discretized with eight-noded quadrilateral shell elements which can carry membrane, bending, and transverse shear loads. This element was chosen in contrast to a fournoded quadrilateral element because of the faster convergence of the numerical process. In a preliminary calculation, 256 elements were distributed over the whole structure with 128 elements confined to the cylindrical shell and 64 elements over each end. The distribution over the curved shell consists of 16 equally spaced elements around the circumference and 8 equally spaced elements along the cylinder. To test the convergence of the calculated dynamic properties, the number of elements was increased to 768--32 equally spaced elements around the circumference, 16 equally spaced elements along the cylindrical shell, and 128 elements over each end. The differences in the results, given in Table 4, were practically negligible (except for those specifically related to the ends) confirming the mathematical convergence of the finite-element procedure. The convergence of the end vibrations was checked according to Price, Randall, and Temarel (1988). In a further analysis involving the 768-element model, the stiffness of the end plates was increased twofold. This caused approximately 0.5 percent difference in the natural frequencies of the cylindrical shell with the exception of the end-related vibrations. For these element distributions, Table 4 displays the calculated natural frequencies obtained from the Nastran finite-element package. The results occur in pairs, apart from those associated with the end plates. That is, to each natural frequency, there exists a pair of mode shapes satisfying the relevant orthogonality condition. These are shown in Fig. 1, with all the dynamic characteristics of the shell scaled to a generalized mass of 0.001 (kg m 2 ) . or Dp(t) = ~=(t) where .~(o~) = A(o)) + A(oo) and ]3(to) = B(to) + B(oo). That is addI) p(t) = - E(t) det D adjD=~(t ) - - (det D)* Idet DI2 (13) where the asterisk denotes a complex conjugate expression. From the resonance behavior of p(t) associated with the minimum value of Idet D I the resonance frequencies of the cylinder forced into vibration in the fluid by the mechanical oscillator may be deduced. Note that equation (11) is not restricted to sinusoidal inputs or outputs but is valid for an arbitrary excitation (that is, random, transient, etc.). Provided the form of h*(~) is 158 JUNE 1992 Table 4 Natural frequencies of thin cylindrical shell (NASTRAN) 256 Elements 768 Elements 768Elements" (Hz.) 197.44 197.44 203.81 203.81 198.6 206.61 350.84 350.84 393.85 393.85 416.15 416.15 (Hz.) 197.21 197.21 203.55 203.55 212.39 220.61 349.56 349.56 391.18 391.18 412.21 412.21 (Hz.) 198.19 198.19 203.79 203.79 276.16 287.59 349.71 349.7 I 392.8 I 392.81 412.86 412.86 Modcs m-n 1-2 I-2 I-3 1-3 End 1 End 2 I-4 1-4 2-3 2-3 2-4 2-4 stiffness of end plains increased by twofold. JOURNAL OF SHIP RESEARCH z m Q~ ~'--; N .--=_ I I ' ~ " ' ~ ~ ' , ~ , ~ . ~ 197.21 (1t.'-) m=|,~2 s2,n=3 END 1 _ m~l,m2 m=l,n=3 391.18(ILL) 212 39 (HI.) =--~ _ ~= 197.21 (Hz.J 203.55(HL) END2 220.61 (Hz.) m~l.n~4 349.56 (Hz.) m=2,t~3 391.1X ttlz.) m=2,~4 412.21(llz.) m.2,wul 412.21(llz.) / i I d -~:---t-~-1 I ~-i L ~ I I c~ 23 z I0 "11 ¢/) _z m m .1- 20355 (Hz.) Fig. 1 m=l,n=4 J ~-i i--~ I-L-t II ?,49.56(Hz.) Distortion modes and natural frequencies of thin cylindrical shell in vacuo using eight-noded quadrilateral shell elements Wet analysis The wet analysis introduces the fluid actions which are applied to the flexible structure and treated as an external loading. A three-dimensional velocity potential analysis is employed with the wetted surface of the structure discretized by panels with an oscillating source potential