CN107122521B - A kind of two-dimensional random load acts on the calculation method of lower fatigue life - Google Patents

Abstract

The invention discloses a calculation method of fatigue life under the action of two-dimensional random loads. The implementation steps include: collecting load spectrum data through a test on a target material, and obtaining the respective distributions of the amplitude value and the mean value of the load spectrum data through statistical analysis. Characteristics and probability density function; calculate the probability density function of the equivalent load of the load spectrum data; use the Miner's rule and the three-parameter empirical formula under the random load to obtain the cumulative fatigue damage calculation model under the two-dimensional random load; according to the cumulative Fatigue damage calculation model Reverse calculation model of fatigue life when cumulative fatigue damage is equal to 1; according to the fatigue life calculation model, the fatigue life of the target material under two-dimensional random load is obtained. The invention can quickly and more accurately predict the fatigue life in the early stage of part design, provide early reference for the durability design of the part, reduce the failure risk of the part in the development process, and shorten the development cycle of the part.

Description

A Calculation Method of Fatigue Life Under Two-Dimensional Random Load

technical field

The invention relates to load spectrum analysis technology, in particular to a calculation method of fatigue life under the action of two-dimensional random load (considering that both the load amplitude and the load mean are random variables).

Background technique

Load spectrum analysis is a key content in the process of automobile life prediction and fatigue durability design. In the study of fatigue durability, there are generally two methods for applying loads, one is to apply cyclic loads, and the other is to apply random loads. Due to the consideration of the statistical characteristics of the load, the random load is more in line with the actual use conditions of the car than the cyclic load. Random loads generally obey a certain continuous probability distribution, such as normal distribution, exponential distribution, lognormal distribution, extreme value distribution, and three-parameter Weibull distribution. Fatigue analysis of components under random loads should comprehensively use probability statistics theory and mechanical analysis methods to solve problems in fatigue analysis and design. When the car is working under random load, the mean value and amplitude of the load change randomly, and the mean value and amplitude should be regarded as binary random variables in the load spectrum. In most cases, the load magnitude X follows a Weibull distribution and the mean Y follows a normal distribution. Since the change of load amplitude and load average will also have a great impact on the fatigue life of the part, in order to make the research results more in line with the actual working state of the part, so as to predict the fatigue life of the part more accurately, the load amplitude based It is very necessary to conduct an in-depth study on the fatigue life under two-dimensional random loads where the mean and mean are random variables. The calculation of fatigue life under two-dimensional random load has become a key technical problem to be solved urgently.

Contents of the invention

The technical problem to be solved by the present invention: Aiming at the above-mentioned problems of the prior art, a method is provided to solve the calculation problem of the fatigue life under the respective statistical characteristics of the load amplitude and the load mean value at the same time. , Predict the fatigue life more accurately, provide early reference for the durability design of parts, reduce the failure risk of parts during the development process, and shorten the development cycle of parts. The calculation method of fatigue life under two-dimensional random load .

In order to solve the problems of the technologies described above, the technical solution adopted in the present invention is:

A calculation method of fatigue life under the action of two-dimensional random load, the implementation steps include:

1) collecting the load spectrum data by carrying out tests on the target material, and obtaining the respective distribution characteristics and probability density functions of the amplitude and the mean value of the load spectrum data through statistical analysis;

2) calculate the probability density function f(S _eq ) of the equivalent load S _eq of the load spectrum data;

3) Using Miner's law under random load shown in formula (1) and the three-parameter empirical formula between fatigue life and load shown in formula (2) in the entire medium and long life interval, it is obtained that under two-dimensional random load The cumulative fatigue damage calculation model;

N _f (SS ₀ ) ^β = α (2)

In formulas (1) and (2), D represents the cumulative fatigue damage of the target material, N represents the total number of cyclic loads on the target material, f(S) represents the probability density function of random loads, and N _f represents the target material under load Fatigue life under the action of S, S represents the load on the target material, S ₀ represents a constant load coefficient, and α and β are constant coefficients;

4) Reversely calculate the fatigue life calculation model when the cumulative fatigue damage is equal to 1 according to the cumulative fatigue damage calculation model;

5) Obtain the fatigue life of the target material under the action of two-dimensional random load according to the fatigue life calculation model.

