1. Introduction
Mechanical components and structures subjected to vibration are affected by dynamic loads, which induce fatigue damage due to the cycling loading application [1,2]. The generated fatigue damage is primarily related to vibration history loading, geometry, and material properties [3]. The generated fatigue damage directly determines the component’s reliability, replacement policy, and warranty costs. Thus, the reliability index characteristic has an important role that is determined mathematically and is used to describe the fatigue damage behavior of mechanical components [4]. That description can be performed by using a probabilistic approach [5]. Then, for mechanical components or systems, according to their application and data complexity, an accurate method must be selected to determine an effective level of service life reliability [6]. Now, since data from fatigue damage are affected by significant scatter, and damage is provoked as a response to random forces, gathering fatigue damage data is generally a difficult activity [7]. One of the more commonly applied methods to work with fatigue damage is the Miner’s rule [8,9]. To consider the random nature of the generated damage, here, we use a probabilistic time-dependent approach [7]. In the fatigue damage accumulation models, the principal factors involved are load sequence, type of load, overloads, plasticization, and type of material. Consequently, for the damage accumulation analysis, we require a probabilistic concept, or a physical quantity related to the probability of occurrence [10]. Thus, the measurement of damage helps us to calculate the probabilities of failure. Since the accumulation of random vibration fatigue damage entails increasing deterioration, an increasing hazard function is required, and the most recommendable cumulative distribution function (cdf) used to estimate the fatigue damage is the Weibull cdf [11,12,13]. In a vibration profile, each row has its own stationary frequency and amplitude which allow for the performance of a vibration analysis with a statistical approach. Since it is possible to estimate the stress from the acceleration responses, here, the Weibull distribution is used. Thus, based on the increasing behavior of the cumulated damage, in this paper, the increasing random damage behavior is used in the Weibull cdf to determine the failure percentile that the observed cumulated damage represents in the used vibration profile. Then, once the damage percentiles are determined based on their corresponding cycles of the S–N curve, the Weibull scale parameter is determined. Similarly, the Weibull shape parameter is determined directly from the cumulated response stress of the used profile (see Equations (9) and (10) and Section 3.2). Thus, in Section 2.3, a probabilistic methodology to characterize the fatigue damage induced by random vibration is developed by using the Weibull distribution, which uses a relation of the scale and shape parameters with the mechanical vibration fatigue damage. The methodology includes probabilistic estimations based on the Weibull scale and shape parameters that are governed by a new way of analyzing the fatigue damage accumulation presented in Section 2.1. Then, the vibration analysis is based on the fatigue damage, which is determined directly from the principal stresses and ; therefore, the methodology is efficient because it uses the damage as the platform to project and represent its random behavior and consequently the component’s fatigue life. With the purpose to assess it, the methodology is applied in Section 3 to a probabilistic failure analysis of a panel support made of cold drawn steel AISI 1025. The mechanical component is submitted to a random vibration loading profile of 10–55 Hertz with an amplitude of 1.5 mm peak to peak during a 2 h period per block. The testing was performed by using an electrodynamic vibration system in which the vibration profile loading was applied 29 times. The component’s physical damage results are shown in Section 3.1. The purpose of the method proposed lies in the use of the accumulated fatigue damage Di instead of the median rank operation. This is illustrated by Equations (12) and (13) which allow the use of the resultant vector Y in the estimation of Weibull parameters that completely reproduces the principal vibration stress values.
The paper is organized as follows. Section 2 includes the generalities of the vibration fatigue damage accumulation and the proposed Weibull fatigue damage analysis method. In Section 3, a numerical application is presented. Section 4 is related to the median rank method comparison. Finally, in Section 5, the conclusions are given.
2. Fatigue Damage
2.1. Fatigue Damage Accumulation
Fatigue damage can be described as a failure mechanism that is manifested when a material tends to fail or break under repeated deflections [14,15,16]. Thus, a nonlinear model to accumulate the random vibration fatigue damage has been proposed [17] with the purpose of evaluating the fatigue damage of different dynamic loads in mechanical components and structural elements. The acceleration response of the analyzed vibration system is determined by stress as in Equation (1). The applied vibration cycles are determined by the rainflow method [18]. From the S–N material’s curve, the corresponding life cycles are determined by using the Basquin Equation [19] as is in Equation (4).
(1)
(2)
(3)
(4)
In Equation (1), and are the dynamic load factor and the acceleration response, respectively. In Equation (2) [20], K is the stress concentration factor in the mechanical component, is the effective mass, C is the distance to the neutral axis, is the distance from the fixed point of the component to the point of application load, A is the constant of gravity, and I is the moment of inertia. In Equation (3), is the frequency applied by the vibration power spectral density (PSD) and is the gravity constant. In Equation (4), is the maximum number of cycles that the material’s component can sustain at a vibration load with stress amplitude and the parameters a and b are constant variables that represent the intercept and the slope of the S–N curve, respectively.
