Helikopter Rotor Pali Dinamik Yüklemesi Altinda Yorulma-çatlak Analizi

Abstract

Hava yapılarında dinamik yükler alanda yorulma ve çatlak oluşumu statik yükler altındaki gerilme ve şekil değişimi problemlerinden daha hayati problemlere sebep olabilmektedir. Gerek başlangıç dizayn çalışmaları, gerekse mevcut sistemlerin modernizasyonu ve modifikasyonunda statik mukavemet analizleri kadar şiddetleri birçok statik yükten daha az olan fakat sürekli tekrarlayan dinamik yükler etkisinde uzun sürede ortaya çıkan yorulma ve nihai kırılmalar aksam tasarımında ön plana çıkmaktadır. Hava yapılarının fonksiyon dışı kalmalarından çok önce emniyetle çalıştıkları ömürlerinin tayini tasarım ve bakım idame şekillerini tayin etmektedir. Hava yapılan içinde uçaklar ve helikopterler dinamik yük etkileri açısından önemli farklılıklar göstermektedir. Uçaklarda yük spektrumlannda uçağın kalkış ve inişi sırasında ve bazı keskin manevraları sırasında ortaya çıkan düşük tekrarlı yüksek şiddetli yüklere karşılık helikopterlerin standart uçuşlarında bile sürekli dinamik yükler bulunmaktadır. Periyodik kontrol girişleri olan ve bu girişlere periyodik hareketlerle cevap veren bir rotor ve bağlantılı olduğu döner tabla sisteminde değişik şiddetlerde periyodik yükler bulunmaktadır. Dinamik aksamların yorulma ve çatlak ilerlemeleri üç temel kısımdan meydana gelmektedir. Yorulma analizlerinin incelenmesinde dinamik yük spektrumlannm tayin ilk adım olup yüklemelerin frekans ve şiddet değişimi olarak gerçekçi bir modellemesi gerekmektedir. Yük spektrumlan genel olarak yüklerin sayısal olarak değişik şartlar için bulunması ya da test uçuşları sırasında ölçülmesiyle tayin edilmektedir. Her iki yöntemin de çeşitli zorluklan ve sımrlamalan olduğu için ikisinin birlikte kullanımı genelde uygulanır. Çatlak analizinde önemli bir diğer adım da malzeme çatlak ilerleme karakteristiklerinin tayini olup malzemenin mikro seviyedeki metalurjik yapısıyla yakından ilgili olmaktadır. Mikro seviyede malzeme modellemeleri yerine daha çok malzeme testlerinden elde edilen istatiki değişimler bu amaçla kullanılmaktadır. Üçüncü önemli adım da yükün etkidiği yapı elemanının geometrisine ve uygulanan yükün tipine göre çatlak oluşan ve gelişen bölgede gerilme yoğunluğu etkilerinin modellenmesidir. Günümüzde çeşitli çatlak büyüme simülasyon programlarında değişik yükler ve geometri kombinasyonlan için modelleme dağarcıklan bulunmaktadır. Bu çalışma çerçevesinde helikopter dinamik aksamlarında yük spektrumlan oluşturulması, aksam yüklemelerinin standart gerilme şekillerinde ve ortaya çıktıktan geometrilere bağlı olarak çatlak simülasyonlan genel metotlarından Walker çatlak ilerleme ve Willenborg geciktirme modeli incelenmektedir, örnek olarak ANKA-1 Mikro-Helikopterinin ana rotor spindle lug aksamı için gerçekleştirilmiş bir dizi çatlak ilerleme simülasyonu gösterilmiştir.

