Reliability analysis of crankshaft for high-speed punch based on Monte-Carlo method

The crankshaft, which has fateful consequence to the performance and reliability, is a key component of high-speed punch. In manufacturing process, the high reliability value of crankshaft contributes to the increasing of the reliability of the whole punch system. The study builds a reliability analysis model of the crankshaft for punching process under the premise of regarding design parameters as random variables. Monte-Carlo method is employed to make reliability analysis for crankshaft based on ANSYS. The numerical results present that the failure probability of strength and stiffness for crankshaft satisfied the use requirement. The impact of every input variable on reliability are obtained from the sensitivity analysis of status function factors. The design variable D4 and PRXY have the greatest impact on the strength of crankshaft and the design variable YOUNG and L3 have the greatest impact on the stiffness of crankshaft. And the result matches up with the response surface graph. The reliability analysis result provides some useful information for the improvement of the reliability of crankshaft for high-speed punch.


Introduction
High-speed punch is an efficient, high-precision and high-automation punching machine. It has been widely used in aeronautics and astronautics, automobile, traffic and transportation, metallurgical chemical industries and so on [1][2]. The crankshaft is a key part of main transmission system for highspeed punch and its reliability has great impact on the reliability of high-speed punch. Modeling and simulation of the crankshaft is obviously not new, as the crankshaft is a very popular mechanism used in many different types of machinery. Zhou [3] constructed a 2-D stress strength interference model, considering the disparity of the load asymmetric factor, to make reliability analysis for diesel engine crankshaft and the analysis result was consistent with the statistical data of actual failure cases. Ye [4][5] analyzed the fatigue reliability of crankshaft for reciprocating pump using the radius vector method and he developed a practical object-oriented application software to make fatigue reliability analysis for improving the design of reciprocating pump. Wang [6] built a quarter finite element model of the entire assembly to make reliability analysis and he got the temperature distribution, thermal deformation and thermal stress distribution of the piston. Taraza [7] developed an algorithm capable to identify faulty cylinders based on the phases of the lowest three harmonic orders of the measured speed. Sun [8] made the fatigue reliability analysis of crankshaft using stochastic response surface method with implicit performance function and some examples were employed to demonstrate the advantages of the proposed method. Wang [9] presented a sensitivitybased algorithm about tool change time and the reliability of the machining process to improve the reliability of both tools and the overall process. Singh [10] provides an optimal model in predicting the reliability for an automobile crankshaft through the time dependent failure analysis based on three parameter Weibull distribution. As stated previously, there is few study on reliability analysis of crankshaft for high-speed punch. This work introduces the reliability analysis theory including the basic assumptions of reliability and the reliability calculation method. Then, the reliability analysis model of crankshaft is built in ANSYS and the reliability analysis based on Monte-Carlo method is carried out.

Basic assumptions of reliability
Reliability is defined as: the probability that a device will perform its intended function during a specified period of time under stated conditions. The stress-strength interference model is usually employed to make reliability analysis for mechanical components. Stress in the model is typically the factors that has an impact on functionality, such as stress, torque, temperature, vibration, corrosion, and so on. Strength in the model is typically the resistance ability of external loads. The influence in strength is corrosion, temperature, materials performance, quality of machining and external loads. So the definition of reliability is described as: the probability of the strength is greater than the stress. Some basic assumptions of reliability should be made to analyze the reliability for mechanical products [11]: • The strength of mechanical components is nonnegative random variable or random process.
• The stress is nonnegative random variable or random process.
• When the strength is greater than the stress, the part is believed to be reliable. Otherwise, the part is believed to be of failure. • For the stress-strength interference model, the part is of failure only because of the stress.
• The formula for calculating the strength and stress still applies and all quantitative symbols are treated as random variable or random process. In the stress-strength interference model, the stress and strength are all regarded as random variable or random process, which could reflect the exact thing that the stress and strength are of objective uncertainty, discreteness and time-varying uncertainty. For this reason, the probabilistic method must be used to calculate the reliability of mechanical components.

Reliability calculation method
The reliability could be calculated in the case that the probability density function or joint probability density function of random variable X is known. That may not always be the case, and the distribution probability is difficult to be determined in the operating condition. There are some approximation methods to calculate the reliability, such as Moment method, Monte-Carlo method, Response Surface method, and so on.

Moment method.
The moment method mainly includes the first-order second-moment method and the higher order moment method. And the first method is widely used in the actual projects. The basic idea is that Taylor expand the nonlinearity performance function according to the first two moments, and remain the once migration, and then approximately calculate the mean value & standard deviation of performance function to get the reliability.

