Lineer olmayan yalpa hareketinin sistematik duyarlılık analizi

Systematic sensibility analysis of the nonlinear roll motion

Tez No: 66736
Yazar: ZAFER ATAR
Danışmanlar: PROF. DR. ALİM YILDIZ
Tez Türü: Yüksek Lisans
Konular: Gemi Mühendisliği, Marine Engineering
Anahtar Kelimeler: Belirtilmemiş.
Yıl: 1997
Dil: Türkçe
Üniversite: İstanbul Teknik Üniversitesi
Enstitü: Fen Bilimleri Enstitüsü
Ana Bilim Dalı: Gemi İnşaat Mühendisliği Ana Bilim Dalı
Bilim Dalı: Belirtilmemiş.
Sayfa Sayısı: 82

Özet

ÖZET Yalpa, geminin boyuna eksen etrafında dönmesi sonucunda oluşan bir harekettir. Denizcilikte istenmeyen bir hareket olan yalpanın yapısı oldukça karışıktır. Yalpa hareketi nedeniyle meydana gelen zorlamalar geminin devrilmesine dahi sebep olabildiğinden, hareketin oluşumu üzerine geliştirilen yaklaşımlar büyük önem taşımaktadır. Yalpa hareketinin yapısının karışık olması, matematiksel modellemesini de güçleştirmektedir. Lineer teori modeline ait formülasyon, büyük genlikli hareketlerde geminin doğadaki yalpa hareketinden oldukça farklılaşmaktadır. Lineer olmayan teori modeli ise, doğadaki yalpa hareketi için daha uyumlu ve sonuçlan da çok daha sağlıklıdır. Bu nedenle son yıllarda yapılan hemen tüm çalışmalarda lineer olmayan teori modeli benimsenmiştir. Bu çalışmada da lineer olmayan model kullanılmış ve yalpa hareketi denklemi sunulmuştur. STAB bilgisayar programı yardımıyla bu denklem çözülmüş ve yalpa hareketlerinin sistematik duyarlılık analizi yapılmıştır. Duyarlılık analizi; üç farklı tipte gemi, GZ-O eğrisinin iki ayrı temsili ve gemi ve deniz koşullarına ilişkin dört parametre değerinin kombinasyonlarıyla gerçekleştirilmiştir. Parametre değerlerinden biri değiştirilirken diğer tüm özellikler sabit tutulmuş ve değişimler eğrilerle gösterilmiştir. Analizler sonucunda oluşturulan eğrilerden; özellikle rezonans frekansı civarındaki yalpa genliği değerlerinde dikkate değer değişimlerin olduğu gözlenmiştir. Rezonans frekansı ve civarında; lineer ve lineer olmayan sönüm katsayıları arttıkça ve gemilerin dizayn metesantr yüksekliği değerine yaklaştıkça, yalpa genliği değerlerinde dikkate değer azalmalar kaydedilmiştir. En büyük dalga dikliği değerlerinin arttırılması durumunda ise yalpa genliği değerlerinde büyük artışlar gözlemlenmektedir. vm

Özet (Çeviri)