Preferably, the detailed steps of step 2) include:

2.1) Obtain the probability density function f _Y (y) of Y=S _a according to the probability density function of the load amplitude S _a in the expression of the equivalent load _Seq shown in formula (3) in the Goodman formula, and according to the load mean value S _m Calculate the probability density function f _X (x) of the quotient X of the difference between the tensile strength σ _b of the material and the load mean value S _m divided by the tensile strength σ _b ;

In formula (3), σ _b represents the tensile strength of the target material, S _a represents the load amplitude, and S _m represents the average value of the load;

2.2) obtain the probability density function f(S _eq ) of equivalent load S _eq according to formula (4);

In formula (4), f _Y/X (z) represents the probability density function of the equivalent load _Seq , f _Y (zx) represents the probability density function of Y, the variable Z is equal to Y divided by X, and Y represents the load amplitude of the target material The value S _a , X represents the quotient of the difference between the tensile strength σ _b of the target material and the load mean value S _m divided by the tensile strength σ _b .

Preferably, step 2.2) is solved to obtain the probability density function f(S _eq ) of the equivalent load S _eq as shown in formula (5);

In formula (5), f(S _eq ) represents the probability density function of the equivalent load S _eq , the load of the target material is that the load amplitude S _a and the load mean S _m both conform to the normal distribution, and the load amplitude S _a and the load Two-dimensional random loads with mean S _m independent of each other, where the load amplitude S _a obeys N(μ _a ,σ _a ² ), the load mean S _m obeys N(μ _m ,σ _m ² ), and μ _a represents the load amplitude S The mean value of _a , σ _a ² represents the variance of the load amplitude S _a , μ _m represents the mean value of the load mean value S _m , σ _m ² represents the variance of the load mean value S _m , a, b, c, σ ₁ , σ ₂ , μ ₁ and μ ₂ are intermediate parameters, the value of the intermediate parameter μ ₂ is the same as the mean value μ _a of the load amplitude S _a , the value of σ ₂ ² is the same as the variance σ _a ² of the load amplitude S _a , the intermediate parameter μ The value of ₁ is μ ₁ =-μ _m /σ _b +1, σ _b represents the tensile strength of the target material, the variable Z is equal to Y divided by X, Y represents the load amplitude S _a of the target material, and X represents the target material The quotient of the difference between the tensile strength σ _b and the load mean value S _m divided by the tensile strength σ _b , q is an integral variable.

Preferably, the cumulative fatigue damage calculation model under the action of two-dimensional random load in step 3) is shown in formula (6);

In formula (6), D represents the cumulative fatigue damage of the target material, N represents the total number of cyclic loads on the target material, f(S _eq ) represents the probability density function of the equivalent load _Seq , S ₀ represents the constant load factor, α and β are constant coefficients.

Preferably, the fatigue life calculation model in step 4) is shown in formula (7);

In formula (7), N _f represents fatigue life, α and β are constant coefficients, f(S _eq ) represents the probability density function of equivalent load _Seq , and S ₀ represents a constant load coefficient.

Preferably, the detailed steps of step 5) include: integrating the fatigue life calculation model through the Gauss-Legendre quadrature formula, so as to calculate the fatigue life of the target material under the action of two-dimensional random load.

Preferably, the detailed steps of step 5) include: calculating the equivalent load S _D corresponding to the fatigue life according to formula (8), and obtaining the corresponding fatigue life according to the equivalent load S _D as the target material under the two-dimensional random load fatigue life;

In formula (8), S _D represents the equivalent load corresponding to the fatigue life, S ₀ represents the constant load factor, N _f represents the fatigue life, and α and β are constant coefficients.

The calculation method of the fatigue life under the two-dimensional random load of the present invention has the following advantages: the calculation method of the present invention has fully considered the respective statistical characteristics of the load amplitude and the load mean value, so that the loading of the load is more in line with the actual working state of the parts, Thus, the fatigue life can be calculated more accurately. Using this method can quickly and more accurately predict the fatigue life in the early stage of part design, provide early reference for the durability design of parts, reduce the failure risk of parts during the development process, and shorten the development cycle of parts.

Description of drawings

Fig. 1 is a schematic flow chart of the basic method of the first embodiment of the present invention.

