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Comparison of added mass modelling for ships Yang, James 1990

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COMPARISON OF ADDED MASS MODELLING FOR SHIPS By James Yang B A.Sc. (Mechanical Engineering), University of British Columbia A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE in THE FACULTY OF GRADUATE STUDIES MECHANICAL ENGINEERING We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA June 1990 © James Yang, 1990 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Mechanical Engineering The University of British Columbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date: Abstract This thesis presents a comparison of added mass modeling techniques that may be used to determine the the vibration response characteristics of ships in water. The mathematical treatment of added mass is reviewed, and a number of numerical approaches are discussed. Experiments to determine the natural frequencies of a ship model in air and in water were performed and were compared with the results obtained from the numerical approaches. It will be shown in this thesis that the use of modal analysis to predict ship vibration responses in water is a satisfactory and less time consuming alternative to a full eigenvalue solution. ii Table of Contents Abstract ii Table of Contents iv List of Tables v List of Figures vii Nomenclature ix Acknowledgement xi 1 Introduction 1 2 Theoretical Background 2 2.1 Strip Theory 2 2.2 Mathematical Treatment of Added Mass 4 2.3 Vibration of a Submerged Beam 9 2.4 Numerical Approaches 13 2.4.1 Finite Element Method 14 2.4.2 Boundary Element Method 20 2.4.3 HYDRO 23 2.4.4 Modal Analysis of Fluid-Structure Vibration 23 2.4.5 Modal Analysis Using Wet - Dry Mode Equivalence 25 iii 3 Finite Element Implementation of Added Mass 28 3.1 Wet Mode Catgeory 29 3.2 Dry Mode Category . - 30 3.2.1 Modal Analysis 32 4 Comparison of Results 35 4.1 Ship Model 35 4.1.1 Experimental Results 36 4.1.2 Finite Element Results 38 4.1.3 Modal Analysis Results 49 4.1.4 PANEL Results 69 4.2 Plate Model 71 4.2.1 Cantilevered Boundary Condition 71 4.2.2 Free-Free Boundary Condition 71 5 Conclusions 77 Bibliography 78 iv List of Tables 4.1 Experimental Natural Frequencies (in Hertz) of Ship Model 36 4.2 Comparison of Natural Frequencies in Air - VAST 38 4.3 Comparison of CPU Times in Hours for S12-8 43 4.4 Comparison of Natural Frequencies in Water - VAST 48 4.5 Comparison of A Frequency in Air and in Water 48 4.6 Comparison of CPU Times in Hours for Ship Model 49 4.7 Comparison of Natural Frequencies in Water - VLS 50 4.8 Comparison of Air Frequencies Between Experimental Model and Simpli-fied Beam Model 57 4.9 Comparison of Water Frequencies Between Experimental Model and Sim-plified Beam Model 57 4.10 Comparison of Air Frequencies Between Glenwright's Experimental Model and Simplified Beam Model 58 4.11 Comparison of Water Frequencies Between Glenwright's Experimental Model and Simplified Beam Model 58 4.12 Comparison of Natural Frequencies - 14 Mode Approximation, S12-8 . . 61 4.13 Comparison of Natural Frequencies - 1 Mode Approximation, S12-8 ... 61 4.14 Natural Frequencies of S12-8, In Ascending Order 62 4.15 Summary of Degrees of Freedom in Generalized Added Mass for S12-8 . . 63 4.16 Ratio of Natural Frequencies 66 4.17 Ratio of Natural Frequencies Calculated from Generalized Added Mass . 67 v List of Tables 4.1 Experimental Natural Frequencies (in Hertz) of Ship Model 36 4.2 Comparison of Natural Frequencies in Air - VAST 38 4.3 Comparison of CPU Times in Hours for S12-8 43 4.4 Comparison of Natural Frequencies in Water - VAST 48 4.5 Comparison of A Frequency in Air and in Water 48 4.6 Comparison of CPU Times in Hours for Ship Model 49 4.7 Comparison of Natural Frequencies in Water - VLS 50 4.8 Comparison of Air Frequencies Between Experimental Model and Simpli-fied Beam Model 57 4.9 Comparison of Water Frequencies Between Experimental Model and Sim-plified Beam Model 57 4.10 Comparison of Air Frequencies Between Glenwright's Experimental Model and Simplified Beam Model 58 4.11 Comparison of Water Frequencies Between Glenwright's Experimental Model and Simplified Beam Model 58 4.12 Comparison of Natural Frequencies - 14 Mode Approximation, S12-8 . . 61 4.13 Comparison of Natural Frequencies - 1 Mode Approximation, S12-8 ... 61 4.14 Natural Frequencies of S12-8, In Ascending Order 62 4.15 Summary of Degrees of Freedom in Generalized Added Mass for S12-8 . . 63 4.16 Ratio of Natural Frequencies 66 4.17 Ratio of Natural Frequencies Calculated from Generalized Added Mass . 67 v 4.18 Generalized Added Mass Calculated from Mode Shapes in Water 68 4.19 Generalized Added Mass Calculated from Mode Shapes in Air 68 4.20 Comparison of Air Natural Frequencies (Hz) Between Glenwright and New Ship Model 69 4.21 Comparison of Water Natural Frequencies (Hz) Between Glenwright and New Ship Model 69 4.22 Comparison of Natural Frequencies - VPS, 1 Mode Approximation ... 70 4.23 Comparison of CPU Times in Seconds for Cantilevered Plate 75 4.24 Comparison of CPU Times in Seconds for Free Free Plate 76 vi 4.18 Generalized Added Mass Calculated from Mode Shapes in Water 68 4.19 Generalized Added Mass Calculated from Mode Shapes in Air ...... 68 4.20 Comparison of Air Natural Frequencies (Hz) Between Glenwright and New Ship Model 69 4.21 Comparison of Water Natural Frequencies (Hz) Between Glenwright and New Ship Model 69 4.22 Comparison of Natural Frequencies - VPS, 1 Mode Approximation ... 70 4.23 Comparison of CPU Times in Seconds for Cantilevered Plate 74 4.24 Comparison of CPU Times in Seconds for Free Free Plate 76 vi List of Figures 2.1 Variation of J Factor for the Fundamental Mode Along the Length of a Circular Beam 12 2.2 Boundary Value Problem 15 3.1 Stages of Evaluation for Ship Vibration Problems 31 4.1 Experimental Ship Model 37 4.2 2 Node Vertical Mode of Ship in Air 39 4.3 3 Node Vertical Mode of Ship in Air 40 4.4 4 Node Vertical Mode of Ship in Air 41 4.5 5 Node Vertical Mode of Ship in Air 42 4.6 2 Node Vertical Mode of Ship in Water 44 4.7 3 Node Vertical Mode of Ship in Water 45 4.8 4 Node Vertical Mode of Ship in Water 46 4.9 5 Node Vertical Mode of Ship in Water 47 4.10 Iterative Solution 51 4.11 Comparison of Mode Shapes, 2 Node Vertical Mode 53 4.12 Comparison of Mode Shapes, 3 Node Vertical Mode 54 4.13 Comparison of Mode Shapes, 4 Node Vertical Mode 55 4.14 Comparison of Mode Shapes, 5 Node Vertical Mode 56 4.15 Comparison of Solution Times for Modal Analysis 60 4.16 Generalized Added Mass for S12-8, Ma 64 4.17 Plate Model 72 vii 4.18 Cantilevered Plate - Comparison of Results Between Finite Element Anal-ysis and Modal Analysis 73 4.19 Free Free Plate - Comparison of Results Between Finite Element Analysis and Modal Analysis 75 viii Nomenclature k Structural Stiffness m — Structural Mass ma — Added Mass u* = Structural Natural Frequency in Air = Structural Natural Frequency in Water J = Longitudinal Inertia Coefficient T = Kinetic Energy of Fluid per Unit Length B Half Beam of Ship Section P = Density of Fluid V = Sectional Velocity 9 = Gravitational Constant C Cross Sectional Inertia Coefficient P = Pressure —* F = Force Vector n = Normal Vector = Velocity Potential r Radius Vector from Center of Rotation Ui = Surge Velocity u2 = Heave Velocity u3 — Sway Velocity u< = Roll Angular Velocity — Yaw Angular Velocity ix UQ = Pitch Angular Velocity rriij = t'th direction Added Mass given unit Acceleration in j'th direction Aj = Amplitude of Sinusoidal Motion in j'th direction dij — Added Mass in the presence of Free Surface b{j — Added Damping in the presence of Free Surface E = Young's Modulus / = Area Moment of Inertia Ni = Shape Functions M, = Structural Mass Matrix Ka = Structural Stiffness Matrix C, = Structural Damping Matrix T = Transformation Matrix Relating Nodal Velocity to Centroidal Velocity C = Transformation Matrix Relating Element Centroid Velocity to Element Source Strength Y = Transformation Matrix Relating Potential Flux to Source Strength A = Added Mass Matrix Calculated from Boundary Element Method B = Added Damping Matrix Calculated from Boundary Element Method <j>a = Natural Mode Shape in Air <f>w = Natural Mode Shape in Water x Acknowledgement I would like to express my thanks to Johnson Chan for his insight into the phenonmena of added mass, to David Glenwright for his invaluable assistance throughout this project, and to Professor Stan Hutton for his supervision and direction on this project. A special thanks goes fellow graduate students Philip Chan, Yetvart Hosepyan, Darcy Montgomery, and Malcolm Smith who constantly remind me of the saying: Numbers, not Insight. Or was it the other way around? Finally, I would like to acknowledge the financial assistance of the Defense Research Establishment Atlantic. xi Chapter 1 Introduction The vibration characteristics of a structure are significantly affected by the medium in which the structure is vibrating. Thus the vibration characteristics of a ship in water cannot be deduced without due attention being paid to the modeling of the added mass effects of the surrounding fluid. From a designer's view point, it is important to have accurate estimates of the fundamental hull modes of a ship at an early stage in order to avoid resonant excitation caused by propellor forces or by wave interaction. Various techniques have been developed for the analysis of fluid effects amongst which the finite element and boundary element element methods have been the most popular. In this report, the fundamental theory governing the interaction between the fluid and a floating structure are reviewed and a discussion is presented of the various numerical procedures that are available to solve the problem. Recommendations are presented as to the most efficient procedure for analysing the effect of the fluid. 1 Chapter 2 Theoretical Background In this section, a review is presented of the various mathematical approaches that have been developed in order to estimate the effect of the fluid on the vibration response of a submerged and floating structure. 