A method for calculating structural reliability of a related failure mode

Abstract

The application discloses a structure reliability calculation method of a related failure mode and belongs to the technical field of structure reliability analysis. The application provides the structure reliability calculation method of the related failure mode, which is suitable for analyzing the structure reliability of a structure with a related failure mode and a strong nonlinear characteristic, is high in calculation precision, low in sampling frequency and high in calculation efficiency. The application comprises the following steps: step one, constructing state equations corresponding to different failure modes according to a model of the structure with the strong nonlinear characteristic; step two, taking a mean point as an initial point and acquiring checking points on different failure boundaries by using a first-order reliability analysis method; step three, establishing hyperplanes through the checking points, determining high-order (m-order) initial points and acquiring high-order (m-order) checking points; step four, establishing fitting functions according to the checking points on the different failure boundaries and the respective high-order (m-order) checking points, and calculating the reliability of the structure with the related failure mode; step five, judging whether the precision requirement is met, if yes, executing step six, otherwise, returning to execute step three; and step six, outputting the structure reliability and verifying the structure reliability by using a Monte Carlo method.

Description

Technical Field

[0001] This invention belongs to the field of structural reliability analysis technology, and in particular relates to a method for calculating the reliability of structures with strongly nonlinear characteristics and related failure modes. Background Technology

[0002] Failure modes in engineering structures are often not unique, and due to the shared load sources and load-bearing structures, various failure modes are correlated. Furthermore, due to the complexity of these structures, their models generally exhibit strong nonlinear characteristics and low failure probabilities. How to accurately and efficiently calculate the reliability of multi-failure-mode structures with strong nonlinear characteristics has always been a research hotspot in the field of reliability analysis.

[0003] Currently, common methods for structural reliability analysis under correlated failure modes include the bounds method, the correlation coefficient method, and stochastic simulation. The bounds method assumes that the reliability of a structure under correlated failure modes lies between the reliability under the assumption of complete independence and the reliability under the assumption of complete correlation, varying within the estimated range depending on the degree of correlation between the failure modes. The correlation coefficient method assumes a linear correlation between various failure modes and calculates the covariance between each failure mode using the verification point method in reliability analysis to obtain the correlation coefficient matrix, then uses multidimensional normal integrals to solve for the structural reliability. Stochastic simulation, primarily using the Monte Carlo method, involves large-sample sampling of test points according to the distribution characteristics of random variables, and then obtaining the reliability of structures with correlated failure modes through statistical sampling results. Here, the bounds method can only obtain a general range of structural reliability under correlated failure modes; the correlation coefficient method relies on the assumption of linearity between failure modes, which can lead to significant calculation errors, especially for structures with strongly nonlinear failure modes; stochastic simulation, represented by the Monte Carlo method, has universal applicability for both linear and nonlinear failure modes. However, random simulation methods are usually based on the results of large-sample experiments, which are computationally expensive and inefficient for complex engineering structures.

[0004] Therefore, as engineering practices place increasingly higher demands on structural reliability, there is a growing need for an accurate, efficient, and computationally cost-effective method for structural reliability analysis based on related failure modes. Summary of the Invention

[0005] To address the aforementioned shortcomings of existing technologies, this invention provides a structural reliability calculation method for relevant failure modes. This method not only boasts high calculation accuracy but also requires fewer sampling steps and offers high computational efficiency.

[0006] To achieve the above objectives, the present invention adopts the following technical solution: a structural reliability calculation method based on relevant failure modes, comprising the following steps:

[0007] Step 1: Based on the model of the strongly nonlinear characteristic structure, construct the state equations corresponding to different failure modes;

[0008] Step 2: Using the mean point as the initial point, use a single reliability analysis method to obtain the verification points on different failure boundaries;

[0009] Step 3: Establish a hyperplane through each verification point, determine the high-order (m-th order) initial point, and obtain the high-order (m-th order) verification points;

[0010] Step 4: Establish fitting functions based on the verification points on different failure boundaries and their respective higher-order (m-th order) verification points, and calculate the reliability of the relevant failure mode structure;

[0011] Step 5: Determine if the accuracy requirement is met. If yes, proceed to Step 6; otherwise, return to Step 3.