of constant strength situated at the center of each panel [Bishop et al (1986)]. The theory includes the influences of the boundary constraints introduced by the free-surface disturbances and finite or infinite depths of water in the derivation of the applicable boundary-value problem [Inglis & Price (1980), Newman (1977)]. This approach is extensively discussed elsewhere [Bishop et al (1986), Inglis & Price (1980), Newman (1977), Price & Wu (1985), Wu (1984)] and is omitted here. The analysis produces the fluid structure interaction effects which are usually described in terms of addedmass, damping, restoring (fluid stiffness) force coefficients and additional fluid or other external excitations. Generalized hydrodynamic coefficients In the generalized equation of motion expressed by equation (12), the quantities associated with the dry structure (that is, generalized mass, structural damping, and stiffness, natural frequencies and mode shapes) are constants independent of the surrounding fluid medium whereas the generalized hydrodynamic coefficients [that is, added masses: A(oo), A(¢o), A(~o) and fluid damping: B(~), B(~o), 1~ (¢o)] vary according to the properties of the fluid (that is, density) and the presence or otherwise of boundaries (free surface, tank wall and bottom, etc.). --79m. ~ In the present calculations of the generalized hydrodynamic coefficients, the wetted surface of the shell was discretized using 768 panels, thus producing a one-to-one correspondence between structural element and hydrodynamic panel. No rigid boundaries--It is well known that when a body oscillates in an unbounded, inviscid fluid the generalized added-mass coefficients are constants independent of frequency and the generalized fluid damping coefficients are zero. In contrast, for a surface-piercing oscillating body or a submerged body oscillating close to the free surface, the generalized hydrodynamic coefficients exhibit frequency dependence because of the free-surface wave disturbances. These results were confirmed and demonstrated through calculations of the generalized hydrodynamic coefficients of the oscillating cylindrical shell immersed at different depths below the free surface of an infinitely deep unbounded fluid. The same mathematical model, incorporating free-surface effects, was adopted in all calculations. In the deeply submerged case (a 9 m depth below the free surface) the calculated added-mass values are constants independent of frequency giving, that is, A~(¢o) = A~(~) # 0, Ars(Oo) 0 and the generalized damping coefficients are all zero. This constant value characteristic is clearly shown in Fig. 2. As the cylinder is raised to 0.36 m from the free surface, the added-mass and damping coefficients exhibit frequency dependence (Figs. 3 and 4), whereas for the cylinder floating in the free surface, frequency dependence in the generalized hydrodynamic coefficients is clearly evident as shown in Figs. 5 and 6. However, in the latter, the numerical calculations display the occurrence of irregular frequen= R = 0.18 m. MODE SHAPE 1-2 MODE SHAPE 1-3 4oE-4] 0.0045" 35E-4 0.0040 ~ 30E-4 0.0035" ~ 25E-4" 0"00301 < ~,~ "20E-4" 0.0025" ooo2oI ~,~ 15E-4" 0.00151 < 10E-4" 5E-4 0.0010/ 2 4 6 8 IO 12 2 4 e} (tad/s) 6 8 IO 12 (e (tad/s) END 1 0.008 MODE SHAPE 1-4 30E-4 0.007 -~ 25£-4 0.006 ~-~20E-4 0.005' < tn ~ i5E-4 0.004' 0.003" ~ 10E-4' 0.002 5E-4' 0.001" 2 Fig. 2 160 JUNE 1992 4 6 e (~ (tad/s) 10 12 ' 2' 4' 6' 8' (o (r,,d/s) 10' 12' Generalized added-mass values [Art(O)) = Ar,(~) + A,,(co)] of thin cylindrical shell submerged at 9.0 m depth from free surface in water of infinite depth JOURNAL OF SHIP RESEARCH ~R 3 6 m T 0.0033" MODE SHAPE 1-3 R = 0.18 m. "MODE SHAPE 1-2 0.0032" 0.00210" E" ~ 0.0031" ~ 