2.1 Strip Theory In 1927, Frank Lewis [25], submitted a paper to the Society of Naval Architects and Marine Engineers which outlined procedures for calculating the inertia of the water sur-rounding a vibrating ship. He recognized the relationship between natural frequency of a vibrating ship and its stiffness and mass. According to Lewis, "... the added mass can be evaluated with fair accuracy by the methods explained in this paper. Accurate knowledge of a ship's stiffness is still lacking. It is recognized that the flexural rigidity of a hull is somewhat less than that based on the nominal I of its cross-section... Until further experiments on hull rigidity have been made, the frequency calculations must rest upon an empirical basis..." Lewis solved the added mass problem in two steps. First, he realized that fluid flow surrounding an arbitrary vibrating ship is complex. The fluid dynamics problem would be much easier to solve if it were two dimensional. Thus a relationship between two dimensional flow and three dimensional flow must be established. Next, he noted 2 Chapter 2. Theoretical Background 3 that added mass for ships with different cross sections is different. Hence a relationship between added mass and ship cross section must also be established. Among the few three dimensional problems in fluid dynamics for which an exact solution has been obtained is that of flow around an ellipse of revolution completely immeresed in an infinite fluid. From the exact solution of this problem, Lewis derived a formula for longitudinal J factors corresponding to the 2 node and 3 node vertical modes. These factors are defined as J = longitudinal inertia coefficient actual K.E. of surrounding fluid K.E. of surrounding fluid if motion is confined to transverse plane That is, J factors indicate the ratio of kinetic energies between three dimensional flow and two dimensional flow. Lewis assumed that J factors are the same for ships with similar length to beam ratios. Thus, fluid flow surrounding any vibrating ship has been reduced to a two dimensional problem. Traditionally, two dimensional fluid dynamic problems are solved by conformal map-ping. The kinetic energy per unit length of fluid surrounding a cylinder of circular cross section moving as a rigid body is given by 2T = irB2pv2 where T = kinetic energy B = half beam perpendicular to the motion p — density of the fluid v = velocity g = gravity constant. Chapter 2. Theoretical Background 4 Then, for all other cross sections with the same beam, the energy can be written as 2T = CnB V * where C is the inertia coefficient for that cross section. Thus the additional inertia mass per unit length of any ship cross section can be written as ma = j-CJitB* it Recall that the J factor is included to account for three dimensional flow. Lewis' work arose from the inability of researchers to calculate fluid effects surrounding a vibrating ship. It is an exact solution, given the assumptions he had to make. J factors were assumed to be the same for 6hips of similar length to beam ratios. They were calculated only for the 2 node and 3 node vertical modes. The free surface effects have been neglected. An arbitrary ship cross section is tranformed to an circular cross section by a series of conformal transformations, from which the sectional inertia coefficient C is found. 2.2 Mathematical Treatment of Added Mass With some restrictions, the added mass of a submerged body can be calculated mathe-matically. Newman [27] derived the rigid body added mass of a structure vibrating in an infinite fluid as well as on a free surface. This derivation differs in many respects to the work by Lewis. • The added mass is derived in integral form. • Sectional inertia coefficients are not required. • The free surface is included in the derivation. Chapter 2. Theoretical Background 5 However, this derivation is also similar to the work by Lewis because both derivations are based on rigid body ship vibrations, with longitudinal J factors included for three dimensional flow. Lewis' approach is useful for ships with simple cross sections where sec-tional coefficients can be calculated and tabulated. Whereas the mathematical approach is more flexible because ship cross sections can be arbitrary. Added Mass For A Three Dimensional Body In An Infinite Fluid Consider a body with surface Tj, surrounded by a fluid of volume V, having an outer imaginary surface r o. The forces acting on the body are represented by the integral of the pressure over the surface of the body, or (2.1) -* where F the sum of forces acting on the body V the pressure of the fluid over the body, and n the normal vector pointing out of the fluid. Substituting Bernoulli's equation, (2.2) into (2.1), the force on the body is given by (2-3) Using the Transport theorem, / / / ( £ • » - ) as (2.4) Gauss' theorem, (2.5) Chapter 2. Theoretical Background 6 and by assuming that the body is moving in an ideal infinite fluid, equation 2.3 can be reduced to If the body is moving (with the motion having six degrees of freedom), the velocity potential must satisfy the boundary conditionat Tj,, ?T = U-n + n-(rxn) (2.7) On where U is the translational velocity denoted by {U\, C/j, Ug), Q is the angular velocity of the rotating body by (C/4, Us, C/6)i a nd the radius vector from the center of rotation. In Naval Architecture terms, {U\, E/j, Us, U4, Us, U*) are identified as (surge, heave, sway, roll, yaw, and pitch). The boundary condition 2.7 suggests that the velocity potential can be written as a sum of 6ix components, or e 4> = T,Ui4>i, and (2.8) = f IT-?* (2 9) Using vector analysis, and substituting equations 2.8 and 2.9, equation 2.6 can be reduced to ^ = - p E t o / / r i A ^ (2.10) Or, Fj = Y.mnUi{t) (2.11) where Oh M I J = P I D T ( 2 1 2 ) is the added ma6s associated with the body moving in an infinite fluid. A property of the added mass matrix is that it is symmetric. That is my = vrtji. (2.13) Chapter 2. Theoretical Background 7 This property can be easily shown by Green's second identity, ///WVV - *>W)dA = JJ(i>^-<f>^)dT. (2.14) If both i}> and <f> are potential functions, then the left hand side vanishes. That is / / < * £ - < 2 1 5 > Or, Added Mass and Added Damping in a Free Surface Application The added mass coefficients m,j of a body moving in an infinite ideal fluid were derived in the previous section. Derivation of added mass and added damping in the more complicated situation of a body moving on a free surface follow the same approach. The motion of the body on or near a free surface is defined to be, Uj(t)= Re{twAiefc"} (2.17) where j = 1, 2, 6 represents the six motions (surge, heave, sway, roll, yaw, and pitch) of the body and Aj is the amplitude of the sinusoidal motion. Ignoring the presence of incident waves the velocity potential can be written as 4>(x,y,z,t)= Re^A^.y,*)^*} (2.18) i = i For example, if a body were forced to oscillate in heave motion with unit amplitude, in calm water, the resulting fluid motion can be represented by the velocity potential <f>2. The pressure over the body, ignoring second order terms, is given as M P = -P\~Qi + 9V} Chapter 2. Theoretical Background 8 Or, p=-p Re{£ Ajhiue*"} - pgy (2.19) Substituting the pressure into equation 2.1, the forces and moments on the body are M ( n ^ = -P9[f ydS-p// R e £ > V * JJs» \rxn j JJs" i=i ( iJ \ r x n fa dS (2.20) Note that equation 2.20 contains two terms: the first being a hydrostatic restoring force which has traditionally been neglected by Naval Architects (because its magnitude is smaller than the other forces) and the second representing a force similar to that derived from equation 2.11. But instead of a simple added mass as before, we now have real and imaginary terms because the velocity potential in this problem is complex. Thu6 the six components of force and moment on the body can be written as, Fi= Re-rxV"1/*}; » = 1,2,...,6 (2.21) where < f& = turn, 1=1,2,3 S = (^»)i-3 t = 4,5,6. The coefficient fa is physically interpreted as the complex force in the direction i, due to a sinusoidal motion of unit amplitude in the direction j. The added mass and added damping coefficients are given by, (2.22) Substituting (2.22) into (2.21) yields, 6 = Re{£ A^Wan - Aje^iubij} Chapter 2. Theoretical Background 9 6 (2.23) 3=1 The coefficient a^- is known as the added mass since it represents the force component proportional to the acceleration of the body. Similarly, the coefficient 6tJ- is known as the added damping; it represents the force component proportional to the velocity of the body. Note that added mass a^- calculated from this sections is different in value than m,j from the previous section. As the vibration frequency approaches 0, for horizontal plane motion (ie sway, surge, and yaw). Equation 2.24 is also valid for vertical plane motion (ie heave, roll, and pitch) as the vibration frequency approaches infinity. Readers are referred to Newman's textbook [27] which offers a concise, physical explanation of equation 2.24. 2.3 Vibration of a Submerged Beam The previous methods described how fluid surrounding a vibrating ship can be quantified as an added mass. However, neither explain specifically how natural frequencies can be predicted from the added mass. J.C. Daidola [7] presented a mathematical technique for the prediction of vertical and lateral natural vibrations of an Euler beam in a fluid. This technique was based on the simultaneous solution of the mechanical equations of motion for a vibrating beam and the coupled three-dimensional equations of motion of the surrounding fluid. He also investigated the applicability of the "J-factor" approach in considering the effects of the fluid and gave specific results for circular Euler beams with different boundary conditions. (2.24) Chapter 2. Theoretical Background 10 For a beam vibrating in the y direction, the Euler equation for a uniform beam is given by (2.25) where a4v d2y X = t = y(M) E I F(x,t) L m = distance along the length of the beam time vertical or lateral deflection of the beam modulus of elasticity of beam moment of inertia of beam any external force acting on the beam length of beam mass per unit length of beam When a beam is vibrating in a fluid, there are significant forces F(x,t) acting on the beam due to the fluid. F(x,t) can be determined from the linearized Bernoulli equation P{x,y,z,t) = -p— (2.26) which defines the pressure P in terms of the velocity potential <f>. Hence for a beam with a circular cross section, the force per unit length exerted on the beam by the fluid in the y direction can be written as F(x,t) = Jt p-^-R Bin9 d9 at (2.27) where JT indicates integration around the contour of a cross section of the beam. Using a series of Fourier Transformations and Inverse Fourier Transforms, the governing equation Chapter 2. Theoretical Background 11 becomes p r ^ _ w » m u = _ ^ r Kd\p\R)eipx dx* y 2 J-oo \V\(KMP\R) + K*[ J. \p\(K0( p\R) + K2(\p\R)) y(x')e-ipx' dx' dp (2.28) where Ko, K\, and K2 represent Bessel functions and p and x' represent dummy integra-tion variables. Note that the right hand side of equation 2.28 represents the hydrodynamic force on the structure. The solution to equation 2.28, from which the mode shapes and natural frequencies of the beam in water can be found, is outlined in Daidola's thesis [7]. Daidola approach to solve equation 2.28 is not important to this report. However, some insight may be gained from his results: 1. The results show that the historical approach of using Lewis added masses and J-factors to predict free vibration of beams in fluid gives reasonable results. The hydrodynamic force acting on the beam is given by ( ) 2 J-oo \p\{K0 + K^J-oo^ ] Hence the added mass can be written as m°{x) - ~ 2yW 7-oc |P|(JT0 + K2) U V{X )e d x d p (2 3 0 ) and the local J factor i6 defined as ™>i-D\?) _ ma(x) \p*R* ~ ^RW)L \P\(K0 + K 2 ) L y { s ) e d x d p ( 2 3 1 ) Figure 2.1 from Daidola [7] shows the variation of the J factor along the length of the beam for the fundamental mode of a circular beam with free ends. This Chapter 2. Theoretical Background 12 0 0 O © 0 © e © « © © © C G © © 0 -.5L -.4L -.3L -.2L -.1L 0.0 © Lewis Ellipsoid Figure 2.1: Variation of J Factor for the Fundamental Mode Along the Length of a Circular Beam Chapter 2. Theoretical Background 13 contrasts the traditional approaches of placing masses uniformly along the length of a ship of uniform cross section. 