FIG. 2 is a comparison diagram of the probability density function curves under different distributions of the parameter μ in Embodiment ₁ of the present invention.

FIG. 3 is a comparison diagram of probability density function curves under different distributions of the parameter σ1 in Embodiment ₁ of the present invention.

FIG. 4 is a comparison diagram of probability density function curves under different distributions of parameter μ ₂ in Embodiment 1 of the present invention.

5 is a comparison diagram of probability density function curves under different distributions of the parameter _σ2 in Embodiment 1 of the present invention.

Detailed ways

Embodiment one:

As shown in Figure 1, the implementation steps of the calculation method of fatigue life under the action of two-dimensional random load in this embodiment include:

1) collecting the load spectrum data by carrying out tests on the target material, and obtaining the respective distribution characteristics and probability density functions of the amplitude and the mean value of the load spectrum data through statistical analysis;

2) calculate the probability density function f(S _eq ) of the equivalent load S _eq of the load spectrum data;

3) Using Miner's law under random load shown in formula (1) and the three-parameter empirical formula between fatigue life and load shown in formula (2) in the entire medium and long life interval, it is obtained that under two-dimensional random load The cumulative fatigue damage calculation model;

N _f (SS ₀ ) ^β = α (2)

In formulas (1) and (2), D represents the cumulative fatigue damage of the target material, N represents the total number of cyclic loads on the target material, f(S) represents the probability density function of random loads, and N _f represents the target material under load Fatigue life under the action of S, S represents the load on the target material, S ₀ represents a constant load coefficient, and α and β are constant coefficients;

4) Reversely calculate the fatigue life calculation model when the cumulative fatigue damage is equal to 1 according to the cumulative fatigue damage calculation model;

5) Obtain the fatigue life of the target material under the action of two-dimensional random load according to the fatigue life calculation model.

In the present embodiment, the detailed steps of step 2) include:

2.1) According to the probability density function of the load amplitude S _a in the expression of the equivalent load _Seq shown in formula (3) in Goodman's formula, find out the probability density function f _Y (y) of Y=S _a , and according to the load mean value S The probability density function of _m is used to obtain the probability density function f _X (x) of the difference between the tensile strength σ _b of the material and the average value of load S _m divided by the quotient X of the tensile strength σ _b ;

In formula (3), σ _b represents the tensile strength of the target material, S _a represents the load amplitude, and S _m represents the average value of the load;

2.2) obtain the probability density function f(S _eq ) of equivalent load S _eq according to formula (4);

In formula (4), f _Y/X (z) represents the probability density function of the equivalent load _Seq , f _Y (zx) represents the probability density function of Y, the variable Z is equal to Y divided by X, and Y represents the load amplitude of the target material The value S _a , X represents the quotient of the difference between the tensile strength σ _b of the target material and the load mean value S _m divided by the tensile strength σ _b .

In order to calculate the fatigue life under the action of two-dimensional random loads, this embodiment provides the probability density function f(S _eq ) of the equivalent load _Seq of two-dimensional random loads in which both the load amplitude and the load mean conform to a normal distribution. In the present embodiment, step 2.2) is solved to obtain the probability density function f(S _eq ) of the equivalent load S _eq as shown in formula (5);

In formula (5), f(S _eq ) represents the probability density function of the equivalent load S _eq , the load of the target material is that the load amplitude S _a and the load mean S _m both conform to the normal distribution, and the load amplitude S _a and the load Two-dimensional random loads with mean S _m independent of each other, where the load amplitude S _a obeys N(μ _a ,σ _a ² ), the load mean S _m obeys N(μ _m ,σ _m ² ), and μ _a represents the load amplitude S The mean value of _a , σ _a ² represents the variance of the load amplitude S _a , μ _m represents the mean value of the load mean value S _m , σ _m ² represents the variance of the load mean value S _m , a, b, c, σ ₁ , σ ₂ , μ ₁ and μ ₂ are intermediate parameters, the value of the intermediate parameter μ ₂ is the same as the mean value μ _a of the load amplitude S _a , the value of σ ₂ ² is the same as the variance σ _a ² of the load amplitude S _a , the intermediate parameter μ The value of ₁ is μ ₁ =-μ _m /σ _b +1, σ _b represents the tensile strength of the target material, the variable Z is equal to Y divided by X, Y represents the load amplitude S _a of the target material, and X represents the target material The quotient of the difference between the tensile strength σ _b and the load mean value S _m divided by the tensile strength σ _b , q is an integral variable.