0.00216" ~ 0.00301 ~0.00214" 0.0029 < 0.0028 ~ 0.00212" < 0.0027 2 4 6 8 co (radls) i0 2 12 4 6 8 co (mdls) I0 12 MODE SHAPE 1-4 END 1 0.001600" 0.0055 0.001595" 0.001590 0.0050 0.001585 0.001580" ~< 0.0045' r~ ,,I 0.001575" t% < 0.0040" 0.001565" 0.001570" 0.001560" 0.0035 0.001555 2 4 6 8 i0 2 12 4 6 co (radls) Fig. 3 10 12 Generalized added-mass values [A,r(~) = A,,(~) + A,,(o=)] of thin cylindrical shell submerged at 0.36 m depth from free surface in water of infinite depth ~200E-4 ' ~ 17.5E-4" _•36,.. R = 0.18 m. MODE SHAPE 1-2 MODE SHAPE 1-3 6E-4 5~-4 ~_" 15.0E-4 Z ----.12.5E-, tO ~ 4E-4 ~m, 10.0E-~ i 3E-4" L) 7.5E-~ 0 Z 5.0E-4 ~. 8 2E-4 2.5E-4" IE-4 < 8 co (radls) Z '~-~' - 4' 6' 8' CO (rod/s) I0' 12' 4" 6' 8' i0' 12' CO (radls) END I 40E-4 "- 35E-, "~I16E-5 Z 30E-~ 14E-5 z ~ 12E-5 ~j 25E-4 MODE SHAPE 1-4 10E-5 "~---10E-5 20E-4" 0 8E-5 O 15E-4" 6E-5" ~- 10E-4 3= ,~< 5E-4 4E-5" < 2E-5" 2 Fig. 4 JUNE 1992 4 6 0 co (tad/s) I0 12 - 6' 8' co (tad/s) 10' 12' Generalized hydrodynamic damping coefficient values [B,,0,') = B,,(,,~) + Bm(oc)] of thin cylindrical shell submerged at 0.36 m depth from free surface in water of infinite depth J O U R N A L OF SHIP RESEARCH 161 R- MODE SHAPE I-2 0.0019" 2 MODE SHAPE 1-3 0.0018: 22.5E-4" 0.0017 20.0E-4" 0.0016 < 0.18 m. 0.0015 ~ ~ ~ B ~ 17.5E-4 ~ 15.0E-4 e~ 0.0014 ~ 12.5E-4 0.0013 e~ < 0.0012' 10.0E-4 <~ 7.5E-4' 0.0011" 2 4' 6 0 m (tad/s) 1o 5.0E-4" 12' 2 4 END 1 35E-4' 6 8 m (tad/s) 10 12 MODE SHAPE 1-4 9.00;:-4" 8.75E-4" 30E-4' % ~ 25E-4 o.soE-4 ~ 8.25E-4 u~ < ~ 8.00E-4 ,'~15E-4 ,,I 7.75E-4 ~ log-4 < 7.50E-4 7.25E-4" 5E-4 • 2 Fig. 5 4 6' 0 m (rad/s) 1o 12 ? JUNE 1992 4 6 8 co (tad/s) 10 12 Generalized added-mass values [,4.(~o) = A,.(co) + A,.(oc)] of thin cylindrical shell at free surface in water of infinite depth cies [Wu & Price (1986), Lee & Sclavounos (1989)]. This was confirmed from the analysis given by Wu and Price (1986), predicting irregular frequencies at 10.2 (rad/s) for heave motion, 10.5 (rad/s) for surge and pitch motions, 13.8 (rad/ s) for sway and roll motions, and 14.0 (rad/s) for yaw motion of the shell structure. In the vicinity of these frequencies, marked variations in the generalized hydrodynamic coefficients may occur (see Figs. 5 and 6), requiring suitable modifications to eliminate these spurious effects. Rigid boundaries and free surface--For a deeply submerged body moving near a rigid boundary (towing tank bottom or wall) in an otherwise unbounded fluid, no surface wave disturbances are created from the motion of the body. Therefore, a potential-flow analysis predicts that the generalized fluid damping is zero and the generalized added mass depends on the relative distance between the body and boundary. For example, in the case of a sphere of radius R moving pear a rigid boundary the generalized added mass, that is, A~(~o) = Ars(Oo), has the form given in Fig. 7, tending to the value 2~pR3/3 as the distance between the body and the wall H --* ~ [see Wu (1984)]. In the case of the cylindrical shell submerged at 9 m below the free surface, the influence of the relative position (defined by the ratios R / H ) on the generalized hydrodynamic coefficient was examined. These predictions are shown in Fig. 8 with the most marked variation occurring as R / H --~ 1, agreeing with the trends observed in Fig. 7. When the fluid domain is bounded vertically by a free surface and a rigid boundary, Figs. 9 and 10 illustrate typical 162 2 results derived for the oscillating submerged cylinder and the frequency dependence of the generalized hydrodynamic coefficients is clearly displayed. Furthermore, by comparing the data shown in these figures for the equivalent generalized hydrodynamic coefficients (Ars, Brs, etc.) the influence of water