2. Natural mode shapes predicted by this approach show that they are not altered by the presence of a fluid. Daidola noted: "... the natural mode shapes of a circular uniform beam do not appear to be altered by the presence of the fluid, regardless of its density ..." It is interesting to note that even though Daidola first approached the problem by not assuming equivalence of wet and dry modes of vibrations, he concluded that "... Utilization of the in-vacuum mode shapes in modal analysis to deter-mine beam response appears to be justifiable by virtue of the equivalence of the fluid mode shapes... " 2.4 Numerical Approaches Ship vibration problems are complex. Unless simplfying assumptions are made (restrict-ing the solution to an Euler beam for example, as Daidola has done) closed form solutions usually are not possible. Thus the problems are generally solved with numerical tech-niques such as finite element or boundary element methods. The application of finite elements in the investigation of added mass vibration phe-nomena is well documented. It has been shown [2], [12], [15], [26] that numerical results are consistent with experimental results. The main drawback of the finite element method is the computation time required to solve a large problem. Implementation of the boundary element method is well suited to the evaluation of the added mass matrix of a vibrating ship. The effect of the fluid on the ship can be represented by a distribution of source singularities along the bounding surface, and a Chapter 2. Theoretical Background 14 reduction in the number of variables is achieved because the governing equations have been reduced by a dimension. This approach is attractive in view of the computation time saved as compared to the finite element method. 2.4.1 Finite Element Method There have been many papers [2], [12], [15], [26], [28] written about using finite element methods to calculate the added mass of a vibrating structure. The fluid-structure interaction problem can be formulated as a boundary value prob-lem as represented by figure 2.2 where the pressure p is to satisfy V2p = 0 (2.32) inside the domain. At every point on the fluid-structure interface, Sn, ^ = -pun on Sh (2.33) On where un is the displacement normal to the element. The equilibrium condition at the free surface is generally used as p = 0 on 5/ (2.34) This is a valid assumption in the frequency range of interest for propellor induced vi-brations (6 to 15Hz), but is not valid at very low frequencies. For example rigid-body motions and sometimes even the first fundamental modes of vibration occur at frequen-cies below 1 Hz for some large ships. In such cases, the linearized free surface boundary condition ^ + -p = 0 (2.35) oy g (which is frequency dependent) should be used. For the infinite boundary condition, 5r, a suitable boundary condition must be imposed so that no waves are reflected; since Chapter 2. Theoretical Background 15 Fluid Domain infinity sea bed 0 in the fluid domain -pii n fluid structure interface 0 at infinity C on the free surface ^ j on sea bed V'p = dP = dn dp_ = an p = Figure 2.2: Boundary Value Problem Chapter 2. Theoretical Background 16 waves generated by the vibrating hull will travel outward and die out at infinity. It was stated [2] that this is equivalent to |£ = 0 on Sr . (2.36) On Note that this condition will have to be tested to see that if the boundary has been put back far enough. Generally, the test for this boundary condition involves creating several fluid meshes and determining (from numerical results) if an increase of the outer boundary has a significant impact on the results. This problem may also be formulated in terms of the velocity potential <f>. Readers are referred to Zienkiewicz [43] for a complete derivation of the formulae. The velocity field v in an ideal fluid can be written as v = -V<f> (2.37) or dv d t = -V* (2.38) Similarly, the pressure is found from Equations 2.38 and 2.39 imply it=--Vp (2.39) P -VJ>= --Vp (2.40) P Hence the relationship between pressure p and velocity potential <j> is then given as P = p4> (2-41) The finite element formulation follows the classical Galerkin approach. The pressure inside the domain is approximated by known shape functions Nm{x,y, z) such that p'(x,y,z,t) = NT(x,y,z)P(t) = 2>.(*.**)ft(0 (2.42) »=i Chapter 2. Theoretical Background 17 The equilibrium conditon is then approximated over the domain, thus leaving a resid-ual R* where V V = R* . (2.43) which, according to the Galerkin procedure is to be made as small as possible. This is accomplished by multiplying the residual by all the shape functions, integrating through the the domain D, and setting the result to 0. Thus, JJJ NiV'p* dxdydz = 0, i = l,2,...,M (2.44) Integrating the above by parts, and taking into account the boundary conditions, we find (for m = 1,2,... ,M) ff „dp* ,„ fffrdNidp" dNidp* 8Nidp\ , , , n ,n t r . JL ~dHdS - l l l ^ - k + + r f 1 * • i y d z - 0 ( 2- 4 5 ) But dp* " dNj dx £1 dx Pi Substituting to get ( 2 - 4 6 ) where H over each finite element is given by Next the hull displacements are approximated with a different set of shape functions N' so that and thus dp — = -pun On 3=1 Chapter 2. Theoretical Background 18 Substituting to get where H P = B_Un (2.47) *y = pjJNiN'jdS Note that the above equation can be partitioned into H„ H.f]\ P.\ \ 0 where the subscript / represent those structural degrees of freedom in contact with the fluid and the subscript a represent those degrees of freedom not in contact with the fluid. Solving first for P_t , P_. = -E-} Ht} P, Thus, we arrive at Hf Pj = B U n where HJ = HJf ~ HJ» HI> Hjf Next, we transform the normal acceleration U_n to cartesian coordinates as U_n = A Uf (2.48) (2.49) where A is a rectangular array of the normal coordinate direction cosines and Uf is a vector of cartesian coordinate accelerations of all nodes on the hull surface in contact with the water. Hence Hf Pf = B&ijf Chapter 2. Theoretical Background 19 or Pf =t HJ1 B A Uf (2.50) Using a standard finite element formulation, the equations of motion for the structure are M. U (*) +C.U (t) + K.U it) = F. + Rj. (0 (2.51) where M, is the mass matrix of the structure with no water, C, is the damping matrix, K, is the stiffness matrix, F, is the vector of external excitation, and Rf, is the vector of hydrodynamic forces acting on the immersed part of the hull. The hydrodynamic force, Rf,, is found to be Rf. = --ATBTPf p = = -Substituting, the governing equations can be reduced to (M. + Ma)U(t) + C. U(t)+ K. U(t) = F, (2.52) where the hydrodynamic mass matrix is given by 0 0 Ma = 0 ^ATBTH71 BA p - - 1 = The hydrodynamic mass matrix Ma in this formulation will be a full symmetric matrix of dimension 3S by 3S, where S is the number of submerged nodes. Entries of M„ are simply added to the entries of M, to form the global mass matrix. Eigenvalue solution time of the added mass vibration problem will be similar to the structural dynamics problem if S << N, where N is the total number of structural nodes. However, if S is of the same order of N then solution time is expected to increase dramatically. For example, experience in analysing a model where N = 574 and S = 147 have shown that the added mass problem took 143.39 hours (22.82 of which was used to formulate the added mass matrix), compared to 11.08 hours for the structural free vibration problem alone. Chapter 2. Theoretical Background 20 2.4.2 Boundary Element Method The added mass of a vibrating ship can also be calculated by the boundary element method. This method is based on Green's second identity / / / W W - *VV) * = //<*£* - (2.53) There have been relatively few boundary element results for added mass problems pub-lished in the literature. A paper by Vernon [35] (which was based upon the formulation of Vorus [37]) included comparisons to both theoretical and experimental values. In general, results showed good agreement. Vernon concluded that "... the CPU time ratios for the finite element method versus the panel (boundary element) method varied from a minimum of approximately 10 in the propellor analysis to a maximum of 70 for the floating cylinder ... in general, a CPU time reduction of at least an order of magnitude can be expected using the panel method ..." A matrix formulation routine, using Green's function (or source distribution method), 16 outlined below for the added mass of a vibrating ship. The resulting added mass matrix i6 independent of any predefined vibratory mode shapes and is superimposed directly in the global mass matrix. The vector 6(t) is defined as the set of unknown displacements of nodal points on the wetted hull surface. These nodal points must be assumed to be taken from the finite ele-ment mesh originally defined by the user. The matrix T is defined as the transformation matrix which expresses the normal velocities at the centroids of the elements in terms of the unknown nodal velocities, that is V = T & (2.54) Chapter 2. Theoretical Background 21 The matrix Q is defined as the transformation matrix which relates the normal velocity of an element v, to the source strengths <ri, over all the elements, or v = Q q ' (2.55) The pressure pj over element j is given in terms of <Tj and &$ as P = ~pP <Z - p U Y q (2.56) where '"-lb0*3 is the potential at element i due to unit source density of element j, U is the ship forward speed, and V is the transformation matrix which relates the potential flux to the source strength; that is, tj>n = V q. Next, let the transformation matrix [S] be defined as one which relates the force on the element to the pressure on the element f = Sp (2.57) And finally, we find the force on the structure as / = ~PS E C _ 1 T I -pUS Y Q' 1! 6 (2.58) Or, f=-4i- 8 6 where the added mass and damping matrices are given by 4 = PSEQ'1! 5 = pUSYG-'T (2.59) (2.60) Several important points are inherent in the above formulation. Chapter 2. Theoretical Background 22 1. The matrix P was derived from the assumption of zero dynamic pressure (and therefore zero velocity potential) on the free surface, which requires that u>2 3> g. 2. Matrices A and B the added mass and damping matrices to be attached to the structural nodes which are in contact with the fluid. Unlike the added mass matrix derived from finite elements, these are generally non- symmetric. Entries of each of the matrices can be physically interpreted as the influence of a unit load on one degree of freedom on another d.o.f. For example, for a certain arrangment of the finite element mesh, the entry cooresponding to the first row and the first column of the added mass matrix represent the force on the x'th d.o.f. of node 1, given a unit acceleration of the x'th d.o.f. of node 1. Similarly, the entry cooresponding to the first row and the fifth column represent the force on the x'th d.o.f. of node 1 given a unit acceleration of the y'th d.o.f. of node 2. Hence for a certain numbering scheme, the sum of the first row of every third entry represents the force on the x'th d.o.f. of node 1 given a rigid body acceleration of the entire structure. 