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1 Reliability calculation model based on linear fatigue cumulative damage theory A mechanical component has only two working states: if it is in a safe state, it is recorded as event A; if it is in a fatigue failure state, it is recorded as event B. Therefore, the probability space of the component can be Expressed as: {,}ABΩ=(1) For reliability design, one goal is to calculate the probability that the component can work safely, ie the reliability is: ()RPA=(2) Obviously (2) is not convenient for quantitative description zero Component reliability and engineering applications. The random variables K and D are now introduced. K is called the fatigue damage strength, reflecting the inherent ability of the component itself to resist fatigue damage. It is only related to the properties of the material and the shape of the component, and has nothing to do with the load. D is called the fatigue cumulative damage. Reflects the degree of fatigue damage of the part, mainly depending on the load spectrum, working life and dimensional error of the part. According to the meaning of the random variables K and D, the two working states of the mechanical component can be described as follows: {}AKD=>,{}BKD=≤(3) At the same time, the probability that the component can work safely is () () RPAPKD=>(4) Equation (4) is a formula for calculating fatigue failure reliability based on random variables K and D. Therefore, by finding such random variables and obtaining their distribution laws and parameters, fatigue reliability design and analysis of mechanical components can be performed.
1.1 Anti-fatigue damage strength of gear contact and bending According to the definition of anti-fatigue damage strength, the anti-fatigue damage strengths jK and WK of gear contact and bending are: '()mjTHTNTLVRWXKNZZZZZZσ=(5)''()mWTFTSTNTrelTRrelTXKNYYYYδσ=(6) Where: XZ is the size factor; LZ is the lubricating oil coefficient; RZ is the roughness coefficient; VZ is the speed coefficient; WZ is the working hardening coefficient; NTZ is the contact strength life coefficient; STY is the stress correction coefficient; relTYδ is the relative tooth root Fillet sensitivity coefficient; RrelTY is the relative tooth surface condition coefficient; XY is the size factor; NTY is the bending strength life coefficient; HTσ and FTσ are the contact stress and bending stress respectively during the experiment; TN and 'TN are pitting corrosion respectively And the number of cycles when the bending fatigue fails; 'm is the slope of the SN curve reflecting the fatigue strength characteristics of the material in the logarithmic coordinate system; the others are the adjustment coefficients.
The contact fatigue limit 1Himσ and the bending fatigue limit 1Fimσ of the material and the cyclic base 0N of the S-N curve are substituted into the equations (5) and (6), respectively, and the logarithm of the two ends of the equation is obtained, which can be obtained: '01ln(lnln jHimLVRWXNTKNmZσ= (7)1'0Reln(lnln)FimWSTrelTltXNTKNmYδσ= (8) Combining equations (7) and (8), we can derive the mean and standard deviation of the logarithmic normal distribution of the fatigue damage strength jK of the gear contact. They are as follows: '101ln(ln)jKHZLZVZWZXZNTZRNmσμ= (9)'21jKHZLZVZRZWZNTmCσ= (10) where: 1Hσμ and 1Hσ are the logarithmic mean and standard deviation of 1Himσ, and other parameters are the normal mean and coefficient of variation of the adjustment coefficient. The value can be selected according to the literature [3].
At the same time, it can be deduced that the mean and standard deviation of the logarithmic normal distribution of the WK of the gear bending are expressed as follows: '101ln(ln)WKFYSTYrelTYRrelTYxYNTNmσμ= (11)'21YRrelTWKFYSTYrelTYXYNTmCσδσ= (12) where: 1Fσμ and 1Fσ They are the logarithmic mean and standard deviation of 1Fimσ, and the other parameters are the normal mean of the adjustment coefficient, which can be calculated according to the national standard GB/T10062-2003 and GB/T6413-2003. The coefficient of variation can be selected according to the literature [3]. .
