İstanbul Teknik Üniversitesi / Fen Bilimleri Enstitüsü / Makine Mühendisliği Anabilim Dalı / Katı Cisimlerin Mekaniği Bilim Dalı
2x400 ton portal krenin matematiksel ve sonlu elemanlar yöntemiyle düzlem titreşim analizi
In-plane vibration analysis of 2x400 tone gantry crane by mathematical and finite element methods
Doğadaki her cisim, "Doğal Titreşim Frekansı" olarak adlandırılan, sonsuz sayıda titreşim frekansı ve genliğine sahiptir. Bu frekansların hesaplanması ve genliğinin bilinmesi, titreşim kaynaklı mühendislik problemlerinin çözülmesinde temel önem taşımaktadır. Basit cisimlerin doğal frekans ve şekillerini analitik olarak hesaplamak mümkündür. Ancak karmaşık şekillerin hesabı numerik yöntemlerle mümkündür. Sonlu elemanlar yöntemi ve bilgisayar hesap kapasitelerindeki gelişmeler, karmaşık yapıların, ancak idealleştirme yapılarak hesaplanabilen doğal frekans ve şekillerinin daha doğru ve anlaşılır hesaplanmasına imkan tanımışlardır. Konstrüksiyonlarda da kimi zaman çevresel etkilerden kimi zaman da nereden kaynaklandığı bilinmeyen titreşimlere rastlanır. Bu titreşimler bazen bir konstrüksiyonun çökmesine kadar giden ciddi sonuçlara sebep olabilir. Bahsedilen bu titreşimlerin belirlenmesinde öncelikli olarak irdelenmesi gereken 'Doğal Frekans' ve 'Rezonans' kavramlarıdır. Doğal frekans, bir cismin sadece kütlesine ve esnekliğine bağlı olan ve cismin o frekansta uyarılırsa yüksek genlikle ve sürekli titreşeceği frekansa denirken; rezonans, bir cismin doğal frekansıyla çakışan bir frekansta uyarılmasıyla ortaya çıkan fiziksel olaya denir ve rezonansa girmiş cisim aşırı şekilde titreşmeye devam eder. Konstrüksiyonların dizaynı sırasında dikkate alınması gereken en önemli sorunlardan birisi olan rezonans sorununun önlenebilmesi için farklı yöntemler kullanılabilmektedir. Karmaşık olmayan sistemler için analitik yaklaşımlar kolaylık sağladığı gibi sayısal yöntemlerle doğrulanması da hesaplamalarda hataların belirlenmesi ve sonradan karşılaşılabilecek sorunların önüne geçilmesi için faydalı olur. Bu çalışmada öncelikli olarak krenlerle ilgili tarihsel gelişim ve kren çeşitleri tanıtılmıştır. Sonrasında, Euler-Bernoulli kirişi enine titreşimi yaklaşımına uygun olarak, portal krenin düzlem titreşiminin matematiksel modelin oluşturulması için uygulanacak metod tanıtılmıştır. Bir sonraki bölümde ise sonlu elemanlar metodu ve ANSYS Workbench 14.5'te modal analizin yapılışı anlatılmıştır. Sonraki bölümde, 2x400 ton portal krenin düzlem titreşim analizi için matematiksel modeli oluşturulmuş, özfonksiyonlar ve özdeğerler yardımıyla bu krene ait düzlem titreşim durumundaki doğal frekanslar elde edilmiştir. Aynı bölümde sonlu elemanlar metodu kullanılarak 2x400 ton portal krenin modal analizleri yapılmış ve doğal frekanslar elde edilmiştir. Son bölümde ise bir önceki bölümde elde edilen doğal frekans değerleri karşılaştırılmış ve karşılaştırma sonucundaki değerlendirmeler sunulmuştur.
Every object in the nature has infinite number of vibration frequency and amplitute as called 'Natural Vibration Frequency'. The calculation of these frequencies and to know the amplitudes of them are essential in solving of the vibration-induced engineering problems. It is possible to calculate the natural frequencies and shapes of simple analytically. However, the calculation of complex shapes is possible with numerical methods. Development at Finite Element Method and computerise capacities allow to be calculated marely idealized calculation of natural frequencies and shapes of complex structures more accurate and understandable. Also at constructions, sometimes couse of environmental impact and sometimes couse of unknown reasons, vibrations are encountered. These vibrations can lead to serious consequences, sometimes leading up to the collapse of a construction. Determination of mentioned vibrations, concepts of 'Natural Frequency' and 'Rezonance' should be examined firstly. Natural frequency is a frequency which is only up to mass and flexibility of an object and if it is induced at that frequency, it will vibrate continuously and at high amplitude. If an object is excited by a frequency coincides with the natural frequency of that object, a physical event occurs and that is called rezonance and resonate body continue to vibrate excessively. Different methods can be used to avoid the resonance problem which is one of the most important issues to be considered during the design of structures. Analytical approaches for non-complex system makes it easy such as verifiying by numerical methods for detecting errors in the calculations and preventing the problems that may be encountered later. In this study a dual-trolley (2x400 tons) heavy duty overhead gantry crane that can carry loads up to 800 tons was analyzed by mathematical and finite element methods. The more precise dynamical analysis of engineering structure is based on the assumption that a structure has distributed masses. In this case, the structure has infinite number degrees of freedom and mathematical model presents a partial differential equation. Additional assumptions allow construction of the different mathematical models of transversal vibration of the beam. The simplest mathematical models consider a plane vibration of uniform beam with, taking into account only, bending moments; shear and inertia of rotation of the cross sections are neglected. The beam upon these assumptions is called as Bernoulli-Euler beam. In this study the mathematical method is based on Euler-Bernoulli transverse vibration approch. It was recognized by the early researchers that the bending effect is the single most important factor in a transversely vibrating beam. The Euler-Bernoulli model includes the strain energy due to the bending and the kinetic energy due to the lateral displacement. The Euler-Bernoulli model dates back to the 18th century. Jacob Bernoulli (1654-1705) first discovered that the curvature of an elastic beam at any point is proportional to the bending moment at that point. Daniel Bernoulli (1700-1782), nephew of Jacob, was the first one who formulated the differential equation of motion of a vibrating