[0012] Step 6: Output the structural reliability and verify it using the Monte Carlo method.

[0013] Step one, which involves constructing state equations for different failure modes based on a model with strongly nonlinear characteristic structures, specifically includes the following steps:

[0014] Step A: Establish a model of the strongly nonlinear characteristic structure. This model can be an analytical model, a simulation model, or an actual physical model.

[0015] Step B: Based on the structural model, construct the state equations corresponding to different failure modes;

[0016] Step C: By using the state equations under different failure modes, different input quantities are given to obtain different response output quantities.

[0017] Step two, which uses the mean point as the initial point and employs a single reliability analysis method to obtain verification points on different failure boundaries, specifically includes the following steps:

[0018] Step A: Select the mean point of the nonlinear structural state equation as the initial sampling point and normalize it to the standard normal space;

[0019] Step B: Calculate the gradient value at the sampling point. For explicit analytical models, the gradient value is obtained directly by taking the partial derivative; for implicit structural models, the gradient value is obtained by the central difference method or the forward difference method.

[0020] Step C: Use a single reliability analysis method to obtain the verification points on different failure boundaries;

[0021] Step D: Calculate the probability of primary failure under different failure modes based on the verification points on different failure boundaries.

[0022] Step three, which involves establishing a hyperplane through each verification point, determining the high-order (m-th order) initial point, and obtaining the high-order (m-th order) verification points, specifically includes the following steps:

[0023] Step A: Establish a first-order linear hyperplane through each verification point, and set the initial value of m to 2;

[0024] Step B: Using each verification point as the center, select an appropriate step size as the radius to establish the m-th hypersphere corresponding to that verification point;

[0025] Step C: Determine the intersection point between the first-order linear hyperplane and the m-th hypersphere at each verification point; this is the m-th initial point.

[0026] Step D: Continue using the reliability analysis method to obtain the m verification points corresponding to the m initial points.

[0027] Step four, which involves establishing a fitting function based on the verification points on different failure boundaries and their respective higher-order (m-order) verification points, and calculating the reliability of the relevant failure mode structure, specifically includes the following steps:

[0028] Step A: Select an appropriate fitting function to fit the verification points on different failure boundaries and their respective higher-order (m-th) verification points;

[0029] Step B: Replace the limit state equations for different failure modes of the nonlinear structure with these fitted functions;

[0030] Step C: Using numerical integration, calculate the m-fold failure probability P of the structure with correlated failure modes using the probability addition or multiplication formula. f (m) .

[0031] If the accuracy requirement is met as described in step five, proceed to step six; otherwise, return to step three, which specifically includes the following steps:

[0032] Step A: Solve for the m-fold failure probability P of the structure with correlated failure modes. f (m) With the failure probability P of m-1 times f (m-1) difference;

[0033] Step B: If P f (m) -P f (m-1)If the value is less than the given precision value, then the precision requirement is met, and proceed to step six; if P f (m) -P f (m -1) If the value is greater than or equal to the given precision value, the precision requirement is not met. After setting m = m + 1, return to step three.

[0034] The reliability of the output structure described in step six is ​​verified using the Monte Carlo method, which specifically includes the following steps:

[0035] Step A: Output the m-fold failure probability P of the relevant failure mode structure. f (m) ;

[0036] Step B: Calculate the failure probability of the nonlinear correlated failure mode structure using the Monte Carlo stochastic simulation method;

[0037] Step C: Compare and verify the calculation results.

[0038] Beneficial effects of the present invention

[0039] 1. The structural reliability calculation method for relevant failure modes described in this invention is based on the classic single-mode reliability analysis method, and its theoretical foundation is relatively mature. Therefore, the calculation process of this method is stable and has good convergence.

[0040] 2. The structural reliability calculation method based on related failure modes described in this invention calculates reliability by fitting the limit state equations of nonlinear structures. Therefore, it has higher calculation accuracy compared to the currently widely used linear correlation coefficient method.

