Reliability assessment of folding wing system deployment performance considering failure correlation

▪ Dynamic model of the folding wing mechanism with joint clearances is developed and solved. ▪ The system reliability models with failure correlation are established. ▪ The new evaluation methods for the system reliability models are proposed. ▪ Variation rules of system reliability with different distribution parameters are analyzed. The reliability of folding wing deployment performance greatly impacts flight vehicle reliability. Based on the dynamic analysis theory, the deployment dynamic model of folding wing mechanism with joint clearances is established and solved. Considering the failure correlation, the system reliability models are developed for both cases, without considering synchronization and considering synchronization. For the former, a solution method combining saddle point approximation and numerical integration is proposed. For the latter, an estimation method based on a combination of the fourth order moment Pearson distribution family and the numerical integration is proposed. The efficiency and accuracy of the proposed methods are verified through examples. In addition, the trend of the system reliability change when the distribution parameters of random variables are different is also analyzed. From the perspective of improving reliability, the above study can provide theoretical guidance and data support for the design, manufacturing and service process of the folding wing mechanism system.


Introduction
As a vital flight control unit, the folding wing mechanism ensures flight safety, stability, and trajectory adjustments.Its deployment performance reliability directly impacts the flight vehicle's mission success.Therefore, it is necessary to evaluate the reliability of the deployment performance of the folding wing system.Generally, successful deployment is achieved when the mechanism's deployment time remains under a threshold.In harsh environments, this also involves synchronization reliability, where the time span between maximum and minimum deployment must stay below a threshold.These At present, many reliability evaluation methods have been derived for various reliability problems.Monte Carlo simulation (MCS) is the most classical and widely used numerical simulation method 13,22.Zhuang et al.48 applied MCS to obtain the dynamic wear reliability of aircraft locking mechanism.However, the efficiency of MCS is difficult to be accepted when dealing with time-consuming engineering simulations (multibody system dynamics).The first-order reliability method (FORM) 5,14 is very efficient.But FORM will have large errors when dealing with high nonlinear problems.In order to balance the efficiency and accuracy of reliability evaluation, the response surface method 3,36 has been proposed.Support vector machine 17,39 has also been widely used in reliability assessment.The nonlinear fitting capability of artificial neural networks is also strong, so they can also be used in surrogate models 7,44.The Kriging model 33,34 also has a good fitting effect on the problems of high To comprehensively address the shortcomings of the above studies and to improve the computational efficiency as much as possible in the reliability assessment process, we took the folding wing deployment mechanism as the object of this study, and conducted an in-depth study on the reliability assessment process including failure correlation by taking into full consideration the problems that have not been considered in the above studies.In addition, to the best of our knowledge, there is no systematic report on the reliability assessment of deployment time and deployment synchronization of the folding wing system with clearance considering failure correlation, which is also the purpose of this study.In this study, we considered failure correlation due to external random loads and joint clearances in the folding wing system.The deployment time reliability and the deployment synchronization reliability with multiple deployments are also analyzed.Furthermore, reliability assessment methods are proposed to efficiently solve the reliability.
In this paper, the contributions are summarized as follows: Section 3 shows the system reliability model of the folding wing mechanism considering failure correlation, and proposes the corresponding reliability evaluation method to evaluate the system reliability.In Section 4, the correctness of the proposed method is illustrated by the numerical examples, and the system reliability with different random variable parameters is calculated.The conclusion is given in Section 5.The deployment actions of the four groups of folding wing mechanisms are all completed parallel to the XOY plane, the main wing can be simplified as a crank, while the auxiliary wing can be simplified as a connecting rod.As shown in Figure 2, taking the first group of folding wing mechanisms as an example, in the unfolding process, the cylinder will start, the driving force will push the slider to move along the slot, so that the crank will rotate and unfold around the rotation axis A.

Dynamic analysis of folding wing mechanism with joint
Finally, the slider reaches the specified locking position, at this time the whole deployment process is finished.