3. The derivation of Q and V is outlined in the Douglas report [14]. 4. Comparison of the Vorus' solutions [37] with those done by F.M. Lewis for estimat-ing the added mass distribution of a vertically vibrating ship showed remarkable agreement. The disadvantages of Lewis' method remain. But, where the vibratory modes of interest are vertical, with mode shapes that can be approximated as one-dimensional, it appears that Lewis' method should provide acceptable engineering accuracy 5. This formulation is capable of evaluating the added mass matrices of complex mul-tidimensional modal patterns. Assumption of specific vibratory modes is not nec-cessary. Chapter 2. Theoretical Background 23 6. Computation time for the added mass and damping matrices is expected to be small because we are working with surface integrals instead of volume integrals (as in the case of finite elements). For example, computation time of A for a large ship model is of the order of 15 minutes interactive time whereas it takes 22.82 hours for the finite element method. 2.4.3 HYDRO HYDRO is a computer written at the University of British Columbia which determines the added mass of a two dimensional section, including the free surface effects. The theory behind HYDRO is explained in the report by Hutton, Glenwright, et al. [10]. HYDRO formulates the following boundary value problem: V2^ = 0, inside the domain; |£ = V^ , on the fluid-structure interface |^  — yd) = 0, on the free surface; limji_00 #'{f^  — tA^ } = 0, on the radiation boundary; and limy_00 |^  = 0, on the sea bed. and solves for the velocity potential <j>. The added mass is then found from equation 2.12. Note that the added mass calculated by HYDRO is a function of vibration frequency because of the frequency dependent linearized free surface boundary condition. 2.4.4 Modal Analysis of Fluid-Structure Vibration This approach uses the principle of modal superposition to calculate fluid-structure dy-namic characteristics and is the basis of the paper by Ohta [28]. The simplest method of incorporating vertical added mass into a finite element model is to find a lumped value representative of the fluid and put it on the diagonal entries of Chapter 2. Theoretical Background 24 the global mass matrix. Hence, instead of, Mx + Kx = 0, the following is solved M„ Mfa Mft + AM„ X, J ) + K» K.f Kf. Kft xf = 0- (2.61) where A Ma = Ami 0 Am,- = the lumped added mass (determined from Lewis forms, for example) to be introduced to the diagonals of the t'th degree of freedom. x, = degrees of freedom of the structure not in contact with the fluid Xf = degrees of freedom on the fluid-structure interface The mode shapes, <j> of equation 2.61 can be normalized such that and 1 T - ( \ M„ M.f U. Mf. Mff + AM, 4>. K„ K.f Kff I h J Mi 0 Then, instead of equation 2.61, the following is solved M„ M.f MSt Mff + Ma + K„ K.f x. Kf. Kff * 0 = M = K = 0 (2.62) (2.63) (2.64) Chapter 2. Theoretical Background 25 where M„ i6 the added mass matrix calculated from either finite elements or boundary elements. We then transform from global coordinates to modal coordinates using ' ' * = <f>q (2.65) N Then, multiplying equation 2.64 by <f? on the left hand side, we get (M + Ma)'q + K g =0 (2.66) where Ma = <t>T Ma 4> from which the final mode shape of the ship can be extracted from q using equation 2.65. 2.4.5 Modal Analysis Using Wet - Dry Mode Equivalence The modal analysis approach assumes that ship mode shapes in water can be calculated by a combination of its mode shapes in air. However, if the assumption of wet and dry mode equivalence is made, modal analysis can be simplified significantly. The governing equations of motion for a structure vibrating in air with no damping can be written as M, x + Kt x = 0, (2.67) where M, is the structural mass matrix, K, is the stiffness matrix and x is a vector of nodal displacements. If we assume sinusoidal motion with x{t) = Xsinwt, equation (2.67) reduces to {Kt -w1 M, }x = 0. (2.68) Equation (2.68) is an eigenvalue problem for which the solution gives eigenvalues u\ and eigenvectors <f>a of the structure vibrating in air. Further, the eigenvectors can be Chapter 2. Theoretical Background 26 normalized such that = 1 (2.69) Next, consider the equations of motion for a structure vibrating in water with no damping. The fluid effects can be accounted for with an added mass matrix M„ 6uch that (M. + M„)|+ K.x = 0. (2.70) Equation (2.70) is also an eigenvalue problem for the structure vibrating in water for which the solution gives eigenvalues u^, and eigenvectors <f>w. As above, the eigenvectors can be normalized such that 4>w K, 4>w = uw <g{M. + M 0)«^ = 1 (2.71) Hence substituting the eigenvectors into equation (2.70) we get {K. - u>l{ M. + M a)}^ =0. Multiplying by <g, -u>l{M. + Ma)}<t>_w =0. It has been shown from experimental and analytical results [7], [10] that the mode shapes for a structure vibrating in water is approximately the same as the mode shape in air. That is <f>_a ~ <t>w-Thus <g{K. -"KM. + Ma))4>_a =0. Chapter 2. Theoretical Background 27 Multiplying the matrices (as per equation 2.71), i-uZ4£(M. + Ma)<j>_a =0. Or, ^-u;Ju(l+<^Mad>o) = 0. And finally, 4 = l+#M 0<k. w - (2.72) Hence we find that the ratio of natural frequency in air to natural frequency in water can be found explicitly by equation (2.72). This is the quickest approach of obtaining approximate results because only dry modes of vibrations are needed: re-analysis is not neccessary. Natural frequencies of the ship in water are functions of the added mass matrix and of the dry mode shapes. By virtue of using equation (2.72), wet dry mode equivalence is assumed to be true. Note that this approach is identical to a one mode modal analysis. Chapter 3 Finite Element Implementation of Added Mass The purpose of this chapter is to illustrate how added mass, which was calculated from methods outlined in the previous chapter, can be implemented into an existing finite element program (VAST) [34]. A hierarchy of approaches, from the most time consuming to the least, will be presented. There are many possible approaches of incorporating an added mass matrix in VAST. These generally fall within two categories: Wet Mode and Dry Mode. The Wet Mode category is denned as those approaches which incorporate the added mass matrix into the global mass matrix before natural frequencies and mode shapes are sought. The Dry Mode category is denned as those approaches which solve the equations of motion without the fluid and then incorporate the added mass to arrive at the final solution. It is important to remember that Dry Mode approaches require prior knowledge of structural behaviour in air before dynamic characteristics in fluid can be found. Below i6 a listing of the possible solution methods. 1. Wet Mode Category • approximate both the fluid and structure with finite elements, to be referred to as VAST for the remainder of the report • approximate the fluid with panel method (boundary elements) and structure with finite elements, to be referred to as PANEL 2. Dry Mode Category 28 Chapter 3. Finite Element Implementation of Added Mass 29 • Finite Elements — use VAST to find dynamic response in air and (for a specific mode of interest) find the lumped added mass (using Lewis formulae) and then solve for the response in water using VAST (VLV) • Modal Superposition — modal analysis using mode shapes in air and lumped masses (VLS) — modal analysis using mode shapes in air and finite element added mass matrix (VFS) — modal analysis using mode shapes in air and boundary element added mass matrix (VPS) • Wet and Dry Mode Equivalence — assume equivalence of modes in air and in water. This approach will be incorporated into both VFS and VPS. This approach is equivalent to modal analysis where only one mode is used in the approximation. Discussions on each of the approaches will include: what are their advantages and dis-advantages and how they can be implemented into VAST. 3.1 Wet Mode Catgeory Solution time for the Wet Mode category is generally much longer than those from the dry mode category. This is mainly because the bandwidth of the global mass matrix is much greater, due to the introduction of a full added mass matrix. The simplest and most common of the Wet Mode category is the finite element method (VAST), where both the structure and the fluid are modelled using finite elements. This method gives accurate results for vertical modes of vibration and is already incoporated Chapter 3. Finite Element Implementation of Added Mass 30 into the current version of VAST. This project will involve evaluating the numerical results of VAST and comparing them to experimental results. A more elegant method in the Wet Mode category is to use boundary elements for the fluid domain (PANEL [32]). This method follows the derivations of Vorus [37] to arrive at an added mass matrix which i6 independent of vibration mode. This is deemed a more desirable approach because computation time required to calculate the added mass matrix is significantly less than the time used by the finite element approach. Figure 3.1 obtained from the report by Vernon [35] shows how PANEL can be implemented into VAST. The computer program PANEL will calculate an added mass matrix compatible with VAST. This project will compare boundary element results to experimental and finite element results. 3.2 Dry Mode Category Unlike the Wet Mode approaches, Dry Mode approaches are relatively quick in solving the coupled fluid-structure interaction problem. A reduction of CPU time is achieved because these methods do not solve the submerged ship vibration problem. Vibration characteristics of the ship in air are first analysed with finite elements. Then, an added mass representative of the fluid surrounding the vibrating ship is calculated. This added mass can be either in the form of a lumped value (calculated for example by HYDRO) or a matrix (calculated by either VAST or PANEL). Finally, re-analysis of the structure with the added mass is done, from which mode shapes and natural frequencies of the ship in water is determined. Re-analysis of the structure can be done using two different methods: full eigenvalue analysis, or modal analysis. It was been shown by Glenwright [10], [11] that for minimal structural property changes, modal analysis will give solutions much more quickly than apter 3. Finite Element Implementation of Added Mass Finite Element Model Generation Structural Stiffness [Kg] Interface/Fluid Model Generation Fluid MAM I I Finite Element Method Assembly [K] [M.