SYSTEMATIC SENSIBILITY ANALYSIS OF THE NONLINEAR ROLL MOTION SUMMARY The elements that cause a ship to roll in a sea are primarily due to the unbalanced moments resulting from a shifting center of buoyancy. As a wave approaches and passes a ship, the waterplane is in a state of motion and is inclined at a rate depending upon the frequency, length, and amplitude of the wave. The center of buoyancy, whose position depends upon the slope of the waterplane at any given draft, will move out of a vertical line through the center of gravity if the waterline is inclined. The transverse component of this inclation results in an inclining arm being set up, where by the ship will heel and tend to align itself so that the centerplane will be perpendicular to the wave surface. This action is, however, modified to some extend by the motion of the ship with the water itself. Halfway up the upper slope of a wave, the water particles are moving vertically upward; at the crest they are moving horizontally with the wave front; halfway down the down slope, they are moving vertically downward; and in the trough, they are moving horizontally against the wave front. Thus, a ship floating in large waves will be influenced by the orbital motion of the water itself. At this time, there exist a centrifugal force that must be considered in addition to the force of gravity and buoyancy. This centrifugal force is opposed by the dynamic force of the water that it produces, and consequently there are two distinct couples acting upon the ship and causing its rolling action [1]: 1- The familiar one between buoyancy and gravity. 2- The one between the centrifugal force of the ship moving in an orbital motion and the dynamic opposing force of the water. The second couple is produced by forces that are entirely a function of the wave motion itself. The ship, and more specifically her center of gravity, moves in a circular orbit with a motion having the same period and orbital radius as the particles of water at the same level outside. The resulting centrifugal force, acting through the center of gravity and opposing the hyrodynamic force acting through the center of buoyancy, produces the new couple that has the periodic motion of the waves in which the ship rolls. Consequently, the resulting period of roll is a function of both the ship's natural roll period and the period of the waves. Such a statement, however, is incomplete without taking note of another fundamental fact. A ship rolling in still water in its own natural period was subjected to only a single initial impulse and allowed to roll ixfreely until the energy from this initial impulse was entirely consumed by the resistance of the water. This motion is known as 'free oscillation'. A ship rolling in waves, however, is subject to nearly periodic impulse, with the result known as 'forced oscillation' or 'forced rolling' [1]. It should be pointed out here that these periodic impulses are the humps of an irregular sea, and that these waves tend toward regularity when they are the components of an underlying wave system. Any suspended object, such as a pendulum, if free to oscillate when subjected to a series of forced impulses of regular frequency, will oscillate in the period of applied impulses. Thus the ship, if rolling in waves of exact uniformity and regular period, would eventually assume a period of roll identical to that of the waves. However, real sea waves are not constant in either period, amplitude, or wavelength and will, therefore, produce a series of more or less nonuniform impulses. Under such condition, when the impulses producing the roll are not fairly regular, it is the ever-present tendency of the ship the revert to its own to natural period of roll a period of roll for the ship that is a combination of her own natural period and the period of the waves producing the rolling moment, where the latter is generally the more predominant period [1]. There are two ways of thinking with regard to the possible approach to the ship stability in waves. Some researchers suggest using the available linear ship motion theory to predict the condition which can lead to the vessel capsize. The others pursue development of the nonlinear theoretical model, particularly for rolling motion, by which roll instability can be predicted. Application of both methods over the last decade increased considerably our knowledge of ship stability safety. Equation (3.1) is valid for nonlinear roll motion with nonlinear representation of damping B(ö) and restoring M(O) moments. The nonlinear model involves quadratic or more complex function, particularly in damping B(Ö) and the restoring M(), the restoring moment in equation (3.1), so that the equation of motion correctly represents the roll responses for large angles of roll. Nonlinear theory of roll motion deals with the conditions for which the differantial equation of motion is nonlinear. Most of the time the state of linearity of the differential equation is violated for the cases where the GZ -curve deviates from the control tangent drawn to the curve from the origin of the coordinates or the resistance to the motion is a quadratic or more complex function of the velocity. Two forms of nonlinearity, the one in restoring and the other in damping, are considered to be the most significant ones to affect ship rolling. Along with these two nonlinearities, it is also possible to have nonlinearities due to inertia and exciting forces. These cases are found respectively when the mass moment of inertia is a function of angle of heel and when the amplitude of the exciting force changes with the angle of heel. All the nonlinearities can occur simultaneously or solely, and effect of each nonlinearity may differ one from another. As mentioned earlier, roll motion is the most difficult motion to model mathematically. Consideration of nonlinearities in restoring and damping terms makes it even harder to evaluate. For this reason, certain assumption must be made to simplify the problem without swaying from the main goal. These assumption may be listed as; 1- The added mass moment of inertia has a constant value which is independent of the angle of heel. 2- The inclinations of rolling ship takes place with constant displacement. 3- Rolling is uncoupled, notably heave and sway are absent. 4- The wave is considered to be a regular sinusoidal beam wave. 5- Representation of the GZ -curve is confined to a cubic and a quintic polynomial. 