Durability analysis of large and medium-sized unmanned aerial vehicles based on load spectrum and probabilistic fracture mechanics

The analysis of unmanned aerial vehicle (UAV) durability is crucial for cost-effective repair and maintenance. Assessing the degree of damage and predicting the economic lifespan of a UAV is a critical component of its design and use. To evaluate the durability of large and medium-sized UAVs, data is gathered, and simulations are performed under real-world working conditions. The analysis includes creating a crack growth model for small cracks and determining the distribution of crack sizes, estimating the distribution of Time to Crack Initiation (TTCI), and calculating the general Equivalent Initial Flaw Size (EIFS) distribution. The results obtained from these analyses meet the UAV’s flight hour requirements and help to predict its economic life. This prediction is also beneficial for managing and maintaining the UAV’s life cycle.


Introduction
The analysis of aircraft durability is closely linked to safety life and damage tolerance analysis, although there are some key differences.The primary objective is to investigate the natural initiation time of fatigue cracks in typical structures while also considering the structures' economic feasibility and ease of maintenance.The durability analysis of typical structures involves determining the initial defect expansion necessary to predict the economic life until structural failure.This analysis is crucial for ensuring the safety and longevity of aircraft structures [1][2][3][4].
Currently, there exist three primary methods for assessing the durability of aircraft structures.The Crack Initiation Approach (CIA) is a method for analyzing the durability of aircraft structures, which has evolved from traditional fatigue analysis.This approach is based on the "crack initiation life" P-S-N curve family, which corresponds to the initiation of cracks in structural details and reaching the economic repair limit.The method also utilizes the linear cumulative damage theory (Miner theory) for life estimation under spectrum load.By establishing the functional relationship between the degree of damage and time, it can predict the economic life of the structure.However, this approach cannot calculate the sequence effect of load in the spectrum load.He et al. [5] employed this method to investigate the durability of aircraft fleet structures.
The Deterministic Crack Growth Approach (DCGA) is a method for assessing aircraft structure durability, which is based on the calculation method of long crack propagation life in damage tolerance design.This approach utilizes an assumed initial flaw size, a relatively small crack growth rate, and a IOP Publishing doi:10.1088/1742-6596/2764/1/012015 2 crack growth calculation program.The Initial Flaw Quantity (IFQ) of structural details is based on an assumed initial flaw size and the corresponding relatively small crack growth rate.However, it is unable to synthesize the damage of each stress zone and provide the function of structural damage degree changing with time.Zhou et al. [6] employed this method to carry out fatigue tests on typical aviation aluminum alloy riveted box structures.
The Probabilistic Fracture Mechanics Approach (PFMA) is a method for assessing aircraft structure durability, which can consider the effect of load sequence and integrate the damage of each stress zone to provide the function of structural damage degree changing over time.Cai et al. [7] have utilized this approach to analyze the durability of a key component of the aircraft structure.This method is unique because it can account for the uncertainties associated with fatigue crack growth, the variability of the material properties, and the uncertainties related to the applied loads.As a result, it provides a more accurate assessment of the structural life and helps ensure the safety of aircraft structures.
In this paper, the Probabilistic Fracture Mechanics Approach (PFMA) is used to analyze the life and durability of a large fixed-wing UAV.The drone is a typical reconnaissance and combat integrated drone layout as shown in Figure 1.The drone system has a take-off weight of 2,000 kilograms, a flight time of 40 hours, and a payload capacity of 500 kilograms.It employs PFMA to establish the distribution of the Equivalent Initial Fracture Size (EIFS), which describes the Initial Fatigue Quality (IFQ) of the structure based on the load spectrum of the UAV during the flight test and the Time to Crack Initiation (TTCI) data set obtained.The principles of probabilistic fracture mechanics are then applied to provide the functional relationship between damage degree and time.Using the specified damage degree requirements (allowable crack exceedance and reliability), the economic life of the structure is predicted.To carry out the durability analysis by using the economic life criterion, the finite element method is used to calculate the structural data under three stress levels, based on real flight data.

Data collection
The airborne vibration environment recording equipment is used to record the vibration data of the whole machine during flight.The recording device weighs 0.5 kilograms and can be directly installed in the internal space of the body.It has a sampling rate of 500-3000 Hz, a range of 100-200g, and uses a low-pass Butterworth 5-level filter.The environmental data collected in this flight subject is shown in Fig. 2.After data processing, the PSD curve is drawn as in Fig. 3.