[0041] 3. The structural reliability calculation method for relevant failure modes described in this invention uses the gradient direction of nonlinear structural sample points as a guide to complete the iterative sampling process. Therefore, compared with the stochastic simulation method based on large sample sampling, it has higher computational efficiency and lower computational cost.

[0042] In summary, this invention is based on a single-step reliability analysis method. It iteratively samples and finds sample points on the limit state surface to obtain the limit state equations for different failure modes of the structure. Then, it uses numerical integration and probabilistic addition or multiplication formulas to calculate the reliability of the structure under relevant failure modes. Therefore, compared with commonly used methods, this method has better convergence, higher computational accuracy, and higher efficiency, making it particularly suitable for the reliability analysis of relevant failure modes of large, complex, and nonlinear engineering structures with high computational demands. Attached Figure Description

[0043] Figure 1This is a flowchart of the reliability calculation method of the present invention.

[0044] Figure 2 This is a schematic diagram of the limit state equations and mean points for a gear with multiple failure modes.

[0045] Figure 3 The flowchart shows the procedure for obtaining verification points using a single reliability analysis method.

[0046] Figure 4 This is a schematic diagram of the fitting of the contact fatigue failure boundary of a gear.

[0047] Figure 5 This is a schematic diagram of the fitting of the gear contact fatigue failure boundary and bending fatigue failure boundary. Detailed implementation method:

[0048] The following uses a cylindrical spur gear as an example to perform reliability calculations for gear structures under relevant failure modes. The gear material is 40Cr steel, heat-treated. The number of teeth on the driving gear z1 is 20, the number of teeth on the driven gear z2 is 32, the module m is 8, the pressure angle α is 20°, the tooth width B is 83mm, and the addendum coefficient h is... a Taking option 1, the gear operates in one direction, the load is stable, one shift per day, and the expected lifespan is 10 years. The flowchart of the reliability calculation method of this invention is as follows: Figure 1 As shown.

[0049] Step 1: Based on the model of the strongly nonlinear characteristic structure, construct the state equations corresponding to different failure modes;

[0050] Transmission gears exhibit various failure modes during operation, such as tooth breakage, tooth surface wear, and plastic deformation. This study focuses on two related failure modes: tooth breakage and tooth surface wear, and constructs state equations for tooth root bending fatigue strength and tooth surface contact fatigue strength.

[0051] Step A: Establish a model of the strongly nonlinear characteristic structure. This model can be an analytical model, a simulation model, or an actual physical model.

[0052] Considering that transmission gears share the same load source and load-bearing body, the two failure modes of gears—tooth breakage and tooth surface wear—are correlated. Furthermore, under the condition of transmitting the same torque, the pinion tooth surface will bear a greater contact load; therefore, the pinion is generally considered the weakest link in the transmission gear system and is used for modeling and analysis. Thus, the commonly used analytical models for tooth root bending fatigue stress and tooth surface contact fatigue stress can be expressed as follows:

[0053]

[0054]

[0055] In the formula, σF For the bending fatigue stress at the tooth root, σ H Let be the tooth surface contact fatigue stress, K be the load factor (taken as 0.725), T be the transmitted torque, d1 be the pinion pitch circle diameter (d1 = m·z1 = 160 mm), i be the transmission ratio (i = z1 / z2 = 1.6), and Y be the transmission torque. Fa Y is the pinion tooth form factor, taken as 2.82. Sa Y is the stress correction factor for the pinion, taken as 1.57. ε The overlap factor for bending fatigue is calculated to be 0.715, Z. H Z is the gear pitch region coefficient, calculated to be 2.49. E The gear spring constant is, according to the table, [value]. Z ε The overlap factor for contact fatigue is calculated to be 0.89.