depth, cylinder's position below the free surface, boundary, etc., may be easily assessed. Generalized excitation In the experiment, the external mechanical forcing oscillator acting at position (Xo, Yo, Zo) on the cylinder creates a generalized excitation ~(t) = E(Xo, Yo, zo)e i~t where ¢odenotes the frequency of forced sinusoidal oscillation and the amplitude ~(Xo, Yo, Zo) is a function of the forcing amplitude vector, F, and amplitudes of the relevant modes at the position of excitation, namely, -=(Xo,Yo, Zo) = Fu(xo, Yo, Zo) where U(Xo, Yo, zo) denotes a mode vector. Resonance frequencies In equation (12), the only remaining unknown is the generalized principal coordinate p(t) which can be easily calculated and, from this information the resonance frequencies of the fluid-structure vibration deduced. From the forms of the terms occurring in equations (12,13) this occurs when Idet DI is a minimum. Alternatively, in a simplified analysis, if it is assumed that the generalized structural damping is negligibly small and therefore ignored, that is, b = 0, free-surface effects disJ O U R N A L OF SHIP RESEARCH .~jh_-.- R 0.18 m. = MODE SHAPE i-3 MODE SHAPE 1-2 O.oOT 0.006" ~ ~ 0.005" FZ N 0.004" w 0.006" Z~ 0.005" : uJ 0.004" ~ 0.o03: 8 i 0.003" ~ 0.002 0.002" < 0.001 0.001" ~ " - - -4" 6' 0' co (rad/s) lO" END 1 0.035" 2 12" 4 6 0 10 O)(tad/s) 12 MODE SHAPE 1-4 .~ 14E-4 % 0.030" 12E-4 u.lZ~0.025" ~ 10E-4 8E-4 0.020 O 0.015 O 6E-4 rD 0.010 Z ~ 2E-4' i0.005 2 Fig. 6 4 6 8 CO(tad/s) 10 12 ] 1/2 Larr~-Arr] (14) ((Dr)dry Calculations of the resonance frequencies were performed using both procedures to assess the influence of the various physical aspects (that is, tethers, boundaries, position, etc.) occurring in the experiment. These results were compared with the experimental data. 2~ A(--)=-]--p R3 3 Ra (1+----8 H3 . ~ A ( - - ) = ~ - pR3 (I + ~3- "R~3) Generalized added mass of a sphere moving in unbounded water by an infinite rigid boundary JUNE 1992 ~ - - -4' 6' 8' 10' 12' Generalized hydrodynamic damping coefficient values [B,,(oa) = Brr(~) + Br,(oc) of thin cylindrical shell at free surface in water of infinite depth arr Fig. 7 " co (radls) carded such that the generalized hydrodynamic coefficients a r e constants with Ars(c~) =/: O, A rs((D) = Brs((D) = Brs(C~) = 0 and all coupling effects ignored, the resonance frequencies are given by the expression ((Or)wet = 4E-4 An analysis of experimental data revealed that the flexible cylinder is very lightly damped. For example, for the modal sequence shown in Fig. 1 (Table 4) and a generalized structural damping formulation [Bishop & Price (1979)] brr = 2 (Dr ~rarr, the damping factor ~r for each mode has values of the order 0.0005, 0.0006, 0.020, 0.023, 0.00045, 0.0004, and 0.00025, respectively. Idealizations and convergence The structural and fluid idealizations are independent of one another, both depending on the complexity of the structure and the convergence of results. In an investigation using the results of the 256 finite-element dry analysis, hydrodynamic panels were distributed over the cylindrical surface as follows: one structural element corresponding to one hydrodynamic panel; two hydrodynamic panels coincident with one structural element; and, finally, four hydrodynamic panels corresponding to one structural element. On the flat end plates, the same number of hydrodynamic panels and structural elements were adopted in the first two cases and two hydrodynamic panels coincided with one structural element in the third idealization. The same number of panels was used around the circumference of the shell and the end plates. Tables 5 and 6 give the predicted resonance frequencies calculated using equation (14) for the neutrally buoyant cylinder