Reliability by Using Weibull Distribution Based on Vibration Fatigue Damage

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 $\beta$ and $\eta$ parameters, and the corresponding component reliability R(t). The vibration fatigue damage is analyzed to accumulate the damage as a stress function for a fatigue life exponent derived with the assistance of the acceleration’s force response. The steps to determine the Weibull $\beta$ and $\eta$ parameters are estimated based only on the principal vibration stresses ${\sigma }_1$ and ${\sigma }_2$ that allow the reproduction of the vibration fatigue damage. The method’s efficiency is based on the probabilistic approach by using the vibration fatigue damage as the Y_i vector that covers the arithmetic mean as well as the $\beta$ parameter. Finally, the procedure proposed is applied in a practical case where a mechanical component is used as a support for telecommunication connections and is submitted to vibration stress. The results show that using the damage accumulated as the Y_i vector to estimate the parameters allows for the analysis of dynamic and individual applications.

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 ${\sigma }_1$ and ${\sigma }_2$; 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 D_i 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).

$${\sigma \left(vib\right)}_i={\sigma }_{dynamic}{\times A}_{res}$$

(1)

$${\sigma }_{dynamic}=\left(\frac{K{m}_e\widehat{L}C}{I}\right)A$$

(2)

$$A_{res}=\frac{2{\pi }^2{F}^2{D}_2}{G}$$

(3)

$$N_i\times {\sigma \left(vib\right)}_i^b=a^b$$

(4)

In Equation (1), ${\sigma }_{dynamic}$ and $A_{res}$ are the dynamic load factor and the acceleration response, respectively. In Equation (2) [20], K is the stress concentration factor in the mechanical component, $m_e$ is the effective mass, C is the distance to the neutral axis, $\widehat{L}$ 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), $F$ is the frequency applied by the vibration power spectral density (PSD) and $G$ is the gravity constant. In Equation (4), $N_i$is the maximum number of cycles that the material’s component can sustain at a vibration load with stress amplitude ${\sigma \left(vib\right)}_i$ 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 $n_i$ represents the applied vibration cycles at the stress level ${\sigma \left(vib\right)}_i$.

$${D=\sum _{i=1}^2{D}_2={\left[\frac{n_2}{N_{2,f}}\right]}^{\left(\frac{N_{2,f}}{N_{1,f}}\right)}}^{\left[\frac{{\sigma \left(vib\right)}_1}{{\sigma \left(vib\right)}_2}\right]}$$

(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.

$$f\left(t\right)=\frac{\beta }{\eta }{\left(\frac{t}{\eta }\right)}^{\beta -1}exp\left\{-{\left(\frac{t}{\eta }\right)}^{\beta }\right\}$$

(6)

$$F\left(t\right)=1-exp\left\{-{\left(\frac{t}{\eta }\right)}^{\beta }\right\}$$

(7)

where, $\beta$ is the shape parameter, $\eta$ is the scale parameter, and t is the selected random variable (damage or fatigue life). The corresponding reliability function R(t) is given as

$$R\left(t\right)=exp\left\{-{\left(\frac{t}{\eta }\right)}^{\beta }\right\}$$

(8)

From [28], the Weibull fatigue damage ${\beta }_D$ and ${\eta }_D$ parameters are determined as

$${\beta }_D=\frac{-4{\mu }_y}{0.995\times ln\left(\frac{{\sigma }_1}{{\sigma }_2}\right)}$$

(9)

$${\eta }_D=exp{(\mu }_x)$$

(10)

where ${\mu }_y$ represents the mean of the Y vector (see Equation (13)) determined by using Equation (5). ${\mathrm{\mu }}_x$ represents the log-mean of the failure-time data, which is determined here directly from the addressed maximum ${\sigma }_1$ and minimum ${\sigma }_2$ stress values of Section 2.1. Thus, ${\mathrm{\mu }}_x$ is determined as

$${\mu }_x=ln{\left({\sigma }_1{\sigma }_2\right)}^{\frac{1}{2}}$$

(11)