[0056] The commonly used analytical models for tooth root bending fatigue strength and tooth surface contact fatigue strength can be expressed as follows:

[0057]

[0058]

[0059] In the formula, For tooth root bending fatigue strength, S represents the tooth surface contact fatigue strength. Fmin S is the minimum safety factor for calculating bending fatigue strength. Hmin Y is the minimum safety factor for calculating contact fatigue strength. ST Y is a stress correction factor, taken as 2. N The life factor for calculating the gear bending strength is taken as 1.05, Y. X The size factor for calculating the gear bending strength is taken as 1.03, Z. N The life factor calculated for gear contact strength is taken as 1.45, Z. X The size factor for calculating gear contact strength is taken as 1.0, Z. W Z is the work hardening factor, taken as 1.0. LVR The influence coefficient of the lubricating oil film is taken as 0.92, σ Flim σ is the bending fatigue limit at the tooth root. HFlim This represents the tooth surface contact fatigue limit.

[0060] Here, σ Flim and σ HFlim Both are related to the Brinell hardness of the gear material. When the gear material is 40Cr, σ Flim and σ HFlim With Brinell hardness γ h The relationships between them can be expressed as follows:

[0061]

[0062]

[0063] In the formula, γ h This represents the Brinell hardness of the gear material, expressed in HBW.

[0064] Step B: Based on the structural model, construct the state equations corresponding to different failure modes;

[0065] Based on equations (1), (3), and (5), the gear tooth root bending fatigue state equation can be constructed as follows:

[0066]

[0067] In the formula, x is a basic random vector, and x = [T, γ] h ].

[0068] Based on equations (2), (4), and (6), the gear tooth surface contact fatigue state equation can be constructed as follows:

[0069]

[0070] Furthermore, as can be seen from equations (2) and (6), the tooth surface contact fatigue state equation exhibits significant nonlinearity. For example... Figure 2 The figure shows the limit state equations and mean points of a gear with multiple failure modes.

[0071] Step C: By using the state equations under different failure modes, different input quantities are given to obtain different response output quantities.

[0072] Here, equation (7) is the state equation under the tooth breakage failure mode, and equation (8) is the state equation under the tooth surface wear failure mode. Both state equations have the same random vector x. Among them, x includes the transmitted torque T and the Brinell hardness γ of the gear material. h Obviously T and γ h They are independent of each other. Therefore, given different input quantities x, different response output quantities g can be obtained according to equations (7) and (8). F (x) and g H (x). Moreover, g F (x) and g H (x) represent different states of the transmission gears. If g F (x) < 0, indicating that the gear has failed due to tooth breakage; if g H (x) < 0, indicating that the gear has failed due to tooth surface wear; if g F (x)>0, meaning the gear will not experience tooth breakage failure; if g H(x)>0, indicating that the gear will not experience tooth surface wear failure; if g F (x) = 0, indicating that the gear is in a critical state of tooth breakage failure; if g H (x) = 0, which means that the gear is in the critical state of tooth surface wear failure.

[0073] Step 2: Using the mean point as the initial point, use a single reliability analysis method to obtain the verification points on different failure boundaries;

[0074] From equations (7) and (8), we can see that the gear state equation g F (x) and g H (x) have the same random vector x, assuming the initial point of the random vector x is the mean point, i.e. Two failure boundaries g can be obtained using a single reliability analysis method. F (x) = 0 and g H The verification point is at (x) = 0. The flowchart for obtaining the verification point using a single reliability analysis method is shown below. Figure 3 As shown.

[0075] Step A: Select the mean point of the nonlinear structural state equation as the initial sampling point and normalize it to the standard normal space;

[0076] Suppose a normally distributed random vector x = [x1, x2, ..., xn] n The variable ] contains n independent random variables, whose mean vector is μ. x The standard deviation vector is σ x The state equation is expressed as

[0077] z X =g(x) =g(x1,x2,…,x) n (9)

[0078] In the formula, when z X When x = 0, the corresponding g(x) = 0 is called the limit state equation.

[0079] Normalizing a normally distributed random vector x to the standard normal space can transform it into a standard normal random vector y, i.e.

[0080]

[0081] Here, vectors x and y satisfy the following relationship:

[0082] F x (x)=Φ(y) (11)

[0083]

[0084] In the formula, F xLet (·) be the probability distribution function of the random vector x, and Φ(·) be the standard normal probability distribution function. For F x The inverse function of (·).