Contact force analysis of joint with clearance
Generally, the rotating pairs of the mechanism are not ideal.As shown in Figure 3, the eccentricity vector  is introduced to describe the relative position between the journal and the bearing, which is expressed as 11: where,   and   are the generalized coordinates of the center of mass for the journal and the bearing components, respectively.  and   are the transformation matrices of the local coordinate system for the journal and the bearing components with respect to the generalized coordinate system, respectively.During the collision phase, there will be a certain penetration depth  between the journal and the bearing, which is calculated as: ∆= (e T e) 1/2 − (  −   ) = (e T e) 1/2 −   (2) where,   and   represent the radii of the bearing and journal, respectively.  is the clearance size.
Denoting the contact points between the journal and the bearing during their collision as   and   .The velocity calculation formulas for contact points   and   can be expressed as follows: where, the superscripted black dots of the vector represents its first derivative concerning time.
The normal and tangential projections of the relative contact velocity are: where,  represents the unit tangent vector.
The expression for the normal contact force   based on the L-N model is given by 10,11,28: where,   represents the restitution coefficient,  ̇ denotes the penetration velocity,  ̇0 represents the initial penetration velocity. represents the stiffness coefficient.It can be expressed as: The calculation equation for the friction force proposed by Ambrósio is given as follows 1: where,   denotes the friction coefficient, () represents the sign function, and   is a dynamic correction parameter that depends on the tangential velocity.
The centroids of the bearing and journal components experience resultant forces: The moments experienced at the centroids of the bearing and journal components can be determined as:

Dynamics of folding wing mechanism with joint clearance
As shown in Figure 5, the origin of the generalized coordinates is located at A. The generalized coordinates of the crank, connecting rod, and slider are denoted as ( 1 ,   According to section 2.2, the contact forces  1  and  2  in the joint B, can be determined: where,  1 and  1 are the unit normal and tangent vectors in joint B. Based on Eq. ( 8), the expressions for the moments  1  and  2  at the crank and connecting rod centroids are given by: where,   1  1 and   2  2 are the radius vectors of the crank and connecting rod.
Similarly, the contact forces in the joint C, denoted as  3

𝑄
and  4  , the corresponding moments exerted at the slider and connecting rod are  3  and  4  .
The folding wing mechanism is also subject to other external forces.The electric cylinder, applies a driving force directly to the center of mass of the slider.The direction always points towards the positive X-axis.The segmental function expression for the variation of the driving force   with time  is: where,   is the peak of the driving force.
During the unfolding process, both the main wing and the auxiliary wing experience aerodynamic loads.The centers of mass of the crank and the connecting rod are subjected to aerodynamic drag forces, denoted as  1 and  2 , which act continuously in the positive X-axis direction: where,   is the air density. 0 is the air drag coefficient.  are the wing thickness.|     | is the projection length of the folded wing on the Y-axis.  is the flight speed of the flight vehicle.
The flight attitude (pitch angle) of the flight vehicle also influences the unfolding dynamics.When the folding wing is not parallel to the ground, the gravitational force can be decomposed into two components along the X-axis and Z-axis directions.The gravitational component along the X-axis will inevitably impact the unfolding behavior.As shown in Figure 6, when the angle between the flight direction and the vertical upward direction of the ground is acute, the pitch angle  is defined as positive.The magnitude of   can be calculated as: where,   is the mass of the ith component,  is the acceleration of gravity.Figure 7 illustrates the force distribution on the crank, connecting rod and the slider.The generalized force vector  can be derived: The dynamic equations of the folding wing mechanism can be formulated in the index-3 form: where, ̈ represents the second derivative of the generalized coordinates.   denotes the transpose of the Jacobian matrix for constraint equations. represents the Lagrange multiplier vector. corresponds to the mass matrix, which can be expressed as: where,   ( = 1,2,3) represents the mass of the ith component.
( = 1,2,3) represents the moment of inertia for the ith component.These parameters can be further described as: where,   represents the density of the ith component, while   denotes its volume.Additionally,   corresponds to the width of the ith component.
The core of the Baumgarte algorithm 2 lies in preventing constraint violation during the numerical solution of Eq. ( 17).
Based on this, Eq. ( 17) can be rewritten as: where Connecting rod Slider ( ) The dynamic simulation parameters are shown in Table 1.In this study, we defined the clearance sizes    and    in joints B and C as equal.This is because it is possible to reduce the number of random variables in one dimension, which accordingly reduces the samples used for the subsequent reliability analysis and reduces the computational burden.At the same time, such a definition does not materially affect the reliability models and assessment methods proposed in this study.It is the same with  .Based on Eq. ( 20), a dynamic numerical simulation is conducted, the rotation angle of the main wing is illustrated in Figure 8.
where, (•) represents the deployment time response implicit function.  ,   and   correspond to the peak of driving force, wing thickness and joint clearance size, respectively, for the ith set of folding wing.
In this study, it is required that   for each set of folding wing is less than the threshold  .The deployment performance reliability   for any set of folding wing can be expressed as: In practical engineering, the components that make up a system often operate under the same random load environment 35.Therefore, it is necessary to consider the failure correlation in the system reliability.Based on Eq. ( 22), it is evident that where,    (  ) and   () are the probability density functions of   and .
Furthermore, after each flight mission, every set of folding wing mechanism needs to be disassembled and replaced.The conditional PDF of  |  , can be defined as ℎ   |  , (  |  , ) .
Then Eq. ( 24) can be rewritten as: When a flight vehicle operates under a specified airspace, its flight velocity   and flight attitude (pitch angle ) are constant.

