Random Vibration Fatigue Life Analysis of Airborne Electrical Control Box

To study the influence of random vibration on the fatigue life of airborne equipment, an aircraft electrical control box was selected as the research object. First, finite element software was used to model the dynamics of the airborne electrical control box to investigate its mode frequencies. The accuracy of finite element simulations was verified by performing mode experiments. Second, the mode superposition method was used to analyze the flight direction (X axis), side direction (Y axis), and altitude direction (Z axis) random vibration responses of the electrical control box. The analysis results were combined with the Miner linear cumulative damage criterion and the Gaussian-distribution Steinberg method to estimate the fatigue life of the electrical control box in the three directions. Finally, the calculation results were verified by performing the random vibration durability test on the electrical control box. The finite element mode analysis results show good agreement with the vibration experiment results, and the maximum error is 13.4%, indicating that the finite element model established in this paper is acceptable. The fatigue life of the electrical control box in the three axes meets the user requirements, and random vibration along the side direction (Y axis) has the greatest impact on the fatigue life, which is consistent with the results of the actual experimental data. The research method can be extended to predict the fatigue life of other airborne equipment and thus has practical significance for structural design and reliability analysis of airborne equipment.


Introduction
An aircraft is subjected to complex and volatile aerodynamic loads, engine loads, and inertial loads during flight. The vibration caused by the complex mechanical setup is mostly random in nature. The long-term effect of load leads to the fatigue failure of airborne equipment. This phenomenon is more pronounced when the vibration frequency of the aircraft and airborne equipment overlap. Therefore, the fatigue problem of structural vibration of airborne equipment has received increased attention in recent years.
The stability and safety of aircraft flight depend to a certain extent on the reliability of airborne equipment. Accurate evaluation of random vibration of airborne equipment is one of the key steps to ensure its normal operation. Random vibration is the motion of a structure excited by a random input. The mathematical theory of random vibration is used for the realistic modeling of structural dynamic systems [1]. By using a group of 7075-T651 aluminum alloy specimens, Wu et al. [2] studied the practicability of the fatigue damage and fatigue life estimation method under random load and verified the method by performing a series of low-cycle fatigue tests. An effective strain damage (ESD) model has been proposed to predict the fatigue life under random load. Tovo [3] proposed a rain flow damage assessment method in the frequency-domain by performing an extensive literature review. For the analytical solution of the expected damage, in addition to the calculation program, an exact approximate solution of fatigue damage under Gaussian load has been

Basic Theory and Solution Method
The fatigue analysis approach of random vibration can be divided into time-domain and frequency-domain approaches. In the time-domain approach, uses a technique called rainflow cycle counting to decompose a variable amplitude time signal of stress into fatigue cycles, and then calculate the fatigue life combined with the stress-life (S-N) curve of the material. A large amount of storage space is used for the processing and calculation time-domain signals in the finite element analysis. This is not convenient for engineering applications.
The frequency-domain approach is established on the power spectral density (PSD) of stress. It is easy to calculate and has strong engineering applicability. The variance, amplitude probability distribution and peak frequency of random stress signal can be obtained by PSD function, and then the cumulative damage can be calculated. Therefore, in this study, the frequency-domain method was used.

Linear Cumulative Damage Theory
According to the fatigue cumulative damage theory, the cyclic action of the load causes structural fatigue damage. The fatigue damage under different stress amplitudes is affected by the number of cycles under this amplitude and the total number of cycles.
Miner theory [11] holds that the fatigue damage of the structure is independent of each other under various stresses, and the total damage can be obtained by the linear superposition of independent fatigue damage. According to the linear cumulative damage theory, the cumulative fatigue damage of components under multiple load cycles can be expressed as follows: where n s is the number of cycles when the stress amplitude is s and can be expressed by the probability density function p(s) of the stress amplitude. The number of failure cycles N s with stress amplitude s can be determined using the S-N curve equation: where E(p) is the expected peak frequency of the random stress response signal, T is the action time of the random response, and m and c are the S-N curve constants related to the characteristics of structural materials. By substituting Equations (2) and (3) into Equation (1), the linear fatigue cumulative damage equation is obtained: When the cumulative damage of the cycle reaches the structural fatigue damage threshold D under various stress levels, the structure will suffer from fatigue failure. Numerous random vibration fatigue tests in engineering applications have demonstrated that the fatigue threshold set by the Miner fatigue criterion is close to 1. The formula is simple in principle and can provide an acceptable accuracy range for engineering applications.