positioned 0.68 m below the free surface in a water of infinite depth and 1.6 m depth, respectively. In contrast to the finite-element data in Table 4, these results show a much more marked dependence on the hydrodynamic panel disJOURNAL OF SHIP RESEARCH 163 0.0054 ~ 0.0052 ..... ~ 0.0050 _ R = 0.18 m. MODE SHAPE 1-2 9,~. MODE SHAPE 1-3 0.00350 '~H ~ 0.0034! , , r ,~l / '-~ 0.0034( ~ o.oons ~< 0.0048 e~ u~ ~K~ 0.00330" < ,,I 0.0046 < 0.00325 0.0044 0.2' 0.3' 0.4' 0.5' 0.6' 0.7' 0.8" 0.9 0.2 0.3 0.4 R/H 0.5 '0.6 0.7 0.8 0.9 R/H MODE SHAPE I-4 END i 0.0085 ~= 0.0080 0.00185 ~"0.00184 N o.oo75 < o.oo183 1 ~ ~ 0.0070 0.00102' e~ < 0.0065 0.00181' 0.2" 0.3' 0.4' 0.5' 0.6' 0.7 0.8' 0.9' 0.2 0.3 0.4 R/H Fig. 8 0.5 0.6 0.7 0.8 0.9 RIB Generalized added-mass values [,4,.(o,) = A,.(oa) + A~(o~)] of thin cylindrical shell vibrating in the vicinity of an infinite rigid boundary in water of infinite depth (90 m below free surface) "f 0.0052 R~6m. R = 0.18 m. -q~o.36m.MODE SHAPE 1-2 0.0050 ...... MODE SHAPE i-3 0.00360 0.00355 I' , . ~ 0.00350 "~ 0.00345' N o.o048 ~ O.OO46" < ~ 0.00335" 0.00330 o.oo3,5 < < 0.00320 0.0044" 0.0031 2 4 6 10 8 2 12 4 6 8 10 12 o1 ~,cad/s) O~(rad/s) MODE SHAPE I-4 END 1 tm 0.00188 0.010 0.00186 ~ 0.009/~ ~ 0.00184 ~ 0.00182' ~ o.oo7I 0.0051 ~. . 2 ~ . . 4 . 6 . . 8 (tad/s) Fig. 9 164 JUNE 1992 0.oo18o 0,00176 10 12 2' 4 6 8' 10' 12 CO (tad/s) Generalized added-mass values [Ar,(~) = A,,(oJ) + A,.(=)] of thin cylindrical shell submerged at 0.36 m depth from free surface in shallow water of 0.72 m depth JOURNAL OF S H I P R E S E A R C H I _~.36m. MODE SHAPE 1-3 R = 0.18 m. MODE SHAPE 1-2 ~8E-4 22.5E-4 ~20.0E-4 Z~ 17.5E-4 ~U 15.8E-4 ~' 12.5E-4" lm O I0.0E-4" L9 7.5E-4" 7E-4" ~ 6E-4" z 5E-4" 4E-4 O 3E-4" ~ ~ 5.0E-4" ,~< 2.5E-4" 2E-4 IE-4 - - -2"- 4' 6' 8' co (tad/s) lO' t2' " --f " 4' 12E-5 10E-5' t~ 0.003' O "tu 8E-5" O [.9 (D 6E-5" Z 4E-5" ZO0.002" < 0.001" 2E-5" .2' 4' s' co (rad/s) 6' 10' 12 Table 6 JUNE 1992 m~ .... f m 6 8 co (tad/s) 4" 1o 12 Generalized hydrodynamic damping coefficient values [B,(w) = B,(w) + B,,(~)] of thin cylindrical shell submerged at 0.36 m depth from free surface in shallow water of 0.72 m depth tribution and though the results may not be fully convergent they show similar trends. However, for the same idealization conditions, they demonstrate the influence of water depth (that is, shallow and infinite) on the resonance frequencies. It is clearly seen that for a fixed modal combination (m, n) Convergence of hydrodynamic results for cylindrical shell 0.68 m from free surface (infinitely deep water) Finite 12 e~ 14E-5 Z ~ O.OO4 Table 5 10 MODE SHAPE I-4 "~ 16E-5 ~0.005 Fig. 10 8 to (tad/s) END 1 0'0061 6 256 Panels 384 Panels 768 Panels Modes Element 256 Elem. (coD,m, (Hz.) 197.4 198.6 203.8 206.6 350.8 393.8 416.2 (coD--, (Hz.) 103.0 87.0 117.0 96.0 252.0 248.0 284.0 (co,)~, (¢o,)w,, (Hz.) 100.0 87.0 109.0 96.0 223.0 226.0 247.0 m-n (Hz.) 102.0 85.0 116.0 93.5 250.0 239.0 275.0 the lower resonance frequency value occurs, as expected, in the shallow water. The reasons for this are given in the generalized addedmass data in Table 7 calculated using 768 finite elements and 768 hydrodynamic panels. This conveys the variation of the added mass associated with the cylinder positioned at various depths below the free surface in water of 1.6 m depth and in infinite depth. The influence of the relative distance between cylinder and boundary is clearly evident and this effects the resonance frequency values as shown in Table 8. Comparisons 1-2 End 1 1-3 End 2 1-4 2-3 2-4 Convergence of hydrodynamic results for cylindrical shell 0.68 m from free surface (1.6 m deep water) Finite Element 256 Elem. (co,)a,-y 256 Panels 384 Panels 768 Panels Modes (co,)~,, (to,),,,, (co,)w,, m-n (Hz.) (Hz.) (Hz.) (Hz.) 197.4 198.6 203.8 206.6 350.8 393.8 416.2 102.0 72.7 119.0 80.4 290.0 251.0 272.