Gemilerin iki boyutlu hidroelastisite teorisi için genel hesaplar

Abstract

ÖZET Dalgalardan dolayı oluşan zorlanmalar, dövünme gibi geçici dinamik yüklerin etkisi ile büyük boyutlara ula şırlar. Bu zorlanmalar, deniz sürekli olarak değişen, an cak olasılık hesapları ile tanımlanabilen bir ortam ol duğundan çoğu hallerde hassas olarak tahmin edilemez. Bu tahminin iyi yapılabilmesi de geminin hidrodinamik kuvvet lerini ve yükleme durumunu iyi bilmeyi gerektirir. Bu çalışmada, ağırlık dağılımı belli olan bir geminin `kuru gemi yapısı analizi` yapılmış ve geminin dinamik yük hesaplarının yapılmasında gerekli olan iki boyutlu eksu kütlesi ve sönüm katsayılarının hesaplanmasına çalışılmış tır. Kuru analiz için gemi yapısı bir `Timoshenko Kiriş` olarak düşünülmüştür. İki boyutlu eksu kütlesi ve sönüm katsayılarının hesaplanmasında ise, Lewis-formu ve konform dönüşüm teknikleri karşılaştırmalı olarak ele alınmıştır. Nümerik hesaplamalar sonucunda çok parametreli kon form dönüşüm metodunun özellikle bulb kesitlerinde ve çene hattına haiz kesitlerde daha iyi sonuçlar verdiği gözlen miştir. Lewis formu metodu ise başlangıç aşamasında iyi bir yaklaşım sağlamaktadır. -vı-