Frequency-Domain Analysis
The steps involved in the frequency-domain analysis are as follows ( The frequency-domain analysis method of the PSD function can be divided into narrow-band and broad-band methods. Bendat [12] proposed a classical narrow-band method. When the effective bandwidth ε < 0.3-0.35, the random process can be regarded as a narrow-band process. The peak and amplitude of the stress cycle response are assumed to be equal; that is, the mean value of   [12] proposed a classical narrow-band method. When the effective bandwidth ε < 0.3-0.35, the random process can be regarded as a narrow-band process. The peak and amplitude of the stress cycle response are assumed to be equal; that is, the mean value of the stress cycle is usually close to zero. The peak probability density function of the narrow-band response random process can be expressed as follows: where σ 2 is the variance of the random process and s is the stress. From Equations (4) and (5), the fatigue cumulative damage formula of narrow-band method can be obtained: where Γ(x) is a gamma function that is usually defined as: The peak frequency is similar to the zero-crossing frequency; thus, it is assumed that E(p) = f 0 is the center frequency of the narrow-band peak.

Broad-Band Method
In the broad-band random process, due to the large difference between peak frequency and crossing frequency in unit time. If the traditional narrow-band method is still used to calculate the fatigue damage will make the result larger. Therefore, Wirsching [13] introduced the correction factor λ to compensate the calculation results of narrow-band process fatigue damage by using the approximate narrow-band method. When the correction factor λ is substituted into Equation (6), the fatigue damage formula for the broad-band stationary process is obtained: where correction factor λ is a function of exponent m of the effective bandwidth ε and S-N curve: The irregular factor γ varies between 0 and 1. When the stress signal approaches the narrow band process (sine wave γ = 1), it approaches 1. When the stress signal is close to the white noise process, it is close to 0.
The n-order spectral moment of the stress power spectral density S(f ) can be expressed as follows:

Steinberg Method
Generally, the stress amplitude and average stress of the random response vary randomly. Steinberg [14] proposed a three-interval method based on Gaussian-distribution and Miner linear cumulative damage theory to analyze the fatigue life from the perspective of probability. The stress can be divided into three ranges in terms of probability, as shown in Figure 2 and Table 1.

Steinberg Method
Generally, the stress amplitude and average stress of the random response vary randomly. Steinberg [14] proposed a three-interval method based on Gaussian-distribution and Miner linear cumulative damage theory to analyze the fatigue life from the perspective of probability. The stress can be divided into three ranges in terms of probability, as shown in Figure 2 and Table 1.  Most of the time, the stress is (−1σ, +1σ), and the probability is 68.3%. Outside (−3σ, +3σ), the probability is 0.27%, which is very small. Thus, the time greater than 3σ will cause almost no damage to the structure. Therefore, the overall fatigue damage can be simplified from Equation (1) to Equation (13) by using the Miner linear cumulative damage theory: V + a represents the average frequency, and T is the action time of the random response.
Most of the time, the stress is (−1σ, +1σ), and the probability is 68.3%. Outside (−3σ, +3σ), the probability is 0.27%, which is very small. Thus, the time greater than 3σ will cause almost no damage to the structure. Therefore, the overall fatigue damage can be simplified from Equation (1) to Equation (13) by using the Miner linear cumulative damage theory: When D = 1, the fatigue life of structural components can be expressed as follows: Fatigue damage to structural components when D ≥ 1. N iσ is the number of cycles under the iσ stress value obtained according to the S-N curve (i = 1, 2, 3).

Finite Element Modeling and Analysis
The finite element model of the electric control box was established using the ANSYS Workbench 2020 R1, and the fatigue life analysis was performed. The analysis process is showed in Figure 3. under the iσ stress value obtained according to the S-N curve (i = 1, 2, 3).

Finite Element Modeling and Analysis
The finite element model of the electric control box was established using the ANSYS Workbench 2020 R1, and the fatigue life analysis was performed. The analysis process is showed in Figure 3.

Model Description of The Electrical Control Box
The electrical control box was installed on the aircraft by using lugs and bolts ( Figure  4). To facilitate the preprocessing for model simplification, meshing, and element definition, the 3D model of the electrical control box was established using CATIA.