(I 87.6 70.5 106.8 78.0 276.0 226.0 268.0 84.4 72.4 93.5 80.0 213.0 194.0 223.0 between The measured s o c i a t e d with a data calculated presented and experimental data in Tables 2 are 1 and as- buoyant, tethered cylinder though, unfortunately, no measurements of the forces in the four tethers were recorded during the experiment. In both experimental tanks, the distances between cylinder and sidewalls were kept as large as possible. For this Table 7 Numerical results for added-mass values in 1.6 m deep shallow water for various depths and infinitely deep water (body is 0.68 m submerged from free surface) 0.68 (m.) 0.93 (m.) 1.18 (m.) 1.38 (m.) Infinite Modes Water I-2 End 1 1-3 End 2 1-4 2-3 2-4 A~ A,, At, Art Ar~ (Kgm 2) 10-2 0.4574 0.3304 0.6445 0.5638 0.1818 0.3080 0.1713 (Kgm 2) 10-2 0.4579 0.3304 0.6612 0.5765 0.1817 0.3081 0.1713 (Kgm 2) 10-2 0.4623 0.3308 0.6821 0.5907 0.1815 0.3087 0.1711 (Kgm 2) 10-2 0.5374 0.3574 0.7222 0.6221 0.1896 0.3377 0.1787 (Kgm 2 ) 10-2 0.2947 0.2150 0.4170 0.3628 0.1576 0.2005 0.1496 m-n 1-2 1-3 End 1 End 2 1-4 2-3 2-4 JOURNAL OF SHIP RESEARCH 165 Table 8 Numerical results for resonant frequencies in 1.6 m deep shallow water for various depths and infinitely deep water (body is 0.68 m submerged from free surface) Finite 0.68 (m.) 0.93 (m.) 1.18 (m.) 1.38 (m.) Infinite Water Modes (to,.),~ (to,),,,,, (to,)~,, (to,)w,t (to,),,..,,, (co,)w,., m-n (Hz.) 197.2 203.5 212.3 220.6 349.5 391.1 412.2 (Hz.) 83.6 98. I 78.0 86.0 208.2 194.0 250.2 (Hz.) 83.5 98.1 77.0 84.8 208.2 193.6 250.3 (Hz.) 83.1 98.0 76.0 84.0 208.3 193.5 250.4 (Hz.) 78.1 95.1 74.0 82.0 205.3 187.0 246.9 (Hz.) 99.2 114.6 93.4 102.5 218.0 225.6 261.0 I-2 1-3 End 1 End 2 1-4 2-3 2-4 Element COUPLED Modes 0.23 (m.) (co,)., fHz.) (to,)., (co,),,,, (co,),,,, (to,)., (co,),,,, (Hz.) 88.2 101.0 78.6 86.1 210.8 200.2 253.0 (Hz.) 83.6 98.1 78.0 86.0 208.2 194.0 250.2 (Hz.) 89.4 103.3 79.0 86.6 213.8 202.6 255.6 (Hz.) 88.0 101.3 78.6 86.0 210.9 200.1 253.1 '(Hz.) 83.4 98.1 77.8 85.6 208.5 193.4 250.7 90.0 102.4 79.1 86.6 213.2 202.7 255.6 166 JUNE 1992 0.68 (m.) 0.21 (m.) 0.23 (m.) 0.25 (in.) (co,)~r Table 9 Calculated resonance frequencies for neutrally buoyant cylinder positioned at various depths from free surface in shallow water of 1.6 m depth 0.21 (m.) UNCOUPLED COUPLED Modes reason, it was assumed that the sidewalls created only minor/negligible influences on the measured and predicted quantities. Therefore, the two parameters of interest are water depth (that is, bottom boundary) and free surface. The results presented in Tables 1 and 2 clearly illustrate these effects of the distances between cylinder and boundaries on the measured frequency values. In other words, the measured frequency value is dependent on free surface, water depth, and their nearness to the cylinder. The theoretical predictions given in Tables 9 and 10 describe the behavior of an idealized, neutrally buoyant cylinder with no tethers attached. In the uncoupled results it is assumed for simplification that -4r~(C0) = 0 = Crs for r s whereas the coupled data relate to the complete calculations, that is, Arc(co) ¢ 0, Cr8 ~ 0 for r ~ s. A comparison of these results shows the influence of coupling in the fluidstructure interaction. To mirror the experiment more closely, the theoretical model was modified to include the influences of buoyancy and tethers. An estimation of the generalized tethering force, dependent on the position of the attachment, was included in the theoretical model through the generalized stiffness term. This generalized tethering force created negligible influence on the resonance behavior of the structure