+MJ I Decomposition Solution Interface Geometry Fluid Mass C M . ] I I Panel Method Figure 3.1: Stages of Evaluation for Ship Vibration Problems Chapter 3. Finite Element Implementation of Added Mass 32 full eigenvalue re- analysis. Hence, only the latter method will be fully explained in this report. 3.2.1 Modal Analysis The method of modal analysis transforms a set of coupled equations (written in any user-defined coordinates) into a set of uncoupled equations in the principal coordinates. Each principal coordinate (or modal coordinate) describes a mode of vibration. The theory of modal analysis has been explained in the previous chapter. A more detailed report is given by Hutton and Baldwin [16]. Hutton and Baldwin wrote the computer program STRUM (for STRUctural Modification) which uses modal condensation to calculate the natural frequencies and mode shapes of modified structures using the mode shapes of the original structure. Modal condensation differs from modal analysis in that only a few mode shapes are used to uncouple the system of equations. The original version of STRUM allowed the user to make modifications to the original (or baseline) structure. These modifications included changes in structural properties as well as additions of lumped added masses. However, it cannot be used if added mass modifications are in the form of added mass matrices. Hence another computer program STRAM (for STRuctural modifictions using Added Masses) was written to complement STRUM for situations were added mass modifications are in matrix form. VLS The VLS approach (which stands for VAST-Lumped mass-STRUM) combines the power of finite elements, the simplicity of boundary elements, and the elegance of modal analysis into a package which is relatively simple to use and yet provides a great deal of insight. First, natural frequencies and mode shapes of a structure in air are found using VAST. Then for a specific mode of interest, the effect of the fluid moving about the structure Chapter 3. Finite Element Implementation of Added Mass 33 is found as a lumped mass using HYDRO. Finally, this mass is added to the original structure and the problem is re-analysed using STRUM, giving an approximate natural frequency for the mode of vibration which we have assumed initially. The current version of HYDRO is fairly restrictive: • hull variation along the length of the 6hip is neglected, • the ship is restricted to only heave motions, and • cross section of the ship must be circular. These restrictions were initially imposed to assess the validity of the approach using a simple problem. Constraints (2) and (3) can be easily resolved with minor changes to the computer code, but the resolution of constraint (1) will require a more significant re-coding. The elegance of the current version of HYDRO lies in its ability to analyse the fluid-structure interaction problem in two dimensions (variation of velocity potential along the length is assumed to be negligible). If the analysis were extended to three dimensions, it would remain to be a lumped mass model, but CPU time would increase drastically. Since the idea behind VLS was to find a satisfactory result for a specific mode in the shortest amount of time, the three dimensional approach has not been pursued. Currently, the executable version of VLS require the following input from the user 1. mode of interest, 2. node numbers to which the lumped masses are to be attached, 3. draft of the structure, 4. density of the fluid in which the structure is vibrating and, 5. radius of the cylindrical cross section. Chapter 3. Finite Element Implementation of Added Mass 34 VPS and VFS When vibratory motion of the structure is complex, it appears that the more numer-ically rigorous methods would better represent the effect of the fluid than VLS. VPS (which stands for VAST-PANEL-STRUM) and VFS (which stands for VAST-FINITE ELEMENT-STRUM) use added mass matrices which account for all the displacments of the interface nodes. Application of VPS and VFS is identical to VLS, with the exception that STRAM is used instead of STRUM. The computer program STRAM is capable of running both VPS and VFS. Calculation of the added mass matrix as a function of nodal displacements has been shown in numerous papers [2], [28], and [37]. It is anticipated that VPS will be quicker to run, partly because the panel method added mass matrix can be found much faster. Note also that there are fewer degrees of freedom in the panel added mass. Hence finding <f>T Mj <f> will also take a shorter amount of time. While it is clear that VPS will take longer to solve than VLS, it should also be noted that it will give all the modes of vibration as compared to the one mode solution of VLS. Chapter 4 Comparison of Results The purpose of this chapter is to summarize results which were obtained using the differ-ent methods outlined in the previous chapter. Several examples (finite element models) are included: 1. a ship model from which experimental results are compared with numerical results, 2. a cantilevered plate model, and 3. a free free plate model. Not all proposed methods are applied to all of the finite element models. For example, the boundary element method was not applied to any of the plate models because the computer program PANEL [32] could not produce a suitable added mass matrix. 4.1 Ship Model This section compares experimental natural frequencies to predicted natural frequencies of a ship model and represents the focus of this report. Most methods of predicting natural frequencies in water will be applied to the experimental ship model. Unfortu-nately, due to hardware constraints, some methods cannot be applied to the model. For example, a full solution (with a finite element fluid added mass matrix) of the structure vibrating in water was found, hence a comparison of finite element natural frequencies can be made with experimental natural frequencies. However, lack of disk space pre-vented obtaining a full solution with a boundary element added mass matrix. Hence 35 Chapter 4. Comparison of Results 36 Mode In Air In Water 2 Node Vertical 20.63 8.75 3 NV 61.0 25.5 4 NV 109.5 47.0 5 NV 139.0 67.5 2 N Horizontal + 2N Torsional 13.75 8.50 3 NH + 3NT 30.75 19.25 4 NH -f 4NT 44.25 31.0 5 NH + 5NT 57.50 40.5 Table 4.1: Experimental Natural Frequencies (in Hertz) of Ship Model comparison of boundary element natural frequencies to experimental values cannot be made. 4.1.1 Experimental Results An experiment was conducted to study the accuracy of VAST [34] in determining free vibration response of a ship. The experimental model which was used is shown in figure 4.1. The model consisted of a long (96 inches) semi-circular (radius 4.196 inches) hull, reinforced with 11 bulkheads and two semi-spherical endcaps. Since the acrylic model is light and will not produce significant draft by itself, two steel blocks (with a combined weight of approximately twenty seven pounds) were placed inside the model to add extra draft. The properties of steel are well known (and can be easily modeled using finite elements), hence numerical results are expected to be close to the experimental results (shown in table 4.1). The final experimental model (including the steel blocks) draws approximately 2.1 inches of water. Chapter 4. Comparison of Results 37 SHIP MODEL WITH : NST=11 NBLK=9 NELM=8 NM = 3Q L0AD=3OLBS. mass as iec=20 STRUCTURAL FINITE ELEMENT MODEL ELEMENT TYPES: ALL 21.357 Figure 4.1: Experimental Ship Model Chapter 4. Comparison of Results 38 Mode Expt. S20-8 Diff. (%) S12-8 Diff. (%) 2 NV 20.63 20.36 -1.31 20.37 -1.26 3 NV 61.0 61.31 0.51 61.4 0.66 4 NV 109.5 108.6 -0.82 108.8 -0.64 -5 NV 139.0 136.9 -1.51 137.8 -0.86 2 NH + 2NT 13.75 13.84 0.65 14.20 3.27 3 NH + 3NT 30.75 32.70 6.34 33.67 9.50 4 NH + 4NT 44.25 46.29 4.61 47.01 6.24 5 NH + 5NT 57.50 57.79 0.50 59.09 2.77 Table 4.2: Comparison of Natural Frequencies in Air - VAST 4.1.2 Finite Element Results Two finite element models were constructed to simulate the experimental ship model. The first model, S20-8, has 20 elements along the length and 8 elements around the cir-cumference. The second model, S12-8 has 12 elements along the length and 8 elements around the circumference. S12-8 was created in hindsight because a complete finite ele-ment run (including fluid modeling) for S20-8 required more disk space than was available from the computer system available. Natural frequencies in air for the experiment and both finite element models are shown in table 4.2. As expected, results for free vibration in air of S20-8 were better than results of S12-8 because there were more degrees of free-dom in S20-8 than S12- 8. It is interesting to note that the coupled torsional-horizontal modes are more sensitive to longitudinal modeling than the vertical modes, and the de-termination of accurate horizontal modes require such a fine mesh. Natural frequencies for all the modes are calculated to within 10% of experimental results. Figures 4.2, 4.3, 4.4, and 4.5 show the vertical modes of vibration of the finite element model. A summary of CPU times for S12-8 is given in table 4.3. Note that a significant portion (approximately 86%) of computer time was used in the eigenvalue analysis. CPU time for decomposition of mass and stiffness matrices into upper and lower triangular matrices represented at 8.57 percent of the total CPU time. Chapter 4. Comparison of Results 39 IP MODEL WITH : NST=11 NBLK=9 NELM=8 NM = 3Q L0AD=3QLBS. mass as iec=20 NATURAL MODE SHAPE MODE NUMBER 9 2.037E+01 CPS MAGNIFICATION FACTOR: 20.00 ELEMENT TYPES: ALL 29.231 Figure 4.2: 2 Node Vertical Mode of Ship in Air Chapter 4. Comparison of Results 40 SHIP MODEL WITH : NST=11 NBLK=9 NELM=8 NM=30 L0AD=30LBS. mass as iec=20 NATURAL MODE SHAPE MODE NUMBER 13 6.136E+01 CPS MAGNIFICATION FACTOR: 20.00 ELEMENT TYPES: ALL + + 29.231 Figure 4.3: 3 Node Vertical Mode of Ship in Air Chapter 4. Comparison of Results 41 SHIP MODEL WITH : NST-11 NBLK=9 NELM=8 NM = 30 L0AD=30LBS. mass as 1ec=20 NATURAL MODE SHAPE MODE NUMBER 18 1.088E+02 CPS MAGNIFICATION FACTOR: 20.00 ELEMENT TYPES: ALL 29.231 Figure 4.4: 4 Node Vertical Mode of Ship in Air Chapter 4. Comparison of Results 42 SHIP MODEL WITH : NST=11 NBLK=9 NELM=8 NM = 30 L0AD=30LBS. mass as iec=20 NATURAL MODE SHAPE MODE NUMBER 20 1.378E+02 CPS MAGNIFICATION FACTOR: 20.00 ELEMENT TYPES: ALL Z ) 29.231 Figure 4.5: 5 Node Vertical Mode of Ship in Air Chapter 4. Comparison of Results 43 Subroutine S12-8 % of Total Time K,M Formation 0.45 4.06 Band Width Reduction 0.02 0.18 K,M Assembly 0.08 0.72 K Addition 0.03 0.27 M Addition 0.02 0.18 Matrix Decomposition 0.95 8.57 Eigenvalue Analysis 9.53 86.02 Total 11.08 100.00% Table 4.3: Comparison of CPU Times in Hours for S12-8 S20-8 was not utilized any further because of hardware constraints imposed by the computer system