Model Description of The Electrical Control Box
The electrical control box was installed on the aircraft by using lugs and bolts ( Figure 4). To facilitate the preprocessing for model simplification, meshing, and element definition, the 3D model of the electrical control box was established using CATIA.

Model Simplification
The airborne equipment was designed to meet the requirements of machining and assembly. To obtain accurate analysis results, we ensured that the load transmission path did not change, the connections of the parts were consistent with the actual situation, and

Model Simplification
The airborne equipment was designed to meet the requirements of machining and assembly. To obtain accurate analysis results, we ensured that the load transmission path did not change, the connections of the parts were consistent with the actual situation, and the integrity of the main structure was not compromised [15]. Accordingly, the following simplifications were performed before modeling.
(1) Treatment of transition fillets: Too many fillets lead to the generation of a complex mesh and increase the amount of meaningless computation. Therefore, the circular angle that exists in the non-stress concentration region and the radius r ≤ 3 mm was directly ignored, and the circular angle that affects the analysis results in the stress concentration region was retained. When the fillet radius was 3 mm < r ≤ 8 mm, the transition was divided by a row of grids. The transition was divided by two rows of grids when the fillet radius was 8 mm < r ≤ 15 mm. (2) Treatment of holes: Excessive perforation of the structure leads to uneven meshing and stress concentration which not only increases the amount of calculation but may also lead to divergence of simulation results. Thus, the non-installed hole with a radius of r ≤ 2 mm was ignored, and the hole was located in the stress concentration area left behind. When the radius of the hole was 2 mm < r ≤ 3 mm, the tetrahedral mesh with a central node was used. (3) Simplification of the bolt connection: The bolt connection is the main fixation way employed for the electrical control box and the aircraft body. The screw thread structure is complex, and the bolt model exhibits a cumbersome nonlinear contact problem, which does not reflect the load and load transmission form of the main structure and results in excessive calculation time. Therefore, the bolt connection was simulated using a common node with the connector.
By using the aforementioned methods, under the premise of satisfying the calculation and analysis accuracy, the simplified electrical control box model was obtained to improve the modeling efficiency.

Finite Element Modeling
For the simplified electric control box, considering the complex geometry of the front and lower panels and the uneven thickness of the component, the solid element was used to discretize the component model, and the tetrahedral mesh with good adaptability was selected. The unit size was 2 mm. Because the upper cover was fabricated using the stamping process, the thickness was much less than the size in the other two directions, so the shell element is used for discrete. To study the strength and stiffness of the shell structure of the electric control box by performing random vibration analysis, the mass element used for the light electronic circuit board in the box was simplified. The finite element model established in this study is depicted in Figure 5.
Tetrahedral and shell elements with intermediate nodes were adopted in the model, with a total of 71,410 nodes and 142,873 elements. The electrical control box housing was made of PA6 (upper cover) and 7075 aluminum alloy (fixed base plate), and the mechanical properties are presented in Table 2. used for the light electronic circuit board in the box was simplified. The finite element model established in this study is depicted in Figure 5.  Table 2.

Finite Element Modeling
In the actual working process of the electric control box, the bottom of the box is in a constraint state. Therefore, four fixed holes at the bottom of the box were constrained in the mode analysis, as shown in Figure 6. The finite element software was used to perform a constrained mode analysis on the electrical control box. The calculations revealed that the structure has 42 natural frequencies in the experimental frequency-domain (20-2000 Hz). Among them, the first and last natural frequencies are 251.5 and 1987.6 Hz, respectively. The frequencies of the first 10 modes were selected, as shown in Table 3.

Finite Element Modeling
In the actual working process of the electric control box, the bottom of the box is in a constraint state. Therefore, four fixed holes at the bottom of the box were constrained in the mode analysis, as shown in Figure 6. Tetrahedral and shell elements with intermediate nodes were adopted with a total of 71,410 nodes and 142,873 elements. The electrical control bo made of PA6 (upper cover) and 7075 aluminum alloy (fixed base plate), an ical properties are presented in Table 2.

