Bir gemi kirişinin serbest titreşimleri

Abstract

Gemi ve uçak yapısı olarak gözönüne alınabilen Timoshenko kirişinin çözümü Galerkin yöntemi kullanılarak yapılmıştır. Timoshenko kirişi, değişken kesite sahip olup kayma etkisinide gözönünde bulundurmaktadır. Kayma etkisi, eğer kirişin uzunluk-derinlik oranı oldukça küçük ise, genellikle çok önemlidir. Kirişin iki uçuda ankestre bağlı olduğunda, Timoshenko kirişinin öz değeri, uniform bir kirişin öz değeri ile tamamen aynıdır. Thomson'un analitik çözümü, hem Euler-Bernoullu kirişinin hem de Timoshenko kirişinin analitik çözümü ile karalaştırılmıştır.

The subject of vibration deals with the oscillatory of bodies. All bodies possesing mass and elasticity are capable of vibration. Most machine and engineering staictures experience vibration in differing degrees, and their destng generally requires consideration for their oscillatory behavior. Oscillatory systems can be broadly characterized as linear or non-linear. For linear systems the principle of superposition holds, and mathematical techniques available for their treatment are well-developed. In contrast, techniques for non linear systems are less known and difficult to apply. Some knowledge of nonlinear systems is desirable since all system tend to become nonlinear with increasing amplitude of oscillation. The time elapsed while the motion repeats itself is called period. The number of complete cycles in a unit of time is designated as the frequency of vibration. The peak value of the motion is called the amplitude. Vibration of linear systems fall into two general classes : free and forced. The system under free vibration vibrates at one or more of its natural frequencies which are the dynamical system. When the frequeny of the exciting force coincides with one of natural frequencies of system, a condition of resonance is encountered, and dangerously large amplitudes may result. Consequently, the calculation of natural frequencies is of interest in all types of vibrating systems. Vibrating systems are all more or less subject to damping because energy is dissipated by friction and other resistance. The primary objectives of the control of vibration in ships are to achieve acceptable environmental conditions for ship personnel and passengers, and to VIII obtain conditions to meet operational requirements. Greater speed, increased power, reduced structural weight, and slimmer lines of ships have created many problems that must be solved in order to achieve acceptable conditions. Machinery and equipment can be affected by excessive vibration, either through maloperations or failure. On passenger ships, passengers and crew may find excessive vibration uncomfortable and fatiguing ; vibration can reduce efficiency of lookouts and of personnel at other duty stations on naval ships. Steady-state vibration usually does not affect the strength of ships, although transiet vibration may. Naval ships and commercial ships that might be requisitioned for service in times of national emergency must be able to withstand shocks generated by air blasts and underwater exploisions in order to minimize damage or maloperation of shipboard machinery and equipment. The equations of Timoshenko's beam theory are derived by integration of the equations of three-dimensional elasticity theory. A new formula for the shear coefficient comes out of the derivation. The Galerkin method is used to perform the free flexural vibration of the Timoshenko beam, which may simulate ship hull, aircraft structure. The Timoshenko beam has variable cross section and includes the effect of shear deformation, which is usually important if span-depth ratio of the beam is relatively small. When the ends copletelly fixed, the eigenvalues of the Timoshenko beam are the same as those of uniform beams. For the clamped- clamped beam analytical solition is compared to that handled both Euler-Bernoullu and the Timoshenko beams. In the analysis of the ship hull vibration, The first mathematical treatment of the flexural vibrations of a ship hull was modelled as a Euler-Bernoullu beam that has constant variable cross section. Nowadays, The Timoshenko beam having IX variable cross section and including rotary inertia and shear deformation has been widely used to represent the structural behaviour of the hull [9,10,1 1,12,13]. Accurate determinations of the lower modes of the ship hull vibrations can be made by modeling the hull as a one-dimensional beam. For higher modes of vibration, however, one-dimensional beam model is inadequate because distortions of the hull occur. For higher modes of vibration, the ship hull has to be regarded as a complicated three- dimensional shell staicture. Charpie and Burroughs [14] studied the Timoshenko mode that include the effects of shear deformation, rotary inertia, and centerline extensibility is compared to other theoretical models and experimental results. The equation of motion is solved by an extension of the classic the Galericin method in the [14]. Finite element fomulation for thin-walled beams with arbitrary cross sections has been developed by Chen and Blandford [15] based on Timoshenko's and Vlasov's beam theories. The elements consider both the warping effect and the shear deformations caused by the bending moments and bimoment in [15]. A paremetric study of the vertical free vibrations of a ship hull is presented by Chen and Bertran [16]. All the effects of the rotary inertia of mass, shear distortion, trust force, shear lag and various damping compenants are taken into account in analysis in [16]. An analytical procedure to estimate both the natural frequencies and mode shapes of thin-walled members is presented in [17]. An analytical solution of a pantoon flexural vibration and vibration of its substructure is presented in [18]. A recent extensive investigation of free flexural vibration of non-uniform beams is made by Abrate [19] using the Ritz method with Euler-Bernoulli beam having variable cross section. In the present work the free flexural vibration of the ship beam is studied by using the Galerkin method. Influence from shear deformation on the ship beam is investigated. The analytical solution taken Thomson [20] and Abrate [19] is compared with the present results, and the displacements are expanded in a series of polynomial approximation functions that can be satisfy boundry conditions at x = 0 and x = L, respectively. Under the class of the method of weighted residuals (MWR), a wide varety of approximate techniques, such as the Collocatin, the Galerkin, the Subdomain methods and method of least squares, method of moments are included. All the MWR assume trial solutions usually in a polynomial form with undetermined coefficients. Each method uses a different approach to evaluate the coefficients through minimization of the error in a weighted form. Comparative studies and representative applications can be found in the [6,7,8]. The Galerkin method [7] is used to study the free fluxeral vibration of the ship beam. GL = 1 8r i(x)dx where 8r is the error function over the domain and