available. The model S12-8 was transformed into S12-F in which the water surrounding S12-8 was modeled using finite elements, thus the F suffix. Water surrounding ship model S12-8 is approximated using two layers of 20- noded fluid finite elements. The outer boundaries of the fluid model extend 8.87 inches away and 9.45 inches below the center of the ship model. Recall that the ship has a radius of 4.2 inches and draws approximately 2.1 inches of water. The free surface is assumed to have zero velocity potential; that is, wave formation on the free surface has been neglected. This is a valid assumption since natural frequencies for all modes of interest, with the lowest being the 2 node vertical mode, are high enough such that wave formation effects can be neglected when it is compared to the fluid inertia effects. The outer most surfaces (including the radiation and sea bed boundaries) of the fluid domain have been given a zero velocity potential boundary condition. Table 4.4 presents a comparison between the finite element and the experimental results and figures 4.6, 4.7, 4.8, and 4.9 show the vertical modes of vibration of the ship in water. An indication of how well fluid finite elements modeled water surrounding the ship can Chapter 4. Comparison of Results 44 SHIP MODEL WITH : N S T . l l NBLK=9 NELM=8 NM = 30 L0AD=30LBS. mass as iec= 20 NATURAL MODE SHAPE MODE NUMBER 9 8.760E+00 CPS MAGNIFICATION FACTOR: 20.00 ELEMENT TYPES: ALL 29.231 Figure 4.6: 2 Node Vertical Mode of Ship in Water Chapter 4. Comparison of Results 45 SHIP MODEL WITH : NST=11 NBLK=9 NELM=8 NM=30 L0AD=30LBS. mass as iec=20 NATURAL MODE SHAPE MODE NUMBER 11 2.643E+01 CPS MAGNIFICATION FACTOR: 20.00 ELEMENT TYPES: ALL •X 29.231 Figure 4.7: 3 Node Vertical Mode of Ship in Water Chapter 4. Comparison of Results 46 SHIP MODEL WITH : NST=11 NBLK=9 NELM=8 NM=30 L0AD=30LBS. mass as iec=20 NATURAL MODE SHAPE MODE NUMBER 14 4.794E+01 CPS MAGNIFICATION FACTOR: 20.00 ELEMENT TYPES: ALL 29.231 Figure 4.8: 4 Node Vertical Mode of Ship in Water Chapter 4. Comparison of Results 47 SHIP MODEL WITH : NST=11 NBLK=9 NELM=8 NM=30 L0AD=30LBS. mass as 1ec=20 NATURAL MODE SHAPE MODE NUMBER 16 7.084E+01 CPS MAGNIFICATION FACTOR: 20.00 ELEMENT TYPES: ALL V I 2 * i + + 29.231 Figure 4.9: 5 Node Vertical Mode of Ship in Water Chapter 4. Comparison of Results 48 Mode Expt. S12-F Diff. (%) 2 NV 8.75 8.76 0.1 3 NV 25.5 26.43 3.6 4 NV 47.0 47.94 2.0 5 NV 67.5 70.80 4.9 2 NH + 2 NT 8.5 8.84 4.0 3 NH + 3 NT 19.25 21.10 9.6 4 NH -f 4 NT 31.0 33.09 6.7 5 NH + 5 NT 40.5 42.71 5.5 Table 4.4: Comparison of Natural Frequencies in Water - VAST Mode A Freq in Air A Freq in Water Difference 2 NV -1.26 0.1 1.37 3 NV 0.66 3.6 3.66 4 NV -0.64 2.0 2.64 5 NV -0.86 4.9 5.76 2 NH -f 2 NT 3.27 4.0 0.73 3 NH + 3 NT 9.50 9.6 0.10 4 NH + 4 NT 6.24 6.7 0.46 5 NH + 5 NT 2.77 5.5 2.73 Table 4.5: Comparison of A Frequency in Air and in Water be illustrated from the difference between predicted natural frequencies and experimental natural frequencies in water and in air. Table 4.5 shows quantitatively that the fluid finite element model has perhaps underestimated the extent of the fluid domain since the difference in predicted natural frequency has increased from the air model to the water model. It is also interesting to note that the vertical modes of vibration have been affected more by fluid modeling than the coupled horizontal-torsional modes. As suggested in the previous section, solution time for 12-F was quite large (see table 4.6). The breakdown of CPU times is significantly different than from table 4.3. The time for decomposition of % and M has increased from 0.95 hours to 41.65 hours, a four thousand percent increase. This is a result of a large increase in bandwidth of the global mass matrix due to the introduction of an added mass matrix. Formulation of the added Chapter 4. Comparison of Results 49 Subroutine S12-F % of Total Time K,M Formation 0.46 0.32 Band Width Reduction 0.02 0.01 K,M Assembly 0.09 0.06 K Addition 0.03 0.02 M Addition 0.33 0.23 Matrix Decomposition 41.65 29.04 Eigenvalue Analysis 78.00 54.40 Fluid M Formation 0.23 0.16 Fluid M Assembly 1.10 0.77 Fluid M Decomposition 0.26 0.18 Fluid Added Matrix 21.24 14.81 Total 143.41 100% Table 4.6: Comparison of CPU Times in Hours for Ship Model mass matrix also represented a large portion of the total CPU time at more than 22 hours. Eigenvalue analysis was the most CPU intensive subroutine, increasing from 9.53 hours to 78 hours. Total time for analysis of S12-F was approximately 143 hours. Note that the time quoted is CPU time. Actual run time depends on how much the central processor is being shared with other users and is often double or even triple the CPU time. Thus the full finite element approach, even though it is fairly accurate, is very time consuming. 4.1.3 Modal Analysis Results It was established in the previous section that a full finite element solution of the ship model was excessively time consuming. A more practical alternative is to solve the structural vibration problem in water using modal analysis; in which ship behavior in air is used to predict its behavior in water. Chapter 4. Comparison of Results 50 Mode Experiment S12-F VLS Diff (%) 2NV 8.75 8.76 7.16 -18.26 3NV 25.5 26.43 23.4 -11.46 4NV 47.0 47.94 42.2 -11.97 5NV 67.5 70.80 62.1 -12.29 Table 4.7: Comparison of Natural Frequencies in Water - VLS Lumped Mass - VLS Approach The VLS approach is an iterative procedure as shown in figure 4.10. 1. Dynamic characteristics of the structure in air are found using finite elements. 2. Fluid effects for a specific mode are quantified as a series of lumped masses. 3. Lumped masses are introduced into the structure as a mass modification. 4. The modified structure is re-analysed. 5. Steps 2 to 5 are repeated until convergence has been achieved. Note that this iteration will only be required if the added mass is frequency dependent. Implementation of a lumped added mass into finite elements is straight forward; the mass is placed on the diagonal entry in the global mass matrix corresponding to the correct degree of freedom. Natural frequencies for the lower vertical beam modes are shown in table 4.7. It was a surprise to see that VLS gave very poor results. These results imply that traditional methods of predicting ship natural frequencies in water (by adding sectional added masses along each section of the ship) are incorrect. However, closer examination into these results showed that the errors were a combination of three factors. 1. HYDRO has over estimated the fluid effects by approximately 21 percent. For example, sectional added mass of the 2 node vertical vibration mode calculated by Chapter 4. Comparison of Results 51 finite element model I VAST f find: 1) eigenvalues, A 2) eigenvectors, (p * 1 fluid modelling t find: added mass t STRUM t Figure 4.10: Iterative Solution Chapter 4. Comparison of Results 52 HYDRO was 0.243 which translated to a total added mass of 62.5 pounds. As a comparison, PNA [30] predicted the added mass to be 0.208 and VAST predicted the rigid body added mass to be 0.194 8^U^S, or a total added mass of 50 pounds. Thus the apparent mass of the ship (which is the sum of the structural mass and the added mass) calculated from HYDRO is 98 pounds as compared to 85.5 pounds from VAST; approximately 15 percent too high. 2. J factors have not been incorporated into the computer model. Thus VLS has over estimated the added mass effect by using a frequency dependent added mass, but assuming that the ship was heaving as a rigid body. 3. Air mode shapes are not the same as water mode shapes. Figures 4.11, 4.12, 4.13, and 4.14 give a comparison of mode shapes (along the centerline of the ship model) in air versus mode shapes in water. When modal analysis is applied to any structure, the difference in mode shapes is a significant source of error. Two finite element beam models were created to test the findings of VLS. The first beam model mimics the experimental ship model of this report. It is created such that its weight is 5.5 pounds and that when two 15 pound weights are placed at the correct locations, its 2 node vertical natural frequency is equivalent to experimental values in air. This is achieved on a trial and error basis by adjusting Young's modulus of the beam model. Table 4.8 compares the experimental air frequencies with the simplified beam model air frequencies. The last column entry of table 4.8 is used to illustrate that there is a significant deviation (the largest being approximately 27%) between the author's experimental results and the simple beam model results. This indicated that a simple beam model can not satisfactorily predict all lower vertical vibration modes for the author's experimental model. Once the in air case is simulated to satisfactory precision Chapter 4. Comparison of Results Station  • Air Mode Shape • Water Mode Shape Figure 4.11: Comparison of Mode Shapes, 2 Node Vertical Mode Chapter 4. Comparison of Results 54 Station • Air Mode Shape • Water Mode Shape Figure 4.12: Comparison of Mode Shapes, 3 Node Vertical Mode Chapter 4. Comparison of Results 55 Station  • Ak Mode Shape • Water Mode Shape Figure 4.13: Comparison of Mode Shapes, 4 Node Vertical Mode Chapter 4. Comparison of Results 56 Station • Air Mode Shape • Water Mode Shape Figure 4.14: Comparison of Mode Shapes, 5 Node Vertical Mode Chapter 4. Comparison of Results 57 Mode Expt. Air Freq Beam Model Air Freq Ratio 2NV 20.6 20.6 1.00 3NV 61.0 73.4 0.83 4NV 109.5 136.9 0.80 5NV 139.0 189.8 0.73 Table 4.8: Comparison of Air Frequencies Between Experimental Model and Simplified Beam Model Mode Expt. Water Freq Beam Model Water Freq Ratio 2NV 8.75 7.01 1.25 3NV 25.5 21.4 1.19 4NV 47.0 38.5 1.22 5NV 67.5 57.1 1.18 Table 4.9: Comparison of Water Frequencies Between Experimental Model and Simplified Beam Model (for the two node vertical mode) sectional added mass derived from HYDRO is then introduced to the beam model. Results (tabulated in table 4.9) from this simulation showed that VLS is correct. VLS predicted the 2 node vertical mode to be 7.16 Hz whereas results from the simple beam model gave 7.01 Hz (recall that experiments showed the 2 node vertical mode in water to be 8.75 Hz). It is thus concluded that both methods have predicted natural frequencies which are significantly different than experimental results. The second beam model approximates the experimental ship model in Glenwright's thesis [10], in which 30 pounds of weight are distributed evenly along the length of the model. Results (tabulated in tables 4.10) and 4.11 from the second beam are encouraging. Experimental frequency for the 2 node vertical in water is 7.75 Hz whereas the beam model predicted a frequency of 7.51 Hz. Thus the second beam model is able to predict ship behavior in water within satisfactory accuracy. Note that the second beam model has also predicted the lower vertical modes satisfactorily. Both the 3 node and 4 node vertical modes have been predicted to