Finite Element Modeling
In the actual working process of the electric control box, the bottom of constraint state. Therefore, four fixed holes at the bottom of the box were the mode analysis, as shown in Figure 6. The finite element software was used to perform a constrained mode electrical control box. The calculations revealed that the structure has 42 na cies in the experimental frequency-domain (20-2000 Hz). Among them, th natural frequencies are 251.5 and 1987.6 Hz, respectively. The frequencies modes were selected, as shown in Table 3. The finite element software was used to perform a constrained mode analysis on the electrical control box. The calculations revealed that the structure has 42 natural frequencies in the experimental frequency-domain (20-2000 Hz). Among them, the first and last natural frequencies are 251.5 and 1987.6 Hz, respectively. The frequencies of the first 10 modes were selected, as shown in Table 3. The first 10 mode shapes are shown in Figure 7.
electrical control box. The calculations revealed that the structure has 42 natural frequencies in the experimental frequency-domain (20-2000 Hz). Among them, the first and last natural frequencies are 251.5 and 1987.6 Hz, respectively. The frequencies of the first 10 modes were selected, as shown in Table 3.

Mode Frequency Testing
The experimental setup included sensors, a vibration test bench, and a control and acquisition system. The natural frequency test of the electrical control box was performed using the MPA409/M437A high-thrust high-frequency electrodynamic vibration table (ETS Solutions Ltd., Beijing, China) and the LMS Test Lab 14A software (Siemens, Munich, Germany). The details of the experimental equipment and the site are presented in Table  4 and Figure 8.

Mode Frequency Testing
The experimental setup included sensors, a vibration test bench, and a control and acquisition system. The natural frequency test of the electrical control box was performed using the MPA409/M437A high-thrust high-frequency electrodynamic vibration table (ETS Solutions Ltd., Beijing, China) and the LMS Test Lab 14A software (Siemens, Munich, Germany). The details of the experimental equipment and the site are presented in Table 4 and Figure 8.  Using the finite element calculation results, the acceleration sensors were affixed to the maximum displacement positions of each mode, and the first 10 order natural frequencies were measured and compared with the finite element results. The comparison results are shown in Figure 9. The first 10 natural frequencies obtained by finite element calculation exhibited good agreement with the test results (Figure 9), and the maximum relative error was 13.4%, which meets the requirements of engineering calculation, thereby indicating that the finite element model can be used for random response and fatigue life analysis.

Random Response Analysis
By using the finite element software and mode superposition method, the random vibration response of the electric control box was analyzed based on mode analysis.
The boundary conditions were not changed during the calculation. The applied random load spectrum is shown in Figure 10; the abscissa f represents the frequency (Hz), and the ordinate Wf represents the acceleration PSD (g 2 /Hz). The loading was performed in the X, Y, and Z directions. The calculated total acceleration root mean square ( ) was 8.42 g. Using the finite element calculation results, the acceleration sensors were affixed to the maximum displacement positions of each mode, and the first 10 order natural frequencies were measured and compared with the finite element results. The comparison results are shown in Figure 9. Using the finite element calculation results, the acceleration sensors were affixed to the maximum displacement positions of each mode, and the first 10 order natural frequencies were measured and compared with the finite element results. The comparison results are shown in Figure 9. The first 10 natural frequencies obtained by finite element calculation exhibited good agreement with the test results (Figure 9), and the maximum relative error was 13.4%, which meets the requirements of engineering calculation, thereby indicating that the finite element model can be used for random response and fatigue life analysis.

Random Response Analysis
By using the finite element software and mode superposition method, the random vibration response of the electric control box was analyzed based on mode analysis.
The boundary conditions were not changed during the calculation. The applied random load spectrum is shown in Figure 10; the abscissa f represents the frequency (Hz), and the ordinate Wf represents the acceleration PSD (g 2 /Hz). The loading was performed in the X, Y, and Z directions. The calculated total acceleration root mean square ( ) was 8.42 g. The first 10 natural frequencies obtained by finite element calculation exhibited good agreement with the test results (Figure 9), and the maximum relative error was 13.4%, which meets the requirements of engineering calculation, thereby indicating that the finite element model can be used for random response and fatigue life analysis.