6- Quadratic damping term is considered. 7- The nonlinear terms of restoring and damping moments are small. 8- Steady-state solution of the equation of motion is sought. The moment of inertia of a vessel is equal to the total mass moment of inertia of the vessel in roll motion times the angular acceleration, . Equation (3.14) relates the requared total damping coefficient b with the given total roll decay coefficient n. However, the proposed roll damping model (equation 3.14) requires that the total damping is split up into a linear component bt, proportional to the angular velocity and nonlinear component b2, proportional to the angular velocity squared, as follows in equation (3.15). Putting n = np + nv, where np is the potential (wave-making nw and lift ni) roll decay coefficient and nv is the viscous (friction % hull eddy-making n,. and bilge keels nbk) roll decay coefficient, it follows equations (3.16) or (3.17). By separating the linear and nonlinear parts of the left hand-side damping and equating them to the corresponding terms of the right hand-side of the equation one can obtain equations XU(3.18), (3.19) and (3.20). Buta = ©«.O*, where Oa is the amplitude of the roll motion. Therefore, with the applications of the ship-model testing, np and nv can be calculated and used to evaluate damping input bi and b2 to the present roll motion simulation procedure. The restoring moment of a vessel is equal to the righting GZ times the displacement. The GZ -curve is an odd order polynomial will be used to represent it. Due to mathematical complexity and the difficulties involved in using higher order polynomials, the roll motion model used in the present study will be confined to a fifth order polynomial representation (3.21). Where the coefficients C x, C 3 and C 5 are evaluated by approximating the actual GZ -curve with the fifth-order polynomial. It has been found that this quintic polynomial fits most of the practical GZ -curves with the highest accuracy as compared with other representations. The coefficients C,, C 3 and C 5 are calculated with three distinct characteristics of the given GZ -curve, namely the initial metacentric height GM, the angle of vanishing stability Ov and the area under the GZ -curve A^ equations (3.21a), (3.21b), (3.21c) and (3.21d). The exciting moment due to regular wave having maximum slope of 0Cm is given by the term on the right hand side of equation (3. 1) as -IxxOC. With using the equations (3.22) and (3.24), the equation of exciting moment (3.25) can be evaluated. Where e>e and Gh, are the frequencies of encaunter and waves, and t is the time, 8 is the phase angle, V is the ship speed and \i is the heading angle. For beam waves the heading angle u = 90°, therefore ©e = o». The maximum wave slope ev can be calculated for a wave of the period equal to the ship's natural period (TW=T()), resonance condition) using equation (3. 23). Lw can be calculated using equation (3.27) and used to evaluate maximum wave slope dm. Where Hw, Lw and d are the wave height, wave length, and water depth (constant). The ship rolling period T$ and the roll natural frequency a$ are provided from equations (3.28) and (3.29). Substituting the expressions discussed in the preceding chapter, for the inertia, damping, restoring and exciting moments into the general equation of roll motion (3.1), the following nonlinear equations of roll motion is obtained: (3.30). The square of the velocity is given by $ I I so that, when the sign of O is changed, this term also changes its sign in order that the damping moment will always oppose the motion. Dividing throughout by (1^ + SI**), and substituting equations (3.21a), (3.21b), (3.21c), (3.31a), (3.31b), (3.31c), (3.31d), (3.31e), (3.31f) and (3.32a) respectively, the equation (3.30) becomes equations (3.31) and (3.32). Where m3 XUIand m J are the coefficients of the nonlinear restoring terms, respectively. The solution to equation (3.30) has been found using the Duffing' s method [11]. The obtain solution provides the relationship between the roll amplitude Oa, the encounter frequency oe. A computer program STAB in Pascal has been used to calculate the roll responses according to equation (3.33) and (3.34) [11]. To solve the equations and eventually to determine the maximum rolling amplitude, Om the folowing parameters should be known for a number of encounter frequencies: 1-The maximum wave slope, ctm. This can be calculated with the given wave height and the wave length, equation (3.23). 2- The added mass moment of inertia Sl^ to calculate, X,equation (3.3 If). 3- The linear and nonlinear damping coefficients b, and b 2, equations (3.3 Id) and (3.3 le). 4- The restoring moment coefficients m3 and m 3, equations (3.3 lb), (3.31c) and (3.32a). The oscillatory motion determined by equation (3.30) may have either an established or an unsteady character. When the oscillations are steady, the rolling motion is periodic with a constant frequency coe which equal to the frequency of the exciting force with a consant amplitude and a fixed phase relationship to the exciting forces. Established nonlinear oscillation are often called 'forced oscillations'. In the unsteady regime, however the oscillatory motion is more complex and nonperiodic in nature. In linear rolling, in the unsteady regime of motion, the forced oscillations are added to natural oscillations. When time passes by, natural oscillations are damped out and steady character of the motion is established. In nonlinear rolling of a ship in regular waves, any added irregular exciting force applied for a short period of time can cause an unsteady regime. This type of a regime of oscillations may be called a 'forced oscillatory motion'. The present study will use three kind of ships which have got different types (a container ship, a tanker and a passenger ferry) for the nonlinear roll motion. There are eight following parameters in this sensebility analysis: X, Oa^, bi, b2, cu, ^v, A$v and GM (these eight parameters were dealt with in the last paragraphs). In addition, the roll motion model used in the present study is confined to third and fifth order polynomial representation. When a parameter is changed in orderly intervals, the other parameters are kept constant and variations of the roll amplitude are observed. Thus, there will be twenty-four different solutions. The obtain solutions are presented in graphs which show the relation between the roll amplitude

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