because the tethers were attached on the structure where the distortional motions are minimum. Therefore, the resonance frequency values given in Tables 9 and 10 are practically unchanged. From all the data presented, we see that changes in the. physical variables--water depth, position of the flexible structure in the fluid, and rigid and free-surface boundary effects--all influence the resonance frequency values of the vibrating cylinder because of the dependence of the fluid loadings on these variables. Differences are observed between measured and predicted resonance frequencies and, in all probability, these reflect inaccuracies in the evaluation of the generalized hydrodynamic coefficients rather than structural property characteristics. However, the hydroelas- UNCOUPLED Table 10 Calculated resonance frequencies for neutrally buoyant cylinder positioned at various depths from free surface in shallow water of 4.0 m depth 0.68 (m.) m-n 1-2 1-3 End 1 End 2 1-4 2-3 2-4 (Hz.) 87.0 I00.0 78.2 85.7 209.7 198.8 251.7 1.50 (m.) 3.50 (m.) 0.25 (m.) 1.50 (m.) 3.50 (m.) (to,)~,, (coD,,, (co,),,,, (co,),,,, (co,),,, (Hz.) 83.5 98. I 76.5 84.5 208.2 194.0 250.1 (Hz.) 83.3 98. I 74.8 84.0 208.3 193.6 250.3 (Hz.) 86.8 I00.0 78.1 85.7 209.5 198.5 252.0 (Hz.) 83.5 98.2 76.4 84.5 208.4 193.4 250.6 (Hz.) 83.3 98. I 74.7 84.0 208.6 193.3 250.8 m-n 1-2 I-3 End I End 2 1-4 2-3 2-4 ticity theory upon which the theoretical predictions are based reflects the trends exhibited in the measured data both qualitatively and quantitatively as well as satisfactorily describing the physical processes and mechanisms observed in the experiments. Note that although the analytical predictions of Warburton (1961) produce closer quantitative agreement with experimental data, this now would appear to be very fortuitous because contributions from many effects (ends, buoyancy, water depth, position, etc.) shown to influence the experimental data are ignored in this model and therefore it does not describe the true fluid-structure dynamic interaction process. Furthermore, this analytical solution applies only to infinitely long cylindrical structures whereas the proposed approach described here is applicable to flexible structures of arbitrary geometrical form [Bishop et al (1986), Price & Wu (1989)] excited by sinusoidal or transient loadings. 6. Conclusions From the evidence presented, the results derived from the numerical method based on a three-dimensional hydroelasticity theory show satisfactory agreement with the experimental results. It is expected that the error between the numerical and experimental results would reduce through more refined or improved calculations of the generalized hydrodynamic coefficients especially in the shallow-water model where, from the evidence presented, there appears a tendency to overestimate the values of the hydrodynamic coefficients. This high-frequency, small cylindrical shell provides a severe test to the numerical procedures which have been proven in previous studies and calculations involving low-frequency fluid-structure interactions of floating full-scale flexible structures [Bishop et al (1986)]. The present study confirms that the theory is applicable to high-frequency structures such as submarine pressure hulls as well as to more flexible ship-like structures [Bishop & Price (1979)]. The natural frequencies obtained from the finite-element analysis show good agreement with the analytical results (Tables 3, 4). This is because the analytical evaluation assumes that the ends of the cylindrical shell stay circular and do not distort; the end plates of the experimental cylindrical shell are stiff and show little distortion during vibrations of the cylindrical shell. In addition, for