Statistical analysis of
. The probability density function of  |  , () is: When   and  are fixed, the probability that the ith set of folding wing successfully unfolds in all  flight missions can be expressed as: The system deployment performance reliability after  flight missions is: Based on Eq. ( 28)-( 29), the expression for the system deployment performance reliability, without considering synchronization, can be obtained using the law of total probability:   (30)

Reliability assessment method
The efficiency of MCS is hindered by the integration operations and unknown conditional cumulative distribution functions.
The surrogate models demand numerous samples for precise global models, they also spend much time in predicting small failure probabilities through large training sets.Numerical integration methods strike a balance between efficiency and accuracy for integration tasks.Additionally, SPA excels in estimating distribution function tails.Thus, this study employs the numerical integration and SPA to solve the reliability issues.
The Gauss-Hermite quadrature formula is utilized to handle the integration operation in Eq. ( 30), and the full factorial numerical integration (FFNI) method is applied to discretize the integration operation in Eq. ( 30), resulting in: where,  1 and  2 represent the number of integration nodes.
The real number saddle point   at  =  can be determined: The expression of the CDF for  is 6: where, ( * ) and ( * ) are the CDF and PDF of the standard normal variable.The parameters  and  are: where,  ̂Ÿ (  ) is the second derivative of the CGF for  at   .
To compute   ′ ( = 1,2,3,4), the use of the FFNI method and Gauss quadrature formula is necessary.Once the conditional CDFs for all nodes are obtained, the system reliability without synchronization can be calculated.The specific process is illustrated in Figure 9.

Input random variable distribution parameters
Using the FFNI/SGNI method to discretize the integration with common cause variables Solving for the first four order origin moments of the conditional response corresponding to all integration nodes Through Eq. (32)-Eq.( 35), the conditional cumulative distribution functions corresponding to all integration nodes are calculated Using Eq. ( 31) to solve the reliability of the folded wing system without considering synchronization Identifying Gaussian integration nodes of other random variables by FFNI method and combining the input samples Using Eq. ( 20) to solve the deployment time response of the folding wing mechanism with joint clearances

Deterministic multibody dynamic analysis
System Reliability Analysis 3.2.System reliability analysis considering deployment synchronization

Reliability model
In some extreme conditions, the deployment time of each folding wing should not differ too much.Otherwise, it will result in a reduction of stability.In this case, the deployment synchronization reliability also needs to be considered.In this study, the difference between the maximum and minimum deployment time within the system is taken as the indicator.The system reliability   * considering synchronization for one flight mission can be expressed as: where, the symbol () in the upper right corner represents the ith flight mission.
By combining Eq. ( 37) and Eq. ( 38), the reliability   *  of the folding wing system for m flight missions can be obtained as follows: ,   () |  , *    (39) The total reliability of the folding wing system considering synchronization is:

Reliability assessment method
When calculating  |  , and  |  , ,  |  , ( = 1,2,3,4) need to be calculated, so the reliability model that introduces synchronization is more complex and difficult to solve.Due to the long computational time required for solving the multibody dynamic with clearances, MCS is difficult to accept.Using large training datasets for surrogate models can significantly extend computational time, especially when predicting responses for multiple flight missions.Therefore, this section presents an efficient approach combining Gaussian integration with the Pearson distribution family to solve the reliability model.It is now illustrated with a folding wing system,   ,   ,  3 and  4 of  |  , can be solved by the FFNI method, then , ,  and  can be calculated.Subsequently, ℎ ̂|  , (  |  , ) will be approximated.It is important to note that we need to solve for both the maximum and minimum values of  |  , . Fortunately where, the second number in the lower right corner represents the number of groups.