beam. Later, Jacob Bernoulli's theory was accepted by Leonhard Euler (1707-1783) in his investigation of the shape of elastic beams under various loading conditions. Many advances on the elastic curves were made by Euler. The Euler-Bernoulli beam theory, sometimes called the classical beam theory, Euler beam theory, Bernoulli beam theory, or Bernoulli-Euler beam theory, is the most commonly used because it is simple and provides reasonable engineering approximations for many problems. However, the Euler-Bernoulli model tends to slightly overestimate the natural frequencies. The procedure of determining eigenfrequencies at complex systems (systems with large number of the freedom degrees) is the most expensive phase of dynamic analysis. Previous studies of determining its eigenfrequencies of complex supporting structures were based on the use of approximate expressions and methods. Accurate determination of eigenfrequencies was limited to the simple supporting structure (simple beam and console). Finding out solutions of frequent equation for complex elastic bodies is very difficult, because it contained the trigonometric and hyperbolic functions. Nowadays, mathematical software enables routine solving of frequency equations for complex elastic bodies oscillation. Methodology of solving frequent equation is illustrated by example of portal crane supporting constuction using Mathematica software. Accurate determination of eigenfrequencies is important from the aspect of optimizing supporting structures. But, in the case of too complex supporting structures, using of method of distributed masses is limited. In this case, to determine eigenfrequencies of supporting structure, we opt for the method of consistent masses or method of directy concentrated masses. Methodology of solving frequent equation for both methods is illustrated by identical example. On the other hand, finite element method is one of the most common numerical methods that can solve many engineering problems in range from solid mechanics to acoustic. By this method, strength analyzes can be made rapidly, reliably and non-destructively. From the first invention in 1941, finite element method developed rapidly and with the help of improving computer technology became very popular. Its popularity comes from his realistic results which were taken from the comparisons between finite element method and analytical approaches. A variety of specializations under the umbrella of the mechanical engineering discipline (such as aeronautical, biomechanical, and automotive industries) commonly use integrated finite element method in design and development of their products. Several modern finite element method packages include specific components such as thermal, electromagnetic, fluid, and structural working environments. In a structural simulation, finite element method helps tremendously in producing stiffness and strength visualizations and also in minimizing weight, materials, and costs. Finite element method allows detailed visualization of where structures bend or twist, and indicates the distribution of stresses and displacements. Finite element method software provides a wide range of simulation options for controlling the complexity of both modeling and analysis of a system. Similarly, the desired level of accuracy required and associated computational time requirements can be managed simultaneously to address most engineering applications. Finite element method allows entire designs to be constructed, refined, and optimized before the design is manufactured. In the first chapter, historical development of cranes and crane types are introduced primarily. Then, according to Euler-Bernoulli transverse vibration approch, the applied method for the creation of the mathematical model of the in-plane vibration of a gantry crane is introduced. For the mathematical model, the differential equations are prepared by using Fourier and Krylov-Duncan Methods. By the methods Fourier and Krylov-Duncan, the differantial equation of the transverse vibration of the uniform Bernoulli-Euler beam changed to uncoupled ordinary differantial equations with respect to unknown functions which are depend on coordinate and time. In the next section, the finite element method and the modal analysis at ANSYS Workbench 14.5 has described. In order to apply this method to the problem, firstly, all parts creating the crane were 3-D modeled by using the SOLIDWORKS drawing program.3-D modeled parts were assembled by using the same drawing program. All 3-D models were created with the help of the draft drawings which were formed by mechanical calculations and the selection of the structural elements. The generated solid model was analyzed by the finite element method with the help of ANSYS Workbench 14.5 which is a commonly used analysis program. Modal mode was used to analyze in ANSYS Workbench 14.5. By modal analysis which is commonly used in engineering applications, the natural frequncies can be obtained. In the next section, mathematical model is created for in-plane vibration analysis of 2x400 tone gantry crane and by the help of eigenfunctions and eigenvalues, the natural frequencies of this crane have been obtained. In the same section, the modal analysis of 2x400 tone gantry crane made by using finite element method and natural frequencies are obtained. Before running the program, the general settings of modal analysis were prepared. Most important parts of the settings are, entering the engineering data, sizing and the tolerence value. After the settings, meshing was generated. Mesh quality is the most important factor that affects the finite element results. Increasing mesh quality increases the accuracy of the finite element method. Although minimizing size of the meshes can be an effective method to increase mesh quality, the solving capacity of the computers limit us. Then, the boundary conditions were applied. It was applied by fix support and displacement commands. In the last section, the obtained values of natural frequencies at previous section are compared and the results of comparison are presented.