Kiriş destekli plak yapılarının serbest titreşimlerinin teorik ve sonlu eleman yöntemiyle analizi
Theoretical and numerical modal analyses of beam supported plates
Yazar: FULDE GÜNDÜZ
Danışman: DOÇ. DR. ADİL YÜCEL
İstanbul Teknik Üniversitesi / Lisansüstü Eğitim Enstitüsü / Makine Ana Bilim Dalı / Katı Cisimlerin Mekaniği Bilim Dalı
Derece: Yüksek Lisans, 2024
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Konsol kirişler = Cantilever beams ; Serbest titreşim analizi = Free vibration analysis ; Titreşim frekansı = Vibrational frequency ; Çelik levhalar = Steel plates

Özet:
Hız ve hafif yapıya eğilimin arttığı günümüz teknolojisinde, makina ve tesislerin titreşim etkilerinin göz önüne alınması büyük önem arz etmektedir. Bu kapsamda titreşim kontrolünde kullanılan çeşitli metotlar arasında dinamik titreşim absorberlerinin özel bir yeri vardır. Dinamik absorberler, klasik kütle yay sistemlerinin sönümlemesine ek olarak kiriş, plak ve kabuk titreşimlerinin sönümlenmesinde de başarıyla kullanılmaktadır. Günümüz modern teknolojisinde ise klasik kütle yay absorberleri yerine farklı yapıda kirişler dinamik absorber olarak kullanılabilmektedir. Bu çalışma, klasik kütle-yay absorberinden sonraki aşamayı, kiriş biçimli dinamik titreşim absorberini araştırmakta; bu absorberlerle plak titreşimlerinin etkin biçimde kontrol edilebileceğini teorik ve numerik olarak göstermektedir.  
Bu çalışmada eğimli kiriş biçimli dinamik titreşim absorberleri ile desteklenmiş; üç kenarı serbest, bir kenarı ankastre mesnetli olan konsol plak modelinin serbest titreşim davranışı incelenmiştir. n sayıda kiriş için farklı açı değerlerinde teorik ve numerik çözümleme gerçekleştirilmiştir. Doğal frekans değerlerinin analitik olarak hesaplanmasında enerji metodu uygulanmıştır. Bu şekilde kiriş biçimli absorberlerin sayısının ve açısının ana sisteme tabii frekans değişimi yönünden etkisi ve rezonans tehlikesini minimize etmek için en uygun kiriş kombinasyonu araştırılmıştır. Analitik çözüm sonucunda kiriş açısı ve sayısına bağlı olarak doğal frekans değerini Ritz enerji metodu kullanarak hesaplanmıştır. Teorik çözümlemeye ek olarak sonlu eleman analizleri ile incelen yapının doğal frekans değerleri hesaplanmıştır.  Elde edilen sonuçlara göre plak teorisi ve enerji metodu kullanılarak yapılan analitik çözüm sonuçlarının ve sonlu elemanlar yöntemi kullanılarak elde edilen analiz sonuçlarının örtüştüğü görülmüştür. Sonlu eleman yöntemleri ile elde edilen sonuçlar kiriş sayısı ve açı değerine bağlı olarak grafikler haline getirilmiştir. Elde edilen grafik sonuçlara göre kiriş sayısının ve mesnetleme açısının plağın titreşim davranışı üzerine etkisi incelenmiştir. 
Plak titreşimi ile ilgili literatürde sönüm elemanı olarak sembolik viskoz damper kullanılarak yapılan birçok çalışma mevcuttur. Bunun yanında dinamik sönüm elemanı olarak kiriş kullanılan yapılarla ilgili çalışmaların yetersiz olduğu görülmüştür. Bu çalışmada sönüm elemanı olarak kullanılan kiriş destekli konsol plakların serbest titreşimleri araştırılarak literatürdeki eksikliğe katkı sağlamak amaçlanmıştır.

Abstract:
Plates are one of the carrier systems frequently preferred in engineering applications. Geometrically, they are planar elements whose thickness is much smaller than the other two dimensions. The plane at a distance of h/2 from the surfaces of a rectangular plate of dimensions axbxh is defined as the middle surface of the plate. While the thickness value is fixed, there are also plate structures with variable values. Plates are one of the carrier systems frequently preferred in engineering applications. Geometrically, they are planar elements whose thickness is much smaller than the other two dimensions. The plane at a distance of h/2 from the surfaces of a rectangular plate of dimensions axbxh is defined as the middle surface of the plate. While the thickness value is fixed, there are also plate structures with variable values.
Plate type structures, which have a wide range of usage in many different areas in the industry, are gaining more importance day by day. Plate elements are preferred as structural components in engineering structures. Plate structures such as dams, bridges, wall panels and channel covers, which we frequently encounter in the construction industry, also have effective use in the ship, space and aviation industries. In addition, it is possible to encounter plate structures in the automotive industry, from the bodywork to the small thin circular elements used in hard disks. There are many applications of plate structures in the machinery industry, from micro to macro dimensions, including aircraft wings.
Numerical analysis of plate structures, which is one of the main issues of strength among load-bearing elements, can be done largely through the equations of elasticity theory. For different plate structures, these differential equations give exact solutions only under certain boundary conditions. In this case, in addition to analytical solutions, different energy methods are applied according to the plate problem and results close to the numerical solution are obtained. Nowadays, numerical analysis can be performed for many different plate problems with the finite element method, mostly using computer software.