within 10 percent of experimental natural frequencies. Chapter 4. Comparison of Results 58 Mode Glenwright's Expt. Air Freq Beam Model Air Freq Ratio 2NV 12.5 12.5 1.00 3NV 33.3 35.8 0.93 4NV 61.8 68.2 0.91 5NV 92.0 107.7 ff.85 Table 4.10: Comparison of Air Frequencies Between Glenwright's Experimental Model and Simplified Beam Model Mode Glenwright's Expt. Water Freq Beam Model Water Freq Ratio 2NV 7.75 7.51 1.03 3NV 21.3 21.6 0.98 4NV 40.8 41.1 0.99 5NV 65.8 64.7 1.02 Table 4.11: Comparison of Water Frequencies Between Glenwright's Experimental Model and Simplified Beam Model Recall that the only difference between the two beam models is in the way the 30 pound load is modeled. Added Mass Matrix - VFS and VPS Approach It was established in the previous section that the VLS approach did not predict the nat-ural frequency satisfactorily partly because the added mass effect was mis-represented. Thus, if fluid dynamics surrounding the ship is modeled more accurately, then the nu-merical results are expected to be much better. The next logical step to take would be to model the fluid domain in three dimensions. This section summarizes results which were obtained from using a fully coupled added mass matrix. The modal analysis program STRUM, written to allow only lumped added mass changes to the global mass matrix, was re-written to include mass changes in matrix form. A new program STRAM (which stands for STRuctural modification using Added Mass matrices) was written to predict structural response in water, given the known structural properties in air and an added mass matrix. STRAM is a flexible program as Chapter 4. Comparison of Results 59 it allows the user several options: 1. Use added mass matrix from finite elements (VFS); the V prefix designates that structural analysis in air is done with VAST and the S suffix designates that pre-diction of structural behavior is done with STRAM. 2. Use added mass matrix from panel method (VPS), 3. Allow specification of degree of complexity. Quick solutions can be obtained using single mode approximations. Complex analysis can be obtained using as many dry modes as the user desires. The solution time for modal analysis increases as the number of modes used increases. Figure 4.15 shows the relationship between the time required to solve a problem with STRAM and the number of modes used for the analysis. Figure 4.15 was derived from a finite element model with 588 degrees of freedom. Clearly, this figure will not be the same for all finite element models. However, the shape of figure 4.15 will be the same for all finite element models: the actual time will be a function of the size of the added mass matrix, as per equation 2.66. Figure 4.15 illustrate that the time required to solve a twenty-mode problem is greater than solving 2 ten-mode problems. Physically this means that it takes longer to solve a 20 by 20 matrix than two 10 by 10 matrices. This result has significant implications. The user may choose to use single mode approximations instead of more complex analysis to save CPU time. The consequences of this approach will result in less accurate numerical results. Thus the user needs to find a compromise between quick solution times and accurate solutions. Results for the different options are tabulated in tables 4.12 and 4.13. Chapter 4. Comparison of Results 5 0 0 to 4 0 0 -1 3 0 0 2 0 0 -4 TOO -4 - 1 1 1 6 12 Number of Modes I 76 I— 2 0 Figure 4.15: Comparison of Solution Times for Modal Analysis Chapter 4. Comparison of Results 61 Mode Experiment S12-F VFS Diff (%) 2NV 8.75 8.76 7.94 -9.4 3NV 25.5 26.43 25.78 -2.5 2NT + 2NH 8.5 8.84 8.49 -4.0 3NT + 3NH 19.25 21.10 20.91 -0.9 4NT.+ 4NH 31.0 33.09 33.36 0.8 5NT + 5NH 40.5 42.71 42.06 -1.5 Table 4.12: Comparison of Natural Frequencies - 14 Mode Approximation, S12-8 Mode Experiment S12-F VFS Diff (%) 2NV 8.75 8.76 7.95 -9.3 3NV 25.5 26.43 26.87 1.67 4NV 47.0 47.94 60.94 27.1 5NV 67.5 70.80 60.79 -14.1 2NT + 2NH 8.5 8.84 8.50 -3.9 3NT + 3NH 19.25 21.10 21.1 0.00 4NT + 4NH 31.0 33.09 33.43 1.0 5NT + 5NH 40.5 42.71 40.48 -5.2 Table 4.13: Comparison of Natural Frequencies - 1 Mode Approximation, S12-8 Results for VFS-14 (table 4.12) were encouraging. Natural frequencies for most beam-like modes have been calculated to within 10 percent of finite element natural frequencies. The 4 node vertical and 5 node vertical modes were not included in the table because they could not be predicted by STRAM. In this case only the first 20 mode shapes of S12-8 were calculated by VAST. However, a floating ship is an unconstrained body, thus the first 6 mode shapes were rigid body modes with zero frequency. Since rigid body modes are not used, only 14 modes can be used by STRAM to approximate ship behavior. Table 4.14 list6 the beam-like modes of S12-8 in ascending order. The 4 node and 5 node vertical modes represented the 18'th and 20'th modes respectively. These higher modes are not expected to give good results because modal analysis uses m vibration modes from the original structure to predict n vibration modes of the modified structure. As a general rule, n is always less than m, with the exception to those models with relatively Chapter 4. Comparison of Results 62 Mode In Air Frequency (Hz) Comment 8 14.2 2NT + 2NH 9 20.37 2NV 10 33.67 3NT + 3NH 11 47.01 4NT + 4NH 12 59.09 5NT + 5NH 13 61.4 3NV 18 108.8 4NV 20 137.8 5NV Table 4.14: Natural Frequencies of S12-8, In Ascending Order small structural modifications. Deeper understanding into the the results of VFS-14 can be gained by looking at orthogonality of air mode shapes with respect to the added mass matrix. It is well known that mode shapes in air are orthogonal to the structural mass matrix; the same is true for water mode shapes with respect to the sum of the structural and added mass matrices. Mathematically, this can be written as 1 ift = j 0 otherwise (4.1) and <f>mT(Mg + Ma)<j>Wi = « (4.2) 1 if» = j 0 otherwise However, air modes cannot be assumed to be orthogonal to the added mass matrix. That is, tat* M a 7 ^ 0 (4.3) For a multi-mode analysis, the governing equation has been shown to be (l + Ma)q + Q q = 0 where (4.4) I = identity matrix Chapter 4. Comparison of Results 63 D.O.F. Mode 1 INT 2 2NT + 2NH 3 2NV 4 3NT + 3NH 5 4NT + 4NH 6 5NT + 5NH 7 3NV 12 4NV 14 5NV Table 4.15: Summary of Degrees of Freedom in Generalized Added Mass for S12-8 Ma = t J M . ^ = generalized added mass matrix 0 = matrix with u>\ on diagonal entries By inspection, this implies that if air mode shapes can be assumed to be orthogonal to the added mass matrix, a multi-mode analysis is identical to a one-mode analysis. Hence the off-diagonal entries of a generalized added mass matrix (with respect to the air modes) provide information as to the contribution of the air modes to the water modes. If ofT-diagonal terms are significant compared to diagonal terms, then a multi-mode analysis is required. However, if diagonal terms are larger than the off-diagonal terms, results for a multi-mode analysis will be similar to a one-mode analysis. The symmetric matrix Mn is shown in figure 4.16. Note that, for identification pur-poses, boxes are drawn on the diagonal degrees of freedom representing vertical vibration modes. Ma is a 14 by 14 matrix, with each degree of freedom representing a specific vibration mode in air. A summary of d.o.f. against global vibration mode in air is given in table 4.15. CPU time for VFS-14 was 4 hours 39 minutes. A complete finite element analysis of Chapter 4. Comparison of Results 64 M. = 0.151 -0.006 0.000 -0.214 0.020 -0.264 -0.024 0.004 -0.006 -0.094 -0.021 -0.001 0.173 0.007 1.795 -0.007 -0.031 0.165 0.039 0.001 0.070 0.052 0.425 0.002 -0.002 -0.008 0.000 |S.564 J 0.000 0.002 -0.014 0.179 0.000 0.679 -0.251 -1.100 0.486 0.011 0.150 1.548 0.086 0.499 0.043 0.018 0.004 -0.019 -0.012 0.000 -0.035 0.002 0.977 -0.108 0.006 0.119 0.052 -0.074 0.005 -0.007 0.021 0.013 1.131 -0.247 -0.019 0.094 0.213 -0.041 0.022 0.222 -0.161 14.2151 -0.003 -1.047 -0.378 0.042 -0.290 -0.059 2.066 0.032 0.000 0.056 0.000 0.000 0.005 0.000 2.198 0.174 -0.903 0.163 0.021 -1.391 1.093 -0.024 0.068 0.121 0.770 0.613 -0.419 0.015 0.115 12.1861 0.004 0.248 1.439 -0.086 [4.140| Figure 4.16: Generalized Added Mass for S12-8, Mu Chapter 4. Comparison of Results 65 S12-F took approximately 143 hours whereas CPU time for modal analysis is approxi-mately 38 hours (11 hours for the structural vibration problem, 22 hours for added mass matrix formulation, and 4.65 hours for modal analysis). This represents a significant saving in cost. The results for a 14-mode approximation are presented in table 4.13 and for a 1-mode approximation in table 4.13. The results for the vertical modes are surprisingly poor. It was first suspected that the computer program STRAM gave incorrect results. A simple check of the program can be done by looking at the equations of motion closely. Recall that the natural frequency of a structure vibrating in air is given as W < , = <t>aTj{.<t>a from which the mode shapes are normalized such that <t>_a T KM <t>a = &\ and <f>_aTM. <j>_a = 1 Moreover, the natural frequency of a structure vibrating in water is given as <f>wT K$ <f>w "W = <j>y,T{M. + Ma)<t>_w Hence u>l _ <J>_*TK.<l>_a <k,T(M. 4- Ma)<t>_w u£ ~ K. «V <f,~TM. i* (4'5) Note that equation 4.5 is an exact correlation; no assumptions have been made. This correlation can be used as a check on the numerical results of STRAM as well as on the errors inherent to the assumption of equivalence of modes. Numerical results from STRAM can be verified by setting <f>a ~ <t>w Chapter 4. Comparison of Results 66 Mode (Hz) vw(Hz) LHS 2NV 20.37 7.95 2.562 3NV 61.4 26.87 2.285 4NV 108.8 60.94 1.785 5NV 137.8 60.79 2.267 2NT + 2NH 14.20 8.50 1.671 3NT + 3NH 33.67 21.1 1.576 4NT + 4NH 47.01 33.43 1.406 5NT 4- 5NH 59.09 40.48 1.460 Table 4.16: Ratio of Natural Frequencies Thus equation 4.5 becomes <j>_aT(M. + Ma)<£a ^ <t>_aTMt$_a y/l + <l>_aTMa ^ (4.6) since the mode shapes have been normalized such that <t>_aTMt<j>_a =1 A computer program, independent of STRAM, was written to find the value of <f>aT Ma <f>a • A check on the results of STRAM can be done by comparing the right hand side of equation 4.6 to the left hand side. Note that in the comparison, u>w is the frequency given by the single-mode analysis and <j>a T Ma <^a is calculated by the inde-pendent program. The results are shown in tables 4.16 and 4.17. Comparisons of the LHS to the RHS show that they are indeed equivalent, which imply that VFS-1 results are correct (under the assumption of mode equivalence). Hence the differences between VAST and STRAM are not caused by numerical errors. Equation 4.6 can also be U6ed to provide an indication of the magnitude of errors introduced by assuming equivalence between wet and dry modes. The natural frequency of any structure can be calculated by the ratio of its potential energy (or generalized Chapter 4. Comparison of Results 67 Mode <t>_aTMa<t>_a RHS2NV 5.564 2.562 3NV 4.215 2.284 4NV 2.186 1.785 