Mid Gear Bending Fatigue design based on Rainflow Principle.doc

Mid Gear Bending Fatigue Design Method based on Rainflow PrincipleWang Guo-jun1,a Zhu Shi-shun2,b Chen Xin3,c1.Department of Automobile Engineering, Academy of Military Transportation, Tianjin 300161,CKeyWords:mid gear;bending fatigue;rainflowAbstract. The growing requirements of light constructions are pushing the needs for more detailed durability analysis and design for transmission. The methodology described in this paper tackles one of the key problem, mid gear bending fatigue design. Generally, gear bending fatigue design was based on the torque transmitted by the gear and the revolution of it. For mid gear, only one coefficient was concerned compared with the gear which was loaded on the single side of the teeth. The modification was not related with the stress-strain relationship of the gear tooth root. In order to solve the problem, defects of the traditional design method was pointed out. For gears with single side loaded, the mean stress was equal to the amplitude, which was same to the gear bending fatigue property test. But the characteristic of load cycle for mid gear was different. For mid gear, mean stress on the root of teeth was zero. On the view of metal fatigue, repeated plastic deformation was the reason of gear teeth fatigue fracture. Rainflow counting method was in accordance with relationship of the stress-strain relationship based on metal fatigue. Based on rainflow principle, a new methodology with modification of mean stress was given for mid gear design. Introdcution Bending fatigue was one of the main falure mode of gears.The method of gear fatigue design was discussed in many papers. Most of them were based on GB/T3480-83.But the study shows that the method only fits for the single side loaded gear.Double sides loaded gears were widely used in vehicle,engineering machine and common speed reduction machines.To solve the fatigue designation problem of mid gear, the mean stress modification method was given which was based on rainflow princple and fatigue demage theory.It is an addition to the traditional original method.Traditonal MethodsReference12 pointed out that when the load spectrum of gear transmission system was known, the equivalent torque was got as follows.The gear ability to resist fatigue demage was shown in Fig.1 as T-N, N was the life of gear. The load spectrum experenced by gear was also shown in Fig.1 two. The equivalent cylce NLeq experenced by the gear was got by Eq.1.NLeq=N1+N2+N3+N4 (1)Ni=60nikhi (2)In Eq.2,Ni was the life cycles related to i level load,ni was the load cylces experence by the gear according to i level load, was the contact numbers on the surface of gear tooth when the gear rotate on cycle, hi was the duration (hour) the gear worked under i level load.N0NNT-N curve of gearN1T1T2T3T4N2N3N4TFig.1 The ability that gear resist fatigue and load spectrum1According to Miner rule, the equivalent load was obtained through Eq.3. (3)In Eq.3, T was the torque apllied on the gear(Nm),d was the diameter of gear(mm), n was rotating speed of gear(r/min).= (4)In Eq.4, Ft was the torque apllied on the gear tooth tangency to refference circle,b was width of gear teeth (mm)；mn was the mode of gear (mm)。 (5)If FFP,the result was acceptable.FP was the acceptable stress.K was modification coefficient.Defects of tranditional methodTraditonal method thought that load cycles caused gear bending fatigue was related to the contact times at the same side with other gears and the rotation of it. For single side loaded gears,such as gear I and gear III in Fig.2, it was one times that gear teeth was contacted when they rotated one revolution.For gear II, there was one time contact for both sides of gear teeth when it rotated one revolution. There was no clear method to solve the fatigue problem for gear II. Fig.2 Gear transmission system Fig.3 The force acted on gear teethFor single side loaded gear, the maximum stress of gear root was got by Eq.4 and Eq.5.The minimum stress was 0.It was accordance to gear bending faigue property test method with pulse load. The test result was the basic data to predict bending fatigue life of gear.For mid gear, which was double sides loaded,the mean stress of it was zero and the minimum stress was -F. It was not accordance to gear bending faigue property test method with pulse load.Bending fatigue design method for mid gearStress analyses for mid gearFor single side loaded gear,such as I and III in Fig.2, the load acted on the gear teeth was simplyfied in Fig.2.It was pulse load applied on the gear. For double sides loaded gear, there was tension stress on one side and compression stress on the same side.If only the stress caused by Ft was considered and the shear stress caused by Fn was ommited, the stress on the root was considered as positive and negative repeated stress cycle. From metal fatigue princle we know that for high cycle fatiuge, under the repeated load cycle, although the gear body was under elastic state, the metal in the local area was in plastic state. According to rainflow princple, if the gear rotate one cycle, then the stress-strian state was shown in Fig.5.timeBending stress F Strain Stress FFig.4 Bending stress-time history for mid gear Fig.5 Bending stress-strain relationshipThe maximum and minimum bending stress of gear was obtained by Eq.5 and Eq.6. (6)The amplititude of stress cycle was obtained by Eq.7.= (7)The mean stress was got by Eq.8.=0 (8)The mean stress of mid gear was zero, but mean stress cycle of gear speciment was Fmax/2.So mean stress of mid gear should be modified before the gear specimen data was used.Modification of Mean Stress m was mean stress of pulse loaded gear. -1 was the fatigue limit with zero mean stress. a was the stress amplitude of pulse loaded gear. b was ultimate strength of steel.Then the relation of a and -1 was (9)Change the form of Eq.9,Eq.10 was got (10)For gear specimen test, m =a,take it into Eq.10,so (11)The gear root stress was got by Eq.9,which was quivalent to pluse loaded gear.Combined with Miner rule and S-N curve of gear specimen, it was easy to get the fatigue life of gear.Eq.12 shows that there was a nolinear relationship between of lg-1 and lga. So the relationship between lg-1 and lgN was nolinear.Steps of fatigue design of mid gearFirstly, Eq.5 and Eq.7 were used to calculate the bending stress amplitude of gear, then -1 was got.Secondly, a was got by Eq.12.Thirdly,different stress amplitude a as sign as 1,2, 3, 4.Take them into Eq.14,the equivalent stress was got. In Eq.14,N1,N2,N3 and N4 were the fatigue life related to 1,2, 3, 4.There was a similar form between Eq.14 and Eq.3.At last, Eq should