Simetrik gemi hareketleri ve mukavemetinin lineer modal analizi

Abstract

Bir gemi tasarımcısının tasarladığı gemiden beklenen özellikler; geminin taşıdığı yükün zarar görmemesi, mürettebatının çalışma koşul larından etkilenmemesi, taşınan yükün elden geldiğince hızlı taşınma sı, bakım masraflarının az olması, çalışma ömrünün uzun olması şeklin de sıralanabilir. Bu yüzden geminin çalıştığı deniz koşullarına, nasıl cevap vere ceği önem taşır. Ayrıca gemilerin denizde elastik davranışlarda bulun duğu da gözardı edilmemelidir. Geminin kendi yapısının karakteristikleri olan doğal frekans ve titreşim şekillerinin saptanması için kuru tekne analizi yaparken tek ne Timoshenko kirişi olarak ele alınmıştır. Daha sonra ıslak analize geçilerek, dilim teorisi yardımıyla, Lewis konform dönüşümü kullanılarak akışkan kuvvetleri saptanmıştır. Teknenin elastik ve rijid hareketlerini veren hareket denklemi matris formunda oluşturulmuştur. Buradan hareketle, genelleştirilmiş koordinatların elde edilme siyle, şekil değiştirmeler, kesme kuvveti, eğilme momenti cevapları is tenen kesitler için bulunmuştur. Son olarak da düzensiz deniz koşulları, dalga spektrumu kullanı larak yaratılıp, bu koşullarda geminin cevapları elde edilmiştir. Yapılan hesaplamalar daha sonra grafik çizim programları aracılı ğıyla grafiğe dökülmüştür. Burada gözönüne alınan gemi bir savaş gemisi olan destroyerdir. Tekne verilerinden hareket edilerek hesaplamalara başlanmıştır. Hesaplamalar İ.T.Ü. Bilgi îşlem Sistemine uyarlanan programlar a racılığıyla yapılmıştır.

