Alttan kesmeye sahip dişlilerde mukavemetin incelenmesi
Evaluation of material strength in gears with an undercut
Yazar: MİTHAT CAN KAYRAK
Danışman: PROF. DR. ATA MUGAN ; ÖĞR. GÖR. ALİ İMRE AYDENİZ
İstanbul Teknik Üniversitesi / Lisansüstü Eğitim Enstitüsü / Makine Mühendisliği Ana Bilim Dalı / Konstrüksiyon Bilim Dalı
Derece: Yüksek Lisans, 2022
Konu: Makine Mühendisliği = Mechanical Engineering
Dizin: Diş kökü = Tooth root ; Dişliler = Gears ; Gerilme analizi = Stress analysis

Özet:
Günlük hayat ve endüstride en yaygın kullanılan güç aktarma organları olan dişliler, üretim metotlarına bağlı olarak geometrilerinin bazı bölgelerinde tanımlandıkları profillerin dışına çıkabilmektedirler. Bu tarz profil bozukluklarının en yaygın ve bilineni ise „alttan kesme‟ adı verilen profil bozukluğudur. Diş dibinde yuvarlanma olarak görülen bu bozukluk, düşük diş sayılarında kendini göstermekte olup, temel dairenin altında meydana gelmektedir. Diş kesitinde daralma meydana getirip dişte çentik etkisi yaratmakta ve. dişlinin mukavemetinde azalmaya sebep olmaktadır. Buna karşılık alttan kesmeye sahip olan bir dişlide maksimum gerilmenin değeri veya artış oranı ve maksimum gerilmenin meydana geldiği kesit tam olarak bilinmemektedir. Tez kapsamında yapılan çalışmada, alttan kesmeye sahip dişlilerde maksimum gerilmenin ulaştığı değer, alttan kesmeye sahip olmayan dişliler ile karşılaştırmıştır. Temel amaç, alttan kesmedeki mukavemet düşüşünü ve gerilmenin oluştuğu bölgeyi belirleyebilmektir. Bunun için matematiksel ifadeler geliştirilmeye çalışılacaktır. Alttan kesmenin pratikte her sistem için uzak durulması gereken bir düzgünsüzlük olmadığını, bazı sistemlerde (mukavemet açısından sorun yaratmadığı sürece) alttan kesmeye izin verilerek hacim ve kütleden tasarruf edilebileceği gösterilmek istenmiştir. Alttan kesme sebebiyle ortaya çıkan diş geometrilerinde, temel dairenin altında diş kesitini daraltan bir oyulma görülmektedir. Diş kesitinin daralması sebebi ile dişlide mukavemet düşüşü ortaya çıkmaktadır. Alttan kesmenin sebep olduğu yuvarlanma bir çember ile tanımlanacak olursa söz konusu çemberin yarıçapı, alttan kesmeye sahip olmayan dişlilerdeki diş dibi yuvarlanmasının yarı çapından çok daha büyüktür. Bu sayede alttan kesmeye sahip dişlilerde diş dibindeki yuvarlanmanın çentik etkisinin, alttan kesmeye sahip olmayan dişlilerdeki diş dibi yuvarlanması ile karşılaştırıldığında daha az olması beklenmektedir. Diş mukavemetini pozitif olarak etkileyecek bu parametre ile, alttan kesmenin sebep olduğu mukavemet azalmasının yüksek oranlarda gerçekleşmeyeceği tahmin edilmektedir. Konunun çalışılması sırasında izlenecek metodoloji şu şekildedir;  m=3, z=7 diş modeli üzerinde çalışılacaktır. Ayrıca yine m=3 modüle sahip z=9, z=11 ve z=13 diş sayılarına sahip dişliler de çalışma kapsamında incelenmiştir.  Alttan kesme sonucu oluşan kesit daralmasının mukavemete etkilerinin
gözlenebilmesi için x=0, x=0.1, x=0.2 ve x=0.4 profil kaydırma oranına sahip dişliler incelenmiştir.xx

Abstract:
Gears, which are the most widely used powertrains in daily life and industry, can face issues such as profile deviations since different manufacturing methods can alter the functionality of the end product. The most problematic and widely encountered type of profile deviation is called gear undercut. Gear undercut is generally faced as dedendum area chamfering which takes place underneath the base circle and generally occurs with low gear count gears. Undercut creates a narrowing in the tooth section and generates a notch effect on the tooth and triggers a decrease in the material strength of the gear. On the other hand, in undercut gears, pieces of information such as the value of the maximum stress, the rate of increase of the maximum stress, and the cross-section where the maximum stress occurs are not known precisely. Within the scope of this study, the maximum stress observed in an undercut gear was compared with gears without an undercut. The main purpose of this study is to determine the strength drop in an undercut gear and the region where the stress occurs. In order to achieve this goal, mathematical expressions will be used and will be developed.
