Free vibration analysis of foam core beam structures via differential transform method

Traditional structures unable to meet the needs of today. The use of composite structure instead of the traditional is increasing day by day. Especially in the aviation sector, to the weight of the airframe is affected the number of passengers will be transported directly and the business items such as fuel consumption. As a result of this the use of composite has become a necessity. Sandwich structures can be classed as composite materials. A comparatively thick and light core, also has decent stiffness in a perpendicular direction to the panel faces, is enclosed by two facings. Lots of alternative forms of sandwich structure might be attained by associating distinct facing and core materials. Steel, aluminium, wood, fiber-reinforced plastic or concrete might make up the facings. And cork, balsa wood, rubber, solid plastic material, rigid foam material, mineral wool plates or honeycombs of metal might make up the core. There are variety of composite sandwich structures in aircraft applications. Aerodynamic fairings, covers, doors, radomes, leading and trailing edge fairings, engine cowlings and landing gear doors are typical examples of sandwich structures on aircrafts. In addition to this, there are composite sandwich control surfaces in Airbus fleet, such as rudder, aileron, spoiler. In the passenger compartment, fairings and floor panels are examples of the sandwich structures inside the aircraft. There are so different necessities to use composite sandwich structures in commercial aircraft. Large scale of working temperature and high aerodynamic forces affect on the external structures. External structures, such as Radomes and leading edge fairings, are affected by swarm of bird and atmospheric phenomena. Furthermore requirement of electromagnetic transmission of radar and avionics is so important. One of the main threat on the lower side of the aircraft is runway debris causing foreign object damage. The strength of the floor panels needs to be reinforced to compensate transverse, discrete loads caused by passengers. Therefore it is clearly seen from the examples, how many diversity and environmental necessities of different sandwich applications. So, being familiar with the different necessities as possible as early in the designing takes more advanced and proper structure. Vibration analysis in aircraft as well as in the entire structure which must under continuous dynamic loading is an important phase of aircraft design. To increase the fatigue life of aircraft structures under these loads, to prevent resonance and to anticipate the frequency and structure relation should be considered carefully. For this reason dynamic behaviour of sandwich structures is intestigated more than half century. The sandwich structures offer the designers higher strength to weight ratio, and this is one of the main reasons that the research are conducted in this area. In aerospace application without compromising the strength, weight saving is an important part of design period. The literature on the free vibration analysis of sandwich structures are mainly about the three-layered elastic beams which are homogeneous and isotropic, but joined together rigidly, and the top and bottom layers are mostly made of strong materials such as aluminium and steel. The middle layer is relatively soft, for instance, rubber or honeycomb structures, in order to provide enough damping and good energy absorption characteristics. Before examining the vibration characteristics of the sandwich beams, extensive research has been done in order to understand the sandwich structures. First it is mentioned about definition of the sandwich beams and reason for the usage of sandwich stuctures. Sector based application areas, the reason of deciding sandwich stucture, the advantages and disadvantages are discussed. Due to the its structure sandwich beams is effected to much by the shear stress. The analysis used in this study is utilized from Shear and Timoshenko beam theory which based on shear deformation. With the help of this theories It is described in detailed way how to find the natural frequency of the sandwich beams with various boundary conditions. The resulting formulations are listed and dissolved by using Mathematica program. In addition to analytical solution, linear and nonlinear ordinary differential equations are obtained by differential transform method which is used for obtaining the solution of partial differential equations. This method involves coefficient calculated differently that used in Taylor series solutions of differential equations. Calculation of high order derivatives in the Taylor series solution is a long and difficult process, but differential conversion method provides to convert differential equations into algebraic expressions so that finding solution will be quick and easy. Differential transformation method which obtains the approximate solutions like other numerical methods has a simpler procedure when compared to other methods. This feature makes it easier to solve the computer-aided solution to linear, nonlinear partial differential equations and also expanding the application areas. This thesis free vibration characteristics of the sandwich beams is compared examining the numerical and analytical solutions work under various boundary conditions. The natural frequencies of the sandwich beams which has different materials and geometric features is obtained by numerical and analytical solutions. Results are presented comparatively in tables and graphs. At results section all calculations and analysis are reviewed.