1.2 Fatigue damage amount of gear contact and bending The load spectrum at the time of gear operation is used to define the fatigue damage amount of the gear. In order to simplify the problem, only the gear under the load spectrum of single-stage load is taken as the research object, and the tooth surface contact stress of the working gear is: 1()mtVHMBHELSKvbmVAVHFuZZZZZZdmKKKKβασ
= × (13) where: MBZ
Is the midpoint region coefficient; HZ is the node region coefficient; LSZ is the load distribution coefficient; Zβ is the helix angle coefficient; KZ is the bevel gear coefficient; 1Vd is the end face equivalent pinion index circle diameter; bml is the tooth midpoint contact line length Vu is the end gear equivalent gear ratio; AK is the use coefficient; EZ is the elastic coefficient; VK is the dynamic load coefficient; HKα is the tooth load distribution coefficient; HKβ is the inter-tooth load distribution coefficient; mtF is the reference point (the tooth width midpoint) ) Indexing the circumferential tangential force. Among the above 14 parameters, the first 9 parameters are the parameters related to the gear geometry. They can only be changed within the tolerance range allowed by the accuracy level. The value space is small and the process can be guaranteed. Therefore, these variables are It can be processed into a deterministic variable, denoted as E; and the other five correction factors EZ, AK, vK, HKβ and HKα are processed into normal random variables, denoted as F, and their mean values ​​are in accordance with national standard GB/T10062-2003 and GB/T6413-2003 is calculated and its variation coefficient can be selected according to the literature [3], so there are: () 1 (1) () VHmtMBHLSKVbnEAVHmtuEFFZZZZZudlZKKKKFβασ = × under the single-stage load, the tooth surface contact fatigue cumulative damage of the gear It is: () '2mjmtDNABF' = (15) where: N is the total number of cycles.
The logarithm of the two ends of the formula (15) can be obtained: 1ln[ln()][ln()]ln2VjMBHLSKVVbmEAVHmtuDNmZZZZZudlmZKKKKFβα
'= ' (16) The tooth root bending stress of the working gear is: mtLSKAvFSnFYYKKKKYYYbmβεααεσ=(17) where: FYα is the tooth profile coefficient; SYα is the stress correction coefficient; Yε is the coincidence coefficient; KY is the bevel gear coefficient; LSY is the load Distribution coefficient; mtF is the reference point (the midpoint of the tooth width) is the circumferential tangential force; b is the tooth width; nm is the normal modulus; AK is the use coefficient; VK is the dynamic load coefficient; FKα is the inter-tooth load coefficient; FKβ is the tooth load coefficient. Among the above 12 parameters, b, nm and Yε can be processed into deterministic variables, denoted as I, and other coefficients can be processed into random distributions of normal variables, denoted as J, and their mean values ​​are calculated or checked according to the method specified by the national standard. The gear manual, its coefficient of variation can be selected according to the literature [3], so there are: () () FmtFSKLSAVFnYIJFFYYYYKKKKbmεαβσ == (18) under the single-stage load, the tooth root bending fatigue cumulative damage of the gear is: mWmtDNIJF=' ''(19) where: N is the total number of cycles.
The logarithm of the two ends is obtained: lnln()ln()WnFSKLSAVFmtYDNmbmmYYYYKKKKFεαβ'= ′ (20)1.3 Reliability calculation limit equation The reliability calculation model based on linear fatigue cumulative damage theory, the reliability of tooth surface contact fatigue is: () jRPKD=>(21) The reliability of tooth root bending fatigue is: () WRPKD=>(22) Since jK, WK, jD and WD are all random variables, it is impossible to use explicit to find the reliability because: ()(lnln)RPKDPKD=>=>(23) Therefore, the following limit state equation can be used to calculate the reliability of the gear: lnjZKD=
(24) lnWZKD=
(25)()RZ=Φ(26) In the above limit state equation, the (14) value of each deterministic variable and the distribution law and distribution parameters of each random variable are known, and the design verification point method can be used to obtain reliable degree.
So far, we have established a gear transmission reliability calculation model based on the linear fatigue cumulative damage theory. The model will be used to calculate and analyze the reliability of the straight bevel gear transmission to verify its feasibility and rationality.
2 Straight-tooth bevel gear transmission reliability calculation is based on the design data of the straight-toothed bevel gear in a mechanical designer's manual. Under the design conditions, the above model (24) to (26) are used to calculate the straight-toothed bevel gear transmission. Tooth surface contact and root bending reliability.