[0085] Therefore, within the standard normal space Y, the state equation of the structure can also be expressed as:

[0086]

[0087] Based on the given conditions, the gear state equation g F (x) and g H If the mean of (x) is used as the initial sampling point, then And μ T =2100 N·m, σ T =210 N·m, According to equations (10) and (13), the random variable x can be normalized to the standard normal space Y, and the corresponding initial sampling point y in the standard normal space Y can be obtained. (0) and state equation G F (y) and G H (y).

[0088] Step B: Calculate the gradient value at the sampling point. For explicit analytical models, the gradient value is obtained directly by taking the partial derivative; for implicit structural models, the gradient value is obtained by the central difference method or the forward difference method.

[0089] Let i Y =0, calculate sampling points The gradient value at that point, then

[0090]

[0091] In the formula, Let be the probability density function of the standard normal distribution. Let x be a random variable i The corresponding probability density function.

[0092] If the state equation g(x) is explicit, the gradient value can be obtained directly by taking the partial derivative; if the state equation g(x) is implicit, the gradient value can be obtained through the central difference method or the forward difference method. The formula for the central difference method is as follows:

[0093]

[0094] The formula for calculating the forward difference method is:

[0095]

[0096] In the formula, △x i Represents the variable x in vector xi The minute changes; and

[0097]

[0098] Due to the gear state equation g F (x) and g H (x) are all explicit analytical models, and the gradient values ​​can be obtained directly by taking the partial derivatives according to equation (14). and

[0099] Step C: Use a single reliability analysis method to obtain the verification points on different failure boundaries;

[0100] In the standard normal space Y, the shortest distance from the origin to the limit state surface is the reliability index β; and the point corresponding to the shortest distance on the limit state surface is the verification point y. * ,and Therefore, the verification point y can be calculated using the following formula. * Corresponding β *

[0101]

[0102] Based on Taylor's expansion method, the limit state equation z shown in equation (13) is... Y =0 at the verification point y * Expand into a linearized equation

[0103]

[0104] Clearly, all terms on the right-hand side of equation (18) are linear combinations of random vectors y, therefore their mean is... and standard deviation for

[0105]

[0106] Therefore, the linearized limit state function z YL The corresponding reliability index β YL for

[0107]

[0108] Based on equations (18), (19), and (20), the verification point y in the standard normal space Y can be derived. * coordinates

[0109]

[0110] Therefore, according to equation (18), the two state equations G for the gear can be obtained. F(y) and G H (y) past Linearized state equations of points and And determined by equation (20) and Corresponding reliability index and Finally, according to equation (21), after repeated iterations, until... and The difference is less than the given precision value ε β At that time, two failure boundaries G close to the gear can be obtained. F (y) = 0 and G H Verification point on (y) = 0 and

[0111] Step D: Calculate the probability of primary failure under different failure modes based on the verification points on different failure boundaries.

[0112] Based on the principle of first-order reliability analysis, the first-order failure probability of a structure can be approximately expressed as:

[0113] P f ≈Φ(-β * In equation (22), β * For the verification point y * The corresponding reliability index.

[0114] Therefore, based on the verification points at the two failure boundaries of the gear and The first-order failure probability under the two failure modes can be calculated using equations (17) and (22). and

[0115] Step 3: Establish a hyperplane through each verification point, determine the high-order (m-th order) initial point, and obtain the high-order (m-th order) verification points;

[0116] Based on Taylor's expansion method, the check points on the two failure boundaries of the gear are... and Establish each of the first-order linear hyperplanes, let m be initially 2, and determine m initial points. and Following the process described in step two, m verification points on the two failure boundaries can be obtained. and

[0117] Step A: Establish a first-order linear hyperplane through each verification point, and set the initial value of m to 2;

[0118] According to equation (18), the check point for the gear can be established. and First-order linearized limit state equation and That is, a linear hyperplane, and let m be initially 2.