Here, notice that the efficiency of the Weibull parameters ${\beta }_D$ and ${\eta }_D$ only depends on the accuracy with which the ${\sigma }_1$ and ${\sigma }_2$ 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 ${\sigma }_1$ and ${\sigma }_2$ 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 $D_i$ 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 $R\left(D_i\right),$ as

$$R\left(D_i\right)=1-D_i$$

(12)

Step 3. By using the $R\left(D_i\right)$elements in the linearized form of the reliability function given in Equation (8), the corresponding $Y_i$ elements are determined as in Equation (13). Then, its corresponding arithmetic mean value is computed as in Equation (14).

$$Y_i=LN(-LN(1-D_i\left)\right)$$

(13)

$${\mu }_y={\sum }_{i=1}^n\frac{Y_i}{n}$$

(14)

Step 4. By plugging the ${\mu }_y$ value and the ${\sigma }_1$ and ${\sigma }_2$ values into Equation (9), the corresponding Weibull shape ${\beta }_D$ parameter is determined. Similarly, by plugging the ${\sigma }_1$ and ${\sigma }_2$ values into Equation (11), the corresponding ${\mu }_x$ value is determined. Then, by using this corresponding ${\mu }_x$ value in Equation (10), the corresponding Weibull scale ${\eta }_D$ parameter is determined. These ${\beta }_D$ and ${\eta }_D$ parameters represent the Weibull fatigue damage family that is used to model the random behavior of the estimated ${\sigma }_1$ and ${\sigma }_2$ principal stress values.

Note 1. Here, notice the random behavior of the ${\sigma }_1$ and ${\sigma }_2$ values. In the proposed Weibull analysis, let us use the ${\sigma }_{1\mathrm{i}}$ 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 ${\beta }_D$ and ${\eta }_D$ parameters, the minimum strength ${\sigma }_{1\mathrm{i}}$ values are determined by using the $t_{0i}$ value that correspond to each $Y_i$ element as

$$t_{0i}={\mathrm{e}\mathrm{x}\mathrm{p}\{Y}_i/{\beta }_D\}$$

(15)

Thus, the ${\sigma }_{2\mathrm{i}}$ value is determined as

$${\sigma }_{2i}={\eta }_D\times t_{0i}$$

(16)

and the ${\sigma }_{1\mathrm{i}}$ value is determined as

$${\sigma }_{1i}={\eta }_D/t_{0i}$$

(17)

Additionally, from Equation (18), by using the known ${\sigma }_1$ value, the $t_{01}$ element that belongs to the ${\sigma }_1$ and ${\sigma }_2$ values determined in Section 2.1 is determined as

$$t_{01}={\eta }_D/{\sigma }_1$$

(18)

Now, the $t_{01}$ and the ${\beta }_D$ values are used to determine the corresponding $Y_1$ value as follows,

$$Y_1=\ln \left(t_{01}\right)\times {\beta }_D$$

(19)

Finally, the reliability index that corresponds to the $Y_1$ value is determined as

$$R\left(t\right)=\exp \left\{-\exp \left\{Y_1\right\}\right\}$$

(20)

Note 2. Here, observe that the R(t) index determined in Equation (20) by using the ${\sigma }_1$ value according to our proposed method corresponds to a component with strength equivalent to the ${\sigma }_1$ value. Thus, if we define the material $S_y$ parameter as the actual strength of the component, then by using this S_y value in Equation (18) and the corresponding $Y_{Sy}$ value of Equation (19) in Equation (20), the minimum expected reliability of a component that presents a strength of $S_y$ 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 $F\left(t\right)$ and the reliability $R\left(t\right)$ 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 $S_y$= 430 MPa, ultimate tensile strength S_ut = 510 MPa, endurance limit S_e = 255 MPa, density $\rho$= 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 ${\sigma }_{dynamic}$ and the vibration stresses ${\sigma \left(vib\right)}_i$, are calculated, respectively. The acceleration responses ${A_{res}}_i$ are obtained from the vibration system but can also be determined by using Equation (3). Then, the vibration cycles applied $n_{\mathrm{i}}$ are determined by the rainflow method and the total cycles $N_i$ are determined by using Equation (4). Finally, the fatigue damage accumulation is obtained by Equation (5).