It is aimed to show that undercutting is not an irregularity that should be avoided in practice for every system as long as it does not affect the strength of the material negatively, and that in particular systems, the volume and mass of the gear can be reduced by allowing undercutting. In gear tooth geometries caused as a result of undercutting, a cavity is observed below the base circle, which narrows the tooth section. Due to this narrowing of the tooth cross-section, a decrease in durability occurs in the gear. If a circle defines the undercut induced roll, the radius of that circle is much larger than the radius of the tread roll in gears without an undercut. Thus, it is expected that the notch effect of undercut gears will be less effective than tooth pitch roll in gears without undercut. With the inclusion of this parameter, which will positively affect the tooth durability, it is expected that the strength reduction caused by the undercut will not occur at high rates. The methodology to be followed within the scope of the study is as follows;
 Tooth models of m=3, z=7 will be studied. In addition, gears with m=3 modulus and z=9, z=11, and z=13 tooth count were also examined within the scope of the study. 
 Gears with a profile deviation ratio of x=0, x=0.1, x=0.2, and x=0.4 were examined in order to observe the effects of section narrowing on strength as a result of undercutting.
 The notch effect caused by section narrowing will be expressed mathematically.
 The notch effect coefficient mentioned in numerous sources does not cover undercut gears. In order to prove the compatibility of the notch effect coefficients with the literature, the study was repeated on a reference gear
mentioned within the table.
 Gears with m=3, z=30 coefficients, and the identical profile deviation ratios are used as the reference. It was aimed to develop a mathematical expression to determine the value of the maximum stress occurring in a gear with an undercut and the cross-section information about the area it occurs.
Bending stress is the only parameter to be taken into consideration in the strength calculations made in gears. Complex gear mechanics are generally avoided during bending stress calculations. The stress generated by the maximum force acting upon each is determined. Correction coefficients are used during the calculations in order to make the analysis more accurete. These coefficients are form coeffient, notch coefficient, contact coefficient and helix coefficient. Due to spur gears being not included in this study, helix coefficient is always set at 1. The other 3 factors were taken into accout since the calculation of material strength of undercut gears was studied. Gears with undercut are design and submitted into finite elements software ANSYS and critical sections are determined. Lever arm is calculated based on
previously obtained critical section information. Obtaining both the lever arm and critical section enabled calculation form factor. Dedendum and base circle informations are used for calculating the contrant ratio and contact factor.
The beams are modeled and analyzed over the sections where the maximum stresses are observed. The results of the simulations on tooth geometries and beams were compared considering the contact factor and form factor. As a result of this comparison, the notch factor was obtained.
The calculations and methodology of this study was compared with the other scientific publications.
After determining the minimum number of teeth where bending stress is managable, the gear was checked for Hertz Pressure. In order to reduce the effect of Hertz pressure, the gear was modified in order to locate the point of contact away from the base circle. Since bending stress does not pose a problem in gears with number of tooth below 14, Hertz pressure control was carried out in the gear. It has been shown that the system remains safe in terms of Hertz pressure, by trying modifications that can lower the Hertz pressure.
As a result of the bending stress calculation, the sample pinion which was reduced to 11 teeth with the assumptions made, was also inspected for Hertz pressure by constructing an example system. The conversion ratio of the sample system is assumed to be i=3. The gear modification applied in the sample system was profile shifting, which is the most applied gear modification method in the industry. The improvements made with profile shifting were found sufficient for this system and it was concluded that the system could work with 11 teeth. Cutting the tooth head of the gear has also been determined as an alternative gear modification in case the profile shift is insufficient, so that the point of contact will be located further away from the pinion base circle.
There are two geometric limits of the system for the profile shift ratio to be applied.
The first limit is identified as the profile shift rate that will sharpen the pinion to tip.
The second limit is identified as the profile shift rate which determines the distance of the tooth head of the gear from the pinion base circle. In addition, the contact ratio is kept above ε=1.1, thus avoiding the constant application of the load in the system to a single tooth.
According to the limitations and the Hertz Pressure calculations made according to
these limitations, the geometric limits have been determined to keep the system safe in terms of Hertz pressure.