[0119] Step B: Using each verification point as the center, select an appropriate step size as the radius to establish the m-th hypersphere corresponding to that verification point;

[0120] Using the verification point y * Centered on the standard normal space Y, and choosing a step size r as the radius, we construct the hypersphere equation in the standard normal space Y.

[0121]

[0122] In the formula: r = α·r R α is the step size coefficient, r R For reference step size, and

[0123] Therefore, according to equation (23), the verification points on the two failure boundaries of the gear are used. and Centered on a point, hyperspheres corresponding to different failure modes can be established. As the step size coefficient α starts from 1 and successively takes positive integer values ​​of m-1, m hyperspheres corresponding to different failure modes can be obtained sequentially.

[0124] Step C: Determine the intersection point between the first-order linear hyperplane and the m-th hypersphere at each verification point; this is the m-th initial point.

[0125] Passing point y * Given any two coordinates y in the n-dimensional standard normal space Y. k and y s (Here, k < s) are paired to establish A point y * coordinate plane

[0126]

[0127] In the formula, j = 1,…,k-1,k+1,…,s-1,s+1,…,n.

[0128] Linearize the first-order limit state equation of the gear and Solving these equations simultaneously with equations (23) and (24) yields the following results: Points on a hyperplane and (here, ), and use it as the initial point for m times.

[0129] Step D: Continue using the reliability analysis method to obtain the m verification points corresponding to the m initial points.

[0130] By performing an iterative search using the process described in step two, we can obtain... Sample points closely approximating the two failure surfaces of the gear. and This is used as a sample point for the m-th iteration search of the gear.

[0131] Step 4: Establish fitting functions based on the verification points on different failure boundaries and their respective higher-order (m-th order) verification points, and calculate the reliability of the relevant failure mode structure;

[0132] Based on the verification points and their respective m-fold verification points at the boundaries of tooth breakage failure and tooth surface wear failure, a polynomial function is selected as the fitting function to fit the gear failure boundary. Then, numerical integration is used to solve for the failure probability in different failure domains of the gear. Finally, the reliability of the multi-failure mode gear is calculated based on the probability addition and multiplication formulas. Figure 4 The diagram shows a fitting schematic of the gear contact fatigue failure boundary; as shown below. Figure 5 The diagram shows the fitting of the gear contact fatigue failure boundary and bending fatigue failure boundary.

[0133] Step A: Select an appropriate fitting function to fit the verification points on different failure boundaries and their respective higher-order (m-th) verification points;

[0134] Based on the verification points at the fatigue fracture failure boundary of gear teeth and m verification points Choosing a polynomial function as the fitting function, an explicit expression for the tooth breakage failure boundary can be fitted as follows:

[0135]

[0136] In the formula, fit(·) is a polynomial fitting function.

[0137] In addition, based on the verification points at the wear failure boundary of the tooth surface and m verification points Choosing a polynomial function as the fitting function, an explicit expression for the tooth surface wear failure boundary can be fitted as follows:

[0138]

[0139] Step B: Replace the limit state equations for different failure modes of the nonlinear structure with these fitted functions;

[0140] Using polynomial fitting function and Limit state equations for alternative failure modes of gear teeth breakage and tooth surface wear.

[0141] Step C: Using numerical integration, calculate the m-fold failure probability P of the structure with correlated failure modes using the probability addition or multiplication formula. f (m) .

[0142] First, numerical integration is used to solve for the failure probability of the gear in different failure domains. When the gear experiences tooth breakage failure, its failure probability is...

[0143]

[0144] In the formula, f Y (·) is the joint probability density function of the random vector y; F F Let be the failure domain in the state space where gear tooth breakage occurs, and

[0145] When a gear fails due to tooth surface wear, its failure probability is:

[0146]

[0147] In the formula, F H Let be the failure domain in the state space where gear tooth surface wear failure occurs, and

[0148] When a gear experiences both tooth breakage and tooth surface wear failure, its failure probability is:

[0149]

[0150] In the formula, F FH This refers to the failure domain in the state space where a gear experiences both tooth breakage and tooth surface wear failure.