As shown by Equations (1)–(4), because a vibration has a nonlinear behavior [21], the generated fatigue damage also presents a nonlinear behavior. Consequently, the fatigue damage is determined by using Equation (5), where the damage is described by a curve that represents the effect under two-level loading conditions [22], where represents the applied vibration cycles at the stress level .
(5)
Now that the random fatigue damage generated by the vibration environment is determined, a Weibull formulation is presented that let us use the generated damage in the Weibull Y vector (see Equation (13)) to determine the reliability of the analyzed element.
2.2. Weibull Analysis
A two-parameter Weibull distribution is used to statistically analyze fatigue behaviors [23,24,25]. It allowed us to perform accurate fatigue failure analysis [26,27]. The probability density function f(t) and cumulative distribution function F(t) are described by Equations (6) and (7), respectively.
(6)
(7)
where, is the shape parameter, is the scale parameter, and t is the selected random variable (damage or fatigue life). The corresponding reliability function R(t) is given as(8)
From [28], the Weibull fatigue damage and parameters are determined as
(9)
(10)
where represents the mean of the Y vector (see Equation (13)) determined by using Equation (5). represents the log-mean of the failure-time data, which is determined here directly from the addressed maximum and minimum stress values of Section 2.1. Thus, is determined as(11)
Here, notice that the efficiency of the Weibull parameters and only depends on the accuracy with which the and values are determined by Equation (1). In this paper, they were determined from an electrodynamic shaker acceleration response and by the method performed in [17], Section 3.1.
2.3. Weibull Fatigue Damage Analysis
The expected behavior of the and parameters is determined by the following steps:
Step 1. By using the fatigue damage accumulation results from the component’s operation, the times that the vibration profile (PSD) loading were applied to the mechanical component are taken as n.
Step 2. The accumulation fatigue damage determined by Equation (5) in the end of each one of the n vibration loadings is used by Equation (12) as the corresponding cumulated failure percentile as
(12)
Step 3. By using the elements in the linearized form of the reliability function given in Equation (8), the corresponding elements are determined as in Equation (13). Then, its corresponding arithmetic mean value is computed as in Equation (14).
(13)
(14)
Step 4. By plugging the value and the and values into Equation (9), the corresponding Weibull shape parameter is determined. Similarly, by plugging the and values into Equation (11), the corresponding value is determined. Then, by using this corresponding value in Equation (10), the corresponding Weibull scale parameter is determined. These and parameters represent the Weibull fatigue damage family that is used to model the random behavior of the estimated and principal stress values.
Note 1. Here, notice the random behavior of the and values. In the proposed Weibull analysis, let us use the values as the minimum required strength that the component’s material must present in order to ensure that the reliability of the component will meet at least (as a minimum) the desired R(t) index.
From the Weibull analysis, by using the and parameters, the minimum strength values are determined by using the value that correspond to each element as
(15)
Thus, the value is determined as
(16)
and the value is determined as(17)
Additionally, from Equation (18), by using the known value, the element that belongs to the and values determined in Section 2.1 is determined as
(18)
Now, the and the values are used to determine the corresponding value as follows,
(19)
Finally, the reliability index that corresponds to the value is determined as
(20)
Note 2. Here, observe that the R(t) index determined in Equation (20) by using the value according to our proposed method corresponds to a component with strength equivalent to the value. Thus, if we define the material parameter as the actual strength of the component, then by using this Sy value in Equation (18) and the corresponding value of Equation (19) in Equation (20), the minimum expected reliability of a component that presents a strength of is determined. Please also notice from the proposed Weibull analysis that any desired strength value can be used to determine its corresponding reliability. The diagram with the steps required to determine the probability of failure and the reliability based on the vibration fatigue damage D is shown in Figure 1.
Next, a numerical application is presented where the proposed method is applied to a mechanical component that is submitted to random vibration stress due to its field application.
3. Numerical Application
3.1. Fatigue Damage Accumulation
The accumulation fatigue damage study case is performed by analyzing a mechanical support component that is used to install a fiber optic panel into a frame. It is shown in Figure 2.