In this study, fatigue crack propagation in ANKA-1 Micro-helicopter main rotor spindle lug holes have been investigated for different materials by using a load spectrum. Basic steps of analysis are as follows. First is the calculation of maximum and minimum stress intensity factors corresponding to the maximum and minimum stress for each segment of stress spectrum. Stress intensity factors are a function of the crack size, together with local structural geometry, crack geometry and the global stress distribution. The next is the determination of an appropriate crack growth equation. These crack growth equations or the crack growth behavior can be described as functions of the stress intensity factors, their range, ratio to maximum stress, threshold stress intensity factors below which crack growth will not take place, nature of the variable amplitude loading and other material properties. The third step is the selection of a damage integration routine that can integrate the crack growth equation step by step to yield a crack growth curve by using a generated stress spectrum. The last step is to use the increasing crack length as a function of flight hours and the critical stress intensity factors to predict the number of flight hours at which failure will take place at the selected critical crack locations. There are three different types of stress intensity factors, Ki, Kn and Km which are defined in linear elastic fracture mechanics corresponding to three different modes in which cracks can grow in a structure. These are called mode I, mode II and mode HI, and represent opening, sliding, and tearing motions, respectively. Stresses at a crack tip are infinite when calculated on the basis of linear elasticity. They are said to be singular with a singularity of 1/r. The stress intensity factors are mathematical descriptions of the strength of the singularity. Physically, Ki, Kn and Km are regarded as the intensity of load transmission through the crack tip when a crack is introduced into the elastic body. Alternatively, they can also be considered as the intensities of linear elastic stress distributions surrounding a crack tip. For example, for a flat plate of width w containing a center crack of length 2a and subjected to a uniaxial stress of magnitude a far away from the crack, the stress intensity factor is given by xiu K = ap>[mx (1) where P is a function of a and w. Most of the available fatigue crack growth equations correspond to the mode I crack growth, typically known as the opening mode. Many fixed wing aircraft parts fail by mode I crack growth. Many helicopter fatigue critical components do not necessarily exhibit only mode I crack growth. Mode II and mode IQ types of crack growth are suggested in many fatigue critical components of a helicopter. In actual practices, the crack growth is in a mixed mode. Very little information is available when mixed mode, mode II or mode IQ behaviors are encountered. Very often a mixed mode crack growth initially leads to a mode I type of growth prior to failure. In this study, locations where mode I crack growth predominantly takes place were selected for damage tolerance analysis. The fatigue crack growth behavior depends on the following defined parameters. AK=Aa0y[m (2) K^v^PJ^ (3) K+^o^PJZ (4) r> ^min °min /c\ K = ~P = ~ (5) max max where AK is stress intensity factor range, omax maximum stress, amin minimum stress, R stress ratio. One way of quantitatively describing a fatigue crack growth behavior is to specify crack growth rate per cycle da/dN. The quantity N represents the number of cycles. Many curves representing crack growth rate as a function of AK are usually plotted for different values of R. A fatigue crack growth analysis needed for damage tolerance analysis can be perforned using numerical or tabular inputs of data from curves. It is to be reiterated that these behavoir is for a constant amplitude loading. Instead of using the tabular inputs of crack growth behavior, many proposed crack growth equations are very convenient to use and yield robust crack length - flight hours variations. In this study crack growth equations used are as follows. 1. Paris Erdoğan law % = CAK' (6) where C and m are experimentally determined parameters. MV 2. Walker equation, to account for the stress ratio. §-C((l-J!)'jJ- (7) where C, m and p are experimentally determined constants. Constants in these equations, that repress fatigue crack growth for constant amplitude cycling loading, are usually determined by least square fits to experimental data. These equations are also known as fatigue crack growtrh equations based on linear elastic fracture mechanics. They are not valid where large scale plasticity, non-linear stress-strain behavior or short cracks are present. It is to be noted that unlike the tensile strength, the yield strength, and modulus of elasticity, fatigue crack growth constants display a certain amount of scatter and are affected by some factors such as the heat treatment of the material, orientation of the crack with respect to the grain boundaries and grain directions, method of production such as extension and forging, frequency, temperature environment, composition and inclusion content Specific crack growth constants used in damage tolerance analysis are required to conform to the service conditions and manufacturing specifications of the particular helicopter. Very often to account for the scatter in crack growth behavior, scatter in the generated stress spectra and the knowledge of the existing crack length distributions a risk analysis is performed the following damage tolerance analysis. Crack growth equations discussed in the previous paragraphs are valid for mode I type of fatigue crack growth and for constant amplitude cyclic loading conditions. Maximum and minimum stresses change from one mission segment to another. If there were a high amplitude load in a sequence and there were no interaction effects due to loading sequenc, one can obtain new crack lengths by means of cycle by cycle integration of crack growth equations. However, crack growth due to variable amplitude loading does exhibit interaction effects due to sequences of high and low loads. A significant effect of high and low load sequence is retardation. When the stress history contains low constant amplitude and discrete high loads, the pattern of residual stresses and plastic deformation in front of the crack tip change due to the high amplitude load immediately