[0151]

[0152] Then, according to the addition and multiplication formulas for probability, the m-fold failure probability P of the gear-related failure mode can be determined. f (m) for

[0153]

[0154] Step 5: Determine if the accuracy requirement is met. If yes, proceed to Step 6; otherwise, return to Step 3.

[0155] Determine the m-fold failure probability P of the gear-related failure modes. f (m)Does the accuracy requirement meet the requirement? If it does, the calculation result can be output; otherwise, iterative calculation of the failure probability result that meets the accuracy requirement needs to continue.

[0156] Step A: Solve for the m-fold failure probability P of the structure with correlated failure modes. f (m) With the failure probability P of m-1 times f (m-1) difference;

[0157] Let m be initially 2, and according to equation (30), the probability of m failures of the gear in the relevant failure mode is calculated as P. f (m) At the same time, the failure probability P of m-1 times can also be obtained. f (m-1) .

[0158] Step B: If P f (m) -P f (m-1) If the value is less than the given precision value, then the precision requirement is met, and proceed to step six; if P f (m) -P f (m -1) If the value is greater than or equal to the given precision value, the precision requirement is not met. After setting m = m + 1, return to step three.

[0159] Let P be the probability of m failures of the gear with the relevant failure mode. f (m) With the failure probability P of m-1 times f (m-1) Take the difference, if P f (m) -P f (m-1) If the result is less than the given precision value, it indicates that the calculation result meets the precision requirements, and step six can be continued. Furthermore, if P... f (m) -P f (m-1) If the result is greater than or equal to the given precision value, it indicates that the calculation result does not meet the precision requirements. In this case, we need to set m = m + 1 and then return to step three.

[0160] Step 6: Output the structural reliability and verify it using the Monte Carlo method.

[0161] The m-fold failure probability of the gear is output as the final result. At the same time, the large-sample Monte Carlo method is used to randomly sample the two failure mode models of the gear. The failure frequency results of the sampling analysis are used as the true values ​​of the failure probability and compared with the m-fold failure probability results to verify the effectiveness of the method proposed in this invention.

[0162] Step A: Output the m-fold failure probability P of the relevant failure mode structure. f (m) ;

[0163] The failure probability P of the gear in the m-th failure mode is... f (m) This will be the final calculation result output. Furthermore, the calculation result will be listed in column 1 of Table 1.

[0164] Step B: Calculate the failure probability of the nonlinear correlated failure mode structure using the Monte Carlo stochastic simulation method;

[0165] The transmitted torque T of the gear and the Brinell hardness γ of the gear material are considered. h Treat it as a random variable that follows a normal distribution, and μ T =2100 N·m, σ T =210 N·m, μ γh =330HBW, σ γh =33HBW, using the normrnd function in Matlab software, two sets of normally distributed data were randomly generated, with a sampling number of 10. 6 Substituting into equations (7) and (8) for calculation, the statistical results are used as the true value P of the gear failure probability of the relevant failure mode. f ,Right now

[0166]

[0167] In the formula, N is the number of random samples, where N = 10. 6 ;I[g F (x),g H [x] is the objective function, and

[0168]

[0169] Based on equations (31) and (32), the true value P of the gear failure probability of the relevant failure mode is calculated. f Meanwhile, the calculation results of the gear failure probabilities for relevant failure modes calculated by various methods are all listed in Table 1.

[0170] Table 1. Statistical table of gear failure probabilities calculated by different methods for relevant failure modes.

[0171]

[0172] Step C: Compare and verify the calculation results.

[0173] Comparing the results in Table 1, it can be seen that the calculation results of the method proposed in this invention are consistent with those of the Monte Carlo simulation.6 The calculation results are relatively close, resulting in high accuracy. Furthermore, with only 36 sampling points, the calculation efficiency is also high. In addition, as shown in Table 1, the boundary method can only obtain the approximate range of gear reliability for related failure modes, while the correlation coefficient method, based on the assumption of linear correlation between failure modes, has a larger calculation error for gear structures with nonlinear failure modes.