Table 1. Vibration Stress and Cycles Results.

Frequency (Hz)	Accel. Response (G)	Dynamic Factor ${\mathit{\sigma }}_{\mathit{d}\mathit{y}\mathit{n}\mathit{a}\mathit{m}\mathit{i}\mathit{c}}$ Equation (2)	Vibration Stress ${\mathit{\sigma }\left(\mathit{v}\mathit{i}\mathit{b}\right)}_{\mathit{i}}$ Equation (1)	Applied Vibration Cycles $\left({\mathit{n}}_{\mathit{i}}\right)$	Total Cycles ${(\mathit{N}}_{\mathit{i}})$ Equation (4)
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

shows the vibration stress and cycle results and

Table 2. 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

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, ${\sigma }_1$ and ${\sigma }_2$, 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 $Y$ 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 D_i value of each individual block of Table 2, and using it in Equation (13), the n = 29 $Y_i$ elements are determined. Thus, from Equation (14), the mean is, ${\mu }_y=-0.6672$. Next, from the results of Table 1, the principal vibration stress values ${\sigma }_1$ = 304.76 MPa and ${\sigma }_2$ = 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 ${\beta }_D$ value is

$${\beta }_D=\frac{(-4)(-0.6672)}{0.995\times \mathrm{l}\mathrm{n}\left(\frac{304.76}{15.99}\right)}=0.9102$$

and from Equation (11), the logarithm average value is ${\mu }_x=\ln \sqrt{304.76\times 15.99}=4.2457$. Thus, from Equation (10), the Weibull parameter ${\mathrm{ƞ}}_D$ is ${\mathrm{ƞ}}_D=\mathrm{e}\mathrm{x}\mathrm{p}\left\{4.2457\right\}=69.8077$. Consequently, the Weibull damage family is W(0.9102, 69.8077 MPa). This Weibull family completely represents the observed principal vibration stresses ${\sigma }_1$ and ${\sigma }_2$ 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 ${\sigma }_1$ and ${\sigma }_2$ 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 ${\sigma }_1$and ${\sigma }_2$ is determined by Equations (16) and (17). Then, by using the ${\beta }_D$ and the $Y_i$ values in Equation (15), the basic Weibull elements [28] $t_{0i}$ for each $Y_i$ are obtained. Whereas by using the ${\eta }_D$ and ${\sigma }_1$ values in Equation (18), the Weibull $t_{01}$value from the ${\sigma }_1$ and ${\sigma }_2$ stress values are reproduced. This value is calculated as

$$t_{01}=\frac{69.8077}{304.76}=0.2291$$

and, by using the ${\beta }_D$ value in Equation (19), the $Y_1$ value that belongs to the $t_{01}$ value is determined as

$$Y_1=\ln \left(0.2291\right)\times 0.9102=-1.3414$$

Next, by substituting the $Y_1$ value in Equation (20), the reliability $R\left(t\right)$that belongs to the $t_{01}$ element is

$$R\left(t\right)=\exp \left\{-\exp \left\{-1.3414\right\}\right\}=0.7699$$

The previous results shown in this section are included in

Table 3. Weibull vibration fatigue damage statistics analysis for the numerical application data.

ni	Damage (Di) Equation (5)	R (Di) Equation (12)	Yi Equation (13)	µy Equation (14)	R(t) Equation (20)	toi Equation (15)	σ2i Equation (16)	σ1i Equation (17)	F(t) Equation (7)
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 S_y 430 MPa.

.