5NV 4.140 2.267 2NT + 2NH 1.795 1.672 3NT + 3NH 1.548 1.596 4NT + 4NH 0.977 1.406 5NT + 5NH 1.131 1.460 Table 4.17: Ratio of Natural Frequencies Calculated from Generalized Added Mass stiffness) to its kinetic energy (or generalized mass). Hence, a comparison of how well the air modes approximate the water modes can only be done if the mode shapes are normalized to the same factor. For example, each mode shape in air have been normalized by VAST such that <f>_a T K, 4>a = and tJM. <f>_a = 1 Similarly, each mode shape in water have been normalized by VAST such that 4>w T KM <f>w = and <K,T(M. + Ma)<f>_w = 1 If the mode shapes in water are normalized such that 4>w T KM <K = and hence <j>wT(M. + Ma)<f>w = 4 Chapter 4. Comparison of Results 68 Mode <K,T{M, + Ma)<j>_w 2NV 5.407 3NV 5.397 4NV 5.151 5NV 3.788 2NT + 2NH 2.580 3NT + 3NH 2.546 4NT + 4NH 2.018 5NT + 5NH 1.914 Table 4.18: Generalized Added Mass Calculated from Mode Shapes in Water Mode <p_aT{M. + Ma)4>_a 2NV 6.564 3NV 5.215 4NV 3.186 5NV 5.140 2NT + 2NH 2.795 3NT + 3NH 2.548 4NT + 4NH 1.977 5NT + 5NH 2.131 Table 4.19: Generalized Added Mass Calculated from Mode Shapes in Air The mode shapes <f>a and <j>w are now normalized to the same factor. Comparisons between <py, T(M, + Ma)<f>y, and <f>a T( M, + Ma) <f>_a will show how well the mode shapes in air have approximated the kinetic energy of the ship in water. Results of the comparison are shown in tables 4.18 and 4.19. Note that ww for this comparison is the natural frequency in water calculated by VAST. Tables 4.18 and 4.19 show that air mode shapes did not approximate water mode shapes as well as expected. The differences can be attributed to the fact that, for this model, mode shapes in air ore different to the mode shapes in water (as shown in figures 4.11, 4.12, 4.13, and 4.14). Furthermore, the difference in mode shapes can be attributed to the fact that weight distribution of the ship is different in air than in water. The apparent kinetic energy of the ship is mis-represented by assuming mode equivalence, as Chapter 4. Comparison of Results 69 Mode Glenwright's Expt. New Ship Model 2NV 12.5 11.93 3NV 33.25 32.81 4NV 61.75 56.98 Table 4.20: Comparison of Air Natural Frequencies (Hz) Between Glenwright and New Ship Model Mode Glenwright's Expt. New Ship Model 2NV 7.75 7.93 3NV 21.25 21.88 4NV 40.75 42.0 Table 4.21: Comparison of Water Natural Frequencies (Hz) Between Glenwright and New Ship Model shown in tables 4.18 and 4.19. An additional step was performed to check the validity of numerical results from STRAM. This step involved creating a ship model similar to that of Glenwright's thesis [10] and, using the air modes from that model, estimating the natural frequencies in water and comparing them with Glenwright's experimental results. Glenwright's experimental model, with 30 pounds of weight uniformly distributed along the length of the hull is more representative of a real ship than the model currently used. The results, shown in tables 4.20 and 4.21, indicate that STRAM is capable of predicting structural natural frequencies in water. The coupled horizontal torsional modes were not included in the tables because, in order to minimize computation time and computer resource require-ments, the uniform weight distribution was modeled as point loads: rotational inertia was not modeled. 4.1.4 PANEL Results The PANEL method, which uses boundary element theory, serves as an alternative to the conventional finite element method, in determining the added mass matrix. Derivation of Chapter 4. Comparison of Results 70 Mode Experiment VFS-1 VPS-1 Diff (%) 2NV 8.75 7.95 6.86 -13.7 3NV 25.5 26.87 23.17 -14.0 4NV 47.0 60.94 51.49 -15.5 5NV 67.5 60.79 53.21 -12.5 2NT + 2NH 8.5 8.50 7.87 -7.4 3NT + 3NH 19.25 21.1 19.38 -8.2 4NT + 4NH 31.0 33.43 30.20 -8.7 5NT + 5NH 40.5 40.48 36.97 -8.7 Table 4.22: Comparison of Natural Frequencies - VPS, 1 Mode Approximation the added mass matrix i6 much faster with boundary elements. Approximately 15 minutes of interactive run time was required to generate an added mass matrix as compared to 22 hours CPU time for S12-8 (fluid finite elements). A full finite element solution with PANEL added mass could not be obtained due to hardware constraints. However, a comparison was made between VPS and VFS (using identical fluid meshes) to determine how close the PANEL added mass matrix is to the finite element added mass matrix. Results for this comparison are shown in table 4.22. The vertical modes were calculated to within approximately 15 percent and the coupled horizontal- -torsional modes to within 9 percent. A simple manipulation of added mass matrices will indicate how poorly PANEL compares to VAST. For rigid body heave motion of the ship model, the total added mass is the sum of y d.o.f. added mass entries for all wetted nodes. Hence a simple computer program was written to extract all y d.o.f entries from both added mass matrices. Heave added mass for boundary elements was 0.195 (or 75.3 pounds) compared to 0.129 (or 49.8 pounds) for finite elements. The apparent mass of the ship is 110.9 pounds for PANEL method model and 85.4 pounds for finite element method model. The panel method has over predicted the added mass by almost thirty percent. This is consistent with results from VPS-1, where heave natural frequencies are approximately fifteen percent too low. Chapter 4. Comparison of Results 71 Recall that one of the conclusions deduced from the finite element fluid modeling was that the fluid domain has not been extended out far enough. This implies that the added mass has been under-predicted. In contrast to the formulation of fluid finite elements, equations used by PANEL implicitly assume that the fluid domain goes to infinity. Thus the comparison between PANEL and VAST can be summarized as the comparison between an added mass which is most likely over-predicted and an added mass which has been under-predicted. 4.2 Plate Model 4.2.1 Cantilevered Boundary Condition Another finite element model, with less degrees of freedom than S12-8, was formed to further test STRAM. The model is a steel plate 8 in. by 8 in. and 0.105 in. thick fully immersed in fluid, see figure 4.17. Figure 4.18 shows a comparison between finite element results and STRAM results. STRAM gave relatively good results. The first 19 of 20 modes calculated were within 8% accuracy. Note that 18 out of 19 modes were calculated to less than 4%. Good results were obtained because mode shapes in air are the same as mode shapes in water. This can be attributed to the fact that properties of the plate (i.e. weight distribution) are the same in air as in water. Table 4.23 gives a breakdown of CPU times for finite elements and STRAM. The only significant difference between finite elements and STRAM is in the amount of time for eigenvalue analysis: a savings of 61.76 seconds is made. 4.2.2 Free-Free Boundary Condition A free-free plate similar to the cantilevered plate, with the exception of the boundary conditions, was modeled by VAST. This model has 45 degrees of freedom more than the Chapter 4. Comparison of Results 8. Figure 4.17: Plate Model Chapter 4. Comparison of Results 73 1< -T--2 J -Mode NUmber Figure 4.18: Cantilevered Plate ysis and Modal Analysis - Comparison of Results Between Finite Element Anal-Chapter 4. Comparison of Results 74 Subroutine VAST Modal Analysis K,M Formation 106.17 106.17 K,M Assembly 16.15 16.15 K Addition 8.41 8.41 M Addition 12.82 5.33 Matrix Decomposition 66.33 66.54 Eigenvalue Analysis 1429.22 1167.54 Fluid M Formation 117.84 117.84 Fluid M Assembly 37.54 37.54 Fluid M Decomposition 53.10 53.10 Fluid Added Matrix 80.45 80.45 STRAM 207.25 Total 1928.08 1866.32 Table 4.23: Comparison of CPU Times in Seconds for Cantilevered Plate cantilevered plate, 9 nodes along the previously cantilevered edge with 5 d.o.f. per node. Results for the free free plate are shown in figure 4.19. Table 4.24 gives a breakdown of CPU times for the free free plate. STRAM seems more attractive in this example because a savings of more than one thousand seconds is achieved, compared to approximately 60 in the cantilevered plate model. This is a combination of several factors. 1. The free free plate is a free free model. The stiffness matrix is singular. Recall that the system of equations for a free vibration problem is given by M x + % x - 0 From which the solution is found from K x = w* M x If the structure is unconstrained (i.e. a free free system) the determinant of the Chapter 4. Comparison of Results 75 « -r •14 - L Mode Number Figure 4.19: Free Free Plate - Comparison of Results Between Finite Element Analysis and Modal Analysis Chapter 4. Comparison of Results 76 Subroutine VAST Modal Analysis K , M Formation 107.03 107.03 K , M Assembly 16.66 16.66 K Addition 8.94 8.94 M Addition 12.57 5.80 Matrix Decomposition 121.45 74.52 Eigenvalue Analysis 2321.53 1208.78 Fluid M Formation 117.22 117.22 Fluid M Assembly 37.62 37.62 Fluid M Decomposition 53.72 53.72 Fluid Added Matrix 80.48 80.48 STRAM 151.33 Total 2877.22 1862.10 Table 4.24: Comparison of CPU Times in Seconds for Free Free Plate system of equations cannot be found. Thus to solve this problem, a matrix propor-tional to the mass matrix is added to both sides of the equation, or ( K +ctM)x = (a;8 + a)M x Hence more CPU time is required to decompose the matrices. 2. The free free plate has an extra 45 degrees of freedom. Studies [42] showed that CPU time required to solve eigenvalue problems is proportional to the square of the size problem. Hence more CPU time is dedicated to solve the eigenvalue problem. Chapter 5 Conclusions The use of modal superposition in ship vibration problems was analysed. Several finite element models were used to test numerical results. It was shown that modal analysis satisfactorily predicted ship natural frequencies. Further, it was able to predict these natural frequencies at a fraction of the time required for finite elements. The only assumption made by this method was that structural mode shapes in water can be predicted by a combination of structural modes shapes in air. This is a valid assumption if structural properties does not change significantly with the introduction of an added mass matrix. In hindsight, the choice of experimental model used for this report was not a good one. Because of the skewed weight distribution, an unforseeable problem was encountered during research. Modal analysis of the ship did not give expected results because mode shapes in air for the structure were different than mode shapes in water. Attempts to find a boundary element formulated added mass matrix which is capable of accurately representing the surrounding fluid have failed. However, this problem is not principal to this report and is thus not pursued any further. Modal analysis should only be used to assist Naval Architects at early design stages, where the need for an approximate answer in a short time is greatest. 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