2.2.2.
Monte-Carlo method.The basis theory of Monte-Carlo method is large numbers law in probability theory. Monte-Carlo method is one of the most intuitive, accurate and the most efficient for high non-liner problem structural reliability analysis approach. The reliability can be got from statistical analysis of a large random sample.

Response surface method.
Response surface method is employed to solve the reliability problem of complex mechanical system. The basic idea of this method is to suppose a simple analytic expression between limit state variable and basic variable firstly, then to determine the unknown parameters using interpolation method, and to get the certain analytic expression finally. The reliability calculation is simplified and the calculation precision could be improved using response surface method. And it can be calculated using finite element analysis program.

Structure of crankshaft
The structure sketch of crankshaft is shown as Figure 1 [12][13]. This type of crankshaft is used for 800KN high-speed punch made by a company. The working conditions of the punch are as follows: Nominal Pressure is 800 KN, Nominal Pressure Stroke is 2 mm, Slider Stroke is 30 mm, number of strokes is 600 SPM (Strokes per Minute), the thickness of punched part is 2 mm, the material of crankshaft is 40Cr. The crankshaft transfer drive torque from the flywheel, which is installed in location 4 with clutch. The four groups of support bearings, which are installed in location 1, 3, 5, 8, form a revolute pair with the punch frame. The two groups of support bearings, which are installed in location 6, 7, form a revolute pair with main connecting rod. One group of support bearing, which is installed in location 2, form a revolute pair with balancing connecting rod.

Design variables of crankshaft
Design variables of crankshaft fall into three types: material characteristics parameters, load parameters and geometry characteristic parameters. The name of parameter, the distribution type, the variation coefficient and the mean are shown in Table 1. The data is from reference [12][13]. The output variables of crankshaft are SUBS and SUBU.
Where U max is the maximum allowable displacement of crankshaft in vertical direction, maxu is the maximum allowable displacement of crankshaft in vertical direction.

Histogram of status function
The histogram of design variable D1 for reliability analysis is shown in Figure 2. As can be seen from the chart, the histogram of design variable approaches the normal distribution probability function curve. The histogram generates a smooth curve without larger gap and the sampling number of simulation is bigger enough to keep the calculating accuracy. The response surface graph of output status function is shown in Figure 3-Figure 4. Figure 3 shows the impacts of input variable PRXY and D4 on output variable SUBS. SUBS decreases with the increase of the D4, and increases with the increase of PRXY. The optimal combination of input variables PRXY and D4 could be found to keep an appropriate SUBS.  Figure 4 shows the impacts of input variable L3 and YOUNG on output variable SUBU. SUBU decreases with the increase of the L3, and decreases with the increase of YOUNG. The optimal combination of input variables L3 and YOUNG could be found to keep an appropriate SUBU. The

Cumulative distribution function ofoutputstatusfunction
Thecumulative distribution function of output status function isshowninFigure 5- Figure 6. The value of any point in the distribution function equals to the probability value of data appeared in this point. As shown in Figure 5, the failure probability of strength for crankshaft is probability value of when SUBS<0. And the probability value of SUBS is 0.94863. As shown in Figure 6, the failure probability of stiffness for crankshaft is probability value of when SUBU<0. And the probability value of SUBU is 0.99953.

Sensitivity analysis of statusfunction factors
The probability sensitivity reflects the impact of every input variable on reliability. The input variable couldbe controlled inthedesignandmanufacturingprocesstogetbetterproduct. The sensitivity graph of SUBS andSUBU isshownin Figure 7-Figure 8. As shown in Figure 7, the input variable D4 andPRXY have the greatest impact on SUBS. D4 isbelow zero graduation line, which means that SUBS decreases with the increase of D4, and PRXY is above zero graduation line, which means that SUBS increases with the increase of PRXY. And it matched up with the response surface graph. As shown in Figure 8, the input variable YOUNG andL3 have the greatest impact on SUBU. They are all below zero graduation line and SUBU decreases with the increase of YOUNG andL3. This paper studies the reliability ofcrankshaft forhigh-speed punch.Thereliability analysis theoryisintroduced, and the basic assumptions of reliability and the reliability calculation method are analyzed.Monte-Carlo method is employedtomakereliabilityanalysisforcrankshaftbasedon ANSYS. The numerical results present thatthe failure probability of strength for crankshaft is 0.94863andthe failure probability of stiffness for crankshaft is 0.99953. The impact of every input variable on reliabilityareobtainedfromthesensitivity analysis of status function factors. The input variableD4 andPRXYhave the greatest impact onSUBSandthe input variableYOUNG andL3 have the greatest impact onSUBU. Thesensitivity analysisresultmatches up with the response surface graph.