THE FUNDAMENTAL CALCULATIONS OF THE 2~D HYDROELASTICITY THEORY FOR SHIPS SUMMARY It will be helpful to discuss the motions and the effects of waves on ships. When the stresses and strains of the hull is considered in a confused sea, the profound consequences of these effects cannot be ignored. These stresses and strains can split the hull and cause loss of human life. Therefore, it is generally assumed that a ship is a flexible structure due to the stresses and strains of the hull. The ship hull which is a flexible structure might be treated as an elastic beam. The esti mation of hydrodynamic actions on a hull depend on the assumption that the hull is beamlike. This suggests the Timoshenko beam theory which is a more precise analysis [Si' [9]. We have firstly to extract the principal modes and natural frequencies of such a beam. Timoshenko beam is free-free at the ends. Since the slioes of the hull ro tate about axes that lay in the planes of the cross-sec tion, rotatory motions associated with inertia effects cannot be ignored. There are many ways of extracting the principal modes and natural frequencies of a non-uniform beam vibrating in flexure. The most usefull is the Prohl- Myklestod method [6} [7] which is familiar in vibration theory and which can be made to cater for a Timoshenko beam. Thus, using the Prohl-Myklestad method, we can ob tain the bending moment and shearing force distributions corresponding to the natural mode shapes of the beam. The hull is imagined to be cut into a number of slices. It is convinient to make all the slices of the same thickness. The number of slices to be employed de pends on the degree of accuracy that is required and on what use is subsequently to be made of the derived modes and frequencies. Generally, speaking, the greater the number of slices used the greater will be the accuracy of subsequent calculations. In practice, twenty slices at least should be used for most ships. We shall first discuss symmetric deflections. That is to say, we shall discuss bodily motions of heaving and pitching and distortions in which motions are confined to a vertical plane, preserving symmetry. The distortions -vxi-will be assumed to conform to the Timoshenko theory of beam analysis. We shall write the equation of motion of such a beam in matrix form, (a + A) p(t) + (b+p) p(t) + (c+C) p(t) = Z(t) Here, a,b and c are the inertia, damping and stiff ness matrices of the dryhull. A,B and C are also inertia damping and stiffness matrices due to fluid actions, respectively. The values of A,B and C depend upon which of the Strip theory is used. The nature of the coeffi - cients A, B, C is of the utmost importance. They appear in the equation of motion along with the coefficients a, b, c of the dryhull, but they are nothing like as straight forward. Unlike the inertia and stiffness co efficients, the A and C do not form a symmetric array, let alone a diagonal one. Further, the B do not form a symmetric or positive definite matrix an(3- they do not even remotely produce a diagonal one. P and Z are column vectors representing the princi pal coordinates of the response and the input loading, respectively. The upward force Z(x,t) per unit length applied to the slice includes contributions from weight, bouyancy and all other fluid forces. The equations of motion contain constant coefficients which specify the hull's dynamical characteristics. These quantities are the generalised masses a, the generalised stiffnesses C and the generalised damping coefficients b. They can be deduced for a given hull. The inertia coefficients a is given by J (raw W + I 9 e ) dx = a 6 o r s y r s rs rs If no allowance is to be made for rotatory inertia, I must be taken as zero. The generalised stiffnesses C are related to the generalised masses and C is given by Crs = w r ars This is obviously a more economical approach than computing the quantities Crs = fl (EI er 6s * kAG W dx -viix-The generalised damping is represented by the co efficients b = /SakAGy Y + B Eie'e' ) dx rs o 'r s r s This expression is impossible to evaluate theoreti cally, hence the below relationship may be used b = 2 a w v rr rr r r according to elementary vibration theory [10 ], [l]. The values of the elements A,B and C are partially determined by the added mass M(x) and damping factors N(x). The estimation of hydrodynamic loading applied to the hull by a sinusoidal wave has been made in recent years by means of `Strip Theory'. While this approach appears to offer as good a chance as any of success in determining generalised forces, it is unlikely to pro ~ vide much accuracy with any but the lowest modes. If, however, strip theory can cope adequately with the r = 0/1/2 and 3 modes of symmetric motion it could well suffice for practical strength analysis. Strip theory is not uniquely defined, generally accepted formulation of hydrodynamics action and alternative forms have been proposed [ll]. The hydrodynamic actions are initially found from potential flow theory in which the fluid is assumed ir- rotational in compressible and inviscid. The term in volving the fluid damping function N(x) is different in the alteranitve strip theory. The difference apparently originates in the interpretation of the momentum of the fluid in the underlying hydrodynamic theories. In this theory, the effect of the outgoing waves has been includ ed in the expression for the momentum. THE REPRESENTATION OF SHIP HULLS BY CONFORMAL MAPPING AND CALCULATION OF THE HYDRODYNAMIC COEFFICIENTS. The cross-sections of the ship are represented by conformal mapping functions whose coefficients are poly nomial functions of the longitudinal coordinate of the ship. Such a representation of a ship hull is desirable for two reasons : Firstly, it is convinient for calculat ing the hydrodynamic and hydrostatic properties of the ship, secondly, it may be used in the design and construc tion process of the ship. The method consists of fitting -lx-each cross- section of the ship by a mapping function in the least square sense. Then the set of coefficients corresponding to a given term in the mapping function is fitted with a lengthwise polynomial. Mapping techniqu es are extensively used in problems of fluid mechanics» The method can be applied to arbit rary hull surface and describes the added mass of an infinitely long cylinder of ship shaped cross-section, oscillating in a fluid of infinite depth which could be obtained from that of a semi-circle of unit radius by means of the conformal transformation [5], [12 ]. Z = x + iy = ale* l=1 a2n-1 `^^ r - İÖ The foregoing expression, £ = ie represents the semi-circle which is mapped into the ship section. The coefficients a-,, a`,...., a are constants which must be determined for a given section. By restricting the number of parameters N to 2, the 'Lewis form' ap proximation is obtained such that, Z = x + iy = a(C + a± C-1 + a3 C-3 With this assumption the oscillatory velocity po tential is found i n terms of an unknown Green's func - tion, the existence of which is also assumed. This po tential is then used to show the symmetry properties of the damping and added mass coefficients. The transforma tion equation may be written in a parametric form, 1 N n+1 X = a! sin 6 + Z_1 (-l)n x a^^sin (2n-l)e p = 2,...., p N Y = alcos 6.*. l^ (.-1) a2n^ cos(2n-l) 9 p = 1,, p-1 By a suitable choice of the parameters P, N and M the hydro-dynamic coefficients for a variety of shapes may be determined to any required accuracy, using con- formal mapping techniques. Better accuracy is generally obtained in the lower, frequency region and with larger -x-values of M. It is found that the Lewis form approxima tion and the multiparameter fit may, under certain cir cumstances, give significantly different hydrodynamics coefficient values over the entire frequency range [5]. COMPUTATIONS A destroyer hull was assumed to be divided into 20 slices in computing the modal properties of the dryhulls. These divisions were also used in the computation of the fluid actions. For each section, the added mass m(x,oj) and fluid damping N(x,w) were calculated for a range of frequency using conformal mapping technique and a Lewis form fit. The calculations have been `made by using com puter programs. These programs consist of the dry. hull analysis, Lewis form fit and conformal mapping technique calculations. CONCLUSIONS The principal modes and natural frequencies might be deduced by estimating that the hull is beamlike. This suggests the Timoshenko beam theory. The Prohl-Myklestad finite difference method is the most useful method which is familiar in vibration theory and which can be made to cater for a Timoshenko beam. The curves of the principal modes which were obtained using Timoshenko beam estima - tion are adequate if they are compared to others. Further The inertia, damping and stiffness matrices of the dry hull were deduced. The values of the elements A, B and C can also be determined by using the added mass and damping factors calculations. The estimation of hydrodynamic loading applied to the hull can be made by means of 'Strip theory' Thus, the equation of motion can be solved. Two general approaches have been mentioned by means of which added masses and damping coefficients may be estimated. First, the Lewis form fit employes a conformal transformation with two coefficients. Secondly, a more comprehensive use may be made of conformal mapping. The added mass and damping coefficients for ship-shaped sec tions are obtained over a range of non-dimensional fre - quency w2$/2g for a number of sections. Where conformal transformations of higher order than those of Lewis's method were used, the transformation coefficients were determined in a least square sense. The multi-parameter fit reproduces the sectional shape with reasonable ac curacy whereas the Lewis form fit give poor representa tions for chine sections, fine sections and bulbous sec tions. Close agreement is fand for triangular and rectan gular ship sections if we compare the two methods. -xx-