In this embodiment, the cumulative fatigue damage calculation model under the action of two-dimensional random load in step 3) is shown in formula (6);

In formula (6), D represents the cumulative fatigue damage of the target material, N represents the total number of cyclic loads on the target material, f(S _eq ) represents the probability density function of the equivalent load _Seq , S ₀ represents the constant load factor, α and β are constant coefficients.

In the present embodiment, the fatigue life calculation model in step 4) is as shown in formula (7);

In formula (7), N _f represents fatigue life, α and β are constant coefficients, f(S _eq ) represents the probability density function of equivalent load _Seq , and S ₀ represents a constant load coefficient.

In this embodiment, the detailed steps of step 5) include: integrating the fatigue life calculation model through the Gauss-Legendre quadrature formula, so as to calculate the fatigue life of the target material under the action of two-dimensional random load.

In this embodiment, different values are selected for the intermediate parameters σ ₁ , σ ₂ , μ ₁ , and μ ₂ respectively, and the probability density function f( _{S eq )} ( In the figure, it is expressed as f _Z (Z)) to generate curves, and Figure 2, Figure 3, Figure 4, and Figure 5 are obtained respectively. Figure ₂ records the values of the probability density function _f (S _eq ) when σ1 is 0.08, _σ2 is 30, and μ2 is 190, and μ1 is 0.5, 0.75, ₁ , 1.25, 1.5 respectively Contrast curves. Figure ₃ records the probability _density function _f ( _{S eq} ) comparison curve. Fig. 4 has recorded that σ ₁ takes a value of 0.08, σ ₂ takes a value of 30, and μ ₁ takes a value of 0.75, and the probability density function _f (S _eq ) comparison curve. Figure 5 records that σ1 takes _a value of 0.08, μ ₁ takes a value of 0.75, and μ ₂ takes a value of 190. When σ2 takes a value of 5, 10, ₂₀ , 30, 40, and 50, respectively, the probability density function f(S _eq ) comparison curve. From the analysis of Figures 2 to 5, it can be concluded that: 1) The shape of the f _Z (z) function is similar to that of the normal distribution function, mainly due to the following three points: a) The function value has a peak point, and z is far from the peak point The farther away, the smaller the function value; b) the function has inflection points on both sides of the peak point; c) the curve takes the horizontal axis as an asymptote. 2) When the other three parameters remain unchanged, the decrease of μ ₁ and the increase of σ ₁ , μ ₂ and σ ₂ make the peak value of the function decrease, and the shape of the function curve becomes smoother. 3) The increase of μ ₂ makes the peak point move to the right, the increase of μ ₁ and σ ₁ makes the peak point move to the left, and the change of σ ₂ does not change the position of the peak point.

Embodiment two:

This embodiment is basically the same as Embodiment 1, and the main differences are: Step 5) The method of obtaining the fatigue life of the target material under the action of two-dimensional random loads according to the fatigue life calculation model is different.

In this embodiment, the research on fatigue life is transformed into the research on equivalent load. The equivalent load is defined as follows: From the perspective of the overall effect, the final result under the action of both random load and cyclic load will cause the damage state of the component to reach a critical value and fail, which can be considered to correspond to a random load process , there must be an equivalent constant-amplitude load, so that the component will be damaged at the same time under the same initial state and after the same action time. Define the constant amplitude load as the equivalent load of the random load, denoted by _SD . The corresponding relationship between equivalent load and fatigue life is shown in the three-parameter empirical formula.

In this embodiment, the detailed steps of step 5) include: calculating the equivalent load S _D corresponding to the fatigue life according to formula (8), and obtaining the corresponding fatigue life according to the equivalent load S _D as the target material under the action of two-dimensional random load under the fatigue life;

In formula (8), S _D represents the equivalent load corresponding to the fatigue life, S ₀ represents the theoretical fatigue limit expressed by the equivalent stress amplitude, N _f represents the fatigue life, and α and β are constant coefficients. Taking 16Mn material as an example, according to existing research, α=3.95×10 ⁸ , S ₀ =261 MPa, and β=2. According to formula (7) and formula (8), the equivalent load of 16Mn material under two-dimensional random load is:

In formula (10), SD represents the equivalent load corresponding to the fatigue life, and _{S eq} represents the equivalent load.