Mathematical example
Consider a series system containing four units as shown in Figure 11.The response of each unit is:

Input random variable distribution parameters
Using the FFNI/SGNI method to discretize the integration with common cause variables Solving for the mean, standard deviation, skewness and kurtosis of the conditional response corresponding to all integration nodes According to Eq. ( 42), fitting the distribution and generating a large number of samples Summarizing all the samples and then obtaining the set of samples corresponding to each integration node Identifying Gaussian integration nodes of other random variables by FFNI method and combining the input samples Using Eq. ( 20) to solve the deployment time response of the folding wing mechanism with joint clearances

System reliability with synchronization
Using Eq. ( 39), Eq. ( 40) and Eq. ( 47), the reliability of the folding wing system with synchronization can be solved Fig. 11.Illustration of the system in Example 1.
The random variables are mutually independent, and the distribution parameters are shown in Table 2.  needs to be less than the given threshold  1 .Figure 12(c) presents the reliability curves for  = 2 and  = 5 , as  1 increases, the system reliability also increases.For any given operating condition, the reliability increase rate gradually diminishes until it reaches a nearly constant level.
The sample sizes for both methods are listed in Table 3.The relative errors compared to the MCS are listed in Figure 12(d)-12(e).In this study, the calculation error of the proposed method is determined by the relative error of the failure probability, the expression is: where,   is the reliability obtained by the proposed method.
is the reliability obtained by MCS.In fact, all the calculation results based on MCS and the proposed method are approximate consistent.At the same time, the number of samples required by the proposed method is also small.The proposed method 75 75

Motion reliability of the four-bar mechanism system
Consider a four-bar mechanism system as shown in Figure 13.The system consists of two sets of four-bar mechanisms, with the driving link rod 1 being shared by both sets.The length of each rod is denoted by   ( = 1,2, ⋯ 7) .The rod length distribution parameters are in Table 4. Taking the first group of four-bar mechanism as an example, output angle  3 is:

Group II
It is defined that the system is failure when  3 and  5 exceed the threshold  2 .The reliability is illustrated in Figure 14(a).Consider this system that operates for a total of  times, and at the kth ( ≥ 2 ) operation, each rod, except rod 1, is removed and replaced.Then the positioning reliability is shown in Figure 14  Fig. 14(a).Reliability curve without synchronization for m=1.

Deployment performance reliability of folding wing system
The random variables of the folding wing mechanism system are shown in Table 6.Under the given  and , the system reliability decreases with an increase in   .When  exceeds 0.57 s, higher values of  result in denser reliability curves corresponding to different   .
It is evident that both the deployment time threshold  and the number of deployment times  significantly influence the system reliability.In the context of considering the deployment synchronization, at any given  and  level, an increase in  leads to lower system reliability.At any given  and  level, an increase in  leads to higher system reliability.As  increases, the system reliability first gradually increases and then reaches a stable state.Taking into consideration the uncertainty of , under the given  ,  and   , a smaller   leads to higher system reliability.Under the given  ,  and   , a higher   results in lower system reliability.
The sample sizes for both methods are listed in Table 7.The relative errors compared to the MCS are listed in Figure 15(f)-15(g).According to the MCS results, the proposed methods are suitable for solving the reliability of the folding wing system.
In order to visualize the differences between the failure independent case and failure correlation case, the differences are shown in Figure 15(h).As can be seen from the figure, in the folding wing series system, the results in the assumed independent case are more conservative compared to the case where failure correlation is considered.Also, the difference between the two cases becomes smaller as the threshold increases.However, when the threshold is a constant value, the difference gradually increases as the number of components n in the system increases, and the results obtained based on the assumed independent case become less accurate.The proposed method 1125 1125 Fig. 15(a).Reliability curves without synchronization for m>1.