Random Response Analysis
By using the finite element software and mode superposition method, the random vibration response of the electric control box was analyzed based on mode analysis.
The boundary conditions were not changed during the calculation. The applied random load spectrum is shown in Figure 10; the abscissa f represents the frequency (Hz), and the ordinate W f represents the acceleration PSD (g 2 /Hz). The loading was performed in the X, Y, and Z directions. The calculated total acceleration root mean square (G RMSin ) was 8.42 g. Figure 10. PSD curve of random excitation. Figure 11 shows the equivalent stress change of the electrical control box at the dangerous point in the X, Y, and Z directions; in the figure, the abscissa is the frequency, and the ordinate is the stress. The second-order mode in the X direction, the fourth-order mode in the Y direction, and the third-order mode in the Z direction exhibited the maximum stress, and the corresponding mode frequencies were = 258.3 Hz, = 377.7 Hz, and = 309.8 Hz, respectively. During the actual flight, random vibrations may cause the local equivalent stress of the component to be too large and may result in the premature plastic deformation of the component and fatigue failure during service. For the electrical control box, when the vibration frequency is 250-400 Hz, the stress is large; thus, this frequency range should be avoided during its operation.

Material Fatigue Life Curve of the Electrical Control box
The main material of the electrical control box is 7075 aviation aluminum alloy. According to the multi-axial fatigue test in Ref. [16], the American material test standard  Figure 11 shows the equivalent stress change of the electrical control box at the dangerous point in the X, Y, and Z directions; in the figure, the abscissa is the frequency, and the ordinate is the stress. The second-order mode in the X direction, the fourth-order mode in the Y direction, and the third-order mode in the Z direction exhibited the maximum stress, and the corresponding mode frequencies were f X = 258.3 Hz, f Y = 377.7 Hz, and f Z = 309.8 Hz, respectively.
Appl. Sci. 2022, 12, x FOR PEER REVIEW 13 of 19 Figure 10. PSD curve of random excitation. Figure 11 shows the equivalent stress change of the electrical control box at the dangerous point in the X, Y, and Z directions; in the figure, the abscissa is the frequency, and the ordinate is the stress. The second-order mode in the X direction, the fourth-order mode in the Y direction, and the third-order mode in the Z direction exhibited the maximum stress, and the corresponding mode frequencies were = 258.3 Hz, = 377.7 Hz, and = 309.8 Hz, respectively. During the actual flight, random vibrations may cause the local equivalent stress of the component to be too large and may result in the premature plastic deformation of the component and fatigue failure during service. For the electrical control box, when the vibration frequency is 250-400 Hz, the stress is large; thus, this frequency range should be avoided during its operation.

Material Fatigue Life Curve of the Electrical Control box
The main material of the electrical control box is 7075 aviation aluminum alloy. According to the multi-axial fatigue test in Ref. [16], the American material test standard During the actual flight, random vibrations may cause the local equivalent stress of the component to be too large and may result in the premature plastic deformation of the component and fatigue failure during service. For the electrical control box, when the vibration frequency is 250-400 Hz, the stress is large; thus, this frequency range should be avoided during its operation.

Material Fatigue Life Curve of the Electrical Control Box
The main material of the electrical control box is 7075 aviation aluminum alloy. According to the multi-axial fatigue test in Ref. [16], the American material test standard ASTM E2077 [17] was adopted. The fatigue test data are presented in Table 5 for the stress concentration factor K t = 2.4 and stress ratio R = −1. According to the experimental data, the S-N curve equation was obtained using the curve fitting method as follows: From Equation (15), the lgS-lgN curve of the material was obtained, as shown in Figure 12. ASTM E2077 [17] was adopted. The fatigue test data are presented in Table 5 for the stress concentration factor Kt = 2.4 and stress ratio R = −1. According to the experimental data, the S-N curve equation was obtained using the curve fitting method as follows: = 20.62 − 7.11 (15) From Equation (15), the lgS-lgN curve of the material was obtained, as shown in Figure 12.

Fatigue Life Analysis of the Electrical Control Box
The equivalent stress of the dangerous point of the electric control box as determined through random vibration response analysis is shown in Figure 13; the confidence interval and confidence level of the stress are presented in Table 6.