The support is made of cold drawn steel AISI 1025, with a modulus of elasticity E = 200 GPa, Poisson’s ratio ɤ = 0.29, yield strength = 430 MPa, ultimate tensile strength Sut = 510 MPa, endurance limit Se = 255 MPa, density = 7.9 g/cm3, length L = 51 mm, width W = 200 mm, and a wall thickness t = 3 mm. During its function, the component supports a static load of 80 N. It is submitted to an operating random vibration with an input PSD consisting of frequencies ranging from 10 to 55 Hz at an amplitude of 1.5 mm peak to peak for a period of 2 h. The testing is carried out physically by using a vibration system, and the results are as follows. By using Equations (1) and (2), the dynamic factor and the vibration stresses , are calculated, respectively. The acceleration responses are obtained from the vibration system but can also be determined by using Equation (3). Then, the vibration cycles applied are determined by the rainflow method and the total cycles are determined by using Equation (4). Finally, the fatigue damage accumulation is obtained by Equation (5). Table 1 shows the vibration stress and cycle results and Table 2 shows the vibration fatigue damage accumulation results, where the failure is presented when D = 1 [29].
Figure 3 shows the mechanical support areas where the fatigue damage accumulated was presented. The analysis and estimation of the damage were determined from the acceleration responses that were the base to calculate the principal vibration stresses, and , that are employed in the Weibull probabilistic analysis.
Now, in the following section, the probabilistic approach is applied to the vibration fatigue damage results obtained in Table 2.
3.2. Weibull Fatigue Damage Analysis
Here, it is remarked that if a different sample with the same features was submitted to the same vibration damage accumulation experiment, due to random behavior, the results would be different. For the damage data given in Table 2, where 29 blocks were tested, the probabilistic analysis is as follows. For this purpose, the vector that includes the fatigue damage as described by Equation (13) is used, instead of using the median rank approach, to determine the Weibull parameters as follows [30,31].
By selecting the fatigue damage accumulated Di value of each individual block of Table 2, and using it in Equation (13), the n = 29 elements are determined. Thus, from Equation (14), the mean is, . Next, from the results of Table 1, the principal vibration stress values = 304.76 MPa and = 15.99 MPa are obtained. With these data, we can proceed to use the Weibull distribution in order to obtain its failure probability, reliability, and random behavior. From Equation (9), the Weibull shape parameter value is
and from Equation (11), the logarithm average value is . Thus, from Equation (10), the Weibull parameter is . Consequently, the Weibull damage family is W(0.9102, 69.8077 MPa). This Weibull family completely represents the observed principal vibration stresses and values.Now, by using the Weibull family results, the random behavior strength can be determined. Since the Weibull parameters only depend on the principal stress values caused by the random vibration and values, the random behavior can be obtained by performing the Weibull analysis using the following steps.
Since the determination of the fatigue damage is based on the random behavior of the σ2 stress value, here, the random behavior of and is determined by Equations (16) and (17). Then, by using the and the values in Equation (15), the basic Weibull elements [28] for each are obtained. Whereas by using the and values in Equation (18), the Weibull value from the and stress values are reproduced. This value is calculated as
and, by using the value in Equation (19), the value that belongs to the value is determined asNext, by substituting the value in Equation (20), the reliability that belongs to the element is
The previous results shown in this section are included in Table 3.
Here, is important to mention that the reliability obtained corresponds to a designed component or structure with minimal strength of = = 304.76 MPa. In relation to the mechanical support, it has a = 430 MPa, then, its reliability is , which is determined as follows. The minimal reliability of the component is obtained when the value that belongs to the value is used in Equation (20). The steps to determine the reliability of the design component when the value is used as in Equation (18) are, the element that belongs to the value is . From Equation (19), the corresponding value is . The reliability index for the value is calculated by using Equation (20), . Thus, we conclude that the reliability of the design component is . Additionally, regarding the material’s component or structure, it is noticed about that the higher the strength Sy value, the higher the reliability will be [28].
Now, as a comparison for the probabilistic cumulative density, the Weibull distribution is used, but in this case, by using the median rank method to estimate the vector , which is the base to determine the reliability and the probability of failure.
4. Median Rank Approach
In this section, the median rank method [30] is applied to the data given in Section 3. By using this method, the corresponding cumulated failure percentile that previously was determined by Equation (12) is now determined as
(21)
Then, by using the elements in the linearized form of the reliability function given in Equation (8), the corresponding elements are determined as in Equation (22),
(22)
By selecting the fatigue damage accumulated Di value of each one block of Table 2 and using it in Equation (22), the n = 29 elements are determined. Thus, from Equation (14), the mean is, . Next, from the results of Table 1, the principal vibration stress values = 304.76 MPa and = 15.99 MPa are obtained. With these data, we can now proceed to use the Weibull distribution in order to obtain its failure probability, reliability, and random behavior. From Equation (9), the Weibull shape parameter value is
Since the principal vibration stress values σ1 = 304.76 MPa and σ2 = 15.99 MPa are maintained, the Weibull parameter value remains.
Next, the remaining steps followed in Section 3 are applied and the results are included in Table 4.
A comparison of the principal results between both the proposed method and the median rank method by following the formulation given in Section 2 is shown in Table 5.