The designer of a ship is expected to satisfy some basic requirements; the ship has to serve the intended function satisfactorily, it has to create the enviroment in which the crew's working conditions allow efficiency, where there is cargo to be transported, the service has to be at the intended rate, maintanence expences have to be reasonable and finally the service life has to be as planned. All these requirements have to be met against the elements faced at sea; extreme wave loading on the struc ture in extreme conditions, deck wetting, slamming, violent motions and accelerations on board, global, and local vibration problems, structural fatigue, etc. Naval architects have long been aware of the fundamental problems of design and many of these have been tackled under what now appears to be a sweeping assump tion that the ship executes rigid motions amongst waves. The structural strenght analysis has been performed in a quasi-static manner. For the analysis of global hull vibrations. The structure has been assumed to be an elas tic beam. In this way, there has been a definitive dis tinction between the rigid and the elastic motions of the hull. This is reflected in the "Seakeeping Theory" and "Ship on a Wave Crest/in a Wave Trough Calculations". In this study, the fundamental theoretical approach employed disregards this rather artificial discrimination between the rigid motions (seakeeping) and the elastic distortions (ship vibrations). Using the linear modal analysis techniques of dynamics, the ship is thought to be under harmonic ex citation by the waves and the resulting motions are mathe matically expressed as a coupled system of equations of motion, encompassing both the rigid and elastic displace ments. Therefore, the solution obtained consists not only of the rigid motions (in this case, heave and pitch) but also the distortions in the vertical plane and the IX associated shear force an bending moment distributions along the beam-like hull. This approach is thought to be essential, because the present day shipbuilding technology makes use of high tensile stell alloy with great reductions in the scant lings, producing highly elastic structures such as ore and obo carriers, tankers, Great Lakes ore carriers etc. The first stage of the linear modal analysis is the free vibration analysis of the dry hull. Here, the hull is assumed to be a uniform free-free Timoshenko beam whose natural modes of vibration include both the rigid and the elastic modes, provided that the principle of orthogonality is valid in both cases. The natural frequencies of the dry hull, together with the natural modes of vibration. The generalised equations of motion are constructed with the generalised mass, structural damping and stiffness matri ces. (See Table 6.3-5). In doing so, all fluid forces (reaction and excitation) are placed on the right hand side of the system of aquations, indicating that the present approach regards all fluid forces as "applied", Further more, the extraction of the natural modes of vibration leads to the determination of the associated modal dist ributions of the shear force and bending moment along the hull. (See figures 6.2-4) The second stage of the analysis, "the wet analysis" is the determination of the fluid forces. In this study, the most advanced form of ship theory of Salvesen, Tuck and Faltinsen (1970) and Vugts (1971) are employed to determine the reaction forces (the hydrodynamic inertia, damping and restoring forces) as well as the wave excitation for a given heading, wave encounter frequency and amplitude, (normally unity in harmonic analysis). The hydro- dynamic coeficents, the added mass and damping, are obtained by the Lewis Conformal Transformation technique [11 ]. There is no theoretical and practical reason for not to employ the more precise "multiparameter conformal transformation" procedures. Lewis Forms are found to be sufficient for the type of ship for which the calculations are performed in the present study. The comparisons of the multiparameter conformal transformations and the Lewis Form calculations may be found in Reference [3] By utilising the results of the "dry" and the "wet" analysis, the generalised set of equations of motion may be written is the matrix form: Here, jd is the diagonal inertia matrix of the dry hull, A is the hydrodynamic mass matrix, b^ is the structural damping matrix, B is the hydrodynamic damping matrix, c is the stiffness matrix of the dry hull, C is the hydrostatic stiffness matrix, E is the wave excitation vector, p is the generalised coodinates (principal coordinates), (Dp is the encounter frequency, all matrices being in the generalised form and all hydrodiynamic force matrices with the generalised coordinates being complex quantities. The solution or the coupled set of equations of mo tion fields the generalised coordinates at as many modes as required for precision. The rigid an J elastic displacements, shear force and bending moment distributions along the hull are calculated by the following expression as functions of either the encounter frequency or the ship lenght/wave length ratio: w(x,t) = e"lüfe f p" w (x) displacement, r=n r r' M(x,t) = e~iwel f pr Mr^x) bending moment, r=o '(x,t) = e"iütet ? pr Vx^ shear force- r=c Here, Pr e e is the generalised coordinate and w (x), M (x), V (x) are respectively the natural mode shape, r r r bending moment and shear force distributions associated with the rth natural mode. When the wave amplitude is unity (e.g lm.) w(x,t), M(x,t) become the complex response xi functions to harmonic excitation and if |w( x, t )|, |M(x, t )| and jV(x,t)|are considered it is seen that the "response ampli tude operators" are obtained. A set of calculated generalised coordinates and the RAO ' s at three positions along the sample ship are given as functions of ship length/wave length ratio in Figures 6.5-7 and 6.8-16 respectively. The last stage of the calculations performed in this study incorporates the ship responses in irregular waves. For this purpose, the ISSC 1967 wave spectrum is choosen as a typical representation of the seaway although many other spectra are commonly used for this type of analysis. For the sample ship travelling withl0.3m/s for ward speed is head waves of significant h,. = 7m and K M 1/3 characteristic period T = lis, the response spectra are obtained for the displacements, bending moment and shear force responses. These results are given in Figures 6.18-23 For the analysis whose procedures are summarised above, the input data describing the ship under investi gation is somewhat different from that required for con ventional seakeeping calculation. Apart from the form definition of the hull, the distribution of mass along the hull as well as the shear and bending rigidity and rotary inertia values for an appropriate number of hull slice are required. The ship investigated in this study is a destroyer whose data had been available. The conditions of operation (head seas, 10,3 m/s ~ 20 knots speed, signi ficant wave height 7m, charecteristic wave period lls) describe a typical extreme case of navigation for this type of ship. The calculated wave spectrum shown in Fig. 6.17 describes a seaway of storm force 8 on Boufort Scala. It also has to be considered that the symmetric responses are the highest when the ship heads directly into the waves. The calculations indicate that the coupling of the elastic modes and rigid modes produce the high structural response where the rigid motions of the hull are dominant (ship's length/wave length ratio approximately unity). For higher values of this ratio the elastic mode dominate and the resonances occur. The resonant encounter frequen cies are smaller than the "dry" natural frequencies because of the increased damping due to the fluid and system is predominantly non-conservative due to both the structural and the hydrodynamic damping. xu The most important result of the analysis concerns ships with hull structures of lower stiffness. It is app arent that such ships are in danger of having low resonant frequencies that may be within the range of the higher frequency energies present in the seaway represented by the tail of the sea spectrum. If this is the case, the time variations of the elastic motions and the structural res ponses (shear force, bending moment) would show the reso nant frequencies superimposed on the encounter frequencies with undesired increase in the amplitude values. The present study is based on the previous work (4,5,14,15]. The available programs are redeveloped and adapted to the computing facilities in the I.T.U. The results are compared with those of the previous work and they are validated. This thesis incorporates the second stage of these studies in the Faculty of Naval Architecture and Ocean Engineering using the present theoretical back ground. This study will continue with the determination of the transient excitation acting on the hull when waves impart impact forces. This later stage of the analysis is more complex with time domain solutions required for the coupled set of equations of motion. This requires a switch from the Lagrangian form of equations to the Hamiltonian forms. In this way, responses to "slamming" can be cal culated without departing from the linear modal analysis.