Plate theory is used to solve static and dynamic problems such as stress, deformation, and vibration in the plate, which is unloaded or under a certain load. Analyzing the vibration behavior of plate structures is very important to be protected from the harmful effects of vibration. Especially in the field of aviation, the vibrations that occur when critical speeds are exceeded can reach dangerous levels, which necessitates analysis on the vibration behavior of the plates. Obtaining the vibration characteristics of a three-dimensional continuous plate involves great mathematical difficulties. For this reason, a practical approach is taken by transforming plate problems into a two-dimensional problem.
The plate structure discussed in this thesis study was analyzed by assuming a thin rigid plate. These are plates with plate thickness in the range of 8…10≤a/h≤80…100, where h is the plate thickness and a is the other section length. Thin plates are the most used type of plates. Thin plate behavior is divided into two according to plate deflection. These are called rigid plate if w/h≤0.2, where w is deflection and h is thickness, and if w/h≥0.3 it is called flexible plate. Basically, in engineering studies, except for exceptional cases, rigid plate is accepted when examining plate behavior. With the rigid plate approach, the problem is reduced to a much simpler form.
Vibration effects are of great importance in plate constructions, which are frequently preferred in engineering applications. Above a certain critical speed, serious damage may occur when vibrations caused by environmental conditions reach dangerous levels. The vibration behavior of the system depends on the source causing the vibration, the system-environment interface, and the system's own internal dynamics. Vibration problems that may occur at the source and interface are solved by taking various isolation measures. The vibration behavior of the system in response to the force mainly occurs depending on the natural frequency value of the system.
When the damages caused by vibration are examined, it is seen that the damage occurs as a result of the system being forced at one of its natural frequency values and the resulting resonance phenomenon. In order to prevent such situations, the natural frequency values of the system are calculated, and it is ensured that the warning frequency is not at one of these values. However, this solution is often not applicable in practice since it is not possible to change the current speed and load characteristics under certain production conditions. Instead, it is important in system design, material selection, etc. A more appropriate solution would be to change the natural frequency of the system with possible changes.
In this thesis study, the vibration behavior of a fixed support plate along one side and a free cantilever plate along the other edges, which has intensive industrial application, was examined. The cantilever plate is supported using beam-shaped dynamic absorbers. The changes in the vibration behavior of the system were analyzed when these beams, used as damping elements, were connected to the console plate at different angles. Natural frequency values were calculated by taking into account the internal damping of this plate-beam structure, which was created with a beam support instead of a classical damping element.
The analysis was done both analytically and numerically using the finite element method. In the created plate beam system, the internal damping of both the beam, which acts as an absorber, and the cantilever plate, which is the main element, are included in the calculation. By comparing the results obtained from two different solution methods, it was seen that the results were compatible. Accordingly, the optimum beam angle and number were evaluated for the most appropriate vibration behavior in this type of constructions to be used in practice.
In the analytical solution, the motion of plates, which are continuous elastic systems, can be expressed mathematically. Using this differential equation, the vibration behavior of different plate systems is investigated. The equation of motion of the thin plate, written mathematically according to the balance of forces, is written depending on the material density and plate thickness. In the problem we are considering, a homogeneous dynamic differential equation is used by arranging the deflection function so that the external force is zero. Using this equation, the free vibration response of thin plates was mathematically analyzed in the Cartesian coordinate system. This differential equation, which is also a boundary value problem, can be obtained by writing appropriate boundary conditions in different support situations, such as natural frequency values, mode shapes and frequency parameters of the plate.
In solving plate problems, the equation expressing plate vibrations was rewritten according to boundary conditions. It is only possible to write the boundary conditions that provide the solution of the differential equation for the cantilever plates, which are the subject of this study, for the recessed edge. . In this case, geometric boundary conditions are equations written only for the fixed support. The boundary conditions written on the other three edges are not geometric, but dynamic or natural boundary conditions that indicate the force and moment state. Since the console plate is such a difficult solution, approximate methods have been developed in the literature to solve this problem.
In this study, the method called the Ritz method, which provides geometric boundary conditions as the solution function and aims to solve the function extremum problem by placing the selected expression depending on the uncertain coefficients into the function that should be extreme, was used. Accordingly, the approximation of the results obtained varies depending on the number of functions. In this case, it can be considered as if there are built-in-freely supported (cantilever) beams in the direction of the x and y axis of deflection functions, and as if there are beams with two free ends in the y axis direction. However, in order to ensure convergence in the plate with free edges on three sides, it was deemed appropriate to take free beam functions in the y-axis direction. Similarly, the motion functions written for the support beams were written into the potential and kinetic energy change equation of the system written according to Hamilton's principle, and the differential equation of the motion of the absorber plate was obtained. Natural frequency values were obtained by differentiating the resulting differential equation according to the C_mn coefficients and setting the determinant of the coefficients to zero.
The same models were modeled with different beam angles and numbers, and mode shapes and frequency values were determined by finite element analysis. According to the outputs obtained from the relevant software, the effect of beam angle and number on vibration behavior was investigated. The outputs were evaluated, and critical frequency values were tabulated according to mode shapes. The results were compared with the data obtained by the analytical method, and it was seen that the results were compatible.
There are many studies in the literature on plate vibration using symbolic viscous dampers as damping elements. In addition, it has been observed that studies on structures using beams as dynamic damping elements are insufficient. In this study, it is aimed to contribute to the deficiency in the literature by investigating the free vibrations of beam-supported cantilever plates used as damping elements.