Under the statistical distribution parameters of different load amplitudes and load averages, the Gauss-Legendre quadrature formula is still used to integrate the function, and some equivalent load data are calculated in Tables 1 to 5. The equivalent loads of dimensional random loads are shown in Table 6. After obtaining the equivalent load, the fatigue life can be obtained by inverting according to formula (8).

Table 1: Equivalent load (MPa) of different μ ₁ and σ ₁ (μ ₂ =210, σ ₂ =40).

μ₁\σ₁ 0.05 0.06 0.07 0.08 0.09 0.1 0.5 >600 >600 >600 >600 >600 >600 0.75 314 316 344 417 526 >600 1 270 271 272 273 274 275 1.25 262 262 262 262 263 263 1.5 261 261 261 261 261 261

Table 2: Equivalent load (MPa) of different μ ₂ and σ ₂ (μ ₁ =0.75, σ ₁ =0.05).

σ₂\μ₂ 150 170 190 210 230 250 10 262 265 274 291 314 340 20 264 270 281 298 319 343 30 269 277 289 306 325 348 40 276 285 298 314 333 354 50 283 294 307 323 341 362

Table 3: Equivalent load (MPa) of different μ ₂ and σ ₂ (μ ₁ =1, σ ₁ =0.05).

σ₂\μ₂ 150 170 190 210 230 250 10 261 261 261 261 263 268 20 261 261 261 263 266 273 30 261 262 263 266 271 279 40 262 264 266 270 277 286 50 264 267 271 276 283 292

Table 4: Equivalent load (MPa) of different μ ₂ and σ ₂ (μ ₁ =1, σ ₁ =0.1).

Table 5: Equivalent load (MPa) of different μ ₂ and σ ₂ (μ ₁ =1.25, σ ₁ =0.1).

σ₂\μ₂ 150 170 190 210 230 250 10 261 261 261 261 261 262 20 261 261 261 261 262 263 30 261 261 261 262 263 264 40 261 261 262 263 264 267 50 262 262 263 265 267 270

Table 6: Equivalent loads (MPa) for one-dimensional random loads.

σ\μ 150 170 190 210 230 250 10 261 261 261 261 261 264 20 261 261 261 262 264 270 30 261 261 262 265 269 277 40 262 263 265 269 275 284 50 264 266 269 274 281 291

According to the analysis results, the average _load has a _great influence on the equivalent load of the two-dimensional normal random load. Equivalent load; when μ _m is equal to 0, the equivalent load considering the load mean value is slightly higher than the equivalent load without considering the load mean value; when μ _m is greater than 0, with the increase of μ ₂ and σ ₂ , the two-dimensional random The equivalent load of the load increases rapidly until it is much larger than that of the one-dimensional random load.

The above descriptions are only preferred implementations of the present invention, and the scope of protection of the present invention is not limited to the above-mentioned embodiments, and all technical solutions under the idea of the present invention belong to the scope of protection of the present invention. It should be pointed out that for those skilled in the art, some improvements and modifications without departing from the principle of the present invention should also be regarded as the protection scope of the present invention.

Claims (6)