In Table 5, notice that although we are using the same = 304.76, = 15.99 and values in both methods, the component’s reliability R(t) is different. This difference occurs because in both methods, the Weibull β parameter shape have different values. This is because the arithmetic mean of both methods is different. Thus, because in the proposed method the addressed damage completely represents the analyzed component, we conclude that the real reliability is the one given by the proposed method. Notic that because the damage is random, the proposed method is dynamic, and its efficiency depends only on the accuracy with which and are determined. Finally, observe from the last row of Table 5 that if R(t) = 0.95 is required, then from the proposed method, the maximum allowed stress is = 1824.3190 MPa. At the same time, using the median rank method, it is = 3591.0886 MPa, implying that by using the median rank approach, we can predict that the component will become overstressed, lowering its life.
5. Discussion
The paper presents an alternative to fit a probabilistic vibration fatigue analysis based on the use of the damage accumulated for two parameters provided by the Weibull distribution model. The main contribution of the proposed method is the probabilistic approach that the Weibull distribution function has in the mechanical industrial field. A considerable number of industrial standards and guidelines employ the Weibull model for their fatigue analysis; thus, reliability engineers are especially familiar with this model.
This paper presents a probabilistic aspect that involves fatigue experiments and the proper use of vibration fatigue damage during the investigation and design of mechanical components. The use of vibration fatigue damage as the cumulative failure percentile in the process allows the transfer to obtain an estimation of the reliability and a prediction of failure, in spite of the significant variability involved in the vibration fatigue.
Regarding the results obtained for the probabilistic prediction of reliability and cumulated failure, based on the principal vibration stress values = 304.76 and = 15.99, the proposed method predicts a reliability of R(t) = 0.826 and a cumulative failure of F(t) = 0.174. Here, it is important to notice that if we use the standard median rank approach, then R(t) = 0.776 and F(t) = 0.224. By their comparison, we have an R(t) variation of 5%. Moreover, it can be observed that because the used cumulated damage depends, among other factors, on the vibration load distribution applied, the material’s strength, and the geometry of the tested mechanical component, then because each analysis is different, the proposed method is completely dynamic, and can be used in any application where the accumulated damage can be measured. Here, we highlight that the only restriction to use the cumulated damage is that the failure must be defined when the damage is equal to one. Therefore, among other applications, the proposed model can be used in the automotive and telecommunications industries, where mechanical supports are used to hold electrical and electronic devices that are subjected to environmental vibration due to their field application.
6. Conclusions
-
A probabilistic alternative of the Weibull distribution to vibration fatigue analysis is developed, which allows it to define reliability and probability of failure.
-
Contrary to other models that use the median rank method as the cumulated failure percentile, the proposed methodology considers the accumulated fatigue damage for a platform in which the component’s probabilistic life, the probability of failure, and the reliability index are estimated.
-
A methodology is developed based on the model presented to permit a probabilistic approach to vibration fatigue damage accumulation which allows the probabilistic failure and reliability estimation for mechanical components and structures subjected to variable amplitude loading, specifically random vibration.
-
The model and methodology included in this work are applied to mechanical components used in the telecommunication industry to assist in real and practical structural fatigue analysis.
-
An application case is selected to illustrate the proposed Weibull model and its parameter estimation methodology based on the cumulated vibration damage included in this paper.
Conceptualization, J.M.B.-C., M.R.P.-M. and R.C.T.-V.; methodology, J.M.B.-C. and M.R.P.-M.; data analysis, J.M.B.-C. and R.C.T.-V.; writing—original draft preparation, J.M.B.-C. and M.R.P.-M.; writing—review and editing, J.M.B.-C., M.R.P.-M. and R.C.T.-V.; supervision, M.R.P.-M.; funding acquisition, J.M.B.-C. and R.C.T.-V. All authors have read and agreed to the published version of the manuscript.
Not applicable.
Not applicable.
Not applicable.
Not applicable.
The authors declare no conflict of interest.
Footnotes
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Vibration Stress and Cycles Results.
Frequency |
Accel. |
Dynamic Factor |
Vibration Stress |
Applied |
Total Cycles |
---|---|---|---|---|---|
10 | 0.72 | 22.22 | 15.99 | 70,384 | 1.36 × 1020 |
20 | 2.65 | 58.86 | 140,195 | 3.04 × 1013 | |
30 | 5.62 | 124.84 | 92,619 | 4.42 × 109 | |
40 | 9.17 | 203.69 | 10,807 | 1.40 × 107 | |
50 | 13.72 | 304.76 | 2921 | 1.23 × 105 | |
55 | 12.36 | 274.55 | 762 | 4.20 × 105 |
Vibration Fatigue Damage Accumulation.