Reliability of deployment performance under standard deviation variations
This section primarily focuses on evaluating the system reliability under different standard deviations.The standard deviations are presented in Table 9. Figure 17 When the standard deviations of random variables change, both the dispersion and mean value of the system response will be affected.Without considering both synchronization and randomness of  , as  increases, the system reliability with smaller standard deviation for   ,  ,   and   will gradually increase and surpass that with larger standard deviation.For   , the reliability curves of both conditions are nearly coincident, and at the right boundary of , the reliability is higher for the smaller standard deviation.As  increases, the difference in reliability, at any  level, always exhibits a pattern of "troughpeak-stabilization".When considering both synchronization and randomness of , at any  level, the system reliability with the smaller standard deviation is always initially lower than the other, and then the former gradually increases and surpasses the latter.Moreover, near the right boundary of  , the system reliability with the smaller standard deviation is consistently higher than the other.

Deployment reliability considering randomness of crank and connecting rod lengths
In this example, the randomness of the crank and connecting rod lengths is taking into account.A total of 28,125 sample points are computed.The distribution parameters of the random variables are listed in Table 10.

Conclusion
In this study, firstly, the dynamic model of the folding wing mechanism with joint clearances is established and solved.
reliability indicators are related to the deployment time, so the dynamic of the folding wing mechanism with clearances needs to be modeled and solved.Many scholars have studied the dynamics of planar multibody systems containing joint clearances.Flores et al. 10,11 examined contact forces in revolute joints with clearances, and solved a crank-slider mechanism dynamic model using numerical integration.Mukras et al. 28 deduced hinge wear depth variation after longterm operation.Zheng et al. 43 built a dynamic model for a flexible multilink high-speed press with joint clearance.Li et al. 23,24 focused on deployment mechanisms.They developed a model for rigid-flexible solar sail system dynamics with joint clearances.Li et al.21 simulated spatial deployment mechanisms, studying the impacts of clearance, damping, friction, gravity, and flexibility on dynamic performance.These studies offer valuable insights for establishing and solving dynamic models for folding wing mechanisms with clearances.Traditional series system reliability calculation involves the assumption of independent component failures.However, in complex engineering scenarios, component failures often exhibit statistical correlation, which makes the independence assumption of traditional models invalid.Failure correlation was first proposed by Epler and has been studied in depth by many scholars.Marshall et al. 26 first proposed a multidimensional exponential distribution model, which is the basis for many subsequent failure correlation analysis models.Fleming 9 proposed the  factor model, which has the advantages of few parameters, simplicity and flexibility.However, this model is only applicable to second-order redundant systems.Vaurio et al. 31 proposed the basic parameter (BP) model, which can be applied to calculate the failure probability of each order directly from the known failure data.Fleming et al. 8 proposed the Multiple Greek Letter (MGL) model on the basis of the  factor model, which is widely used in failure correlation analysis.Mosleh et al. 27 proposed the  factor model, which is more accurate than the  factor model.Currently, polynomial multilevel binomial failure rate 15 and parametric mixture 18 models are also proposed for failure correlation analysis.Zhang et al. 38 proposed a unit conditional probability based on safety, failure, and hybrid information to approximate the reliability calculation problem for series, parallel, and voting systems when considering failure correlation.Although the above methods are able to deal with the failure correlation problem, they fail to deeply analyze the causes of system failure correlation and the interaction law.Xie et al. 35 proposed a system-level load-strength interference model, which not only avoids the assumption of "failure independence", but also does not rely on the correlation coefficient.This research can provide a theoretical basis for the establishment of reliability models for similar engineering systems.
nonlinearity and local response mutation.The maximum entropy method 19,20,47 is a method to approximate the output response probability density function.The saddle point approximation (SPA) method 4,45 can solve the probability density function (PDF) of the output response quickly and accurately.In addition, the hybrid dimension reduction method 37 can still analyze the mechanism reliability.The Pearson frequency curve method based on the first four moment is to approximate the PDF by the Pearson distribution family 25,40,41,42.All these types of methods mentioned above have achieved good results in different reliability problems.Currently, the majority of research centers on the dynamic attributes of deployment mechanisms, with limited attention to their reliability evaluation.Fewer still delve into the reliability assessment of folding wing deployment mechanisms.Gao et al. 12 analyzed the reliability of deployment synchronization.However, this study does not consider not only the mechanism joint clearance but also the failure correlation under external random aerodynamic loads.Pang et al. 29 calculated the deployment performance and impact resistance reliability of the folding tail system by the response surface method and MCS.However, the influence of joint clearance and the number of deployments are not considered.Pang et al. 30 also conducted an in-depth study on the synchronization reliability.But this method does not consider the correlation caused by load-sharing characteristics.Wang et al. 32 considered the failure correlation of the folding wing system and solved the deployment reliability through a system-level load-strength interference model.Nonetheless, this study did not analyze the synchronization, and the conditional deployment time distribution assumed normal, limiting applicability.