1. A calculation method of fatigue life under the action of a two-dimensional random load, characterized in that the steps of implementation include: 1) collecting the load spectrum data by carrying out tests on the target material, and obtaining the respective distribution characteristics and probability density functions of the amplitude and the mean value of the load spectrum data through statistical analysis; 2) calculate the probability density function f(S _eq ) of the equivalent load S _eq of the load spectrum data; 3) Using Miner's law under random load shown in formula (1) and the three-parameter empirical formula between fatigue life and load shown in formula (2) in the entire medium and long life interval, it is obtained that under two-dimensional random load The cumulative fatigue damage calculation model of , and the cumulative fatigue damage calculation model under the action of two-dimensional random load is shown in formula (6); N _f (SS ₀ ) ^β = α (2) In formulas (1) and (2), D represents the cumulative fatigue damage of the target material, N represents the total number of cyclic loads on the target material, f(S) represents the probability density function of random loads, and N _f represents the target material under load Fatigue life under the action of S, S represents the load on the target material, S ₀ represents a constant load coefficient, and α and β are constant coefficients; In formula (6), D represents the cumulative fatigue damage of the target material, N represents the total number of cyclic loads on the target material, f(S _eq ) represents the probability density function of the equivalent load _Seq , S ₀ represents the constant load factor, α and β are constant coefficients in formula (2); 4) Reversely calculate the fatigue life calculation model when the cumulative fatigue damage is equal to 1 according to the cumulative fatigue damage calculation model; 5) Obtain the fatigue life of the target material under the action of two-dimensional random load according to the fatigue life calculation model. 2. the calculation method of fatigue life under the action of two-dimensional random load according to claim 1, is characterized in that, the detailed steps of step 2) comprise: 2.1) Obtain the probability density function f _Y (y) of Y=S _a according to the probability density function of the load amplitude S _a in the expression of the equivalent load _Seq shown in formula (3) in the Goodman formula, and according to the load mean value S _m Calculate the probability density function f _X (x) of the quotient X of the difference between the tensile strength σ _b of the material and the load mean value S _m divided by the tensile strength σ _b ; In formula (3), σ _b represents the tensile strength of the target material, S _a represents the load amplitude, and S _m represents the average value of the load; 2.2) obtain the probability density function f(S _eq ) of equivalent load S _eq according to formula (4); In formula (4), f _Y/X (z) represents the probability density function of the equivalent load _Seq , f _Y (zx) represents the probability density function of Y, the variable Z is equal to Y divided by X, and Y represents the load amplitude of the target material The value S _a , X represents the quotient of the difference between the tensile strength σ _b of the target material and the load mean value S _m divided by the tensile strength σ _b . 3. the calculating method of fatigue life under the action of two-dimensional random load according to claim 2 is characterized in that, step 2.2) obtains the probability density function f (S _eq ) of equivalent load S _eq as shown in formula (5) ; In formula (5), f(S _eq ) represents the probability density function of the equivalent load S _eq , the load of the target material is that the load amplitude S _a and the load mean S _m both conform to the normal distribution, and the load amplitude S _a and the load Two-dimensional random loads with mean S _m independent of each other, where the load amplitude S _a obeys N(μ _a ,σ _a ² ), the load mean S _m obeys N(μ _m ,σ _m ² ), and μ _a represents the load amplitude S The mean value of _a , σ _a ² represents the variance of the load amplitude S _a , μ _m represents the mean value of the load mean value S _m , σ _m ² represents the variance of the load mean value S _m , a, b, c, σ ₁ , σ ₂ , μ ₁ and μ ₂ are intermediate parameters, the value of the intermediate parameter μ ₂ is the same as the mean value μ _a of the load amplitude S _a , the value of σ ₂ ² is the same as the variance σ _a ² of the load amplitude S _a , the intermediate parameter μ The value of ₁ is μ ₁ =-μ _m /σ _b +1, σ _b represents the tensile strength of the target material, the variable Z is equal to Y divided by X, Y represents the load amplitude S _a of the target material, and X represents the target material The quotient of the difference between the tensile strength σ _b and the load mean value S _m divided by the tensile strength σ _b , q is an integral variable. 4. the calculation method of fatigue life under the action of two-dimensional random load according to claim 3 is characterized in that, the fatigue life calculation model in step 4) is as shown in formula (7); In formula (7), N _f represents fatigue life, α and β are constant coefficients, f(S _eq ) represents the probability density function of equivalent load _Seq , and S ₀ represents a constant load coefficient. 5. according to the calculation method of fatigue life under the action of any one of claim 1～4, it is characterized in that, the detailed steps of step 5) comprise: through Gauss-Legendre described fatigue life calculation model The integral operation is performed by the quadrature formula, so as to calculate the fatigue life of the target material under the action of two-dimensional random load. 6. The method for calculating fatigue life under any one of claims 1 to 4, wherein the detailed steps of step 5) include: calculating the corresponding equation of fatigue life according to formula (8) Effective load S _D , according to the equivalent load S _D , the corresponding fatigue life is obtained as the fatigue life of the target material under the action of two-dimensional random load; In formula (8), S _D represents the equivalent load corresponding to the fatigue life, S ₀ represents the constant load factor, N _f represents the fatigue life, and α and β are constant coefficients.