Simulation results
Based on the strength calculation finite element model, the working conditions are established.The frequency range is set, and the coupled mass matrix method is used for eigenvalue analysis to obtain the stress level of the whole plane.Among them, the main load-transferred area of the wing, the front fuselage, and the middle and rear fuselage are the most serious.The stress nephogram of the main loadtransferred area of the wing is shown in Figure 4, the stress nephogram of the load-transferred area of the front fuselage is shown in Figure 5, and the stress nephogram of the load-transferred area of the middle and rear fuselage is shown in Figure 6.Table 1 shows the typical stress values of components in each load-transferred area.
Table 1.Typical stress values of parts.The front beam of the wing The bolt hole of the front beam edge strip 168.7 176.9 186.8 The rear beam of the wing The According to the calculation results, in the main Load-transferred area of the wing, the stress of the joint flange fastening hole of the wing front beam joint is large; in the load-transferred area of the front fuselage, the stress of the bolt hole of the front lifting and retracting cylinder support is large; in the load-transferred area of the middle and rear fuselage, the stress of the flange bolt hole of the main lifting buffer joint is large, and the load of the tail wing load-transferred path is small, which is not considered here.Only these three details are considered in the durability analysis.
4 Initial fatigue quality evaluation of structure

Crack growth model and crack size distribution of small cracks
In the small crack stage, the crack growth rate [8] can be expressed as Equation ( 1): Where: , Q b is the crack growth rate parameter;   a t is the crack size at the time of t.
For the parameters Q and b in the above equation, the logarithm taken on both sides is determined by linear regression.In the general durability analysis, it is recommended to use the case of b = 1 for data processing.The fatigue test is conducted under the i-th stress level, and L-effective fractures are obtained.Assuming that the k-th effective fracture has m pairs of (a, t) data, the crack growth rate parameters of this fracture Q k can be obtained by the least square method as expressed in Equations (2-4).
  ) In the equation, the reference crack size a r is a selected crack size which is easy to observe.The lower limit needs to ensure that the crack can be observed reliably.Its upper bound guarantees the validity of the selected crack growth equation.The value of a r has a direct impact on the TTCI value of crack formation time.The larger the a r is, the longer the TTCI is.Therefore, several discrete reference values should be selected within the value range of a r , and the a r value should be selected by optimizing the parameters of TTCI distribution.We make T=ε to be the lower bound of TTCI, and then the upper bound x u of EIFS can be obtained as expressed in Equation (5).
x a e    (5)

TTCI (Time to Crack Initiation) distribution parameter estimation
For fastening holes, the random variable TTCI obeys the three-parameter Weibull distribution under the specified load spectrum and a r .If t is used to represent the value of the random variable TTCI, as expressed in Equation ( 6), its probability density function is: The cumulative distribution function is as expressed in Equation ( 7): The parameters required to obtain TTCI distribution under various stress levels through preliminary calculation are shown in Table 2 7, which is the TTCI distribution of the main lifting buffer joint under the loadtransferred area of the middle and rear fuselage.

General EIFS (Equivalent Initial Flaw Size) distribution
EIFS is the hypothetical initial flaw size contained in the structural details before use, so it is not directly available.EIFS is a random variable.Under the specified load spectrum stress level and reference crack size, EIFS is a function of TTCI, so the EIFS distribution can be derived from the TTCI distribution.
The EIFS distribution can be derived from the TTCI distribution [9], and the value of EIFS is represented by x, so the probability density function of the EIFS distribution is as Equation ( 8): The cumulative distribution function [9] is shown in Equation ( 9): The criterion of parameter optimization is to determine the general EIFS distribution parameters according to the TTCI data by selecting different values of a r , x u , so that the cumulative distribution probability of the TTCI value corresponding to the obtained values of Qβ, α are as close as possible to the fracture observation results, that is to make Sum of Squares for Error (SSE) between the predicted value and the fracture observation results to be the smallest.
The EIFS distribution can be obtained from the TTCI distribution.The general EIFS distribution parameters are α=5.0126,Qβ=8.699, and b=1.We make the upper limit of the EIFS distribution x u =0.7 and refer to the crack size a r =1.08 and economic repair size a e =0.9, then the general EIFS distribution is shown in Figure 8.