( 1 )
The deployment dynamic model of the folding wing mechanism with joint clearances is established and solved.(2) The reliability models considering failure correlation are established which categorized as with or without synchronization.(3) Proposing estimation methods for each reliability model, the efficiency and accuracy of the proposed methods are verified by examples.In addition, this paper also studies the reliability variation trend when the random variable distribution parameters are different.It can provide theoretical guidance and data support for the design, manufacture, and work of similar folding wing deployment mechanisms from the perspective of reliability.The organization of this paper is as follows.Section 2 presents the establishment and solution process of the dynamic model of the folding wing mechanism with joint clearances.
position vectors of the journal center and the bearing center in the local coordinate system, respectively.

Fig. 3 .
Fig. 3. Relative position of bearing and journal.When the journal and the bearing collide with each other, as shown in Figure4, define the unit vector  as the normal vector
where,   and   are the modulus of elasticity of the bearing and journal, respectively.  and   are the Poisson's ratio of the bearing and journal, respectively.  and   are the radii of the inner circle of the bearing and the outer circle of the journal, respectively.

Fig. 5 .
Fig. 5.A set of folding wing mechanism with joint clearances.
the obtained data shows that   and  of the flight vehicle in these airspaces follow the normal distribution.In this study, according to the task description and regulations given in the engineering project, the flight vehicle will complete multiple missions, not only that, in multiple missions, the flight vehicle needs to continue to work in the same airspace, which means that in the process of completing multiple missions, the folding wing mechanisms on the flight vehicle will be deployed in the same airspace.However, during the first mission, the airspace in which the flight vehicle is operating is random, and therefore   and  of the vehicle are random during the first mission.Upon completion of the first mission, the airspace in which the vehicle operates can be determined, which in turn allows for the determination of   and  under subsequent missions.That is,   and  during the first mission are the observation values under the respective distributions, while   and  during the subsequent missions are equal to the observation values.Let  |  ,  ,  = 1,2, ⋯ ,  be defined as the deployment time of the ith set of folding wing under the fixed   and  during the jth deployment event.By arranging  |  ,

𝑑𝛿𝑑𝑉 𝑞 ( 28 )
(  )   ,   () [   |  , (|  , )] 4 Considering the uncertainty associated with the actual number of completed missions  during the designated operational period, it is typical to establish a predetermined total number of flight missions   to be accomplished.The probability of successfully completing each individual mission is denoted by   , and the outcomes of mission completion are mutually independent.Consequently, the actual number of completed missions  in   planned flights follows a binomial distribution, which can be expressed as follows: { = } =        (1 −   )   −  = 0,1,2, ⋯ ,   (29)

.
Additionally, numerical simulation techniques are also applied.The Pearson distribution family allows the representation of the distribution's parameters as functions of the first four moments.Let  = () be the performance function, then the mean   , standard deviation   , skewness  3 and kurtosis  4 of  are: {   = ∫ ()   = √∫( −   ) 2 ()The PDF  of the standardized variable   satisfies the given differential equation 40,41,42: The coefficients , ,  and  are related to the first four moments.