Claims

1. A method for calculating the structural reliability of related failure modes, characterized in that... It includes the following steps: Step 1: Based on the model of the strongly nonlinear characteristic structure, construct the state equations for tooth root bending fatigue strength and tooth surface contact fatigue strength. Step 2: Using the mean point as the initial point, use a single reliability analysis method to obtain the verification points on different failure boundaries; Step 3: Establish a hyperplane through each verification point, determine m initial points, and obtain m verification points; Step 4: Establish a fitting function based on the verification points on different failure boundaries and their respective m-th verification points, and calculate the reliability of the relevant failure mode structure; Step 5: Determine if the accuracy requirement is met. If yes, proceed to Step 6; otherwise, return to Step 3. This specifically includes the following two steps: Step A: Solve for the m-fold failure probability of the structure with correlated failure modes. With m-1 failure probabilities difference; Step B: If - If the value is less than the given precision value, then the precision requirement is met, and proceed to step six; if... - If the value is greater than or equal to the given precision value, the precision requirement is not met. Let m = m + 1, then return to step three. Step 6: Output the structural reliability and verify it using the Monte Carlo method.

2. The structural reliability calculation method based on relevant failure modes according to claim 1, characterized in that... Step one, which involves constructing the state equations for tooth root bending fatigue strength and tooth surface contact fatigue strength based on the model of the strongly nonlinear characteristic structure, specifically includes the following steps: Step A: Establish a model of the strongly nonlinear characteristic structure. This model can be an analytical model, a simulation model, or an actual physical model. Step B: Based on the structural model, construct the state equations for tooth root bending fatigue strength and tooth surface contact fatigue strength. Step C: By using the state equations for tooth root bending fatigue strength and tooth surface contact fatigue strength, different input quantities are given to obtain different response output quantities.

3. The structural reliability calculation method based on relevant failure modes according to claim 1, characterized in that... Step two, which uses the mean point as the initial point and employs a single reliability analysis method to obtain verification points on different failure boundaries, specifically includes the following steps: Step A: Select the mean point of the nonlinear structural state equation as the initial sampling point and normalize it to the standard normal space; Step B: Calculate the gradient value at the sampling point. For explicit analytical models, the gradient value is obtained directly by taking the partial derivative; for implicit structural models, the gradient value is obtained by the central difference method or the forward difference method. Step C: Use a single reliability analysis method to obtain the verification points on different failure boundaries; Step D: Calculate the probability of primary failure under different failure modes based on the verification points on different failure boundaries.

4. The structural reliability calculation method based on relevant failure modes according to claim 1, characterized in that... Step 3, which involves establishing a hyperplane through each verification point, determining m initial points, and obtaining m verification points, specifically includes the following steps: Step A: Establish a first-order linear hyperplane through each verification point, and set the initial value of m to 2; Step B: Using each verification point as the center, select an appropriate step size as the radius to establish the m-th hypersphere corresponding to that verification point; Step C: Determine the intersection point between the first-order linear hyperplane coordinates of each verification point and the m-th hypersphere; this is the m-th initial point. Step D: Continue using the reliability analysis method to obtain the m verification points corresponding to the m initial points.

5. The structural reliability calculation method for related failure modes according to claim 1, characterized in that... Step four, which involves establishing a fitting function based on the verification points on different failure boundaries and their respective m-fold verification points, and calculating the reliability of the relevant failure mode structure, specifically includes the following steps: Step A: Select an appropriate fitting function to fit the verification points on different failure boundaries and their respective m-fold verification points; Step B: Replace the limit state equations for different failure modes of the nonlinear structure with these fitted functions; Step C: Using numerical integration, calculate the m-fold failure probability of the structure with correlated failure modes using the probability addition or multiplication formula. .

6. The structural reliability calculation method based on relevant failure modes according to claim 1, characterized in that... The reliability of the output structure described in step six is ​​verified using the Monte Carlo method, which specifically includes the following steps: Step A: Output the m-fold failure probability of the relevant failure mode structure. ; Step B: Calculate the failure probability of the nonlinear correlated failure mode structure using the Monte Carlo stochastic simulation method; Step C: Compare and verify the calculation results.

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