10 Hz | 20 Hz | 30 Hz | 40 Hz | 50 Hz | 55 Hz | |
---|---|---|---|---|---|---|
Block No. | D1 | D1+2 | D1+2+3 | D1+2+3+4 | D1+2+3+4+5 | D1+2+3+4+5+6 |
1 (2 h) | 5.18 × 10−16 | 5.18 × 10−16 | 5.20 × 10−16 | 5.73 × 10−16 | 2.37 × 10−2 | 2.00 × 10−2 |
2 (2 h) | 2.00 × 10−2 | 2.42 × 10−2 | 2.42 × 10−2 | 2.42 × 10−2 | 4.84 × 10−2 | 5.00 × 10−2 |
3 (2 h) | 5.00 × 10−2 | 4.91 × 10−2 | 4.93 × 10−2 | 5.01 × 10−2 | 7.38 × 10−2 | 7.00 × 10−2 |
4 (2 h) | 7.00 × 10−2 | 7.49 × 10−2 | 7.51 × 10−2 | 7.63 × 10−2 | 1.00 × 10−1 | 1.00 × 10−1 |
5 (2 h) | 1.00 × 10−1 | 1.01 × 10−1 | 1.02 × 10−1 | 1.03 × 10−1 | 1.27 × 10−1 | 1.30 × 10−1 |
6 (2 h) | 1.30 × 10−1 | 1.29 × 10−1 | 1.29 × 10−1 | 1.31 × 10−1 | 1.55 × 10−1 | 1.60 × 10−1 |
7 (2 h) | 1.60 × 10−1 | 1.56 × 10−1 | 1.57 × 10−1 | 1.59 × 10−1 | 1.83 × 10−1 | 1.90 × 10−1 |
8 (2 h) | 1.90 × 10−1 | 1.85 × 10−1 | 1.86 × 10−1 | 1.89 × 10−1 | 2.12 × 10−1 | 2.10 × 10−1 |
9 (2 h) | 2.10 × 10−1 | 2.14 × 10−1 | 2.15 × 10−1 | 2.18 × 10−1 | 2.42 × 10−1 | 2.40 × 10−1 |
10 (2 h) | 2.40 × 10−1 | 2.45 × 10−1 | 2.45 × 10−1 | 2.49 × 10−1 | 2.73 × 10−1 | 2.80 × 10−1 |
11 (2 h) | 2.80 × 10−1 | 2.76 × 10−1 | 2.76 × 10−1 | 2.81 × 10−1 | 3.04 × 10−1 | 3.10 × 10−1 |
12 (2 h) | 3.10 × 10−1 | 3.07 × 10−1 | 3.08 × 10−1 | 3.13 × 10−1 | 3.37 × 10−1 | 3.40 × 10−1 |
13 (2 h) | 3.40 × 10−1 | 3.40 × 10−1 | 3.41 × 10−1 | 3.46 × 10−1 | 3.70 × 10−1 | 3.70 × 10−1 |
14 (2 h) | 3.70 × 10−1 | 3.73 × 10−1 | 3.74 × 10−1 | 3.80 × 10−1 | 4.03 × 10−1 | 4.10 × 10−1 |
15 (2 h) | 4.10 × 10−1 | 4.07 × 10−1 | 4.08 × 10−1 | 4.14 × 10−1 | 4.38 × 10−1 | 4.40 × 10−1 |
16 (2 h) | 4.40 × 10−1 | 4.42 × 10−1 | 4.43 × 10−1 | 4.50 × 10−1 | 4.73 × 10−1 | 4.80 × 10−1 |
17 (2 h) | 4.80 × 10−1 | 4.77 × 10−1 | 4.79 × 10−1 | 4.86 × 10−1 | 5.10 × 10−1 | 5.10 × 10−1 |
18 (2 h) | 5.10 × 10−1 | 5.14 × 10−1 | 5.15 × 10−1 | 5.23 × 10−1 | 5.47 × 10−1 | 5.50 × 10−1 |
19 (2 h) | 5.50 × 10−1 | 5.51 × 10−1 | 5.53 × 10−1 | 5.61 × 10−1 | 5.85 × 10−1 | 5.90 × 10−1 |
20 (2 h) | 5.90 × 10−1 | 5.89 × 10−1 | 5.91 × 10−1 | 6.00 × 10−1 | 6.24 × 10−1 | 6.30 × 10−1 |
21 (2 h) | 6.30 × 10−1 | 6.29 × 10−1 | 6.30 × 10−1 | 6.40 × 10−1 | 6.63 × 10−1 | 6.70 × 10−1 |
22 (2 h) | 6.70 × 10−1 | 6.68 × 10−1 | 6.70 × 10−1 | 6.80 × 10−1 | 7.04 × 10−1 | 7.10 × 10−1 |
23 (2 h) | 7.10 × 10−1 | 7.09 × 10−1 | 7.11 × 10−1 | 7.22 × 10−1 | 7.45 × 10−1 | 7.50 × 10−1 |
24 (2 h) | 7.50 × 10−1 | 7.51 × 10−1 | 7.53 × 10−1 | 7.64 × 10−1 | 7.88 × 10−1 | 7.90 × 10−1 |
25 (2 h) | 7.90 × 10−1 | 7.94 × 10−1 | 7.96 × 10−1 | 8.08 × 10−1 | 8.31 × 10−1 | 8.40 × 10−1 |
26 (2 h) | 8.40 × 10−1 | 8.37 × 10−1 | 8.40 × 10−1 | 8.52 × 10−1 | 8.76 × 10−1 | 8.80 × 10−1 |
27 (2 h) | 8.80 × 10−1 | 8.82 × 10−1 | 8.85 × 10−1 | 8.97 × 10−1 | 9.21 × 10−1 | 9.30 × 10−1 |