Fig. 10 . 4 .
Fig. 10.Reliability assessment process with synchronization.4.Examples of reliability analysisIn this section, the first three examples are used to validate the accuracy of the proposed method by MCS.It should be noted that obtaining high-precision dynamic responses requires a significant amount of simulation time.Consequently, using MCS to compute the reliability in the fourth and fifth examples is not practical.Furthermore, the purpose of the fourth and fifth examples is to explore the variation trends of the system reliability under different random variable conditions.The sixth example is to explore the variation law of the reliability considering the randomness of the folding wing lengths.So only the proposed method is used in the fourth, fifth and sixth

𝑋 3 ,
5 , 7 , 9 Normal 0.5 0.1 Defining the response threshold as  1 .The system reliability at one working time is shown in Figure 12(a).When the system works  times, the system works for the kth ( 2 k  ) time and the observations of  1 are the same as for the first time.The calculated results of system reliability are shown in Figure 12(b).As  increases, the dispersion of the system response

5 𝜑
(b).As  increases, the reliability and the system response dispersion become smaller.Introducing   to describe the difference in output angles.The synchronization threshold is  2 .The system reliability curves for  = 2 and  = 5 are given in Figure14(c).A higher value of  2 leads to higher reliability.With the continuous increase of  2 , the increment of reliability becomes smaller until it reaches a plateau.The MCS results also confirm the accuracy of the proposed method in all cases.The sample sizes for both methods are listed in Table5.The relative errors compared to the MCS are listed in Figure14(d)-14(e).

Figure 15 (
Figure 15(a) depicts the reliability under  ( = 2,5,8,10) flight missions.As  increases, the reliability and system response dispersion decreases.The case where synchronization is included is shown in Figure 15(b).When  is greater than 0.57 s, the size of  becomes the main factor influencing reliability.When  is less than 0.545 s, the size of  becomes the main factor.Figure 15(c) presents the reliability for different values of , , and .When  is greater than 0.57 s, the larger the value of  is, the denser the reliability curves corresponding to different times  are.Considering the uncertainty of , as shown in Figure 15(d),   is 0.5 and the expected number of planned flights is   (  = 2,4,6,8,10 ).For the given  and  ( ∈ [0.545,0.57]), the reliability increases with decreasing   .When  exceeds 0.57 s, a higher value of  results in denser reliability curves corresponding to different   .Figure 15(e) illustrates the reliability under different   , with   set to 10.

Fig. 15 (
Fig. 15(h).Differences between the failure independent case and failure correlation for Example 3.

4. 4 .
Reliability of deployment performance under mean value variationsThis section assesses system reliability changes under varied mean values of random variables.The mean values listed in Table8.Figure16(a), 16(c), 16(e), 16(g) and 16(i) depict the reliability without synchronization for   ,  ,   ,   and   .
Fig. 17(d).Reliability curves considering both synchronization and randomness of  for conditions 13 and 14.

Figure 18 (
Figure 18(a) depicts the reliability under  ( = 2,5,8,10) flight missions.As  increases, the reliability decreases.The case where synchronization is included is shown in Figure 18(b).It can be found that the reliability curves gradually remain in a stable state when  is greater than 0.47s.At a constant value of the number of deployments, the larger the  is, the higher the reliability is.At a constant value of , the more the number of deployments, the lower the reliability.In addition, the final reliability difference between the two operating conditions gradually decreases as the  grows.Obviously, these laws are consistent with the previous numerical examples.The differences between the failure independent case and failure correlation case are shown in Figure18(c).The standard deviations of the common cause random variables   and  are 25.It is clear that the greater the dispersion of the common cause random variables, the greater the difference, the larger the error in the results under the assumption of independence.The other laws are the same as in Example 3.
Subsequently, under the consideration of failure correlation, two reliability models for the deployment performance are developed: one without considering deployment synchronization and the other with deployment synchronization taken into account.Finally, the proposed methods are applied to solve the aforementioned reliability models, and their validity is verified through three illustrative examples.Additionally, an analysis of the system reliability variation under different distribution parameters of random variables is conducted.The conclusions are as follows: 1.The proposed methods are employed to calculate the first three cases.Remarkably, the results from MCS are found to be Nomenclature   ,   the generalized coordinates of the center of mass for the journal and the bearing components   ,   the transformation matrices for the journal and bearing components   ′ ,   ′ the position vectors of the journal and bearing centers in the local 1 ,  1 ) , ( 2 ,  2 ,  2 ) and ( 3 ,  3 ,  3 ) , respectively.The crank and connecting rod have lengths of  1 and  2 .The distance between the slider and the X-axis is denoted as  .Considering the

Table 1 .
Simulation parameters of folded wing mechanism.