after its application. The region in which changes have taken place in front of the crack tip is usually known as the perturbed zone. As the crack propagates through the perturbed zone due to low amplitude cycles, crack growth is slow or retarded. Following the propagation through the perturbed zone, crack growth rates return to their regular rate. Models to explain retardation are based on the compressive residual stresses, plastic deformation zone or a combination of both. The magnitude of the retardation effect depends on the ratio of the high stress to the low stress. It has also been observed that if a high tensile stress is followed by a compressive stress the beneficial effect of low rate of crack growth due to high tensile stress is reduced. This effect is known as annihilation of retardation. XV Some mathematical equations or models have been developed to quantatively account for retardation effects. Even though these models can be further refined to provide accurate results, the models have been found to yield conservative retardation effects. The first model is due to Wheeler and the crack growth equation of the previous equation are modified as follows : fda\ (da\ KdNJ, p\dNJ Following a high tensile stress, crack growth rates (da/dN) discussed in the previous paragraph are multiplied by a retardation factor, Cp. c,= Ag + 'kc) (9) In this equation, Aa : the difference distance between the last two crack length, ry(C) : the plastic zone at the current cycle, r^oi) : the plastic zone at the overload cycle, y : shape parameter determined experimentally. Just like the Wheeler 's model is related to the plastic zone size, another model due to Willenborg also relates to the plastic zone. The magnitude of retardation hasbeen established by Wheeler in terms of effective stress intensity factor. The effective stress intensity factor replaces the stress intensity factors used in the crack growth equations of the previous paragraph. *£ = *--** (io) a ? - a where aoi : the crack size at the overload stress, ry(oi) : the plastic zone due to overload, Kmax(oi) : maximum stress intensity factors. According to this Willenborg equation retardation effects continue until the plastic zone size due to low amplitude stresses reach the boundary of the larger plastic zone created by high amplitude loads. This model, however, considers that residual stress effects are the same irrespective of the stresses involved. A modified Willenborg model due to Gallegher and Hughes uses an experimentally determined multiplication factor § such that, *£=*«-«« (12) xvi v-th i max max, s-1 (13) when K^ is the threshold stress intensity factor where no crack growth takes place and s is a material constant related to a value of high stress at which crack arrest takes place in the given material. In this study a modified Willenborg model was used as the basis for fatigue crack growth propagation analysis. In order to account the crack growth propagation rate a lot of crack growth computer programs had been developped. MODGRO, one of these crack growth programs, is capable of using the generated stress spectra and a local structural geometry around a crack and integrating the crack growth equation with retardation effects. The output of this program consists of a history crack length vs. time. This crack growth program is required to: 1. calculate the stress intensity factor, K, as the crack length increases, 2. calculate the corresponding range of the stress intensity factors, K, 3. correct the K range for retardation effects, 4. integrate the crack growth rate equation to obtain new crack lengths and 5. verify if the crack lengths have reached critical levels. Crack growth program also need user specified data to define the crack model and quantify the loading conditions. Inputs take the form of : 1. applied stress spectrum, 2. local geometry surrounding the crack, where the applied stress spectrum is considered to be valid, 3. material properties such as constants in the crack growth equation, critical stress intensity factors, and retardation models, and 4. cycles or incremental steps of crack propagation length used to integrate the crack growth rate equation. The computational details of the program begin with the calculation of stress intensity factor, K. K is a functional the applied load, crack length and local geometry. Under oscillatory loading values of Ls and K», correspond to the highest and lowest stresses applied during one cycle. The stress ratio, R, is simple the ratio of Kmi,, to K^a. The stress intensity range factor, AK, is the difference between K^ and Knm,. For the analysis of retardation effects under the varying amplitude of the stress spectrum, these quantities are modified to obtain effective values, Re» and AK«.ff. A threshold value for the stress intensity factor range, AKth, is given as the limit below which no crack growth occurs. Similarly, a critical stress intensity factor range, AKc, is defined as the failure criteria for a specific material. Physically, it is the point where the crack grows without bound and results in a catastrophic failure xvu of the model. If the calculated value of AK exceeds AKc failure has occured and the program will stop. Otherwise, the procedure continues to the next step. The crack growth rate, da/dN is expressed as a function of R and AK. By using their calculated values along with the specified material properties, the incremental crack growth is obtained by integrating the da/dN equation. Limits of integration can be specified on the basis of a selected step size or number of cycles. A new crack length for the next loading period is found by adding the incremental growth to the previous length. The entire process repeats with the calculation of new R and AK values. In this study, we first examine the crack growth of a selected component of ANKA.-1 microhelicopter with known load spectrum. The second chapter is devoted to the laws and techniques of linear elastic fracture mechanics and of fatigue which constitutes a basis for this thesis. In the third chapter, the Walker method and the Willenborg retardation method of fatigue crack growth are explained. In addition, a software using these methods is explored. The fourth chapter presents applications of the two fatigue crack growth method on the spindle lug hole of the ANKA-1 microhelicopter' s main rotor whose load spectrum has been known. The results and their interpretation are given in the last chapter.