28 (2 h) | 9.30 × 10−1 | 9.28 × 10−1 | 9.30 × 10−1 | 9.44 × 10−1 | 9.67 × 10−1 | 9.70 × 10−1 |
29 (2 h) | 9.70 × 10−1 | 9.74 × 10−1 | 9.77 × 10−1 | 9.91 × 10−1 | 1.01 × 100 | 1.02 × 100 |
Weibull vibration fatigue damage statistics analysis for the numerical application data.
ni | Damage (Di) |
R (Di) |
Yi |
µy |
R(t) |
toi |
σ2i |
σ1i |
F(t) |
---|---|---|---|---|---|---|---|---|---|
1 | 0.0242 | 0.9758 | −3.7106 | −0.1280 | 0.9758 | 0.0170 | 1.1843 | 4114.9115 | 0.0242 |
2 | 0.0491 | 0.9509 | −2.9881 | −0.1030 | 0.9509 | 0.0375 | 2.6192 | 1860.5420 | 0.0491 |
0.0500 | 0.9500 | −2.9702 | −0.1024 | 0.9500 | 0.0383 | 2.6712 | 1824.3190 | 0.0500 | |
3 | 0.0749 | 0.9251 | −2.5535 | −0.0881 | 0.9251 | 0.0605 | 4.2219 | 1154.2363 | 0.0749 |
4 | 0.1013 | 0.8987 | −2.2366 | −0.0771 | 0.8987 | 0.0857 | 5.9806 | 814.8266 | 0.1013 |
5 | 0.1285 | 0.8715 | −1.9838 | −0.0684 | 0.8715 | 0.1131 | 7.8951 | 617.2292 | 0.1285 |
0.1740 | 0.8260 | −1.6548 | −1.8180 | 0.8260 | 0.1623 | 11.3328 | 430.0000 | 0.1740 | |
6 | 0.1564 | 0.8436 | −1.7713 | −0.0611 | 0.8436 | 0.1428 | 9.9713 | 488.7123 | 0.1564 |
7 | 0.1851 | 0.8149 | −1.5863 | −0.0547 | 0.8149 | 0.1750 | 12.2184 | 398.8335 | 0.1851 |
8 | 0.2145 | 0.7855 | −1.4212 | −0.0490 | 0.7855 | 0.2098 | 14.6488 | 332.6637 | 0.2145 |
0.2301 | 0.7699 | −1.3414 | −1.4738 | 0.7699 | 0.2291 | 15.9900 | 304.7600 | 0.2301 | |
9 | 0.2446 | 0.7554 | −1.2709 | −0.0438 | 0.7554 | 0.2475 | 17.2778 | 282.0444 | 0.2446 |
10 | 0.2756 | 0.7244 | −1.1321 | −0.0390 | 0.7244 | 0.2883 | 20.1243 | 242.1505 | 0.2756 |
11 | 0.3072 | 0.6928 | −1.0023 | −0.0346 | 0.6928 | 0.3325 | 23.2107 | 209.9507 | 0.3072 |
12 | 0.3397 | 0.6603 | −0.8794 | −0.0303 | 0.6603 | 0.3805 | 26.5641 | 183.4471 | 0.3397 |
13 | 0.3729 | 0.6271 | −0.7621 | −0.0263 | 0.6271 | 0.4329 | 30.2170 | 161.2704 | 0.3729 |
14 | 0.4069 | 0.5931 | −0.6492 | −0.0224 | 0.5931 | 0.4900 | 34.2089 | 142.4515 | 0.4069 |
15 | 0.4418 | 0.5582 | −0.5396 | −0.0186 | 0.5582 | 0.5528 | 38.5881 | 126.2854 | 0.4418 |
16 | 0.4774 | 0.5226 | −0.4323 | −0.0149 | 0.5226 | 0.6219 | 43.4144 | 112.2465 | 0.4774 |
17 | 0.5139 | 0.4861 | −0.3266 | −0.0113 | 0.4861 | 0.6985 | 48.7630 | 99.9346 | 0.5139 |
18 | 0.5513 | 0.4487 | −0.2215 | −0.0076 | 0.4487 | 0.7840 | 54.7301 | 89.0389 | 0.5513 |
19 | 0.5895 | 0.4105 | −0.1162 | −0.0040 | 0.4105 | 0.8802 | 61.4414 | 79.3132 | 0.5895 |
20 | 0.6285 | 0.3715 | −0.0097 | −0.0003 | 0.3715 | 0.9894 | 69.0649 | 70.5585 | 0.6285 |
0.6321 | 0.3679 | 0.0000 | 0.0000 | 0.3679 | 1.0000 | 69.8065 | 69.8065 | 0.6321 | |
21 | 0.6685 | 0.3315 | 0.0990 | 0.0034 | 0.3315 | 1.1150 | 77.8327 | 62.6101 | 0.6685 |