Here, is important to mention that the reliability obtained $R\left(t\right)=0.7699$ corresponds to a designed component or structure with minimal strength of $S_y$ = ${\sigma }_1$ = 304.76 MPa. In relation to the mechanical support, it has a $S_y$ = 430 MPa, then, its reliability is $R\left(t\right)=0.8260$, which is determined as follows. The minimal reliability of the component is obtained when the $Y_1$ value that belongs to the $S_y$ value is used in Equation (20). The steps to determine the reliability of the design component when the $S_y$ value is used as ${\sigma }_1$ in Equation (18) are, the $t_{Sy}$ element that belongs to the $S_y$ value is $t_{sy}=\frac{69.8077}{430}=0.1623$. From Equation (19), the corresponding $Y_{sy}$ value is $Y_{sy}=\ln \left(0.1623\right)\times 0.9102=-1.6548$. The reliability index for the $Y_{sy}$ value is calculated by using Equation (20), $R\left(t\right)=\exp \left\{-\exp \left\{-1.6548\right\}\right\}=0.8260$. Thus, we conclude that the reliability of the design component is $R\left(t\right)=0.8260$. Additionally, regarding the material’s component or structure, it is noticed about that the higher the strength S_y value, the higher the reliability $R\left(t\right)$ 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 $Y$, 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 $F\left(t_i\right)$that previously was determined by Equation (12) is now determined as

$$F\left(t_i\right)=\frac{i-0.3}{n+0.4}$$

(21)

Then, by using the $F\left(t_i\right)$ elements in the linearized form of the reliability function given in Equation (8), the corresponding $Y_i$ elements are determined as in Equation (22),

$$Y_i=LN(-LN(1-\left(\right(i-0.3)/(n+0.4\left)\right)\left)\right)$$

(22)

By selecting the fatigue damage accumulated D_i value of each one block of Table 2 and using it in Equation (22), the n = 29 $Y_i$ elements are determined. Thus, from Equation (14), the mean is, ${\mu }_y=-0.5525$. Next, from the results of Table 1, the principal vibration stress values ${\sigma }_1$ = 304.76 MPa and ${\sigma }_2$ = 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 ${\beta }_D$ value is

$${\beta }_D=\frac{(-4)(-0.5525)}{0.995\times \mathrm{l}\mathrm{n}\left(\frac{304.76}{15.99}\right)}=0.7538$$

Since the principal vibration stress values σ₁ = 304.76 MPa and σ₂ = 15.99 MPa are maintained, the Weibull parameter value ${\eta }_D=69.8077$ remains.

Next, the remaining steps followed in Section 3 are applied and the results are included in

Table 4. Weibull vibration fatigue damage statistics analysis by using median rank method.

ni	Yi Equation (21)	µy Equation (14)	R(t) Equation (20)	toi Equation (15)	σ2i Equation (16)	σ1i Equation (17)	F(t) Equation (7)
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 S_y 430 MPa.

.

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. The proposed method and median rank method comparison results.

Feature	Proposed Method	Median Rank Method
Weibull Shape Parameter Equation (9)	β = 0.9102	β = 0.7538
Weibull Scale Parameter Equation (10)	ƞ = 69.8077	ƞ = 69.8077
Principal Stresses, Equations (17) and (16) σ1 = 304.7600 MPa σ2 = 15.9900 MPa Material strength, Sy = 430 MPa	R(t) = 0.8260	R(t) = 0.7757
R(t) = 0.95 Equation (8)	σ1 = 1824.3190 MPa σ2 = 2.6712 MPa	σ1 = 3591.0886 MPa σ2 = 1.3570 MPa

.

In Table 5, notice that although we are using the same ${\sigma }_1$ = 304.76, ${\sigma }_2$= 15.99 and $S_y$ 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 ${\sigma }_1$ and ${\sigma }_2$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 ${\sigma }_1$ = 1824.3190 MPa. At the same time, using the median rank method, it is ${\sigma }_1$ = 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 ${\sigma }_1$ = 304.76 and ${\sigma }_2$ = 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.