Fatigue Life Analysis of the Electrical Control Box
The equivalent stress of the dangerous point of the electric control box as determined through random vibration response analysis is shown in Figure 13; the confidence interval and confidence level of the stress are presented in Table 6. Based on the GJB150.16A-2009 [18] standard, simulation was carried out of long-life random vibration fatigue test. The total fatigue damage of the structure under random excitation load of 1.5 h (0.5 h in each direction) in X, Y, and Z directions was investigated. By using the Miner linear cumulative damage criterion and fatigue life Steinberg method, the fatigue life of the electric control box was estimated by substituting the stress of the dangerous point into Equation (15). The data [19] for the working environment of the electrical control box revealed that the average frequency of random vibration V + a = 18 Hz and vibration time T = 1.8 × 10 4 s. The cumulative damage was calculated using Equation (13).   Table 7. Its structural fatigue life is much higher than the 1.5 h required by GJB150.16A-2009 standard, and the total damage of the electrical control box is less than 1, indicating that the structural fatigue resistance of the electrical control box meets the design requirements. The vibration in the Y direction has the greatest influence on the fatigue life of the electric control box, followed by that in the Z and X directions. In the product design stage, attention must be paid to the strength of the maximum stress position of the electrical control box, such as the rear floor and the upper floor fixed bolts, to improve the fatigue strength of the electrical control box.

Experimental Procedures
The vibration test fixture in X, Y and Z directions of the electrical control box were de-signed and manufactured based on GJB150.16 A-2009 standard before the test, and in-stalled on the vibration test bench, as shown in Figure 14. (1) The random vibration function of the vibration table was debugged, and the electrical control box was fixed on the surface of the vibration table according to the installation requirements.
(2) Two calibrated acceleration sensors were attached to the fixture, as close as possible (1 ) The random vibration function of the vibration table was debugged, and the electrical  control box was fixed on the surface of the vibration table according to the installation  requirements. (2) Two calibrated acceleration sensors were attached to the fixture, as close as possible to the diagonal position of the fixed bolt of the electrical control box, for accurate two-point average random vibration control. The frequency response under each axis excitation is illustrated in Figure 15. (1 ) The random vibration function of the vibration table was debugged, and the electrical  control box was fixed on the surface of the vibration table according to the installation  requirements. (2) Two calibrated acceleration sensors were attached to the fixture, as close as possible to the diagonal position of the fixed bolt of the electrical control box, for accurate twopoint average random vibration control. The frequency response under each axis excitation is illustrated in Figure 15. In the Figure 15, the green line is the target spectrum, and the blue line is the actual vibration spectrum. When the vibration is between the two orange lines, the random vibration is within the control error range, and the red line is the interrupt mark, indicating that, when the vibration PSD value reaches or exceeds the red line, the vibration is too large. In order to protect the specimen and equipment, the vibration control system sends out instructions and the random vibration stops. In the Figure 15, the green line is the target spectrum, and the blue line is the actual vibration spectrum. When the vibration is between the two orange lines, the random vibration is within the control error range, and the red line is the interrupt mark, indicating that, when the vibration PSD value reaches or exceeds the red line, the vibration is too large. In order to protect the specimen and equipment, the vibration control system sends out instructions and the random vibration stops.

Test Results
Based on the GJB150.16A-2009 standard, the electrical control box experienced random vibration under given load spectrum excitation for 1.5 h (0.5 h per direction), and its dimensions, weight, and internal components were not abnormal. The electrical control box passed the durability test, indicating the correctness of the random vibration fatigue life analysis results and proving that the anti-fatigue performance of the electrical control box structure meets the design requirements.

Conclusions
An aircraft electric control box was employed as the research object, and the modal and random vibration response analyses of the electric control box were performed using the finite element software. In addition, the fatigue life of the electric control box was estimated using the frequency-domain analysis theory of fatigue damage and the Gaussiandistribution Steinberg method. Moreover, the finite element analysis results were verified by performing the random vibration test on the electric control box, and the following conclusions were obtained: (1) A comparison of the results of finite element calculation and mode analysis revealed that the mechanical model of the electrical control box could be established using the finite element software. The random vibration response analysis of the electrical control box in the X, Y, and Z directions was performed using this model. Furthermore, the fatigue life estimation of the electrical control box under random excitation was performed. (2) A comparison of the fatigue damage under random vibration excitation in the X, Y, and Z directions revealed that random excitation in the Y direction has the greatest influence on the fatigue life of the electrical control box, followed by random excitation in the Z and X directions. Informed Consent Statement: Not applicable.

Data Availability Statement:
The data presented in this study cannot be shared at this time as the data also forms part of an ongoing study.

Conflicts of Interest:
The authors declare that they have no conflict of interest in this work. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript, or in the decision to publish the results.