22 | 0.7094 | 0.2906 | 0.2116 | 0.0073 | 0.2906 | 1.2617 | 88.0781 | 55.3272 | 0.7094 |
23 | 0.7511 | 0.2489 | 0.3299 | 0.0114 | 0.2489 | 1.4368 | 100.3032 | 48.5838 | 0.7511 |
24 | 0.7938 | 0.2062 | 0.4569 | 0.0158 | 0.2062 | 1.6519 | 115.3161 | 42.2588 | 0.7938 |
25 | 0.8375 | 0.1625 | 0.5972 | 0.0206 | 0.1625 | 1.9273 | 134.5426 | 36.2199 | 0.8375 |
26 | 0.8821 | 0.1179 | 0.7599 | 0.0262 | 0.1179 | 2.3045 | 160.8711 | 30.2920 | 0.8821 |
27 | 0.9277 | 0.0723 | 0.9659 | 0.0333 | 0.0723 | 2.8898 | 201.7269 | 24.1570 | 0.9277 |
28 | 0.9743 | 0.0257 | 1.2979 | 0.0448 | 0.0257 | 4.1616 | 290.5145 | 16.7741 | 0.9743 |
0.9782 | 0.0218 | 1.3414 | 1.4738 | 0.0218 | 4.3657 | 304.7600 | 15.9900 | 0.9782 | |
29 | 0.9900 | 0.0100 | 1.5272 | 0.0527 | 0.0100 | 5.3540 | 373.7498 | 13.0384 | 0.9900 |
β = 0.9102 | ƞ = 69.8065 | µy = −0.6672 | σ1 = 304.7600 | σ2 = 15.9900 |
BOLD: The principal vibration stresses above and below the ƞ parameter, the R(t) index of 95%, and the R(t) index of the Sy 430 MPa.
Weibull vibration fatigue damage statistics analysis by using median rank method.
ni | Yi |
µy |
R(t) |
toi |
σ2i |
σ1i |
F(t) |
---|---|---|---|---|---|---|---|
1 | −3.7256 | −0.1285 | 0.9762 | 0.0071 | 0.4981 | 9783.4030 | 0.0238 |
−2.9702 | −0.1024 | 0.9500 | 0.0194 | 1.3570 | 3591.0886 | 0.0500 | |
2 | −2.8207 | −0.0973 | 0.9422 | 0.0237 | 1.6546 | 2945.1760 | 0.0578 |
3 | −2.3400 | −0.0807 | 0.9082 | 0.0449 | 3.1311 | 1556.3556 | 0.0918 |
4 | −2.0062 | −0.0692 | 0.8741 | 0.0698 | 4.8755 | 999.5045 | 0.1259 |
5 | −1.7476 | −0.0603 | 0.8401 | 0.0984 | 6.8706 | 709.2668 | 0.1599 |
6 | −1.5347 | −0.0529 | 0.8061 | 0.1305 | 9.1130 | 534.7438 | 0.1939 |
−1.3704 | −1.8180 | 0.7757 | 0.1623 | 11.3328 | 430.0000 | 0.2243 | |
7 | −1.3524 | −0.0466 | 0.7721 | 0.1663 | 11.6070 | 419.8408 | 0.2279 |
8 | −1.1918 | −0.0411 | 0.7381 | 0.2058 | 14.3630 | 339.2823 | 0.2619 |
−1.1109 | −1.4738 | 0.7194 | 0.2291 | 15.9900 | 304.7500 | 0.2806 | |
9 | −1.0474 | −0.0361 | 0.7041 | 0.2492 | 17.3959 | 280.1293 | 0.2959 |
10 | −0.9154 | −0.0316 | 0.6701 | 0.2969 | 20.7257 | 235.1238 | 0.3299 |
11 | −0.7930 | −0.0273 | 0.6361 | 0.3492 | 24.3773 | 199.9034 | 0.3639 |
12 | −0.6784 | −0.0234 | 0.6020 | 0.4066 | 28.3815 | 171.7005 | 0.3980 |
13 | −0.5699 | −0.0197 | 0.5680 | 0.4695 | 32.7756 | 148.6811 | 0.4320 |
14 | −0.4663 | −0.0161 | 0.5340 | 0.5387 | 37.6055 | 129.5852 | 0.4660 |
15 | −0.3665 | −0.0126 | 0.5000 | 0.6149 | 42.9271 | 113.5207 | 0.5000 |
16 | −0.2697 | −0.0093 | 0.4660 | 0.6992 | 48.8095 | 99.8395 | 0.5340 |
17 | −0.1751 | −0.0060 | 0.4320 | 0.7927 | 55.3389 | 88.0595 | 0.5680 |
18 | −0.0819 | −0.0028 | 0.3980 | 0.8971 | 62.6241 | 77.8153 | 0.6020 |
