Dişli çarklarda yük taşıma kabiliyetinin ve diş gibi gerilmelerinin incelenmesi
Investigation of root stresses and fatique strength of spur gears
- Tez No: 39212
- Danışmanlar: PROF.DR. MUSTAFA AKKURT
- Tez Türü: Doktora
- Konular: Makine Mühendisliği, Mechanical Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1993
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 124
Özet
ÖZET Günümüzde en çok kullanılan mekanik güç ve hareket iletim elemanlarından olan dişli çarkların yük taşıma kabiliyetini belirleyebilmek için bunu etkileyen faktörleri ve diş dibinde meydana gelen gerilmeleri incelemek için, önce konuyla ilgili hesap esasları gösterilmiş, daha önce konuyla ilgili yapılmış çeşitli çalışmalar özetlenmiş ve değerlendirilmiş, işlenecek konular sıralanmış, daha sonra Türkiye'de de önem taşıdığı için bu konudaki Alman Standardı olan DİN 3990 ' a göre yük taşıma kabiliyetini etkileyen faktörler ve diş dibi kırılma mukavemetleri incelenmiş, daha sonra dişli çarklardaki yorulma sonucu kırılmanın mekanizması incelenmiş, bu hesapların daha sağlıklı yapılabilmesi için dişlerin profili bilinmesi gerektiğinden bunu en çok etkileyen dişin açılmasında kullanılan takımlar ve bunların oluşturduğu diş formlarına etkisi belirlenmiş, daha sonra da modern gerilme ve şekil değişimi inceleme yöntemlerinden Sonlu Elemanlar Yöntemi yardımıyla dişdibi gerilmeleri incelenmiş ve başka hesap ve deney sonuçlatıyla karşılaştınlmıştır. IX
Özet (Çeviri)
SUMMARY INVESTIGATION OF ROOT STRESSES AND FATIGUE STRENGTH OF SPUR GEARS Main couses of gear tooth failures are, breakage under stresses, pitting, wear types of adhesion, abrasion, scuffing, spalling etc., heat, noise of over load, over speed and assemblying failures. One of the primary couse of gear tooth failure is the presence of large tensile stresses in the root fillets of loaded gear teeth. These stresses tend to reduce overall gear life and can result in catastrophic tooth failure under peak loading conditions. Many attempts have been made by earlier investigators to relate tensile fillet stresses observed in statically loaded gear teeth to the geometric appearence of the tooth. The aim of this study is to investigate and evaluation of stresses at gear teeth and to determine some geometrical factors which influence the fatigue strength. In part one the subjects are introduced, and then explained how the stresses can occur. When the applying load corresponds to the fatigue limit, a crack appears only at the compressive side and grow to some extent, then after a certain number of load cycles it ceases its growth, which is called a non propagating crack. If the applying load is below and closely approaching to the fatigue limit, no sign of sliplines can be observed at the tensile side, but on the contrary at the compressive side sliplines can be observed clearly. But no cracks initiate yet at both sides. When the applying load exceeds the fatigue limit, both slip lines and cracks appear first at the compressive side of the tooth, later at the tensile side these cracks grow with the load application. Then the crack which is at a certain distance from the tooth end, propagates toward both ends with an increasing number of load cycles. Since the crack grows more rapidly at the tensile side than the compressive side, cyclic life of the gear is governed by this event.As early as 1 893, Wilfred Lewis applied elementary beam theory to symmetrical tooth profiles by inscribing a parabola to represent a beam of uniform strength. As a result of the uniform strength assumption the Lewis formula was unable to deal effectively with abrupt changes in tooth section that occur in the tooth fillet region. Furthermore, for teeth with high pressure angles a radial component of load exists that tends to modify the stress produced by the applied bending moment. The introduction of the photoelastic technique by Baud and Timoshenko gave investigators the first real opportunity to examine in detail the stress raising effect of gear tooth fillets. Dolan and Broghamer studied the subject in depth and introduced a combined stress correction and stress concentration factor to be used in conjunction with the Lewis stress formula. This factor related the increase in observed fillet stress over the nominal bending stress to load height, fillet radius and pressure angle. Niemann and Glaubitz defined a nominal stress which is not related to the point of applying load, based on their photoelastic experiments. The shear stress and a u factor together were added on this stress. Later Heywood used the photoelastic technique to develop a fillet stress formula that accounted for some pressure angle unbalance and for the proximity of the point of load application to the tension fillet. Kelley and Pedersen improved this formula by employing more realistic tooth shapes in their photoelastic models and by adding a term to account for the effect of the angle of loading relative to the direction of the principal stresses. Then Albert in 1965 think the tooth as a key and describe an equivalent key profile. Here the key boundaries are tangential to the fillet radiuses. The stresses is determining at intersection of this key lines and dedendum circle by aid of Airy ' s stress functions. Neuber, Aida /Terauchi and Baronet / Tordion describe a conform picture which is resemble a tooth. For calculation the stresses at the tooth fillet were used Kolossow and Muskelisvili ' s complex stress function. Then Neuber who saws the difficulty of this solution under condition of linear elasticity, describe a new and simplier method CETIM (Cente Technique des Industries Mechaniques) and Chabert, Coleman / Wilcox, Neuber / Bart and other investigators who use Finite Element Method briefly are showned. XICalculation of load capacity and the related factors of spur gear teeth by using German standard DIN 3990 is shown in detail. These factors can be investigate in two groups. The first group factors are related to the teeth geometry and must calculate by suitable