the cylinder in water and all boundary influences ignored, good agreement exists between numerical and analytical results as demonstrated through the results presented in Table 3. These are associated with the cylinder positioned at 9 m depth from the free surface of water of infinite depth, such that the free surface has negligible influence in this part of the study. The predicted frequency values of the end vibrations do not compare with the experimental results as well as those for the other mode shapes. The hatch plates at the ends give JOURNAL OF SHIP RESEARCH considerable stiffness (of unknown magnitude) to the end plates and therefore this was not modeled correctly in the mathematical model. The natural and resonance frequencies of the ends are much higher than those expected from predictions relating to the idealized structural model. The plates at both ends appear as internal supports to the cylindrical shell. According to the sensitivity analysis in which the stiffness of the end plates was increased twofold (see Table 4), the natural frequencies of the cylindrical shell increased by approximately 0.5 percent except for the end modes. This suggests that significant changes in the stiffness of the end plates create a greatly reduced variation in the values of the frequencies of the shell. It implies that the end plates behave like internal simple supports to the thin cylindrical shell. The experimental results seen in Tables 1 and 2 are comparable to the numerical results in Tables 9 and 10. The differences lie within the limits one would expect when comparing experimental results with numerical calculations. On the other hand, the cylindrical shell was forced to remain at fixed positions by tethers attached to the tank floor because the displacement force on the cylindrical shell was larger than its weight during the experimental work; that is, the cylinder was buoyant. The increase in the generalized stiffness due to these tethering arrangements was negligible and did not create significant changes in the resonance frequency values. The coupling created by the off-diagonal contributions of the generalized hydrodynamic coefficients in the resonance frequency calculations of the neutrally buoyant cylinder is shown to be negligibly small (Tables 9, 10). The generalized added-mass and the damping coefficients calculated by the numerical method behave as theory predicts. As the body approaches the free surface the coefficients exhibit frequency dependence (Figs. 3, 4) but unfortunately, for the cylinder piercing the free surface, irregular frequencies occurred in the calculations, creating local unrealistic results. For a deeply submerged body near a rigid boundary, the generalized hydrodynamic coefficients show dependence on the distance between itself and the boundary. This is confirmed in Fig. 8. As seen from Figs. 9 and 10, when the fluid domain is of shallow depth, that is, bounded by a free surface and rigid boundary, the generalized added-mass and damping coefficients vary with frequency, cylinder position in the fluid, nearness of free surface, depth of water, etc. (compare data with Fig. 3). 7. Acknowledgment One of us, A. Ergin, gratefully acknowledges the financial support of Istanbul Technical University, Turkey, during his Ph.D. study. 8. References ARNOLD, R. N. AND WARBURTON, G.B. 1953 The flexural vibrations of thin cylinders. Proceedings, Instn. Mech. Engrs., 167(A), 62-80. BISHOP, R. E. D., PARKINSON,A. G., AND PRICE, W.G. 1977 On the nature of slow motion derivatives. Journal of Sound and Vibration, 51, 1, 111-116. BISHOP, R. E. D., PRICE, W. G., AND TEMAREL, P. 1979 Wave-induced antisymmetric response of flexible ship. 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