0.0000 | 0.0000 | 0.3679 | 1.0000 | 69.8065 | 69.8065 | 0.6321 | |
19 | 0.0107 | 0.0004 | 0.3639 | 1.0143 | 70.8051 | 68.8243 | 0.6361 |
20 | 0.1033 | 0.0036 | 0.3299 | 1.1469 | 80.0653 | 60.8642 | 0.6701 |
21 | 0.1969 | 0.0068 | 0.2959 | 1.2986 | 90.6512 | 53.7567 | 0.7041 |
22 | 0.2925 | 0.0101 | 0.2619 | 1.4741 | 102.9040 | 47.3559 | 0.7381 |
23 | 0.3913 | 0.0135 | 0.2279 | 1.6805 | 117.3143 | 41.5390 | 0.7721 |
24 | 0.4950 | 0.0171 | 0.1939 | 1.9285 | 134.6219 | 36.1985 | 0.8061 |
25 | 0.6062 | 0.0209 | 0.1599 | 2.2349 | 156.0158 | 31.2347 | 0.8401 |
26 | 0.7288 | 0.0251 | 0.1259 | 2.6298 | 183.5827 | 26.5445 | 0.8741 |
27 | 0.8703 | 0.0300 | 0.0918 | 3.1730 | 221.4969 | 22.0008 | 0.9082 |
28 | 1.0474 | 0.0361 | 0.0578 | 4.0133 | 280.1599 | 17.3940 | 0.9422 |
1.1109 | 1.4738 | 0.0480 | 4.3656 | 304.7500 | 15.9900 | 0.9520 | |
29 | 1.3185 | 0.0455 | 0.0238 | 5.7498 | 401.3795 | 12.1409 | 0.9762 |
BOLD: The principal vibration stresses above and below the ƞ parameter, the R(t) index of 95%, and the R(t) index of the Sy 430 MPa.
The proposed method and median rank method comparison results.
Feature | Proposed Method | Median Rank Method |
---|---|---|
Weibull Shape Parameter |
β = 0.9102 | β = 0.7538 |
Weibull Scale Parameter |
ƞ = 69.8077 | ƞ = 69.8077 |
Principal Stresses, |
R(t) = 0.8260 | R(t) = 0.7757 |
R(t) = 0.95 |
σ1 = 1824.3190 MPa |
σ1 = 3591.0886 MPa |
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Abstract
In this paper, a Weibull probabilistic methodology is proposed with an approach to model vibration fatigue damage accumulation using two parameters: Weibull distribution and a nonlinear fatigue damage accumulation model. The damage is cumulated based on the application of a vibration stress profile and is used to determine both the Weibull
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Neither ProQuest nor its licensors make any representations or warranties with respect to the translations. The translations are automatically generated "AS IS" and "AS AVAILABLE" and are not retained in our systems. PROQUEST AND ITS LICENSORS SPECIFICALLY DISCLAIM ANY AND ALL EXPRESS OR IMPLIED WARRANTIES, INCLUDING WITHOUT LIMITATION, ANY WARRANTIES FOR AVAILABILITY, ACCURACY, TIMELINESS, COMPLETENESS, NON-INFRINGMENT, MERCHANTABILITY OR FITNESS FOR A PARTICULAR PURPOSE. Your use of the translations is subject to all use restrictions contained in your Electronic Products License Agreement and by using the translation functionality you agree to forgo any and all claims against ProQuest or its licensors for your use of the translation functionality and any output derived there from. Hide full disclaimer