formula. The second group are which contain many effects and / or non - related factors such as K\, Kv, K^a > Kgy, Kpa and strength coefficients. Calculations of these factors are made according to ISO ' s A, B, C, D methods. Investigation of strength factors by using DIN 3990 and ISO ' s A, B, C, D methods for spur gear teeth is shown at part four. Furthermore to examine some methods which increase the bending fatigue strength of gears such as profile shifting (modifying of addendum), shot-peening and case hardening. The root stresses decrease with an increasing addendum modification factor x.It is possible to raise the bending fatigue strength limit for cast iron and cast steel gears by more than %25 over standard (non profile shifted) gears. The bending fatigue strength of profile shifted gears can be estimated with fairly high accuracy, by introducing the addendum factor Bx=l+0,5 x into the bending strength of standard spur gear. The contact ratio of the profile shifted gears decreases with increasing x, and x2 addendum modification factors. The root stresses decrease with increasing pressure angle. In example, the fatigue bending limit load of a=27° is higher %15 than those of gears with ot=20°. The effect of case depth on the strength is not remarkable. Effect of the tools at the determining the tooth form is investigated at the sixth part.Matematical principals are investigate, and a program is given for determining the tooth form by using tool forms with protuberance and without protuberance. Summarizing, it would appear that a primary difficulty associated with experimental investigations is in constucting and preparing models and the accuracy with the fillet stresses can be determinated. Analytical techniques, on the other hand, invariably make some assumption about the location of the maximum fillet stress and are limited to symmetric tooth profiles with relatively high pressure angles. The intent of this part of this study to show how the Finite Elements Method can be applied to determination of the exact stress distribution in the fillet regions of gear teeth and to determine the maximum surface stress in the tensile fillet. XIIWhile performing this study the computing speed and memory capacities of my computer was not enough, so my computing results are not high accurate, but in nearby with the new computers it will be shown that the finite element method is accurate, flexible and does not require previous knowledge about stress conditions in the gear tooth. In the conventional numerical method based on matrix analysis, the structure is idealized as a network of small elastic elements connected only at a finite number of nodes. Although the stiffness properties of each element are well known, this method assumes an absence of of load sharing along the sides of the elements. For the case of open structures such as bridges and aircraft frames this limitation is unimportant ; however, in the case of a continuous structure, such as a gear tooth, the conventional method would be entirely inadequate. With the relatively new finite element analysis, the continuous structural shape is idealized by a finite number of elastic, plate - like elements connected not only at the nodes but also along the inter - element boundaries (Figure 7.4). The variation of stress across the individual elements is predetermined by the mathematical form of the stress - strain relationship assumed in the elastic model of element.. The final state of stress in a fully loaded, idealized structure, is determined by minimizing the total potential energy of the structure with respect to nodal displacements. As a consequence of this minimum energy principle, the solution will converge monotonically on the exact solution of the stresses as the density of elements in the structure is increased. Convergency is further facilitated by constructing the finite element mesh with high densities of elements in regions of high stress gradients and low densities in regions of low stress gradients. The considerations necessary, therefore, for applying the finite element method to the two dimensional problem of plane stress can be summarized as follows: Selection of an element with a stress - strain relationship that is compatible with anticipated stress gradients ; Refinement of finite element mesh in regions of anticipated high stress gradients ; Selection a computer with adequate computational speed and memory core. The first requirement of the finite element method is the representation of the tooth shape by a network made up of finite elements. Points on the boundary of the network must have a one to one correspondence with points on the boundary of the tooth shape being analyzed. The location of interior points of the network are less restricted but in general, reflect the shape of the tooth boundary. At the boundary of the fillet region the element spacing is approximately two percent of the tooth depth, both radially and tangentially. In the interior of the tooth where the stress gradients are generally lower than in the fillet region the grid spacing is approximately ten percent of the tooth depth. X1UThe finite element program list has given and the results has shown in various figures.The output of the program list the principal stresses for a mesh element. Because the stress - strain relationship within an element is assumed to be constant, the exact location of a stress value within an element is unknown. Near the fillet surface of the tooth the stress gradient can be very large in the direction normal to the surface. Consequently, an uncertainty in the location of stresses within the elements near the fillet surfaces can result in some error when projecting stress gradients to the fillet surfaces This uncertainty of elemental stress locations can be overcome to a large degree by using statistical methods to average out the stress error over several elements and by using small elements in this region. XIV
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