3. Reliability analysis of folding wing mechanism system 3.1. System reliability analysis without considering deployment synchronization 3.1.1. Reliability model
,  ,  and   are treated as normal random the clearance sizes obey the normal distribution.It is important to note that   ,  and   for each set of folding wing are independent and identically distributed.Defining   ( = 1,2,3,4) as the actual deployment time for the ith set of folding wing.We can derive the following expression:   = (  , ,   ,   ,   )  = 1,2,3,4 is solely determined by   ,   and   , then the failure events are considered to be mutually independent.The system conditional reliability  |  , can be determined as:  |  , = ∏  { |  , ≤ }  |  , = (  ,   ,   |  , ) is the deployment time of the ith set of folding wing under the condition that   and  are   = ∫    (  )  () [∏  { |  , ≤ } 3,4)are collectively influenced by   and  .The introduction of   and  results in interdependencies among the failures of the folding wings.When   and  are constant, (  )  () [∏ ∫ ℎ   |  , (  |  , ) ) (  |  , ) =  [   |  , (  |  , )]   |  , (  |  , ) (26) where,    |  , (  |  , ) represents the conditional cumulative distribution function (CDF) of  |  , . ℎ 1 and   2 denote the integration weights.  1 and   2 represent the integration nodes.   and   are the means.   and   are the standard deviations.
For    |  , , it can be solved using SPA.Let  = (), an approximate method 16,46 can be used to estimate the cumulant-generating function (CGF) of  , given by the (  )  () [ {  |  , ≤ ,   |  , − where,  is the given difference value threshold.Let  |  , =   |  , −   |  , ,  |  , =   |  , .When   and  are fixed, the conditional reliability  |  , *  of the folding wing system for  flight missions can be expressed as follows: When the flight vehicle performs  missions, it is repeated  times to generate the sample set of  |  , .The sample set is

Table 2 .
Random variables and distributed parameters in example 1.

Table 3 .
The sample sizes for both methods in Example 1.

Table 4 .
Distribution parameters for each rod length.

Table 5 .
The sample sizes for both methods in Example 2.

Table 6
Distribution parameters of random variables for folding

Table 7 .
The sample sizes for both methods in Example 3.

Table 8 .
Mean values of random variables under different operating conditions.

Table 10 .
Distribution parameters of random variables for folding wing system in example 6.  1 , 2 , 3 , 4 N Normal 1100 10  1 , 2 , 3 , 4  1 , 2 , 3 , 4 mm Normal 1.5 0.05  11 , 12 , 13 , 14 mm Normal 1500 2  21 , 22 , 23 , 24 mm Normal 1300 2 ̇  , ̇The velocities of contact points for the journal and bearing   , The normal and tangential projections of the relative contact velocity   restitution coefficient  stiffness coefficient   ,   the normal contact force and friction force   ,   the contact resultant forces for the bearing and journal components    ,    the moments for the bearing and journal components   the peak of the driving force   air density  0 air drag coefficient  1 ,  2 the thickness of the main wing and auxiliary wing  1 ,  2 the lengths of the main wing and auxiliary wing   flight speed  pitch angle  1 ,  2 the widths of the main wing and auxiliary wing  3 the volume of the slider  1 ,  2 ,  3 the densities of the main wing, auxiliary wing, and slider  1 ,  2 Elastic modulus of aluminum alloy and steel  1 ,  2   ( = 1,2,3,4) the actual deployment time for the ith set of folding wing   ( = 1,2,3,4) the deployment performance reliability for the ith set of folding wing  |  , the system conditional reliability when   and  are constant values  |  , the deployment time of ith set of folding wing when   and  are fixed   the system reliability    |  , (  |  , ) the conditional cumulative distribution function of  |  ,   * the system reliability with synchronization  |  , *  the system conditional reliability for  flight missions when   and  are fixed with synchronization