2.1 Calculation example The basic parameters of the design of the straight bevel gear are shown in the figure.
Straight bevel gear design basic parameters basic parameter value basic parameter value z119b28z259 material 20CrT1114 heat treatment carburizing quenching n11000 surface hardness 58HRCm (mn) 3 service life 5000h
2.2 Tooth surface contact fatigue reliability calculation According to the data in the designer's manual and the literature [3], the required parameters take the value of the total number of cycles N calculated using the large gear speed, X is the midpoint of the area map.
According to the data given in the paper, the mean value of the logarithmic normal distribution of the tooth surface contact, jj1jKμ and the standard deviation 1jKσ, can be calculated according to equations (9) and (10): 1128.3869jKμ=, 120.1159jKσ = tooth surface contact fatigue reliability calculation parameter value table parameter value parameter value parameter value Vu9.643HZ2.49457MBZ
0.98736EZ189.8117Zβ1.0KZ0.8XZ1.0LZ0.96580ZLC0.91VZ0.96821ZVC0.93RZ0.93641ZRC0.08WZ1.0NTZ1.07bml27.13795mtF4709.0781Vd50.866AK1.25VK1.03415HKα0.99121HKβ1.65LSZ1.0m'15.385690N910N79.6610×X925 Tooth The cumulative damage amount jD of the surface contact can be calculated by the formula (16): ln39.0205 15.38569ln0.5(lnln)jEAVHDZKαβ=× × Finally, the gear reliability calculation is performed according to the limit state equation, that is, the equation (24) :ln39.020515.38569ln0.5(lnlnjEAVHZKDKZKαβ==× Iteratively calculate the above limit state equation, and the tooth surface contact fatigue reliability of the straight bevel gear can be obtained by equation (26): ()2.893299.81RZ=Φ=( )
2.3 Root root bending fatigue reliability calculation According to the data in the designer's manual and the literature [3], the required parameters take the value of the total number of cycles N calculated using the large gear speed, X is the midpoint of the area map. Root bending fatigue reliability calculation parameter value table parameter value parameter value parameter value parameter value YST2.0CYST0.033Yδre/T1.00327CYδre/T0.03YRre/T1.02420CYRre/T0.033YX1.0CYX0YFα2.24964CYFα0.033YSα1.88909CYSα0.04Yε0 .7173YK1.00024YLS1.0Fmt4709.078KA1.25KV1.03415CKV0.011KFα0.99121CKFα0.033KFβ1.65CKFβ0.05m'8.69565N03×106YNT0.933N9.66×107X250 According to the data given, the anti-fatigue damage strength WK logarithm of gear bending The mean value of the normal distribution is 1Kwμ and the standard deviation is 1KWσ, which can be calculated by equations (11) and (12) respectively: 172.6963Kwμ=, 11.8048KWσ=the amount of fatigue damage WD of the root bending under the action of single-stage load can be (20) Calculated: () ln50.5093 8.69565lnWFSKLSAVFDYYYYKKKKαβ=× Similarly, the calculation of gear reliability is based on the limit state equation.
3 Conclusions It can be seen from the calculation results of the example that the tooth root bending fatigue reliability of the design gear is slightly higher than the tooth surface contact fatigue reliability. Generally, the tooth surface pitting does not cause the drive to fail immediately, and the broken tooth will make the gear unable to work immediately. Therefore, the reliability requirement of the tooth breaking should be higher than that of the tooth surface, so the design of the gear is reasonable.
In this paper, the randomness is introduced into the linear fatigue cumulative damage theory and the fatigue failure of the component is taken as a random event to establish the linear fatigue cumulative damage reliability theory. Based on this theory, the reliability calculation model of the straight-toothed bevel gear transmission is established. According to this model, it can be well tested whether the gear design can achieve higher reliability requirements. Through the design calculation of the actual example, the root bending fatigue strength and the tooth surface contact fatigue strength of the example satisfy the requirements, indicating that the results are in good agreement with the actual operation. The method of this paper can be applied to the load spectrum of multi-stage load and distributed load, but there is a slight difference in the calculation of fatigue damage quantity. The reliability calculation model is similar, and the description is not repeated here. It can be seen that this paper proposes an effective method for reliability design of mechanical components.