When calculating the reliability using the traditional analytical method and the numerical method, it is necessary to know the stress S, the distribution type of the intensity, and the distribution parameters. In mechanical design, there are many physical and geometric factors affecting stress and strength, such as elastic modulus E, Poisson's ratio, geometrical coefficient of parts, surface quality, etc. Since these parameters are random variables and have distribution characteristics, There are certain complications and errors in the calculation. In addition, when the stress and intensity distribution are complicated and the distribution types are various, it is difficult to calculate the reliability by analytical method or numerical method. In this case, it can generally only be solved by means of numerical simulation methods. The Monte Carlo method is a widely used method for solving reliability using computer simulation. The basic idea of ​​the MonteCarlo method is to first establish a probability model, so that the solution to the problem is exactly the parameters of the model or other related feature quantities, and then the percentage of the occurrence of an event is counted by simulation statistics, ie, multiple random sampling experiments. As long as the number of experiments is large, the percentage is similar to the probability of an event occurring.
1 Gear reliability design example The original data in the text is taken from the traditional design example of the 7th edition of the mechanical design of the Higher Education Press. 10-2. The data coefficients used in the calculation are directly from the tables, line graphs and formulas in the manual. Checked or calculated.
CM6150 lathe gearbox, known input power P1 = 10kW, pinion speed n1 = 960r / min, gear ratio u = 3.2, driven by the motor. The working life is 15a, and the working is 300d per year. The two-shift system works smoothly and the steering is unchanged. The gearless material is 40Cr tempered, the hardness is 280HBS, the large gear material is 45 steel and the hardness is 240HBS. The main geometric parameters of the gear transmission obtained by the traditional design: the normal modulus of the gear is mn=2mm, the number of teeth Z1=31, Z2=99, helix angle=1425, center distance a=134mm, ruler width b=65mm, accuracy grade 7. Try to find the reliability of the gear transmission.
Solution: Considering that the bending strength of the gear is rich, the reliability of the gear transmission mainly depends on the contact strength of the gear.
1) Determine the mean and standard deviation of the strength from the mechanical parts manual to find Hlim1=600MPa; Hlim2=550MPa.
The allowable contact stress formula for the tooth surface is HP=Hlim ZN ZR ZV ZW ZL ZXSHmin.
In the formula: Since ZR, ZV, ZL and ZX have little effect on the contact stress of the tooth surface, the formula can be simplified to HP=Hlim ZN ZWSHmin. The coefficient of variation of the contact strength of the tooth surface is C=0.1, and the contact strength between the pinion and the large gear The standard deviation is SHlim1=0.1 600=60MPa, SHlim2=0.1 550=55MPa.
General gear transmission, contact strength minimum safety factor SHlim=1, that is, standard deviation SSHlim=0. From N1=60n1t=6.2 109, N2=N1i=1.9 109, look up table ZN1=ZN2=1, SZN1=0. The coefficient of variation of the coefficient is C=0.07, then SZN2=10.07=007.
Because the pinion is a soft tooth surface, without grinding, so take ZW=1, then SZW=0.
Substituting the above parameters into the simplified tooth surface allowable contact stress formula, the mean and variance of the allowable contact stress of the pinion and the large gear are obtained by the multiplication theorem of the independent random variable: HP1=600MPa, SHP1=55MPa, HP2=550MPa, SHP2=67.246MPa. Therefore, the mean and standard deviation of strength are HP=HP1 HP22=600 5502MPa=575MPa, SHP=12(S2HP1 S2HP2)1/2=45.061MPa.
2) Determine the mean value of the stress and the standard deviation gear contact stress formula is Hca=ZEZHZFt(u 1)bd1uKAKVKK, where: the index circle diameter d1, the tooth width b, the gear ratio u, the node area coefficient ZH, etc., belong to the gear Geometric size related parameters, they can only be changed within the tolerance range allowed by the accuracy level, the value range is small, and the process can be guaranteed. For the sake of simplicity, they are treated as fixed values, except for the above parameters. Parameters are processed as random variables.
Let Ftc=FtKAKVKK, apply the independent random variable multiplication theorem to obtain the mean and standard deviation of Ftc as Ftc=3669.138, SFtc=267.923; let Z=Ftc(u 1)bd1u, apply the constant theorem of independent random variable to obtain Z The mean and standard deviation are: Z=1166, SZ=0.085; let M=Z, applying the square root formula of independent random variables, M=1.179, SM=0.036.
From the manual, the mean value of the material elastic coefficient ZE is found to be Z=189.8, and the deviation ZE=(10, the standard deviation is SZE=10/3=3.33. The contact strength coincidence degree is 1.60 in the manual, then Z=1 /=0.791, take the deviation Z=0.03, then the standard deviation SZ=0.033=0.01; from the manual, we can find the mean value of the joint area coefficient ZH=2.44, according to the fixed value, then SZH=0.
Let Hca=ZEZHZM, then apply n times of two independent random variable multiplication theorems, the mean and standard deviation of the obtained stress are: Hca=431.176MPa, SHca=14.337MPa.
3) Calculation reliability If the tooth surface contact stress and the tooth surface fatigue strength limit obey the normal distribution, the reliability coefficient is ZR=HP-HCS2HP S2Hca=575-431.17645.0612 14.3372=3.041.
From the standard normal distribution table, the reliability of the tooth surface contact fatigue strength is found as: R(t)=(ZR)=99.8817.
2 Monte Carlo method in gear reliability design The above calculation is based on the fact that all random variables follow a normal distribution and are independent of each other. When each random variable is not all normally distributed, it is impossible to calculate reliable. Degree, in this case, you can use the Monte Carlo method to solve. An important step in the Monte Carlo method is to generate random numbers. The basis for generating random variables is to generate uniformly distributed random numbers over the interval [0,1]. Other types of distributions such as normal distribution, lognormal distribution, Poisson distribution, etc. are all transformed by uniform distribution over the interval [0,1]. After obtaining a uniformly distributed random number, a random variable based on the random number can be constructed in a variety of ways. The commonly used method is the inverse function method, that is, the random variable is derived by using the distribution function F-1(x) of the random variable x. The basic algorithm is: first generate a uniform distribution random number r, then let x = F-1 (x), return.
The normal distribution random variable is generated as follows, the density function of the normal distribution is fN(x)=12exp-12x-(2,-)
The flow of gear drive reliability calculation using the Monte Carlo method is shown in Figure 1.
According to the practice of the machine, compared with the R(t)=99.8817 obtained by the traditional calculation method, the results obtained by the two methods are relatively close, which indicates that the Monte Carlo method is feasible and correct for reliability calculation.
3 Conclusion Compared with the traditional algebraic method to calculate the reliability, the Monte Carlo method only needs to know the stress and intensity distribution types and probability parameters, and can obtain the reliability of mechanical parts without complicated integral calculation, and with the simulation As the number of tests increases, the accuracy of the simulation results also increases. Applying this method to mechanical CAD saves time and labor, greatly improves work efficiency, and ensures efficient and accurate calculations, ensuring reliability design.
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