Crack master curve during service life
For a given stress area, the Equivalent Initial Flaw Size (EIFS) describing the detail Initial Fatigue Quality (IFQ) expands with time t.At a given time t, the size distribution of the detail group in the stress area is a distribution with the same shape as the EIFS distribution, but the minimum and maximum values are different (greater than the minimum and maximum values of the EIFS distribution), as shown in Figure 9.The reference crack size x 1 is specified.At time t, the probability that the detail crack size of a given stress area i exceeds x 1 (called the probability of crack exceedance number) is p(i, t).At time t, the value of the EIFS exceedance distribution function corresponding to the equivalent initial flaw size y 1i (t) of details with crack size equal to x 1 at time t=0 is also equal to p(i, t).
Obviously, for different time t, the probability p(i, t) of crack size exceeding x 1 is different, and the corresponding y 1i (t) is also different.Therefore, y 1i (t) is a function of time t and is related to the size of reference crack size x 1 .When x 1 is specified, y 1i (t) decreases with the increase of time t.The timedependent curve of y 1i (t) in a given stress zone is called the Service Crack Growth Master Curve (SCGMC).
According to the general EIFS distribution parameters calculated above, the economic repair limit of 0.9 mm is made, and the maximum stress corresponding to three stress levels σ and EIFS master curve parameter Q values are shown in Table 3.
Then the SCGMC of each stress zone is as Equation (10).

Number of crack excesses
The reference crack size x 1 is specified.At time t, the probability that the detail crack size of a given stress zone i exceeds x 1 is called the probability of crack exceedance number, which is expressed in p(i, t), and its mathematical expression is as Equation ( 11): The p(i, t) corresponding to the specified time t can be calculated from the SCGMC of the stress region and the general EIFS distribution function expression of the detail group.The method is as follows: the specified time t, the selected economic repair limit, and the SCGMC parameters of the given stress region (i) are brought into the SCGMC expression of the stress region [10] to calculate y 1i (t), taking b=1 [8], then as shown in Equation (12).
Its mathematical expectation (average crack exceedance) and standard deviation are respectively as expressed in Equation (13) and Equation ( 14): ,

Number of crack excesses of structural detail group
The specified detail group of the structure contains several stress areas (j=1,2,…,m).The number of details of the crack size a e in the detail group of the structure, and the crack exceeding number of the detail group are normal variables, and its mathematical expectation and standard deviation are as Equations (15-16): Taking t=10000 hours, the crack exceedance probability p(i, t) and the mean and standard deviation of the crack exceedance number in each stress region can be obtained from the y 1i (t) calculated by the above equation in Table 4.
The average value of the damage degree is equal to the average value of the number of excesses of the cracks in the detail group, as expressed in Equation (20).The load-transferred area of the middle and rear fuselage with the highest stress level is taken for analysis, and the allowable damage degree is the crack exceeding number i=2.0 corresponding to the reliability R=95%.It is predicted that the minimum service life of the middle and rear fuselage loadtransferred area is 11820 flight hours.Table 5 shows the flying hours in the main wing load-transferred area and the front fuselage load-transferred area by the same method.This paper presents a method for predicting the economic life of large and medium-sized UAVs.The load spectrum is obtained by collecting data on typical cruising conditions of UAVs.This load spectrum is then used as input for simulation calculation, which supports the initial fatigue quality assessment.
The following conclusions are drawn:  The analysis input uses the static model, with the main force transfer areas of the UAV wing, front middle fuselage, and rear fuselage identified as the calculation targets;  The TTCI distribution is calculated for the middle and rear fuselage load-transferred areas under three stress levels, and the EIFS distribution is established;  By evaluating the damage degree, the economic life of this type of UAV is predicted.The results demonstrate that the UAV structure can meet the requirements of flight hours.

Figure 1 .
Figure 1.Schematic figure of a certain type of drone.

Figure 2 . 3 Figure 3 .
Figure 2. Schematic figure of a certain type of drone.

Figure 4 .
Figure 4. Stress nephogram of the main load-transferred area of the wing.

Figure 5 .
Figure 5. Stress nephogram of the load-transferred area of the front fuselage.

Figure 6 .
Figure 6.Stress nephogram of the load-transferred area of the middle and rear fuselage.Table1shows the typical stress values of components in each load-transferred area.Table1.Typical stress values of parts.

Figure 7 .
Figure 7. Distribution of Time to Crack Initiation (TTCI) of the main lifting buffer joint.
21) D m and D R curves are shown in Figure 10.

Table 2 .
. Typical stress values of parts.Then the TTCI distribution diagram under each stress level can be obtained, as shown in the following Figure

Table 3 .
Comparison table of σ and Q 16)5.4 Damage degree evaluation and economic life predictionDamage degree is a quantitative measure of the durability damage of a structure when it reaches the specified time t.It is usually expressed by the crack exceedance percentage of the structural detail group.
The damage degree is a function of time t.Its expression is as Equations (17-19):

Table 4 .
Calculation of crack exceedance probability, average valueand standard deviation of crack exceedance.The upper bound of damage degree corresponding to reliability R=95% is equal to the number of detail group crack excesses corresponding to reliability R=95% as expressed in Equation (21).

Table 5 .
The economic life of each region.